mpdl-xml-content: texts/archimedesOldCVSRepository/archimedes/raw/heath_mathe_02_en

annotate texts/archimedesOldCVSRepository/archimedes/raw/heath_mathe_02_en_1981.raw @ 11:35edd67cabf9

Appendix Version 2009-02-14

author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Thu, 02 May 2013 11:12:52 +0200
parents	22d6a63640c6
children

rev	line source
6 22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2 <table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3 <caption align=top><B>CONTENTS OF VOL II</B></caption>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4 <tr><td>XII. ARISTARCHUS OF SAMOS</td><td align=right>PAGES 1-15</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5 <tr><td>XIII. ARCHIMEDES</td><td align=right>16-109</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6 <tr><td>Traditions</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7 <tr><td>(<G>a</G>) Astronomy</td><td align=right>17-18</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8 <tr><td>(<G>b</G>) Mechanics</td><td align=right>18</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9 <tr><td>Summary of main achievements</td><td align=right>19-20</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10 <tr><td>Character of treatises</td><td align=right>20-22</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13 <tr><td>The text of Archimedes</td><td align=right>25-27</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14 <tr><td>Contents of <I>The Method</I></td><td align=right>27-34</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15 <tr><td><I>On the Sphere and Cylinder</I>, I, II</td><td align=right>34-50</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17 <tr><td>(i) Archimedes's own solution</td><td align=right>45-46</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19 <tr><td>(iii) Diocles's solution of original problem</td><td align=right>47-49</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20 <tr><td><I>Measurement of a Circle</I></td><td align=right>50-56</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22 <tr><td><I>On Spirals</I></td><td align=right>64-75</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	23 <tr><td><I>On Plane Equilibriums</I>, I, II</td><td align=right>75-81</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	24 <tr><td><I>The Sand-reckoner</I> (<I>Psammites</I> or <I>Arenarius</I>)</td><td align=right>81-85</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	25 <tr><td><I>The Quadrature of the Parabola</I></td><td align=right>85-91</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	27 <tr><td>The problem of the crown</td><td align=right>92-94</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	29 <tr><td>(<G>a</G>) The Cattle-Problem</td><td align=right>97-98</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	35 <tr><td>XIV. CONIC SECTIONS. APOLLONIUS OF PERGA</td><td align=right>110-196</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	36 <tr><td>A. HISTORY OF CONICS UP TO APOLLONIUS</td><td align=right>110-126</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	39 <tr><td>Works by Aristaeus and Euclid</td><td align=right>116-117</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	40 <tr><td>‘Solid loci’ and ‘solid problems’</td><td align=right>117-118</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	46 <pb n=vi><head>CONTENTS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	47 <tr><td>XIV. CONTINUED.</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	48 <tr><td>B. APOLLONIUS OF PERGA</td><td align=right>PAGES 126-196</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	50 <tr><td>Apollonius's own account of the <I>Conics</I></td><td align=right>128-133</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	56 oblique cone</td><td align=right>134-138</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	63 <tr><td>Book II</td><td align=right>148-150</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	64 <tr><td>Book III</td><td align=right>150-157</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	136 <MATH>sin(<G>q</G>-<G>f</G>)=sin<G>q</G>cos<G>f</G>-cos<G>q</G>sin<G>f</G></MATH></td><td align=right>278-280</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	161 Mensurae</I></td><td align=right>316-320</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	209 <tr><td>Book III. Section (1). On the problem of the two
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	215 <tr><td>Book IV. Section (1). Extension of theorem of
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	233 Archimedes</td><td align=right>394</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	235 <I>On the Sphere and Cylinder</I></td><td align=right>394-395</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	238 <tr><td>Book VI.</td><td align=right>396-399</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	266 the axes</td><td align=right>437-438</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	267 <tr><td>Problem of seven hexagons in a circle</td><td align=right>438-439</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	268 <tr><td>Construction of toothed wheels and indented screws</td><td align=right>439</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	269 <tr><td>XX. ALGEBRA: DIOPHANTUS OF ALEXANDRIA</td><td align=right>440-517</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	270 <tr><td>Beginnings learnt from Egypt</td><td align=right>440</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	271 <tr><td>‘Hau’-calculations</td><td align=right>440-441</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	272 <tr><td>Arithmetical epigrams in the Greek Anthology</td><td align=right>441-443</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	274 <tr><td>Indeterminate equations of second degree before Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	275 phantus</td><td align=right>443-444</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	276 <tr><td>Indeterminate equations in Heronian collections</td><td align=right>444-447</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	277 <tr><td>Numerical solution of quadratic equations</td><td align=right>448</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	278 <tr><td>Works of Diophantus</td><td align=right>448-450</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	279 <tr><td>The <I>Arithmetica</I></td><td align=right>449-514</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	280 <tr><td>The seven lost Books and their place</td><td align=right>449-450</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	281 <tr><td>Relation of ‘Porisms’ to <I>Arithmetica</I></td><td align=right>451-452</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	282 <tr><td>Commentators from Hypatia downwards</td><td align=right>453</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	283 <tr><td>Translations and editions</td><td align=right>453-455</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	284 <tr><td>Notation and definitions</td><td align=right>455-461</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	285 <tr><td>Sign for unknown (= <I>x</I>) and its origin</td><td align=right>456-457</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	286 <tr><td>Signs for powers of unknown &c.</td><td align=right>458-459</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	287 <tr><td>The sign (<FIG>) for <I>minus</I> and its meaning</td><td align=right>459-460</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	288 <tr><td>The methods of Diophantus</td><td align=right>462-479</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	289 <tr><td>I. Diophantus's treatment of equations</td><td align=right>462-476</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	290 <tr><td>(A) Determinate equations</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	291 <tr><td>(1) Pure determinate equations</td><td align=right>462-463</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	292 <tr><td>(2) Mixed quadratic equations</td><td align=right>463-465</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	294 <tr><td>(4) Cubic equation</td><td align=right>465</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	295 <tr><td>(B) Indeterminate equations</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	296 <tr><td>(<I>a</I>) Indeterminate equations of the second degree</td><td align=right>466-473</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	298 <tr><td>(2) Double equation</td><td align=right>468-473</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	299 <tr><td>1. Double equations of first degree</td><td align=right>469-472</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	300 <tr><td>2. Double equations of second degree</td><td align=right>472-473</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	301 <tr><td>(<I>b</I>) Indeterminate equations of degree higher
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	302 than second</td><td align=right>473-476</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	303 <tr><td>(1) Single equations</td><td align=right>473-475</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	304 <tr><td>(2) Double equations</td><td align=right>475-476</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	305 <tr><td>II. Method of limits</td><td align=right>476-477</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	306 <tr><td>III. Method of approximation to limits</td><td align=right>477-479</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	307 <tr><td>Porisms and propositions in the Theory of Numbers</td><td align=right>479-484</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	308 <tr><td>(<G>a</G>) Theorems on the composition of numbers as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	309 sum of two squares</td><td align=right>481-483</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	311 <tr><td>(<G>g</G>) Composition of numbers as the sum of four squares</td><td align=right>483-484</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	312 <tr><td>Conspectus of <I>Arithmetica</I>, with typical solutions</td><td align=right>484-514</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	313 <tr><td>The treatise on Polygonal Numbers</td><td align=right>514-517</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	314 <pb n=xi><head>CONTENTS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	315 <tr><td>XXI. COMMENTATORS AND BYZANTINES</td><td align=right>PAGES 518-555</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	319 <tr><td>Theon of Alexandria</td><td align=right>526-528</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	320 <tr><td>Commentary on the <I>Syntaxis</I></td><td align=right>526-527</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	321 <tr><td>Edition of Euclid's <I>Elements</I></td><td align=right>527-528</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	322 <tr><td>Edition of the <I>Optics</I> of Euclid</td><td align=right>528</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	323 <tr><td>Hypatia</td><td align=right>528-529</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	324 <tr><td>Porphyry. Iamblichus</td><td align=right>529</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	325 <tr><td>Proclus</td><td align=right>529-537</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	326 <tr><td>Commentary on Euclid, Book I</td><td align=right>530-535</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	329 <tr><td><I>Hypotyposis of Astronomical Hypotheses</I></td><td align=right>535-536</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	330 <tr><td>Commentary on the <I>Republic</I></td><td align=right>536-537</td></tr>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	347 <tr><td>Rule for approximating to square root of a non-square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	348 number</td><td align=right>553-554</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	349 <tr><td>Ioannes Pediasimus</td><td align=right>554</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	350 <tr><td>Barlaam</td><td align=right>554-555</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	351 <tr><td>Isaac Argyrus</td><td align=right>555</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	352 <tr><td>APPENDIX. On Archimedes's proof of the subtangent-property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	353 of a spiral</td><td align=right>556-561</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	354 <tr><td>INDEX OF GREEK WORDS</td><td align=right>563-569</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	355 <tr><td>ENGLISH INDEX</td><td align=right>570-586</td></tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	356 </table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	357 <pb><C>XII</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	358 <C>ARISTARCHUS OF SAMOS</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	359 <p>HISTORIANS of mathematics have, as a rule, given too little
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	360 attention to Aristarchus of Samos. The reason is no doubt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	361 that he was an astronomer, and therefore it might be supposed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	362 that his work would have no sufficient interest for the mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	363 matician. The Greeks knew better; they called him Aristar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	364 chus ‘the mathematician’, to distinguish him from the host
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	365 of other Aristarchuses; he is also included by Vitruvius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	366 among the few great men who possessed an equally profound
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	367 knowledge of all branches of science, geometry, astronomy,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	368 music, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	369 <p>‘Men of this type are rare, men such as were, in times past,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	370 Aristarchus of Samos, Philolaus and Archytas of Tarentum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	371 Apollonius of Perga, Eratosthenes of Cyrene, Archimedes and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	372 Scopinas of Syracuse, who left to posterity many mechanical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	373 and gnomonic appliances which they invented and explained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	374 on mathematical (lit. ‘numerical’) principles.’<note>Vitruvius, <I>De architectura,</I> i. 1. 16.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	375 <p>That Aristarchus was a very capable geometer is proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	376 his extant work <I>On the sizes and distances of the Sun and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	377 Moon</I> which will be noticed later in this chapter: in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	378 mechanical line he is credited with the discovery of an im-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	379 proved sun-dial, the so-called <G>ska/fh</G>, which had, not a plane,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	380 but a concave hemispherical surface, with a pointer erected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	381 vertically in the middle throwing shadows and so enabling
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	382 the direction and the height of the sun to be read off by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	383 of lines marked on the surface of the hemisphere. He also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	384 wrote on vision, light and colours. His views on the latter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	385 subjects were no doubt largely influenced by his master, Strato
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	386 of Lampsacus; thus Strato held that colours were emanations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	387 from bodies, material molecules, as it were, which imparted to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	388 the intervening air the same colour as that possessed by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	389 body, while Aristarchus said that colours are ‘shapes or forms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	390 <pb n=2><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	391 stamping the air with impressions like themselves, as it were’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	392 that ‘colours in darkness have no colouring’, and that ‘light
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	393 is the colour impinging on a substratum’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	394 <p>Two facts enable us to fix Aristarchus's date approximately.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	395 In 281/280 B.C. he made an observation of the summer
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	396 solstice; and a book of his, presently to be mentioned, was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	397 published before the date of Archimedes's <I>Psammites</I> or <I>Sand-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	398 reckoner,</I> a work written before 216 B.C. Aristarchus, there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	399 fore, probably lived <I>circa</I> 310-230 B.C., that is, he was older
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	400 than Archimedes by about 25 years.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	401 <p>To Aristarchus belongs the high honour of having been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	402 the first to formulate the Copernican hypothesis, which was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	403 then abandoned again until it was revived by Copernicus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	404 himself. His claim to the title of ‘the ancient Copernicus’ is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	405 still, in my opinion, quite unshaken, notwithstanding the in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	406 genious and elaborate arguments brought forward by Schia-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	407 parelli to prove that it was Heraclides of Pontus who first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	408 conceived the heliocentric idea. Heraclides is (along with one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	409 Ecphantus, a Pythagorean) credited with having been the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	410 to hold that the earth revolves about its own axis every 24
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	411 hours, and he was the first to discover that Mercury and Venus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	412 revolve, like satellites, about the sun. But though this proves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	413 that Heraclides came near, if he did not actually reach, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	414 hypothesis of Tycho Brahe, according to which the earth was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	415 in the centre and the rest of the system, the sun with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	416 planets revolving round it, revolved round the earth, it does
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	417 not suggest that he moved the earth away from the centre.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	418 The contrary is indeed stated by Aëtius, who says that ‘Hera-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	419 clides and Ecphantus make the earth move, <I>not in the sense of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	420 translation,</I> but by way of turning on an axle, like a wheel,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	421 from west to east, about its own centre’.<note>Aët. iii. 13. 3, <I>Vors.</I> i<SUP>3</SUP>, p. 341. 8.</note> None of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	422 champions of Heraclides have been able to meet this positive
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	423 statement. But we have conclusive evidence in favour of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	424 claim of Aristarchus; indeed, ancient testimony is unanimous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	425 on the point. Not only does Plutarch tell us that Cleanthes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	426 held that Aristarchus ought to be indicted for the impiety of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	427 ‘putting the Hearth of the Universe in motion’<note>Plutarch, <I>De faciè in orbe lunae,</I> c. 6, pp. 922 F-923 A.</note>; we have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	428 best possible testimony in the precise statement of a great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	429 <pb n=3><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	430 contemporary, Archimedes. In the <I>Sand-reckoner</I> Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	431 medes has this passage.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	432 <p>‘You [King Gelon] are aware that “universe” is the name
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	433 given by most astronomers to the sphere the centre of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	434 is the centre of the earth, while its radius is equal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	435 straight line between the centre of the sun and the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	436 the earth. This is the common account, as you have heard
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	437 from astronomers. But Aristarchus brought out <I>a book con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	438 sisting of certain hypotheses,</I> wherein it appears, as a conse-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	439 quence of the assumptions made, that the universe is many
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	440 times greater than the “universe” just mentioned. His hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	441 theses are that <I>the fixed stars and the sun remain unmoved,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	442 that the earth revolves about the sun in the circumference of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	443 circle, the sun lying in the middle of the orbit,</I> and that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	444 sphere of the fixed stars, situated about the same centre as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	445 sun, is so great that the circle in which he supposes the earth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	446 to revolve bears such a proportion to the distance of the fixed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	447 stars as the centre of the sphere bears to its surface.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	448 <p>(The last statement is a variation of a traditional phrase, for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	449 which there are many parallels (cf. Aristarchus's Hypothesis 2
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	450 ‘that the earth is in the relation of a point and centre to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	451 sphere in which the moon moves’), and is a method of saying
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	452 that the ‘universe’ is infinitely great in relation not merely to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	453 the size of the sun but even to the orbit of the earth in its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	454 revolution about it; the assumption was necessary to Aris-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	455 tarchus in order that he might not have to take account of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	456 parallax.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	457 <p>Plutarch, in the passage referred to above, also makes it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	458 clear that Aristarchus followed Heraclides in attributing to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	459 the earth the daily rotation about its axis. The bold hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	460 thesis of Aristarchus found few adherents. Seleucus, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	461 Seleucia on the Tigris, is the only convinced supporter of it of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	462 whom we hear, and it was speedily abandoned altogether,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	463 mainly owing to the great authority of Hipparchus. Nor do
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	464 we find any trace of the heliocentric hypothesis in Aris-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	465 tarchus's extant work <I>On the sizes and distances of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	466 Sun and Moon.</I> This is presumably because that work was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	467 written before the hypothesis was formulated in the book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	468 referred to by Archimedes. The geometry of the treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	469 is, however, unaffected by the difference between the hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	470 theses.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	471 <pb n=4><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	472 <p>Archimedes also says that it was Aristarchus who dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	473 covered that the apparent angular diameter of the sun is about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	474 1/720th part of the zodiac circle, that is to say, half a degree.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	475 We do not know how he arrived at this pretty accurate figure:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	476 but, as he is credited with the invention of the <G>ska/fh</G>, he may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	477 have used this instrument for the purpose. But here again
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	478 the discovery must apparently have been later than the trea-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	479 tise <I>On sizes and distances,</I> for the value of the subtended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	480 angle is there assumed to be 2° (Hypothesis 6). How Aris-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	481 tarchus came to assume a value so excessive is uncertain. As
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	482 the mathematics of his treatise is not dependent on the actual
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	483 value taken, 2° may have been assumed merely by way of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	484 illustration; or it may have been a guess at the apparent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	485 diameter made before he had thought of attempting to mea-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	486 sure it. Aristarchus assumed that the angular diameters of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	487 the sun and moon at the centre of the earth are equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	488 <p>The method of the treatise depends on the just observation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	489 which is Aristarchus's third ‘hypothesis’, that ‘when the moon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	490 appears to us halved, the great circle which divides the dark
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	491 and the bright portions of the moon is in the direction of our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	492 eye’; the effect of this (since the moon receives its light from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	493 the sun), is that at the time of the dichotomy the centres of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	494 the sun, moon and earth form a triangle right-angled at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	495 centre of the moon. Two other assumptions were necessary:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	496 first, an estimate of the size of the angle of the latter triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	497 at the centre of the earth at the moment of dichotomy: this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	498 Aristarchus assumed (Hypothesis 4) to be ‘less than a quad-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	499 rant by one-thirtieth of a quadrant’, i.e. 87°, again an inaccu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	500 rate estimate, the true value being 89° 50′; secondly, an esti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	501 mate of the breadth of the earth's shadow where the moon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	502 traverses it: this he assumed to be ‘the breadth of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	503 moons’ (Hypothesis 5).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	504 <p>The inaccuracy of the assumptions does not, however, detract
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	505 from the mathematical interest of the succeeding investigation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	506 Here we find the logical sequence of propositions and the abso-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	507 lute rigour of demonstration characteristic of Greek geometry;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	508 the only remaining drawback would be the practical difficulty
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	509 of determining the exact moment when the moon ‘appears to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	510 us halved’. The form and style of the book are thoroughly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	511 classical, as befits the period between Euclid and Archimedes;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	512 <pb n=5><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	513 the Greek is even remarkably attractive. The content from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	514 the mathematical point of view is no less interesting, for we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	515 have here the first specimen extant of pure geometry used
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	516 with a <I>trigonometrical</I> object, in which respect it is a sort of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	517 forerunner of Archimedes's <I>Measurement of a Circle.</I> Aristar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	518 chus does not actually evaluate the trigonometrical ratios
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	519 on which the ratios of the sizes and distances to be obtained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	520 depend; he finds limits between which they lie, and that by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	521 means of certain propositions which he assumes without proof,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	522 and which therefore must have been generally known to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	523 mathematicians of his day. These propositions are the equi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	524 valents of the statements that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	525 <p>(1) if <G>a</G> is what we call the circular measure of an angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	526 and <G>a</G> is less than 1/2 <G>p</G>, then the ratio sin <G>a/a</G> <I>decreases,</I> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	527 ratio tan <G>a/a</G> <I>increases,</I> as <G>a</G> increases from 0 to 1/2 <G>p</G>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	528 <p>(2) if <G>b</G> be the circular measure of another angle less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	529 1/2 <G>p</G>, and <G>a</G> > <G>b</G>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	530 <MATH>(sin<G>a</G>)/(sin<G>b</G>)<<G>a/b</G><(tan<G>a</G>)/(tan<G>b</G>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	531 <p>Aristarchus of course deals, not with actual circular measures,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	532 sines and tangents, but with angles (expressed not in degrees
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	533 but as fractions of right angles), arcs of circles and their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	534 chords. Particular results obtained by Aristarchus are the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	535 equivalent of the following:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	536 <MATH>1/18>sin3°>1/20</MATH>, [Prop. 7]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	537 <MATH>1/45>sin1°>1/60</MATH>, [Prop. 11]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	538 <MATH>1>cos1°>89/90</MATH>, [Prop. 12]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	539 <MATH>1>cos<SUP>2</SUP>1°>44/45</MATH>. [Prop. 13]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	540 <p>The book consists of eighteen propositions. Beginning with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	541 six hypotheses to the effect already indicated, Aristarchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	542 declares that he is now in a position to prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	543 <p>(1) that the distance of the sun from the earth is greater than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	544 eighteen times, but less than twenty times, the distance of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	545 moon from the earth;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	546 <p>(2) that the diameter of the sun has the same ratio as afore-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	547 said to the diameter of the moon;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	548 <pb n=6><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	549 <p>(3) that the diameter of the sun has to the diameter of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	550 earth a ratio greater than 19:3, but less than 43:6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	551 <p>The propositions containing these results are Props. 7, 9
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	552 and 15.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	553 <p>Prop. 1 is preliminary, proving that two equal spheres are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	554 comprehended by one cylinder, and two unequal spheres by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	555 one cone with its vertex in the direction of the lesser sphere,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	556 and the cylinder or cone touches the spheres in circles at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	557 right angles to the line of centres. Prop. 2 proves that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	558 a sphere be illuminated by another sphere larger than itself,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	559 the illuminated portion is greater than a hemisphere. Prop. 3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	560 shows that the circle in the moon which divides the dark from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	561 the bright portion is least when the cone comprehending the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	562 sun and the moon has its vertex at our eye. The ‘dividing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	563 circle’, as we shall call it for short, which was in Hypothesis 3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	564 spoken of as a great circle, is proved in Prop. 4 to be, not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	565 a great circle, but a small circle not perceptibly different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	566 from a great circle. The proof is typical and is worth giving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	567 along with that of some connected propositions (11 and 12).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	568 <p><I>B</I> is the centre of the moon, <I>A</I> that of the earth, <I>CD</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	569 diameter of the ‘dividing circle in the moon’, <I>EF</I> the parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	570 diameter in the moon. <I>BA</I> meets the circular section of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	571 moon through <I>A</I> and <I>EF</I> in <I>G,</I> and <I>CD</I> in <I>L. GH, GK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	572 are arcs each of which is equal to half the arc <I>CE.</I> By
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	573 Hypothesis 6 the angle <I>CAD</I> is ‘one-fifteenth of a sign’ = 2°,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	574 and the angle <I>BAC</I> = 1°.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	575 <p>Now, says Aristarchus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	576 <MATH>1°:45°[>tan 1°:tan 45°]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	577 ><I>BC</I>:<I>CA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	578 and, <I>a fortiori,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	579 <MATH><I>BC</I>:<I>BA</I> or <I>BG</I>:<I>BA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	580 <1:45</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	581 that is, <MATH><I>BG</I><1/45<I>BA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	582 <1/44<I>GA</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	583 therefore, <I>a fortiori,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	584 <MATH><I>BH</I><1/44<I>HA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	585 <pb n=7><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	586 <p>Now <MATH><I>BH</I>:<I>HA</I>[=sin<I>HAB</I>:sin<I>HBA</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	587 >&angle;<I>HAB</I>:&angle;<I>HBA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	588 whence <MATH>&angle;<I>HAB</I><1/44&angle;<I>HBA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	589 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	590 and (taking the doubles) <MATH>&angle;<I>HAK</I><1/44&angle;<I>HBK</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	591 <p>But <MATH>&angle;<I>HBK</I>=&angle;<I>EBC</I>=1/90<I>R</I> (where <I>R</I> is a right angle)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	592 therefore <MATH>&angle;<I>HAK</I><1/3960<I>R</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	593 <p>But ‘a magnitude (arc <I>HK</I>) seen under such an angle is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	594 imperceptible to our eye’;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	595 therefore, <I>a fortiori,</I> the arcs <I>CE, DF</I> are severally imper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	596 ceptible to our eye. Q.E.D.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	597 <p>An easy deduction from the same figure is Prop. 12, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	598 shows that the ratio of <I>CD,</I> the diameter of the ‘dividing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	599 circle’, to <I>EF,</I> the diameter of the moon, is < 1 but > 89/90.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	600 <p>We have <MATH>&angle;<I>EBC</I>=&angle;<I>BAC</I>=1°</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	601 therefore <MATH>(arc <I>EC</I>)=1/90 (arc <I>EG</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	602 and accordingly <MATH>(arc <I>CG</I>):(arc <I>GE</I>)=89:90</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	603 <p>Doubling the arcs, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	604 <MATH>(arc <I>CGD</I>):(arc <I>EGF</I>)=89:90</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	605 <p>But <MATH><I>CD</I>:<I>EF</I>>(arc <I>CGD</I>):(arc <I>EGF</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	606 [equivalent to sin<G>a</G>/sin<G>b</G>><G>a/b</G>, where &angle;<I>CBD</I>=2<G>a</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	607 and 2<G>b</G>=<G>p</G></MATH>];
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	608 therefore <MATH><I>CD</I>:<I>EF</I> [=cos 1°]>89:90</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	609 while obviously <MATH><I>CD</I>:<I>EF</I><1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	610 <p>Prop. 11 finds limits to the ratio <I>EF</I>:<I>BA</I> (the ratio of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	611 diameter of the moon to the distance of its centre from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	612 the centre of the earth); the ratio is < 2:45 but > 1:30.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	613 <pb n=8><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	614 <p>The first part follows from the relation found in Prop. 4,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	615 namely <MATH><I>BC</I>:<I>BA</I><1:45</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	616 for <MATH><I>EF</I>=2<I>BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	617 <p>The second part requires the use of the circle drawn with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	618 centre <I>A</I> and radius <I>AC.</I> This circle is that on which the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	619 ends of the diameter of the ‘dividing circle’ move as the moon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	620 moves in a circle about the earth. If <I>r</I> is the radius <I>AC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	621 of this circle, a chord in it equal to <I>r</I> subtends at the centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	622 <I>A</I> an angle of 2/3<I>R</I> or 60°; and the arc <I>CD</I> subtends at <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	623 an angle of 2°.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	624 <p>But <MATH>(arc subtended by <I>CD</I>):(arc subtended by <I>r</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	625 <<I>CD</I>:<I>r</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	626 or <MATH>2:60<<I>CD</I>:<I>r</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	627 that is, <MATH><I>CD</I>:<I>CA</I>>1:30</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	628 <p>And, by similar triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	629 <MATH><I>CL</I>:<I>CA</I>=<I>CB</I>:<I>BA</I>, or <I>CD</I>:<I>CA</I>=2<I>CB</I>:<I>BA</I>=<I>FE</I>:<I>BA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	630 <p>Therefore <MATH><I>FE</I>:<I>BA</I>>1:30</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	631 <p>The proposition which is of the greatest interest on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	632 whole is Prop. 7, to the effect that <I>the distance of the sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	633 from the earth is greater than 18 times, but less than 20
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	634 times, the distance of the moon from the earth.</I> This result
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	635 represents a great improvement on all previous attempts to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	636 estimate the relative distances. The first speculation on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	637 subject was that of Anaximander (<I>circa</I> 611-545 B.C.), who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	638 seems to have made the distances of the sun and moon from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	639 the earth to be in the ratio 3:2. Eudoxus, according to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	640 Archimedes, made the diameter of the sun 9 times that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	641 the moon, and Phidias, Archimedes's father, 12 times; and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	642 assuming that the angular diameters of the two bodies are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	643 equal, the ratio of their distances would be the same.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	644 <p>Aristarchus's proof is shortly as follows. <I>A</I> is the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	645 the sun, <I>B</I> that of the earth, and <I>C</I> that of the moon at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	646 moment of dichotomy, so that the angle <I>ACB</I> is right. <I>ABEF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	647 is a square, and <I>AE</I> is a quadrant of the sun's circular orbit.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	648 Join <I>BF,</I> and bisect the angle <I>FBE</I> by <I>BG,</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	649 <MATH>&angle;<I>GBE</I>=1/4<I>R</I> or 22 1/2°</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	650 <pb n=9><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	651 <p>I. Now, by Hypothesis 4, <MATH>&angle;<I>ABC</I>=87°</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	652 so that <MATH>&angle;<I>HBE</I>=&angle;<I>BAC</I>=3°</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	653 therefore <MATH>&angle;<I>GBE</I>:&angle;<I>HBE</I>=(1/4)<I>R</I>:(1/30)<I>R</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	654 =15:2</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	655 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	656 so that <MATH><I>GE</I>:<I>HE</I>[=tan <I>GBE</I>:tan <I>HBE</I>]>&angle;<I>GBE</I>:&angle;<I>HBE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	657 >15:2</MATH>. (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	658 <p>The ratio which has to be proved > 18:1 is <I>AB</I>:<I>BC</I> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	659 <I>FE</I>:<I>EH.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	660 <p>Now <MATH><I>FG</I>:<I>GE</I>=<I>FB</I>:<I>BE</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	661 whence <MATH><I>FG</I><SUP>2</SUP>:<I>GE</I><SUP>2</SUP>=<I>FB</I><SUP>2</SUP>:<I>BE</I><SUP>2</SUP>=2:1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	662 and <MATH><I>FG</I>:<I>GE</I>=√2:1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	663 >7:5</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	664 (this is the approximation to √2 mentioned by Plato and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	665 known to the Pythagoreans).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	666 <pb n=10><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	667 <p>Therefore <MATH><I>FE</I>:<I>EG</I>>12:5 or 36:15</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	668 <p>Compounding this with (1) above, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	669 <MATH><I>FE</I>:<I>EH</I>>36:2 or 18:1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	670 <p>II. To prove <MATH><I>BA</I><20 <I>BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	671 <p>Let <I>BH</I> meet the circle <I>AE</I> in <I>D,</I> and draw <I>DK</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	672 to <I>EB.</I> Circumscribe a circle about the triangle <I>BKD,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	673 let the chord <I>BL</I> be equal to the radius (<I>r</I>) of the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	674 <p>Now <MATH>&angle;<I>BDK</I>=&angle;<I>DBE</I>=1/30<I>R</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	675 so that arc <MATH><I>BK</I>=1/60 (circumference of circle)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	676 <p>Thus <MATH>(arc <I>BK</I>):(arc <I>BL</I>)=(1/60):(1/6),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	677 =1:10</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	678 <p>And <MATH>(arc <I>BK</I>):(arc <I>BL</I>)<<I>BK</I>:<I>r</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	679 [this is equivalent to <G>a/b</G><sin<G>a</G>/sin<G>b</G>, where <G>a</G><<G>b</G><1/2<G>p</G>],
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	680 so that <MATH><I>r</I><10 <I>BK</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	681 and <MATH><I>BD</I><20 <I>BK</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	682 <p>But <MATH><I>BD</I>:<I>BK</I>=<I>AB</I>:<I>BC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	683 therefore <MATH><I>AB</I><20<I>BC</I></MATH>. Q.E.D.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	684 <p>The remaining results obtained in the treatise can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	685 visualized by means of the three figures annexed, which have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	686 reference to the positions of the sun (centre A), the earth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	687 (centre B) and the moon (centre C) during an eclipse. Fig. 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	688 shows the middle position of the moon relatively to the earth's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	689 shadow which is bounded by the cone comprehending the sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	690 and the earth. <I>ON</I> is the arc with centre <I>B</I> along which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	691 move the extremities of the diameter of the dividing circle in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	692 the moon. Fig. 3 shows the same position of the moon in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	693 middle of the shadow, but on a larger scale. Fig. 2 shows
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	694 the moon at the moment when it has just entered the shadow;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	695 and, as the breadth of the earth's shadow is that of two moons
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	696 (Hypothesis 5), the moon in the position shown touches <I>BN</I> at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	697 <I>N</I> and <I>BL</I> at <I>L,</I> where <I>L</I> is the middle point of the arc <I>ON.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	698 It should be added that, in Fig. 1, <I>UV</I> is the diameter of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	699 circle in which the sun is touched by the double cone with <I>B</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	700 as vertex, which comprehends both the sun and the moon,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	701 <pb n=11><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	702 while <I>Y, Z</I> are the points in which the perpendicular through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	703 <I>A,</I> the centre of the sun, to <I>BA</I> meets the cone enveloping the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	704 sun and the earth.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	705 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	706 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	707 <p>This being premised, the main results obtained are as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	708 follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	709 Prop. 13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	710 <p>(1) <MATH><I>ON</I>:(diam. of moon)<2:1</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	711 but <MATH>>88:45</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	712 <pb n=12><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	713 <p>(2) <MATH><I>ON</I>:(diam. of sun)<1:9</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	714 but <MATH>>22:225</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	715 <p>(3) <MATH><I>ON</I>:<I>YZ</I>>979:10125</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	716 Prop. 14 (Fig. 3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	717 <MATH><I>BC</I>:<I>CS</I>>675:1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	718 Prop. 15.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	719 <MATH>(Diam. of sun):(diam. of earth)>19:3</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	720 but <MATH><43:6</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	721 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	722 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	723 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	724 <CAP>FIG. 3.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	725 Prop. 17.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	726 <MATH>(Diam. of earth):(diam. of moon)>108:43</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	727 but <MATH><60:19</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	728 <p>It is worth while to show how these results are proved.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	729 Prop. 13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	730 <p>(1) In Fig. 2 it is clear that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	731 <MATH><I>ON</I><2<I>LN</I> and, <I>a fortiori,</I> <2<I>LP</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	732 <p>The triangles <I>LON, CLN</I> being similar,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	733 <MATH><I>ON</I>:<I>NL</I>=<I>NL</I>:<I>LC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	734 therefore <MATH><I>ON</I>:<I>NL</I>=<I>NL</I>:1/2<I>LP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	735 >89:45</MATH>. (by Prop. 12)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	736 <pb n=13><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	737 <p>Hence <MATH><I>ON</I>:<I>LC</I>=<I>ON</I><SUP>2</SUP>:<I>NL</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	738 >89<SUP>2</SUP>:45<SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	739 therefore <MATH><I>ON</I>:<I>LP</I>>7921:4050
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	740 >88:45, says Aristarchus</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	741 <p>[If 7921/4050 be developed as a continued fraction, we easily
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	742 obtain 1+1/(1+1/(21+1/2)), which is in fact 88/45.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	743 <p>(2) <MATH><I>ON</I><2(diam. of moon)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	744 <p>But <MATH>(diam. of moon)<1/18(diam. of sun)</MATH>; (Prop. 7)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	745 therefore <MATH><I>ON</I><1/9(diam. of sun)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	746 <p>Again <MATH><I>ON</I>:(diam. of moon)>88:45</MATH>, from above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	747 and <MATH>(diam. of moon):(diam. of sun)>1:20</MATH>; (Prop. 7)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	748 therefore, <I>ex aequali,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	749 <MATH><I>ON</I>:(diam. of sun)>88:900
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	750 >22:225</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	751 <p>(3) Since the same cone comprehends the sun and the moon,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	752 the triangle <I>BUV</I> (Fig. 1) and the triangle <I>BLN</I> (Fig. 2) are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	753 similar, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	754 <MATH><I>LN</I>:<I>LP</I>=<I>UV</I>:(diam. of sun)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	755 =<I>WU</I>:<I>UA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	756 =<I>UA</I>:<I>AS</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	757 <<I>UA</I>:<I>AY</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	758 <p>But <MATH><I>LN</I>:<I>LP</I>>89:90</MATH>; (Prop. 12)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	759 therefore, <I>a fortiori,</I> <MATH><I>UA</I>:<I>AY</I>>89:90</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	760 <p>And <MATH><I>UA</I>:<I>AY</I>=2<I>UA</I>:<I>YZ</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	761 =(diam. of sun):<I>YZ</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	762 <p>But <MATH><I>ON</I>:(diam. of sun)>22:225</MATH>; (Prop. 13)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	763 therefore, <I>ex aequali,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	764 <MATH><I>ON</I>:<I>YZ</I>>89X22:90X225
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	765 >979:10125</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	766 <pb n=14><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	767 Prop. 14 (Fig. 3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	768 <p>The arcs <I>OM, ML, LP, PN</I> are all equal; therefore so are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	769 the chords. <I>BM, BP</I> are tangents to the circle <I>MQP,</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	770 <I>CM</I> is perpendicular to <I>BM,</I> while <I>BM</I> is perpendicular to <I>OL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	771 Therefore the triangles <I>LOS, CMR</I> are similar.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	772 <p>Therefore <MATH><I>SO</I>:<I>MR</I>=<I>SL</I>:<I>RC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	773 <p>But <MATH><I>SO</I><2<I>MR,</I> since <I>ON</I><2<I>MP</I></MATH>; (Prop. 13)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	774 therefore <MATH><I>SL</I><2<I>RC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	775 and, <I>a fortiori,</I> <MATH><I>SR</I><2<I>RC,</I> or <I>SC</I><3<I>RC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	776 that is, <MATH><I>CR</I>:<I>CS</I>>1:3</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	777 <p>Again, <MATH><I>MC</I>:<I>CR</I>=<I>BC</I>:<I>CM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	778 >45:1</MATH>; (see Prop. 11)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	779 therefore, <I>ex aequali,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	780 <MATH><I>CM</I>:<I>CS</I>>15:1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	781 <p>And <MATH><I>BC</I>:<I>CM</I>>45:1</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	782 therefore <MATH><I>BC</I>:<I>CS</I>>675:1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	783 Prop. 15 (Fig. 1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	784 <p>We have <MATH><I>NO</I>:(diam. of sun)<1:9</MATH>, (Prop. 13)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	785 and, <I>a fortiori,</I> <MATH><I>YZ</I>:<I>NO</I>>9:1</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	786 therefore, by similar triangles, if <I>YO, ZN</I> meet in <I>X,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	787 <MATH><I>AX</I>:<I>XR</I>>9:1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	788 and <I>convertendo,</I> <MATH><I>XA</I>:<I>AR</I><9:8</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	789 <p>But <MATH><I>AB</I>>18<I>BC,</I> and, <I>a fortiori,</I> >18<I>BR</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	790 whence <MATH><I>AB</I>>18(<I>AR-AB</I>), or 19<I>AB</I>>18<I>AR</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	791 that is, <MATH><I>AR</I>:<I>AB</I><19:18</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	792 <p>Therefore, <I>ex aequali,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	793 <MATH><I>XA</I>:<I>AB</I><19:16</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	794 and, <I>convertendo,</I> <MATH><I>AX</I>:<I>XB</I>>19:3</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	795 therefore <MATH>(diam. of sun):(diam. of earth)>19:3</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	796 <p>Lastly, since <MATH><I>CB</I>:<I>CR</I>>675:1</MATH>, (Prop. 14)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	797 <MATH><I>CB</I>:<I>BR</I><675:674</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	798 <pb n=15><head>ARISTARCHUS OF SAMOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	799 <p>But <MATH><I>AB</I>:<I>BC</I><20:1</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	800 therefore, <I>ex aequali,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	801 <MATH><I>AB</I>:<I>BR</I><13500:674
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	802 <6750:337</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	803 whence, by inversion and <I>componendo,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	804 <MATH><I>RA</I>:<I>AB</I>>7087:6750</MATH>. (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	805 <p>But <MATH><I>AX</I>:<I>XR</I>=<I>YZ</I>:<I>NO</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	806 <10125:979</MATH>; (Prop. 13)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	807 therefore, <I>convertendo,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	808 <MATH><I>XA</I>:<I>AR</I>>10125:9146</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	809 <p>From this and (1) we have, <I>ex aequali,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	810 <MATH><I>XA</I>:<I>AB</I>>10125X7087:9146X6750
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	811 >71755875:61735500
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	812 >43:37, <I>a fortiori</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	813 <p>[It is difficult not to see in 43:37 the expression 1+1/(6+1/6),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	814 which suggests that 43:37 was obtained by developing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	815 ratio as a continued fraction.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	816 <p>Therefore, <I>convertendo,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	817 <MATH><I>XA</I>:<I>XB</I><43:6</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	818 whence <MATH>(diam. of sun):(diam. of earth)<43:6</MATH>. Q.E.D.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	819 <pb><C>XIII</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	820 <C>ARCHIMEDES</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	821 <p>THE siege and capture of Syracuse by Marcellus during the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	822 second Punic war furnished the occasion for the appearance of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	823 Archimedes as a personage in history; it is with this histori-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	824 cal event that most of the detailed stories of him are con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	825 nected; and the fact that he was killed in the sack of the city
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	826 in 212 B. C., when he is supposed to have been 75 years of age,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	827 enables us to fix his date at about 287-212 B.C. He was the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	828 son of Phidias, the astronomer, and was on intimate terms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	829 with, if not related to, King Hieron and his son Gelon. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	830 appears from a passage of Diodorus that he spent some time
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	831 in Egypt, which visit was the occasion of his discovery of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	832 so-called Archimedean screw as a means of pumping water.<note>Diodorus, v. 37. 3.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	833 It may be inferred that he studied at Alexandria with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	834 successors of Euclid. It was probably at Alexandria that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	835 made the acquaintance of Conon of Samos (for whom he had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	836 the highest regard both as a mathematician and a friend) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	837 of Eratosthenes of Cyrene. To the former he was in the habit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	838 of communicating his discourses before their publication;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	839 while it was to Eratosthenes that he sent <I>The Method,</I> with an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	840 introductory letter which is of the highest interest, as well as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	841 (if we may judge by its heading) the famous Cattle-Problem.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	842 <C>Traditions.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	843 <p>It is natural that history or legend should say more of his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	844 mechanical inventions than of his mathematical achievements,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	845 which would appeal less to the average mind. His machines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	846 were used with great effect against the Romans in the siege
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	847 of Syracuse. Thus he contrived (so we are told) catapults so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	848 ingeniously constructed as to be equally serviceable at long or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	849 <pb n=17><head>TRADITIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	850 short range, machines for discharging showers of missiles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	851 through holes made in the walls, and others consisting of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	852 long movable poles projecting beyond the walls which either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	853 dropped heavy weights on the enemy's ships, or grappled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	854 their prows by means of an iron hand or a beak like that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	855 a crane, then lifted them into the air and let them fall again.<note>Polybius, <I>Hist.</I> viii. 7, 8; Livy xxiv. 34; Plutarch, <I>Marcellus,</I> cc. 15-17.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	856 Marcellus is said to have derided his own engineers with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	857 words, ‘Shall we not make an end of fighting against this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	858 geometrical Briareus who uses our ships like cups to ladle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	859 water from the sea, drives off our <I>sambuca</I> ignominiously
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	860 with cudgel-blows, and by the multitude of missiles that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	861 hurls at us all at once outdoes the hundred-handed giants of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	862 mythology?’; but all to no purpose, for the Romans were in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	863 such abject terror that, ‘if they did but see a piece of rope
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	864 or wood projecting above the wall, they would cry “there it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	865 is”, declaring that Archimedes was setting some engine in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	866 motion against them, and would turn their backs and run
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	867 away’.<note><I>Ib.,</I> c. 17.</note> These things, however, were merely the ‘diversions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	868 of geometry at play’,<note><I>Ib.,</I> c. 14.</note> and Archimedes himself attached no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	869 importance to them. According to Plutarch,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	870 ‘though these inventions had obtained for him the renown of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	871 more than human sagacity, he yet would not even deign to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	872 leave behind him any written work on such subjects, but,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	873 regarding as ignoble and sordid the business of mechanics and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	874 every sort of art which is directed to use and profit, he placed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	875 his whole ambition in those speculations the beauty and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	876 subtlety of which is untainted by any admixture of the com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	877 mon needs of life.’<note><I>Ib.,</I> c. 17.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	878 <C>(<G>a</G>) <I>Astronomy.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	879 <p>Archimedes did indeed write one mechanical book, <I>On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	880 Sphere-making,</I> which is lost; this described the construction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	881 of a sphere to imitate the motions of the sun, moon and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	882 planets.<note>Carpus in Pappus, viii, p. 1026. 9; Proclus on Eucl. I, p. 41. 16.</note> Cicero saw this contrivance and gives a description
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	883 of it; he says that it represented the periods of the moon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	884 and the apparent motion of the sun with such accuracy that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	885 it would even (over a short period) show the eclipses of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	886 sun and moon.<note>Cicero, <I>De rep.</I> i. 21, 22, <I>Tusc.</I> i. 63, <I>De nat. deor.</I> ii. 88.</note> As Pappus speaks of ‘those who understand
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	887 <pb n=18><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	888 the making of spheres and produce a model of the heavens by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	889 means of the circular motion of water’, it is possible that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	890 Archimedes's sphere was moved by water. In any case Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	891 medes was much occupied with astronomy. Livy calls him
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	892 ‘unicus spectator caeli siderumque’.<note>Livy xxiv. 34. 2.</note> Hipparchus says, ‘From
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	893 these observations it is clear that the differences in the years
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	894 are altogether small, but, as to the solstices, I almost think
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	895 that Archimedes and I have both erred to the extent of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	896 quarter of a day both in the observation and in the deduction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	897 therefrom’.<note>Ptolemy, <I>Syntaxis,</I> III. 1, vol. i, p. 194. 23.</note> Archimedes then had evidently considered the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	898 length of the year. Macrobius says he discovered the dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	899 tances of the planets,<note>Macrobius, <I>In Somn. Scip.</I> ii. 3; cf. the figures in Hippolytus, <I>Refut.,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	900 p. 66. 52 sq., ed. Duncker.</note> and he himself describes in his <I>Sand-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	901 reckoner</I> the apparatus by which he measured the apparent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	902 angular diameter of the sun.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	903 <C>(<G>b</G>) <I>Mechanics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	904 <p>Archimedes wrote, as we shall see, on theoretical mechanics,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	905 and it was by theory that he solved the problem <I>To move a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	906 given weight by a given force,</I> for it was in reliance ‘on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	907 irresistible cogency of his proof’ that he declared to Hieron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	908 that any given weight could be moved by any given force
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	909 (however small), and boasted that, ‘if he were given a place to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	910 stand on, he could move the earth’ (<G>pa= bw=, kai\ kinw= ta\n ga=n</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	911 as he said in his Doric dialect). The story, told by Plutarch,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	912 is that, ‘when Hieron was struck with amazement and asked
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	913 Archimedes to reduce the problem to practice and to give an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	914 illustration of some great weight moved by a small force, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	915 fixed upon a ship of burden with three masts from the king's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	916 arsenal which had only been drawn up with great labour by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	917 many men, and loading her with many passengers and a full
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	918 freight, himself the while sitting far off, with no great effort
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	919 but only holding the end of a compound pulley (<G>polu/spastos</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	920 quietly in his hand and pulling at it, he drew the ship along
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	921 smoothly and safely as if she were moving through the sea.’<note>Plutarch, <I>Marcellus,</I> c. 14.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	922 <p>The story that Archimedes set the Roman ships on fire by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	923 an arrangement of burning-glasses or concave mirrors is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	924 found in any authority earlier than Lucian; but it is quite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	925 <pb n=19><head>MECHANICS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	926 likely that he discovered some form of burning-mirror, e.g. a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	927 paraboloid of revolution, which would reflect to one point all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	928 rays falling on its concave surface in a direction parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	929 its axis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	930 <p>Archimedes's own view of the relative importance of his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	931 many discoveries is well shown by his request to his friends
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	932 and relatives that they should place upon his tomb a represen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	933 tation of a cylinder circumscribing a sphere, with an inscrip-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	934 tion giving the ratio which the cylinder bears to the sphere;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	935 from which we may infer that he regarded the discovery of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	936 this ratio as his greatest achievement. Cicero, when quaestor
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	937 in Sicily, found the tomb in a neglected state and repaired it<note>Cicero, <I>Tusc.</I> v. 64 sq.</note>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	938 but it has now disappeared, and no one knows where he was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	939 buried.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	940 <p>Archimedes's entire preoccupation by his abstract studies is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	941 illustrated by a number of stories. We are told that he would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	942 forget all about his food and such necessities of life, and would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	943 be drawing geometrical figures in the ashes of the fire or, when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	944 anointing himself, in the oil on his body.<note>Plutarch, <I>Marcellus,</I> c. 17.</note> Of the same sort
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	945 is the tale that, when he discovered in a bath the solution of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	946 the question referred to him by Hieron, as to whether a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	947 crown supposed to have been made of gold did not in fact con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	948 tain a certain proportion of silver, he ran naked through the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	949 street to his home shouting <G>eu(/rhka, eu(/rhka</G>.<note>Vitruvius, <I>De architectura,</I> ix. 1. 9, 10.</note> He was killed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	950 in the sack of Syracuse by a Roman soldier. The story is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	951 told in various forms; the most picturesque is that found in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	952 Tzetzes, which represents him as saying to a Roman soldier
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	953 who found him intent on some diagrams which he had drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	954 in the dust and came too close, ‘Stand away, fellow, from my
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	955 diagram’, whereat the man was so enraged that he killed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	956 him.<note>Tzetzes, <I>Chiliad.</I> ii. 35. 135.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	957 <C>Summary of main achievements.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	958 <p>In geometry Archimedes's work consists in the main of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	959 original investigations into the quadrature of curvilinear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	960 plane figures and the quadrature and cubature of curved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	961 surfaces. These investigations, beginning where Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	962 Book XII left off, actually (in the words of Chasles) ‘gave
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	963 <pb n=20><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	964 birth to the calculus of the infinite conceived and brought to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	965 perfection successively by Kepler, Cavalieri, Fermat, Leibniz
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	966 and Newton’. He performed in fact what is equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	967 <I>integration</I> in finding the area of a parabolic segment, and of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	968 a spiral, the surface and volume of a sphere and a segment of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	969 a sphere, and the volumes of any segments of the solids of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	970 revolution of the second degree. In arithmetic he calculated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	971 approximations to the value of <G>p</G>, in the course of which cal-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	972 culation he shows that he could approximate to the value of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	973 square roots of large or small non-square numbers; he further
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	974 invented a system of arithmetical terminology by which he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	975 could express in language any number up to that which we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	976 should write down with 1 followed by 80,000 million million
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	977 ciphers. In mechanics he not only worked out the principles of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	978 the subject but advanced so far as to find the centre of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	979 of a segment of a parabola, a semicircle, a cone, a hemisphere,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	980 a segment of a sphere, a right segment of a paraboloid and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	981 a spheroid of revolution. His mechanics, as we shall see, has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	982 become more important in relation to his geometry since the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	983 discovery of the treatise called <I>The Method</I> which was formerly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	984 supposed to be lost. Lastly, he invented the whole science of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	985 hydrostatics, which again he carried so far as to give a most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	986 complete investigation of the positions of rest and stability of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	987 a right segment of a paraboloid of revolution floating in a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	988 fluid with its base either upwards or downwards, but so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	989 the base is either wholly above or wholly below the surface of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	990 the fluid. This represents a sum of mathematical achieve-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	991 ment unsurpassed by any one man in the world's history.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	992 <C>Character of treatises.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	993 <p>The treatises are, without exception, monuments of mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	994 matical exposition; the gradual revelation of the plan of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	995 attack, the masterly ordering of the propositions, the stern
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	996 elimination of everything not immediately relevant to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	997 purpose, the finish of the whole, are so impressive in their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	998 perfection as to create a feeling akin to awe in the mind of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	999 the reader. As Plutarch said, ‘It is not possible to find in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1000 geometry more difficult and troublesome questions or proofs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1001 set out in simpler and clearer propositions’.<note>Plutarch, <I>Marcellus,</I> c. 17.</note> There is at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1002 <pb n=21><head>CHARACTER OF TREATISES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1003 same time a certain mystery veiling the way in which he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1004 arrived at his results. For it is clear that they were not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1005 <I>discovered</I> by the steps which lead up to them in the finished
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1006 treatises. If the geometrical treatises stood alone, Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1007 medes might seem, as Wallis said, ‘as it were of set purpose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1008 to have covered up the traces of his investigation, as if he had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1009 grudged posterity the secret of his method of inquiry, while
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1010 he wished to extort from them assent to his results’. And
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1011 indeed (again in the words of Wallis) ‘not only Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1012 but nearly all the ancients so hid from posterity their method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1013 of Analysis (though it is clear that they had one) that more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1014 modern mathematicians found it easier to invent a new
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1015 Analysis than to seek out the old’. A partial exception is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1016 now furnished by <I>The Method</I> of Archimedes, so happily dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1017 covered by Heiberg. In this book Archimedes tells us how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1018 he discovered certain theorems in quadrature and cubature,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1019 namely by the use of mechanics, weighing elements of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1020 figure against elements of another simpler figure the mensura-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1021 tion of which was already known. At the same time he is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1022 careful to insist on the difference between (1) the means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1023 which may be sufficient to suggest the truth of theorems,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1024 although not furnishing scientific proofs of them, and (2) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1025 rigorous demonstrations of them by orthodox geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1026 methods which must follow before they can be finally accepted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1027 as established:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1028 <p>‘certain things’, he says, ‘first became clear to me by a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1029 mechanical method, although they had to be demonstrated by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1030 geometry afterwards because their investigation by the said
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1031 method did not furnish an actual demonstration. But it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1032 of course easier, when we have previously acquired, by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1033 method, some knowledge of the questions, to supply the proof
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1034 than it is to find it without any previous knowledge.’ ‘This’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1035 he adds, ‘is a reason why, in the case of the theorems that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1036 the volumes of a cone and a pyramid are one-third of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1037 volumes of the cylinder and prism respectively having the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1038 same base and equal height, the proofs of which Eudoxus was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1039 the first to discover, no small share of the credit should be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1040 given to Democritus who was the first to state the fact,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1041 though without proof.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1042 <p>Finally, he says that the very first theorem which he found
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1043 out by means of mechanics was that of the separate treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1044 <pb n=22><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1045 on the <I>Quadrature of the parabola,</I> namely that <I>the area of any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1046 segment of a section of a right-angled cone</I> (<I>i. e. a parabola</I>) <I>is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1047 four-thirds of that of the triangle which has the same base and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1048 height.</I> The mechanical proof, however, of this theorem in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1049 <I>Quadrature of the Parabola</I> is different from that in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1050 <I>Method,</I> and is more complete in that the argument is clinched
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1051 by formally applying the method of exhaustion.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1052 <C>List of works still extant.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1053 <p>The extant works of Archimedes in the order in which they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1054 appear in Heiberg's second edition, following the order of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1055 manuscripts so far as the first seven treatises are concerned,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1056 are as follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1057 <p>(5) <I>On the Sphere and Cylinder</I>: two Books.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1058 <p>(9) <I>Measurement of a Circle.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1059 <p>(7) <I>On Conoids and Spheroids.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1060 <p>(6) <I>On Spirals.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1061 <p>(1) <I>On Plane Equilibriums,</I> Book I.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1062 <p>(3) ” ” ” Book II.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1063 <p>(10) <I>The Sand-reckoner</I> (<I>Psammites</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1064 <p>(2) <I>Quadrature of the Parabola.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1065 <p>(8) <I>On Floating Bodies</I>: two Books.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1066 <p>? <I>Stomachion</I> (a fragment).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1067 <p>(4) <I>The Method.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1068 <p>This, however, was not the order of composition; and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1069 judging (<I>a</I>) by statements in Archimedes's own prefaces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1070 certain of the treatises and (<I>b</I>) by the use in certain treatises
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1071 of results obtained in others, we can make out an approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1072 mate chronological order, which I have indicated in the above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1073 list by figures in brackets. The treatise <I>On Floating Bodies</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1074 was formerly only known in the Latin translation by William
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1075 of Moerbeke, but the Greek text of it has now been in great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1076 part restored by Heiberg from the Constantinople manuscript
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1077 which also contains <I>The Method</I> and the fragment of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1078 <I>Stomachion.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1079 <p>Besides these works we have a collection of propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1080 (<I>Liber assumptorum</I>) which has reached us through the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1081 Arabic. Although in the title of the translation by Thãbit b.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1082 <pb n=23><head>LIST OF EXTANT WORKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1083 Qurra the book is attributed to Archimedes, the propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1084 cannot be his in their present form, since his name is several
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1085 times mentioned in them; but it is quite likely that some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1086 of them are of Archimedean origin, notably those about the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1087 geometrical figures called <G>a)/rbhlos</G> (‘shoemaker's knife’) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1088 <G>sa/linon</G> (probably ‘salt-cellar’) respectively and Prop. 8 bear-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1089 ing on the trisection of an angle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1090 <p>There is also the <I>Cattle-Problem</I> in epigrammatic form,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1091 which purports by its heading to have been communicated by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1092 Archimedes to the mathematicians at Alexandria in a letter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1093 to Eratosthenes. Whether the epigrammatic form is due to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1094 Archimedes himself or not, there is no sufficient reason for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1095 doubting the possibility that the substance of it was set as a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1096 problem by Archimedes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1097 <C>Traces of lost works.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1098 <p>Of works which are lost we have the following traces.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1099 <p>1. Investigations relating to <I>polyhedra</I> are referred to by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1100 Pappus who, after alluding to the five regular polyhedra,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1101 describes thirteen others discovered by Archimedes which are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1102 semi-regular, being contained by polygons equilateral and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1103 equiangular but not all similar.<note>Pappus, v, pp. 352-8.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1104 <p>2. There was a book of arithmetical content dedicated to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1105 Zeuxippus. We learn from Archimedes himself that it dealt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1106 with the <I>naming of numbers</I> (<G>katono/maxis tw=n a)riqmw=n</G>)<note>Archimedes, vol. ii, pp. 216. 18, 236. 17-22; cf. p. 220. 4.</note> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1107 expounded the system, which we find in the <I>Sand-reckoner,</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1108 expressing numbers higher than those which could be written
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1109 in the ordinary Greek notation, numbers in fact (as we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1110 said) up to the enormous figure represented by 1 followed by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1111 80,000 million million ciphers.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1112 <p>3. One or more works on mechanics are alluded to contain-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1113 ing propositions not included in the extant treatise <I>On Plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1114 Equilibriums.</I> Pappus mentions a work <I>On Balances</I> or <I>Levers</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1115 (<G>peri\ zugw=n</G>) in which it was proved (as it also was in Philon's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1116 and Heron's <I>Mechanics</I>) that ‘greater circles overpower lesser
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1117 circles when they revolve about the same centre’.<note>Pappus, viii, p. 1068.</note> Heron, too,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1118 speaks of writings of Archimedes ‘which bear the title of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1119 <pb n=24><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1120 “works on the lever”’.<note>Heron, <I>Mechanics,</I> i. 32.</note> Simplicius refers to <I>problems on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1121 centre of gravity,</I> <G>kentrobarika/</G>, such as the many elegant
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1122 problems solved by Archimedes and others, the object of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1123 is to show how to find the centre of gravity, that is, the point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1124 in a body such that if the body is hung up from it, the body
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1125 will remain at rest in any position.<note>Simpl. on Arist. <I>De caelo,</I> ii, p. 508 a 30, Brandis; p. 543. 24, Heib.</note> This recalls the assump-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1126 tion in the <I>Quadrature of the Parabola</I> (6) that, if a body hangs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1127 at rest from a point, the centre of gravity of the body and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1128 point of suspension are in the same vertical line. Pappus has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1129 a similar remark with reference to a point of <I>support,</I> adding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1130 that the centre of gravity is determined as the intersection of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1131 two straight lines in the body, through two points of support,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1132 which straight lines are vertical when the body is in equilibrium
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1133 so supported. Pappus also gives the characteristic of the centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1134 of gravity mentioned by Simplicius, observing that this is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1135 the most fundamental principle of the theory of the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1136 gravity, the elementary propositions of which are found in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1137 Archimedes's <I>On Equilibriums</I> (<G>peri\ i)sorropiw=n</G>) and Heron's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1138 <I>Mechanics.</I> Archimedes himself cites propositions which must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1139 have been proved elsewhere, e.g. that the centre of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1140 of a cone divides the axis in the ratio 3:1, the longer segment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1141 being that adjacent to the vertex<note><I>Method,</I> Lemma 10.</note>; he also says that ‘it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1142 proved in the <I>Equilibriums</I>’ that the centre of gravity of any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1143 segment of a right-angled conoid (i. e. paraboloid of revolution)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1144 divides the axis in such a way that the portion towards the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1145 vertex is double of the remainder.<note><I>On Floating Bodies,</I> ii. 2.</note> It is possible that there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1146 was originally a larger work by Archimedes <I>On Equilibriums</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1147 of which the surviving books <I>On Plane Equilibriums</I> formed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1148 only a part; in that case <G>peri\ zugw=n</G> and <G>kentrobarika/</G> may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1149 only be alternative titles. Finally, Heron says that Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1150 medes laid down a certain procedure in a book bearing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1151 title ‘Book on Supports’.<note>Heron, <I>Mechanics,</I> i. 25.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1152 <p>4. Theon of Alexandria quotes a proposition from a work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1153 of Archimedes called <I>Catoptrica</I> (properties of mirrors) to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1154 effect that things thrown into water look larger and still
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1155 larger the farther they sink.<note>Theon on Ptolemy's <I>Syntaxis,</I> i, p. 29, Halma.</note> Olympiodorus, too, mentions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1156 <pb n=25><head>TRACES OF LOST WORKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1157 that Archimedes proved the phenomenon of refraction ‘by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1158 means of the ring placed in the vessel (of water)’.<note>Olympiodorus on Arist. <I>Meteorologica,</I> ii, p. 94, Ideler; p. 211. 18,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1159 Busse.</note> A scholiast
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1160 to the Pseudo-Euclid's <I>Catoptrica</I> quotes a proof, which he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1161 attributes to Archimedes, of the equality of the angles of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1162 incidence and of reflection in a mirror.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1163 <C>The text of Archimedes.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1164 <p>Heron, Pappus and Theon all cite works of Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1165 which no longer survive, a fact which shows that such works
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1166 were still extant at Alexandria as late as the third and fourth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1167 centuries A.D. But it is evident that attention came to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1168 concentrated on two works only, the <I>Measurement of a Circle</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1169 and <I>On the Sphere and Cylinder.</I> Eutocius (<I>fl.</I> about A. D. 500)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1170 only wrote commentaries on these works and on the <I>Plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1171 Equilibriums,</I> and he does not seem even to have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1172 acquainted with the <I>Quadrature of the Parabola</I> or the work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1173 <I>On Spirals,</I> although these have survived. Isidorus of Miletus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1174 revised the commentaries of Eutocius on the <I>Measurement
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1175 of a Circle</I> and the two Books <I>On the Sphere and Cylinder,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1176 and it would seem to have been in the school of Isidorus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1177 that these treatises were turned from their original Doric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1178 into the ordinary language, with alterations designed to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1179 them more intelligible to elementary pupils. But neither in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1180 Isidorus's time nor earlier was there any collected edition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1181 of Archimedes's works, so that those which were less read
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1182 tended to disappear.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1183 <p>In the ninth century Leon, who restored the University
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1184 of Constantinople, collected together all the works that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1185 could find at Constantinople, and had the manuscript written
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1186 (the archetype, Heiberg's A) which, through its derivatives,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1187 was, up to the discovery of the Constantinople manuscript (C)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1188 containing <I>The Method,</I> the only source for the Greek text.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1189 Leon's manuscript came, in the twelfth century, to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1190 Norman Court at Palermo, and thence passed to the House
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1191 of Hohenstaufen. Then, with all the library of Manfred, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1192 was given to the Pope by Charles of Anjou after the battle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1193 of Benevento in 1266. It was in the Papal Library in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1194 years 1269 and 1311, but, some time after 1368, passed into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1195 <pb n=26><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1196 private hands. In 1491 it belonged to Georgius Valla, who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1197 translated from it the portions published in his posthumous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1198 work <I>De expetendis et fugiendis rebus</I> (1501), and intended to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1199 publish the whole of Archimedes with Eutocius's commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1200 taries. On Valla's death in 1500 it was bought by Albertus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1201 Pius, Prince of Carpi, passing in 1530 to his nephew, Rodolphus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1202 Pius, in whose possession it remained till 1544. At some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1203 time between 1544 and 1564 it disappeared, leaving no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1204 trace.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1205 <p>The greater part of A was translated into Latin in 1269
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1206 by William of Moerbeke at the Papal Court at Viterbo. This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1207 translation, in William's own hand, exists at Rome (Cod.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1208 Ottobon. lat. 1850, Heiberg's B), and is one of our prime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1209 sources, for, although the translation was hastily done and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1210 the translator sometimes misunderstood the Greek, he followed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1211 its wording so closely that his version is, for purposes of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1212 collation, as good as a Greek manuscript. William used also,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1213 for his translation, another manuscript from the same library
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1214 which contained works not included in A. This manuscript
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1215 was a collection of works on mechanics and optics; William
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1216 translated from it the two Books <I>On Floating Bodies,</I> and it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1217 also contained the <I>Plane Equilibriums</I> and the <I>Quadrature
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1218 of the Parabola,</I> for which books William used both manu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1219 scripts.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1220 <p>The four most important extant Greek manuscripts (except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1221 C, the Constantinople manuscript discovered in 1906) were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1222 copied from A. The earliest is E, the Venice manuscript
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1223 (Marcianus 305), which was written between the years 1449
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1224 and 1472. The next is D, the Florence manuscript (Laurent.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1225 XXVIII. 4), which was copied in 1491 for Angelo Poliziano,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1226 permission having been obtained with some difficulty in con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1227 sequence of the jealousy with which Valla guarded his treasure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1228 The other two are G (Paris. 2360) copied from A after it had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1229 passed to Albertus Pius, and H (Paris. 2361) copied in 1544
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1230 by Christopherus Auverus for Georges d'Armagnac, Bishop
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1231 of Rodez. These four manuscripts, with the translation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1232 William of Moerbeke (B), enable the readings of A to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1233 inferred.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1234 <p>A Latin translation was made at the instance of Pope
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1235 Nicholas V about the year 1450 by Jacobus Cremonensis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1236 <pb n=27><head>THE TEXT OF ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1237 It was made from A, which was therefore accessible to Pope
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1238 Nicholas though it does not seem to have belonged to him.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1239 Regiomontanus made a copy of this translation about 1468
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1240 and revised it with the help of E (the Venice manuscript of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1241 the Greek text) and a copy of the same translation belonging
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1242 to Cardinal Bessarion, as well as another ‘old copy’ which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1243 seems to have been B.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1244 <p>The <I>editio princeps</I> was published at Basel (<I>apud Herva-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1245 gium</I>) by Thomas Gechauff Venatorius in 1544. The Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1246 text was based on a Nürnberg MS. (Norimberg. Cent. V,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1247 app. 12) which was copied in the sixteenth century from A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1248 but with interpolations derived from B; the Latin transla-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1249 tion was Regiomontanus's revision of Jacobus Cremonensis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1250 (Norimb. Cent. V, 15).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1251 <p>A translation by F. Commandinus published at Venice in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1252 1558 contained the <I>Measurement of a Circle, On Spirals,</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1253 <I>Quadrature of the Parabola, On Conoids and Spheroids,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1254 the <I>Sand-reckoner.</I> This translation was based on the Basel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1255 edition, but Commandinus also consulted E and other Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1256 manuscripts.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1257 <p>Torelli's edition (Oxford, 1792) also followed the <I>editio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1258 princeps</I> in the main, but Torelli also collated E. The book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1259 was brought out after Torelli's death by Abram Robertson,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1260 who also collated five more manuscripts, including D, G
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1261 and H. The collation, however, was not well done, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1262 edition was not properly corrected when in the press.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1263 <p>The second edition of Heiberg's text of all the works of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1264 Archimedes with Eutocius's commentaries, Latin translation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1265 apparatus criticus, &c., is now available (1910-15) and, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1266 course, supersedes the first edition (1880-1) and all others.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1267 It naturally includes <I>The Method,</I> the fragment of the <I>Stoma-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1268 chion,</I> and so much of the Greek text of the two Books <I>On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1269 Floating Bodies</I> as could be restored from the newly dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1270 covered Constantinople manuscript.<note><I>The Works of Archimedes,</I> edited in modern notation by the present
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1271 writer in 1897, was based on Heiberg's first edition, and the Supplement
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1272 (1912) containing <I>The Method,</I> on the original edition of Heiberg (in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1273 <I>Hermes,</I> xlii, 1907) with the translation by Zeuthen (<I>Bibliotheca Mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1274 matica,</I> vii<SUB>3</SUB>. 1906/7).</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1275 <C>Contents of <I>The Method.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1276 <p>Our description of the extant works of Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1277 may suitably begin with <I>The Method</I> (the full title is <I>On</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1278 <pb n=28><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1279 <I>Mechanical Theorems, Method</I> (communicated) <I>to Eratosthenes</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1280 Premising certain propositions in mechanics mostly taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1281 from the <I>Plane Equilibriums,</I> and a lemma which forms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1282 Prop. 1 of <I>On Conoids and Spheroids,</I> Archimedes obtains by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1283 his mechanical method the following results. The area of any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1284 segment of a section of a right-angled cone (parabola) is 4/3 of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1285 the triangle with the same base and height (Prop. 1). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1286 right cylinder circumscribing a sphere or a spheroid of revolu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1287 tion and with axis equal to the diameter or axis of revolution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1288 of the sphere or spheroid is 1 1/2 times the sphere or spheroid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1289 respectively (Props. 2, 3). Props. 4, 7, 8, 11 find the volume of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1290 any segment cut off, by a plane at right angles to the axis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1291 from any right-angled conoid (paraboloid of revolution),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1292 sphere, spheroid, and obtuse-angled conoid (hyperboloid) in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1293 terms of the cone which has the same base as the segment and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1294 equal height. In Props. 5, 6, 9, 10 Archimedes uses his method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1295 to find the centre of gravity of a segment of a paraboloid of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1296 revolution, a sphere, and a spheroid respectively. Props.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1297 12-15 and Prop. 16 are concerned with the cubature of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1298 special solid figures. (1) Suppose a prism with a square base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1299 to have a cylinder inscribed in it, the circular bases of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1300 cylinder being circles inscribed in the squares which are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1301 the bases of the prism, and suppose a plane drawn through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1302 one side of one base of the prism and through that diameter of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1303 the circle in the opposite base which is parallel to the said
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1304 side. This plane cuts off a solid bounded by two planes and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1305 by part of the curved surface of the cylinder (a solid shaped
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1306 like a hoof cut off by a plane); and Props. 12-15 prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1307 its volume is one-sixth of the volume of the prism. (2) Sup-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1308 pose a cylinder inscribed in a cube, so that the circular bases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1309 of the cylinder are circles inscribed in two opposite faces of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1310 the cube, and suppose another cylinder similarly inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1311 with reference to two other opposite faces. The two cylinders
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1312 enclose a certain solid which is actually made up of eight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1313 ‘hoofs’ like that of Prop. 12. Prop. 16 proves that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1314 volume of this solid is two-thirds of that of the cube. Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1315 medes observes in his preface that a remarkable fact about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1316 <pb n=29><head><I>THE METHOD</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1317 these solids respectively is that each of them is equal to a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1318 solid enclosed by <I>planes,</I> whereas the volume of curvilinear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1319 solids (spheres, spheroids, &c.) is generally only expressible in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1320 terms of other curvilinear solids (cones and cylinders). In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1321 accordance with his dictum that the results obtained by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1322 mechanical method are merely indicated, but not actually
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1323 proved, unless confirmed by the rigorous methods of pure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1324 geometry, Archimedes proved the facts about the two last-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1325 named solids by the orthodox method of exhaustion as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1326 regularly used by him in his other geometrical treatises; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1327 proofs, partly lost, were given in Props. 15 and 16.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1328 <p>We will first illustrate the method by giving the argument
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1329 of Prop. 1 about the area of a parabolic segment.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1330 <p>Let <I>ABC</I> be the segment, <I>BD</I> its diameter, <I>CF</I> the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1331 at <I>C.</I> Let <I>P</I> be any point on the segment, and let <I>AKF,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1332 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1333 <I>OPNM</I> be drawn parallel to <I>BD.</I> Join <I>CB</I> and produce it to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1334 meet <I>MO</I> in <I>N</I> and <I>FA</I> in <I>K,</I> and let <I>KH</I> be made equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1335 <I>KC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1336 <p>Now, by a proposition ‘proved in a lemma’ (cf. <I>Quadrature
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1337 of the Parabola,</I> Prop. 5)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1338 <MATH><I>MO</I>:<I>OP</I>=<I>CA</I>:<I>AO</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1339 =<I>CK</I>:<I>KN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1340 =<I>HK</I>:<I>KN</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1341 <p>Also, by the property of the parabola, <MATH><I>EB</I>=<I>BD</I></MATH>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1342 <MATH><I>MN</I>=<I>NO</I></MATH> and <MATH><I>FK</I>=<I>KA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1343 <p>It follows that, if <I>HC</I> be regarded as the bar of a balance,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1344 a line <I>TG</I> equal to <I>PO</I> and placed with its middle point at <I>H</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1345 balances, about <I>K,</I> the straight line <I>MO</I> placed where it is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1346 i. e. with its middle point at <I>N.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1347 <p>Similarly with <I>all</I> lines, as <I>MO, PO,</I> in the triangle <I>CFA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1348 and the segment <I>CBA</I> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1349 <p>And there are the same number of these lines. Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1350 <pb n=30><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1351 the whole segment of the parabola acting at <I>H</I> balances the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1352 triangle <I>CFA</I> placed where it is.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1353 <p>But the centre of gravity of the triangle <I>CFA</I> is at <I>W,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1354 where <MATH><I>CW</I>=2<I>WK</I></MATH> [and the whole triangle may be taken as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1355 acting at <I>W</I>].
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1356 <p>Therefore <MATH>(segment <I>ABC</I>):&utri;<I>CFA</I>=<I>WK</I>:<I>KH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1357 =1:3</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1358 so that <MATH>(segment <I>ABC</I>)=1/3&utri;<I>CFA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1359 =4/3&utri;<I>ABC</I></MATH>. Q.E.D.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1360 <p>It will be observed that Archimedes takes the segment and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1361 the triangle to be <I>made up</I> of parallel lines indefinitely close
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1362 together. In reality they are made up of indefinitely narrow
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1363 strips, but the width (<I>dx,</I> as we might say) being the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1364 for the elements of the triangle and segment respectively,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1365 divides out. And of course the weight of each element in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1366 both is proportional to the area. Archimedes also, without
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1367 mentioning <I>moments,</I> in effect assumes that the sum of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1368 moments of each particle of a figure, acting where it is, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1369 equal to the moment of the whole figure applied as one mass
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1370 at its centre of gravity.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1371 <p>We will now take the case of any segment of a spheroid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1372 of revolution, because that will cover several propositions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1373 Archimedes as particular cases.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1374 <p>The ellipse with axes <I>AA</I>′, <I>BB</I>′ is a section made by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1375 plane of the paper in a spheroid with axis <I>AA</I>′. It is required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1376 to find the volume of any right segment <I>ADC</I> of the spheroid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1377 in terms of the right cone with the same base and height.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1378 <p>Let <I>DC</I> be the diameter of the circular base of the segment.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1379 Join <I>AB, AB</I>′, and produce them to meet the tangent at <I>A</I>′ to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1380 the ellipse in <I>K, K</I>′, and <I>DC</I> produced in <I>E, F.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1381 <p>Conceive a cylinder described with axis <I>AA</I>′ and base the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1382 circle on <I>KK</I>′ as diameter, and cones described with <I>AG</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1383 axis and bases the circles on <I>EF, DC</I> as diameters.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1384 <p>Let <I>N</I> be any point on <I>AG,</I> and let <I>MOPQNQ</I>′<I>P</I>′<I>O</I>′<I>M</I>′ be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1385 drawn through <I>N</I> parallel to <I>BB</I>′ or <I>DC</I> as shown in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1386 figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1387 <p>Produce <I>A</I>′<I>A</I> to <I>H</I> so that <MATH><I>HA</I>=<I>AA</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1388 <pb n=31><head><I>THE METHOD</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1389 <p>Now <MATH><I>HA</I>:<I>AN</I>=<I>A</I>′<I>A</I>:<I>AN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1390 =<I>KA</I>:<I>AQ</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1391 =<I>MN</I>:<I>NQ</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1392 =<I>MN</I><SUP>2</SUP>:<I>MN.NQ.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1393 <p>It is now necessary to prove that <MATH><I>MN.NQ</I>=<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1394 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1395 <p>By the property of the ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1396 <MATH><I>AN.NA</I>′:<I>NP</I><SUP>2</SUP>=(1/2<I>AA</I>′)<SUP>2</SUP>:(1/2<I>BB</I>′)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1397 =<I>AN</I><SUP>2</SUP>:<I>NQ</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1398 therefore <MATH><I>NQ</I><SUP>2</SUP>:<I>NP</I><SUP>2</SUP>=<I>AN</I><SUP>2</SUP>:<I>AN.NA</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1399 =<I>NQ</I><SUP>2</SUP>:<I>NQ.QM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1400 whence <MATH><I>NP</I><SUP>2</SUP>=<I>MQ.QN.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1401 <p>Add <I>NQ</I><SUP>2</SUP> to each side, and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1402 <MATH><I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>=<I>MN.NQ.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1403 <p>Therefore, from above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1404 <MATH><I>HA</I>:<I>AN</I>=<I>MN</I><SUP>2</SUP>:(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>).</MATH> (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1405 <p>But <I>MN</I><SUP>2</SUP>, <I>NP</I><SUP>2</SUP>, <I>NQ</I><SUP>2</SUP> are to one another as the areas of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1406 circles with <I>MM</I>′, <I>PP</I>′, <I>QQ</I>′ respectively as diameters, and these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1407 <pb n=32><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1408 circles are sections made by the plane though <I>N</I> at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1409 angles to <I>AA</I>′ in the cylinder, the spheroid and the cone <I>AEF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1410 respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1411 <p>Therefore, if <I>HAA</I>′ be a lever, and the sections of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1412 spheroid and cone be both placed with their centres of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1413 at <I>H</I>, these sections placed at <I>H</I> balance, about <I>A,</I> the section
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1414 <I>MM</I>′ of the cylinder where it is.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1415 <p>Treating all the corresponding sections of the segment of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1416 the spheroid, the cone and the cylinder in the same way,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1417 we find that the cylinder with axis <I>AG,</I> where it is, balances,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1418 about <I>A,</I> the cone <I>AEF</I> and the segment <I>ADC</I> together, when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1419 both are placed with their centres of gravity at <I>H</I>; and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1420 if <I>W</I> be the centre of gravity of the cylinder, i.e. the middle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1421 point of <I>AG,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1422 <MATH><I>HA</I>:<I>AW</I>=(cylinder, axis <I>AG</I>):(cone <I>AEF</I>+segmt. <I>ADC</I>).</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1423 <p>If we call <I>V</I> the volume of the cone <I>AEF,</I> and <I>S</I> that of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1424 segment of the spheroid, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1425 <MATH>(cylinder):(<I>V</I>+<I>S</I>)=3<I>V.</I>(<I>AA</I>′<SUP>2</SUP>)/(<I>AG</I><SUP>2</SUP>):(<I>V</I>+<I>S</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1426 while <MATH><I>HA</I>:<I>AW</I>=<I>AA</I>′:1/2<I>AG.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1427 <p>Therefore <MATH><I>AA</I>′:1/2<I>AG</I>=3<I>V.</I>(<I>AA</I>′<SUP>2</SUP>)/(<I>AG</I><SUP>2</SUP>):(<I>V</I>+<I>S</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1428 and <MATH>(<I>V</I>+<I>S</I>)=3/2<I>V.</I>(<I>AA</I>′)/(<I>AG</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1429 whence <MATH><I>S</I>=<I>V</I>((3<I>AA</I>′)/(2<I>AG</I>)-1)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1430 <p>Again, let <I>V</I>′ be the volume of the cone <I>ADC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1431 <p>Then <MATH><I>V</I>:<I>V</I>′=<I>EG</I><SUP>2</SUP>:<I>DG</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1432 =(<I>BB</I>′<SUP>2</SUP>)/(<I>AA</I>′<SUP>2</SUP>).<I>AG</I><SUP>2</SUP>:<I>DG</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1433 <p>But <MATH><I>DG</I><SUP>2</SUP>:<I>AG.GA</I>′=<I>BB</I>′<SUP>2</SUP>:<I>AA</I>′<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1434 <p>Therefore <MATH><I>V</I>:<I>V</I>′=<I>AG</I><SUP>2</SUP>:<I>AG.GA</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1435 =<I>AG</I>:<I>GA</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1436 <pb n=33><head><I>THE METHOD</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1437 <p>It follows that <MATH><I>S</I>=<I>V</I>′.<I>AG/GA</I>′(3<I>AA</I>′)/(2<I>AG</I>)-1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1438 =<I>V</I>′.(3/2<I>AA</I>′-<I>AG</I>)/(<I>A</I>′<I>G</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1439 =<I>V</I>′.(1/2<I>AA</I>′+<I>A</I>′<I>G</I>)/(<I>A</I>′<I>G</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1440 which is the result stated by Archimedes in Prop. 8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1441 <p>The result is the same for the segment of a sphere. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1442 proof, of course slightly simpler, is given in Prop. 7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1443 <p>In the particular case where the segment is half the sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1444 or spheroid, the relation becomes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1445 <MATH><I>S</I>=2<I>V</I>′</MATH>, (Props. 2, 3)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1446 and it follows that the volume of the whole sphere or spheroid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1447 is 4<I>V</I>′, where <I>V</I>′ is the volume of the cone <I>ABB</I>′; i.e. the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1448 volume of the sphere or spheroid is two-thirds of that of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1449 circumscribing cylinder.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1450 <p>In order now to find the centre of gravity of the segment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1451 of a spheroid, we must have the segment acting <I>where it is,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1452 not at <I>H.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1453 <p>Therefore formula (1) above will not serve. But we found
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1454 that <MATH><I>MN.NQ</I>=(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1455 whence <MATH><I>MN</I><SUP>2</SUP>:(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>)=(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>):<I>NQ</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1456 therefore <MATH><I>HA</I>:<I>AN</I>=(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>):<I>NQ</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1457 <p>(This is separately proved by Archimedes for the sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1458 in Prop. 9.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1459 <p>From this we derive, as usual, that the cone <I>AEF</I> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1460 segment <I>ADC</I> both acting <I>where they are</I> balance a volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1461 equal to the cone <I>AEF</I> placed with its centre of gravity at <I>H.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1462 <p>Now the centre of gravity of the cone <I>AEF</I> is on the line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1463 <I>AG</I> at a distance 3/4<I>AG</I> from <I>A.</I> Let <I>X</I> be the required centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1464 of gravity of the segment. Then, taking moments about <I>A,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1465 we have <MATH><I>V.HA</I>=<I>S.AX</I>+<I>V.</I>3/4<I>AG</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1466 or <MATH><I>V</I>(<I>AA</I>′-3/4<I>AG</I>)=<I>S.AX</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1467 =<I>V</I>(3/2<I>AA</I>′)/<I>AG</I>-1)<I>AX</I></MATH>, from above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1468 <pb n=34><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1469 <p>Therefore <MATH><I>AX</I>:<I>AG</I>=(<I>AA</I>′-3/4<I>AG</I>):(3/2<I>AA</I>′-<I>AG</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1470 =(4<I>AA</I>′-3<I>AG</I>):(6<I>AA</I>′-4<I>AG</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1471 whence <MATH><I>AX</I>:<I>XG</I>=(4<I>AA</I>′-3<I>AG</I>):(2<I>AA</I>′-<I>AG</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1472 =(<I>AG</I>+4<I>A</I>′<I>G</I>):(<I>AG</I>+2<I>A</I>′<I>G</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1473 which is the result obtained by Archimedes in Prop. 9 for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1474 sphere and in Prop. 10 for the spheroid.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1475 <p>In the case of the hemi-spheroid or hemisphere the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1476 <I>AX</I>:<I>XG</I> becomes 5:3, a result obtained for the hemisphere in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1477 Prop. 6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1478 <p>The cases of the paraboloid of revolution (Props. 4, 5) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1479 the hyperboloid of revolution (Prop. 11) follow the same course,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1480 and it is unnecessary to reproduce them.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1481 <p>For the cases of the two solids dealt with at the end of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1482 treatise the reader must be referred to the propositions them-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1483 selves. Incidentally, in Prop. 13, Archimedes finds the centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1484 of gravity of the half of a cylinder cut by a plane through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1485 the axis, or, in other words, the centre of gravity of a semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1486 circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1487 <p>We will now take the other treatises in the order in which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1488 they appear in the editions.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1489 <C>On the Sphere and Cylinder, I, II.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1490 <p>The main results obtained in Book I are shortly stated in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1491 a prefatory letter to Dositheus. Archimedes tells us that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1492 they are new, and that he is now publishing them for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1493 first time, in order that mathematicians may be able to ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1494 amine the proofs and judge of their value. The results are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1495 (1) that the surface of a sphere is four times that of a great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1496 circle of the sphere, (2) that the surface of any segment of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1497 sphere is equal to a circle the radius of which is equal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1498 straight line drawn from the vertex of the segment to a point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1499 on the circumference of the base, (3) that the volume of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1500 cylinder circumscribing a sphere and with height equal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1501 diameter of the sphere is 3/2 of the volume of the sphere,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1502 (4) that the surface of the circumscribing cylinder including
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1503 its bases is also 3/2 of the surface of the sphere. It is worthy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1504 of note that, while the first and third of these propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1505 appear in the book in this order (Props. 33 and 34 respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1506 <pb n=35><head>ON THE SPHERE AND CYLINDER, I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1507 tively), this was not the order of their discovery; for Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1508 medes tells us in <I>The Method</I> that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1509 <p>‘from the theorem that a sphere is four times as great as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1510 cone with a great circle of the sphere as base and with height
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1511 equal to the radius of the sphere I conceived the notion that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1512 the surface of any sphere is four times as great as a great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1513 circle in it; for, judging from the fact that any circle is equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1514 to a triangle with base equal to the circumference and height
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1515 equal to the radius of the circle, I apprehended that, in like
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1516 manner, any sphere is equal to a cone with base equal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1517 surface of the sphere and height equal to the radius’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1518 <p>Book I begins with definitions (of ‘concave in the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1519 direction’ as applied to curves or broken lines and surfaces, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1520 a ‘solid sector’ and a ‘solid rhombus’) followed by five
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1521 Assumptions, all of importance. <I>Of all lines which have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1522 same extremities the straight line is the least</I>, and, if there are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1523 two curved or bent lines in a plane having the same extremi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1524 ties and concave in the same direction, but one is wholly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1525 included by, or partly included by and partly common with,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1526 the other, then that which is included is the lesser of the two.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1527 Similarly with plane surfaces and surfaces concave in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1528 same direction. Lastly, Assumption 5 is the famous ‘Axiom
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1529 of Archimedes’, which however was, according to Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1530 himself, used by earlier geometers (Eudoxus in particular), to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1531 the effect that <I>Of unequal magnitudes the greater exceeds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1532 the less by such a magnitude as, when added to itself, can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1533 made to exceed any assigned magnitude of the same kind</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1534 the axiom is of course practically equivalent to Eucl. V, Def. 4,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1535 and is closely connected with the theorem of Eucl. X. 1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1536 <p>As, in applying the method of exhaustion, Archimedes uses
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1537 both circumscribed and inscribed figures with a view to <I>com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1538 pressing</I> them into coalescence with the curvilinear figure to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1539 be measured, he has to begin with propositions showing that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1540 given two unequal magnitudes, then, however near the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1541 of the greater to the less is to 1, it is possible to find two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1542 straight lines such that the greater is to the less in a still less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1543 ratio (>1), and to circumscribe and inscribe similar polygons to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1544 a circle or sector such that the perimeter or the area of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1545 circumscribed polygon is to that of the inner in a ratio less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1546 than the given ratio (Props. 2-6): also, just as Euclid proves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1547 <pb n=36><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1548 that, if we continually double the number of the sides of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1549 regular polygon inscribed in a circle, segments will ultimately be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1550 left which are together less than any assigned area, Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1551 has to supplement this (Prop. 6) by proving that, if we increase
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1552 the number of the sides of a <I>circumscribed</I> regular polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1553 sufficiently, we can make the excess of the area of the polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1554 over that of the circle less than any given area. Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1555 then addresses himself to the problems of finding the <I>surface</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1556 any right cone or cylinder, problems finally solved in Props. 13
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1557 (the cylinder) and 14 (the cone). Circumscribing and inscrib-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1558 ing regular polygons to the bases of the cone and cylinder, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1559 erects pyramids and prisms respectively on the polygons as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1560 bases and circumscribed or inscribed to the cone and cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1561 respectively. In Props. 7 and 8 he finds the surface of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1562 pyramids inscribed and circumscribed to the cone, and in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1563 Props. 9 and 10 he proves that the surfaces of the inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1564 and circumscribed pyramids respectively (excluding the base)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1565 are less and greater than the surface of the cone (excluding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1566 the base). Props. 11 and 12 prove the same thing of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1567 prisms inscribed and circumscribed to the cylinder, and finally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1568 Props. 13 and 14 prove, by the method of exhaustion, that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1569 surface of the cone or cylinder (excluding the bases) is equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1570 to the circle the radius of which is a mean proportional
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1571 between the ‘side’ (i.e. generator) of the cone or cylinder and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1572 the radius or diameter of the base (i.e. is equal to <G>p</G><I>rs</I> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1573 case of the cone and 2<G>p</G><I>rs</I> in the case of the cylinder, where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1574 <I>r</I> is the radius of the base and <I>s</I> a generator). As Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1575 here applies the method of exhaustion for the first time, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1576 will illustrate by the case of the cone (Prop. 14).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1577 <p>Let <I>A</I> be the base of the cone, <I>C</I> a straight line equal to its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1578 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1579 radius, <I>D</I> a line equal to a generator of the cone, <I>E</I> a mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1580 proportional to <I>C, D</I>, and <I>B</I> a circle with radius equal to <I>E</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1581 <pb n=37><head>ON THE SPHERE AND CYLINDER, I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1582 <p>If <I>S</I> is the surface of the cone, we have to prove that <I>S</I>=<I>B</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1583 For, if <I>S</I> is not equal to <I>B</I>, it must be either greater or less.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1584 <p>I. Suppose <I>B</I><<I>S</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1585 <p>Circumscribe a regular polygon about <I>B</I>, and inscribe a similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1586 polygon in it, such that the former has to the latter a ratio less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1587 than <I>S</I>:<I>B</I> (Prop. 5). Describe about <I>A</I> a similar polygon and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1588 set up from it a pyramid circumscribing the cone.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1589 <p>Then <MATH>(polygon about <I>A</I>):(polygon about <I>B</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1590 =<I>C</I><SUP>2</SUP>:<I>E</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1591 =<I>C</I>:<I>D</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1592 =(polygon about <I>A</I>):(surface of pyramid)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1593 <p>Therefore (surface of pyramid)=(polygon about <I>B</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1594 <p>But (polygon about <I>B</I>):(polygon in <I>B</I>)<<I>S</I>:<I>B</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1595 therefore (surface of pyramid):(polygon in <I>B</I>)<<I>S</I>:<I>B</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1596 <p>But this is impossible, since (surface of pyramid)><I>S</I>, while
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1597 (polygon in <I>B</I>)<<I>B</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1598 therefore <I>B</I> is not less than <I>S</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1599 <p>II. Suppose <I>B</I>><I>S</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1600 <p>Circumscribe and inscribe similar regular polygons to <I>B</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1601 such that the former has to the latter a ratio < <I>B</I>:<I>S</I>. Inscribe
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1602 in <I>A</I> a similar polygon, and erect on <I>A</I> the inscribed pyramid.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1603 <p>Then <MATH>(polygon in <I>A</I>):(polygon in <I>B</I>)=<I>C</I><SUP>2</SUP>:<I>E</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1604 =<I>C</I>:<I>D</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1605 >(polygon in <I>A</I>):(surface of pyramid)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1606 <p>(The latter inference is clear, because the ratio of <I>C</I>:<I>D</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1607 greater than the ratio of the perpendiculars from the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1608 <I>A</I> and from the vertex of the pyramid respectively on any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1609 side of the polygon in <I>A</I>; in other words, if <MATH><G>b</G><<G>a</G><1/2<G>p</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1610 sin<G>a</G>>sin<G>b</G></MATH>.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1611 <p>Therefore (surface of pyramid)>(polygon in <I>B</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1612 <p>But (polygon about <I>B</I>):(polygon in <I>B</I>)<<I>B</I>:<I>S</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1613 whence (<I>a fortiori</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1614 <p>(polygon about <I>B</I>):(surface of pyramid)<<I>B</I>:<I>S</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1615 which is impossible, for (polygon about <I>B</I>)><I>B</I>, while (surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1616 of pyramid) < <I>S</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1617 <pb n=38><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1618 <p>Therefore <I>B</I> is not greater than <I>S</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1619 <p>Hence <I>S</I>, being neither greater nor less than <I>B</I>, is equal to <I>B</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1620 <p>Archimedes next addresses himself to the problem of finding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1621 the surface and volume of a sphere or a segment thereof, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1622 has to interpolate some propositions about ‘solid rhombi’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1623 (figures made up of two right cones, unequal or equal, with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1624 bases coincident and vertices in opposite directions) the neces-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1625 sity of which will shortly appear.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1626 <p>Taking a great circle of the sphere or a segment of it, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1627 inscribes a regular polygon of an even number of sides bisected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1628 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1629 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1630 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1631 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1632 by the diameter <I>AA</I>′, and approximates to the surface and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1633 volume of the sphere or segment by making the polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1634 revolve about <I>AA</I>′ and measuring the surface and volume of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1635 solid so inscribed (Props. 21-7). He then does the same for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1636 a circumscribed solid (Props. 28-32). Construct the inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1637 polygons as shown in the above figures. Joining <I>BB</I>′, <I>CC</I>′, ...,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1638 <I>CB</I>′, <I>DC</I>′ ... we see that <I>BB</I>′, <I>CC</I>′ ... are all parallel, and so are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1639 <I>AB, CB</I>′, <I>DC</I>′ ....
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1640 <p>Therefore, by similar triangles, <MATH><I>BF</I>:<I>FA</I>=<I>A</I>′<I>B</I>:<I>BA</I></MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1641 <MATH><I>BF</I>:<I>FA</I>=<I>B</I>′<I>F</I>:<I>FK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1642 =<I>CG</I>:<I>GK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1643 =<I>C</I>′<I>G</I>:<I>GL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1644 . . . . . . .
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1645 =<I>E</I>′<I>I</I>:<I>IA</I>′ in Fig. 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1646 (=<I>PM</I>:<I>MN</I> in Fig. 2)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1647 <pb n=39><head>ON THE SPHERE AND CYLINDER, I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1648 whence, adding antecedents and consequents, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1649 (Fig. 1) <MATH>(<I>BB</I>′+<I>CC</I>′+ ... +<I>EE</I>′):<I>AA</I>′=<I>A</I>′<I>B</I>:<I>BA</I></MATH>, (Prop. 21)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1650 (Fig. 2) <MATH>(<I>BB</I>′+<I>CC</I>′+ ... +1/2<I>PP</I>′):<I>AM</I>=<I>A</I>′<I>B</I>:<I>BA</I></MATH>. (Prop. 22)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1651 <p>When we make the polygon revolve about <I>AA</I>′, the surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1652 of the inscribed figure so obtained is made up of the surfaces
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1653 of cones and frusta of cones; Prop. 14 has proved that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1654 surface of the cone <I>ABB</I>′ is what we should write <G>p</G>.<I>AB.BF</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1655 and Prop. 16 has proved that the surface of the frustum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1656 <I>BCC</I>′<I>B</I>′ is <MATH><G>p</G>.<I>BC</I> (<I>BF</I>+<I>CG</I>)</MATH>. It follows that, since <I>AB</I>=
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1657 <I>BC</I>= ..., the surface of the inscribed solid is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1658 <MATH><G>p</G>.<I>AB</I> {1/2<I>BB</I>′+1/2(<I>BB</I>′+<I>CC</I>′)+ ...}</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1659 that is, <MATH><G>p</G>.<I>AB</I>(<I>BB</I>′+<I>CC</I>′+ ... +<I>EE</I>′)</MATH> (Fig. 1), (Prop. 24)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1660 or <MATH><G>p</G>.<I>AB</I>(<I>BB</I>′+<I>CC</I>′+ ... +1/2<I>PP</I>′)</MATH> (Fig. 2). (Prop. 35)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1661 <p>Hence, from above, the surface of the inscribed solid is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1662 <G>p</G>.<I>A</I>′<I>B.AA</I>′ or <G>p</G>.<I>A</I>′<I>B.AM</I>, and is therefore less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1663 <G>p</G>.<I>AA</I>′<SUP>2</SUP> (Prop. 25) or <G>p</G>.<I>A</I>′<I>A.AM</I>, that is, <G>p</G>.<I>AP</I><SUP>2</SUP> (Prop. 37).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1664 <p>Similar propositions with regard to surfaces formed by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1665 revolution about <I>AA</I>′ of regular circumscribed solids prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1666 that their surfaces are greater than <G>p</G>.<I>AA</I>′<SUP>2</SUP> and <G>p</G>.<I>AP</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1667 respectively (Props. 28-30 and Props. 39-40). The case of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1668 segment is more complicated because the circumscribed poly-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1669 gon with its sides parallel to <I>AB, BC ... DP</I> circumscribes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1670 the <I>sector POP</I>′. Consequently, if the segment is less than a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1671 semicircle, as <I>CAC</I>′, the base of the circumscribed polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1672 (<I>cc</I>′) is on the side of <I>CC</I>′ towards <I>A</I>, and therefore the circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1673 scribed polygon leaves over a small strip of the inscribed. This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1674 complication is dealt with in Props. 39-40. Having then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1675 arrived at circumscribed and inscribed figures with surfaces
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1676 greater and less than <G>p</G>.<I>AA</I>′<SUP>2</SUP> and <G>p</G>.<I>AP</I><SUP>2</SUP> respectively, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1677 having proved (Props. 32, 41) that the surfaces of the circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1678 scribed and inscribed figures are to one another in the duplicate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1679 ratio of their sides, Archimedes proceeds to prove formally, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1680 the method of exhaustion, that the surfaces of the sphere and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1681 segment are equal to these circles respectively (Props. 33 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1682 42); <G>p</G>.<I>AA</I>′<SUP>2</SUP> is of course equal to four times the great circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1683 of the sphere. The segment is, for convenience, taken to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1684 <pb n=40><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1685 less than a hemisphere, and Prop. 43 proves that the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1686 formula applies also to a segment greater than a hemisphere.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1687 <p>As regards the volumes different considerations involving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1688 ‘solid rhombi’ come in. For convenience Archimedes takes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1689 in the case of the whole sphere, an inscribed polygon of 4<I>n</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1690 sides (Fig. 1). It is easily seen that the solid figure formed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1691 by its revolution is made up of the following: first, the solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1692 rhombus formed by the revolution of the quadrilateral <I>ABOB</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1693 (the volume of this is shown to be equal to the cone with base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1694 equal to the surface of the cone <I>ABB</I>′ and height equal to <I>p</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1695 the perpendicular from <I>O</I> on <I>AB</I>, Prop. 18); secondly, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1696 extinguisher-shaped figure formed by the revolution of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1697 triangle <I>BOC</I> about <I>AA</I>′ (this figure is equal to the difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1698 between two solid rhombi formed by the revolution of <I>TBOB</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1699 and <I>TCOC</I>′ respectively about <I>AA</I>′, where <I>T</I> is the point of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1700 intersection of <I>CB, C</I>′<I>B</I>′ produced with <I>A</I>′<I>A</I> produced, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1701 this difference is proved to be equal to a cone with base equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1702 to the surface of the frustum of a cone described by <I>BC</I> in its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1703 revolution and height equal to <I>p</I> the perpendicular from <I>O</I> on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1704 <I>BC</I>, Prop. 20); and so on; finally, the figure formed by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1705 revolution of the triangle <I>COD</I> about <I>AA</I>′ is the difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1706 between a cone and a solid rhombus, which is proved equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1707 a cone with base equal to the surface of the frustum of a cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1708 described by <I>CD</I> in its revolution and height <I>p</I> (Prop. 19).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1709 Consequently, by addition, the volume of the whole solid of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1710 revolution is equal to the cone with base equal to its whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1711 surface and height <I>p</I> (Prop. 26). But the whole of the surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1712 of the solid is less than 4<G>p</G><I>r</I><SUP>2</SUP>, and <I>p</I> < <I>r</I>; therefore the volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1713 of the inscribed solid is less than four times the cone with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1714 base <G>p</G><I>r</I><SUP>2</SUP> and height <I>r</I> (Prop. 27).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1715 <p>It is then proved in a similar way that the revolution of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1716 the similar circumscribed polygon of 4<I>n</I> sides gives a solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1717 the volume of which is <I>greater</I> than four times the same cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1718 (Props. 28-31 Cor.). Lastly, the volumes of the circumscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1719 and inscribed figures are to one another in the triplicate ratio of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1720 their sides (Prop. 32); and Archimedes is now in a position to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1721 apply the method of exhaustion to prove that the volume of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1722 the sphere is 4 times the cone with base <G>p</G><I>r</I><SUP>2</SUP> and height <I>r</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1723 (Prop. 34).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1724 <p>Dealing with the segment of a sphere, Archimedes takes, for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1725 <pb n=41><head>ON THE SPHERE AND CYLINDER, I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1726 convenience, a segment less than a hemisphere and, by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1727 same chain of argument (Props. 38, 40 Corr., 41 and 42), proves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1728 (Prop. 44) that the volume of the <I>sector</I> of the sphere bounded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1729 by the surface of the segment is equal to a cone with base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1730 equal to the surface of the segment and height equal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1731 radius, i.e. the cone with base <G>p</G>.<I>AP</I><SUP>2</SUP> and height <I>r</I> (Fig. 2).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1732 <p>It is noteworthy that the proportions obtained in Props. 21,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1733 22 (see p. 39 above) can be expressed in trigonometrical form.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1734 If 4<I>n</I> is the number of the sides of the polygon inscribed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1735 the circle, and 2<I>n</I> the number of the sides of the polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1736 inscribed in the segment, and if the angle <I>AOP</I> is denoted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1737 by <G>a</G>, the trigonometrical equivalents of the proportions are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1738 respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1739 (1) <MATH>sin<G>p</G>/(2<I>n</I>)+sin(2<G>p</G>)/(2<I>n</I>)+ ... +sin(2<I>n</I>-1) <G>p</G>/(2<I>n</I>)=cot<G>p</G>/(4<I>n</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1740 (2) <MATH>2 {sin<G>a</G>/<I>n</I>+sin(2<G>a</G>)/<I>n</I>+ ... +sin(<I>n</I>-1)<G>a</G>/<I>n</I>} +sin<G>a</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1741 =(1-cos<G>a</G>) cot<G>a</G>/(2<I>n</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1742 Thus the two proportions give in effect a summation of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1743 series
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1744 <MATH>sin<G>q</G>+sin2<G>q</G>+ ... +sin(<I>n</I>-1)<G>q</G></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1745 both generally where <I>n</I><G>q</G> is equal to any angle <G>a</G> less than <G>p</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1746 and in the particular case where <I>n</I> is even and <MATH><G>q</G>=<G>p</G>/<I>n</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1747 Props. 24 and 35 prove that the areas of the circles equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1748 the surfaces of the solids of revolution described by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1749 polygons inscribed in the sphere and segment are the above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1750 series multiplied by 4<G>p</G><I>r</I><SUP>2</SUP>sin<G>p</G>/(4<I>n</I>) and <G>p</G><I>r</I><SUP>2</SUP>.2 sin<G>a</G>/(2<I>n</I>) respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1751 and are therefore 4<G>p</G><I>r</I><SUP>2</SUP>cos <G>p</G>/(4<I>n</I>) and <G>p</G><I>r</I><SUP>2</SUP>.2 cos<G>a</G>/(2<I>n</I>) (1-cos<G>a</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1752 respectively. Archimedes's results for the surfaces of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1753 sphere and segment, 4<G>p</G><I>r</I><SUP>2</SUP> and 2<G>p</G><I>r</I><SUP>2</SUP>(1-cos<G>a</G>), are the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1754 limiting values of these expressions when <I>n</I> is indefinitely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1755 increased and when therefore cos<G>p</G>/(4<I>n</I>) and cos<G>a</G>/(2<I>n</I>) become
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1756 unity. And the two series multiplied by 4<G>p</G><I>r</I><SUP>2</SUP>sin<G>p</G>/(4<I>n</I>) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1757 <pb n=42><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1758 <G>p</G><I>r</I><SUP>2</SUP>.2 sin <G>a</G>/(2<I>n</I>) respectively are (when <I>n</I> is indefinitely increased)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1759 precisely what we should represent by the integrals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1760 <MATH>4<G>p</G><I>r</I><SUP>2</SUP>.(1/2)&int,<<SUP><G>p</G></SUP><SUB>0</SUB>>sin<G>q</G><I>d</I><G>q</G>, or 4<G>p</G><I>r</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1761 and <MATH><G>p</G><I>r</I><SUP>2</SUP>.&int,<<SUP><G>a</G></SUP><SUB>0</SUB>> 2 sin<G>q</G><I>d</I><G>q</G>, or 2<G>p</G><I>r</I><SUP>2</SUP>(1-cos<G>a</G>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1762 <p>Book II contains six problems and three theorems. Of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1763 theorems Prop. 2 completes the investigation of the volume of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1764 any segment of a sphere, Prop. 44 of Book I having only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1765 brought us to the volume of the corresponding sector. If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1766 <I>ABB</I>′ be a segment of a sphere cut off by a plane at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1767 angles to <I>AA</I>′, we learnt in I. 44 that the volume of the <I>sector</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1768 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1769 <I>OBAB</I>′ is equal to the cone with base equal to the surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1770 of the segment and height equal to the radius, i.e. 1/3<G>p</G>.<I>AB</I><SUP>2</SUP>.<I>r</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1771 where <I>r</I> is the radius. The volume of the segment is therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1772 <MATH>1/3<G>p</G>.<I>AB</I><SUP>2</SUP>.<I>r</I>-1/3<G>p</G>.<I>BM</I><SUP>2</SUP>.<I>OM</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1773 <p>Archimedes wishes to express this as a cone with base the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1774 same as that of the segment. Let <I>AM</I>, the height of the seg-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1775 ment, =<I>h</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1776 <p>Now <MATH><I>AB</I><SUP>2</SUP>:<I>BM</I><SUP>2</SUP>=<I>A</I>′<I>A</I>:<I>A</I>′<I>M</I>=2<I>r</I>:(2<I>r</I>-<I>h</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1777 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1778 <MATH>1/3<G>p</G>(<I>AB</I><SUP>2</SUP>.<I>r</I>-<I>BM</I><SUP>2</SUP>.<I>OM</I>)=1/3<G>p</G>.<I>BM</I><SUP>2</SUP>{(2<I>r</I><SUP>2</SUP>)/(2<I>r</I>-<I>h</I>)-(<I>r</I>-<I>h</I>)}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1779 =1/3<G>p</G>.<I>BM</I><SUP>2</SUP>.<I>h</I>((3<I>r</I>-<I>h</I>)/(2<I>r</I>-<I>h</I>))</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1780 <p>That is, the segment is equal to the cone with the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1781 base as that of the segment and height <MATH><I>h</I>(3<I>r</I>-<I>h</I>)/(2<I>r</I>-<I>h</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1782 <pb n=43><head>ON THE SPHERE AND CYLINDER, II</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1783 This is expressed by Archimedes thus. If <I>HM</I> is the height
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1784 of the required cone,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1785 <MATH><I>HM</I>:<I>AM</I>=(<I>OA</I>′+<I>A</I>′<I>M</I>):<I>A</I>′<I>M</I></MATH>, (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1786 and similarly the cone equal to the segment <I>A</I>′<I>BB</I>′ has the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1787 height <I>H</I>′<I>M</I>, where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1788 <MATH><I>H</I>′<I>M</I>:<I>A</I>′<I>M</I>=(<I>OA</I>+<I>AM</I>):<I>AM</I></MATH>. (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1789 His proof is, of course, not in the above form but purely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1790 geometrical.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1791 <p>This proposition leads to the most important proposition in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1792 the Book, Prop. 4, which solves the problem <I>To cut a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1793 sphere by a plane in such a way that the volumes of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1794 segments are to one another in a given ratio</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1795 <C><I>Cubic equation arising out of II. 4</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1796 <p>If <I>m</I>:<I>n</I> be the given ratio of the cones which are equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1797 the segments and the heights of which are <I>h, h</I>′, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1798 <MATH><I>h</I>((3<I>r</I>-<I>h</I>)/(2<I>r</I>-<I>h</I>))=<I>(m/n)h</I>′((3<I>r</I>-<I>h</I>′)/(2<I>r</I>-<I>h</I>′))</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1799 and, if we eliminate <I>h</I>′ by means of the relation <I>h</I>+<I>h</I>′=2<I>r</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1800 we easily obtain the following cubic equation in <I>h</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1801 <MATH><I>h</I><SUP>3</SUP>-3<I>h</I><SUP>2</SUP><I>r</I>+(4<I>m</I>)/(<I>m</I>+<I>n</I>)<I>r</I><SUP>3</SUP>=0</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1802 <p>Archimedes in effect reduces the problem to this equation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1803 which, however, he treats as a particular case of the more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1804 general problem corresponding to the equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1805 <MATH>(<I>r</I>+<I>h</I>):<I>b</I>=<I>c</I><SUP>2</SUP>:(2<I>r</I>-<I>h</I>)<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1806 where <I>b</I> is a given length and <I>c</I><SUP>2</SUP> any given area,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1807 or <MATH><I>x</I><SUP>2</SUP>(<I>a</I>-<I>x</I>)=<I>bc</I><SUP>2</SUP></MATH>, where <MATH><I>x</I>=2<I>r</I>-<I>h</I></MATH> and <MATH>3<I>r</I>=<I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1808 <p>Archimedes obtains his cubic equation with one unknown
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1809 by means of a <I>geometrical</I> elimination of <I>H, H</I>′ from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1810 equation <MATH><I>HM</I>=<I>(m/n).H</I>′<I>M</I></MATH>, where <I>HM, H</I>′<I>M</I> have the values
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1811 determined by the proportions (1) and (2) above, after which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1812 the one variable point <I>M</I> remaining corresponds to the one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1813 unknown of the cubic equation. His method is, first, to find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1814 <pb n=44><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1815 values for each of the ratios <I>A</I>′<I>H</I>′:<I>H</I>′<I>M</I> and <I>H</I>′<I>H</I>:<I>A</I>′<I>H</I>′ which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1816 are alike independent of <I>H, H</I>′ and then, secondly, to equate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1817 the ratio compounded of these two to the known value of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1818 ratio <I>HH</I>′:<I>H</I>′<I>M</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1819 (<G>a</G>) We have, from (2),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1820 <MATH><I>A</I>′<I>H</I>′:<I>H</I>′<I>M</I>=<I>OA</I>:(<I>OA</I>+<I>AM</I>)</MATH>. (3)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1821 (<G>b</G>) From (1) and (2), <I>separando</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1822 <MATH><I>AH</I>:<I>AM</I>=<I>OA</I>′:<I>A</I>′<I>M</I></MATH>, (4)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1823 <MATH><I>A</I>′<I>H</I>′:<I>A</I>′<I>M</I>=<I>OA</I>:<I>AM</I></MATH>. (5)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1824 <p>Equating the values of the ratio <I>A</I>′<I>M</I>:<I>AM</I> given by (4), (5),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1825 we have <MATH><I>OA</I>′:<I>AH</I>=<I>A</I>′<I>H</I>′:<I>OA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1826 =<I>OH</I>′:<I>OH</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1827 whence <MATH><I>HH</I>′:<I>OH</I>′=<I>OH</I>′:<I>A</I>′<I>H</I>′</MATH>, (since <MATH><I>OA</I>=<I>OA</I>′</MATH>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1828 or <MATH><I>HH</I>′.<I>A</I>′<I>H</I>′=<I>OH</I>′<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1829 so that <MATH><I>HH</I>′:<I>A</I>′<I>H</I>′=<I>OH</I>′<SUP>2</SUP>:<I>A</I>′<I>H</I>′<SUP>2</SUP></MATH>. (6)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1830 <p>But, by (5), <MATH><I>OA</I>′:<I>A</I>′<I>H</I>′=<I>AM</I>:<I>A</I>′<I>M</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1831 and, <I>componendo</I>, <MATH><I>OH</I>′:<I>A</I>′<I>H</I>′=<I>AA</I>′:<I>A</I>′<I>M</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1832 <p>By substitution in (6),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1833 <MATH><I>HH</I>′:<I>A</I>′<I>H</I>′=<I>AA</I>′<SUP>2</SUP>:<I>A</I>′<I>M</I><SUP>2</SUP></MATH>. (7)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1834 <p>Compounding with (3), we obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1835 <MATH><I>HH</I>′:<I>H</I>′<I>M</I>=(<I>AA</I>′<SUP>2</SUP>:<I>A</I>′<I>M</I><SUP>2</SUP>).(<I>OA</I>:<I>OA</I>+<I>AM</I>)</MATH>. (8)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1836 <p>[The algebraical equivalent of this is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1837 <MATH>(<I>m</I>+<I>n</I>)/<I>n</I>=(4<I>r</I><SUP>3</SUP>)/((2<I>r</I>-<I>h</I>)<SUP>2</SUP> (<I>r</I>+<I>h</I>))</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1838 which reduces to <MATH>(<I>m</I>+<I>n</I>)/<I>m</I>=(4<I>r</I><SUP>3</SUP>)/(3<I>h</I><SUP>2</SUP><I>r</I>-<I>h</I><SUP>3</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1839 or <MATH><I>h</I><SUP>3</SUP>-3<I>h</I><SUP>2</SUP><I>r</I>+(4<I>m</I>)/(<I>m</I>+<I>n</I>)<I>r</I><SUP>3</SUP>=0, as above.]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1840 <p>Archimedes expresses the result (8) more simply by pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1841 ducing <I>OA</I> to <I>D</I> so that <MATH><I>OA</I>=<I>AD</I></MATH>, and then dividing <I>AD</I> at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1842 <pb n=45><head>ON THE SPHERE AND CYLINDER, II</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1843 <I>E</I> so that <MATH><I>AD</I>:<I>DE</I>=<I>HH</I>′:<I>H</I>′<I>M</I></MATH> or (<I>m</I>+<I>n</I>):<I>n</I>. We have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1844 then <MATH><I>OA</I>=<I>AD</I></MATH> and <MATH><I>OA</I>+<I>AM</I>=<I>MD</I></MATH>, so that (8) reduces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1845 <MATH><I>AD</I>:<I>DE</I>=(<I>AA</I>′<SUP>2</SUP>:<I>A</I>′<I>M</I><SUP>2</SUP>).(<I>AD</I>:<I>MD</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1846 or <MATH><I>MD</I>:<I>DE</I>=<I>AA</I>′<SUP>2</SUP>:<I>A</I>′<I>M</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1847 <p>Now, says Archimedes, <I>D</I> is given, since <MATH><I>AD</I>=<I>OA</I></MATH>. Also,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1848 <I>AD</I>:<I>DE</I> being a given ratio, <I>DE</I> is given. Hence the pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1849 blem reduces itself to that of dividing <I>A</I>′<I>D</I> into two parts at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1850 <I>M</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1851 <MATH><I>MD</I>:(a given length)=(a given area):<I>A</I>′<I>M</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1852 <p>That is, the generalized equation is of the form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1853 <MATH><I>x</I><SUP>2</SUP>(<I>a</I>-<I>x</I>)=<I>bc</I><SUP>2</SUP></MATH>, as above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1854 <C>(i) Archimedes's own solution of the cubic.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1855 <p>Archimedes adds that, ‘if the problem is propounded in this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1856 general form, it requires a <G>diorismo/s</G> [i.e. it is necessary to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1857 investigate the limits of possibility], but if the conditions are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1858 added which exist in the present case [i.e. in the actual
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1859 problem of Prop. 4], it does not require a <G>diorismo/s</G>’ (in other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1860 words, a solution is always possible). He then promises to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1861 give ‘at the end’ an analysis and synthesis of both problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1862 [i.e. the <G>diorismo/s</G> and the problem itself]. The promised
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1863 solutions do not appear in the treatise as we have it, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1864 Eutocius gives solutions taken from ‘an old book’ which he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1865 managed to discover after laborious search, and which, since it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1866 was partly written in Archimedes's favourite Doric, he with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1867 fair reason assumed to contain the missing <I>addendum</I> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1868 Archimedes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1869 <p>In the Archimedean fragment preserved by Eutocius the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1870 above equation, <MATH><I>x</I><SUP>2</SUP>(<I>a</I>-<I>x</I>)=<I>bc</I><SUP>2</SUP></MATH>, is solved by means of the inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1871 section of a parabola and a rectangular hyperbola, the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1872 of which may be written thus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1873 <MATH><I>x</I><SUP>2</SUP>=(<I>c</I><SUP>2</SUP>/<I>a</I>)<I>y</I>, (<I>a</I>-<I>x</I>)<I>y</I>=<I>ab</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1874 <p>The <G>diorismo/s</G> takes the form of investigating the maximum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1875 possible value of <MATH><I>x</I><SUP>2</SUP>(<I>a</I>-<I>x</I>)</MATH>, and it is proved that this maximum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1876 value for a real solution is that corresponding to the value
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1877 <MATH><I>x</I>=2/3<I>a</I></MATH>. This is established by showing that, if <MATH><I>bc</I><SUP>2</SUP>=(4/27)<I>a</I><SUP>3</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1878 <pb n=46><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1879 the curves touch at the point for which <MATH><I>x</I>=2/3<I>a</I></MATH>. If on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1880 other hand <MATH><I>bc</I><SUP>2</SUP><(4/27)<I>a</I><SUP>3</SUP></MATH>, it is proved that there are two real
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1881 solutions. In the particular case arising in Prop. 4 it is clear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1882 that the condition for a real solution is satisfied, for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1883 expression corresponding to <I>bc</I><SUP>2</SUP> is <MATH><I>m</I>/(<I>m</I>+<I>n</I>)4<I>r</I><SUP>3</SUP></MATH>, and it is only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1884 necessary that <MATH><I>m</I>/(<I>m</I>+<I>n</I>)4<I>r</I><SUP>3</SUP></MATH> should be not greater than <MATH>(4/27)<I>a</I><SUP>3</SUP></MATH> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1885 4<I>r</I><SUP>3</SUP>, which is obviously the case.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1886 <C>(ii) Solution of the cubic by Dionysodorus.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1887 <p>It is convenient to add here that Eutocius gives, in addition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1888 to the solution by Archimedes, two other solutions of our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1889 problem. One, by Dionysodorus, solves the cubic equation in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1890 the less general form in which it is required for Archimedes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1891 proposition. This form, obtained from (8) above, by putting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1892 <MATH><I>A</I>′<I>M</I>=<I>x</I></MATH>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1893 <MATH>4<I>r</I><SUP>2</SUP>:<I>x</I><SUP>2</SUP>=(3<I>r</I>-<I>x</I>):<I>n</I>/(<I>m</I>+<I>n</I>)<I>r</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1894 and the solution is obtained by drawing the parabola and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1895 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1896 the rectangular hyperbola which we should represent by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1897 equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1898 <MATH><I>n</I>/(<I>m</I>+<I>n</I>)<I>r</I>(3<I>r</I>-<I>x</I>)=<I>y</I><SUP>2</SUP> and <I>n</I>/(<I>m</I>+<I>n</I>)2<I>r</I><SUP>2</SUP>=<I>xy</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1899 referred to <I>A</I>′<I>A</I> and the perpendicular to it through <I>A</I> as axes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1900 of <I>x, y</I> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1901 <p>(We make <I>FA</I> equal to <I>OA</I>, and draw the perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1902 <I>AH</I> of such a length that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1903 <MATH><I>FA</I>:<I>AH</I>=<I>CE</I>:<I>ED</I>=(<I>m</I>+<I>n</I>):<I>n</I></MATH>.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1904 <pb n=47><head>ON THE SPHERE AND CYLINDER, II</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1905 <C>(iii) Solution of the original problem of II. 4 by Diocles.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1906 <p>Diocles proceeded in a different manner, satisfying, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1907 a geometrical construction, not the derivative cubic equation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1908 but the three simultaneous relations which hold in Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1909 medes's proposition, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1910 <MATH><BRACE><I>HM</I>:<I>H</I>′<I>M</I>=<I>m</I>:<I>n</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1911 <I>HA</I>:<I>h</I>=<I>r</I>:<I>h</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1912 <I>H</I>′<I>A</I>′:<I>h</I>′=<I>r</I>:<I>h</I></BRACE></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1913 with the slight generalization that he substitutes for <I>r</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1914 these equations another length <I>a</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1915 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1916 The problem is, given a straight line <I>AA</I>′, a ratio <I>m</I>:<I>n</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1917 another straight line <I>AK</I> (= <I>a</I>), to divide <I>AA</I>′ at a point <I>M</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1918 and at the same time to find two points <I>H, H</I>′ on <I>AA</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1919 produced such that the above relations (with <I>a</I> in place
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1920 of <I>r</I>) hold.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1921 <p>The analysis leading to the construction is very ingenious.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1922 Place <I>AK</I> (= <I>a</I>) at right angles to <I>AA</I>′, and draw <I>A</I>′<I>K</I>′ equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1923 and parallel to it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1924 <p>Suppose the problem solved, and the points <I>M, H, H</I>′ all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1925 found.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1926 <p>Join <I>KM</I>, produce it, and complete the rectangle <I>KGEK</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1927 <pb n=48><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1928 Draw <I>QMN</I> through <I>M</I> parallel to <I>AK</I>. Produce <I>K</I>′<I>M</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1929 meet <I>KG</I> produced in <I>F</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1930 <p>By similar triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1931 <MATH><I>FA</I>:<I>AM</I>=<I>K</I>′<I>A</I>′:<I>A</I>′<I>M</I>, or <I>FA</I>:<I>h</I>=<I>a</I>:<I>h</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1932 whence <MATH><I>FA</I>=<I>AH</I> (<I>k</I>, suppose)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1933 Similarly <MATH><I>A</I>′<I>E</I>=<I>A</I>′<I>H</I>′ (<I>k</I>′, suppose)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1934 <p>Again, by similar triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1935 <MATH>(<I>FA</I>+<I>AM</I>):(<I>A</I>′<I>K</I>′+<I>A</I>′<I>M</I>)=<I>AM</I>:<I>A</I>′<I>M</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1936 =(<I>AK</I>+<I>AM</I>):(<I>EA</I>′+<I>A</I>′<I>M</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1937 or <MATH>(<I>k</I>+<I>h</I>):(<I>a</I>+<I>h</I>′)=(<I>a</I>+<I>h</I>):(<I>k</I>′+<I>h</I>′)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1938 i.e. <MATH>(<I>k</I>+<I>h</I>) (<I>k</I>′+<I>h</I>′)=(<I>a</I>+<I>h</I>) (<I>a</I>+<I>h</I>′)</MATH>. (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1939 <p>Now, by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1940 <MATH><I>m</I>:<I>n</I>=(<I>k</I>+<I>h</I>):(<I>k</I>′+<I>h</I>′)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1941 =(<I>k</I>+<I>h</I>) (<I>k</I>′+<I>h</I>′):(<I>k</I>′+<I>h</I>′)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1942 =(<I>a</I>+<I>h</I>) (<I>a</I>+<I>h</I>′):(<I>k</I>′+<I>h</I>′)<SUP>2</SUP> [by (1)]</MATH>. (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1943 <p>Measure <I>AR, A</I>′<I>R</I>′ on <I>AA</I>′ produced both ways equal to <I>a</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1944 Draw <I>RP, R</I>′<I>P</I>′ at right angles to <I>RR</I>′ as shown in the figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1945 Measure along <I>MN</I> the length <I>MV</I> equal to <I>MA</I>′ or <I>h</I>′, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1946 draw <I>PP</I>′ through <I>V, A</I>′ to meet <I>RP, R</I>′<I>P</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1947 <p>Then <MATH><I>QV</I>=<I>k</I>′+<I>h</I>′, <I>P</I>′<I>V</I>=√2(<I>a</I>+<I>h</I>′),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1948 <I>PV</I>=√2(<I>a</I>+<I>h</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1949 whence <MATH><I>PV.P</I>′<I>V</I>=2(<I>a</I>+<I>h</I>) (<I>a</I>+<I>h</I>′)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1950 and, from (2) above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1951 <MATH>2<I>m</I>:<I>n</I>=2(<I>a</I>+<I>h</I>) (<I>a</I>+<I>h</I>′):(<I>k</I>′+<I>h</I>′)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1952 =<I>PV.P</I>′<I>V</I>:<I>QV</I><SUP>2</SUP></MATH>. (3)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1953 <p>Therefore <I>Q</I> is on an ellipse in which <I>PP</I>′ is a diameter, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1954 <I>QV</I> is an ordinate to it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1955 <p>Again, &rect; <I>GQNK</I> is equal to &rect; <I>AA</I>′<I>K</I>′<I>K</I>, whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1956 <MATH><I>GQ.QN</I>=<I>AA</I>′.<I>A</I>′<I>K</I>′=(<I>h</I>+<I>h</I>′)<I>a</I>=2<I>ra</I></MATH>, (4)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1957 and therefore <I>Q</I> is on the rectangular hyperbola with <I>KF</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1958 <I>KK</I>′ as asymptotes and passing through <I>A</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1959 <pb n=49><head>ON THE SPHERE AND CYLINDER, II</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1960 <p>How this ingenious analysis was suggested it is not possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1961 to say. It is the equivalent of reducing the four unknowns
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1962 <I>h, h</I>′, <I>k, k</I>′ to two, by putting <I>h</I>=<I>r</I>+<I>x, h</I>′=<I>r</I>-<I>x</I> and <I>k</I>′=<I>y</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1963 and then reducing the given relations to two equations in <I>x, y</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1964 which are coordinates of a point in relation to <I>Ox, Oy</I> as axes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1965 where <I>O</I> is the middle point of <I>AA</I>′, and <I>Ox</I> lies along <I>OA</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1966 while <I>Oy</I> is perpendicular to it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1967 <p>Our original relations (p. 47) give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1968 <MATH><I>y</I>=<I>k</I>′=(<I>ah</I>′)/<I>h</I>=<I>a</I>(<I>r</I>-<I>x</I>)/(<I>r</I>+<I>x</I>), <I>k</I>=(<I>ah</I>)/<I>h</I>′=<I>a</I>(<I>r</I>+<I>x</I>)/(<I>r</I>-<I>x</I>)</MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1969 <MATH><I>m</I>/<I>n</I>=(<I>h</I>+<I>k</I>)/(<I>h</I>′+<I>k</I>′)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1970 <p>We have at once, from the first two equations,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1971 <MATH><I>ky</I>=<I>a</I>(<I>r</I>+<I>x</I>)/(<I>r</I>-<I>x</I>)<I>y</I>=<I>a</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1972 whence <MATH>(<I>r</I>+<I>x</I>)<I>y</I>=<I>a</I>(<I>r</I>-<I>x</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1973 and <MATH>(<I>x</I>+<I>r</I>) (<I>y</I>+<I>a</I>)=2<I>ra</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1974 which is the rectangular hyperbola (4) above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1975 <p>Again, <MATH><I>m</I>/<I>n</I>=(<I>h</I>+<I>k</I>)/(<I>h</I>′+<I>k</I>′)=((<I>r</I>+<I>x</I>)(1+<I>a</I>/(<I>r</I>-<I>x</I>)))/((<I>r</I>-<I>x</I>) (1+<I>a</I>/(<I>r</I>+<I>x</I>)))</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1976 whence we obtain a cubic equation in <I>x</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1977 <MATH>(<I>r</I>+<I>x</I>)<SUP>2</SUP>(<I>r</I>+<I>a</I>-<I>x</I>)=(<I>m/n</I>)(<I>r</I>-<I>x</I>)<SUP>2</SUP>(<I>r</I>+<I>a</I>+<I>x</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1978 which gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1979 <MATH>(<I>m/n</I>)(<I>r</I>-<I>x</I>)<SUP>2</SUP>((<I>r</I>+<I>a</I>+<I>x</I>)/(<I>r</I>+<I>x</I>))<SUP>2</SUP>=(<I>r</I>+<I>a</I>)<SUP>2</SUP>-<I>x</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1980 <p>But <MATH><I>y</I>/(<I>r</I>-<I>x</I>)=<I>a</I>/(<I>r</I>+<I>x</I>)</MATH>, whence <MATH>(<I>y</I>+<I>r</I>-<I>x</I>)/(<I>r</I>-<I>x</I>)=(<I>r</I>+<I>a</I>+<I>x</I>)/(<I>r</I>+<I>x</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1981 and the equation becomes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1982 <MATH>(<I>m/n</I>)(<I>y</I>+<I>r</I>-<I>x</I>)<SUP>2</SUP>=(<I>r</I>+<I>a</I>)<SUP>2</SUP>-<I>x</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1983 which is the ellipse (3) above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1984 <pb n=50><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1985 <p>To return to Archimedes. Book II of our treatise contains
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1986 further problems: To find a sphere equal to a given cone or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1987 cylinder (Prop. 1), solved by reduction to the finding of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1988 mean proportionals; to cut a sphere by a plane into two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1989 segments having their surfaces in a given ratio (Prop. 3),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1990 which is easy (by means of I. 42, 43); given two segments of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1991 spheres, to find a third segment of a sphere similar to one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1992 of the given segments and having its surface equal to that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1993 the other (Prop. 6); the same problem with volume substituted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1994 for surface (Prop. 5), which is again reduced to the finding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1995 of two mean proportionals; from a given sphere to cut off
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1996 a segment having a given ratio to the cone with the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1997 base and equal height (Prop. 7). The Book concludes with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1998 two interesting theorems. If a sphere be cut by a plane into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	1999 two segments, the greater of which has its surface equal to <I>S</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2000 and its volume equal to <I>V,</I> while <I>S</I>′, <I>V</I>′ are the surface and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2001 volume of the lesser, then <I>V</I>:<I>V</I>′<<I>S</I><SUP>2</SUP>:<I>S</I>′<SUP>2</SUP> but > <I>S</I><SUP>3/2</SUP>:<I>S</I>′<SUP>3/2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2002 (Prop. 8): and, of all segments of spheres which have their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2003 surfaces equal, the hemisphere is the greatest in volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2004 (Prop. 9).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2005 <C>Measurement of a Circle.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2006 <p>The book on the <I>Measurement of a Circle</I> consists of three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2007 propositions only, and is not in its original form, having lost
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2008 (as the treatise <I>On the Sphere and Cylinder</I> also has) prac-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2009 tically all trace of the Doric dialect in which Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2010 wrote; it may be only a fragment of a larger treatise. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2011 three propositions which survive prove (1) that the area of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2012 a circle is equal to that of a right-angled triangle in which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2013 the perpendicular is equal to the radius, and the base to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2014 circumference, of the circle, (2) that the area of a circle is to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2015 the square on its diameter as 11 to 14 (the text of this proposition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2016 is, however, unsatisfactory, and it cannot have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2017 placed by Archimedes before Prop. 3, on which it depends),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2018 (3) <I>that the ratio of the circumference of any circle to its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2019 diameter</I> (i. e. <G>p</G>) <I>is</I> < 3 1/7 <I>but</I> > 3 10/71. Prop. 1 is proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2020 the method of exhaustion in Archimedes's usual form: he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2021 approximates to the area of the circle in both directions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2022 (<I>a</I>) by inscribing successive regular polygons with a number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2023 <pb n=51><head>MEASUREMENT OF A CIRCLE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2024 sides continually doubled, beginning from a square, (<I>b</I>) by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2025 circumscribing a similar set of regular polygons beginning
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2026 from a square, it being shown that, if the number of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2027 sides of these polygons be continually doubled, more than half
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2028 of the portion of the polygon outside the circle will be taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2029 away each time, so that we shall ultimately arrive at a circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2030 scribed polygon greater than the circle by a space less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2031 any assigned area.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2032 <p>Prop. 3, containing the arithmetical approximation to <G>p</G>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2033 the most interesting. The method amounts to calculating
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2034 approximately the perimeter of two regular polygons of 96
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2035 sides, one of which is circumscribed, and the other inscribed,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2036 to the circle; and the calculation starts from a greater and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2037 a lesser limit to the value of √3, which Archimedes assumes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2038 without remark as known, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2039 <MATH>265/153<√3<1351/780</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2040 <p>How did Archimedes arrive at these particular approximations?
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2041 No puzzle has exercised more fascination upon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2042 writers interested in the history of mathematics. De Lagny,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2043 Mollweide, Buzengeiger, Hauber, Zeuthen, P. Tannery, Heilermann,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2044 Hultsch, Hunrath, Wertheim, Bobynin: these are the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2045 names of some of the authors of different conjectures. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2046 simplest supposition is certainly that of Hunrath and Hultsch,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2047 who suggested that the formula used was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2048 <MATH><I>a</I>±<I>b</I>/2<I>a</I>>√(<I>a</I><SUP>2</SUP>±<I>b</I>)><I>a</I>±<I>b</I>/(2<I>a</I>±1)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2049 where <I>a</I><SUP>2</SUP> is the nearest square number above or below <I>a</I><SUP>2</SUP>±<I>b</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2050 as the case may be. The use of the first part of this formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2051 by Heron, who made a number of such approximations, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2052 proved by a passage in his <I>Metrica</I><note>Heron, <I>Metrica,</I> i. 8.</note>, where a rule equivalent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2053 to this is applied to √720; the second part of the formula is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2054 used by the Arabian Alkarkh&imacr; (eleventh century) who drew
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2055 from Greek sources, and one approximation in Heron may be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2056 obtained in this way.<note><I>Stereom.</I> ii, p. 184. 19, Hultsch; p. 154. 19, Heib. <MATH>√54=7 1/3=7 5/15</MATH> instead of 7 5/14.</note> Another suggestion (that of Tannery
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2057 <pb n=52><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2058 and Zeuthen) is that the successive solutions in integers of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2059 the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2060 <MATH><BRACE><I>x</I><SUP>2</SUP>-3<I>y</I><SUP>2</SUP>=1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2061 <I>x</I><SUP>2</SUP>-3<I>y</I><SUP>2</SUP>=-2</BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2062 may have been found in a similar way to those of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2063 equations <MATH><I>x</I><SUP>2</SUP>-2<I>y</I><SUP>2</SUP>=±1</MATH> given by Theon of Smyrna after
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2064 the Pythagoreans. The rest of the suggestions amount for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2065 most part to the use of the method of continued fractions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2066 more or less disguised.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2067 <p>Applying the above formula, we easily find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2068 <MATH>2-1/4>√3>2-1/3</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2069 or <MATH>7/4>√3>5/3</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2070 <p>Next, clearing of fractions, we consider 5 as an approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2071 mation to √(3.3<SUP>2</SUP>) or √27, and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2072 <MATH>5+2/10>3√3>5+2/11</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2073 whence <MATH>26/15>√3>19/11</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2074 <p>Clearing of fractions again, and taking 26 as an approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2075 mation to √(3.15<SUP>2</SUP>) or √675, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2076 <MATH>26-1/52>15√3>26-1/51</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2077 which reduces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2078 <MATH>1351/780>√3>265/153</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2079 <p>Archimedes first takes the case of the circumscribed polygon.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2080 Let <I>CA</I> be the tangent at <I>A</I> to a circular arc with centre <I>O.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2081 Make the angle <I>AOC</I> equal to one-third of a right angle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2082 Bisect the angle <I>AOC</I> by <I>OD,</I> the angle <I>AOD</I> by <I>OE,</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2083 angle <I>AOE</I> by <I>OF,</I> and the angle <I>AOF</I> by <I>OG.</I> Produce <I>GA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2084 to <I>AH,</I> making <I>AH</I> equal to <I>AG.</I> The angle <I>GOH</I> is then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2085 equal to the angle <I>FOA</I> which is 1/24th of a right angle, so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2086 that <I>GH</I> is the side of a circumscribed regular polygon with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2087 96 sides.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2088 <p>Now <MATH><I>OA</I>:<I>AC</I>[=√3:1]>265:153</MATH>, (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2089 and <MATH><I>OC</I>:<I>CA</I>=2:1=306:153</MATH>. (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2090 <pb n=53><head>MEASUREMENT OF A CIRCLE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2091 <p>And, since <I>OD</I> bisects the angle <I>COA,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2092 <MATH><I>CO</I>:<I>OA</I>=<I>CD</I>:<I>DA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2093 so that <MATH>(<I>CO</I>+<I>OA</I>):<I>OA</I>=<I>CA</I>:<I>DA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2094 or <MATH>(<I>CO</I>+<I>OA</I>):<I>CA</I>=<I>OA</I>:<I>AD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2095 <p>Hence <MATH><I>OA</I>:<I>AD</I>>571:153</MATH>, by (1) and (2).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2096 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2097 <p>And <MATH><I>OD</I><SUP>2</SUP>:<I>AD</I><SUP>2</SUP>=(<I>OA</I><SUP>2</SUP>+<I>AD</I><SUP>2</SUP>):<I>AD</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2098 >(571<SUP>2</SUP>+153<SUP>2</SUP>):153<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2099 >349450:23409</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2100 <p>Therefore, says Archimedes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2101 <MATH><I>OD</I>:<I>DA</I>>591 1/8:153</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2102 <p>Next, just as we have found the limit of <I>OD</I>:<I>AD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2103 from <I>OC</I>:<I>CA</I> and the limit of <I>OA</I>:<I>AC</I>, we find the limits
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2104 of <I>OA</I>:<I>AE</I> and <I>OE</I>:<I>AE</I> from the limits of <I>OD</I>:<I>DA</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2105 <I>OA</I>:<I>AD</I>, and so on. This gives ultimately the limit of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2106 <I>OA</I>:<I>AG.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2107 <p>Dealing with the inscribed polygon, Archimedes gets a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2108 similar series of approximations. <I>ABC</I> being a semicircle, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2109 angle <I>BAC</I> is made equal to one-third of a right angle. Then,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2110 if the angle <I>BAC</I> is bisected by <I>AD,</I> the angle <I>BAD</I> by <I>AE,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2111 the angle <I>BAE</I> by <I>AF,</I> and the angle <I>BAF</I> by <I>AG,</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2112 straight line <I>BG</I> is the side of an inscribed polygon with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2113 96 sides.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2114 <pb n=54><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2115 <p>Now the triangles <I>ADB, BDd, ACd</I> are similar;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2116 therefore <MATH><I>AD</I>:<I>DB</I>=<I>BD</I>:<I>Dd</I>=<I>AC</I>:<I>Cd</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2117 =<I>AB</I>:<I>Bd,</I> since <I>AD</I> bisects &angle;<I>BAC,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2118 =(<I>AB</I>+<I>AC</I>):(<I>Bd</I>+<I>Cd</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2119 =(<I>AB</I>+<I>AC</I>):<I>BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2120 <p>But <MATH><I>AC</I>:<I>CB</I><1351:780</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2121 while <MATH><I>BA</I>:<I>BC</I>=2:1=1560:780</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2122 <p>Therefore <MATH><I>AD</I>:<I>DB</I><2911:780</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2123 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2124 <p>Hence <MATH><I>AB</I><SUP>2</SUP>:<I>BD</I><SUP>2</SUP><(2911<SUP>2</SUP>+780<SUP>2</SUP>):780<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2125 <9082321:608400</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2126 and, says Archimedes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2127 <MATH><I>AB</I>:<I>BD</I><3013 3/4:780</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2128 <p>Next, just as a limit is found for <I>AD</I>:<I>DB</I> and <I>AB</I>:<I>BD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2129 from <I>AB</I>:<I>BC</I> and the limit of <I>AC</I>:<I>CB,</I> so we find limits for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2130 <I>AE</I>:<I>EB</I> and <I>AB</I>:<I>BE</I> from the limits of <I>AB</I>:<I>BD</I> and <I>AD</I>:<I>DB,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2131 and so on, and finally we obtain the limit of <I>AB</I>:<I>BG.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2132 <p>We have therefore in both cases two series of terms <I>a</I><SUB>0</SUB>, <I>a</I><SUB>1</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2133 <I>a</I><SUB>2</SUB> ... <I>a</I><SUB>n</SUB> and <I>b</I><SUB>0</SUB>, <I>b</I><SUB>1</SUB>, <I>b</I><SUB>2</SUB> ... <I>b</I><SUB>n</SUB>, for which the rule of formation is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2134 <MATH><I>a</I><SUB>1</SUB>=<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>, <I>a</I><SUB>2</SUB>=<I>a</I><SUB>1</SUB>+<I>b</I><SUB>1</SUB>,...</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2135 where <MATH><I>b</I><SUB>1</SUB>=√(<I>a</I><SUB>1</SUB><SUP>2</SUP>+<I>c</I><SUP>2</SUP>), <I>b</I><SUB>2</SUB>=√(<I>a</I><SUB>2</SUB><SUP>2</SUP>+<I>c</I><SUP>2</SUP>) ...;</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2136 and in the first case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2137 <MATH><I>a</I><SUB>0</SUB>=265, <I>b</I><SUB>0</SUB>=306, <I>c</I>=153</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2138 while in the second case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2139 <MATH><I>a</I><SUB>0</SUB>=1351, <I>b</I><SUB>0</SUB>1560, <I>c</I>=780</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2140 <pb n=55><head>MEASUREMENT OF A CIRCLE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2141 <p>The series of values found by Archimedes are shown in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2142 following table:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2143 <MATH></MATH><note>Here the ratios of <I>a</I> to <I>c</I> are in the first instance reduced to lower terms.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2144 and, bearing in mind that in the first case the final ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2145 <MATH><I>a</I><SUB>4</SUB>:<I>c</I></MATH> is the ratio <MATH><I>OA</I>:<I>AG</I>=2<I>OA</I>:<I>GH</I></MATH>, and in the second case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2146 the final ratio <I>b</I><SUB>4</SUB>:<I>c</I> is the ratio <I>AB</I>:<I>BG</I>, while <I>GH</I> in the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2147 figure and <I>BG</I> in the second are the sides of regular polygons
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2148 of 96 sides circumscribed and inscribed respectively, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2149 finally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2150 <MATH>96X153/(4673 1/2)><G>p</G>>96X66/(2017 1/4)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2151 <p>Archimedes simply infers from this that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2152 <MATH>3 1/7><G>p</G>>3 10/71</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2153 <p>As a matter of fact <MATH>96X153/(4673 1/2)=3 (667 1/2)/(4673 1/2)</MATH>, and <MATH>(667 1/2)/(4672 1/2)=1/7</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2154 It is also to be observed that <MATH>3 10/71=3+1/(7+1/10)</MATH>, and it may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2155 have been arrived at by a method equivalent to developing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2156 the fraction 6336/(2017 1/4) in the form of a continued fraction.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2157 <p>It should be noted that, in the text as we have it, the values
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2158 of <I>b</I><SUB>1</SUB>, <I>b</I><SUB>2</SUB>, <I>b</I><SUB>3</SUB>, <I>b</I><SUB>4</SUB> are simply stated in their final form without
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2159 the intermediate step containing the radical except in the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2160 <pb n=56><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2161 case of all, where we are told that <MATH><I>OD</I><SUP>2</SUP>:<I>AD</I><SUP>2</SUP>>349450:23409</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2162 and then that <MATH><I>OD</I>:<I>DA</I>>591 1/8:153</MATH>. At the points marked
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2163 * and <FIG> in the table Archimedes simplifies the ratio <I>a</I><SUB>2</SUB>:<I>c</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2164 <I>a</I><SUB>3</SUB>:<I>c</I> before calculating <I>b</I><SUB>2</SUB>, <I>b</I><SUB>3</SUB> respectively, by multiplying each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2165 term in the first case by 4/13 and in the second case by 11/40.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2166 He gives no explanation of the exact figure taken as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2167 approximation to the square root in each case, or of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2168 method by which he obtained it. We may, however, be sure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2169 that the method amounted to the use of the formula <MATH>(<I>a</I>±<I>b</I>)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2170 =<I>a</I><SUP>2</SUP>±2<I>ab</I>+<I>b</I><SUP>2</SUP></MATH>, much as our method of extracting the square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2171 root also depends upon it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2172 <p>We have already seen (vol. i, p. 232) that, according to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2173 Heron, Archimedes made a still closer approximation to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2174 value of <G>p</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2175 <C>On Conoids and Spheroids.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2176 <p>The main problems attacked in this treatise are, in Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2177 medes's manner, stated in his preface addressed to Dositheus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2178 which also sets out the premisses with regard to the solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2179 figures in question. These premisses consist of definitions and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2180 obvious inferences from them. The figures are (1) the <I>right-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2181 angled conoid</I> (paraboloid of revolution), (2) the <I>obtuse-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2182 conoid</I> (hyperboloid of revolution), and (3) the <I>spheroids</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2183 (<I>a</I>) the <I>oblong,</I> described by the revolution of an ellipse about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2184 its ‘greater diameter’ (major axis), (<I>b</I>) the <I>flat,</I> described by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2185 the revolution of an ellipse about its ‘lesser diameter’ (minor
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2186 axis). Other definitions are those of the <I>vertex</I> and <I>axis</I> of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2187 figures or segments thereof, the vertex of a segment being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2188 the point of contact of the tangent plane to the solid which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2189 is parallel to the base of the segment. The <I>centre</I> is only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2190 recognized in the case of the spheroid; what corresponds to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2191 the centre in the case of the hyperboloid is the ‘vertex of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2192 the enveloping cone’ (described by the revolution of what
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2193 Archimedes calls the ‘nearest lines to the section of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2194 obtuse-angled cone’, i.e. the asymptotes of the hyperbola),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2195 and the line between this point and the vertex of the hyper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2196 boloid or segment is called, not the axis or diameter, but (the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2197 line) ‘adjacent to the axis’. The axis of the segment is in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2198 the case of the paraboloid the line through the vertex of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2199 segment parallel to the axis of the paraboloid, in the case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2200 <pb n=57><head>ON CONOIDS AND SPHEROIDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2201 of the hyperboloid the portion within the solid of the line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2202 joining the vertex of the enveloping cone to the vertex of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2203 the segment and produced, and in the case of the spheroids the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2204 line joining the points of contact of the two tangent planes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2205 parallel to the base of the segment. Definitions are added of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2206 a ‘segment of a cone’ (the figure cut off towards the vertex by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2207 an elliptical, not circular, section of the cone) and a ‘frustum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2208 of a cylinder’ (cut off by two parallel elliptical sections).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2209 <p>Props. 1 to 18 with a Lemma at the beginning are preliminary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2210 to the main subject of the treatise. The Lemma and Props. 1, 2
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2211 are general propositions needed afterwards. They include
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2212 propositions in summation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2213 <MATH>2{<I>a</I>+2<I>a</I>+3<I>a</I>+...+<I>na</I>}><I>n.na</I>>2{<I>a</I>+2<I>a</I>+...+(<I>n</I>-1)<I>a</I>}</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2214 (Lemma)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2215 (this is clear from <MATH><I>S</I><SUB><I>n</I></SUB>=1/2<I>n</I>(<I>n</I>+1)<I>a</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2216 <MATH>(<I>n</I>+1)(<I>na</I>)<SUP>2</SUP>+<I>a</I>(<I>a</I>+2<I>a</I>+3<I>a</I>+...+<I>na</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2217 =3{<I>a</I><SUP>2</SUP>+(2<I>a</I>)<SUP>2</SUP>+(3<I>a</I>)<SUP>2</SUP>+...+(<I>na</I>)<SUP>2</SUP>};</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2218 (Lemma to Prop. 2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2219 whence (Cor.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2220 <MATH>3{<I>a</I><SUP>2</SUP>+(2<I>a</I>)<SUP>2</SUP>+(3<I>a</I>)<SUP>2</SUP>+...+(<I>na</I>)<SUP>2</SUP>}><I>n</I>(<I>na</I>)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2221 >3{<I>a</I><SUP>2</SUP>+(2<I>a</I>)<SUP>2</SUP>+...+(<I>n</I>-1<I>a</I>)<SUP>2</SUP>};</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2222 lastly, Prop. 2 gives limits for the sum of <I>n</I> terms of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2223 series <MATH><I>ax</I>+<I>x</I><SUP>2</SUP>, <I>a.</I>2<I>x</I>+(2<I>x</I>)<SUP>2</SUP>, <I>a.</I>3<I>x</I>+(3<I>x</I>)<SUP>2</SUP>,...</MATH>, in the form of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2224 inequalities of ratios, thus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2225 <MATH><I>n</I>{<I>a.nx</I>+(<I>nx</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>a.rx</I>+(<I>rx</I>)<SUP>2</SUP>}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2226 >(<I>a</I>+<I>nx</I>):(1/2<I>a</I>+1/3<I>nx</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2227 ><I>n</I>{<I>a.nx</I>+(<I>nx</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>a.rx</I>+(<I>rx</I>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2228 Prop. 3 proves that, if <I>QQ</I>′ be a chord of a parabola bisected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2229 at <I>V</I> by the diameter <I>PV,</I> then, if <I>PV</I> be of constant length,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2230 the areas of the triangle <I>PQQ</I>′ and of the segment <I>PQQ</I>′ are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2231 also constant, whatever be the direction of <I>QQ</I>′; to prove it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2232 Archimedes assumes a proposition ‘proved in the conics’ and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2233 by no means easy, namely that, if <I>QD</I> be perpendicular to <I>PV,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2234 and if <I>p, p<SUB>a</SUB></I> be the parameters corresponding to the ordinates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2235 parallel to <I>QQ</I>′ and the principal ordinates respectively, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2236 <MATH><I>QV</I><SUP>2</SUP>:<I>QD</I><SUP>2</SUP>=<I>p</I>:<I>p<SUB>a</SUB></I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2237 Props. 4-6 deal with the area of an ellipse, which is, in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2238 <pb n=58><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2239 first of the three propositions, proved to be to the area of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2240 the auxiliary circle as the minor axis to the major; equilateral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2241 polygons of 4<I>n</I> sides are inscribed in the circle and compared
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2242 with corresponding polygons inscribed in the ellipse, which are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2243 determined by the intersections with the ellipse of the double
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2244 ordinates passing through the angular points of the polygons
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2245 inscribed in the circle, and the method of exhaustion is then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2246 applied in the usual way. Props. 7, 8 show how, given an ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2247 with centre <I>C</I> and a straight line <I>CO</I> in a plane perpendicular to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2248 that of the ellipse and passing through an axis of it, (1) in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2249 case where <I>OC</I> is perpendicular to that axis, (2) in the case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2250 where it is not, we can find an (in general oblique) circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2251 cone with vertex <I>O</I> such that the given ellipse is a section of it,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2252 or, in other words, how we can find the circular sections of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2253 cone with vertex <I>O</I> which passes through the circumference of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2254 the ellipse; similarly Prop. 9 shows how to find the circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2255 sections of a cylinder with <I>CO</I> as axis and with surface passing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2256 through the circumference of an ellipse with centre <I>C,</I> where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2257 <I>CO</I> is in the plane through an axis of the ellipse and perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2258 dicular to its plane, but is not itself perpendicular to that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2259 axis. Props. 11-18 give simple properties of the conoids and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2260 spheroids, easily derivable from the properties of the respective
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2261 conics; they explain the nature and relation of the sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2262 made by planes cutting the solids respectively in different ways
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2263 (planes through the axis, parallel to the axis, through the centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2264 or the vertex of the enveloping cone, perpendicular to the axis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2265 or cutting it obliquely, respectively), with especial reference to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2266 the elliptical sections of each solid, the similarity of parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2267 elliptical sections, &c. Then with Prop. 19 the real business
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2268 of the treatise begins, namely the investigation of the volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2269 of segments (right or oblique) of the two conoids and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2270 spheroids respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2271 <p>The method is, in all cases, to circumscribe and inscribe to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2272 the segment solid figures made up of cylinders or ‘frusta of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2273 cylinders’, which can be made to differ as little as we please
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2274 from one another, so that the circumscribed and inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2275 figures are, as it were, compressed together and into coincidence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2276 with the segment which is intermediate between them.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2277 <p>In each diagram the plane of the paper is a plane through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2278 the axis of the conoid or spheroid at right angles to the plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2279 <pb n=59><head>ON CONOIDS AND SPHEROIDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2280 of the section which is the base of the segment, and which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2281 is a circle or an ellipse according as the said base is or is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2282 at right angles to the axis; the plane of the paper cuts the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2283 base in a diameter of the circle or an axis of the ellipse as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2284 the case may be.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2285 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2286 <p>The nature of the inscribed and circumscribed figures will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2287 be seen from the above figures showing segments of a para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2288 boloid, a hyperboloid and a spheroid respectively, cut off
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2289 <pb n=60><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2290 by planes obliquely inclined to the axis. The base of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2291 segment is an ellipse in which <I>BB</I>′ is an axis, and its plane is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2292 at right angles to the plane of the paper, which passes through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2293 the axis of the solid and cuts it in a parabola, a hyperbola, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2294 an ellipse respectively. The axis of the segment is cut into a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2295 number of equal parts in each case, and planes are drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2296 through each point of section parallel to the base, cutting the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2297 solid in ellipses, similar to the base, in which <I>PP</I>′, <I>QQ</I>′, &c., are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2298 axes. Describing frusta of cylinders with axis <I>AD</I> and passing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2299 through these elliptical sections respectively, we draw the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2300 circumscribed and inscribed solids consisting of these frusta.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2301 It is evident that, beginning from <I>A,</I> the first inscribed frustum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2302 is equal to the first circumscribed frustum, the second to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2303 second, and so on, but there is one more circumscribed frustum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2304 than inscribed, and the difference between the circumscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2305 and inscribed solids is equal to the <I>last frustum</I> of which <I>BB</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2306 is the base, and <I>ND</I> is the axis. Since <I>ND</I> can be made as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2307 small as we please, the difference between the circumscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2308 and inscribed solids can be made less than any assigned solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2309 whatever. Hence we have the requirements for applying the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2310 method of exhaustion.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2311 <p>Consider now separately the cases of the paraboloid, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2312 hyperboloid and the spheroid.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2313 <p>I. The <I>paraboloid</I> (Props. 20-22).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2314 <p>The frustum the base of which is the ellipse in which <I>PP</I>′ is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2315 an axis is proportional to <I>PP</I>′<SUP>2</SUP> or <I>PN</I><SUP>2</SUP>, i.e. proportional to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2316 <I>AN.</I> Suppose that the axis <I>AD</I> (=<I>c</I>) is divided into <I>n</I> equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2317 parts. Archimedes compares each frustum in the inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2318 and circumscribed figure with the frustum of the whole cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2319 <I>BF</I> cut off by the same planes. Thus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2320 <MATH>(first frustum in <I>BF</I>):(first frustum in inscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2321 =<I>BD</I><SUP>2</SUP>:<I>PN</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2322 =<I>AD</I>:<I>AN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2323 =<I>BD</I>:<I>TN</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2324 Similarly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2325 <MATH>(second frustum in <I>BF</I>):(second in inscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2326 =<I>HN</I>:<I>SM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2327 and so on. The last frustum in the cylinder <I>BF</I> has none to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2328 <pb n=61><head>ON CONOIDS AND SPHEROIDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2329 correspond to it in the inscribed figure, and we should write
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2330 the ratio as (<I>BD</I>:zero).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2331 <p>Archimedes concludes, by means of a lemma in proportions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2332 forming Prop. 1, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2333 <MATH>(frustum <I>BF</I>):(inscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2334 =(<I>BD</I>+<I>HN</I>+...):(<I>TN</I>+<I>SM</I>+...+<I>XO</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2335 =<I>n</I><SUP>2</SUP><I>k</I>:(<I>k</I>+2<I>k</I>+3<I>k</I>+...+&horbar;(<I>n</I>-1)<I>k</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2336 where <MATH><I>XO</I>=<I>k,</I> so that <I>BD</I>=<I>nk</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2337 <p>In like manner, he concludes that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2338 <MATH>(frustum <I>BF</I>):(circumscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2339 =<I>n</I><SUP>2</SUP><I>k</I>:(<I>k</I>+2<I>k</I>+3<I>k</I>+...+<I>nk</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2340 <p>But, by the Lemma preceding Prop. 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2341 <MATH><I>k</I>+2<I>k</I>+3<I>k</I>+...+&horbar;(<I>n</I>-1)<I>k</I><1/2<I>n</I><SUP>2</SUP><I>k</I><<I>k</I>+2<I>k</I>+3<I>k</I>+...+<I>nk</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2342 whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2343 <MATH>(frustum <I>BF</I>):(inscr.fig.)>2>(frustum <I>BF</I>):(circumscr. fig.)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2344 <p>This indicates the desired result, which is then confirmed by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2345 the method of exhaustion, namely that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2346 <MATH>(frustum <I>BF</I>)=2(segment of paraboloid)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2347 or, if <I>V</I> be the volume of the ‘segment of a cone’, with vertex
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2348 <I>A</I> and base the same as that of the segment,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2349 <MATH>(volume of segment)=3/2<I>V</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2350 <p>Archimedes, it will be seen, proves in effect that, if <I>k</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2351 indefinitely diminished, and <I>n</I> indefinitely increased, while <I>nk</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2352 remains equal to <I>c,</I> then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2353 limit of <MATH><I>k</I>{<I>k</I>+2<I>k</I>+3<I>k</I>+...+(<I>n</I>-1)<I>k</I>}=1/2<I>c</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2354 that is, in our notation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2355 <MATH>∫<SUP><I>c</I></SUP><SUB>0</SUB><I>xdx</I>=1/2<I>c</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2356 <p>Prop. 23 proves that the volume is constant for a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2357 length of axis <I>AD,</I> whether the segment is cut off by a plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2358 perpendicular or not perpendicular to the axis, and Prop. 24
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2359 shows that the volumes of two segments are as the squares on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2360 their axes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2361 <pb n=62><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2362 <p>II. In the case of the <I>hyperboloid</I> (Props. 25, 26) let the axis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2363 <I>AD</I> be divided into <I>n</I> parts, each of length <I>h</I>, and let <MATH><I>AA</I>′=<I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2364 Then the ratio of the volume of the frustum of a cylinder on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2365 the ellipse of which any double ordinate <I>QQ</I>′ is an axis to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2366 volume of the corresponding portion of the whole frustum <I>BF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2367 takes a different form; for, if <I>AM</I>=<I>rh,</I> we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2368 <MATH>(frustum in <I>BF</I>):(frustum on base <I>QQ</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2369 =<I>BD</I><SUP>2</SUP>:<I>QM</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2370 =<I>AD.A</I>′<I>D</I>:<I>AM.A</I>′<I>M</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2371 ={<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2372 By means of this relation Archimedes proves that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2373 <MATH>(frustum <I>BF</I>):(inscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2374 =<I>n</I>{<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2375 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2376 <MATH>(frustum <I>BF</I>):(circumscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2377 =<I>n</I>{<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2378 But, by Prop. 2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2379 <MATH><I>n</I>{<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}>(<I>a</I>+<I>nh</I>):(1/2<I>a</I>+1/3<I>nh</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2380 ><I>n</I>{<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2381 <p>From these relations it is inferred that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2382 <MATH>(frustum <I>BF</I>):(volume of segment)=(<I>a</I>+<I>nh</I>):(1/2<I>a</I>+1/3<I>nh</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2383 or <MATH>(volume of segment):(volume of cone <I>ABB</I>′)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2384 =(<I>AD</I>+3<I>CA</I>):(<I>AD</I>+2<I>CA</I>);</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2385 and this is confirmed by the method of exhaustion.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2386 <p>The result obtained by Archimedes is equivalent to proving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2387 that, if <I>h</I> be indefinitely diminished while <I>n</I> is indefinitely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2388 increased but <I>nh</I> remains always equal to <I>b,</I> then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2389 limit of <MATH><I>n</I>(<I>ab</I>+<I>b</I><SUP>2</SUP>)/<I>S</I><SUB><I>n</I></SUB>=(<I>a</I>+<I>b</I>)/(1/2<I>a</I>+1/3<I>b</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2390 or limit of <MATH><I>b/n S<SUB>n</SUB></I>=<I>b</I><SUP>2</SUP>(1/2<I>a</I>+1/3<I>b</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2391 where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2392 <MATH><I>S<SUB>n</SUB></I>=<I>a</I>(<I>h</I>+2<I>h</I>+3<I>h</I>+...+<I>nh</I>)+{<I>h</I><SUP>2</SUP>+(2<I>h</I>)<SUP>2</SUP>+(3<I>h</I>)<SUP>2</SUP>+...+(<I>nh</I>)<SUP>2</SUP>}</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2393 <pb n=63><head>ON CONOIDS AND SPHEROIDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2394 so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2395 <MATH><I>hS<SUB>n</SUB></I>=<I>ah</I>(<I>h</I>+2<I>h</I>+...+<I>nh</I>)+<I>h</I>{<I>h</I><SUP>2</SUP>+(2<I>h</I>)<SUP>2</SUP>+...+(<I>nh</I>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2396 <p>The limit of this latter expression is what we should write
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2397 <MATH>∫<SUP><I>b</I></SUP><SUB>0</SUB>(<I>ax</I>+<I>x</I><SUP>2</SUP>)<I>dx</I>=<I>b</I><SUP>2</SUP>(1/2<I>a</I>+1/3<I>b</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2398 and Archimedes's procedure is the equivalent of this integration.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2399 <p>III. In the case of the <I>spheroid</I> (Props. 29, 30) we take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2400 a segment less than half the spheroid.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2401 <p>As in the case of the hyperboloid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2402 <MATH>(frustum in <I>BF</I>):(frustum on base <I>QQ</I>′)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2403 =<I>BD</I><SUP>2</SUP>:<I>QM</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2404 =<I>AD.A</I>′<I>D</I>:<I>AM.A</I>′<I>M</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2405 but, in order to reduce the summation to the same as that in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2406 Prop. 2, Archimedes expresses <I>AM.A</I>′<I>M</I> in a different form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2407 equivalent to the following.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2408 <p>Let <I>AD</I> (=<I>b</I>) be divided into <I>n</I> equal parts of length <I>h,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2409 and suppose that <MATH><I>AA</I>′=<I>a, CD</I>=1/2<I>c.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2410 <p>Then <MATH><I>AD.A</I>′<I>D</I>=1/4<I>a</I><SUP>2</SUP>-1/4<I>c</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2411 and <MATH><I>AM.A</I>′<I>M</I>=1/4<I>a</I><SUP>2</SUP>-(1/2<I>c</I>+<I>rh</I>)<SUP>2</SUP> (<I>DM</I>=<I>rh</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2412 =<I>AD.A</I>′<I>D</I>-{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2413 =<I>cb</I>+<I>b</I><SUP>2</SUP>-{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2414 <p>Thus in this case we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2415 <MATH>(frustum <I>BF</I>):(inscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2416 =<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>):[<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>)-<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2417 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2418 <MATH>(frustum <I>BF</I>):(circumscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2419 =<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>):[<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>)-<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}]</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2420 <p>And, since <I>b</I>=<I>nh,</I> we have, by means of Prop. 2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2421 <MATH><I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>):[<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>)-<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2422 >(<I>c</I>+<I>b</I>):{<I>c</I>+<I>b</I>-(1/2<I>c</I>+1/3<I>b</I>)}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2423 ><I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>):[<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>)-<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}]</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2424 <pb n=64><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2425 <p>The conclusion, confirmed as usual by the method of ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2426 haustion, is that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2427 <MATH>(frustum <I>BF</I>):(segment of spheroid)=(<I>c</I>+<I>b</I>):{<I>c</I>+<I>b</I>-(1/2<I>c</I>+1/3<I>b</I>)}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2428 =(<I>c</I>+<I>b</I>):(1/2<I>c</I>+2/3<I>b</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2429 whence (volume of segment):(volume of cone <I>ABB</I>′)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2430 =(3/2<I>c</I>+2<I>b</I>):(<I>c</I>+<I>b</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2431 =(3<I>CA</I>-<I>AD</I>):(2<I>CA</I>-<I>AD</I>), since <I>CA</I>=1/2<I>c</I>+<I>b</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2432 <p>As a particular case (Props. 27, 28), half the spheroid is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2433 double of the corresponding cone.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2434 <p>Props. 31, 32, concluding the treatise, deduce the similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2435 formula for the volume of the greater segment, namely, in our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2436 figure,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2437 <MATH>(greater segmt.):(cone or segmt.of cone with same base and axis)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2438 =(<I>CA</I>+<I>AD</I>):<I>AD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2439 <C>On Spirals.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2440 <p>The treatise <I>On Spirals</I> begins with a preface addressed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2441 Dositheus in which Archimedes mentions the death of Conon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2442 as a grievous loss to mathematics, and then summarizes the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2443 main results of the treatises <I>On the Sphere and Cylinder</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2444 <I>On Conoids and Spheroids,</I> observing that the last two pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2445 positions of Book II of the former treatise took the place
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2446 of two which, as originally enunciated to Dositheus, were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2447 wrong; lastly, he states the main results of the treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2448 <I>On Spirals,</I> premising the definition of a spiral which is as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2449 follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2450 <p>‘If a straight line one extremity of which remains fixed be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2451 made to revolve at a uniform rate in a plane until it returns
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2452 to the position from which it started, and if, at the same time
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2453 as the straight line is revolving, a point move at a uniform
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2454 rate along the straight line, starting from the fixed extremity,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2455 the point will describe a spiral in the plane.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2456 <p>As usual, we have a series of propositions preliminary to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2457 the main subject, first two propositions about uniform motion,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2458 <pb n=65><head>ON SPIRALS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2459 then two simple geometrical propositions, followed by pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2460 positions (5-9) which are all of one type. Prop. 5 states that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2461 given a circle with centre <I>O,</I> a tangent to it at <I>A,</I> and <I>c,</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2462 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2463 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2464 circumference of any circle whatever, it is possible to draw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2465 a straight line <I>OPF</I> meeting the circle in <I>P</I> and the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2466 in <I>F</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2467 <MATH><I>FP</I>:<I>OP</I><(arc <I>AP</I>):<I>c.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2468 <p>Archimedes takes <I>D</I> a straight line greater than <I>c,</I> draws
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2469 <I>OH</I> parallel to the tangent at <I>A</I> and then says ‘let <I>PH</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2470 placed equal to <I>D verging</I> (<G>neu/ousa</G>) towards <I>A</I>’. This is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2471 usual phraseology of the type of problem known as <G>neu=sis</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2472 where a straight line of given length has to be placed between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2473 two lines or curves in such a position that, if produced, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2474 passes through a given point (this is the meaning of <I>verging</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2475 Each of the propositions 5-9 depends on a <G>neu=sis</G> of this kind,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2476 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2477 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2478 which Archimedes assumes as ‘possible’ without showing how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2479 it is effected. Except in the case of Prop. 5, the theoretical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2480 solution cannot be effected by means of the straight line and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2481 circle; it depends in general on the solution of an equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2482 of the fourth degree, which can be solved by means of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2483 <pb n=66><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2484 points of intersection of a certain rectangular hyperbola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2485 and a certain parabola. It is quite possible, however, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2486 such problems were in practice often solved by a mechanical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2487 method, namely by placing a ruler, by trial, in the position of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2488 the required line: for it is only necessary to place the ruler
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2489 so that it passes through the given point and then turn it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2490 round that point as a pivot till the intercept becomes of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2491 given length. In Props. 6-9 we have a circle with centre <I>O,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2492 a chord <I>AB</I> less than the diameter in it, <I>OM</I> the perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2493 from <I>O</I> on <I>AB, BT</I> the tangent at <I>B, OT</I> the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2494 through <I>O</I> parallel to <I>AB; D</I>:<I>E</I> is any ratio less or greater,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2495 as the case may be, than the ratio <I>BM</I>:<I>MO</I>. Props. 6, 7
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2496 (Fig. 2) show that it is possible to draw a straight line <I>OFP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2497 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2498 <CAP>FIG. 3.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2499 meeting <I>AB</I> in <I>F</I> and the circle in <I>P</I> such that <I>FP</I>:<I>PB</I>=<I>D</I>:<I>E</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2500 (<I>OP</I> meeting <I>AB</I> in the case where <I>D</I>:<I>E</I><<I>BM</I>:<I>MO,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2501 meeting <I>AB</I> produced when <I>D</I>:<I>E</I>><I>BM</I>:<I>MO</I>). In Props. 8, 9
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2502 (Fig. 3) it is proved that it is possible to draw a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2503 <I>OFP</I> meeting <I>AB</I> in <I>F,</I> the circle in <I>P</I> and the tangent at <I>B</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2504 <I>G,</I> such that <I>FP</I>:<I>BG</I>=<I>D</I>:<I>E</I> (<I>OP</I> meeting <I>AB</I> itself in the case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2505 where <I>D</I>:<I>E</I><<I>BM</I>:<I>MO,</I> and meeting <I>AB</I> produced in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2506 case where <I>D</I>:<I>E</I>><I>BM</I>:<I>MO</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2507 <p>We will illustrate by the constructions in Props. 7, 8,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2508 as it is these propositions which are actually cited later.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2509 Prop. 7. If <I>D</I>:<I>E</I> is any ratio><I>BM</I>:<I>MO,</I> it is required (Fig. 2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2510 to draw <I>OP′F</I>′ meeting the circle in <I>P</I>′ and <I>AB</I> produced in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2511 <I>F</I>′ so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2512 <MATH><I>F</I>′<I>P</I>′:<I>P</I>′<I>B</I>=<I>D</I>:<I>E</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2513 <p>Draw <I>OT</I> parallel to <I>AB,</I> and let the tangent to the circle at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2514 <I>B</I> meet <I>OT</I> in <I>T.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2515 <pb n=67><head>ON SPIRALS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2516 <p>Then <MATH><I>D</I>:<I>E</I>><I>BM</I>:<I>MO</I></MATH>, by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2517 <MATH>><I>OB</I>:<I>BT</I></MATH>, by similar triangles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2518 <p>Take a straight line <I>P</I>′<I>H</I>′ (less than <I>BT</I>) such that <MATH><I>D</I>:<I>E</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2519 =<I>OB</I>:<I>P</I>′<I>H</I>′</MATH>, and place <I>P</I>′<I>H</I>′ between the circle and <I>OT</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2520 ‘verging towards <I>B</I>’ (construction assumed).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2521 <p>Then <MATH><I>F</I>′<I>P</I>′:<I>P</I>′<I>B</I>=<I>OP</I>′:<I>P</I>′<I>H</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2522 =<I>OB</I>:<I>P</I>′<I>H</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2523 =<I>D</I>:<I>E</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2524 <p>Prop. 8. If <I>D</I>:<I>E</I> is any given ratio < <I>BM</I>:<I>MO</I>, it is required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2525 to draw <I>OFPG</I> meeting <I>AB</I> in <I>F</I>, the circle in <I>P</I>, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2526 tangent at <I>B</I> to the circle in <I>G</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2527 <MATH><I>FP</I>:<I>BG</I>=<I>D</I>:<I>E</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2528 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2529 <p>If <I>OT</I> is parallel to <I>AB</I> and meets the tangent at <I>B</I> in <I>T</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2530 <MATH><I>BM</I>:<I>MO</I>=<I>OB</I>:<I>BT</I></MATH>, by similar triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2531 whence <MATH><I>D</I>:<I>E</I><<I>OB</I>:<I>BT</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2532 <p>Produce <I>TB</I> to <I>C,</I> making <I>BC</I> of such length that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2533 <MATH><I>D</I>:<I>E</I>=<I>OB</I>:<I>BC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2534 so that <I>BC</I>><I>BT.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2535 <p>Describe a circle through the three points <I>O, T, C</I> and let <I>OB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2536 produced meet this circle in <I>K</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2537 <p>‘Then, since <I>BC</I>><I>BT</I>, and <I>OK</I> is perpendicular to <I>CT</I>, it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2538 possible to place <I>QG</I> [between the circle <I>TKC</I> and <I>BC</I>] equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2539 <I>BK</I> and verging towards <I>O</I>’ (construction assumed).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2540 <pb n=68><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2541 <p>Let <I>QGO</I> meet the original circle in <I>P</I> and <I>AB</I> in <I>F</I>. Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2542 <I>OFPG</I> is the straight line required.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2543 <p>For <MATH><I>CG.GT</I> = <I>OG.GQ</I> = <I>OG.BK</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2544 <p>But <MATH><I>OF</I>:<I>OG</I> = <I>BT</I>:<I>GT</I></MATH>, by parallels,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2545 whence <MATH><I>OF.GT</I> = <I>OG.BT</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2546 <p>Therefore <MATH><I>CG.GT</I>:<I>OF.GT</I> = <I>OG.BK</I>:<I>OG.BT</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2547 whence <MATH><I>CG</I>:<I>OF</I> = <I>BK</I>:<I>BT</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2548 = <I>BC</I>:<I>OB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2549 = <I>BC</I>:<I>OP</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2550 <p>Therefore <MATH><I>OP</I>:<I>OF</I> = <I>BC</I>:<I>CG</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2551 and hence <MATH><I>PF</I>:<I>OP</I> = <I>BG</I>:<I>BC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2552 or <MATH><I>PF</I>:<I>BG</I> = <I>OB</I>:<I>BC</I> = <I>D</I>:<I>E</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2553 <p>Pappus objects to Archimedes's use of the <G>neu=sis</G> assumed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2554 Prop. 8, 9 in these words:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2555 <p>‘it seems to be a grave error into which geometers fall
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2556 whenever any one discovers the solution of a plane problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2557 by means of conics or linear (higher) curves, or generally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2558 solves it by means of a foreign kind, as is the case e.g. (1) with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2559 the problem in the fifth Book of the Conics of Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2560 relating to the parabola, and (2) when Archimedes assumes in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2561 his work on the spiral a <G>neu=sis</G> of a “solid” character with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2562 reference to a circle; for it is possible without calling in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2563 aid of anything solid to find the proof of the theorem given by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2564 Archimedes, that is, to prove that the circumference of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2565 circle arrived at in the first revolution is equal to the straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2566 line drawn at right angles to the initial line to meet the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2567 to the spiral (i.e. the subtangent).’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2568 <p>There is, however, this excuse for Archimedes, that he only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2569 assumes that the problem <I>can</I> be solved and does not assume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2570 the actual solution. Pappus<note>Pappus, iv, pp. 298-302.</note> himself gives a solution of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2571 particular <G>neu=sis</G> by means of conics. Apollonius wrote two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2572 Books of <G>neu/seis</G>, and it is quite possible that by Archimedes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2573 time there may already have been a collection of such problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2574 to which tacit reference was permissible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2575 <p>Prop. 10 repeats the result of the Lemma to Prop. 2 of <I>On</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2576 <pb n=69><head>ON SPIRALS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2577 <I>Conoids and Spheroids</I> involving the summation of the series
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2578 <MATH>1<SUP>2</SUP> + 2<SUP>2</SUP> + 3<SUP>2</SUP> + ... + <I>n</I><SUP>2</SUP></MATH>. Prop 11 proves another proposition in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2579 summation, namely that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2580 <MATH>(<I>n</I>-1)(<I>na</I>)<SUP>2</SUP>:{<I>a</I><SUP>2</SUP>+(2<I>a</I>)<SUP>2</SUP>+(3<I>a</I>)<SUP>2</SUP>+...+(&horbar;(<I>n</I>-1))<I>a</I>)<SUP>2</SUP>}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2581 >(<I>na</I>)<SUP>2</SUP>:{<I>na.a</I>+1/3(<I>na</I>-<I>a</I>)<SUP>2</SUP>}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2582 >(<I>n</I>-1)(<I>na</I>)<SUP>2</SUP>:{(2<I>a</I>)<SUP>2</SUP>+(3<I>a</I>)<SUP>2</SUP>+...+(<I>na</I>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2583 The same proposition is also true if the terms of the series
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2584 are <MATH><I>a</I><SUP>2</SUP>, (<I>a</I>+<I>b</I>)<SUP>2</SUP>, (<I>a</I>+2<I>b</I>)<SUP>2</SUP> ... (<I>a</I>+&horbar;(<I>n</I>-1)<I>b</I>)<SUP>2</SUP></MATH>, and it is assumed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2585 the more general form in Props. 25, 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2586 <p>Archimedes now introduces his Definitions, of the <I>spiral</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2587 itself, the <I>origin</I>, the <I>initial line</I>, the <I>first distance</I> (= the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2588 radius vector at the end of one revolution), the <I>second distance</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2589 (= the equal length added to the radius vector during the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2590 second complete revolution), and so on; the <I>first area</I> (the area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2591 bounded by the spiral described in the first revolution and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2592 the ‘first distance’), the <I>second area</I> (that bounded by the spiral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2593 described in the second revolution and the ‘second distance’),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2594 and so on; the <I>first circle</I> (the circle with the ‘first distance’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2595 as radius), the <I>second circle</I> (the circle with radius equal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2596 sum of the ‘first’ and ‘second distances’, or twice the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2597 distance), and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2598 <p>Props. 12, 14, 15 give the fundamental property of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2599 spiral connecting the length of the radius vector with the angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2600 through which the initial line has revolved from its original
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2601 position, and corresponding to the equation in polar coordinates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2602 <MATH><I>r</I> = <I>a</I><G>q</G></MATH>. As Archimedes does not speak of angles greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2603 than <G>p</G>, or 2<G>p</G>, he has, in the case of points on any turn after
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2604 the first, to use multiples of the circumference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2605 of a circle as well as arcs of it. He uses the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2606 ‘first circle’ for this purpose. Thus, if <I>P, Q</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2607 are two points on the first turn,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2608 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2609 <MATH><I>OP</I>:<I>OQ</I> = (arc <I>AKP</I>′):(arc <I>AKQ</I>′)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2610 if <I>P, Q</I> are points on the <I>n</I>th turn of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2611 spiral, and <I>c</I> is the circumference of the first circle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2612 <MATH><I>OP</I>:<I>OQ</I> = {(<I>n</I> - 1)<I>c</I> + arc <I>AKP</I>′}:{(<I>n</I> - 1)<I>c</I> + arc <I>AKQ</I>′}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2613 <p>Prop. 13 proves that, if a straight line touches the spiral, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2614 <pb n=70><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2615 touches it at one point only. For, if possible, let the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2616 at <I>P</I> touch the spiral at another point <I>Q</I>. Then, if we bisect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2617 the angle <I>POQ</I> by <I>OL</I> meeting <I>PQ</I> in <I>L</I> and the spiral in <I>R</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2618 <MATH><I>OP</I> + <I>OQ</I> = 2<I>OR</I></MATH> by the property of the spiral. But by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2619 the property of the triangle (assumed, but easily proved)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2620 <MATH><I>OP</I> + <I>OQ</I> > 2<I>OL</I></MATH>, so that <MATH><I>OL</I> < <I>OR</I></MATH>, and some point of <I>PQ</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2621 lies within the spiral. Hence <I>PQ</I> cuts the spiral, which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2622 contrary to the hypothesis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2623 <p>Props. 16, 17 prove that the angle made by the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2624 at a point with the radius vector to that point is obtuse on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2625 ‘forward’ side, and acute on the ‘backward’ side, of the radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2626 vector.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2627 <p>Props. 18-20 give the fundamental proposition about the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2628 tangent, that is to say, they give the length of the <I>subtangent</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2629 at any point <I>P</I> (the distance between <I>O</I> and the point of inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2630 section of the tangent with the perpendicular from <I>O</I> to <I>OP</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2631 Archimedes always deals first with the first turn and then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2632 with any subsequent turn, and with each complete turn before
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2633 parts or points of any particular turn. Thus he deals with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2634 tangents in this order, (1) the tangent at <I>A</I> the end of the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2635 turn, (2) the tangent at the end of the second and any subse-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2636 quent turn, (3) the tangent at any intermediate point of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2637 first or any subsequent turn. We will take as illustrative
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2638 the case of the tangent at any intermediate point <I>P</I> of the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2639 turn (Prop. 20).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2640 <p>If <I>OA</I> be the initial line, <I>P</I> any point on the first turn, <I>PT</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2641 the tangent at <I>P</I> and <I>OT</I> perpendicular to <I>OP</I>, then it is to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2642 proved that, if <I>ASP</I> be the circle through <I>P</I> with centre <I>O</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2643 meeting <I>PT</I> in <I>S</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2644 <MATH>(subtangent <I>OT</I>) = (arc <I>ASP</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2645 <p>I. If possible, let <I>OT</I> be greater than the arc <I>ASP</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2646 <p>Measure off <I>OU</I> such that <I>OU</I> > arc <I>ASP</I> but < <I>OT</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2647 <p>Then the ratio <I>PO</I>:<I>OU</I> is greater than the ratio <I>PO</I>:<I>OT</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2648 i.e. greater than the ratio of 1/2<I>PS</I> to the perpendicular from <I>O</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2649 on <I>PS</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2650 <p>Therefore (Prop. 7) we can draw a straight line <I>OQF</I> meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2651 <I>TP</I> produced in <I>F</I>, and the circle in <I>Q</I>, such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2652 <MATH><I>FQ</I>:<I>PQ</I> = <I>PO</I>:<I>OU</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2653 <pb n=71><head>ON SPIRALS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2654 <p>Let <I>OF</I> meet the spiral in <I>Q</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2655 <p>Then we have, <I>alternando</I>, since <MATH><I>PO</I> = <I>QO</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2656 <I>FQ</I>:<I>QO</I> = <I>PQ</I>:<I>OU</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2657 < (arc <I>PQ</I>):(arc <I>ASP</I>)</MATH>, by hypothesis and <I>a fortiori.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2658 Componendo</I>, <MATH><I>FO</I>:<I>QO</I> < (arc <I>ASQ</I>):(arc <I>ASP</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2659 < <I>OQ</I>′:<I>OP</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2660 <p>But <MATH><I>QO</I> = <I>OP</I></MATH>; therefore <MATH><I>FO</I> < <I>OQ</I>′</MATH>; which is impossible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2661 <p>Therefore <I>OT</I> is not greater than the arc <I>ASP</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2662 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2663 <p>II. Next suppose, if possible, that <I>OT</I> < arc <I>ASP</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2664 <p>Measure <I>OV</I> along <I>OT</I> such that <I>OV</I> is greater than <I>OT</I> but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2665 less than the arc <I>ASP</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2666 <p>Then the ratio <I>PO</I>:<I>OV</I> is less than the ratio <I>PO</I>:<I>OT</I>, i.e.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2667 than the ratio of 1/2<I>PS</I> to the perpendicular from <I>O</I> on <I>PS</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2668 therefore it is possible (Prop. 8) to draw a straight line <I>OF</I>′<I>RG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2669 meeting <I>PS</I>, the circle <I>PSA</I>, and the tangent to the circle at <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2670 in <I>F</I>′, <I>R, G</I> respectively, and such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2671 <MATH><I>F</I>′<I>R</I>:<I>GP</I> = <I>PO</I>:<I>OV</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2672 <pb n=72><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2673 <p>Let <I>OF</I>′<I>G</I> meet the spiral in <I>R</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2674 <p>Then, since <I>PO</I> = <I>RO</I>, we have, <I>alternando</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2675 <MATH><I>F</I>′<I>R</I>:<I>RO</I> = <I>GP</I>:<I>OV</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2676 > (arc <I>PR</I>):(arc <I>ASP</I>)</MATH>, <I>a fortiori</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2677 whence <MATH><I>F</I>′<I>O</I>:<I>RO</I> < (arc <I>ASR</I>):(arc <I>ASP</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2678 < <I>OR</I>′:<I>OP</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2679 so that <I>F</I>′<I>O</I> < <I>OR</I>′; which is impossible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2680 <p>Therefore <I>OT</I> is not less than the arc <I>ASP</I>. And it was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2681 proved not greater than the same arc. Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2682 <MATH><I>OT</I> = (arc <I>ASP</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2683 <p>As particular cases (separately proved by Archimedes), if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2684 <I>P</I> be the extremity of the first turn and <I>c</I><SUB>1</SUB> the circumference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2685 of the first circle, the subtangent = <I>c</I><SUB>1</SUB>; if <I>P</I> be the extremity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2686 of the second turn and <I>c</I><SUB>2</SUB> the circumference of the ‘second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2687 circle’, the subtangent = 2<I>c</I><SUB>2</SUB>; and generally, if <I>c<SUB>n</SUB></I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2688 circumference of the <I>n</I>th circle (the circle with the radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2689 vector to the extremity of the <I>n</I>th turn as radius), the sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2690 tangent to the tangent at the extremity of the <I>n</I>th turn = <I>nc<SUB>n</SUB></I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2691 <p>If <I>P</I> is a point on the <I>n</I>th turn, not the extremity, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2692 circle with <I>O</I> as centre and <I>OP</I> as radius cuts the initial line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2693 in <I>K</I>, while <I>p</I> is the circumference of the circle, the sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2694 tangent to the tangent at <MATH><I>P</I> = (<I>n</I> - 1)<I>p</I> + arc <I>KP</I></MATH> (measured
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2695 ‘forward’).<note>On the whole course of Archimedes's proof of the property of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2696 subtangent, see note in the Appendix.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2697 <p>The remainder of the book (Props. 21-8) is devoted to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2698 finding the areas of portions of the spiral and its several
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2699 turns cut off by the initial line or any two radii vectores.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2700 We will illustrate by the general case (Prop. 26). Take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2701 <I>OB, OC</I>, two bounding radii vectores, including an arc <I>BC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2702 of the spiral. With centre <I>O</I> and radius <I>OC</I> describe a circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2703 Divide the angle <I>BOC</I> into any number of equal parts by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2704 radii of this circle. The spiral meets these radii in points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2705 <I>P, Q ... Y, Z</I> such that the radii vectores <I>OB, OP, OQ ... OZ, OC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2706 <pb n=73><head>ON SPIRALS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2707 are in arithmetical progression. Draw arcs of circles with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2708 radii <I>OB, OP, OQ</I> ... as shown; this produces a figure circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2709 scribed to the spiral and consisting of the sum of small sectors
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2710 of circles, and an inscribed figure of the same kind. As the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2711 first sector in the circumscribed figure is equal to the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2712 sector in the inscribed, it is easily seen that the areas of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2713 circumscribed and inscribed figures differ by the difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2714 between the sectors <I>OzC</I> and <I>OBp</I>′; therefore, by increasing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2715 the number of divisions of the angle <I>BOC</I>, we can make the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2716 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2717 difference between the areas of the circumscribed and in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2718 scribed figures as small as we please; we have, therefore, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2719 elements necessary for the application of the method of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2720 exhaustion.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2721 <p>If there are <I>n</I> radii <I>OB, OP ... OC</I>, there are (<I>n</I> - 1) parts of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2722 the angle <I>BOC</I>. Since the angles of all the small sectors are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2723 equal, the sectors are as the square on their radii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2724 <p>Thus <MATH>(whole sector <I>Ob</I>′<I>C</I>):(circumscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2725 = (<I>n</I> - 1)<I>OC</I><SUP>2</SUP>:(<I>OP</I><SUP>2</SUP> + <I>OQ</I><SUP>2</SUP> + ... + <I>OC</I><SUP>2</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2726 and <MATH>(whole sector <I>Ob</I>′<I>C</I>):(inscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2727 = (<I>n</I> - 1)<I>OC</I><SUP>2</SUP>:(<I>OB</I><SUP>2</SUP> + <I>OP</I><SUP>2</SUP> + <I>OQ</I><SUP>2</SUP> + ... + <I>OZ</I><SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2728 <pb n=74><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2729 <p>And <I>OB, OP, OQ, ... OZ, OC</I> is an arithmetical progression
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2730 of <I>n</I> terms; therefore (cf. Prop. 11 and Cor.),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2731 <MATH>(<I>n</I> - 1)<I>OC</I><SUP>2</SUP>:(<I>OP</I><SUP>2</SUP> + <I>OQ</I><SUP>2</SUP> + ... + <I>OC</I><SUP>2</SUP>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2732 < <I>OC</I><SUP>2</SUP>:{<I>OC.OB</I> + 1/3(<I>OC</I> - <I>OB</I>)<SUP>2</SUP>}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2733 < (<I>n</I> - 1)<I>OC</I><SUP>2</SUP>:(<I>OB</I><SUP>2</SUP> + <I>OP</I><SUP>2</SUP> + ... + <I>OZ</I><SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2734 <p>Compressing the circumscribed and inscribed figures together
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2735 in the usual way, Archimedes proves by exhaustion that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2736 <MATH>(sector <I>Ob</I>′<I>C</I>):(area of spiral <I>OBC</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2737 = <I>OC</I><SUP>2</SUP>:{<I>OC.OB</I> + 1/3(<I>OC</I> - <I>OB</I>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2738 <p>If <MATH><I>OB</I> = <I>b, OC</I> = <I>c</I>, and (<I>c</I> - <I>b</I>) = (<I>n</I> - 1)<I>h</I></MATH>, Archimedes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2739 result is the equivalent of saying that, when <I>h</I> diminishes and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2740 <I>n</I> increases indefinitely, while <I>c</I> - <I>b</I> remains constant,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2741 limit of <MATH><I>h</I>{<I>b</I><SUP>2</SUP> + (<I>b</I> + <I>h</I>)<SUP>2</SUP> + (<I>b</I> + 2<I>h</I>)<SUP>2</SUP> + ... + (<I>b</I> + &horbar;(<I>n</I> - 2)<I>h</I>)<SUP>2</SUP>}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2742 = (<I>c</I> - <I>b</I>) {<I>cb</I> + 1/3(<I>c</I> - <I>b</I>)<SUP>2</SUP>}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2743 = 1/3(<I>c</I><SUP>3</SUP> - <I>b</I><SUP>3</SUP>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2744 that is, with our notation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2745 <MATH>∫<SUP>c</SUP><SUB>b</SUB><I>x</I><SUP>2</SUP><I>dx</I> = 1/3(<I>c</I><SUP>3</SUP> - <I>b</I><SUP>3</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2746 <p>In particular, the area included by the first turn and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2747 initial line is bounded by the radii vectores 0 and 2<G>p</G><I>a</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2748 the area, therefore, is to the circle with radius 2<G>p</G><I>a</I> as 1/3(2<G>p</G><I>a</I>)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2749 to (2<G>p</G><I>a</I>)<SUP>2</SUP>, that is to say, it is 1/3 of the circle or 1/3<G>p</G>(2<G>p</G><I>a</I>)<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2750 This is separately proved in Prop. 24 by means of Prop. 10
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2751 and Corr. 1, 2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2752 <p>The area of the ring added while the radius vector describes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2753 the second turn is the area bounded by the radii vectores 2<G>p</G><I>a</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2754 and 4<G>p</G><I>a</I>, and is to the circle with radius 4<G>p</G><I>a</I> in the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2755 of <MATH>{<I>r</I><SUB>2</SUB><I>r</I><SUB>1</SUB> + 1/3(<I>r</I><SUB>2</SUB> - <I>r</I><SUB>1</SUB>)<SUP>2</SUP>} to <I>r</I><SUP>2</SUP><SUB>2</SUB>, where <I>r</I><SUB>1</SUB> = 2<G>p</G><I>a</I> and <I>r</I><SUB>2</SUB> =
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2756 4<G>p</G><I>a</I></MATH>; the ratio is 7:12 (Prop. 25).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2757 <p>If <I>R</I><SUB>1</SUB> be the area of the first turn of the spiral bounded by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2758 the initial line, <I>R</I><SUB>2</SUB> the area of the ring added by the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2759 complete turn, <I>R</I><SUB>3</SUB> that of the ring added by the third turn,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2760 and so on, then (Prop. 27)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2761 <MATH><I>R</I><SUB>3</SUB> = 2<I>R</I><SUB>2</SUB>, <I>R</I><SUB>4</SUB> = 3<I>R</I><SUB>2</SUB>, <I>R</I><SUB>5</SUB> = 4<I>R</I><SUB>2</SUB>, ... <I>R<SUB>n</SUB></I> = (<I>n</I> - 1)<I>R</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2762 Also <MATH><I>R</I><SUB>2</SUB> = 6<I>R</I><SUB>1</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2763 <pb n=75><head>ON SPIRALS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2764 <p>Lastly, if <I>E</I> be the portion of the sector <I>b</I>′<I>OC</I> bounded by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2765 <I>b</I>′<I>B</I>, the arc <I>b</I>′<I>zC</I> of the circle and the arc <I>BC</I> of the spiral, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2766 <I>F</I> the portion cut off between the arc <I>BC</I> of the spiral, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2767 radius <I>OC</I> and the arc intercepted between <I>OB</I> and <I>OC</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2768 the circle with centre <I>O</I> and radius <I>OB</I>, it is proved that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2769 <MATH><I>E</I>:<I>F</I> = {<I>OB</I> + 2/3(<I>OC</I> - <I>OB</I>)}:{<I>OB</I> + 1/3(<I>OC</I> - <I>OB</I>)}</MATH> (Prop. 28).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2770 <C>On Plane Equilibriums, I, II.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2771 <p>In this treatise we have the fundamental principles of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2772 mechanics established by the methods of geometry in its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2773 strictest sense. There were doubtless earlier treatises on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2774 mechanics, but it may be assumed that none of them had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2775 been worked out with such geometrical rigour. Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2776 begins with seven Postulates including the following prin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2777 ciples. Equal weights at equal distances balance; if unequal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2778 weights operate at equal distances, the larger weighs down
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2779 the smaller. If when equal weights are in equilibrium some-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2780 thing be added to, or subtracted from, one of them, equilibrium
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2781 is not maintained but the weight which is increased or is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2782 diminished prevails. When equal and similar plane figures
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2783 coincide if applied to one another, their centres of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2784 similarly coincide; and in figures which are unequal but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2785 similar the centres of gravity will be ‘similarly situated’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2786 In any figure the contour of which is concave in one and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2787 same direction the centre of gravity must be within the figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2788 Simple propositions (1-5) follow, deduced by <I>reductio ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2789 absurdum</I>; these lead to the fundamental theorem, proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2790 first for commensurable and then by <I>reductio ad absurdum</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2791 for incommensurable magnitudes, that <I>Two magnitudes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2792 whether commensurable or incommensurable, balance at dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2793 tances reciprocally proportional to the magnitudes</I> (Props.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2794 6, 7). Prop. 8 shows how to find the centre of gravity of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2795 a part of a magnitude when the centres of gravity of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2796 other part and of the whole magnitude are given. Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2797 then addresses himself to the main problems of Book I, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2798 to find the centres of gravity of (1) a parallelogram (Props.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2799 9, 10), (2) a triangle (Props. 13, 14), and (3) a parallel-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2800 trapezium (Prop. 15), and here we have an illustration of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2801 extraordinary rigour which he requires in his geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2802 <pb n=76><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2803 proofs. We do not find him here assuming, as in <I>The Method</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2804 that, if all the lines that can be drawn in a figure parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2805 (and including) one side have their middle points in a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2806 line, the centre of gravity must lie somewhere on that straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2807 line; he is not content to regard the figure as <I>made up</I> of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2808 infinity of such parallel lines; pure geometry realizes that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2809 the parallelogram is made up of elementary parallelograms,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2810 indefinitely narrow if you please, but still parallelograms, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2811 the triangle of elementary <I>trapezia</I>, not straight lines, so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2812 that to assume directly that the centre of gravity lies on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2813 straight line bisecting the parallelograms would really be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2814 a <I>petitio principii</I>. Accordingly the result, no doubt dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2815 covered in the informal way, is clinched by a proof by <I>reductio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2816 ad absurdum</I> in each case. In the case of the parallelogram
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2817 <I>ABCD</I> (Prop. 9), if the centre of gravity is not on the straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2818 line <I>EF</I> bisecting two opposite sides, let it be at <I>H</I>. Draw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2819 <I>HK</I> parallel to <I>AD</I>. Then it is possible by bisecting <I>AE, ED</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2820 then bisecting the halves, and so on, ultimately to reach
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2821 a length less than <I>KH</I>. Let this be done, and through the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2822 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2823 points of division of <I>AD</I> draw parallels to <I>AB</I> or <I>DC</I> making
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2824 a number of equal and similar parallelograms as in the figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2825 The centre of gravity of each of these parallelograms is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2826 similarly situated with regard to it. Hence we have a number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2827 of equal magnitudes with their centres of gravity at equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2828 distances along a straight line. Therefore the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2829 gravity of the whole is on the line joining the centres of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2830 of the two middle parallelograms (Prop. 5, Cor. 2). But this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2831 is impossible, because <I>H</I> is outside those parallelograms.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2832 Therefore the centre of gravity cannot but lie on <I>EF</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2833 <p>Similarly the centre of gravity lies on the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2834 bisecting the other opposite sides <I>AB, CD</I>; therefore it lies at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2835 the intersection of this line with <I>EF</I>, i.e. at the point of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2836 intersection of the diagonals.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2837 <pb n=77><head>ON PLANE EQUILIBRIUMS, I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2838 <p>The proof in the case of the triangle is similar (Prop. 13).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2839 Let <I>AD</I> be the median through <I>A</I>. The centre of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2840 must lie on <I>AD</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2841 <p>For, if not, let it be at <I>H</I>, and draw <I>HI</I> parallel to <I>BC</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2842 Then, if we bisect <I>DC</I>, then bisect the halves, and so on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2843 we shall arrive at a length <I>DE</I> less than <I>IH</I>. Divide <I>BC</I> into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2844 lengths equal to <I>DE</I>, draw parallels to <I>DA</I> through the points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2845 of division, and complete the small parallelograms as shown in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2846 the figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2847 <p>The centres of gravity of the whole parallelograms <I>SN, TP,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2848 FQ</I> lie on <I>AD</I> (Prop. 9); therefore the centre of gravity of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2849 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2850 figure formed by them all lies on <I>AD</I>; let it be <I>O</I>. Join <I>OH</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2851 and produce it to meet in <I>V</I> the parallel through <I>C</I> to <I>AD</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2852 <p>Now it is easy to see that, if <I>n</I> be the number of parts into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2853 which <I>DC, AC</I> are divided respectively,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2854 <MATH>(sum of small &utri;s <I>AMR, MLS ... ARN, NUP</I> ...):(&utri;<I>ABC</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2855 = <I>n.AN</I><SUP>2</SUP>:<I>AC</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2856 = 1:<I>n</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2857 whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2858 <MATH>(sum of small &utri;s):(sum of parallelograms) = 1:(<I>n</I> - 1)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2859 <p>Therefore the centre of gravity of the figure made up of all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2860 the small triangles is at a point <I>X</I> on <I>OH</I> produced such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2861 <MATH><I>XH</I> = (<I>n</I>-1)<I>OH</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2862 <p>But <MATH><I>VH</I>:<I>HO</I> < <I>CE</I>:<I>ED</I> or (<I>n</I>-1):1</MATH>; therefore <I>XH</I> > <I>VH</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2863 It follows that the centre of gravity of all the small
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2864 triangles taken together lies at <I>X</I> notwithstanding that all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2865 the triangles lie on one side of the parallel to <I>AD</I> drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2866 through <I>X</I>: which is impossible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2867 <pb n=78><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2868 <p>Hence the centre of gravity of the whole triangle cannot
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2869 but lie on <I>AD</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2870 <p>It lies, similarly, on either of the other two medians; so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2871 that it is at the intersection of any two medians (Prop. 14).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2872 <p>Archimedes gives alternative proofs of a direct character,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2873 both for the parallelogram and the triangle, depending on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2874 postulate that the centres of gravity of similar figures are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2875 ‘similarly situated’ in regard to them (Prop. 10 for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2876 parallelogram, Props. 11, 12 and part 2 of Prop. 13 for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2877 triangle).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2878 <p>The geometry of Prop. 15 deducing the centre of gravity of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2879 a trapezium is also interesting. It is proved that, if <I>AD, BC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2880 are the parallel sides (<I>AD</I> being the smaller), and <I>EF</I> is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2881 straight line joining their middle points, the centre of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2882 is at a point <I>G</I> on <I>EF</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2883 <MATH><I>GE</I>:<I>GF</I> = (2<I>BC</I> + <I>AD</I>):(2<I>AD</I> + <I>BC</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2884 <p>Book II of the treatise is entirely devoted to finding the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2885 centres of gravity of a parabolic segment (Props. 1-8) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2886 of a portion of it cut off by a parallel to the base (Props. 9, 10).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2887 Prop. 1 (really a particular case of I. 6, 7) proves that, if <I>P, P</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2888 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2889 be the areas of two parabolic segments and <I>D, E</I> their centres
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2890 of gravity, the centre of gravity of both taken together is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2891 at a point <I>C</I> on <I>DE</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2892 <MATH><I>P</I>:<I>P</I>′ = <I>CE</I>:<I>CD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2893 <pb n=79><head>ON PLANE EQUILIBRIUMS, I, II</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2894 This is merely preliminary. Then begins the real argument,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2895 the course of which is characteristic and deserves to be set out.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2896 Archimedes uses a series of figures inscribed to the segment,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2897 as he says, ‘in the recognized manner’ (<G>gnwri/mws</G>). The rule
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2898 is as follows. Inscribe in the segment the triangle <I>ABB</I>′ with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2899 the same base and height; the vertex <I>A</I> is then the point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2900 of contact of the tangent parallel to <I>BB</I>′. Do the same with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2901 the remaining segments cut off by <I>AB, AB</I>′, then with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2902 segments remaining, and so on. If <I>BRQPAP</I>′<I>Q</I>′<I>R</I>′<I>B</I>′ is such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2903 a figure, the diameters through <I>Q, Q</I>′, <I>P, P</I>′, <I>R, R</I>′ bisect the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2904 straight lines <I>AB, AB</I>′, <I>AQ, AQ</I>′, <I>QB, Q</I>′<I>B</I>′ respectively, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2905 <I>BB</I>′ is divided by the diameters into parts which are all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2906 equal. It is easy to prove also that <I>PP</I>′, <I>QQ</I>′, <I>RR</I>′ are all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2907 parallel to <I>BB</I>′, and that <MATH><I>AL</I>:<I>LM</I>:<I>MN</I>:<I>NO</I> = 1:3:5:7</MATH>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2908 same relation holding if the number of sides of the polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2909 is increased; i.e. the segments of <I>AO</I> are always in the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2910 of the successive odd numbers (Lemmas to Prop. 2). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2911 centre of gravity of the inscribed figure lies on <I>AO</I> (Prop. 2).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2912 If there be two parabolic segments, and two figures inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2913 in them ‘in the recognized manner’ with an equal number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2914 sides, the centres of gravity divide the respective axes in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2915 same proportion, for the ratio depends on the same ratio of odd
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2916 numbers 1:3:5:7 ... (Prop. 3). The centre of gravity of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2917 parabolic segment itself lies on the diameter <I>AO</I> (this is proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2918 in Prop. 4 by <I>reductio ad absurdum</I> in exactly the same way
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2919 as for the triangle in I. 13). It is next proved (Prop. 5) that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2920 the centre of gravity of the segment is nearer to the vertex <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2921 than the centre of gravity of the inscribed figure is; but that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2922 it is possible to inscribe in the segment in the recognized
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2923 manner a figure such that the distance between the centres of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2924 gravity of the segment and of the inscribed figure is less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2925 any assigned length, for we have only to increase the number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2926 of sides sufficiently (Prop. 6). Incidentally, it is observed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2927 Prop. 4 that, if in any segment the triangle with the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2928 base and equal height is inscribed, the triangle is greater than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2929 half the segment, whence it follows that, each time we increase
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2930 the number of sides in the inscribed figure, we take away
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2931 more than half of the segments remaining over; and in Prop. 5
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2932 that corresponding segments on opposite sides of the axis, e.g.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2933 <I>QRB, Q</I>′<I>R</I>′<I>B</I>′ have their axes equal and therefore are equal in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2934 <pb n=80><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2935 area. Lastly (Prop. 7), if there be two parabolic segments,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2936 their centres of gravity divide their diameters in the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2937 ratio (Archimedes enunciates this of similar segments only,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2938 but it is true of any two segments and is required of any two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2939 segments in Prop. 8). Prop. 8 now finds the centre of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2940 of any segment by using the last proposition. It is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2941 geometrical equivalent of the solution of a simple equation in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2942 the ratio (<I>m</I>, say) of <I>AG</I> to <I>AO</I>, where <I>G</I> is the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2943 gravity of the segment.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2944 <p>Since the segment = 4/3(&utri;<I>ABB</I>′), the sum of the two seg-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2945 ments <MATH><I>AQB, AQ</I>′<I>B</I>′ = 1/3(&utri;<I>ABB</I>′)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2946 <p>Further, if <I>QD, Q</I>′<I>D</I>′ are the diameters of these segments,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2947 <I>QD, Q</I>′<I>D</I>′ are equal, and, since the centres
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2948 of gravity <I>H, H</I>′ of the segments divide
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2949 <I>QD, Q</I>′<I>D</I>′ proportionally, <I>HH</I>′ is parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2950 to <I>QQ</I>′, and the centre of gravity of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2951 two segments together is at <I>K</I>, the point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2952 where <I>HH</I>′ meets <I>AO</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2953 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2954 <p>Now <MATH><I>AO</I> = 4<I>AV</I></MATH> (Lemma 3 to Prop.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2955 2), and <MATH><I>QD</I> = 1/2<I>AO</I> - <I>AV</I> = <I>AV</I></MATH>. But
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2956 <I>H</I> divides <I>QD</I> in the same ratio as <I>G</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2957 divides <I>AO</I> (Prop. 7); therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2958 <MATH><I>VK</I> = <I>QH</I> = <I>m.QD</I> = <I>m.AV</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2959 <p>Taking moments about <I>A</I> of the segment, the triangle <I>ABB</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2960 and the sum of the small segments, we have (dividing out by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2961 <I>AV</I> and &utri;<I>ABB</I>′)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2962 <MATH>1/3(1 + <I>m</I>) + 2/3.4 = 4/3.4<I>m</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2963 or <MATH>15<I>m</I> = 9</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2964 and <MATH><I>m</I> = 3/5</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2965 <p>That is, <MATH><I>AG</I> = 3/5<I>AO</I>, or <I>AG</I>:<I>GO</I> = 3:2</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2966 <p>The final proposition (10) finds the centre of gravity of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2967 portion of a parabola cut off between two parallel chords <I>PP</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2968 <I>BB</I>′. If <I>PP</I>′ is the shorter of the chords and the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2969 bisecting <I>PP</I>′, <I>BB</I>′ meets them in <I>N, O</I> respectively, Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2970 medes proves that, if <I>NO</I> be divided into five equal parts of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2971 which <I>LM</I> is the middle one (<I>L</I> being nearer to <I>N</I> than <I>M</I> is),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2972 <pb n=81><head>ON PLANE EQUILIBRIUMS, II</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2973 the centre of gravity <I>G</I> of the portion of the parabola between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2974 <I>PP</I>′ and <I>BB</I>′ divides <I>LM</I> in such a way that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2975 <MATH><I>LG</I>:<I>GM</I> = <I>BO</I><SUP>2</SUP>.(2<I>PN</I> + <I>BO</I>):<I>PN</I><SUP>2</SUP>.(2<I>BO</I> + <I>PN</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2976 <p>The geometrical proof is somewhat difficult, and uses a very
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2977 remarkable Lemma which forms Prop. 9. If <I>a, b, c, d, x, y</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2978 straight lines satisfying the conditions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2979 <MATH><BRACE><I>a</I>/<I>b</I> = <I>b</I>/<I>c</I> = <I>c</I>/<I>d</I>(<I>a</I> > <I>b</I> > <I>c</I> > <I>d</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2980 <I>d</I>/(<I>a</I> - <I>d</I>) = <I>x</I>/(3/5(<I>a</I> - <I>c</I>)),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2981 and (2<I>a</I> + 4<I>b</I> + 6<I>c</I> + 3<I>d</I>)/(5<I>a</I> + 10<I>b</I> + 10<I>c</I> + 5<I>d</I>) = <I>y</I>/(<I>a</I> - <I>c</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2982 then must <I>x</I> + <I>y</I> = 2/5<I>a</I></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2983 <p>The proof is entirely geometrical, but amounts of course to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2984 the elimination of three quantities <I>b, c, d</I> from the above four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2985 equations.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2986 <C>The Sand-reckoner (<I>Psammites</I> or <I>Arenarius</I>).</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2987 <p>I have already described in a previous chapter the remark-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2988 able system, explained in this treatise and in a lost work,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2989 ’<I>A<G>rxai/</G>, Principles</I>, addressed to Zeuxippus, for expressing very
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2990 large numbers which were beyond the range of the ordinary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2991 Greek arithmetical notation. Archimedes showed that his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2992 system would enable any number to be expressed up to that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2993 which in our notation would require 80,000 million million
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2994 ciphers and then proceeded to prove that this system more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2995 than sufficed to express the number of grains of sand which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2996 it would take to fill the universe, on a reasonable view (as it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2997 seemed to him) of the size to be attributed to the universe.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2998 Interesting as the book is for the course of the argument by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	2999 which Archimedes establishes this, it is, in addition, a docu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3000 ment of the first importance historically. It is here that we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3001 learn that Aristarchus put forward the Copernican theory of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3002 the universe, with the sun in the centre and the planets
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3003 including the earth revolving round it, and that Aristarchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3004 further discovered the angular diameter of the sun to be 1/(720)th
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3005 of the circle of the zodiac or half a degree. Since Archimedes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3006 in order to calculate a safe figure (not too small) for the size
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3007 <pb n=82><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3008 of the universe, has to make certain assumptions as to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3009 sizes and distances of the sun and moon and their relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3010 to the size of the universe, he takes the opportunity of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3011 quoting earlier views. Some have tried, he says, to prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3012 that the perimeter of the earth is about 300,000 stades; in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3013 order to be quite safe he will take it to be about ten times
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3014 this, or 3,000,000 stades, and not greater. The diameter of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3015 the earth, like most earlier astronomers, he takes to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3016 greater than that of the moon but less than that of the sun.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3017 Eudoxus, he says, declared the diameter of the sun to be nine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3018 times that of the moon, Phidias, his own father, twelve times,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3019 while Aristarchus tried to prove that it is greater than 18 but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3020 less than 20 times the diameter of the moon; he will again be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3021 on the safe side and take it to be 30 times, but not more. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3022 position is rather more difficult as regards the ratio of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3023 distance of the sun to the size of the universe. Here he seizes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3024 upon a dictum of Aristarchus that the sphere of the fixed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3025 stars is so great that the circle in which he supposes the earth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3026 to revolve (round the sun) ‘bears such a proportion to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3027 distance of the fixed stars as the centre of the sphere bears to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3028 its surface’. If this is taken in a strictly mathematical sense,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3029 it means that the sphere of the fixed stars is infinite in size,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3030 which would not suit Archimedes's purpose; to get another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3031 meaning out of it he presses the point that Aristarchus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3032 words cannot be taken quite literally because the centre, being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3033 without magnitude, cannot be in any ratio to any other mag-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3034 nitude; hence he suggests that a reasonable interpretation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3035 the statement would be to suppose that, if we conceive a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3036 sphere with radius equal to the distance between the centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3037 of the sun and the centre of the earth, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3038 (diam. of earth):(diam. of said sphere)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3039 = (diam. of said sphere):(diam. of sphere of fixed stars).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3040 This is, of course, an arbitrary interpretation; Aristarchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3041 presumably meant no such thing, but merely that the size of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3042 the earth is negligible in comparison with that of the sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3043 of the fixed stars. However, the solution of Archimedes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3044 problem demands some assumption of the kind, and, in making
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3045 this assumption, he was no doubt aware that he was taking
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3046 a liberty with Aristarchus for the sake of giving his hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3047 thesis an air of authority.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3048 <pb n=83><head>THE <I>SAND-RECKONER</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3049 <p>Archimedes has, lastly, to compare the diameter of the sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3050 with the circumference of the circle described by its centre.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3051 Aristarchus had made the apparent diameter of the sun 1/(720)th
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3052 of the said circumference; Archimedes will prove that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3053 said circumference cannot contain as many as 1,000 sun's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3054 diameters, or that the diameter of the sun is greater than the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3055 side of a regular chiliagon inscribed in the circle. First he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3056 made an experiment of his own to determine the apparent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3057 diameter of the sun. With a small cylinder or disc in a plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3058 at right angles to a long straight stick and moveable along it,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3059 he observed the sun at the moment when it cleared the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3060 horizon in rising, moving the disc till it just covered and just
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3061 failed to cover the sun as he looked along the straight stick.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3062 He thus found the angular diameter to lie between 1/(164)<I>R</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3063 1/(200)<I>R</I>, where <I>R</I> is a right angle. But as, under his assump-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3064 tions, the size of the earth is not negligible in comparison with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3065 the sun's circle, he had to allow for parallax and find limits
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3066 for the angle subtended by the sun at the centre of the earth.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3067 This he does by a geometrical argument very much in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3068 manner of Aristarchus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3069 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3070 <p>Let the circles with centres <I>O, C</I> represent sections of the sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3071 and earth respectively, <I>E</I> the position of the observer observing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3072 <pb n=84><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3073 the sun when it has just cleared the horizon. Draw from <I>E</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3074 two tangents <I>EP, EQ</I> to the circle with centre <I>O</I>, and from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3075 <I>C</I> let <I>CF, CG</I> be drawn touching the same circle. With centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3076 <I>C</I> and radius <I>CO</I> describe a circle: this will represent the path
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3077 of the centre of the sun round the earth. Let this circle meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3078 the tangents from <I>C</I> in <I>A, B</I>, and join <I>AB</I> meeting <I>CO</I> in <I>M</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3079 <p>Archimedes's observation has shown that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3080 <MATH>(1/164)<I>R</I> > &angle;<I>PEQ</I> > 1/200 <I>R</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3081 and he proceeds to prove that <I>AB</I> is less than the side of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3082 regular polygon of 656 sides inscribed in the circle <I>AOB</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3083 but greater than the side of an inscribed regular polygon of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3084 1,000 sides, in other words, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3085 <MATH>(1/164)<I>R</I> > &angle;<I>FCG</I> > (1/250)<I>R</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3086 The first relation is obvious, for, since <I>CO</I> > <I>EO</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3087 <MATH>&angle;<I>PEQ</I> > &angle;<I>FCG</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3088 <p>Next, the perimeter of any polygon inscribed in the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3089 <I>AOB</I> is less than 44/7 <I>CO</I> (i.e. 22/7 times the diameter);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3090 Therefore <MATH><I>AB</I> < (1/656).(44/7) <I>CO</I> or (11/1148)<I>CO</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3091 and, <I>a fortiori</I>, <MATH><I>AB</I> < (1/100)<I>CO</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3092 <p>Now, the triangles <I>CAM, COF</I> being equal in all respects,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3093 <MATH><I>AM</I> = <I>OF</I>, so that <I>AB</I> = 2<I>OF</I> = (diameter of sun) > <I>CH</I> + <I>OK</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3094 since the diameter of the sun is greater than that of the earth;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3095 therefore <MATH><I>CH</I> + <I>OK</I> < (1/100)<I>CO</I>, and <I>HK</I> > (99/100)<I>CO</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3096 <p>And <MATH><I>CO</I> > <I>CF</I>, while <I>HK</I> < <I>EQ</I>, so that <I>EQ</I> > (99/100)<I>CF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3097 <p>We can now compare the angles <I>OCF, OEQ</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3098 for <MATH>(&angle;<I>OCF</I>)/(&angle;<I>OEQ</I>)[ > (tan <I>OCF</I>)/(tan <I>OEQ</I>)]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3099 > <I>EQ</I>/<I>CF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3100 > 99/100</MATH>, <I>a fortiori</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3101 <p>Doubling the angles, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3102 <MATH>&angle;<I>FCG</I> > (99/100).&angle;<I>PEQ</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3103 > (99/20000)<I>R</I>, since &angle;<I>PEQ</I> > (1/200)<I>R</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3104 > (1/203)<I>R</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3105 <pb n=85><head>THE <I>SAND-RECKONER</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3106 <p>Hence <I>AB</I> is greater than the side of a regular polygon of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3107 812 sides, and <I>a fortiori</I> greater than the side of a regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3108 polygon of 1,000 sides, inscribed in the circle <I>AOB</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3109 <p>The perimeter of the chiliagon, as of any regular polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3110 with more sides than six, inscribed in the circle <I>AOB</I> is greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3111 than 3 times the diameter of the sun's orbit, but is less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3112 1,000 times the diameter of the sun, and <I>a fortiori</I> less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3113 30,000 times the diameter of the earth;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3114 <MATH>therefore (diameter of sun's orbit) < 10,000 (diam. of earth)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3115 < 10,000,000,000 stades.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3116 <p>But <MATH>(diam. of earth):(diam. of sun's orbit)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3117 = (diam. of sun's orbit):(diam. of universe);</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3118 therefore the universe, or the sphere of the fixed stars, is less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3119 than 10,000<SUP>3</SUP> times the sphere in which the sun's orbit is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3120 great circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3121 <p>Archimedes takes a quantity of sand not greater than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3122 a poppy-seed and assumes that it contains not more than 10,000
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3123 grains; the diameter of a poppy-seed he takes to be not less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3124 than 1/(40)th of a finger-breadth; thus a sphere of diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3125 1 finger-breadth is not greater than 64,000 poppy-seeds and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3126 therefore contains not more than 640,000,000 grains of sand
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3127 (‘6 units of <I>second order</I> + 40,000,000 units of <I>first order</I>’)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3128 and <I>a fortiori</I> not more than 1,000,000,000 (‘10 units of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3129 <I>second order</I> of numbers’). Gradually increasing the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3130 of the sphere by multiplying it each time by 100 (making the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3131 sphere 1,000,000 times larger each time) and substituting for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3132 10,000 finger-breadths a stadium (< 10,000 finger-breadths),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3133 he finds the number of grains of sand in a sphere of diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3134 10,000,000,000 stadia to be less than ‘1,000 units of <I>seventh
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3135 order</I> of numbers’ or 10<SUP>51</SUP>, and the number in a sphere 10,000<SUP>3</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3136 times this size to be less than ‘10,000,000 units of the <I>eighth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3137 order</I> of numbers’ or 10<SUP>63</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3138 <C>The Quadrature of the Parabola.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3139 <p>In the preface, addressed to Dositheus after the death of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3140 Conon, Archimedes claims originality for the solution of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3141 problem of finding the area of a segment of a parabola cut off
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3142 by any chord, which he says he first discovered by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3143 mechanics and then confirmed by means of geometry, using
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3144 the lemma that, if there are two unequal areas (or magnitudes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3145 <pb n=86><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3146 generally), then however small the excess of the greater over
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3147 the lesser, it can by being continually added to itself be made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3148 to exceed the greater; in other words, he confirmed the solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3149 by the method of exhaustion. One solution by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3150 mechanics is, as we have seen, given in <I>The Method</I>; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3151 present treatise contains a solution by means of mechanics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3152 confirmed by the method of exhaustion (Props. 1-17), and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3153 then gives an entirely independent solution by means of pure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3154 geometry, also confirmed by exhaustion (Props. 18-24).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3155 <p>I. The mechanical solution depends upon two properties of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3156 the parabola proved in Props. 4, 5. If <I>Qq</I> be the base, and <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3157 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3158 the vertex, of a parabolic segment, <I>P</I> is the point of contact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3159 of the tangent parallel to <I>Qq</I>, the diameter <I>PV</I> through <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3160 bisects <I>Qq</I> in <I>V</I>, and, if <I>VP</I> produced meets the tangent at <I>Q</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3161 in <I>T</I>, then <I>TP</I> = <I>PV</I>. These properties, along with the funda-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3162 mental property that <I>QV</I><SUP>2</SUP> varies as <I>PV</I>, Archimedes uses to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3163 prove that, if <I>EO</I> be any parallel to <I>TV</I> meeting <I>QT, QP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3164 (produced, if necessary), the curve, and <I>Qq</I> in <I>E, F, R, O</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3165 respectively, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3166 <MATH><I>QV</I>:<I>VO</I> = <I>OF</I>:<I>FR</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3167 and <MATH><I>QO</I>:<I>Oq</I> = <I>ER</I>:<I>RO</I></MATH>. (Props. 4, 5.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3168 <p>Now suppose a parabolic segment <I>QR</I><SUB>1</SUB><I>q</I> so placed in relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3169 to a horizontal straight line <I>QA</I> through <I>Q</I> that the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3170 bisecting <I>Qq</I> is at right angles to <I>QA</I>, i.e. vertical, and let the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3171 tangent at <I>Q</I> meet the diameter <I>qO</I> through <I>q</I> in <I>E</I>. Produce
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3172 <I>QO</I> to <I>A</I>, making <I>OA</I> equal to <I>OQ</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3173 <p>Divide <I>Qq</I> into any number of equal parts at <I>O</I><SUB>1</SUB>, <I>O</I><SUB>2</SUB> ... <I>O<SUB>n</SUB></I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3174 and through these points draw parallels to <I>OE</I>, i.e. vertical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3175 lines meeting <I>OQ</I> in <I>H</I><SUB>1</SUB>, <I>H</I><SUB>2</SUB>, ..., <I>EQ</I> in <I>E</I><SUB>1</SUB>, <I>E</I><SUB>2</SUB>, ..., and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3176 <pb n=87><head>THE QUADRATURE OF THE PARABOLA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3177 curve in <I>R</I><SUB>1</SUB>, <I>R</I><SUB>2</SUB>, .... Join <I>QR</I><SUB>1</SUB>, and produce it to meet <I>OE</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3178 <I>F, QR</I><SUB>2</SUB> meeting <I>O</I><SUB>1</SUB><I>E</I><SUB>1</SUB> in <I>F</I><SUB>1</SUB>, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3179 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3180 <p>Now Archimedes has proved in a series of propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3181 (6-13) that, if a trapezium such as <I>O</I><SUB>1</SUB><I>E</I><SUB>1</SUB><I>E</I><SUB>2</SUB><I>O</I><SUB>2</SUB> is suspended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3182 from <I>H</I><SUB>1</SUB><I>H</I><SUB>2</SUB>, and an area <I>P</I> suspended at <I>A</I> balances <I>O</I><SUB>1</SUB><I>E</I><SUB>1</SUB><I>E</I><SUB>2</SUB><I>O</I><SUB>2</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3183 so suspended, it will take a greater area than <I>P</I> suspended at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3184 <I>A</I> to balance the same trapezium suspended from <I>H</I><SUB>2</SUB> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3185 a less area than <I>P</I> to balance the same trapezium suspended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3186 from <I>H</I><SUB>1</SUB>. A similar proposition holds with regard to a triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3187 such as <I>E<SUB>n</SUB>H<SUB>n</SUB>Q</I> suspended where it is and suspended at <I>Q</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3188 <I>H<SUB>n</SUB></I> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3189 <p>Suppose (Props. 14, 15) the triangle <I>QqE</I> suspended where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3190 it is from <I>OQ</I>, and suppose that the trapezium <I>EO</I><SUB>1</SUB>, suspended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3191 where it is, is balanced by an area <I>P</I><SUB>1</SUB> suspended at <I>A</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3192 trapezium <I>E</I><SUB>1</SUB><I>O</I><SUB>2</SUB>, suspended where it is, is balanced by <I>P</I><SUB>2</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3193 suspended at <I>A</I>, and so on, and finally the triangle <I>E<SUB>n</SUB>O<SUB>n</SUB>Q</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3194 suspended where it is, is balanced by <I>P</I><SUB><I>n</I> + 1</SUB> suspended at <I>A</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3195 then <MATH><I>P</I><SUB>1</SUB> + <I>P</I><SUB>2</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB></MATH> at <I>A</I> balances the whole triangle, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3196 <MATH><I>P</I><SUB>1</SUB> + <I>P</I><SUB>2</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB> = 1/3&utri;<I>EqQ</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3197 since the whole triangle may be regarded as suspended from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3198 the point on <I>OQ</I> vertically above its centre of gravity.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3199 <p>Now <MATH><I>AO</I>:<I>OH</I><SUB>1</SUB> = <I>QO</I>:<I>OH</I><SUB>1</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3200 = <I>Qq</I>:<I>qQ</I><SUB>1</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3201 = <I>E</I><SUB>1</SUB><I>O</I><SUB>1</SUB>:<I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB>, by Prop. 5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3202 = (trapezium <I>EO</I><SUB>1</SUB>):(trapezium <I>FO</I><SUB>1</SUB>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3203 <pb n=88><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3204 that is, it takes the trapezium <I>FO</I><SUB>1</SUB> suspended at <I>A</I> to balance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3205 the trapezium <I>EO</I><SUB>1</SUB> suspended at <I>H</I><SUB>1</SUB>. And <I>P</I><SUB>1</SUB> balances <I>EO</I><SUB>1</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3206 where it is.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3207 <p>Therefore <MATH>(<I>FO</I><SUB>1</SUB>) > <I>P</I><SUB>1</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3208 <p>Similarly <MATH>(<I>F</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) > <I>P</I><SUB>2</SUB></MATH>, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3209 <p>Again <MATH><I>AO</I>:<I>OH</I><SUB>1</SUB> = <I>E</I><SUB>1</SUB><I>O</I><SUB>1</SUB>:<I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3210 = (trapezium <I>E</I><SUB>1</SUB><I>O</I><SUB>2</SUB>):(trapezium <I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3211 that is, (<I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) at <I>A</I> will balance (<I>E</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) suspended at <I>H</I><SUB>1</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3212 while <I>P</I><SUB>2</SUB> at <I>A</I> balances (<I>E</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) suspended where it is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3213 whence <MATH><I>P</I><SUB>2</SUB> > <I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3214 <p>Therefore <MATH>(<I>F</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) > <I>P</I><SUB>2</SUB> > (<I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3215 (<I>F</I><SUB>2</SUB><I>O</I><SUB>3</SUB>) > <I>P</I><SUB>3</SUB> > <I>R</I><SUB>2</SUB><I>O</I><SUB>3</SUB></MATH>, and so on;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3216 and finally, <MATH>&utri;<I>E<SUB>n</SUB>O<SUB>n</SUB>Q</I> > <I>P</I><SUB><I>n</I> + 1</SUB> > &utri;<I>R<SUB>n</SUB>O<SUB>n</SUB>Q</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3217 <p>By addition,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3218 <MATH>(<I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) + (<I>R</I><SUB>2</SUB><I>O</I><SUB>3</SUB>) + ... + (&utri;<I>R<SUB>n</SUB>O<SUB>n</SUB>Q</I>) < <I>P</I><SUB>2</SUB> + <I>P</I><SUB>3</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3219 therefore, <I>a fortiori</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3220 <MATH>(<I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) + (<I>R</I><SUB>2</SUB><I>O</I><SUB>3</SUB>) + ... + &utri;<I>R<SUB>n</SUB>O<SUB>n</SUB>Q</I> < <I>P</I><SUB>1</SUB> + <I>P</I><SUB>2</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3221 < (<I>FO</I><SUB>1</SUB>) + (<I>F</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) + ... + &utri;<I>E<SUB>n</SUB>O<SUB>n</SUB>Q</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3222 <p>That is to say, we have an inscribed figure consisting of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3223 trapezia and a triangle which is less, and a circumscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3224 figure composed in the same way which is greater, than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3225 <MATH><I>P</I><SUB>1</SUB> + <I>P</I><SUB>2</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB>, i.e. 1/3&utri;<I>EqQ</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3226 <p>It is therefore inferred, and proved by the method of ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3227 haustion, that the segment itself is <I>equal</I> to 1/3&utri;<I>EqQ</I> (Prop. 16).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3228 <p>In order to enable the method to be applied, it has only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3229 to be proved that, by increasing the number of parts in <I>Qq</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3230 sufficiently, the difference between the circumscribed and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3231 inscribed figures can be made as small as we please. This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3232 can be seen thus. We have first to show that all the parts, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3233 <I>qF</I>, into which <I>qE</I> is divided are equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3234 <p>We have <MATH><I>E</I><SUB>1</SUB><I>O</I><SUB>1</SUB>:<I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB> = <I>QO</I>:<I>OH</I><SUB>1</SUB> = (<I>n</I> + 1):1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3235 or <MATH><I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB> = 1/(<I>n</I> + 1).<I>E</I><SUB>1</SUB><I>O</I><SUB>1</SUB></MATH>, whence also <MATH><I>O</I><SUB>2</SUB><I>S</I> = 1/(<I>n</I> + 1).<I>O</I><SUB>2</SUB><I>E</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3236 <pb n=89><head>THE QUADRATURE OF THE PARABOLA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3237 <p>And <MATH><I>E</I><SUB>2</SUB><I>O</I><SUB>2</SUB>:<I>O</I><SUB>2</SUB><I>R</I><SUB>2</SUB> = <I>QO</I>:<I>OH</I><SUB>2</SUB> = (<I>n</I> + 1):2</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3238 or <MATH><I>O</I><SUB>2</SUB><I>R</I><SUB>2</SUB> = 2/(<I>n</I> + 1).<I>O</I><SUB>2</SUB><I>E</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3239 <p>It follows that <I>O</I><SUB>2</SUB><I>S</I> = <I>SR</I><SUB>2</SUB>, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3240 <p>Consequently <I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB>, <I>O</I><SUB>2</SUB><I>R</I><SUB>2</SUB>, <I>O</I><SUB>3</SUB><I>R</I><SUB>3</SUB> ... are divided into 1, 2, 3 ...
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3241 equal parts respectively by the lines from <I>Q</I> meeting <I>qE</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3242 <p>It follows that the difference between the circumscribed and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3243 inscribed figures is equal to the triangle <I>FqQ</I>, which can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3244 made as small as we please by increasing the number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3245 divisions in <I>Qq</I>, i.e. in <I>qE</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3246 <p>Since the area of the segment is equal to 1/3&utri;<I>EqQ</I>, and it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3247 easily proved (Prop. 17) that &utri;<I>EqQ</I> = 4 (triangle with same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3248 base and equal height with segment), it follows that the area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3249 of the segment = 4/3 times the latter triangle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3250 <p>It is easy to see that this solution is essentially the same as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3251 that given in <I>The Method</I> (see pp. 29-30, above), only in a more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3252 orthodox form (geometrically speaking). For there Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3253 medes took the sum of all the <I>straight lines</I>, as <I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB>, <I>O</I><SUB>2</SUB><I>R</I><SUB>2</SUB> ...,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3254 as making up the segment notwithstanding that there are an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3255 infinite number of them and straight lines have no breadth.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3256 Here he takes inscribed and circumscribed trapezia propor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3257 tional to the straight lines and having finite breadth, and then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3258 compresses the figures together into the segment itself by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3259 increasing indefinitely the number of trapezia in each figure,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3260 i.e. diminishing their breadth indefinitely.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3261 <p>The procedure is equivalent to an integration, thus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3262 <p>If <I>X</I> denote the area of the triangle <I>FqQ</I>, we have, if <I>n</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3263 the number of parts in <I>Qq</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3264 <MATH>(circumscribed figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3265 = sum of &utri;s <I>QqF, QR</I><SUB>1</SUB><I>F</I><SUB>1</SUB>, <I>QR</I><SUB>2</SUB><I>F</I><SUB>2</SUB>, ...
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3266 = sum of &utri;s <I>QqF, QO</I><SUB>1</SUB><I>R</I><SUB>1</SUB>, <I>QO</I><SUB>2</SUB><I>S</I>, ...
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3267 = <I>X</I> {1 + ((<I>n</I> - 1)<SUP>2</SUP>)/(<I>n</I><SUP>2</SUP>) + ((<I>n</I> - 2)<SUP>2</SUP>)/(<I>n</I><SUP>2</SUP>) + ... + 1/(<I>n</I><SUP>2</SUP>)}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3268 = 1/(<I>n</I><SUP>2</SUP><I>X</I><SUP>2</SUP>).<I>X</I> (<I>X</I><SUP>2</SUP> + 2<SUP>2</SUP><I>X</I><SUP>2</SUP> + 3<SUP>2</SUP><I>X</I><SUP>2</SUP> + ... + <I>n</I><SUP>2</SUP><I>X</I><SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3269 Similarly, we find that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3270 <MATH>(inscribed figure) = 1/(<I>n</I><SUP>2</SUP><I>X</I><SUP>2</SUP>).<I>X</I> {<I>X</I><SUP>2</SUP> + 2<SUP>2</SUP><I>X</I><SUP>2</SUP> + ... + (<I>n</I> - 1)<SUP>2</SUP><I>X</I><SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3271 <pb n=90><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3272 <p>Taking the limit, we have, if <I>A</I> denote the area of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3273 triangle <I>EqQ,</I> so that <MATH><I>A</I>=<I>nX,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3274 <MATH>area of segment=1/(<I>A</I><SUP>2</SUP>)∫<SUP>A</SUP><SUB>0</SUB><I>X</I><SUP>2</SUP><I>dX</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3275 =1/3<I>A.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3276 <p>II. The purely geometrical method simply <I>exhausts</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3277 parabolic segment by inscribing successive figures ‘in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3278 recognized manner’ (see p. 79, above). For this purpose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3279 it is necessary to find, in terms of the triangle with the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3280 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3281 base and height, the area added to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3282 inscribed figure by doubling the number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3283 sides other than the base of the segment.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3284 <p>Let <I>QPq</I> be the triangle inscribed ‘in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3285 recognized manner’, <I>P</I> being the point of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3286 contact of the tangent parallel to <I>Qq,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3287 <I>PV</I> the diameter bisecting <I>Qq.</I> If <I>QV, Vq</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3288 be bisected in <I>M, m,</I> and <I>RM, rm</I> be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3289 parallel to <I>PV</I> meeting the curve in <I>R, r,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3290 the latter points are vertices of the next
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3291 figure inscribed ‘in the recognized manner’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3292 for <I>RY, ry</I> are diameters bisecting <I>PQ, Pq</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3293 respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3294 <p>Now <MATH><I>QV</I><SUP>2</SUP>=4<I>RW</I><SUP>2</SUP></MATH>, so that <MATH><I>PV</I>=4<I>PW,</I></MATH> or <MATH><I>RM</I>=3<I>PW.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3295 <p>But <MATH><I>YM</I>=1/2<I>PV</I>=2<I>PW,</I></MATH> so that <MATH><I>YM</I>=2<I>RY.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3296 <p>Therefore <MATH>&utri;<I>PRQ</I>=1/2&utri;<I>PQM</I>=1/4&utri;<I>PQV.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3297 <p>Similarly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3298 <MATH>&utri;<I>Prq</I>=1/4&utri;<I>PVq</I></MATH>; whence <MATH>(&utri;<I>PRQ</I>+&utri;<I>Prq</I>)=1/4<I>PQq.</I></MATH> (Prop. 21.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3299 <p>In like manner it can be proved that the next addition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3300 to the inscribed figure adds 1/4 of the sum of &utri;s<I>PRQ, Prq,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3301 and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3302 <p><MATH>Therefore the area of the inscribed figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3303 ={1+1/4+(1/4)<SUP>2</SUP>+ ...}.&utri;<I>PQq.</I></MATH> (Prop. 22.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3304 <p>Further, each addition to the inscribed figure is greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3305 than half the segments of the parabola left over before the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3306 addition is made. For, if we draw the tangent at <I>P</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3307 complete the parallelogram <I>EQqe</I> with side <I>EQ</I> parallel to <I>PV,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3308 <pb n=91><head>THE QUADRATURE OF THE PARABOLA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3309 the triangle <I>PQq</I> is half of the parallelogram and therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3310 more than half the segment. And so on (Prop. 20).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3311 <p>We now have to sum <I>n</I> terms of the above geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3312 series. Archimedes enunciates the problem in the form, Given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3313 a series of areas <I>A, B, C, D</I> ... <I>Z,</I> of which <I>A</I> is the greatest, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3314 each is equal to four times the next in order, then (Prop. 23)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3315 <MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>Z</I>+1/3<I>Z</I>=4/3<I>A.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3316 <p>The algebraical equivalent of this is of course
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3317 <MATH>1+1/4+(1/4)<SUP>2</SUP>+ ... +(1/4)<SUP><I>n</I>-1</SUP>=4/3-1/3(1/4)<SUP><I>n</I>-1</SUP>=(1-(1/4)<SUP><I>n</I></SUP>)/(1-1/4)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3318 <p>To find the area of the segment, Archimedes, instead of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3319 taking the limit, as we should, uses the method of <I>reductio ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3320 absurdum.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3321 <p>Suppose <MATH><I>K</I>=4/3.&utri;<I>PQq.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3322 <p>(1) If possible, let the area of the segment be greater than <I>K.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3323 <p>We then inscribe a figure ‘in the recognized manner’ such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3324 that the segment exceeds it by an area less than the excess of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3325 the segment over <I>K.</I> Therefore the inscribed figure must be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3326 greater than <I>K,</I> which is impossible since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3327 <MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>Z</I><4/3<I>A,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3328 where <MATH><I>A</I>=&utri;<I>PQq</I></MATH> (Prop. 23).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3329 <p>(2) If possible, let the area of the segment be less than <I>K.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3330 <p>If then <MATH>&utri;<I>PQq</I>=<I>A, B</I>=1/4<I>A, C</I>=1/4<I>B,</I></MATH> and so on, until we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3331 arrive at an area <I>X</I> less than the excess of <I>K</I> over the area of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3332 the segment, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3333 <MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>X</I>+1/3<I>X</I>=4/3<I>A</I>=<I>K.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3334 <p>Thus <I>K</I> exceeds <MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>X</I></MATH> by an area less than <I>X,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3335 and exceeds the segment by an area greater than <I>X.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3336 <p>It follows that <MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>X</I> > (the segment)</MATH>; which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3337 is impossible (Prop. 22).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3338 <p>Therefore the area of the segment, being neither greater nor
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3339 less than <I>K,</I> is equal to <I>K</I> or 4/3&utri;<I>PQq.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3340 <C>On Floating Bodies, I, II.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3341 <p>In Book I of this treatise Archimedes lays down the funda-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3342 mental principles of the science of hydrostatics. These are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3343 <pb n=92><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3344 deduced from Postulates which are only two in number. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3345 first which begins Book I is this:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3346 <p>‘let it be assumed that a fluid is of such a nature that, of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3347 parts of it which lie evenly and are continuous, that which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3348 pressed the less is driven along by that which is pressed the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3349 more; and each of its parts is pressed by the fluid which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3350 perpendicularly above it except when the fluid is shut up in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3351 anything and pressed by something else’;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3352 <p>the second, placed after Prop. 7, says
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3353 <p>‘let it be assumed that, of bodies which are borne upwards in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3354 a fluid, each is borne upwards along the perpendicular drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3355 through its centre of gravity’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3356 <p>Prop. 1 is a preliminary proposition about a sphere, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3357 then Archimedes plunges <I>in medias res</I> with the theorem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3358 (Prop. 2) that ‘<I>the surface of any fluid at rest is a sphere the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3359 centre of which is the same as that of the earth</I>’, and in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3360 whole of Book I the surface of the fluid is always shown in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3361 the diagrams as spherical. The method of proof is similar to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3362 what we should expect in a modern elementary textbook, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3363 main propositions established being the following. A solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3364 which, size for size, is of equal weight with a fluid will, if let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3365 down into the fluid, sink till it is just covered but not lower
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3366 (Prop. 3); a solid lighter than a fluid will, if let down into it,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3367 be only partly immersed, in fact just so far that the weight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3368 of the solid is equal to the weight of the fluid displaced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3369 (Props. 4, 5), and, if it is forcibly immersed, it will be driven
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3370 upwards by a force equal to the difference between its weight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3371 and the weight of the fluid displaced (Prop. 6).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3372 <p>The important proposition follows (Prop. 7) that a solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3373 heavier than a fluid will, if placed in it, sink to the bottom of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3374 the fluid, and the solid will, when weighed in the fluid, be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3375 lighter than its true weight by the weight of the fluid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3376 displaced.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3377 <C><I>The problem of the Crown.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3378 <p>This proposition gives a method of solving the famous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3379 problem the discovery of which in his bath sent Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3380 home naked crying <G>en(/rhka, en(/rhka</G>, namely the problem of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3381 <pb n=93><head>ON FLOATING BODIES, I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3382 determining the proportions of gold and silver in a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3383 crown.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3384 <p>Let <I>W</I> be the weight of the crown, <I>w</I><SUB>1</SUB> and <I>w</I><SUB>2</SUB> the weights of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3385 the gold and silver in it respectively, so that <MATH><I>W</I>=<I>w</I><SUB>1</SUB>+<I>w</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3386 <p>(1) Take a weight <I>W</I> of pure gold and weigh it in the fluid.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3387 The apparent loss of weight is then equal to the weight of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3388 fluid displaced; this is ascertained by weighing. Let it be <I>F</I><SUB>1</SUB>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3389 <p>It follows that the weight of the fluid displaced by a weight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3390 <I>w</I><SUB>1</SUB> of gold is <MATH>(<I>w</I><SUB>1</SUB>/<I>W</I>).<I>F</I><SUB>1</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3391 <p>(2) Take a weight <I>W</I> of silver, and perform the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3392 operation. Let the weight of the fluid displaced be <I>F</I><SUB>2</SUB>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3393 Then the weight of the fluid displaced by a weight <I>w</I><SUB>2</SUB> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3394 silver is <MATH><I>w</I><SUB>2</SUB>/<I>W</I>.<I>F</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3395 <p>(3) Lastly weigh the crown itself in the fluid, and let <I>F</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3396 loss of weight or the weight of the fluid displaced.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3397 <p>We have then <MATH>(<I>w</I><SUB>1</SUB>/<I>W</I>).<I>F</I><SUB>1</SUB>+(<I>w</I><SUB>2</SUB>/<I>W</I>).<I>F</I><SUB>2</SUB>=<I>F,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3398 that is, <MATH><I>w</I><SUB>1</SUB><I>F</I><SUB>1</SUB>+<I>w</I><SUB>2</SUB><I>F</I><SUB>2</SUB>=(<I>w</I><SUB>1</SUB>+<I>w</I><SUB>2</SUB>)<I>F,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3399 whence <MATH><I>w</I><SUB>1</SUB>/<I>w</I><SUB>2</SUB>=(<I>F</I><SUB>2</SUB>-<I>F</I>)/(<I>F</I>-<I>F</I><SUB>1</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3400 <p>According to the author of the poem <I>de ponderibus et men-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3401 suris</I> (written probably about A.D. 500) Archimedes actually
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3402 used a method of this kind. We first take, says our authority,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3403 two equal weights of gold and silver respectively and weigh
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3404 them against each other when both are immersed in water;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3405 this gives the relation between their weights in water, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3406 therefore between their losses of weight in water. Next we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3407 take the mixture of gold and silver and an equal weight of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3408 silver, and weigh them against each other in water in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3409 same way.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3410 <p>Nevertheless I do not think it probable that this was the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3411 way in which the solution of the problem was <I>discovered.</I> As
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3412 we are told that Archimedes discovered it in his bath, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3413 that he noticed that, if the bath was full when he entered it,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3414 so much water overflowed as was displaced by his body, he is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3415 more likely to have discovered the solution by the alternative
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3416 <pb n=94><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3417 method attributed to him by Vitruvius,<note><I>De architectura,</I> ix. 3.</note> namely by measuring
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3418 successively the <I>volumes</I> of fluid displaced by three equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3419 weights, (1) the crown, (2) an equal weight of gold, (3) an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3420 equal weight of silver respectively. Suppose, as before, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3421 the weight of the crown is <I>W</I> and that it contains weights
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3422 <I>w</I><SUB>1</SUB> and <I>w</I><SUB>2</SUB> of gold and silver respectively. Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3423 <p>(1) the crown displaces a certain volume of the fluid, <I>V,</I> say;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3424 <p>(2) the weight <I>W</I> of gold displaces a volume <I>V</I><SUB>1</SUB>, say, of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3425 fluid;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3426 <p>therefore a weight <I>w</I><SUB>1</SUB> of gold displaces a volume <MATH>(<I>w</I><SUB>1</SUB>/<I>W</I>).<I>V</I><SUB>1</SUB></MATH> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3427 the fluid;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3428 <p>(3) the weight <I>W</I> of silver displaces <I>V</I><SUB>2</SUB>, say, of the fluid;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3429 therefore a weight <I>w</I><SUB>2</SUB> of silver displaces <MATH>(<I>w</I><SUB>2</SUB>/<I>W</I>).<I>V</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3430 <p>It follows that <MATH><I>V</I>=(<I>w</I><SUB>1</SUB>/<I>W</I>).<I>V</I><SUB>1</SUB>+(<I>w</I><SUB>2</SUB>/<I>W</I>).<I>V</I><SUB>2</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3431 whence we derive (since <MATH><I>W</I>=<I>w</I><SUB>1</SUB>+<I>w</I><SUB>2</SUB>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3432 <I>w</I><SUB>1</SUB>/<I>w</I><SUB>2</SUB>=(<I>V</I><SUB>2</SUB>-<I>V</I>)/(<I>V</I>-<I>V</I><SUB>1</SUB>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3433 the latter ratio being obviously equal to that obtained by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3434 other method.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3435 <p>The last propositions (8 and 9) of Book I deal with the case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3436 of any segment of a sphere lighter than a fluid and immersed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3437 in it in such a way that either (1) the curved surface is down-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3438 wards and the base is entirely outside the fluid, or (2) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3439 curved surface is upwards and the base is entirely submerged,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3440 and it is proved that in either case the segment is in stable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3441 equilibrium when the axis is vertical. This is expressed here
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3442 and in the corresponding propositions of Book II by saying
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3443 that, ‘if the figure be forced into such a position that the base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3444 of the segment touches the fluid (at one point), the figure will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3445 not remain inclined but will return to the upright position’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3446 <p>Book II, which investigates fully the conditions of stability
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3447 of a right segment of a paraboloid of revolution floating in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3448 a fluid for different values of the specific gravity and different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3449 ratios between the axis or height of the segment and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3450 <pb n=95><head>ON FLOATING BODIES, I, II</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3451 principal parameter of the generating parabola, is a veritable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3452 <I>tour de force</I> which must be read in full to be appreciated.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3453 Prop. 1 is preliminary, to the effect that, if a solid lighter than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3454 a fluid be at rest in it, the weight of the solid will be to that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3455 of the same volume of the fluid as the immersed portion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3456 the solid is to the whole. The results of the propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3457 about the segment of a paraboloid may be thus summarized.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3458 Let <I>h</I> be the axis or height of the segment, <I>p</I> the principal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3459 parameter of the generating parabola, <I>s</I> the ratio of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3460 specific gravity of the solid to that of the fluid (<I>s</I> always<1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3461 The segment is supposed to be always placed so that its base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3462 is either entirely above, or entirely below, the surface of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3463 fluid, and what Archimedes proves in each case is that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3464 the segment is so placed with its axis inclined to the vertical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3465 at any angle, it will not rest there but will return to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3466 position of stability.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3467 <p>I. If <I>h</I> is not greater than 3/4<I>p,</I> the position of stability is with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3468 the axis vertical, whether the curved surface is downwards or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3469 upwards (Props. 2, 3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3470 <p>II. If <I>h</I> is greater than 3/4<I>p,</I> then, in order that the position of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3471 stability may be with the axis vertical, <I>s</I> must be not less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3472 than <MATH>(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>/<I>h</I><SUP>2</SUP></MATH> if the curved surface is downwards, and not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3473 greater than <MATH>{<I>h</I><SUP>2</SUP>-(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>}/<I>h</I><SUP>2</SUP></MATH> if the curved surface is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3474 upwards (Props. 4, 5).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3475 <p>III. If <MATH><I>h</I>>3/4<I>p,</I></MATH> but <MATH><I>h</I>/(1/2<I>p</I>)<15/4</MATH>, the segment, if placed with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3476 one point of the base touching the surface, will never remain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3477 there whether the curved surface be downwards or upwards
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3478 (Props. 6, 7). (The segment will move in the direction of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3479 bringing the axis nearer to the vertical position.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3480 <p>IV. If <MATH><I>h</I>>3/4<I>p,</I></MATH> but <MATH><I>h</I>/(1/2<I>p</I>)<15/4</MATH>, and if <I>s</I> is less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3481 <MATH>(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>/<I>h</I><SUP>2</SUP></MATH> in the case where the curved surface is down-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3482 wards, but greater than <MATH>{<I>h</I><SUP>2</SUP>-(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>}/<I>h</I><SUP>2</SUP></MATH> in the case where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3483 the curved surface is upwards, then the position of stability is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3484 one in which the axis is not vertical but inclined to the surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3485 of the fluid at a certain angle (Props. 8, 9). (The angle is drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3486 in an auxiliary figure. The construction for it in Prop. 8 is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3487 equivalent to the solution of the following equation in <G>q</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3488 <MATH>1/4<I>p</I>cot<SUP>2</SUP><G>q</G>=2/3(<I>h</I>-<I>k</I>)-1/2<I>p,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3489 <pb n=96><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3490 where <I>k</I> is the axis of the segment of the paraboloid cut off by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3491 the surface of the fluid.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3492 <p>V. Prop. 10 investigates the positions of stability in the cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3493 where <MATH><I>h</I>/(1/2<I>p</I>)>15/4</MATH>, the base is entirely above the surface, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3494 <I>s</I> has values lying between five pairs of ratios respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3495 Only in the case where <I>s</I> is not less than <MATH>(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>/<I>h</I><SUP>2</SUP></MATH> is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3496 position of stability that in which the axis is vertical.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3497 <p><I>BAB</I><SUB>1</SUB> is a section of the paraboloid through the axis <I>AM.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3498 <I>C</I> is a point on <I>AM</I> such that <MATH><I>AC</I>=2<I>CM, K</I></MATH> is a point on <I>CA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3499 such that <MATH><I>AM</I>:<I>CK</I>=15:4</MATH>. <I>CO</I> is measured along <I>CA</I> such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3500 that <MATH><I>CO</I>=1/2<I>p</I></MATH>, and <I>R</I> is a point on <I>AM</I> such that <MATH><I>MR</I>=3/2<I>CO.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3501 <I>A</I><SUB>2</SUB> is the point in which the perpendicular to <I>AM</I> from <I>K</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3502 meets <I>AB,</I> and <I>A</I><SUB>3</SUB> is the middle point of <I>AB.</I> <I>BA</I><SUB>2</SUB><I>B</I><SUB>2</SUB>, <I>BA</I><SUB>3</SUB><I>M</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3503 are parabolic segments on <I>A</I><SUB>2</SUB><I>M</I><SUB>2</SUB>, <I>A</I><SUB>3</SUB><I>M</I><SUB>3</SUB> (parallel to <I>AM</I>) as axes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3504 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3505 and similar to the original segment. (The parabola <I>BA</I><SUB>2</SUB><I>B</I><SUB>2</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3506 is proved to pass through <I>C</I> by using the above relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3507 <MATH><I>AM</I>:<I>CK</I>=15:4</MATH> and applying Prop. 4 of the <I>Quadrature of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3508 the Parabola.</I>) The perpendicular to <I>AM</I> from <I>O</I> meets the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3509 parabola <I>BA</I><SUB>2</SUB><I>B</I><SUB>2</SUB> in two points <I>P</I><SUB>2</SUB>, <I>Q</I><SUB>2</SUB>, and straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3510 through these points parallel to <I>AM</I> meet the other para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3511 bolas in <I>P</I><SUB>1</SUB>, <I>Q</I><SUB>1</SUB> and <I>P</I><SUB>3</SUB>, <I>Q</I><SUB>3</SUB> respectively. <I>P</I><SUB>1</SUB><I>T</I> and <I>Q</I><SUB>1</SUB><I>U</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3512 tangents to the original parabola meeting the axis <I>MA</I> pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3513 duced in <I>T, U.</I> Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3514 <p>(i) if <I>s</I> is not less than <MATH><I>AR</I><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH> or <MATH>(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>:<I>h</I><SUP>2</SUP></MATH>, there is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3515 stable equilibrium when <I>AM</I> is vertical;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3516 <pb n=97><head>THE CATTLE-PROBLEM</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3517 <p>(ii) if <MATH><I>s</I><<I>AR</I><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH> but <MATH>><I>Q</I><SUB>1</SUB><I>Q</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, the solid will not rest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3518 with its base touching the surface of the fluid in one point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3519 only, but in a position with the base entirely out of the fluid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3520 and the axis making with the surface an angle greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3521 than <I>U</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3522 <p>(iiia) if <MATH><I>s</I>=<I>Q</I><SUB>1</SUB><I>Q</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, there is stable equilibrium with one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3523 point of the base touching the surface and <I>AM</I> inclined to it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3524 at an angle equal to <I>U</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3525 <p>(iiib) if <MATH><I>s</I>=<I>P</I><SUB>1</SUB><I>P</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, there is stable equilibrium with one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3526 point of the base touching the surface and with <I>AM</I> inclined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3527 to it at an angle equal to <I>T</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3528 <p>(iv) if <MATH><I>s</I>><I>P</I><SUB>1</SUB><I>P</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH> but <MATH><<I>Q</I><SUB>1</SUB><I>Q</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, there will be stable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3529 equilibrium in a position in which the base is more submerged;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3530 <p>(v) if <MATH><I>s</I><<I>P</I><SUB>1</SUB><I>P</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, there will be stable equilibrium with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3531 the base entirely out of the fluid and with the axis <I>AM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3532 inclined to the surface at an angle less than <I>T.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3533 <p>It remains to mention the traditions regarding other in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3534 vestigations by Archimedes which have reached us in Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3535 or through the Arabic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3536 <C>(<G>a</G>) <I>The Cattle-Problem.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3537 <p>This is a difficult problem in indeterminate analysis. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3538 required to find the number of bulls and cows of each of four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3539 colours, or to find 8 unknown quantities. The first part of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3540 the problem connects the unknowns by seven simple equations;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3541 and the second part adds two more conditions to which the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3542 unknowns must be subject. If <I>W, w</I> be the numbers of white
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3543 bulls and cows respectively and (<I>X, x</I>), (<I>Y, y</I>), (<I>Z, z</I>) represent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3544 the numbers of the other three colours, we have first the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3545 following equations:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3546 <MATH>(I) <I>W</I>=(1/2+1/3)<I>X</I>+<I>Y,</I> (<G>a</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3547 <I>X</I>=(1/4+1/5)<I>Z</I>+<I>Y,</I> (<G>b</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3548 <I>Z</I>=(1/6+1/7)<I>W</I>+<I>Y</I>, (<G>g</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3549 (II) <I>w</I>=(1/3+1/4)(<I>X</I>+<I>x</I>), (<G>d</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3550 <I>x</I>=(1/4+1/5)(<I>Z</I>+<I>z</I>), (<G>e</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3551 <I>z</I>=(1/5+1/6)(<I>Y</I>+<I>y</I>), (<G>z</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3552 <I>y</I>=(1/6+1/7)(<I>W</I>+<I>w</I>). (<G>h</G>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3553 <pb n=98><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3554 <p>Secondly, it is required that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3555 <MATH><I>W</I>+<I>X</I>=a square, (<G>q</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3556 <I>Y</I>+<I>Z</I>=a triangular number. (<G>i</G>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3557 There is an ambiguity in the text which makes it just possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3558 that <MATH><I>W</I>+<I>X</I></MATH> need only be the product of two whole numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3559 instead of a square as in (<G>q</G>). Jul. Fr. Wurm solved the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3560 in the simpler form to which this change reduces it. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3561 complete problem is discussed and partly solved by Amthor.<note><I>Zeitschrift für Math. u. Physik</I> (Hist.-litt. Abt.) xxv. (1880), pp.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3562 156 sqq.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3563 <p>The general solution of the first seven equations is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3564 <MATH><I>W</I>=2 . 3 . 7 . 53 . 4657<I>n</I>=10366482<I>n,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3565 <I>X</I>=2 . 3<SUP>2</SUP> . 89 . 4657<I>n</I>=7460514<I>n,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3566 <I>Y</I>=3<SUP>4</SUP> . 11 . 4657<I>n</I>=4149387<I>n,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3567 <I>Z</I>=2<SUP>2</SUP> . 5 . 79 . 4657<I>n</I>=7358060<I>n,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3568 <I>w</I>=2<SUP>3</SUP> . 3 . 5 . 7 . 23 . 373<I>n</I>=7206360<I>n,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3569 <I>x</I>=2.3<SUP>2</SUP> . 17 . 15991<I>n</I>=4893246<I>n,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3570 <I>y</I>=3<SUP>2</SUP> . 13 . 46489<I>n</I>=5439213<I>n,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3571 <I>z</I>=2<SUP>2</SUP> . 3 . 5 . 7 . 11 . 761<I>n</I>=3515820<I>n.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3572 <p>It is not difficult to find such a value of <I>n</I> that <MATH><I>W</I>+<I>X</I>=a</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3573 square number; it is <MATH><I>n</I>=3 . 11 . 29 . 4657<G>x</G><SUP>2</SUP>=4456749<G>x</G><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3574 where <G>x</G> is any integer. We then have to make <MATH><I>Y</I>+<I>Z</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3575 a triangular number, i.e. a number of the form <MATH>1/2<I>q</I>(<I>q</I>+1)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3576 This reduces itself to the solution of the ‘Pellian’ equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3577 <MATH><I>t</I><SUP>2</SUP>-4729494<I>u</I><SUP>2</SUP>=1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3578 which leads to prodigious figures; one of the eight unknown
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3579 quantities alone would have more than 206,500 digits!
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3580 <C>(<G>b</G>) <I>On semi-regular polyhedra.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3581 <p>In addition, Archimedes investigated polyhedra of a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3582 type. This we learn from Pappus.<note>Pappus, v, pp. 352-8.</note> The polyhedra in question
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3583 are semi-regular, being contained by equilateral and equi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3584 <pb n=99><head>ON SEMI-REGULAR POLYHEDRA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3585 angular, but not similar, polygons; those discovered by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3586 Archimedes were 13 in number. If we for convenience
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3587 designate a polyhedron contained by <I>m</I> regular polygons
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3588 of <G>a</G> sides, <I>n</I> regular polygons of <G>b</G> sides, &c., by (<I>m</I><SUB><G>a</G></SUB>, <I>n</I><SUB><G>b</G></SUB> ...),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3589 the thirteen Archimedean polyhedra, which we will denote by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3590 <I>P</I><SUB>1</SUB>, <I>P</I><SUB>2</SUB> ... <I>P</I><SUB>13</SUB>, are as follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3591 <MATH>Figure with 8 faces: <I>P</I><SUB>1</SUB>&equalse3;(4<SUB>3</SUB>, 4<SUB>6</SUB>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3592 Figures with 14 faces: <I>P</I><SUB>2</SUB>&equalse3;(8<SUB>3</SUB>, 6<SUB>4</SUB>), <I>P</I><SUB>3</SUB>&equalse3;(6<SUB>4</SUB>, 8<SUB>6</SUB>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3593 <I>P</I><SUB>4</SUB>&equalse3;(8<SUB>3</SUB>, 6<SUB>8</SUB>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3594 Figures with 26 faces: <I>P</I><SUB>5</SUB>&equalse3;(8<SUB>3</SUB>, 18<SUB>4</SUB>), <I>P</I><SUB>6</SUB>&equalse3;(12<SUB>4</SUB>, 8<SUB>6</SUB>, 6<SUB>8</SUB>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3595 Figures with 32 faces: <I>P</I><SUB>7</SUB>&equalse3;(20<SUB>3</SUB>, 12<SUB>5</SUB>), <I>P</I><SUB>8</SUB>&equalse3;(12<SUB>5</SUB>, 20<SUB>6</SUB>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3596 <I>P</I><SUB>9</SUB>&equalse3;(20<SUB>3</SUB>, 12<SUB>10</SUB>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3597 Figure with 38 faces: <I>P</I><SUB>10</SUB>&equalse3;(32<SUB>3</SUB>, 6<SUB>4</SUB>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3598 Figures with 62 faces: <I>P</I><SUB>11</SUB>&equalse3;(20<SUB>3</SUB>, 30<SUB>4</SUB>, 12<SUB>5</SUB>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3599 <I>P</I><SUB>12</SUB>&equalse3;(30<SUB>4</SUB>, 20<SUB>6</SUB>, 12<SUB>10</SUB>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3600 Figure with 92 faces: <I>P</I><SUB>13</SUB>&equalse3;(80<SUB>3</SUB>, 12<SUB>5</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3601 <p>Kepler<note>Kepler, <I>Harmonice mundi</I> in <I>Opera</I> (1864), v, pp. 123-6.</note> showed how these figures can be obtained. A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3602 method of obtaining some of them is indicated in a fragment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3603 of a scholium to the Vatican MS. of Pappus. If a solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3604 angle of one of the regular solids be cut off symmetrically by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3605 a plane, i.e. in such a way that the plane cuts off the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3606 length from each of the edges meeting at the angle, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3607 section is a regular polygon which is a triangle, square or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3608 pentagon according as the solid angle is formed of three, four,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3609 or five plane angles. If certain equal portions be so cut off
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3610 from all the solid angles respectively, they will leave regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3611 polygons inscribed in the faces of the solid; this happens
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3612 (A) when the cutting planes bisect the sides of the faces and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3613 so leave in each face a polygon of the same kind, and (B) when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3614 the cutting planes cut off a smaller portion from each angle in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3615 such a way that a regular polygon is left in each face which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3616 has double the number of sides (as when we make, say, an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3617 octagon out of a square by cutting off the necessary portions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3618 <pb n=100><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3619 symmetrically, from the corners). We have seen that, accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3620 ing to Heron, two of the semi-regular solids had already been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3621 discovered by Plato, and this would doubtless be his method.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3622 The methods (A) and (B) applied to the five regular solids
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3623 give the following out of the 13 semi-regular solids. We
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3624 obtain (1) from the tetrahedron, <I>P</I><SUB>1</SUB> by cutting off angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3625 so as to leave hexagons in the faces; (2) from the cube, <I>P</I><SUB>2</SUB> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3626 leaving squares, and <I>P</I><SUB>4</SUB> by leaving octagons, in the faces;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3627 (3) from the octahedron, <I>P</I><SUB>2</SUB> by leaving triangles, and <I>P</I><SUB>3</SUB> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3628 leaving hexagons, in the faces; (4) from the icosahedron,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3629 <I>P</I><SUB>7</SUB> by leaving triangles, and <I>P</I><SUB>8</SUB> by leaving hexagons, in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3630 faces; (5) from the dodecahedron, <I>P</I><SUB>7</SUB> by leaving pentagons,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3631 and <I>P</I><SUB>9</SUB> by leaving decagons in the faces.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3632 <p>Of the remaining six, four are obtained by cutting off all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3633 the edges symmetrically and equally by planes parallel to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3634 edges, and then cutting off angles. Take first the cube.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3635 (1) Cut off from each four parallel edges portions which leave
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3636 an octagon as the section of the figure perpendicular to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3637 edges; then cut off equilateral triangles from the corners
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3638 (see Fig. 1); this gives <I>P</I><SUB>5</SUB> containing 8 equilateral triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3639 and 18 squares. (<I>P</I><SUB>5</SUB> is also obtained by bisecting all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3640 edges of <I>P</I><SUB>2</SUB> and cutting off corners.) (2) Cut off from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3641 edges of the cube a smaller portion so as to leave in each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3642 face a square such that the octagon described in it has its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3643 side equal to the breadth of the section in which each edge is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3644 cut; then cut off hexagons from each angle (see Fig. 2); this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3645 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3646 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3647 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3648 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3649 gives 6 octagons in the faces, 12 squares under the edges and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3650 8 hexagons at the corners; that is, we have <I>P</I><SUB>6</SUB>. An exactly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3651 <pb n=101><head>ON SEMI-REGULAR POLYHEDRA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3652 similar procedure with the icosahedron and dodecahedron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3653 produces <I>P</I><SUB>11</SUB> and <I>P</I><SUB>12</SUB> (see Figs. 3, 4 for the case of the icosa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3654 hedron).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3655 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3656 <CAP>FIG. 3.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3657 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3658 <CAP>FIG. 4.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3659 <p>The two remaining solids <I>P</I><SUB>10</SUB>, <I>P</I><SUB>13</SUB> cannot be so simply pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3660 duced. They are represented in Figs. 5, 6, which I have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3661 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3662 <CAP>FIG. 5.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3663 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3664 <CAP>FIG. 6.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3665 taken from Kepler. <I>P</I><SUB>10</SUB> is the <I>snub cube</I> in which each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3666 solid angle is formed by the angles of four equilateral triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3667 and one square; <I>P</I><SUB>13</SUB> is the <I>snub dodecahedron,</I> each solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3668 angle of which is formed by the angles of four equilateral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3669 triangles and <I>one regular pentagon.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3670 <p>We are indebted to Arabian tradition for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3671 <C>(<G>g</G>) <I>The Liber Assumptorum.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3672 <p>Of the theorems contained in this collection many are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3673 so elegant as to afford a presumption that they may really
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3674 be due to Archimedes. In three of them the figure appears
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3675 which was called <G>a)/rbhlos</G>, a shoemaker's knife, consisting of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3676 three semicircles with a common diameter as shown in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3677 annexed figure. If <I>N</I> be the point at which the diameters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3678 <pb n=102><head>ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3679 of the two smaller semicircles adjoin, and <I>NP</I> be drawn at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3680 right angles to <I>AB</I> meeting the external semicircle in <I>P,</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3681 area of the <G>a)/rbhlos</G> (included between the three semicircular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3682 arcs) is equal to the circle on <I>PN</I> as diameter (Prop. 4). In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3683 Prop. 5 it is shown that, if a circle be described in the space
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3684 between the arcs <I>AP, AN</I> and the straight line <I>PN</I> touching
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3685 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3686 all three, and if a circle be similarly described in the space
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3687 between the arcs <I>PB, NB</I> and the straight line <I>PN</I> touching
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3688 all three, the two circles are equal. If one circle be described
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3689 in the <G>a)/rbhlos</G> touching all three semicircles, Prop. 6 shows
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3690 that, if the ratio of <I>AN</I> to <I>NB</I> be given, we can find the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3691 relation between the diameter of the circle inscribed to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3692 <G>a)/rbhlos</G> and the straight line <I>AB</I>; the proof is for the parti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3693 cular case <MATH><I>AN</I>=3/2<I>BN</I></MATH>, and shows that the diameter of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3694 inscribed <MATH>circle=6/19<I>AB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3695 <p>Prop. 8 is of interest in connexion with the problem of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3696 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3697 trisecting any angle. If <I>AB</I> be any chord of a circle with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3698 centre <I>O,</I> and <I>BC</I> on <I>AB</I> produced be made equal to the radius,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3699 draw <I>CO</I> meeting the circle in <I>D, E</I>; then will the arc <I>BD</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3700 one-third of the arc <I>AE</I> (or <I>BF,</I> if <I>EF</I> be the chord through <I>E</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3701 parallel to <I>AB</I>). The problem is by this theorem reduced to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3702 a <G>neu=sis</G> (cf. vol. i, p. 241).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3703 <pb n=103><head>THE <I>LIBER ASSUMPTORUM</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3704 <p>Lastly, we may mention the elegant theorem about the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3705 area of the <I>Salinon</I> (presumably ‘salt-cellar’) in Prop. 14.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3706 <I>ACB</I> is a semicircle on <I>AB</I> as diameter, <I>AD, EB</I> are equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3707 lengths measured from <I>A</I> and <I>B</I> on <I>AB.</I> Semicircles are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3708 drawn with <I>AD, EB</I> as diameters on the side towards <I>C,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3709 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3710 a semicircle with <I>DE</I> as diameter is drawn on the other side of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3711 <I>AB. CF</I> is the perpendicular to <I>AB</I> through <I>O,</I> the centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3712 of the semicircles <I>ACB, DFE.</I> Then is the area bounded by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3713 all the semicircles (the <I>Salinon</I>) equal to the circle on <I>CF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3714 as diameter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3715 <p>The Arabians, through whom the Book of Lemmas has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3716 reached us, attributed to Archimedes other works (1) on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3717 Circle, (2) on the Heptagon in a Circle, (3) on Circles touch-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3718 ing one another, (4) on Parallel Lines, (5) on Triangles, (6) on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3719 the properties of right-angled triangles, (7) a book of Data,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3720 (8) De clepsydris: statements which we are not in a position
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3721 to check. But the author of a book on the finding of chords
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3722 in a circle<note>See <I>Bibliotheca mathematica,</I> xi<SUB>3</SUB>, pp. 11-78.</note> Ab&umacr;'l Raih&amacr;n Muh. al-B&imacr;r&umacr;n&imacr;, quotes some alterna-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3723 tive proofs as coming from the first of these works.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3724 <C>(<G>d</G>) <I>Formula for area of triangle.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3725 <p>More important, however, is the mention in this same work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3726 of Archimedes as the discoverer of two propositions hitherto
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3727 attributed to Heron, the first being the problem of finding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3728 the perpendiculars of a triangle when the sides are given, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3729 the second the famous formula for the area of a triangle in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3730 terms of the sides,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3731 <MATH>√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3732 <pb n=104><head>ERATOSTHENES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3733 <p>Long as the present chapter is, it is nevertheless the most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3734 appropriate place for ERATOSTHENES of Cyrene. It was to him
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3735 that Archimedes dedicated <I>The Method,</I> and the <I>Cattle-Problem</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3736 purports, by its heading, to have been sent through him to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3737 the mathematicians of Alexandria. It is evident from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3738 preface to <I>The Method</I> that Archimedes thought highly of his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3739 mathematical ability. He was, indeed, recognized by his con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3740 temporaries as a man of great distinction in all branches of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3741 knowledge, though in each subject he just fell short of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3742 highest place. On the latter ground he was called Beta, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3743 another nickname applied to him, <I>Pentathlos,</I> has the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3744 implication, representing as it does an all-round athlete who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3745 was not the first runner or wrestler but took the second prize
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3746 in these contests as well as in others. He was very little
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3747 younger than Archimedes; the date of his birth was probably
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3748 284 B.C. or thereabouts. He was a pupil of the philosopher
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3749 Ariston of Chios, the grammarian Lysanias of Cyrene, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3750 the poet Callimachus; he is said also to have been a pupil of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3751 Zeno the Stoic, and he may have come under the influence of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3752 Arcesilaus at Athens, where he spent a considerable time.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3753 Invited, when about 40 years of age, by Ptolemy Euergetes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3754 to be tutor to his son (Philopator), he became librarian at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3755 Alexandria; his obligation to Ptolemy he recognized by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3756 column which he erected with a graceful epigram inscribed on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3757 it. This is the epigram, with which we are already acquainted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3758 (vol. i, p. 260), relating to the solutions, discovered up to date,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3759 of the problem of the duplication of the cube, and commend-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3760 ing his own method by means of an appliance called <G>meso/labon</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3761 itself represented in bronze on the column.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3762 <p>Eratosthenes wrote a book with the title <G>*platwniko/s</G>, and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3763 whether it was a sort of commentary on the <I>Timaeus</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3764 Plato, or a dialogue in which the principal part was played by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3765 Plato, it evidently dealt with the fundamental notions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3766 mathematics in connexion with Plato's philosophy. It was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3767 naturally one of the important sources of Theon of Smyrna's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3768 work on the mathematical matters which it was necessary for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3769 the student of Plato to know; and Theon cites the work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3770 twice by name. It seems to have begun with the famous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3771 problem of Delos, telling the story quoted by Theon how the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3772 god required, as a means of stopping a plague, that the altar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3773 <pb n=105><head><I>PLATONICUS</I> AND <I>ON MEANS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3774 there, which was cubical in form, should be doubled in size.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3775 The book evidently contained a disquisition on <I>proportion</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3776 (<G>a)nalogi/a</G>); a quotation by Theon on this subject shows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3777 Eratosthenes incidentally dealt with the fundamental defini-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3778 tions of geometry and arithmetic. The principles of music
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3779 were discussed in the same work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3780 <p>We have already described Eratosthenes's solution of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3781 problem of Delos, and his contribution to the theory of arith-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3782 metic by means of his <I>sieve</I> (<G>ko/skinon</G>) for finding successive
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3783 prime numbers.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3784 <p>He wrote also an independent work <I>On means.</I> This was in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3785 two Books, and was important enough to be mentioned by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3786 Pappus along with works by Euclid, Aristaeus and Apol-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3787 lonius as forming part of the <I>Treasury of Analysis</I><note>Pappus, vii, p. 636. 24.</note>; this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3788 proves that it was a systematic geometrical treatise. Another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3789 passage of Pappus speaks of certain loci which Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3790 called ‘loci with reference to means’ (<G>to/poi pro\s meso/thtas</G>)<note><I>Ib.,</I> p. 662. 15 sq.</note>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3791 these were presumably discussed in the treatise in question.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3792 What kind of loci these were is quite uncertain; Pappus (if it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3793 is not an interpolator who speaks) merely says that these loci
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3794 ‘belong to the aforesaid classes of loci’, but as the classes are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3795 numerous (including ‘plane’, ‘solid’, ‘linear’, ‘loci on surfaces’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3796 &c.), we are none the wiser. Tannery conjectured that they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3797 were loci of points such that their distances from three fixed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3798 straight lines furnished a ‘médiété’, i.e. loci (straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3799 and conics) which we should represent in trilinear coordinates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3800 by such equations as <MATH>2<I>y</I>=<I>x</I>+<I>z, y</I><SUP>2</SUP>=<I>xz, y</I>(<I>x</I>+<I>z</I>)=2<I>xz,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3801 <I>x</I>(<I>x</I>-<I>y</I>)=<I>z</I>(<I>y</I>-<I>z</I>), <I>x</I>(<I>x</I>-<I>y</I>)=<I>y</I>(<I>y</I>-<I>z</I>)</MATH>, the first three equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3802 representing the arithmetic, geometric and harmonic means,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3803 while the last two represent the ‘subcontraries’ to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3804 harmonic and geometric means respectively. Zeuthen has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3805 a different conjecture.<note>Zeuthen, <I>Die Lehre von den Kegelschnitten im Altertum,</I> 1886, pp.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3806 320, 321.</note> He points out that, if <I>QQ</I>′ be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3807 polar of a given point <I>C</I> with reference to a conic, and <I>CPOP</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3808 be drawn through <I>C</I> meeting <I>QQ</I>′ in <I>O</I> and the conic in <I>P, P</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3809 then <I>CO</I> is the harmonic mean to <I>CP, CP</I>′; the locus of <I>O</I> for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3810 all transversals <I>CPP</I>′ is then the straight line <I>QQ</I>′. If <I>A, G</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3811 are points on <I>PP</I>′ such that <I>CA</I> is the arithmetic, and <I>CG</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3812 <pb n=106><head>ERATOSTHENES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3813 geometric mean between <I>CP, CP</I>′, the loci of <I>A, G</I> respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3814 are conics. Zeuthen therefore suggests that these loci and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3815 the corresponding loci of the points on <I>CPP</I>′ at a distance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3816 from <I>C</I> equal to the subcontraries of the geometric and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3817 harmonic means between <I>CP</I> and <I>CP</I>′ are the ‘loci with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3818 reference to means’ of Eratosthenes; the latter two loci are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3819 ‘linear’, i.e. higher curves than conics. Needless to say, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3820 have no confirmation of this conjecture.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3821 <C><I>Eratosthenes's measurement of the Earth.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3822 <p>But the most famous scientific achievement of Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3823 was his measurement of the earth. Archimedes mentions, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3824 we have seen, that some had tried to prove that the circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3825 ference of the earth is about 300,000 stades. This was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3826 evidently the measurement based on observations made at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3827 Lysimachia (on the Hellespont) and Syene. It was observed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3828 that, while both these places were on one meridian, the head
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3829 of Draco was in the zenith at Lysimachia, and Cancer in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3830 zenith at Syene; the arc of the meridian separating the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3831 in the heavens was taken to be 1/15th of the complete circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3832 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3833 The distance between the two towns
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3834 was estimated at 20,000 stades, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3835 accordingly the whole circumference of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3836 the earth was reckoned at 300,000
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3837 stades. Eratosthenes improved on this.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3838 He observed (1) that at Syene, at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3839 noon, at the summer solstice, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3840 sun cast no shadow from an upright
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3841 gnomon (this was confirmed by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3842 observation that a well dug at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3843 same place was entirely lighted up at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3844 the same time), while (2) at the same moment the gnomon fixed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3845 upright at Alexandria (taken to be on the same meridian with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3846 Syene) cast a shadow corresponding to an angle between the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3847 gnomon and the sun's rays of 1/50th of a complete circle or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3848 four right angles. The sun's rays are of course assumed to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3849 parallel at the two places represented by <I>S</I> and <I>A</I> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3850 annexed figure. If <G>a</G> be the angle made at <I>A</I> by the sun's rays
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3851 with the gnomon (<I>OA</I> produced), the angle <I>SOA</I> is also equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3852 <pb n=107><head>MEASUREMENT OF THE EARTH</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3853 <G>a</G>, or 1/50th of four right angles. Now the distance from <I>S</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3854 to <I>A</I> was known by measurement to be 5,000 stades; it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3855 followed that the circumference of the earth was 250,000
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3856 stades. This is the figure given by Cleomedes, but Theon of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3857 Smyrna and Strabo both give it as 252,000 stades. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3858 reason of the discrepancy is not known; it is possible that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3859 Eratosthenes corrected 250,000 to 252,000 for some reason,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3860 perhaps in order to get a figure divisible by 60 and, inci-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3861 dentally, a round number (700) of stades for one degree. If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3862 Pliny is right in saying that Eratosthenes made 40 stades
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3863 equal to the Egyptian <G>sxoi=nos</G>, then, taking the <G>sxoi=nos</G> at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3864 12,000 Royal cubits of 0.525 metres, we get 300 such cubits,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3865 or 157.5 metres, i.e. 516.73 feet, as the length of the stade.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3866 On this basis 252,000 stades works out to 24,662 miles, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3867 the diameter of the earth to about 7,850 miles, only 50 miles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3868 shorter than the true polar diameter, a surprisingly close
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3869 approximation, however much it owes to happy accidents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3870 in the calculation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3871 <p>We learn from Heron's <I>Dioptra</I> that the measurement of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3872 the earth by Eratosthenes was given in a separate work <I>On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3873 the Measurement of the Earth.</I> According to Galen<note>Galen, <I>Instit. Logica,</I> 12 (p. 26 Kalbfleisch).</note> this work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3874 dealt generally with astronomical or mathematical geography,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3875 treating of ‘the size of the equator, the distance of the tropic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3876 and polar circles, the extent of the polar zone, the size and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3877 distance of the sun and moon, total and partial eclipses of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3878 these heavenly bodies, changes in the length of the day
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3879 according to the different latitudes and seasons’. Several
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3880 details are preserved elsewhere of results obtained by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3881 Eratosthenes, which were doubtless contained in this work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3882 He is supposed to have estimated the distance between the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3883 tropic circles or twice the obliquity of the ecliptic at 11/83rds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3884 of a complete circle or 47° 42′ 39″; but from Ptolemy's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3885 language on this subject it is not clear that this estimate was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3886 not Ptolemy's own. What Ptolemy says is that he himself
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3887 found the distance between the tropic circles to lie always
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3888 between 47° 40′ and 47° 45′, ‘from which we obtain <I>about</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3889 (<G>sxedo/n</G>) the same ratio as that of Eratosthenes, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3890 Hipparchus also used. For the distance between the tropics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3891 becomes (or <I>is found to be,</I> <G>gi/netai</G>) very nearly 11 parts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3892 <pb n=108><head>ERATOSTHENES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3893 out of 83 contained in the whole meridian circle’.<note>Ptolemy, <I>Syntaxis,</I> i. 12, pp. 67. 22-68. 6.</note> The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3894 mean of Ptolemy's estimates, 47° 42′ 30″, is of course nearly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3895 11/83rds of 360°. It is consistent with Ptolemy's language
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3896 to suppose that Eratosthenes adhered to the value of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3897 obliquity of the ecliptic discovered before Euclid's time,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3898 namely 24°, and Hipparchus does, in his extant <I>Commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3899 on the Phaenomena of Aratus and Eudoxus,</I> say that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3900 summer tropic is ‘very nearly 24° north of the equator’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3901 <p>The <I>Doxographi</I> state that Eratosthenes estimated the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3902 distance of the moon from the earth at 780,000 stades and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3903 the distance of the sun from the earth at 804,000,000 stades
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3904 (the versions of Stobaeus and Joannes Lydus admit 4,080,000
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3905 as an alternative for the latter figure, but this obviously
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3906 cannot be right). Macrobius<note>Macrobius, <I>In Somn. Scip.</I> i. 20. 9.</note> says that Eratosthenes made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3907 the ‘measure’ of the sun to be 27 times that of the earth.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3908 It is not certain whether measure means ‘solid content’ or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3909 ‘diameter’ in this case; the other figures on record make the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3910 former more probable, in which case the diameter of the sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3911 would be three times that of the earth. Macrobius also tells
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3912 us that Eratosthenes's estimates of the distances of the sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3913 and moon were obtained by means of lunar eclipses.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3914 <p>Another observation by Eratosthenes, namely that at Syene
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3915 (which is under the summer tropic) and throughout a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3916 round it with a radius of 300 stades the upright gnomon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3917 throws no shadow at noon, was afterwards made use of by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3918 Posidonius in his calculation of the size of the sun. Assuming
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3919 that the circle in which the sun apparently moves round the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3920 earth is 10,000 times the size of a circular section of the earth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3921 through its centre, and combining with this hypothesis the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3922 datum just mentioned, Posidonius arrived at 3,000,000 stades
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3923 as the diameter of the sun.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3924 <p>Eratosthenes wrote a poem called <I>Hermes</I> containing a good
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3925 deal of descriptive astronomy; only fragments of this have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3926 survived. The work <I>Catasterismi</I> (literally ‘placings among
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3927 the stars’) which is extant can hardly be genuine in the form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3928 in which it has reached us; it goes back, however, to a genuine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3929 work by Eratosthenes which apparently bore the same name;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3930 alternatively it is alluded to as <G>*kata/logoi</G> or by the general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3931 <pb n=109><head>ASTRONOMY, ETC.</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3932 word <G>*)astronomi/a</G> (Suidas), which latter word is perhaps a mis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3933 take for <G>)astroqesi/a</G> corresponding to the title <G>)astroqesi/ai
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3934 zw\|di/wn</G> found in the manuscripts. The work as we have it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3935 contains the story, mythological and descriptive, of the con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3936 stellations, &c., under forty-four heads; there is little or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3937 nothing belonging to astronomy proper.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3938 <p>Eratosthenes is also famous as the first to attempt a scientific
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3939 chronology beginning from the siege of Troy; this was the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3940 subject of his <G>*xronografi/ai</G>, with which must be connected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3941 the separate <G>*)olumpioni=kai</G> in several books. Clement of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3942 Alexandria gives a short <I>résumé</I> of the main results of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3943 former work, and both works were largely used by Apollo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3944 dorus. Another lost work was on the Octaëteris (or eight-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3945 years' period), which is twice mentioned, by Geminus and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3946 Achilles; from the latter we learn that Eratosthenes re-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3947 garded the work on the same subject attributed to Eudoxus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3948 as not genuine. His <I>Geographica</I> in three books is mainly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3949 known to us through Suidas's criticism of it. It began with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3950 a history of geography down to his own time; Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3951 then proceeded to mathematical geography, the spherical form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3952 of the earth, the negligibility in comparison with this of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3953 unevennesses caused by mountains and valleys, the changes of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3954 features due to floods, earthquakes and the like. It would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3955 appear from Theon of Smyrna's allusions that Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3956 estimated the height of the highest mountain to be 10 stades
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3957 or about 1/8000th part of the diameter of the earth.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3958 <pb><C>XIV</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3959 <C>CONIC SECTIONS. APOLLONIUS OF PERGA</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3960 <C>A. HISTORY OF CONICS UP TO APOLLONIUS</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3961 <C>Discovery of the conic sections by Menaechmus.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3962 <p>WE have seen that Menaechmus solved the problem of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3963 two mean proportionals (and therefore the duplication of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3964 the cube) by means of conic sections, and that he is credited
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3965 with the discovery of the three curves; for the epigram of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3966 Eratosthenes speaks of ‘the <I>triads</I> of Menaechmus’, whereas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3967 of course only two conics, the parabola and the rectangular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3968 hyperbola, actually appear in Menaechmus's solutions. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3969 question arises, how did Menaechmus come to think of obtain-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3970 ing curves by cutting a cone? On this we have no informa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3971 tion whatever. Democritus had indeed spoken of a section of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3972 a cone parallel and very near to the base, which of course
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3973 would be a circle, since the cone would certainly be the right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3974 circular cone. But it is probable enough that the attention
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3975 of the Greeks, whose observation nothing escaped, would be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3976 attracted to the shape of a section of a cone or a cylinder by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3977 a plane obliquely inclined to the axis when it occurred, as it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3978 often would, in real life; the case where the solid was cut
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3979 right through, which would show an ellipse, would presum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3980 ably be noticed first, and some attempt would be made to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3981 investigate the nature and geometrical measure of the elonga-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3982 tion of the figure in relation to the circular sections of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3983 same solid; these would in the first instance be most easily
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3984 ascertained when the solid was a right cylinder; it would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3985 then be a natural question to investigate whether the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3986 arrived at by cutting the cone had the same property as that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3987 obtained by cutting the cylinder. As we have seen, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3988 <pb n=111><head>DISCOVERY OF THE CONIC SECTIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3989 observation that an ellipse can be obtained from a cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3990 as well as a cone is actually made by Euclid in his <I>Phaeno-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3991 mena</I>: ‘if’, says Euclid, ‘a cone or a cylinder be cut by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3992 a plane not parallel to the base, the resulting section is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3993 section of an acute-angled cone which is similar to a <G>qureo/s</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3994 (shield).’ After this would doubtless follow the question
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3995 what sort of curves they are which are produced if we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3996 cut a cone by a plane which does not cut through the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3997 cone completely, but is either parallel or not parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3998 a generator of the cone, whether these curves have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	3999 same property with the ellipse and with one another, and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4000 if not, what exactly are their fundamental properties respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4001 tively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4002 <p>As it is, however, we are only told how the first writers on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4003 conics obtained them in actual practice. We learn on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4004 authority of Geminus<note>Eutocius, <I>Comm. on Conics</I> of Apollonius.</note> that the ancients defined a cone as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4005 surface described by the revolution of a right-angled triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4006 about one of the sides containing the right angle, and that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4007 they knew no cones other than right cones. Of these they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4008 distinguished three kinds; according as the vertical angle of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4009 the cone was less than, equal to, or greater than a right angle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4010 they called the cone acute-angled, right-angled, or obtuse-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4011 angled, and from each of these kinds of cone they produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4012 one and only one of the three sections, the section being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4013 always made perpendicular to one of the generating lines of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4014 the cone; the curves were, on this basis, called ‘section of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4015 acute-angled cone’ (= an ellipse), ‘section of a right-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4016 cone’ (= a parabola), and ‘section of an obtuse-angled cone’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4017 (= a hyperbola) respectively. These names were still used
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4018 by Euclid and Archimedes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4019 <C><I>Menaechmus's probable procedure.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4020 <p>Menaechmus's constructions for his curves would presum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4021 ably be the simplest and the most direct that would show the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4022 desired properties, and for the parabola nothing could be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4023 simpler than a section of a right-angled cone by a plane at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4024 angles to one of its generators. Let <I>OBC</I> (Fig. 1) represent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4025 <pb n=112><head>CONIC SECTIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4026 a section through the axis <I>OL</I> of a right-angled cone, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4027 conceive a section through <I>AG</I> (perpendicular to <I>OA</I>) and at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4028 right angles to the plane of the paper.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4029 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4030 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4031 <p>If <I>P</I> is any point on the curve, and <I>PN</I> perpendicular to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4032 <I>AG</I>, let <I>BC</I> be drawn through <I>N</I> perpendicular to the axis of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4033 the cone. Then <I>P</I> is on the circular section of the cone about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4034 <I>BC</I> as diameter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4035 <p>Draw <I>AD</I> parallel to <I>BC</I>, and <I>DF, CG</I> parallel to <I>OL</I> meet-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4036 ing <I>AL</I> produced in <I>F, G.</I> Then <I>AD, AF</I> are both bisected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4037 by <I>OL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4038 <p>If now <MATH><I>PN</I>=<I>y</I>, <I>AN</I>=<I>x</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4039 <I>y</I><SUP>2</SUP>=<I>PN</I><SUP>2</SUP>=<I>BN.NC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4040 <p>But <I>B, A, C, G</I> are concyclic, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4041 <MATH><I>BN.NC</I>=<I>AN.NG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4042 =<I>AN.AF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4043 =<I>AN</I>.2<I>AL</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4044 <p>Therefore <MATH><I>y</I><SUP>2</SUP>=<I>AN</I>.2<I>AL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4045 =2<I>AL.x</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4046 and 2<I>AL</I> is the ‘parameter’ of the principal ordinates <I>y.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4047 <p>In the case of the hyperbola Menaechmus had to obtain the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4048 <pb n=113><head>MENAECHMUS'S PROCEDURE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4049 particular hyperbola which we call rectangular or equilateral,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4050 and also to obtain its property with reference to its asymp-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4051 totes, a considerable advance on what was necessary in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4052 case of the parabola. Two methods of obtaining the particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4053 hyperbola were possible, namely (1) to obtain the hyperbola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4054 arising from the section of any obtuse-angled cone by a plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4055 at right angles to a generator, and then to show how a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4056 rectangular hyperbola can be obtained as a particular case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4057 by finding the vertical angle which the cone must have to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4058 give a rectangular hyperbola when cut in the particular way,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4059 or (2) to obtain the rectangular hyperbola direct by cutting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4060 another kind of cone by a section not necessarily perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4061 dicular to a generator.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4062 <p>(1) Taking the first method, we draw (Fig. 2) a cone with its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4063 vertical angle <I>BOC</I> obtuse. Imagine a section perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4064 to the plane of the paper and passing through <I>AG</I> which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4065 perpendicular to <I>OB</I>. Let <I>GA</I> produced meet <I>CO</I> produced in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4066 <I>A</I>′, and complete the same construction as in the case of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4067 the parabola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4068 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4069 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4070 <p>In this case we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4071 <MATH><I>PN</I><SUP>2</SUP>=<I>BN.NC</I>=<I>AN.NG</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4072 <pb n=114><head>CONIC SECTIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4073 <p>But, by similar triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4074 <MATH><I>NG</I>:<I>AF</I>=<I>NC</I>:<I>AD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4075 =<I>A</I>′<I>N</I>:<I>AA</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4076 <p>Hence <MATH><I>PN</I><SUP>2</SUP>=<I>AN.A</I>′<I>N.AF</I>/<I>AA</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4077 =(2<I>AL</I>)/<I>AA</I>′.<I>AN.A</I>′<I>N</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4078 which is the property of the hyperbola, <I>AA</I>′ being what we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4079 call the transverse axis, and 2 <I>AL</I> the parameter of the principal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4080 ordinates.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4081 <p>Now, in order that the hyperbola may be rectangular, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4082 must have 2 <I>AL</I>:<I>AA</I>′ equal to 1. The problem therefore now
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4083 is: given a straight line <I>AA</I>′, and <I>AL</I> along <I>A</I>′<I>A</I> produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4084 equal to 1/2<I>AA</I>′, to find a cone such that <I>L</I> is on its axis and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4085 the section through <I>AL</I> perpendicular to the generator through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4086 <I>A</I> is a rectangular hyperbola with <I>A</I>′<I>A</I> as transverse axis. In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4087 other words, we have to find a point <I>O</I> on the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4088 through <I>A</I> perpendicular to <I>AA</I>′ such that <I>OL</I> bisects the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4089 angle which is the supplement of the angle <I>A</I>′<I>OA.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4090 <p>This is the case if <MATH><I>A</I>′<I>O</I>:<I>OA</I>=<I>A</I>′<I>L</I>:<I>LA</I>=3:1</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4091 therefore <I>O</I> is on the circle which is the locus of all points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4092 such that their distances from the two fixed points <I>A</I>′, <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4093 are in the ratio 3:1. This circle is the circle on <I>KL</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4094 diameter, where <MATH><I>A</I>′<I>K</I>:<I>KA</I>=<I>A</I>′<I>L</I>:<I>LA</I>=3:1</MATH>. Draw this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4095 circle, and <I>O</I> is then determined as the point in which <I>AO</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4096 drawn perpendicular to <I>AA</I>′ intersects the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4097 <p>It is to be observed, however, that this deduction of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4098 particular from a more general case is not usual in early
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4099 Greek mathematics; on the contrary, the particular usually
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4100 led to the more general. Notwithstanding, therefore, that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4101 orthodox method of producing conic sections is said to have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4102 been by cutting the generator of each cone perpendicularly,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4103 I am inclined to think that Menaechmus would get his rect-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4104 angular hyperbola directly, and in an easier way, by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4105 a different cone differently cut. Taking the right-angled cone,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4106 already used for obtaining a parabola, we have only to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4107 a section parallel to the axis (instead of perpendicular to a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4108 generator) to get a rectangular hyperbola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4109 <pb n=115><head>MENAECHMUS'S PROCEDURE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4110 <p>For, let the right-angled cone <I>HOK</I> (Fig. 3) be cut by a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4111 plane through <I>A</I>′<I>AN</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4112 to the axis <I>OM</I> and cutting the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4113 sides of the axial triangle <I>HOK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4114 in <I>A</I>′, <I>A, N</I> respectively. Let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4115 <I>P</I> be the point on the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4116 for which <I>PN</I> is the principal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4117 ordinate. Draw <I>OC</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4118 to <I>HK.</I> We have at once
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4119 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4120 <CAP>FIG. 3.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4121 <MATH><I>PN</I><SUP>2</SUP>=<I>HN.NK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4122 =<I>MK</I><SUP>2</SUP>-<I>MN</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4123 =<I>CN</I><SUP>2</SUP>-<I>CA</I><SUP>2</SUP></MATH>, since <MATH><I>MK</I>=<I>OM</I></MATH>, and <MATH><I>MN</I>=<I>OC</I>=<I>CA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4124 This is the property of the rectangular hyperbola having <I>A</I>′<I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4125 as axis. To obtain a particular rectangular hyperbola with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4126 axis of given length we have only to choose the cutting plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4127 so that the intercept <I>A</I>′<I>A</I> may have the given length.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4128 <p>But Menaechmus had to prove the asymptote-property of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4129 his rectangular hyperbola. As he can hardly be supposed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4130 have got as far as Apollonius in investigating the relations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4131 the hyperbola to its asymptotes, it is probably safe to assume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4132 that he obtained the particular property in the simplest way,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4133 i.e. directly from the property of the curve in relation to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4134 its axes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4135 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4136 <CAP>FIG. 4.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4137 <p>If (Fig. 4) <I>CR, CR</I>′ be the asymptotes (which are therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4138 <pb n=116><head>CONIC SECTIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4139 at right angles) and <I>A</I>′<I>A</I> the axis of a rectangular hyperbola,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4140 <I>P</I> any point on the curve, <I>PN</I> the principal ordinate, draw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4141 <I>PK, PK</I>′ perpendicular to the asymptotes respectively. Let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4142 <I>PN</I> produced meet the asymptotes in <I>R, R</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4143 <p>Now, by the axial property,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4144 <MATH><I>CA</I><SUP>2</SUP>=<I>CN</I><SUP>2</SUP>-<I>PN</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4145 =<I>RN</I><SUP>2</SUP>-<I>PN</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4146 =<I>RP.PR</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4147 =2<I>PK.PK</I>′</MATH>, since &angle;<I>PRK</I> is half a right angle;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4148 therefore <MATH><I>PK.PK</I>′=1/2<I>CA</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4149 <C>Works by Aristaeus and Euclid.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4150 <p>If Menaechmus was really the discoverer of the three conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4151 sections at a date which we must put at about 360 or 350 B.C.,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4152 the subject must have been developed very rapidly, for by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4153 end of the century there were two considerable works on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4154 conics in existence, works which, as we learn from Pappus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4155 were considered worthy of a place, alongside the <I>Conics</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4156 Apollonius, in the <I>Treasury of Analysis.</I> Euclid flourished
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4157 about 300 B.C., or perhaps 10 or 20 years earlier; but his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4158 <I>Conics</I> in four books was preceded by a work of Aristaeus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4159 which was still extant in the time of Pappus, who describes it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4160 as ‘five books of <I>Solid Loci</I> connected (or continuous, <G>sunexh=</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4161 with the conics’. Speaking of the relation of Euclid's <I>Conics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4162 in four books to this work, Pappus says (if the passage is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4163 genuine) that Euclid gave credit to Aristaeus for his dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4164 coveries in conics and did not attempt to anticipate him or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4165 wish to construct anew the same system. In particular,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4166 Euclid, when dealing with what Apollonius calls the three-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4167 and four-line locus, ‘wrote so much about the locus as was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4168 possible by means of the conics of Aristaeus, without claiming
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4169 completeness for his demonstrations’.<note>Pappus, vii, p. 678. 4.</note> We gather from these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4170 remarks that Euclid's <I>Conics</I> was a compilation and rearrange-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4171 ment of the geometry of the conics so far as known in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4172 <pb n=117><head>WORKS BY ARISTAEUS AND EUCLID</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4173 time, whereas the work of Aristaeus was more specialized and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4174 more original.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4175 <C>‘<I>Solid loci</I>’ <I>and</I> ‘<I>solid problems</I>’.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4176 <p>‘Solid loci’ are of course simply conics, but the use of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4177 title ‘Solid loci’ instead of ‘conics’ seems to indicate that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4178 the work was in the main devoted to conics regarded as loci.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4179 As we have seen, ‘solid loci’ which are conics are distinguished
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4180 from ‘plane loci’, on the one hand, which are straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4181 and circles, and from ‘linear loci’ on the other, which are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4182 curves higher than conics. There is some doubt as to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4183 real reason why the term ‘solid loci’ was applied to the conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4184 sections. We are told that ‘plane’ loci are so called because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4185 they are generated in a plane (but so are some of the higher
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4186 curves, such as the <I>quadratrix</I> and the spiral of Archimedes),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4187 and that ‘solid loci’ derived their name from the fact that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4188 they arise as sections of solid figures (but so do some higher
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4189 curves, e.g. the spiric curves which are sections of the <G>spei=ra</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4190 or <I>tore</I>). But some light is thrown on the subject by the corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4191 sponding distinction which Pappus draws between ‘plane’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4192 ‘solid’ and ‘linear’ <I>problems.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4193 <p>‘Those problems’, he says, ‘which can be solved by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4194 of a straight line and a circumference of a circle may pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4195 perly be called <I>plane</I>; for the lines by means of which such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4196 problems are solved have their origin in a plane. Those,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4197 however, which are solved by using for their discovery one or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4198 more of the sections of the cone have been called <I>solid</I>; for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4199 their construction requires the use of surfaces of solid figures,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4200 namely those of cones. There remains a third kind of pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4201 blem, that which is called <I>linear</I>; for other lines (curves)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4202 besides those mentioned are assumed for the construction, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4203 origin of which is more complicated and less natural, as they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4204 are generated from more irregular surfaces and intricate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4205 movements.’<note>Pappus, iv, p. 270. 5-17.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4206 <p>The true significance of the word ‘plane’ as applied to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4207 problems is evidently, not that straight lines and circles have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4208 their origin in a plane, but that the problems in question can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4209 be solved by the ordinary plane methods of transformation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4210 <pb n=118><head>CONIC SECTIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4211 areas, manipulation of simple equations between areas and, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4212 particular, the application of areas; in other words, plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4213 problems were those which, if expressed algebraically, depend
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4214 on equations of a degree not higher than the second.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4215 Problems, however, soon arose which did not yield to ‘plane’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4216 methods. One of the first was that of the duplication of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4217 cube, which was a problem of geometry in three dimensions or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4218 solid geometry. Consequently, when it was found that this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4219 problem could be solved by means of conics, and that no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4220 higher curves were necessary, it would be natural to speak of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4221 them as ‘solid’ loci, especially as they were in fact produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4222 from sections of a solid figure, the cone. The propriety of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4223 term would be only confirmed when it was found that, just as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4224 the duplication of the cube depended on the solution of a pure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4225 cubic equation, other problems such as the trisection of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4226 angle, or the cutting of a sphere into two segments bearing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4227 a given ratio to one another, led to an equation between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4228 volumes in one form or another, i. e. a mixed cubic equation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4229 and that this equation, which was also a solid problem, could
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4230 likewise be solved by means of conics.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4231 <C>Aristaeus's <I>Solid Loci.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4232 <p>The <I>Solid Loci</I> of Aristaeus, then, presumably dealt with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4233 loci which proved to be conic sections. In particular, he must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4234 have discussed, however imperfectly, the locus with respect to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4235 three or four lines the synthesis of which Apollonius says that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4236 he found inadequately worked out in Euclid's <I>Conics.</I> The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4237 theorems relating to this locus are enunciated by Pappus in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4238 this way:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4239 <p>‘If three straight lines be given in position and from one and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4240 the same point straight lines be drawn to meet the three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4241 straight lines at given angles, and if the ratio of the rectangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4242 contained by two of the straight lines so drawn to the square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4243 on the remaining one be given, then the point will lie on a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4244 solid locus given in position, that is, on one of the three conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4245 sections. And if straight lines be so drawn to meet, at given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4246 angles, four straight lines given in position, and the ratio of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4247 the rectangle contained by two of the lines so drawn to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4248 rectangle contained by the remaining two be given, then in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4249 <pb n=119><head>ARISTAEUS'S <I>SOLID LOCI</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4250 the same way the point will lie on a conic section given in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4251 position.’<note>Pappus, vii, p. 678. 15-24.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4252 <p>The reason why Apollonius referred in this connexion to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4253 Euclid and not to Aristaeus was probably that it was Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4254 work that was on the same lines as his own.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4255 <p>A very large proportion of the standard properties of conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4256 admit of being stated in the form of locus-theorems; if a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4257 certain property holds with regard to a certain point, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4258 that point lies on a conic section. But it may be assumed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4259 that Aristaeus's work was not merely a collection of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4260 ordinary propositions transformed in this way; it would deal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4261 with new locus-theorems not implied in the fundamental
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4262 definitions and properties of the conics, such as those just
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4263 mentioned, the theorems of the three- and four-line locus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4264 But one (to us) ordinary property, the focus-directrix property,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4265 was, as it seems to me, in all probability included.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4266 <C>Focus-directrix property known to Euclid.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4267 <p>It is remarkable that the directrix does not appear at all in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4268 Apollonius's great treatise on conics. The focal properties of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4269 the central conics are given by Apollonius, but the foci are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4270 obtained in a different way, without any reference to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4271 directrix; the focus of the parabola does not appear at all.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4272 We may perhaps conclude that neither did Euclid's <I>Conics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4273 contain the focus-directrix property; for, according to Pappus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4274 Apollonius based his first four books on Euclid's four books,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4275 while filling them out and adding to them. Yet Pappus gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4276 the proposition as a lemma to Euclid's <I>Surface-Loci</I>, from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4277 which we cannot but infer that it was assumed in that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4278 treatise without proof. If, then, Euclid did not take it from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4279 his own <I>Conics</I>, what more likely than that it was contained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4280 in Aristaeus's <I>Solid Loci</I>?
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4281 <p>Pappus's enunciation of the theorem is to the effect that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4282 locus of a point such that its distance from a given point is in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4283 a given ratio to its distance from a fixed straight line is a conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4284 section, and is an ellipse, a parabola, or a hyperbola according
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4285 as the given ratio is less than, equal to, or greater than unity.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4286 <pb n=120><head>CONIC SECTIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4287 <C><I>Proof from Pappus.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4288 <p>The proof in the case where the given ratio is different from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4289 unity is shortly as follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4290 <p>Let <I>S</I> be the fixed point, <I>SX</I> the perpendicular from <I>S</I> on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4291 the fixed line. Let <I>P</I> be any point on the locus and <I>PN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4292 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4293 perpendicular to <I>SX</I>, so that <I>SP</I> is to <I>NX</I> in the given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4294 ratio (<I>e</I>);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4295 thus <MATH><I>e</I><SUP>2</SUP>=(<I>PN</I><SUP>2</SUP>+<I>SN</I><SUP>2</SUP>):<I>NX</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4296 <p>Take <I>K</I> on <I>SX</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4297 <MATH><I>e</I><SUP>2</SUP>=<I>SN</I><SUP>2</SUP>:<I>NK</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4298 then, if <I>K</I>′ be another point on <I>SN</I>, produced if necessary,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4299 such that <MATH><I>NK</I>=<I>NK</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4300 <MATH><I>e</I><SUP>2</SUP>:1=(<I>PN</I><SUP>2</SUP>+<I>SN</I><SUP>2</SUP>):<I>NX</I><SUP>2</SUP>=<I>SN</I><SUP>2</SUP>:<I>NK</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4301 =<I>PN</I><SUP>2</SUP>:(<I>NX</I><SUP>2</SUP>-<I>NK</I><SUP>2</SUP>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4302 =<I>PN</I><SUP>2</SUP>:<I>XK.XK</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4303 <p>The positions of <I>N, K, K</I>′ change with the position of <I>P.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4304 If <I>A, A</I>′ be the points on which <I>N</I> falls when <I>K, K</I>′ coincide
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4305 with <I>X</I> respectively, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4306 <MATH><I>SA</I>:<I>AX</I>=<I>SN</I>:<I>NK</I>=<I>e</I>:1=<I>SN</I>:<I>NK</I>′=<I>SA</I>′:<I>A</I>′<I>X</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4307 <p>Therefore <MATH><I>SX</I>:<I>SA</I>=<I>SK</I>:<I>SN</I>=(1+<I>e</I>):<I>e</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4308 whence <MATH>(1+<I>e</I>):<I>e</I>=(<I>SX</I>-<I>SK</I>):(<I>SA</I>-<I>SN</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4309 =<I>XK</I>:<I>AN</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4310 <pb n=121><head>FOCUS-DIRECTRIX PROPERTY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4311 <p>Similarly it can be shown that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4312 <MATH>(1-<I>e</I>):<I>e</I>=<I>XK</I>′:<I>A</I>′<I>N</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4313 <p>By multiplication, <MATH><I>XK.XK</I>′:<I>AN.A</I>′<I>N</I>=(1-<I>e</I><SUP>2</SUP>):<I>e</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4314 and it follows from above, <I>ex aequali</I>, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4315 <MATH><I>PN</I><SUP>2</SUP>:<I>AN.A</I>′<I>N</I>=(1-<I>e</I><SUP>2</SUP>):1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4316 which is the property of a central conic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4317 <p>When <I>e</I><1, <I>A</I> and <I>A</I>′ lie on the same side of <I>X</I>, while
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4318 <I>N</I> lies on <I>AA</I>′, and the conic is an ellipse; when <I>e</I>>1, <I>A</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4319 <I>A</I>′ lie on opposite sides of <I>X</I>, while <I>N</I> lies on <I>A</I>′<I>A</I> produced,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4320 and the conic is a hyperbola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4321 <p>The case where <I>e</I>=1 and the curve is a parabola is easy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4322 and need not be reproduced here.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4323 <p>The treatise would doubtless contain other loci of types
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4324 similar to that which, as Pappus says, was used for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4325 trisection of an angle: I refer to the proposition already
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4326 quoted (vol. i, p. 243) that, if <I>A, B</I> are the base angles of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4327 a triangle with vertex <I>P</I>, and <MATH>&angle;<I>B</I>=2&angle;<I>A</I></MATH>, the locus of <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4328 is a hyperbola with eccentricity 2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4329 <C>Propositions included in Euclid's <I>Conics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4330 <p>That Euclid's <I>Conics</I> covered much of the same ground as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4331 the first three Books of Apollonius is clear from the language
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4332 of Apollonius himself. Confirmation is forthcoming in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4333 quotations by Archimedes of propositions (1) ‘proved in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4334 the elements of conics’, or (2) assumed without remark as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4335 already known. The former class include the fundamental
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4336 ordinate properties of the conics in the following forms:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4337 <p>(1) for the ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4338 <MATH><I>PN</I><SUP>2</SUP>:<I>AN.A</I>′<I>N</I>=<I>P</I>′<I>N</I>′<SUP>2</SUP>:<I>AN</I>′.<I>A</I>′<I>N</I>′=<I>BC</I><SUP>2</SUP>:<I>AC</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4339 <p>(2) for the hyperbola,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4340 <MATH><I>PN</I><SUP>2</SUP>:<I>AN.A</I>′<I>N</I>=<I>P</I>′<I>N</I>′<SUP>2</SUP>:<I>AN</I>′.<I>A</I>′<I>N</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4341 <p>(3) for the parabola, <MATH><I>PN</I><SUP>2</SUP>=<I>p<SUB>a</SUB>.AN</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4342 the principal tangent properties of the parabola;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4343 the property that, if there are two tangents drawn from one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4344 point to any conic section whatever, and two intersecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4345 <pb n=122><head>CONIC SECTIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4346 chords drawn parallel to the tangents respectively, the rect-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4347 angles contained by the segments of the chords respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4348 are to one another as the squares of the parallel tangents;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4349 the by no means easy proposition that, if in a parabola the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4350 diameter through <I>P</I> bisects the chord <I>QQ</I>′ in <I>V</I>, and <I>QD</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4351 drawn perpendicular to <I>PV</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4352 <MATH><I>QV</I><SUP>2</SUP>:<I>QD</I><SUP>2</SUP>=<I>p</I>:<I>p<SUB>a</SUB></I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4353 where <I>p<SUB>a</SUB></I> is the parameter of the principal ordinates and <I>p</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4354 the parameter of the ordinates to the diameter <I>PV.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4355 <C>Conic sections in Archimedes.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4356 <p>But we must equally regard Euclid's <I>Conics</I> as the source
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4357 from which Archimedes took most of the other ordinary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4358 properties of conics which he assumes without proof. Before
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4359 summarizing these it will be convenient to refer to Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4360 medes's terminology. We have seen that the axes of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4361 ellipse are not called axes but <I>diameters</I>, greater and lesser;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4362 the axis of a parabola is likewise its <I>diameter</I> and the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4363 diameters are ‘lines parallel to the diameter’, although in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4364 a segment of a parabola the diameter bisecting the base is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4365 the ‘diameter’ of the segment. The two ‘diameters’ (axes)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4366 of an ellipse are <I>conjugate.</I> In the case of the hyperbola the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4367 ‘diameter’ (axis) is the portion of it within the (single-branch)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4368 hyperbola; the centre is not called the ‘centre’, but the point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4369 in which the ‘nearest lines to the section of an obtuse-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4370 cone’ (the asymptotes) meet; the half of the axis (<I>CA</I>) is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4371 ‘the line adjacent to the axis’ (of the hyperboloid of revolution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4372 obtained by making the hyperbola revolve about its ‘diameter’),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4373 and <I>A</I>′<I>A</I> is double of this line. Similarly <I>CP</I> is the line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4374 ‘adjacent to the axis’ of a segment of the hyperboloid, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4375 <I>P</I>′<I>P</I> double of this line. It is clear that Archimedes did not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4376 yet treat the two branches of a hyperbola as forming one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4377 curve; this was reserved for Apollonius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4378 <p>The main properties of conics assumed by Archimedes in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4379 addition to those above mentioned may be summarized thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4380 <C><I>Central Conics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4381 <p>1. The property of the ordinates to any diameter <I>PP</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4382 <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I>=<I>Q</I>′<I>V</I>′<SUP>2</SUP>:<I>PV</I>′.<I>P</I>′<I>V</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4383 <pb n=123><head>CONIC SECTIONS IN ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4384 <p>In the case of the hyperbola Archimedes does not give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4385 any expression for the constant ratios <MATH><I>PN</I><SUP>2</SUP>:<I>AN.A</I>′<I>N</I></MATH> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4386 <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I></MATH> respectively, whence we conclude that he had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4387 no conception of diameters or radii of a hyperbola not meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4388 the curve.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4389 <p>2. The straight line drawn from the centre of an ellipse, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4390 the point of intersection of the asymptotes of a hyperbola,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4391 through the point of contact of any tangent, bisects all chords
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4392 parallel to the tangent.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4393 <p>3. In the ellipse the tangents at the extremities of either of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4394 conjugate diameters are both parallel to the other diameter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4395 <p>4. If in a hyperbola the tangent at <I>P</I> meets the transverse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4396 axis in <I>T</I>, and <I>PN</I> is the principal ordinate, <I>AN</I>><I>AT.</I> (It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4397 is not easy to see how this could be proved except by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4398 of the general property that, if <I>PP</I>′ be any diameter of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4399 a hyperbola, <I>QV</I> the ordinate to it from <I>Q</I>, and <I>QT</I> the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4400 at <I>Q</I> meeting <I>P</I>′<I>P</I> in <I>T</I>, then <MATH><I>TP</I>:<I>TP</I>′=<I>PV</I>:<I>P</I>′<I>V</I></MATH>.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4401 <p>5. If a cone, right or oblique, be cut by a plane meeting all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4402 the generators, the section is either a circle or an ellipse.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4403 <p>6. If a line between the asymptotes meets a hyperbola and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4404 is bisected at the point of concourse, it will touch the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4405 hyperbola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4406 <p>7. If <I>x, y</I> are straight lines drawn, in fixed directions respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4407 tively, from a point on a hyperbola to meet the asymptotes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4408 the rectangle <I>xy</I> is constant.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4409 <p>8. If <I>PN</I> be the principal ordinate of <I>P</I>, a point on an ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4410 and if <I>NP</I> be produced to meet the auxiliary circle in <I>p</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4411 ratio <I>pN</I>:<I>PN</I> is constant.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4412 <p>9. The criteria of similarity of conics and segments of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4413 conics are assumed in practically the same form as Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4414 gives them.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4415 <C><I>The Parabola.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4416 <p>1. The fundamental properties appear in the alternative forms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4417 <MATH><I>PN</I><SUP>2</SUP>:<I>P</I>′<I>N</I>′<SUP>2</SUP>=<I>AN</I>:<I>AN</I>′, or <I>PN</I><SUP>2</SUP>=<I>p<SUB>a</SUB>.AN</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4418 <I>QV</I><SUP>2</SUP>:<I>Q</I>′<I>V</I>′<SUP>2</SUP>=<I>PV</I>:<I>PV</I>′, or <I>QV</I><SUP>2</SUP>=<I>p.PV</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4419 <p>Archimedes applies the term <I>parameter</I> (<G>a( par) a(\v du/nantai
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4420 ai( a)po\ ta=s toma=s</G>) to the parameter of the principal ordinates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4421 <pb n=124><head>CONIC SECTIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4422 only: <I>p</I> is simply the line to which the rectangle equal to <I>QV</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4423 and of width equal to <I>PV</I> is applied.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4424 <p>2. Parallel chords are bisected by one straight line parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4425 the axis, which passes through the point of contact of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4426 tangent parallel to the chords.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4427 <p>3. If the tangent at <I>Q</I> meet the diameter <I>PV</I> in <I>T</I>, and <I>QV</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4428 the ordinate to the diameter, <MATH><I>PV</I>=<I>PT</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4429 <p>By the aid of this proposition a tangent to the parabola can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4430 be drawn (<I>a</I>) at a point on it, (<I>b</I>) parallel to a given chord.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4431 <p>4. Another proposition assumed is equivalent to the property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4432 of the subnormal, <MATH><I>NG</I>=1/2<I>p<SUB>a</SUB></I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4433 <p>5. If <I>QQ</I>′ be a chord of a parabola perpendicular to the axis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4434 and meeting the axis in <I>M</I>, while <I>QVq</I> another chord parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4435 to the tangent at <I>P</I> meets the diameter through <I>P</I> in <I>V</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4436 <I>RHK</I> is the principal ordinate of any point <I>R</I> on the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4437 meeting <I>PV</I> in <I>H</I> and the axis in <I>K</I>, then <I>PV</I>:<I>PH</I>> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4438 =<I>MK</I>:<I>KA</I>; ‘for this is proved’ (<I>On Floating Bodies</I>, II. 6).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4439 <p>Where it was proved we do not know; the proof is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4440 altogether easy.<note>See <I>Apollonius of Perga</I>, ed. Heath, p. liv.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4441 <p>6. All parabolas are similar.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4442 <p>As we have seen, Archimedes had to specialize in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4443 parabola for the purpose of his treatises on the <I>Quadrature
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4444 of the Parabola, Conoids and Spheroids, Floating Bodies</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4445 Book II, and <I>Plane Equilibriums</I>, Book II; consequently he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4446 had to prove for himself a number of special propositions, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4447 have already been given in their proper places. A few others
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4448 are assumed without proof, doubtless as being easy deductions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4449 from the propositions which he does prove. They refer mainly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4450 to similar parabolic segments so placed that their bases are in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4451 one straight line and have one common extremity.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4452 <p>1. If any three similar and similarly situated parabolic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4453 segments <I>BQ</I><SUB>1</SUB>, <I>BQ</I><SUB>2</SUB>, <I>BQ</I><SUB>3</SUB> lying along the same straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4454 as bases (<I>BQ</I><SUB>1</SUB><<I>BQ</I><SUB>2</SUB><<I>BQ</I><SUB>3</SUB>), and if <I>E</I> be any point on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4455 tangent at <I>B</I> to one of the segments, and <I>EO</I> a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4456 through <I>E</I> parallel to the axis of one of the segments and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4457 meeting the segments in <I>R</I><SUB>3</SUB>, <I>R</I><SUB>2</SUB>, <I>R</I><SUB>1</SUB> respectively and <I>BQ</I><SUB>3</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4458 in <I>O</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4459 <MATH><I>R</I><SUB>3</SUB><I>R</I><SUB>2</SUB>:<I>R</I><SUB>2</SUB><I>R</I><SUB>1</SUB>=(<I>Q</I><SUB>2</SUB><I>Q</I><SUB>3</SUB>:<I>BQ</I><SUB>3</SUB>).(<I>BQ</I><SUB>1</SUB>:<I>Q</I><SUB>1</SUB><I>Q</I><SUB>2</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4460 <pb n=125><head>CONIC SECTIONS IN ARCHIMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4461 <p>2. If two similar parabolic segments with bases <I>BQ</I><SUB>1</SUB>, <I>BQ</I><SUB>2</SUB> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4462 placed as in the last proposition, and if <I>BR</I><SUB>1</SUB><I>R</I><SUB>2</SUB> be any straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4463 line through <I>B</I> meeting the segments in <I>R</I><SUB>1</SUB>, <I>R</I><SUB>2</SUB> respectively,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4464 <MATH><I>BQ</I><SUB>1</SUB>:<I>BQ</I><SUB>2</SUB>=<I>BR</I><SUB>1</SUB>:<I>BR</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4465 <p>These propositions are easily deduced from the theorem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4466 proved in the <I>Quadrature of the Parabola</I>, that, if through <I>E</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4467 a point on the tangent at <I>B</I>, a straight line <I>ERO</I> be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4468 parallel to the axis and meeting the curve in <I>R</I> and any chord
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4469 <I>BQ</I> through <I>B</I> in <I>O</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4470 <MATH><I>ER</I>:<I>RO</I>=<I>BO</I>:<I>OQ</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4471 <p>3. On the strength of these propositions Archimedes assumes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4472 the solution of the problem of placing, between two parabolic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4473 segments similar to one another and placed as in the above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4474 propositions, a straight line of a given length and in a direction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4475 parallel to the diameters of either parabola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4476 <p>Euclid and Archimedes no doubt adhered to the old method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4477 of regarding the three conics as arising from sections of three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4478 kinds of right circular cones (right-angled, obtuse-angled and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4479 acute-angled) by planes drawn in each case at right angles to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4480 a generator of the cone. Yet neither Euclid nor Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4481 was unaware that the ‘section of an acute-angled cone’, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4482 ellipse, could be otherwise produced. Euclid actually says in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4483 his <I>Phaenomena</I> that ‘if a cone or cylinder (presumably right)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4484 be cut by a plane not parallel to the base, the resulting section
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4485 is a section of an acute-angled cone which is similar to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4486 a <G>qureo/s</G> (shield)’. Archimedes knew that the non-circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4487 sections even of an oblique circular cone made by planes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4488 cutting all the generators are ellipses; for he shows us how,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4489 given an ellipse, to draw a cone (in general oblique) of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4490 it is a section and which has its vertex outside the plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4491 of the ellipse on any straight line through the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4492 ellipse in a plane at right angles to the ellipse and passing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4493 through one of its axes, whether the straight line is itself
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4494 perpendicular or not perpendicular to the plane of the ellipse;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4495 drawing a cone in this case of course means finding the circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4496 sections of the surface generated by a straight line always
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4497 passing through the given vertex and all the several points of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4498 the given ellipse. The method of proof would equally serve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4499 <pb n=126><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4500 for the other two conics, the hyperbola and parabola, and we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4501 can scarcely avoid the inference that Archimedes was equally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4502 aware that the parabola and the hyperbola could be found
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4503 otherwise than by the old method.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4504 <p>The first, however, to base the theory of conics on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4505 production of all three in the most general way from any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4506 kind of circular cone, right or oblique, was Apollonius, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4507 whose work we now come.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4508 <C>B. APOLLONIUS OF PERGA</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4509 <p>Hardly anything is known of the life of Apollonius except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4510 that he was born at Perga, in Pamphylia, that he went
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4511 when quite young to Alexandria, where he studied with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4512 successors of Euclid and remained a long time, and that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4513 he flourished (<G>ge/gone</G>) in the reign of Ptolemy Euergetes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4514 (247-222 B.C). Ptolemaeus Chennus mentions an astronomer
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4515 of the same name, who was famous during the reign of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4516 Ptolemy Philopator (222-205 B.C.), and it is clear that our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4517 Apollonius is meant. As Apollonius dedicated the fourth and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4518 following Books of his <I>Conics</I> to King Attalus I (241-197 B.C.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4519 we have a confirmation of his approximate date. He was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4520 probably born about 262 B.C., or 25 years after Archimedes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4521 We hear of a visit to Pergamum, where he made the acquain-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4522 tance of Eudemus of Pergamum, to whom he dedicated the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4523 first two Books of the <I>Conics</I> in the form in which they have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4524 come down to us; they were the first two instalments of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4525 second edition of the work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4526 <C>The text of the <I>Conics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4527 <p>The <I>Conics</I> of Apollonius was at once recognized as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4528 authoritative treatise on the subject, and later writers regu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4529 larly cited it when quoting propositions in conics. Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4530 wrote a number of lemmas to it; Serenus wrote a commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4531 tary, as also, according to Suidas, did Hypatia. Eutocius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4532 (fl. A.D. 500) prepared an edition of the first four Books and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4533 wrote a commentary on them; it is evident that he had before
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4534 him slightly differing versions of the completed work, and he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4535 may also have had the first unrevised edition which had got
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4536 into premature circulation, as Apollonius himself complains in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4537 the Preface to Book I.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4538 <pb n=127>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4539 <head>THE TEXT OF THE <I>CONICS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4540 <p>The edition of Eutocius suffered interpolations which were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4541 probably made in the ninth century when, under the auspices
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4542 of Leon, mathematical studies were revived at Constantinople;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4543 for it was at that date that the uncial manuscripts were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4544 written, from which our best manuscripts, V (= Cod. Vat. gr.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4545 206 of the twelfth to thirteenth century) for the <I>Conics</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4546 W (= Cod. Vat. gr. 204 of the tenth century) for Eutocius,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4547 were copied.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4548 <p>Only the first four Books survive in Greek; the eighth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4549 Book is altogether lost, but the three Books V-VII exist in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4550 Arabic. It was A&hdot;mad and al-&Hdot;asa&ndot;, two sons of Mu&hdot;. b.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4551 M&umacr;s&amacr; b. Sh&amacr;kir, who first contemplated translating the <I>Conics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4552 into Arabic. They were at first deterred by the bad state of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4553 their manuscripts; but afterwards A&hdot;mad obtained in Syria
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4554 a copy of Eutocius's edition of Books I-IV and had them
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4555 translated by Hil&amacr;l b. Ab&imacr; Hil&amacr;l al-&Hdot;im⋅&imacr; (died 883/4).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4556 Books V-VII were translated, also for A&hdot;mad, by Th&amacr;bit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4557 b. Qurra ( 826-901) from another manuscript. Na⋅&imacr;radd&imacr;n's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4558 recension of this translation of the seven Books, made in 1248,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4559 is represented by two copies in the Bodleian, one of the year
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4560 1301 (No. 943) and the other of 1626 containing Books V-VII
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4561 only (No. 885).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4562 <p>A Latin translation of Books I-IV was published by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4563 Johannes Baptista Memus at Venice in 1537; but the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4564 important edition was the translation by Commandinus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4565 (Bologna, 1566), which included the lemmas of Pappus and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4566 the commentary of Eutocius, and was the first attempt to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4567 make the book intelligible by means of explanatory notes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4568 For the Greek text Commandinus used Cod. Marcianus 518
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4569 and perhaps also Vat. gr. 205, both of which were copies of V,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4570 but not V itself.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4571 <p>The first published version of Books V-VII was a Latin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4572 translation by Abraham Echellensis and Giacomo Alfonso
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4573 Borelli (Florence, 1661) of a reproduction of the Books written
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4574 in 983 by Ab&umacr; 'l Fat&hdot; al-I⋅fah&amacr;n&imacr;.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4575 <p>The <I>editio princeps</I> of the Greek text is the monumental
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4576 work of Halley (Oxford, 1710). The original intention was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4577 that Gregory should edit the four Books extant in Greek, with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4578 Eutocius's commentary and a Latin translation, and that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4579 Halley should translate Books V-VII from the Arabic into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4580 <pb n=128>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4581 <head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4582 Latin. Gregory, however, died while the work was proceeding,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4583 and Halley then undertook responsibility for the whole. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4584 Greek manuscripts used were two, one belonging to Savile
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4585 and the other lent by D. Baynard; their whereabouts cannot
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4586 apparently now be traced, but they were both copies of Paris.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4587 gr. 2356, which was copied in the sixteenth century from Paris.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4588 gr. 2357 of the sixteenth century, itself a copy of V. For the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4589 three Books in Arabic Halley used the Bodleian MS. 885, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4590 also consulted (<I>a</I>) a compendium of the three Books by ‘Abdel-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4591 melik al-Sh&imacr;r&amacr;z&imacr; (twelfth century), also in the Bodleian (913),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4592 (<I>b</I>) Borelli's edition, and (<I>c</I>) Bodl. 943 above mentioned, by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4593 of which he revised and corrected his translation when com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4594 pleted. Halley's edition is still, so far as I know, the only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4595 available source for Books V-VII, except for the beginning of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4596 Book V (up to Prop. 7) which was edited by L. Nix (Leipzig,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4597 1889).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4598 <p>The Greek text of Books I-IV is now available, with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4599 commentaries of Eutocius, the fragments of Apollonius, &c.,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4600 in the definitive edition of Heiberg (Teubner, 1891-3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4601 <C>Apollonius's own account of the <I>Conics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4602 <p>A general account of the contents of the great work which,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4603 according to Geminus, earned for him the title of the ‘great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4604 geometer’ cannot be better given than in the words of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4605 writer himself. The prefaces to the several Books contain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4606 interesting historical details, and, like the prefaces of Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4607 medes, state quite plainly and simply in what way the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4608 treatise differs from those of his predecessors, and how much
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4609 in it is claimed as original. The strictures of Pappus (or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4610 more probably his interpolator), who accuses him of being a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4611 braggart and unfair towards his predecessors, are evidently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4612 unfounded. The prefaces are quoted by v. Wilamowitz-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4613 Moellendorff as specimens of admirable Greek, showing how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4614 perfect the style of the great mathematicians could be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4615 when they were free from the trammels of mathematical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4616 terminology.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4617 <C>Book I. General Preface.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4618 <p>Apollonius to Eudemus, greeting.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4619 <p>If you are in good health and things are in other respects
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4620 as you wish, it is well; with me too things are moderately
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4621 <pb n=129>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4622 <head>THE <I>CONICS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4623 well. During the time I spent with you at Pergamum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4624 I observed your eagerness to become acquainted with my
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4625 work in conics; I am therefore sending you the first book,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4626 which I have corrected, and I will forward the remaining
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4627 books when I have finished them to my satisfaction. I dare
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4628 say you have not forgotten my telling you that I undertook
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4629 the investigation of this subject at the request of Naucrates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4630 the geometer, at the time when he came to Alexandria and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4631 stayed with me, and, when I had worked it out in eight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4632 books, I gave them to him at once, too hurriedly, because he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4633 was on the point of sailing; they had therefore not been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4634 thoroughly revised, indeed I had put down everything just as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4635 it occurred to me, postponing revision till the end. Accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4636 ingly I now publish, as opportunities serve from time to time,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4637 instalments of the work as they are corrected. In the mean-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4638 time it has happened that some other persons also, among
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4639 those whom I have met, have got the first and second books
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4640 before they were corrected; do not be surprised therefore if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4641 you come across them in a different shape.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4642 <p>Now of the eight books the first four form an elementary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4643 introduction. The first contains the modes of producing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4644 three sections and the opposite branches (of the hyperbola),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4645 and the fundamental properties subsisting in them, worked
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4646 out more fully and generally than in the writings of others.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4647 The second book contains the properties of the diameters and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4648 the axes of the sections as well as the asymptotes, with other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4649 things generally and necessarily used for determining limits
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4650 of possibility (<G>diorismoi/</G>); and what I mean by diameters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4651 and axes respectively you will learn from this book. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4652 third book contains many remarkable theorems useful for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4653 the syntheses of solid loci and for <I>diorismi</I>; the most and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4654 prettiest of these theorems are new, and it was their discovery
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4655 which made me aware that Euclid did not work out the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4656 synthesis of the locus with respect to three and four lines, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4657 only a chance portion of it, and that not successfully; for it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4658 was not possible for the said synthesis to be completed without
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4659 the aid of the additional theorems discovered by me. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4660 fourth book shows in how many ways the sections of cones
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4661 can meet one another and the circumference of a circle; it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4662 contains other things in addition, none of which have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4663 discussed by earlier writers, namely the questions in how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4664 many points a section of a cone or a circumference of a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4665 can meet [a double-branch hyperbola, or two double-branch
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4666 hyperbolas can meet one another].
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4667 <p>The rest of the books are more by way of surplusage
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4668 (<G>periousiastikw/tera</G>): one of them deals somewhat fully with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4669 <pb n=130>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4670 <head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4671 <I>minima</I> and <I>maxima</I>, another with equal and similar sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4672 of cones, another with theorems of the nature of determina-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4673 tions of limits, and the last with determinate conic problems.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4674 But of course, when all of them are published, it will be open
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4675 to all who read them to form their own judgement about them,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4676 according to their own individual tastes. Farewell.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4677 <p>The preface to Book II merely says that Apollonius is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4678 sending the second Book to Eudemus by his son Apollonius,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4679 and begs Eudemus to communicate it to earnest students of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4680 subject, and in particular to Philonides the geometer whom
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4681 Apollonius had introduced to Eudemus at Ephesus. There is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4682 no preface to Book III as we have it, although the preface to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4683 Book IV records that it also was sent to Eudemus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4684 <C>Preface to Book IV.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4685 <p>Apollonius to Attalus, greeting.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4686 <p>Some time ago I expounded and sent to Eudemus of Per-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4687 gamum the first three books of my conics which I have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4688 compiled in eight books, but, as he has passed away, I have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4689 resolved to dedicate the remaining books to you because of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4690 your earnest desire to possess my works. I am sending you
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4691 on this occasion the fourth book. It contains a discussion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4692 the question, in how many points at most it is possible for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4693 sections of cones to meet one another and the circumference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4694 of a circle, on the assumption that they do not coincide
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4695 throughout, and further in how many points at most a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4696 section of a cone or the circumference of a circle can meet the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4697 hyperbola with two branches, [or two double-branch hyper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4698 bolas can meet one another]; and, besides these questions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4699 the book considers a number of others of a similar kind.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4700 Now the first question Conon expounded to Thrasydaeus, with-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4701 out, however, showing proper mastery of the proofs, and on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4702 this ground Nicoteles of Cyrene, not without reason, fell foul
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4703 of him. The second matter has merely been mentioned by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4704 Nicoteles, in connexion with his controversy with Conon,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4705 as one capable of demonstration; but I have not found it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4706 demonstrated either by Nicoteles himself or by any one else.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4707 The third question and the others akin to it I have not found
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4708 so much as noticed by any one. All the matters referred to,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4709 which I have not found anywhere, required for their solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4710 many and various novel theorems, most of which I have,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4711 as a matter of fact, set out in the first three books, while the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4712 rest are contained in the present book. These theorems are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4713 of considerable use both for the syntheses of problems and for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4714 <pb n=131>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4715 <head>THE <I>CONICS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4716 <I>diorismi.</I> Nicoteles indeed, on account of his controversy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4717 with Conon, will not have it that any use can be made of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4718 discoveries of Conon for the purpose of <I>diorismi</I>; he is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4719 however, mistaken in this opinion, for, even if it is possible,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4720 without using them at all, to arrive at results in regard to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4721 limits of possibility, yet they at all events afford a readier
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4722 means of observing some things, e.g. that several or so many
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4723 solutions are possible, or again that no solution is possible;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4724 and such foreknowledge secures a satisfactory basis for in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4725 vestigations, while the theorems in question are again useful
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4726 for the analyses of <I>diorismi.</I> And, even apart from such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4727 usefulness, they will be found worthy of acceptance for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4728 sake of the demonstrations themselves, just as we accept
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4729 many other things in mathematics for this reason and for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4730 no other.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4731 <p>The prefaces to Books V-VII now to be given are repro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4732 duced for Book V from the translation of L. Nix and for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4733 Books VI, VII from that of Halley.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4734 <C>Preface to Book V.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4735 <p>Apollonius to Attalus, greeting.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4736 <p>In this fifth book I have laid down propositions relating to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4737 <I>maximum</I> and <I>minimum</I> straight lines. You must know
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4738 that my predecessors and contemporaries have only super-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4739 ficially touched upon the investigation of the shortest lines,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4740 and have only proved what straight lines touch the sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4741 and, conversely, what properties they have in virtue of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4742 they are tangents. For my part, I have proved these pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4743 perties in the first book (without however making any use, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4744 the proofs, of the doctrine of the shortest lines), inasmuch as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4745 I wished to place them in close connexion with that part
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4746 of the subject in which I treat of the production of the three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4747 conic sections, in order to show at the same time that in each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4748 of the three sections countless properties and necessary results
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4749 appear, as they do with reference to the original (transverse)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4750 diameter. The propositions in which I discuss the shortest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4751 lines I have separated into classes, and I have dealt with each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4752 individual case by careful demonstration; I have also con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4753 nected the investigation of them with the investigation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4754 the greatest lines above mentioned, because I considered that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4755 those who cultivate this science need them for obtaining
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4756 a knowledge of the analysis, and determination of limits of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4757 possibility, of problems as well as for their synthesis: in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4758 addition to which, the subject is one of those which seem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4759 worthy of study for their own sake. Farewell.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4760 <pb n=132>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4761 <head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4762 <C>Preface to Book VI.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4763 <p>Apollonius to Attalus, greeting.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4764 <p>I send you the sixth book of the conics, which embraces
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4765 propositions about conic sections and segments of conics equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4766 and unequal, similar and dissimilar, besides some other matters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4767 left out by those who have preceded me. In particular, you
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4768 will find in this book how, in a given right cone, a section can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4769 be cut which is equal to a given section, and how a right cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4770 can be described similar to a given cone but such as to contain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4771 a given conic section. And these matters in truth I have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4772 treated somewhat more fully and clearly than those who wrote
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4773 before my time on these subjects. Farewell.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4774 <C>Preface to Book VII.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4775 <p>Apollonius to Attalus, greeting.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4776 <p>I send to you with this letter the seventh book on conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4777 sections. In it are contained a large number of new proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4778 tions concerning diameters of sections and the figures described
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4779 upon them; and all these propositions have their uses in many
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4780 kinds of problems, especially in the determination of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4781 limits of their possibility. Several examples of these occur
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4782 in the determinate conic problems solved and demonstrated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4783 by me in the eighth book, which is by way of an appendix,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4784 and which I will make a point of sending to you as soon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4785 as possible. Farewell.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4786 <C><I>Extent of claim to originality.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4787 <p>We gather from these prefaces a very good idea of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4788 plan followed by Apollonius in the arrangement of the sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4789 ject and of the extent to which he claims originality. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4790 first four Books form, as he says, an elementary introduction,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4791 by which he means an exposition of the elements of conics,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4792 that is, the definitions and the fundamental propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4793 which are of the most general use and application; the term
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4794 ‘elements’ is in fact used with reference to conics in exactly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4795 the same sense as Euclid uses it to describe his great work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4796 The remaining Books beginning with Book V are devoted to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4797 more specialized investigation of particular parts of the sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4798 ject. It is only for a very small portion of the <I>content</I> of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4799 treatise that Apollonius claims originality; in the first three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4800 Books the claim is confined to certain propositions bearing on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4801 the ‘locus with respect to three or four lines’; and in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4802 fourth Book (on the number of points at which two conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4803 <pb n=133>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4804 <head>THE <I>CONICS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4805 may intersect, touch, or both) the part which is claimed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4806 as new is the extension to the intersections of the parabola,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4807 ellipse, and circle with the double-branch hyperbola, and of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4808 two double-branch hyperbolas with one another, of the in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4809 vestigations which had theretofore only taken account of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4810 single-branch hyperbola. Even in Book V, the most remark-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4811 able of all, Apollonius does not say that normals as ‘the shortest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4812 lines’ had not been considered before, but only that they had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4813 been superficially touched upon, doubtless in connexion with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4814 propositions dealing with the tangent properties. He explains
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4815 that he found it convenient to treat of the tangent properties,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4816 without any reference to normals, in the first Book in order
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4817 to connect them with the chord properties. It is clear, there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4818 fore, that in treating normals as <I>maxima</I> and <I>minima</I>, and by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4819 themselves, without any reference to tangents, as he does in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4820 Book V, he was making an innovation; and, in view of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4821 extent to which the theory of normals as maxima and minima
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4822 is developed by him (in 77 propositions), there is no wonder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4823 that he should devote a whole Book to the subject. Apart
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4824 from the developments in Books III, IV, V, just mentioned,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4825 and the numerous new propositions in Book VII with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4826 problems thereon which formed the lost Book VIII, Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4827 only claims to have treated the whole subject more fully and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4828 generally than his predecessors.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4829 <C><I>Great generality of treatment from the beginning.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4830 <p>So far from being a braggart and taking undue credit to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4831 himself for the improvements which he made upon his prede-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4832 cessors, Apollonius is, if anything, too modest in his descrip-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4833 tion of his personal contributions to the theory of conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4834 sections. For the ‘more fully and generally’ of his first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4835 preface scarcely conveys an idea of the extreme generality
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4836 with which the whole subject is worked out. This character-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4837 istic generality appears at the very outset.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4838 <C>Analysis of the <I>Conics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4839 <C>Book I.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4840 <p>Apollonius begins by describing a double oblique circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4841 cone in the most general way. Given a circle and any point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4842 outside the plane of the circle and in general not lying on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4843 <pb n=134>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4844 <head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4845 straight line through the centre of the circle perpendicular to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4846 its plane, a straight line passing through the point and pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4847 duced indefinitely in both directions is made to move, while
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4848 always passing through the fixed point, so as to pass succes-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4849 sively through all the points of the circle; the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4850 thus describes a double cone which is in general oblique or, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4851 Apollonius calls it, <I>scalene.</I> Then, before proceeding to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4852 geometry of a cone, Apollonius gives a number of definitions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4853 which, though of course only required for conics, are stated as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4854 applicable to any curve.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4855 <p>‘In any curve,’ says Apollonius, ‘I give the name <I>diameter</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4856 any straight line which, drawn from the curve, bisects all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4857 straight lines drawn in the curve (chords) parallel to any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4858 straight line, and I call the extremity of the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4859 (i.e. the diameter) which is at the curve a <I>vertex</I> of the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4860 and each of the parallel straight lines (chords) an ordinate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4861 (lit. drawn ordinate-wise, <G>tetagme/nws kath=xqai</G>) to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4862 diameter.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4863 <p>He then extends these terms to a pair of curves (the primary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4864 reference being to the double-branch hyperbola), giving the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4865 name <I>transverse diameter</I> to any straight line bisecting all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4866 chords in both curves which are parallel to a given straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4867 line (this gives two vertices where the diameter meets the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4868 curves respectively), and the name <I>erect diameter</I> (<G>o)rqi/a</G>) to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4869 any straight line which bisects all straight lines drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4870 between one curve and the other which are parallel to any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4871 straight line; the <I>ordinates</I> to any diameter are again the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4872 parallel straight lines bisected by it. <I>Conjugate diameters</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4873 any curve or pair of curves are straight lines each of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4874 bisects chords parallel to the other. <I>Axes</I> are the particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4875 diameters which cut at right angles the parallel chords which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4876 they bisect; and <I>conjugate axes</I> are related in the same way
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4877 as conjugate diameters. Here we have practically our modern
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4878 definitions, and there is a great advance on Archimedes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4879 terminology.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4880 <C><I>The conics obtained in the most general way from an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4881 oblique cone.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4882 <p>Having described a cone (in general oblique), Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4883 defines the <I>axis</I> as the straight line drawn from the vertex to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4884 <pb n=135>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4885 <head>THE <I>CONICS</I>, BOOK I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4886 the centre of the circular base. After proving that all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4887 sections parallel to the base are also circles, and that there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4888 is another set of circular sections subcontrary to these, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4889 proceeds to consider sections of the cone drawn in any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4890 manner. Taking any triangle through the axis (the base of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4891 the triangle being consequently a diameter of the circle which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4892 is the base of the cone), he is careful to make his section cut
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4893 the base in a straight line perpendicular to the particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4894 diameter which is the base of the axial triangle. (There is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4895 no loss of generality in this, for, if any section is taken,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4896 without reference to any axial triangle, we have only to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4897 select the particular axial triangle the base of which is that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4898 diameter of the circular base which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4899 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4900 at right angles to the straight line in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4901 which the section of the cone cuts the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4902 base.) Let <I>ABC</I> be any axial triangle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4903 and let any section whatever cut the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4904 base in a straight line <I>DE</I> at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4905 angles to <I>BC</I>; if then <I>PM</I> be the in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4906 tersection of the cutting plane and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4907 axial triangle, and if <I>QQ</I>′ be any chord
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4908 in the section parallel to <I>DE</I>, Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4909 proves that <I>QQ</I>′ is bisected by <I>PM.</I> In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4910 other words, <I>PM</I> is a <I>diameter</I> of the section. Apollonius is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4911 careful to explain that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4912 <p>‘if the cone is a right cone, the straight line in the base (<I>DE</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4913 will be at right angles to the common section (<I>PM</I>) of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4914 cutting plane and the triangle through the axis, but, if the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4915 cone is scalene, it will not in general be at right angles to <I>PM</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4916 but will be at right angles to it only when the plane through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4917 the axis (i.e. the axial triangle) is at right angles to the base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4918 of the cone’ (I. 7).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4919 <p>That is to say, Apollonius works out the properties of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4920 conics in the most general way with reference to a diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4921 which is not one of the principal diameters or axes, but in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4922 general has its ordinates obliquely inclined to it. The axes do
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4923 not appear in his exposition till much later, after it has been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4924 shown that each conic has the same property with reference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4925 to any diameter as it has with reference to the original
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4926 diameter arising out of the construction; the axes then appear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4927 <pb n=136>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4928 <head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4929 as particular cases of the new diameter of reference. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4930 three sections, the parabola, hyperbola, and ellipse are made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4931 in the manner shown in the figures. In each case they pass
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4932 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4933 through a straight line <I>DE</I> in the plane of the base which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4934 is at right angles to <I>BC</I>, the base of the axial triangle, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4935 to <I>BC</I> produced. The diameter <I>PM</I> is in the case of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4936 <pb n=137>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4937 <head>THE <I>CONICS</I>, BOOK I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4938 parabola parallel to <I>AC</I>; in the case of the hyperbola it meets
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4939 the other half of the double cone in <I>P</I>′; and in the case of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4940 ellipse it meets the cone itself again in <I>P</I>′. We draw, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4941 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4942 the cases of the hyperbola and ellipse, <I>AF</I> parallel to <I>PM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4943 to meet <I>BC</I> or <I>BC</I> produced in <I>F.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4944 <p>Apollonius expresses the properties of the three curves by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4945 means of a certain straight line <I>PL</I> drawn at right angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4946 to <I>PM</I> in the plane of the section.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4947 <p>In the case of the parabola, <I>PL</I> is taken such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4948 <MATH><I>PL</I>:<I>PA</I>=<I>BC</I><SUP>2</SUP>:<I>BA.AC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4949 and in the case of the hyperbola and ellipse such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4950 <MATH><I>PL</I>:<I>PP</I>′=<I>BF.FC</I>:<I>AF</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4951 <p>In the latter two cases we join <I>P</I>′<I>L</I>, and then draw <I>VR</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4952 parallel to <I>PL</I> to meet <I>P</I>′<I>L</I>, produced if necessary, in <I>R.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4953 <p>If <I>HK</I> be drawn through <I>V</I> parallel to <I>BC</I> and meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4954 <I>AB, AC</I> in <I>H, K</I> respectively, <I>HK</I> is the diameter of the circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4955 section of the cone made by a plane parallel to the base.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4956 <p>Therefore <MATH><I>QV</I><SUP>2</SUP>=<I>HV.VK</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4957 <p>Then (1) for the parabola we have, by parallels and similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4958 triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4959 <MATH><I>HV</I>:<I>PV</I>=<I>BC</I>:<I>CA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4960 and <MATH><I>VK</I>:<I>PA</I>=<I>BC</I>:<I>BA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4961 <pb n=138>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4962 <head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4963 <p>Therefore <MATH><I>QV</I><SUP>2</SUP>:<I>PV.PA</I>=<I>HV.VK</I>:<I>PV.PA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4964 =<I>BC</I><SUP>2</SUP>:<I>BA.AC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4965 =<I>PL</I>:<I>PA</I>, by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4966 =<I>PL.PV</I>:<I>PV.PA</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4967 whence <I>QV</I><SUP>2</SUP>=<I>PL.PV</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4968 <p>(2) In the case of the hyperbola and ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4969 <MATH><I>HV</I>:<I>PV</I>=<I>BF</I>:<I>FA</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4970 <I>VK</I>:<I>P</I>′<I>V</I>=<I>FC</I>:<I>AF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4971 <p>Therefore <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I>=<I>HV.VK</I>:<I>PV.P</I>′<I>V</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4972 =<I>BF.FC</I>:<I>AF</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4973 =<I>PL</I>:<I>PP</I>′, by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4974 =<I>RV</I>:<I>P</I>′<I>V</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4975 =<I>PV.VR</I>:<I>PV.P</I>′<I>V</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4976 whence <I>QV</I><SUP>2</SUP>=<I>PV.VR</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4977 <C><I>New names, ‘parabola’, ‘ellipse’, ‘hyperbola’.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4978 <p>Accordingly, in the case of the parabola, the square of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4979 ordinate (<I>QV</I><SUP>2</SUP>) is equal to the rectangle <I>applied</I> to <I>PL</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4980 with width equal to the abscissa (<I>PV</I>);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4981 in the case of the hyperbola the rectangle applied to <I>PL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4982 which is equal to <I>QV</I><SUP>2</SUP> and has its width equal to the abscissa
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4983 <I>PV overlaps</I> or <I>exceeds</I> (<G>u(perba/llei</G>) by the small rectangle <I>LR</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4984 which is similar and similarly situated to the rectangle con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4985 tained by <I>PL, PP</I>′;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4986 in the case of the ellipse the corresponding rectangle <I>falls
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4987 short</I> (<G>e)llei/pei</G>) by a rectangle similar and similarly situated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4988 to the rectangle contained by <I>PL, PP</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4989 <p>Here then we have the properties of the three curves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4990 expressed in the precise language of the Pythagorean applica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4991 tion of areas, and the curves are named accordingly: <I>parabola</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4992 (<G>parabolh/</G>) where the rectangle is exactly <I>applied, hyperbola</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4993 (<G>u(perbolh/</G>) where it <I>exceeds</I>, and <I>ellipse</I> (<G>e)/lleiyis</G>) where it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4994 <I>falls short.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4995 <pb n=139>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4996 <head>THE <I>CONICS</I>, BOOK I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4997 <p><I>PL</I> is called the <I>latus rectum</I> (<G>o)rqi/a</G>) or the <I>parameter of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4998 the ordinates</I> (<G>par) h(\n du/nantai ai( katago/menai tetagme/nws</G>) in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	4999 each case. In the case of the central conics, the diameter <I>PP</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5000 is the <I>transverse</I> (<G>h( plagi/a</G>) or <I>transverse diameter</I>; while,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5001 even more commonly, Apollonius speaks of the diameter and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5002 the corresponding parameter together, calling the latter the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5003 <I>latus rectum or erect side</I> (<G>o)rqi/a pleura/</G>) and the former
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5004 the <I>transverse side</I> of the <I>figure</I> (<G>ei)=dos</G>) <I>on</I>, or <I>applied to</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5005 diameter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5006 <C><I>Fundamental properties equivalent to Cartesian equations.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5007 <p>If <I>p</I> is the parameter, and <I>d</I> the corresponding diameter,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5008 the properties of the curves are the equivalent of the Cartesian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5009 equations, referred to the diameter and the tangent at its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5010 extremity as axes (in general oblique),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5011 <MATH><I>y</I><SUP>2</SUP>=<I>px</I></MATH> (the parabola),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5012 <MATH><I>y</I><SUP>2</SUP>=<I>px</I>±<I>(p/d)x</I><SUP>2</SUP></MATH> (the hyperbola and ellipse respectively).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5013 <p>Thus Apollonius expresses the fundamental property of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5014 central conics, like that of the parabola, as an equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5015 between areas, whereas in Archimedes it appears as a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5016 proportion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5017 <MATH><I>y</I><SUP>2</SUP>:(<I>a</I><SUP>2</SUP>±<I>x</I><SUP>2</SUP>)=<I>b</I><SUP>2</SUP>:<I>a</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5018 which, however, is equivalent to the Cartesian equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5019 referred to axes with the centre as origin. The latter pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5020 perty with reference to the original diameter is separately
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5021 proved in I. 21, to the effect that <I>QV</I><SUP>2</SUP> varies as <I>PV.P</I>′<I>V</I>, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5022 is really evident from the fact that <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I>=<I>PL</I>:<I>PP</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5023 seeing that <I>PL</I>:<I>PP</I>′ is constant for any fixed diameter <I>PP</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5024 <p>Apollonius has a separate proposition (I. 14) to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5025 the opposite branches of a hyperbola have the same diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5026 and equal <I>latera recta</I> corresponding thereto. As he was the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5027 first to treat the double-branch hyperbola fully, he generally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5028 discusses the <I>hyperbola</I> (i.e. the single branch) along with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5029 the ellipse, and <I>the opposites</I>, as he calls the double-branch
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5030 hyperbola, separately. The properties of the single-branch
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5031 hyperbola are, where possible, included in one enunciation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5032 with those of the ellipse and circle, the enunciation beginning,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5033 <pb n=140>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5034 <head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5035 ‘If in a hyperbola, an ellipse, or the circumference of a circle’;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5036 sometimes, however, the double-branch hyperbola and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5037 ellipse come in one proposition, e.g. in I. 30: ‘If in an ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5038 or the opposites (i.e. the double hyperbola) a straight line be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5039 drawn through the centre meeting the curve on both sides of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5040 the centre, it will be bisected at the centre.’ The property of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5041 conjugate diameters in an ellipse is proved in relation to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5042 the original diameter of reference and its conjugate in I. 15,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5043 where it is shown that, if <I>DD</I>′ is the diameter conjugate to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5044 <I>PP</I>′ (i.e. the diameter drawn ordinate-wise to <I>PP</I>′), just as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5045 <I>PP</I>′ bisects all chords parallel to <I>DD</I>′, so <I>DD</I>′ bisects all chords
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5046 parallel to <I>PP</I>′; also, if <I>DL</I>′ be drawn at right angles to <I>DD</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5047 and such that <MATH><I>DL</I>′.<I>DD</I>′=<I>PP</I>′<SUP>2</SUP></MATH> (or <I>DL</I>′ is a third proportional
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5048 to <I>DD</I>′, <I>PP</I>′), then the ellipse has the same property in rela-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5049 tion to <I>DD</I>′ as diameter and <I>DL</I>′ as parameter that it has in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5050 relation to <I>PP</I>′ as diameter and <I>PL</I> as the corresponding para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5051 meter. Incidentally it appears that <MATH><I>PL.PP</I>′=<I>DD</I>′<SUP>2</SUP></MATH>, or <I>PL</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5052 a third proportional to <I>PP</I>′, <I>DD</I>′, as indeed is obvious from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5053 property of the curve <MATH><I>QV</I><SUP>2</SUP>:<I>PV.PV</I>′=<I>PL</I>:<I>PP</I>′=<I>DD</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5054 The next proposition, I. 16, introduces the <I>secondary diameter</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5055 of the double-branch hyperbola (i.e. the diameter conjugate to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5056 the transverse diameter of reference), which does not meet the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5057 curve; this diameter is defined as that straight line drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5058 through the centre parallel to the ordinates of the transverse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5059 diameter which is bisected at the centre and is of length equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5060 to the mean proportional between the ‘sides of the figure’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5061 i.e. the transverse diameter <I>PP</I>′ and the corresponding para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5062 meter <I>PL.</I> The <I>centre</I> is defined as the middle point of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5063 diameter of reference, and it is proved that all other diameters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5064 are bisected at it (I. 30).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5065 <p>Props. 17-19, 22-9, 31-40 are propositions leading up to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5066 and containing the tangent properties. On lines exactly like
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5067 those of Eucl. III. 16 for the circle, Apollonius proves that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5068 a straight line is drawn through the vertex (i.e. the extremity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5069 of the diameter of reference) parallel to the ordinates to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5070 diameter, it will fall outside the conic, and no other straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5071 line can fall between the said straight line and the conic;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5072 therefore the said straight line touches the conic (I. 17, 32).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5073 Props. I. 33, 35 contain the property of the tangent at any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5074 point on the parabola, and Props. I. 34, 36 the property of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5075 <pb n=141>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5076 <head>THE <I>CONICS</I>, BOOK I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5077 the tangent at any point of a central conic, in relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5078 to the original diameter of reference; if <I>Q</I> is the point of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5079 contact, <I>QV</I> the ordinate to the diameter through <I>P</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5080 if <I>QT</I>, the tangent at <I>Q</I>, meets the diameter produced in <I>T</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5081 then (1) for the parabola <MATH><I>PV</I>=<I>PT</I></MATH>, and (2) for the central
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5082 conic <MATH><I>TP</I>:<I>TP</I>′=<I>PV</I>:<I>VP</I>′</MATH>. The method of proof is to take a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5083 point <I>T</I> on the diameter produced satisfying the respective
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5084 relations, and to prove that, if <I>TQ</I> be joined and produced,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5085 any point on <I>TQ</I> on either side of <I>Q</I> is outside the curve: the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5086 form of proof is by <I>reductio ad absurdum</I>, and in each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5087 case it is again proved that no other straight line can fall
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5088 between <I>TQ</I> and the curve. The fundamental property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5089 <MATH><I>TP</I>:<I>TP</I>′=<I>PV</I>:<I>VP</I>′</MATH> for the central conic is then used to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5090 prove that <MATH><I>CV.CT</I>=<I>CP</I><SUP>2</SUP></MATH> and <MATH><I>QV</I><SUP>2</SUP>:<I>CV.VT</I>=<I>p</I>:<I>PP</I>′</MATH> (or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5091 <I>CD</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP>) and the corresponding properties with reference to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5092 the diameter <I>DD</I>′ conjugate to <I>PP</I>′ and <I>v, t</I>, the points where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5093 <I>DD</I>′ is met by the ordinate to it from <I>Q</I> and by the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5094 at <I>Q</I> respectively (Props. I. 37-40).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5095 <C><I>Transition to new diameter and tangent at its extremity.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5096 <p>An important section of the Book follows (I. 41-50), con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5097 sisting of propositions leading up to what amounts to a trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5098 formation of coordinates from the original diameter and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5099 tangent at its extremity to <I>any</I> diameter and the tangent at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5100 its extremity; what Apollonius proves is of course that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5101 <I>any</I> other diameter be taken, the ordinate-property of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5102 conic with reference to that diameter is of the same form as it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5103 is with reference to the original diameter. It is evident that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5104 this is vital to the exposition. The propositions leading up to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5105 the result in I. 50 are not usually given in our text-books of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5106 geometrical conics, but are useful and interesting.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5107 <p>Suppose that the tangent at any point <I>Q</I> meets the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5108 of reference <I>PV</I> in <I>T</I>, and that the tangent at <I>P</I> meets the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5109 diameter through <I>Q</I> in <I>E.</I> Let <I>R</I> be any third point on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5110 the curve; let the ordinate <I>RW</I> to <I>PV</I> meet the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5111 through <I>Q</I> in <I>F</I>, and let <I>RU</I> parallel to the tangent at <I>Q</I> meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5112 <I>PV</I> in <I>U.</I> Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5113 <p>(1) in the parabola, the triangle <I>RUW</I>=the parallelogram
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5114 <I>EW</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5115 <pb n=142>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5116 <head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5117 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5118 <pb n=143>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5119 <head>THE <I>CONICS</I>, BOOK I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5120 <p>(2) in the hyperbola or ellipse, &utri;<I>RUW</I>=the difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5121 between the triangles <I>CFW</I> and <I>CPE.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5122 <p>(1) In the parabola <MATH>&utri;<I>RUW</I>:&utri;<I>QTV</I>=<I>RW</I><SUP>2</SUP>:<I>QV</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5123 =<I>PW</I>:<I>PV</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5124 =&rect;<I>EW</I>:&rect;<I>EV</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5125 <p>But, since <MATH><I>TV</I>=2<I>PV</I>,&utri;<I>QTV</I>=&rect;<I>EV</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5126 therefore <MATH>&utri;<I>RUW</I>=&rect;<I>EW</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5127 <p>(2) The proof of the proposition with reference to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5128 central conic depends on a Lemma, proved in I. 41, to the effect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5129 that, if <I>PX, VY</I> be similar parallelograms on <I>CP, CV</I> as bases,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5130 and if <I>VZ</I> be an equiangular parallelogram on <I>QV</I> as base and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5131 such that, if the ratio of <I>CP</I> to the other side of <I>PX</I> is <I>m</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5132 ratio of <I>QV</I> to the other side of <I>VZ</I> is <I>m.p/PP</I>′, then <I>VZ</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5133 equal to the difference between <I>VY</I> and <I>PX.</I> The proof of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5134 Lemma by Apollonius is difficult, but the truth of it can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5135 easily seen thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5136 <p>By the property of the curve, <MATH><I>QV</I><SUP>2</SUP>:<I>CV</I><SUP>2</SUP>-<I>CP</I><SUP>2</SUP>=<I>p</I>:<I>PP</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5137 therefore <MATH><I>CV</I><SUP>2</SUP>-<I>CP</I><SUP>2</SUP>=<I>PP</I>′/<I>p.QV</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5138 <p>Now <MATH>&rect;<I>PX</I>=<G>m</G>.<I>CP</I><SUP>2</SUP>/<I>m</I></MATH>, where <G>m</G> is a constant depending
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5139 on the angle of the parallelogram.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5140 <p>Similarly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5141 <MATH>&rect;<I>VY</I>=<G>m</G>.<I>CV</I><SUP>2</SUP>/<I>m</I>, and &rect;<I>VZ</I>=<G>m</G>.(<I>PP</I>′/<I>p</I>)<I>QV</I><SUP>2</SUP>/<I>m</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5142 <p>It follows that <MATH>&rect;<I>VY</I>-&rect;<I>PX</I>=&rect;<I>VZ</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5143 <p>Taking now the triangles <I>CFW, CPE</I> and <I>RUW</I> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5144 ellipse or hyperbola, we see that <I>CFW, CPE</I> are similar, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5145 <I>RUW</I> has one angle (at <I>W</I>) equal or supplementary to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5146 angles at <I>P</I> and <I>V</I> in the other two triangles, while we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5147 <MATH><I>QV</I><SUP>2</SUP>:<I>CV.VT</I>=<I>p</I>:<I>PP</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5148 whence <MATH><I>QV</I>:<I>VT</I>=(<I>p</I>:<I>PP</I>′).(<I>CV</I>:<I>QV</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5149 and, by parallels,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5150 <MATH><I>RW</I>:<I>WU</I>=(<I>p</I>:<I>PP</I>′).(<I>CP</I>:<I>PE</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5151 <pb n=144><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5152 <p>Therefore <I>RUW, CPE, CFW</I> are the halves of parallelograms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5153 related as in the lemma;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5154 therefore <MATH>&utri;<I>RUW</I>=&utri;<I>CFW</I>-&utri;<I>CPE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5155 <p>The same property with reference to the diameter <I>secondary</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5156 to <I>CPV</I> is proved in I. 45.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5157 <p>It is interesting to note the exact significance of the property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5158 thus proved for the central conic. The proposition, which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5159 the foundation of Apollonius's method of transformation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5160 coordinates, amounts to this. If <I>CP, CQ</I> are fixed semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5161 diameters and <I>R</I> a variable point, the area of the quadrilateral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5162 <I>CFRU</I> is constant for all positions of <I>R</I> on the conic. Suppose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5163 now that <I>CP, CQ</I> are taken as axes of <I>x</I> and <I>y</I> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5164 If we draw <I>RX</I> parallel to <I>CQ</I> to meet <I>CP</I> and <I>RY</I> parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5165 <I>CP</I> to meet <I>CQ</I>, the proposition asserts that (subject to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5166 proper convention as to sign)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5167 <MATH>&utri;<I>RYF</I>+&rect;<I>CXRY</I>+&utri;<I>RXU</I>=(const.)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5168 <p>But since <I>RX, RY, RF, RU</I> are in fixed directions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5169 <p>&utri;<I>RYF</I> varies as <I>RY</I><SUP>2</SUP> or <I>x</I><SUP>2</SUP>, &rect;<I>CXRY</I> as <I>RX.RY</I> or <I>xy</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5170 and &utri;<I>RXU</I> as <I>RX</I><SUP>2</SUP> or <I>y</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5171 <p>Hence, if <I>x, y</I> are the coordinates of <I>R</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5172 <MATH><G>a</G><I>x</I><SUP>2</SUP>+<G>b</G><I>xy</I>+<G>g</G><I>y</I><SUP>2</SUP>=<I>A</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5173 which is the Cartesian equation of the conic referred to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5174 centre as origin and any two diameters as axes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5175 <p>The properties so obtained are next used to prove that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5176 if <I>UR</I> meets the curve again in <I>R</I>′ and the diameter through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5177 <I>Q</I> in <I>M</I>, then <I>RR</I>′ is bisected at <I>M.</I> (I. 46-8).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5178 <p>Taking (1) the case of the parabola, we have,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5179 <MATH>&utri;<I>RUW</I>=&rect;<I>EW</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5180 and <MATH>&utri;<I>R</I>′<I>UW</I>′=&rect;<I>EW</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5181 <p>By subtraction, <MATH>(<I>RWW</I>′<I>R</I>′)=&rect;<I>F</I>′<I>W</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5182 whence <MATH>&utri;<I>RFM</I>=&utri;<I>R</I>′<I>F</I>′<I>M</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5183 and, since the triangles are similar, <MATH><I>RM</I>=<I>R</I>′<I>M</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5184 <p>The same result is easily obtained for the central conic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5185 <p>It follows that <I>EQ</I> produced in the case of the parabola,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5186 <pb n=145><head>THE <I>CONICS</I>, BOOK I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5187 or <I>CQ</I> in the case of the central conic, bisects all chords as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5188 <I>RR</I>′ parallel to the tangent at <I>Q.</I> Consequently <I>EQ</I> and <I>CQ</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5189 are <I>diameters</I> of the respective conics.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5190 <p>In order to refer the conic to the new diameter and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5191 corresponding ordinates, we have only to determine the <I>para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5192 meter</I> of these ordinates and to show that the property of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5193 conic with reference to the new parameter and diameter is in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5194 the same form as that originally found.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5195 <p>The propositions I. 49, 50 do this, and show that the new
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5196 parameter is in all the cases <I>p</I>′, where (if <I>O</I> is the point of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5197 intersection of the tangents at <I>P</I> and <I>Q</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5198 <MATH><I>OQ</I>:<I>QE</I>=<I>p</I>′:2<I>QT</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5199 <p>(1) In the case of the parabola, we have <MATH><I>TP</I>=<I>PV</I>=<I>EQ</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5200 whence <MATH>&utri;<I>EOQ</I>=&utri;<I>POT</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5201 <p>Add to each the figure <I>POQF</I>′<I>W</I>′;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5202 therefore <MATH><I>QTW</I>′<I>F</I>′=&rect;<I>EW</I>′=&utri;<I>R</I>′<I>UW</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5203 whence, subtracting <I>MUW</I>′<I>F</I>′ from both, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5204 <MATH>&utri;<I>R</I>′<I>MF</I>′=&rect;<I>QU</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5205 <p>Therefore <MATH><I>R</I>′<I>M.MF</I>′=2<I>QT.QM</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5206 <p>But <MATH><I>R</I>′<I>M</I>:<I>MF</I>′=<I>OQ</I>:<I>QE</I>=<I>p</I>′:2<I>QT</I></MATH>, by hypothesis;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5207 therefore <MATH><I>R</I>′<I>M</I><SUP>2</SUP>.<I>R</I>′<I>M.MF</I>′=<I>p</I>′.<I>QM</I>:2<I>QT.QM</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5208 <p>And <MATH><I>R</I>′<I>M.MF</I>′=2<I>QT.QM</I></MATH>, from above;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5209 therefore <MATH><I>R</I>′<I>M</I><SUP>2</SUP>=<I>p</I>′.<I>QM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5210 which is the desired property.<note>The proposition that, in the case of the parabola, if <I>p</I> be the para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5211 meter of the ordinates to the diameter through <I>Q</I>, then (see the first figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5212 on p. 142)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5213 <MATH><I>OQ</I>:<I>QE</I>=<I>p</I>:2<I>QT</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5214 has an interesting application; for it enables us to prove the proposition,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5215 assumed without proof by Archimedes (but not easy to prove otherwise),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5216 that, if in a parabola the diameter through <I>P</I> bisects the chord <I>QQ</I>′ in <I>V</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5217 and <I>QD</I> is drawn perpendicular to <I>PV</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5218 <MATH><I>QV</I><SUP>2</SUP>:<I>QD</I><SUP>2</SUP>=<I>p</I>:<I>p<SUB>a</SUB></I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5219 where <I>p<SUB>a</SUB></I> is the parameter of the principal ordinates and <I>p</I> the para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5220 meter of the ordinates to the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5221 <I>PV.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5222 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5223 <p>If the tangent at the vertex <I>A</I> meets
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5224 <I>VP</I> produced in <I>E</I>, and <I>PT</I>, the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5225 at <I>P</I>, in <I>O</I>, the proposition of Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5226 proves that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5227 <MATH><I>OP</I>:<I>PE</I>=<I>p</I>:2<I>PT</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5228 <p>But <MATH><I>OP</I>=1/2<I>PT</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5229 therefore <MATH><I>PT</I><SUP>2</SUP>=<I>p.PE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5230 =<I>p.AN</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5231 <p>Thus <MATH><I>QV</I><SUP>2</SUP>:<I>QD</I><SUP>2</SUP>=<I>PT</I><SUP>2</SUP>:<I>PN</I><SUP>2</SUP></MATH>, by similar triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5232 <MATH>=<I>p.AN</I>:<I>p<SUB>a</SUB>.AN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5233 =<I>p</I>:<I>p<SUB>a</SUB></I></MATH>.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5234 <pb n=146><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5235 <p>(2) In the case of the central conic, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5236 <MATH>&utri;<I>R</I>′<I>UW</I>′=&utri;<I>CF</I>′<I>W</I>′-&utri;<I>CPE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5237 (Apollonius here assumes what he does not prove till III. 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5238 namely that <MATH>&utri;<I>CPE</I>=&utri;<I>CQT</I></MATH>. This is proved thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5239 <p>We have <MATH><I>CV</I>:<I>CT</I>=<I>CV</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP>; (I. 37, 39.)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5240 therefore <MATH>&utri;<I>CQV</I>:&utri;<I>CQT</I>=&utri;<I>CQV</I>:&utri;<I>CPE</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5241 so that <MATH>&utri;<I>CQT</I>=&utri;<I>CPE</I></MATH>.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5242 <p>Therefore <MATH>&utri;<I>R</I>′<I>UW</I>′=&utri;<I>CF</I>′<I>W</I>′-&utri;<I>CQT</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5243 and it is easy to prove that in all cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5244 <MATH>&utri;<I>R</I>′<I>MF</I>′=<I>QTUM</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5245 <p>Therefore <MATH><I>R</I>′<I>M.MF</I>′=<I>QM</I>(<I>QT</I>+<I>MU</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5246 <p>Let <I>QL</I> be drawn at right angles to <I>CQ</I> and equal to <I>p</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5247 Join <I>Q</I>′<I>L</I> and draw <I>MK</I> parallel to <I>QL</I> to meet <I>Q</I>′<I>L</I> in <I>K</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5248 Draw <I>CH</I> parallel to <I>Q</I>′<I>L</I> to meet <I>QL</I> in <I>H</I> and <I>MK</I> in <I>N.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5249 <p>Now <MATH><I>R</I>′<I>M</I>:<I>MF</I>′=<I>OQ</I>:<I>QE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5250 =<I>QL</I>:2<I>QT</I></MATH>, by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5251 <MATH>=<I>QH</I>:<I>QT</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5252 <p>But <MATH><I>QT</I>:<I>MU</I>=<I>CQ</I>:<I>CM</I>=<I>QH</I>:<I>MN</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5253 so that <MATH>(<I>QH</I>+<I>MN</I>):(<I>QT</I>+<I>MU</I>)=<I>QH</I>:<I>QT</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5254 =<I>R</I>′<I>M</I>:<I>MF</I>′</MATH>, from above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5255 <pb n=147><head>THE <I>CONICS</I>, BOOK I</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5256 <p>It follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5257 <MATH><I>QM</I>(<I>QH</I>+<I>MN</I>):<I>QM</I>(<I>QT</I>+<I>MU</I>)=<I>R</I>′<I>M</I><SUP>2</SUP>:<I>R</I>′<I>M.MF</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5258 but, from above, <MATH><I>QM</I>(<I>QT</I>+<I>MU</I>)=<I>R</I>′<I>M.MF</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5259 therefore <MATH><I>R</I>′<I>M</I><SUP>2</SUP>=<I>QM</I>(<I>QH</I>+<I>MN</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5260 =<I>QM.MK</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5261 which is the desired property.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5262 <p>In the case of the hyperbola, the same property is true for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5263 the opposite branch.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5264 <p>These important propositions show that the ordinate property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5265 of the three conics is of the same form whatever diameter is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5266 taken as the diameter of reference. It is therefore a matter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5267 of indifference to which particular diameter and ordinates the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5268 conic is referred. This is stated by Apollonius in a summary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5269 which follows I. 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5270 <C><I>First appearance of principal axes.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5271 <p>The <I>axes</I> appear for the first time in the propositions next
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5272 following (I. 52-8), where Apollonius shows how to construct
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5273 each of the conics, given in each case (1) a diameter, (2) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5274 length of the corresponding parameter, and (3) the inclination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5275 of the ordinates to the diameter. In each case Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5276 first assumes the angle between the ordinates and the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5277 to be a right angle; then he reduces the case where the angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5278 is oblique to the case where it is right by his method of trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5279 formation of coordinates; i.e. from the given diameter and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5280 parameter he <I>finds</I> the <I>axis</I> of the conic and the length of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5281 corresponding parameter, and he then constructs the conic as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5282 in the first case where the ordinates are at right angles to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5283 diameter. Here then we have a case of the proof of <I>existence</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5284 by means of <I>construction.</I> The conic is in each case con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5285 structed by finding the cone of which the given conic is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5286 section. The problem of finding the axis of a parabola and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5287 the centre and the axes of a central conic when the conic (and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5288 not merely the elements, as here) is given comes later (in II.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5289 44-7), where it is also proved (II. 48) that no central conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5290 can have more than two axes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5291 <pb n=148><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5292 <p>It has been my object, by means of the above detailed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5293 account of Book I, to show not merely what results are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5294 obtained by Apollonius, but the way in which he went to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5295 work; and it will have been realized how entirely scientific
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5296 and general the method is. When the foundation is thus laid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5297 and the fundamental properties established, Apollonius is able
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5298 to develop the rest of the subject on lines more similar to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5299 those followed in our text-books. My description of the rest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5300 of the work can therefore for the most part be confined to a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5301 summary of the contents.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5302 <p>Book II begins with a section devoted to the properties of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5303 the asymptotes. They are constructed in II. 1 in this way.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5304 Beginning, as usual, with <I>any</I> diameter of reference and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5305 corresponding parameter and inclination of ordinates, Apol-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5306 lonius draws at <I>P</I> the vertex (the extremity of the diameter)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5307 a tangent to the hyperbola and sets off along it lengths <I>PL, PL</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5308 on either side of <I>P</I> such that <MATH><I>PL</I><SUP>2</SUP>=<I>PL</I>′<SUP>2</SUP>=1/4<I>p.PP</I>′[=<I>CD</I><SUP>2</SUP>]</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5309 where <I>p</I> is the parameter. He then proves that <I>CL, CL</I>′ pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5310 duced will not meet the curve in any finite point and are there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5311 fore <I>asymptotes.</I> II. 2 proves further that no straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5312 through <I>C</I> within the angle between the asymptotes can itself
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5313 be an asymptote. II. 3 proves that the intercept made by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5314 asymptotes on the tangent at any point <I>P</I> is bisected at <I>P</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5315 that the square on each half of the intercept is equal to one-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5316 fourth of the ‘figure’ corresponding to the diameter through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5317 <I>P</I> (i.e. one-fourth of the rectangle contained by the ‘erect’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5318 side, the <I>latus rectum</I> or parameter corresponding to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5319 diameter, and the diameter itself); this property is used as a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5320 means of drawing a hyperbola when the asymptotes and one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5321 point on the curve are given (II. 4). II. 5-7 are propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5322 about a tangent at the extremity of a diameter being parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5323 to the chords bisected by it. Apollonius returns to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5324 asymptotes in II. 8, and II. 8-14 give the other ordinary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5325 properties with reference to the asymptotes (II. 9 is a con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5326 verse of II. 3), the equality of the intercepts between the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5327 asymptotes and the curve of any chord (II. 8), the equality of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5328 the rectangle contained by the distances between either point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5329 in which the chord meets the curve and the points of inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5330 section with the asymptotes to the square on the parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5331 semi-diameter (II. 10), the latter property with reference to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5332 <pb n=149><head>THE <I>CONICS</I>, BOOK II</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5333 the portions of the asymptotes which include between them
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5334 a branch of the conjugate hyperbola (II. 11), the constancy of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5335 the rectangle contained by the straight lines drawn from any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5336 point of the curve in fixed directions to meet the asymptotes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5337 (equivalent to the Cartesian equation with reference to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5338 asymptotes, <MATH><I>xy</I> = const</MATH>.) (II. 12), and the fact that the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5339 and the asymptotes proceed to infinity and approach con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5340 tinually nearer to one another, so that the distance separating
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5341 them can be made smaller than any given length (II. 14). II. 15
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5342 proves that the two opposite branches of a hyperbola have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5343 same asymptotes and II. 16 proves for the chord connecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5344 points on two branches the property of II. 8. II. 17 shows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5345 ‘conjugate opposites’ (two conjugate double-branch hyper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5346 bolas) have the same asymptotes. Propositions follow about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5347 conjugate hyperbolas; any tangent to the conjugate hyper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5348 bola will meet both branches of the original hyperbola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5349 and will be bisected at the point of contact (II. 19); if <I>Q</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5350 any point on a hyperbola, and <I>CE</I> parallel to the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5351 at <I>Q</I> meets the conjugate hyperbola in <I>E</I>, the tangent at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5352 <I>E</I> will be parallel to <I>CQ</I> and <I>CQ, CE</I> will be conjugate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5353 diameters (II. 20), while the tangents at <I>Q, E</I> will meet on one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5354 of the asymptotes (II. 21); if a chord <I>Qq</I> in one branch of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5355 a hyperbola meet the asymptotes in <I>R, r</I> and the conjugate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5356 hyperbola in <I>Q</I>′, <I>q</I>′, then <MATH><I>Q</I>′<I>Q.Qq</I>′=2<I>CD</I><SUP>2</SUP> (II. 23)</MATH>. Of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5357 rest of the propositions in this part of the Book the following
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5358 may be mentioned: if <I>TQ, TQ</I>′ are two tangents to a conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5359 and <I>V</I> is the middle point of <I>QQ</I>′, <I>TV</I> is a diameter (II. 29,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5360 30, 38); if <I>tQ, tQ</I>′ be tangents to opposite branches of a hyper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5361 bola, <I>RR</I>′ the chord through <I>t</I> parallel to <I>QQ</I>′, <I>v</I> the middle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5362 point of <I>QQ</I>′, then <I>vR, vR</I>′ are tangents to the hyperbola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5363 (II. 40); in a conic, or a circle, or in conjugate hyperbolas, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5364 two chords not passing through the centre intersect, they do not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5365 bisect each other (II. 26, 41, 42). II. 44-7 show how to find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5366 a diameter of a conic and the centre of a central conic, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5367 axis of a parabola and the axes of a central conic. The Book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5368 concludes with problems of drawing tangents to conics in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5369 certain ways, through any point on or outside the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5370 (II. 49), making with the axis an angle equal to a given acute
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5371 angle (II. 50), making a given angle with the diameter through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5372 the point of contact (II. 51, 53); II. 52 contains a <G>diorismo/s</G> for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5373 <pb n=150><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5374 the last problem, proving that, if the tangent to an ellipse at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5375 any point <I>P</I> meets the major axis in <I>T</I>, the angle <I>CPT</I> is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5376 greater than the angle <I>ABA</I>′, where <I>B</I> is one extremity of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5377 minor axis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5378 <p>Book III begins with a series of propositions about the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5379 equality of certain areas, propositions of the same kind as, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5380 easily derived from, the propositions (I. 41-50) by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5381 which, as already shown, the transformation of coordinates is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5382 effected. We have first the proposition that, if the tangents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5383 at any points <I>P, Q</I> of a conic meet in <I>O</I>, and if they meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5384 the diameters through <I>Q, P</I> respectively in <I>E, T</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5385 <MATH>&utri;<I>OPT</I>=&utri;<I>OQE</I> (III. 1, 4)</MATH>; and, if <I>P, Q</I> be points on adjacent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5386 branches of conjugate hyperbolas, <MATH>&utri;<I>CPE</I>=&utri;<I>CQT</I> (III. 13)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5387 With the same notation, if <I>R</I> be any other point on the conic,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5388 and if we draw <I>RU</I> parallel to the tangent at <I>Q</I> meeting the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5389 diameter through <I>P</I> in <I>U</I> and the diameter through <I>Q</I> in <I>M</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5390 and <I>RW</I> parallel to the tangent at <I>P</I> meeting <I>QT</I> in <I>H</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5391 the diameters through <I>Q, P</I> in <I>F, W</I>, then <MATH>&utri;<I>HQF</I>=quadri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5392 lateral <I>HTUR</I> (III. 2. 6)</MATH>; this is proved at once from the fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5393 that <MATH>&utri;<I>RMF</I>=quadrilateral <I>QTUM</I> (see I. 49, 50, or pp. 145-6
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5394 above)</MATH> by subtracting or adding the area <I>HRMQ</I> on each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5395 side. Next take any other point <I>R</I>′, and draw <I>R</I>′<I>U</I>′, <I>F</I>′<I>H</I>′<I>R</I>′<I>W</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5396 in the same way as before; it is then proved that, if <I>RU, R</I>′<I>W</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5397 meet in <I>I</I> and <I>R</I>′<I>U</I>′, <I>RW</I> in <I>J</I>, the quadrilaterals <I>F</I>′<I>IRF, IUU</I>′<I>R</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5398 are equal, and also the quadrilaterals <I>FJR</I>′<I>F</I>′, <I>JU</I>′<I>UR</I> (III. 3,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5399 7, 9, 10). The proof varies according to the actual positions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5400 of the points in the figures.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5401 <p>In Figs. 1, 2 <MATH>&utri;<I>HFQ</I>=quadrilateral <I>HTUR</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5402 &utri;<I>H</I>′<I>F</I>′<I>Q</I>=<I>H</I>′<I>TU</I>′<I>R</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5403 <p>By subtraction, <MATH><I>FHH</I>′<I>F</I>′=<I>IUU</I>′<I>R</I>′&mnplus;(<I>IH</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5404 whence, if <I>IH</I> be added or subtracted, <MATH><I>F</I>′<I>IRF</I>=<I>IUU</I>′<I>R</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5405 and again, if <I>IJ</I> be added to both, <MATH><I>FJR</I>′<I>F</I>′=<I>JU</I>′<I>UR</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5406 <p>In Fig. 3 <MATH>&utri;<I>R</I>′<I>U</I>′<I>W</I>′=&utri;<I>CF</I>′<I>W</I>′-&utri;<I>CQT</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5407 so that <MATH>&utri;<I>CQT</I>=<I>CU</I>′<I>R</I>′<I>F</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5408 <pb n=151><head>THE <I>CONICS</I>, BOOK III</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5409 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5410 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5411 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5412 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5413 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5414 <CAP>FIG. 3.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5415 <pb n=152><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5416 <p>Adding the quadrilateral <I>CF</I>′<I>H</I>′<I>T</I>, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5417 <MATH>&utri;<I>H</I>′<I>F</I>′<I>Q</I>=<I>H</I>′<I>TU</I>′<I>R</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5418 and similarly <MATH>&utri;<I>HFQ</I>=<I>HTUR</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5419 <p>By subtraction, <MATH><I>F</I>′<I>H</I>′<I>HF</I>=<I>H</I>′<I>TU</I>′<I>R</I>′-<I>HTUR</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5420 <p>Adding <I>H</I>′<I>IRH</I> to each side, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5421 <MATH><I>F</I>′<I>IRF</I>=<I>IUU</I>′<I>R</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5422 <p>If each of these quadrilaterals is subtracted from <I>IJ</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5423 <MATH><I>FJR</I>′<I>F</I>′=<I>JU</I>′<I>UR</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5424 <p>The corresponding results are proved in III. 5, 11, 12, 14
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5425 for the case where the ordinates through <I>RR</I>′ are drawn to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5426 a <I>secondary</I> diameter, and in III. 15 for the case where <I>P, Q</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5427 are on the original hyperbola and <I>R, R</I>′ on the conjugate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5428 hyperbola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5429 <p>The importance of these propositions lies in the fact that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5430 they are immediately used to prove the well-known theorems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5431 about the rectangles contained by the segments of intersecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5432 chords and the harmonic properties of the pole and polar.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5433 The former question is dealt with in III. 16-23, which give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5434 a great variety of particular cases. We will give the proof
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5435 of one case, to the effect that, if <I>OP, OQ</I> be two tangents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5436 to any conic and <I>Rr, R</I>′<I>r</I>′ be any two chords parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5437 them respectively and intersecting in <I>J</I>, an internal or external
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5438 point,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5439 then <MATH><I>RJ.Jr</I>:<I>R</I>′<I>J.Jr</I>′=<I>OP</I><SUP>2</SUP>:<I>OQ</I><SUP>2</SUP>=(const.)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5440 <p>We have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5441 <MATH><I>RJ.Jr</I>=<I>RW</I><SUP>2</SUP>-<I>JW</I><SUP>2</SUP></MATH>, and <MATH><I>RW</I><SUP>2</SUP>:<I>JW</I><SUP>2</SUP>=&utri;<I>RUW</I>:&utri;<I>JU</I>′<I>W</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5442 therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5443 <MATH><I>RJ.Jr</I>:<I>RW</I><SUP>2</SUP>=(<I>RW</I><SUP>2</SUP>-<I>JW</I><SUP>2</SUP>):<I>RW</I><SUP>2</SUP>=<I>JU</I>′<I>UR</I>:&utri;<I>RUW</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5444 <p>But <MATH><I>RW</I><SUP>2</SUP>:<I>OP</I><SUP>2</SUP>=&utri;<I>RUW</I>:&utri;<I>OPT</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5445 therefore, <I>ex aequali</I>, <MATH><I>RJ.Jr</I>:<I>OP</I><SUP>2</SUP>=<I>JU</I>′<I>UR</I>:&utri;<I>OPT</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5446 <pb n=153><head>THE <I>CONICS</I>, BOOK III</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5447 <p>Similarly <MATH><I>R</I>′<I>M</I>′<SUP>2</SUP>:<I>JM</I>′<SUP>2</SUP>=&utri;<I>R</I>′<I>F</I>′<I>M</I>′:&utri;<I>JFM</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5448 whence <MATH><I>R</I>′<I>J.Jr</I>′:<I>R</I>′<I>M</I>′<SUP>2</SUP>=<I>FJR</I>′<I>F</I>′:&utri;<I>R</I>′<I>F</I>′<I>M</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5449 <p>But <MATH><I>R</I>′<I>M</I>′<SUP>2</SUP>:<I>OQ</I><SUP>2</SUP>=&utri;<I>R</I>′<I>F</I>′<I>M</I>′:&utri;<I>OQE</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5450 therefore, <I>ex aequali</I>, <MATH><I>R</I>′<I>J.Jr</I>′:<I>OQ</I><SUP>2</SUP>=<I>FJR</I>′<I>F</I>′:&utri;<I>OQE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5451 <p>It follows, since <MATH><I>FJR</I>′<I>F</I>′=<I>JU</I>′<I>UR</I></MATH>, and <MATH>&utri;<I>OPT</I>=&utri;<I>OQE</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5452 that <MATH><I>RJ.Jr</I>:<I>OP</I><SUP>2</SUP>=<I>R</I>′<I>J.Jr</I>′:<I>OQ</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5453 or <MATH><I>RJ.Jr</I>:<I>R</I>′<I>J.Jr</I>′=<I>OP</I><SUP>2</SUP>:<I>OQ</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5454 <p>If we had taken chords <I>Rr</I><SUB>1</SUB>, <I>R</I>′<I>r</I><SUB>1</SUB>′ parallel respectively to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5455 <I>OQ, OP</I> and intersecting in <I>I</I>, an internal or external point,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5456 we should have in like manner
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5457 <MATH><I>RI.Ir</I><SUB>1</SUB>:<I>R</I>′<I>I.Ir</I><SUB>1</SUB>′=<I>OQ</I><SUP>2</SUP>:<I>OP</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5458 <p>As a particular case, if <I>PP</I>′ be a diameter, and <I>Rr, R</I>′<I>r</I>′ be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5459 chords parallel respectively to the tangent at <I>P</I> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5460 diameter <I>PP</I>′ and intersecting in <I>I</I>, then (as is separately
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5461 proved)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5462 <MATH><I>RI.Ir</I>:<I>R</I>′<I>I.Ir</I>′=<I>p</I>:<I>PP</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5463 The corresponding results are proved in the cases where certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5464 of the points lie on the conjugate hyperbola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5465 <p>The six following propositions about the segments of inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5466 secting chords (III. 24-9) refer to two chords in conjugate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5467 hyperbolas or in an ellipse drawn parallel respectively to two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5468 conjugate diameters <I>PP</I>′, <I>DD</I>′, and the results in modern form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5469 are perhaps worth quoting. If <I>Rr, R</I>′<I>r</I>′ be two chords so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5470 drawn and intersecting in <I>O</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5471 <p>(<I>a</I>) in the conjugate hyperbolas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5472 <MATH><I>RO.Or</I>/<I>CP</I><SUP>2</SUP>±<I>R</I>′<I>O.Or</I>′/<I>CD</I><SUP>2</SUP>=2</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5473 and <MATH>(<I>RO</I><SUP>2</SUP>+<I>Or</I><SUP>2</SUP>):(<I>R</I>′<I>O</I><SUP>2</SUP>+<I>Or</I>′<SUP>2</SUP>)=<I>CP</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5474 <p>(<I>b</I>) in the ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5475 <MATH>(<I>RO</I><SUP>2</SUP>+<I>Or</I><SUP>2</SUP>)/<I>CP</I><SUP>2</SUP>+(<I>R</I>′<I>O</I><SUP>2</SUP>+<I>Or</I>′<SUP>2</SUP>)/<I>CD</I><SUP>2</SUP>=4</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5476 <pb n=154><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5477 <p>The general propositions containing the harmonic properties
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5478 of the pole and polar of a conic are III. 37-40, which prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5479 that in any conic, if <I>TQ, Tq</I> be tangents, and if <I>Qq</I> the chord
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5480 of contact be bisected in <I>V</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5481 <p>(1) if any straight line through <I>T</I> meet the conic in <I>R</I>′, <I>R</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5482 <I>Qq</I> in <I>I</I>, then (Fig. 1) <MATH><I>RT</I>:<I>TR</I>′=<I>RI</I>:<I>IR</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5483 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5484 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5485 <p>(2) if any straight line through <I>V</I> meet the conic in <I>R, R</I>′ and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5486 the parallel through <I>T</I> to <I>Qq</I> in <I>O</I>, then (Fig. 2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5487 <MATH><I>RO</I>:<I>OR</I>′=<I>RV</I>:<I>VR</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5488 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5489 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5490 <p>The above figures represent theorem (1) for the parabola and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5491 theorem (2) for the ellipse.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5492 <pb n=155><head>THE <I>CONICS</I>, BOOK III</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5493 <p>To prove (1) we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5494 <MATH><I>R</I>′<I>I</I><SUP>2</SUP>:<I>IR</I><SUP>2</SUP>=<I>H</I>′<I>Q</I><SUP>2</SUP>:<I>QH</I><SUP>2</SUP>=&utri;<I>H</I>′<I>F</I>′<I>Q</I>:&utri;<I>HFQ</I>=<I>H</I>′<I>TU</I>′<I>R</I>′:<I>HTUR</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5495 (III. 2, 3, &c.)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5496 <p>Also <MATH><I>R</I>′<I>T</I><SUP>2</SUP>:<I>TR</I><SUP>2</SUP>=<I>R</I>′<I>U</I>′<SUP>2</SUP>:<I>UR</I><SUP>2</SUP>=&utri;<I>R</I>′<I>U</I>′<I>W</I>′:&utri;<I>RUW</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5497 and <MATH><I>R</I>′<I>T</I><SUP>2</SUP>:<I>TR</I><SUP>2</SUP>=<I>TW</I>′<SUP>2</SUP>:<I>TW</I><SUP>2</SUP>=&utri;<I>TH</I>′<I>W</I>′:&utri;<I>THW</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5498 so that <MATH><I>R</I>′<I>T</I><SUP>2</SUP>:<I>TR</I><SUP>2</SUP>=&utri;<I>TH</I>′<I>W</I>′-&utri;<I>R</I>′<I>U</I>′<I>W</I>′:&utri;<I>THW</I>-&utri;<I>RUW</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5499 =<I>H</I>′<I>TU</I>′<I>R</I>′:<I>HTUR</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5500 =<I>R</I>′<I>I</I><SUP>2</SUP>:<I>IR</I><SUP>2</SUP></MATH>, from above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5501 <p>To prove (2) we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5502 <MATH><I>RV</I><SUP>2</SUP>:<I>VR</I>′<SUP>2</SUP>=<I>RU</I><SUP>2</SUP>:<I>R</I>′<I>U</I>′<SUP>2</SUP>=&utri;<I>RUW</I>:&utri;<I>R</I>′<I>U</I>′<I>W</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5503 and also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5504 <MATH>=<I>HQ</I><SUP>2</SUP>:<I>QH</I>′<SUP>2</SUP>=&utri;<I>HFQ</I>:&utri;<I>H</I>′<I>F</I>′<I>Q</I>=<I>HTUR</I><note>Where a quadrilateral, as <I>HTUR</I> in the figure, is a cross-quadri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5505 lateral, the area is of course the difference between the two triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5506 which it forms, as <I>HTW-RUW.</I></note>:<I>H</I>′<I>TU</I>′<I>R</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5507 so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5508 <MATH><I>RV</I><SUP>2</SUP>:<I>VR</I>′<SUP>2</SUP>=<I>HTUR</I>±&utri;<I>RUW</I>:<I>H</I>′<I>TU</I>′<I>R</I>′±&utri;<I>R</I>′<I>U</I>′<I>W</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5509 =&utri;<I>THW</I>:&utri;<I>TH</I>′<I>W</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5510 =<I>TW</I><SUP>2</SUP>:<I>TW</I>′<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5511 =<I>RO</I><SUP>2</SUP>:<I>OR</I>′<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5512 <p>Props. III. 30-6 deal separately with the particular cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5513 in which (<I>a</I>) the transversal is parallel to an asymptote of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5514 hyperbola or (<I>b</I>) the chord of contact is parallel to an asymp-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5515 tote, i.e. where one of the tangents is an asymptote, which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5516 the tangent at infinity.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5517 <p>Next we have propositions about intercepts made by two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5518 tangents on a third: If the tangents at three points of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5519 parabola form a triangle, all three tangents will be cut by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5520 points of contact in the same proportion (III. 41); if the tan-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5521 gents at the extremities of a diameter <I>PP</I>′ of a central conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5522 are cut in <I>r, r</I>′ by any other tangent, <MATH><I>Pr.P</I>′<I>r</I>′=<I>CD</I><SUP>2</SUP> (III. 42)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5523 if the tangents at <I>P, Q</I> to a hyperbola meet the asymptotes in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5524 <pb n=156><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5525 <I>L, L</I>′ and <I>M, M</I>′ respectively, then <I>L</I>′<I>M, LM</I>′ are both parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5526 to <I>PQ</I> (III. 44).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5527 <p>The first of these propositions asserts that, if the tangents at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5528 three points <I>P, Q, R</I> of a parabola form a triangle <I>pqr</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5529 <MATH><I>Pr</I>:<I>rq</I>=<I>rQ</I>:<I>Qp</I>=<I>qp</I>:<I>pR</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5530 <p>From this property it is easy to deduce the Cartesian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5531 equation of a parabola referred to two fixed tangents as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5532 coordinate axes. Taking.<I>qR, qP</I> as fixed coordinate axes, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5533 find the locus of <I>Q</I> thus. Let <I>x, y</I> be the coordinates of <I>Q.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5534 Then, if <MATH><I>qp</I>=<I>x</I><SUB>1</SUB>, <I>qr</I>=<I>y</I><SUB>1</SUB>, <I>qR</I>=<I>h, qP</I>=<I>k</I></MATH>, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5535 <MATH><I>x</I>/(<I>x</I><SUB>1</SUB>-<I>x</I>)=<I>rQ</I>/<I>Qp</I>=(<I>y</I><SUB>1</SUB>-<I>y</I>)/<I>y</I>=(<I>k-y</I><SUB>1</SUB>)/<I>y</I><SUB>1</SUB>=<I>x</I><SUB>1</SUB>/(<I>h-x</I><SUB>1</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5536 <p>From these equations we derive
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5537 <MATH><I>x</I><SUB>1</SUB><SUP>2</SUP>=<I>hx, y</I><SUB>1</SUB><SUP>2</SUP>=<I>ky</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5538 also, since <MATH><I>x</I><SUB>1</SUB>/<I>x</I>=<I>y</I><SUB>1</SUB>/(<I>y</I><SUB>1</SUB>-<I>y</I>)</MATH>, we have <MATH><I>x</I>/<I>x</I><SUB>1</SUB>+<I>y</I>/<I>y</I><SUB>1</SUB>=1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5539 <p>By substituting for <I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB> the values √(<I>hx</I>), √(<I>ky</I>) we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5540 obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5541 <MATH>(<I>x</I>/<I>h</I>)<SUP>1/2</SUP>+(<I>y</I>/<I>k</I>)<SUP>1/2</SUP>=1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5542 <p>The focal properties of central conics are proved in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5543 III. 45-52 without any reference to the directrix; there is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5544 no mention of the focus of a parabola. The foci are called
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5545 ‘the points arising out of the application’ (<G>ta\ e)k th=s para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5546 bolh=s gino/mena shmei=a</G>), the meaning being that <I>S,S</I>′ are taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5547 on the axis <I>AA</I>′ such that <MATH><I>AS.SA</I>′=<I>AS</I>′.<I>S</I>′<I>A</I>′=1/4<I>P<SUB>a</SUB>.AA</I>′</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5548 or <I>CB</I><SUP>2</SUP>, that is, in the phraseology of application of areas,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5549 a rectangle is applied to <I>AA</I>′ as base equal to one-fourth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5550 part of the ‘figure’, and in the case of the hyperbola ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5551 ceeding, but in the case of the ellipse falling short, by a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5552 square figure. The foci being thus found, it is proved that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5553 if the tangents <I>Ar, A</I>′<I>r</I>′ at the extremities of the axis are met
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5554 by the tangent at any point <I>P</I> in <I>r, r</I>′ respectively, <I>rr</I>′ subtends
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5555 a right angle at <I>S, S</I>′, and the angles <I>rr</I>′<I>S, A</I>′<I>r</I>′<I>S</I>′ are equal, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5556 also are the angles <I>r</I>′<I>rS</I>′, <I>ArS</I> (III. 45, 46). It is next shown
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5557 that, if <I>O</I> be the intersection of <I>rS</I>′, <I>r</I>′<I>S</I>, then <I>OP</I> is perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5558 dicular to the tangent at <I>P</I> (III. 47). These propositions are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5559 <pb n=157><head>THE <I>CONICS</I>, BOOK III</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5560 used to prove that the focal distances of <I>P</I> make equal angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5561 with the tangent at <I>P</I> (III. 48). In III. 49-52 follow the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5562 other ordinary properties, that, if <I>SY</I> be perpendicular to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5563 the tangent at <I>P</I>, the locus of <I>Y</I> is the circle on <I>AA</I>′ as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5564 diameter, that the lines from <I>C</I> drawn parallel to the focal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5565 distances to meet the tangent at <I>P</I> are equal to <I>CA</I>, and that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5566 the sum or difference of the focal distances of any point is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5567 equal to <I>AA</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5568 <p>The last propositions of Book III are of use with reference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5569 to the locus with respect to three or four lines. They are as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5570 follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5571 <p>1. If <I>PP</I>′ be a diameter of a central conic, and if <I>PQ, P</I>′<I>Q</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5572 drawn to any other point <I>Q</I> of the conic meet the tangents at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5573 <I>P</I>′, <I>P</I> in <I>R</I>′, <I>R</I> respectively, then <MATH><I>PR.P</I>′<I>R</I>′=4<I>CD</I><SUP>2</SUP> (III. 53)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5574 <p>2. If <I>TQ, TQ</I>′ be two tangents to a conic, <I>V</I> the middle point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5575 of <I>QQ</I>′, <I>P</I> the point of contact of the tangent parallel to <I>QQ</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5576 and <I>R</I> any other point on the conic, let <I>Qr</I> parallel to <I>TQ</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5577 meet <I>Q</I>′<I>R</I> in <I>r</I>, and <I>Q</I>′<I>r</I>′ parallel to <I>TQ</I> meet <I>QR</I> in <I>r</I>′; then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5578 <MATH><I>Qr.Q</I>′<I>r</I>′:<I>QQ</I>′<SUP>2</SUP>=(<I>PV</I><SUP>2</SUP>:<I>PT</I><SUP>2</SUP>).(<I>TQ.TQ</I>′:<I>QV</I><SUP>2</SUP>). (III. 54, 56.)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5579 <p>3. If the tangents are tangents to opposite branches of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5580 hyperbola and meet in <I>t</I>, and if <I>R, r, r</I>′ are taken as before,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5581 while <I>tq</I> is half the chord through <I>t</I> parallel to <I>QQ</I>′, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5582 <MATH><I>Qr.Q</I>′<I>r</I>′:<I>QQ</I>′<SUP>2</SUP>=<I>tQ.tQ</I>′:<I>tq</I><SUP>2</SUP>. (III. 55.)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5583 <p>The second of these propositions leads at once to the three-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5584 line locus, and from this we easily obtain the Cartesian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5585 equation to a conic with reference to two fixed tangents as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5586 axes, where the lengths of the tangents are <I>h, k</I>, viz.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5587 <MATH>(<I>x</I>/<I>h</I>+<I>y</I>/<I>k</I>-1)<SUP>2</SUP>=2<G>l</G>(<I>xy</I>/<I>hk</I>)<SUP>1/2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5588 <p>Book IV is on the whole dull, and need not be noticed at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5589 length. Props. 1-23 prove the converse of the propositions in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5590 Book III about the harmonic properties of the pole and polar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5591 for a large number of particular cases. One of the proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5592 tions (IV. 9) gives a method of drawing two tangents to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5593 a conic from an external point <I>T.</I> Draw any two straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5594 lines through <I>T</I> cutting the conic in <I>Q, Q</I>′ and in <I>R, R</I>′ respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5595 <pb n=158><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5596 tively. Take <I>O</I> on <I>QQ</I>′ and <I>O</I>′ on <I>RR</I>′ so that <I>TQ</I>′, <I>TR</I>′ are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5597 harmonically divided. The intersections of <I>OO</I>′ produced with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5598 the conic give the two points of contact required.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5599 <p>The remainder of the Book (IV.24-57) deals with intersecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5600 conics, and the number of points in which, in particular cases,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5601 they can intersect or touch. IV. 24 proves that no two conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5602 can meet in such a way that part of one of them is common
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5603 to both, while the rest is not. The rest of the propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5604 can be divided into five groups, three of which can be brought
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5605 under one general enunciation. Group I consists of particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5606 cases depending on the more elementary considerations affect-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5607 ing conics: e.g. two conics having their concavities in oppo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5608 site directions will not meet in more than two points (IV. 35);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5609 if a conic meet one branch of a hyperbola, it will not meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5610 the other branch in more points than two (IV. 37); a conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5611 touching one branch of a hyperbola with its concave side
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5612 will not meet the opposite branch (IV. 39). IV. 36, 41, 42, 45,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5613 54 belong to this group. Group II contains propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5614 (IV. 25, 38, 43, 44, 46, 55) showing that no two conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5615 (including in the term the double-branch hyperbola) can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5616 intersect in more than four points. Group III (IV. 26, 47, 48,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5617 49, 50, 56) are particular cases of the proposition that two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5618 conics which touch at one point cannot intersect at more than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5619 two other points. Group IV (IV. 27, 28, 29, 40, 51, 52, 53, 57)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5620 are cases of the proposition that no two conics which touch
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5621 each other at two points can intersect at any other point.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5622 Group V consists of propositions about double contact. A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5623 parabola cannot touch another parabola in more points than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5624 one (IV. 30); this follows from the property <MATH><I>TP</I>=<I>PV</I></MATH>. A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5625 parabola, if it fall outside a hyperbola, cannot have double
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5626 contact with it (IV. 31); it is shown that for the hyperbola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5627 <MATH><I>PV</I>><I>PT</I></MATH>, while for the parabola <MATH><I>P</I>′<I>V</I>=<I>P</I>′<I>T</I></MATH>; therefore the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5628 hyperbola would fall outside the parabola, which is impossible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5629 A parabola cannot have internal double contact with an ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5630 or circle (IV. 32). A hyperbola cannot have double contact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5631 with another hyperbola having the same centre (IV. 33);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5632 proved by means of <MATH><I>CV.CT</I>=<I>CP</I><SUP>2</SUP></MATH>. If an ellipse have double
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5633 contact with an ellipse or a circle, the chord of contact will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5634 pass through the centre (IV. 34).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5635 <p>Book V is of an entirely different order, indeed it is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5636 <pb n=159><head>THE <I>CONICS</I>, BOOKS IV-V</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5637 most remarkable of the extant Books. It deals with normals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5638 to conics regarded as <I>maximum</I> and <I>minimum</I> straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5639 drawn from particular points to the curve. Included in it are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5640 a series of propositions which, though worked out by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5641 purest geometrical methods, actually lead immediately to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5642 determination of the evolute of each of the three conics; that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5643 is to say, the Cartesian equations to the evolutes can be easily
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5644 deduced from the results obtained by Apollonius. There can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5645 be no doubt that the Book is almost wholly original, and it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5646 a veritable geometrical <I>tour de force.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5647 <p>Apollonius in this Book considers various points and classes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5648 of points with reference to the maximum or minimum straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5649 lines which it is possible to draw from them to the conics,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5650 i.e. as the feet of normals to the curve. He begins naturally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5651 with points on the axis, and he takes first the point <I>E</I> where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5652 <I>AE</I> measured along the axis from the vertex <I>A</I> is 1/2<I>p, p</I> being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5653 the principal parameter. The first three propositions prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5654 generally and for certain particular cases that, if in an ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5655 or a hyperbola <I>AM</I> be drawn at right angles to <I>AA</I>′ and equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5656 to 1/2<I>p</I>, and if <I>CM</I> meet the ordinate <I>PN</I> of any point <I>P</I> of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5657 curve in <I>H</I>, then <MATH><I>PN</I><SUP>2</SUP>=2 (quadrilateral <I>MANH</I>)</MATH>; this is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5658 lemma used in the proofs of later propositions, V. 5, 6, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5659 Next, in V. 4, 5, 6, he proves that, if <MATH><I>AE</I>=1/2<I>p</I></MATH>, then <I>AE</I> is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5660 <I>minimum</I> straight line from <I>E</I> to the curve, and if <I>P</I> be any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5661 other point on it, <I>PE</I> increases as <I>P</I> moves farther away from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5662 <I>A</I> on either side; he proves in fact that, if <I>PN</I> be the ordinate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5663 from <I>P</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5664 <p>(1) in the case of the parabola <MATH><I>PE</I><SUP>2</SUP>=<I>AE</I><SUP>2</SUP>+<I>AN</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5665 <p>(2) in the case of the hyperbola or ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5666 <MATH><I>PE</I><SUP>2</SUP>=<I>AE</I><SUP>2</SUP>+<I>AN</I><SUP>2</SUP>.(<I>AA</I>′±<I>p</I>)/<I>AA</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5667 where of course <MATH><I>p</I>=<I>BB</I>′<SUP>2</SUP>/<I>AA</I>′</MATH>, and therefore <MATH>(<I>AA</I>′±<I>p</I>)/<I>AA</I>′</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5668 is equivalent to what we call <I>e</I><SUP>2</SUP>, the square of the eccentricity.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5669 It is also proved that <I>EA</I>′ is the <I>maximum</I> straight line from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5670 <I>E</I> to the curve. It is next proved that, if <I>O</I> be any point on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5671 the axis between <I>A</I> and <I>E, OA</I> is the minimum straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5672 from <I>O</I> to the curve and, if <I>P</I> is any other point on the eurve,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5673 <I>OP</I> increases as <I>P</I> moves farther from <I>A</I> (V. 7).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5674 <pb n=160><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5675 <p>Next Apollonius takes points <I>G</I> on the axis at a distance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5676 from <I>A</I> greater than 1/2<I>p</I>, and he proves that the <I>minimum</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5677 straight line from <I>G</I> to the curve (i.e. the normal) is <I>GP</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5678 where <I>P</I> is such a point that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5679 <p>(1) in the case of the parabola <MATH><I>NG</I>=1/2<I>p</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5680 <p>(2) in the case of the central conic <MATH><I>NG</I>:<I>CN</I>=<I>p</I>:<I>AA</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5681 and, if <I>P</I>′ is any other point on the conic, <I>P</I>′<I>G</I> increases as <I>P</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5682 moves away from <I>P</I> on either side; this is proved by show-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5683 ing that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5684 <p>(1) for the parabola <MATH><I>P</I>′<I>G</I><SUP>2</SUP>=<I>PG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5685 <p>(2) for the central conic <MATH><I>P</I>′<I>G</I><SUP>2</SUP>=<I>PG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP>.(<I>AA</I>′±<I>p</I>)/<I>AA</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5686 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5687 <p>As these propositions contain the fundamental properties of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5688 the subnormals, it is worth while to reproduce Apollonius's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5689 proofs.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5690 <p>(1) In the parabola, if <I>G</I> be any point on the axis such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5691 <MATH><I>AG</I>>1/2<I>p</I></MATH>, measure <I>GN</I> towards <I>A</I> equal to 1/2<I>p</I>. Let <I>PN</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5692 the ordinate through <I>N, P</I>′ any other point on the curve.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5693 Then shall <I>PG</I> be the minimum line from <I>G</I> to the curve, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5694 <pb n=161><head>THE <I>CONICS</I>, BOOK V</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5695 <p>We have <MATH><I>P</I>′<I>N</I>′<SUP>2</SUP>=<I>p.AN</I>′=2<I>NG.AN</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5696 and <MATH><I>N</I>′<I>G</I><SUP>2</SUP>=<I>NN</I>′<SUP>2</SUP>+<I>NG</I><SUP>2</SUP>±2<I>NG.NN</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5697 according to the position of <I>N</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5698 <p>Therefore <MATH><I>P</I>′<I>G</I><SUP>2</SUP>=2<I>NG.AN</I>+<I>NG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5699 =<I>PN</I><SUP>2</SUP>+<I>NG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5700 =<I>PG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5701 and the proposition is proved.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5702 <p>(2) In the case of the central conic, take <I>G</I> on the axis such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5703 that <MATH><I>AG</I> > 1/2<I>p</I></MATH>, and measure <I>GN</I> towards <I>A</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5704 <MATH><I>NG</I>:<I>CN</I>=<I>p</I>:<I>AA</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5705 Draw the ordinate <I>PN</I> through <I>N</I>, and also the ordinate <I>P</I>′<I>N</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5706 from any other point <I>P</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5707 <p>We have first to prove the lemma (V. 1, 2, 3) that, if <I>AM</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5708 drawn perpendicular to <I>AA</I>′ and equal to 1/2<I>p</I>, and if <I>CM</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5709 produced if necessary, meet <I>PN</I> in <I>H</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5710 <MATH><I>PN</I><SUP>2</SUP>=2(quadrilateral <I>MANH</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5711 <p>This is easy, for, if <MATH><I>AL</I>(=2<I>AM</I>)</MATH> be the parameter, and <I>A</I>′<I>L</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5712 meet <I>PN</I> in <I>R</I>, then, by the property of the curve,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5713 <MATH><I>PN</I><SUP>2</SUP>=<I>AN.NR</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5714 =<I>AN</I>(<I>NH</I>+<I>AM</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5715 =2(quadrilateral <I>MANH</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5716 <p>Let <I>GH</I>, produced if necessary, meet <I>P</I>′<I>N</I>′ in <I>H</I>′. From <I>H</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5717 draw <I>HI</I> perpendicular to <I>P</I>′<I>H</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5718 <p>Now, since, by hypothesis, <MATH><I>NG</I>:<I>CN</I>=<I>p</I>:<I>AA</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5719 =<I>AM</I>:<I>AC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5720 =<I>HN</I>:<I>NC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5721 <MATH><I>NH</I>=<I>NG</I></MATH>, whence also <MATH><I>H</I>′<I>N</I>′=<I>N</I>′<I>G</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5722 <p>Therefore <MATH><I>NG</I><SUP>2</SUP>=2&utri;<I>HNG, N</I>′<I>G</I><SUP>2</SUP>=2&utri;<I>H</I>′<I>N</I>′<I>G</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5723 <p>And <MATH><I>PN</I><SUP>2</SUP>=2(<I>MANH</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5724 therefore <MATH><I>PG</I><SUP>2</SUP>=<I>NG</I><SUP>2</SUP>+<I>PN</I><SUP>2</SUP>=2(<I>AMHG</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5725 <pb n=162><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5726 <p>Similarly, if <I>CM</I> meets <I>P</I>′<I>N</I>′ in <I>K</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5727 <MATH><I>P</I>′<I>G</I><SUP>2</SUP>=<I>N</I>′<I>G</I><SUP>2</SUP>+<I>P</I>′<I>N</I>′<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5728 =2&utri;<I>H</I>′<I>N</I>′<I>G</I>+2(<I>AMKN</I>′)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5729 =2(<I>AMHG</I>)+2&utri;<I>HH</I>′<I>K</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5730 <p>Therefore, by subtraction,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5731 <MATH><I>P</I>′<I>G</I><SUP>2</SUP>-<I>PG</I><SUP>2</SUP>=2&utri;<I>HH</I>′<I>K</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5732 =<I>HI</I>.(<I>H</I>′<I>I</I>±<I>IK</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5733 =<I>HI</I>.(<I>HI</I>±<I>IK</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5734 =<I>HI</I><SUP>2</SUP>.(<I>CA</I>±<I>AM</I>)/<I>CA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5735 =<I>NN</I>′<SUP>2</SUP>.(<I>AA</I>′±<I>p</I>)/<I>AA</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5736 which proves the proposition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5737 <p>If <I>O</I> be any point on <I>PG, OP</I> is the minimum straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5738 from <I>O</I> to the eurve, and <I>OP</I>′ increases as <I>P</I>′ moves away from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5739 <I>P</I> on either side; this is proved in V. 12. (Since <MATH><I>P</I>′<I>G</I> > <I>PG</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5740 <MATH>&angle;<I>GPP</I>′ > &angle;<I>GP</I>′<I>P</I></MATH>; therefore, <I>a fortiori</I>, <MATH>&angle;<I>OPP</I>′ > &angle;<I>OP</I>′<I>P</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5741 and <MATH><I>OP</I>′ > <I>OP</I></MATH>.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5742 <p>Apollonius next proves the corresponding propositions with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5743 reference to points on the <I>minor</I> axis of an ellipse. If <I>p</I>′ be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5744 the parameter of the ordinates to the minor axis, <MATH><I>p</I>′=<I>AA</I>′<SUP>2</SUP>/<I>BB</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5745 or <MATH>1/2<I>p</I>′=<I>CA</I><SUP>2</SUP>/<I>CB</I></MATH>. If now <I>E</I>′ be so taken that <MATH><I>BE</I>′=1/2<I>p</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5746 then <I>BE</I>′ is the <I>maximum</I> straight line from <I>E</I>′ to the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5747 and, if <I>P</I> be any other point on it, <I>E</I>′<I>P</I> diminishes as <I>P</I> moves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5748 farther from <I>B</I> on either side, and <I>E</I>′<I>B</I>′ is the <I>minimum</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5749 straight line from <I>E</I>′ to the curve. It is, in fact, proved that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5750 <MATH><I>E</I>′<I>B</I><SUP>2</SUP>-<I>E</I>′<I>P</I><SUP>2</SUP>=<I>Bn</I><SUP>2</SUP>.(<I>p</I>′-<I>BB</I>′)/<I>BB</I>′</MATH>, where <I>Bn</I> is the abscissa of <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5751 (V. 16-18). If <I>O</I> be any point on the minor axis such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5752 <MATH><I>BO</I> > <I>BE</I>′</MATH>, then <I>OB</I> is the <I>maximum</I> straight line from <I>O</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5753 the curve, &c. (V. 19).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5754 <p>If <I>g</I> be a point on the minor axis such that <MATH><I>Bg</I> > <I>BC</I></MATH>, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5755 <MATH><I>Bg</I> < 1/2<I>p</I>′</MATH>, and if <I>Cn</I> be measured towards <I>B</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5756 <MATH><I>Cn</I>:<I>ng</I>=<I>BB</I>′:<I>p</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5757 then <I>n</I> is the foot of the ordinates of two points <I>P</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5758 <I>Pg</I> is the <I>maximum</I> straight line from <I>g</I> to the curve. Also,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5759 <pb n=163><head>THE <I>CONICS</I>, BOOK V</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5760 if <I>P</I>′ be any other point on it, <I>P</I>′<I>g</I> diminishes as <I>P</I>′ moves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5761 farther from <I>P</I> on either side to <I>B</I> or <I>B</I>′, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5762 <MATH><I>Pg</I><SUP>2</SUP>-<I>P</I>′<I>g</I><SUP>2</SUP>=<I>nn</I>′<SUP>2</SUP>.(<I>p</I>′-<I>BB</I>′)/<I>BB</I>′</MATH> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5763 <MATH><I>nn</I>′<SUP>2</SUP>.(<I>CA</I><SUP>2</SUP>-<I>CB</I><SUP>2</SUP>)/<I>CB</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5764 If <I>O</I> be any point on <I>Pg</I> produced beyond the minor axis, <I>PO</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5765 is the <I>maximum</I> straight line from <I>O</I> to the same part of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5766 ellipse for which <I>Pg</I> is a maximum, i.e. the semi-ellipse <I>BPB</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5767 &c. (V. 20-2).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5768 <p>In V. 23 it is proved that, if <I>g</I> is on the minor axis, and <I>gP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5769 a maximum straight line to the curve, and if <I>Pg</I> meets <I>AA</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5770 in <I>G</I>, then <I>GP</I> is the <I>minimum</I> straight line from <I>G</I> to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5771 curve; this is proved by similar triangles. Only one normal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5772 can be drawn from any one point on a conic (V. 24-6). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5773 normal at any point <I>P</I> of a conic, whether regarded as a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5774 minimum straight line from <I>G</I> on the major axis or (in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5775 case of the ellipse) as a <I>maximum</I> straight line from <I>g</I> on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5776 minor axis, is perpendicular to the tangent at <I>P</I> (V. 27-30);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5777 in general (1) if <I>O</I> be any point within a conic, and <I>OP</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5778 a maximum or a minimum straight line from <I>O</I> to the conic,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5779 the straight line through <I>P</I> perpendicular to <I>PO</I> touches the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5780 conic, and (2) if <I>O</I>′ be any point on <I>OP</I> produced outside the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5781 conic, <I>O</I>′<I>P</I> is the minimum straight line from <I>O</I>′ to the conic,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5782 &c. (V. 31-4).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5783 <C><I>Number of normals from a point.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5784 <p>We now come to propositions about two or more normals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5785 meeting at a point. If the normal at <I>P</I> meet the axis of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5786 a parabola or the axis <I>AA</I>′ of a hyperbola or ellipse in <I>G</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5787 angle <I>PGA</I> increases as <I>P</I> or <I>G</I> moves farther away from <I>A</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5788 but in the case of the hyperbola the angle will always be less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5789 than the complement of half the angle between the asymptotes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5790 Two normals at points on the same side of <I>AA</I>′ will meet on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5791 the opposite side of that axis; and two normals at points on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5792 the same quadrant of an ellipse as <I>AB</I> will meet at a point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5793 within the angle <I>ACB</I>′ (V. 35-40). In a parabola or an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5794 ellipse any normal <I>PG</I> will meet the curve again; in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5795 hyperbola, (1) if <I>AA</I>′ be not greater than <I>p</I>, no normal can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5796 meet the curve at a second point on the same branch, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5797 <pb n=164><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5798 (2) if <MATH><I>AA</I>′ > <I>p</I></MATH>, some normals will meet the same branch again
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5799 and others not (V. 41-3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5800 <p>If <I>P</I><SUB>1</SUB><I>G</I><SUB>1</SUB>, <I>P</I><SUB>2</SUB><I>G</I><SUB>2</SUB> be normals at points on one side of the axis of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5801 a conic meeting in <I>O</I>, and if <I>O</I> be joined to any other point <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5802 on the conic (it being further supposed in the case of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5803 ellipse that all three lines <I>OP</I><SUB>1</SUB>, <I>OP</I><SUB>2</SUB>, <I>OP</I> cut the same half of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5804 the axis), then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5805 <p>(1) <I>OP</I> cannot be a normal to the curve;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5806 <p>(2) if <I>OP</I> meet the axis in <I>K</I>, and <I>PG</I> be the normal at <I>P, AG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5807 is less or greater than <I>AK</I> according as <I>P</I> does or does not lie
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5808 between <I>P</I><SUB>1</SUB> and <I>P</I><SUB>2</SUB>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5809 <p>From this proposition it is proved that (1) three normals at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5810 points on one quadrant of an ellipse cannot meet at one point,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5811 and (2) four normals at points on one semi-ellipse bounded by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5812 the major axis cannot meet at one point (V. 44-8).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5813 <p>In any conic, if <I>M</I> be any point on the axis such that <I>AM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5814 is not greater than 1/2<I>p</I>, and if <I>O</I> be any point on the double
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5815 ordinate through <I>M</I>, then no straight line drawn to any point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5816 on the curve on the other side of the axis from <I>O</I> and meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5817 the axis between <I>A</I> and <I>M</I> can be a normal (V. 49, 50).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5818 <C><I>Propositions leading immediately to the determination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5819 of the</I> evolute <I>of a conic.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5820 <p>These great propositions are V. 51, 52, to the following
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5821 effect:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5822 <p>If <I>AM</I> measured along the axis be greater than 1/2<I>p</I> (but in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5823 the case of the ellipse less than <I>AC</I>), and if <I>MO</I> be drawn per-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5824 pendicular to the axis, then a certain length (<I>y</I>, say) can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5825 assigned such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5826 <p>(<I>a</I>) if <MATH><I>OM</I> > <I>y</I></MATH>, no normal can be drawn through <I>O</I> which cuts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5827 the axis; but, if <I>OP</I> be any straight line drawn to the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5828 cutting the axis in <MATH><I>K, NK</I> < <I>NG</I></MATH>, where <I>PN</I> is the ordinate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5829 and <I>PG</I> the normal at <I>P</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5830 <p>(<I>b</I>) if <MATH><I>OM</I>=<I>y</I></MATH>, only one normal can be so drawn through <I>O</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5831 and, if <I>OP</I> be any other straight line drawn to the curve and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5832 cutting the axis in <MATH><I>K, NK</I> < <I>NG</I></MATH>, as before;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5833 <p>(<I>c</I>) if <MATH><I>OM</I> < <I>y</I></MATH>, two normals can be so drawn through <I>O</I>, and, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5834 <I>OP</I> be any other straight line drawn to the curve, <I>NK</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5835 <pb n=165><head>THE <I>CONICS</I>, BOOK V</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5836 greater or less than <I>NG</I> according as <I>OP</I> is or is not inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5837 mediate between the two normals (V. 51, 52).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5838 <p>The proofs are of course long and complicated. The length
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5839 <I>y</I> is determined in this way:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5840 <p>(1) In the case of the parabola, measure <I>MH</I> towards the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5841 vertex equal to 1/2<I>p</I>, and divide <I>AH</I> at <I>N</I><SUB>1</SUB> so that <MATH><I>HN</I><SUB>1</SUB>=2<I>N</I><SUB>1</SUB><I>A</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5842 The length <I>y</I> is then taken such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5843 <MATH><I>y</I>:<I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB>=<I>N</I><SUB>1</SUB><I>H</I>:<I>HM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5844 where <I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB> is the ordinate passing through <I>N</I><SUB>1</SUB>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5845 <p>(2) In the case of the hyperbola and ellipse, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5846 <MATH><I>AM</I> > 1/2<I>p</I></MATH>, so that <MATH><I>CA</I>:<I>AM</I> < <I>AA</I>′:<I>p</I></MATH>; therefore, if <I>H</I> be taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5847 on <I>AM</I> such that <MATH><I>CH</I>:<I>HM</I>=<I>AA</I>′:<I>p, H</I></MATH> will fall between <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5848 and <I>M.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5849 <p>Take two mean proportionals <I>CN</I><SUB>1</SUB>, <I>CI</I> between <I>CA</I> and <I>CH</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5850 and let <I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB> be the ordinate through <I>N</I><SUB>1</SUB>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5851 <p>The length <I>y</I> is then taken such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5852 <MATH><I>y</I>:<I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB>=(<I>CM</I>:<I>MH</I>).(<I>HN</I><SUB>1</SUB>:<I>N</I><SUB>1</SUB><I>C</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5853 <p>In the case (<I>b</I>), where <MATH><I>OM</I>=<I>y, O</I></MATH> is the point of intersection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5854 of consecutive normals, i.e. <I>O</I> is the centre of curvature at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5855 point <I>P</I>; and, by considering the coordinates of <I>O</I> with reference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5856 to two coordinate axes, we can derive the Cartesian equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5857 of the evolutes. E. g. (1) in the case of the parabola let the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5858 coordinate axes be the axis and the tangent at the vertex.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5859 Then <MATH><I>AM</I>=<I>x</I></MATH>, <MATH><I>OM</I>=<I>y</I></MATH>. Let <MATH><I>p</I>=4<I>a</I></MATH>; then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5860 <MATH><I>HM</I>=2<I>a</I></MATH>, <MATH><I>N</I><SUB>1</SUB><I>H</I>=2/3(<I>x</I>-2<I>a</I>)</MATH>, and <MATH><I>AN</I><SUB>1</SUB>=1/3(<I>x</I>-2<I>a</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5861 <p>But <MATH><I>y</I><SUP>2</SUP>:<I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB><SUP>2</SUP>=<I>N</I><SUB>1</SUB><I>H</I><SUP>2</SUP>:<I>HM</I><SUP>2</SUP></MATH>, by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5862 or <MATH><I>y</I><SUP>2</SUP>:4<I>a.AN</I><SUB>1</SUB>=<I>N</I><SUB>1</SUB><I>H</I><SUP>2</SUP>:4<I>a</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5863 therefore <MATH><I>ay</I><SUP>2</SUP>=<I>AN</I><SUB>1</SUB>.<I>N</I><SUB>1</SUB><I>H</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5864 =4/27(<I>x</I>-2<I>a</I>)<SUP>3</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5865 or <MATH>27<I>ay</I><SUP>2</SUP>=4(<I>x</I>-2<I>a</I>)<SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5866 <p>(2) In the case of the hyperbola or ellipse we naturally take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5867 <I>CA, CB</I> as axes of <I>x</I> and <I>y.</I> The work is here rather more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5868 complicated, but there is no difficulty in obtaining, as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5869 locus of <I>O</I>, the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5870 <MATH>(<I>ax</I>)<SUP>2/3</SUP>&mnplus;(<I>by</I>)<SUP>2/3</SUP>=(<I>a</I><SUP>2</SUP>±<I>b</I><SUP>2</SUP>)<SUP>2/3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5871 <pb n=166><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5872 <p>The propositions V. 53, 54 are particular cases of the pre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5873 ceding propositions.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5874 <C><I>Construction of normals.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5875 <p>The next section of the Book (V. 55-63) relates to the con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5876 struction of normals through various points according to their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5877 position within or without the conic and in relation to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5878 axes. It is proved that one normal can be drawn through any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5879 internal point and through any external point which is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5880 on the axis through the vertex <I>A.</I> In particular, if <I>O</I> is any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5881 point below the axis <I>AA</I>′ of an ellipse, and <I>OM</I> is perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5882 dicular to <I>AA</I>′, then, if <MATH><I>AM</I> > <I>AC</I></MATH>, one normal can always be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5883 drawn through <I>O</I> cutting the axis between <I>A</I> and <I>C</I>, but never
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5884 more than one such normal (V. 55-7). The points on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5885 curve at which the straight lines through <I>O</I> are normals are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5886 determined as the intersections of the conic with a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5887 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5888 rectangular hyperbola. The procedure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5889 of Apollonius is equivalent to the fol-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5890 lowing analytical method. Let <I>AM</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5891 the axis of a conic, <I>PGO</I> one of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5892 normals which passes through the given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5893 point <I>O, PN</I> the ordinate at <I>P</I>; and let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5894 <I>OM</I> be drawn perpendicular to the axis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5895 Take as axes of coordinates the axes in the central conic and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5896 in the case of the parabola, the axis and the tangent at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5897 vertex.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5898 <p>If then (<I>x, y</I>) be the coordinates of <I>P</I> and (<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) those of <I>O</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5899 we have <MATH><I>y</I>/(-<I>y</I><SUB>1</SUB>)=<I>NG</I>/(<I>x</I><SUB>1</SUB>-<I>x</I>-<I>NG</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5900 <p>Therefore (1) for the parabola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5901 <MATH><I>y</I>/(-<I>y</I><SUB>1</SUB>)=1/2<I>p</I>/(<I>x</I><SUB>1</SUB>-<I>x</I>-1/2<I>p</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5902 or <MATH><I>xy</I>-(<I>x</I><SUB>1</SUB>-1/2<I>p</I>)<I>y</I>-<I>y</I><SUB>1</SUB>.1/2<I>p</I>=0</MATH>; (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5903 <p>(2) in the ellipse or hyperbola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5904 <MATH><I>xy</I>(1&mnplus;<I>b</I><SUP>2</SUP>/<I>a</I><SUP>2</SUP>)-<I>x</I><SUB>1</SUB><I>y</I>±<I>b</I><SUP>2</SUP>/<I>a</I><SUP>2</SUP>.<I>y</I><SUB>1</SUB><I>x</I>=0</MATH>. (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5905 <p>The intersections of these rectangular hyperbolas respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5906 <pb n=167><head>THE <I>CONICS</I>, BOOKS V, VI</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5907 tively with the conics give the points at which the normals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5908 passing through <I>O</I> are normals.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5909 <p>Pappus criticizes the use of the rectangular hyperbola in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5910 the case of the parabola as an unnecessary resort to a ‘<I>solid</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5911 locus’; the meaning evidently is that the same points of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5912 intersection can be got by means of a certain circle taking
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5913 the place of the rectangular hyperbola. We can, in fact, from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5914 the equation (1) above combined with <MATH><I>y</I><SUP>2</SUP>=<I>px</I></MATH>, obtain the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5915 circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5916 <MATH>(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>)-(<I>x</I><SUB>1</SUB>+1/2<I>p</I>)<I>x</I>-1/2<I>y</I><SUB>1</SUB><I>y</I>=0</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5917 <p>The Book concludes with other propositions about maxima
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5918 and minima. In particular V. 68-71 compare the lengths of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5919 tangents <I>TQ, TQ</I>′, where <I>Q</I> is nearer to the axis than <I>Q</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5920 V. 72, 74 compare the lengths of two normals from a point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5921 <I>O</I> from which only two can be drawn and the lengths of other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5922 straight lines from <I>O</I> to the curve; V. 75-7 compare the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5923 lengths of three normals to an ellipse drawn from a point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5924 <I>O</I> below the major axis, in relation to the lengths of other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5925 straight lines from <I>O</I> to the curve.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5926 <p>Book VI is of much less interest. The first part (VI. 1-27)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5927 relates to equal (i.e. congruent) or similar conics and segments
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5928 of conics; it is naturally preceded by some definitions includ-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5929 ing those of ‘equal’ and ‘similar’ as applied to conics and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5930 segments of conics. Conics are said to be similar if, the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5931 number of ordinates being drawn to the axis at proportional
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5932 distances from the vertices, all the ordinates are respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5933 proportional to the corresponding abscissae. The definition of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5934 similar segments is the same with diameter substituted for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5935 axis, and with the additional condition that the angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5936 between the base and diameter in each are equal. Two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5937 parabolas are equal if the ordinates to a diameter in each are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5938 inclined to the respective diameters at equal angles and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5939 corresponding parameters are equal; two ellipses or hyper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5940 bolas are equal if the ordinates to a diameter in each are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5941 equally inclined to the respective diameters and the diameters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5942 as well as the corresponding parameters are equal (VI. 1. 2).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5943 Hyperbolas or ellipses are similar when the ‘figure’ on a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5944 diameter of one is similar (instead of equal) to the ‘figure’ on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5945 a diameter of the other, and the ordinates to the diameters in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5946 <pb n=168><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5947 each make equal angles with them; all parabolas are similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5948 (VI. 11, 12, 13). No conic of one of the three kinds (para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5949 bolas, hyperbolas or ellipses) can be equal or similar to a conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5950 of either of the other two kinds (VI. 3, 14, 15). Let <I>QPQ</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5951 <I>qpq</I>′ be two segments of similar conics in which <I>QQ</I>′, <I>qq</I>′ are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5952 the bases and <I>PV, pv</I> are the diameters bisecting them; then,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5953 if <I>PT, pt</I> be the tangents at <I>P, p</I> and meet the axes at <I>T, t</I> at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5954 equal angles, and if <MATH><I>PV</I>:<I>PT</I>=<I>pv</I>:<I>pt</I></MATH>, the segments are similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5955 and similarly situated, and conversely (VI. 17, 18). If two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5956 ordinates be drawn to the axes of two parabolas, or the major or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5957 conjugate axes of two similar central conics, as <I>PN, P</I>′<I>N</I>′ and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5958 <I>pn, p</I>′<I>n</I>′ respectively, such that the ratios <I>AN</I>:<I>an</I> and <I>AN</I>′:<I>an</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5959 are each equal to the ratio of the respective <I>latera recta</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5960 segments <I>PP</I>′, <I>pp</I>′ will be similar; also <I>PP</I>′ will not be similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5961 to any segment in the other conic cut off by two ordinates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5962 other than <I>pn, p</I>′<I>n</I>′, and conversely (VI. 21, 22). If any cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5963 be cut by two parallel planes making hyperbolic or elliptic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5964 sections, the sections will be similar but not equal (VI. 26, 27).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5965 <p>The remainder of the Book consists of problems of con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5966 struction; we are shown how in a given right cone to find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5967 a parabolic, hyperbolic or elliptic section equal to a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5968 parabola, hyperbola or ellipse, subject in the case of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5969 hyperbola to a certain <G>diorismo/s</G> or condition of possibility
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5970 (VI. 28-30); also how to find a right cone similar to a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5971 cone and containing a given parabola, hyperbola or ellipse as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5972 a section of it, subject again in the case of the hyperbola to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5973 a certain <G>diorismo/s</G> (VI. 31-3). These problems recall the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5974 somewhat similar problems in I. 51-9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5975 <p>Book VII begins with three propositions giving expressions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5976 for <MATH><I>AP</I><SUP>2</SUP>(=<I>AN</I><SUP>2</SUP>+<I>PN</I><SUP>2</SUP>)</MATH> in the same form as those for <I>PN</I><SUP>2</SUP> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5977 the statement of the ordinary property. In the parabola <I>AH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5978 is measured along the axis produced (i.e. in the opposite direc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5979 tion to <I>AN</I>) and of length equal to the <I>latus rectum</I>, and it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5980 proved that, for any point <MATH><I>P, AP</I><SUP>2</SUP>=<I>AN.NH</I></MATH> (VII. 1). In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5981 the case of the central conics <I>AA</I>′ is divided at <I>H</I>, internally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5982 for the hyperbola and externally for the ellipse (<I>AH</I> being the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5983 segment adjacent to <I>A</I>) so that <MATH><I>AH</I>:<I>A</I>′<I>H</I>=<I>p</I>:<I>AA</I>′</MATH>, where <I>p</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5984 is the parameter corresponding to <I>AA</I>′, or <MATH><I>p</I>=<I>BB</I>′<SUP>2</SUP>/<I>AA</I>′</MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5985 it is proved that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5986 <MATH><I>AP</I><SUP>2</SUP>:<I>AN.NH</I>=<I>AA</I>′:<I>A</I>′<I>H</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5987 <pb n=169><head>THE <I>CONICS</I>, BOOKS VI, VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5988 The same is true if <I>AA</I>′ is the minor axis of an ellipse and <I>p</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5989 the corresponding parameter (VII. 2, 3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5990 <p>If <I>AA</I>′ be divided at <I>H</I>′ as well as <I>H</I> (internally for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5991 hyperbola and externally for the ellipse) so that <I>H</I> is adjacent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5992 to <I>A</I> and <I>H</I>′ to <I>A</I>′, and if <MATH><I>A</I>′<I>H</I>:<I>AH</I>=<I>AH</I>′:<I>A</I>′<I>H</I>′=<I>AA</I>′:<I>p</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5993 the lines <I>AH, A</I>′<I>H</I>′ (corresponding to <I>p</I> in the proportion) are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5994 called by Apollonius <I>homologues</I>, and he makes considerable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5995 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5996 use of the auxiliary points <I>H, H</I>′ in later propositions from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5997 VII. 6 onwards. Meantime he proves two more propositions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5998 which, like VII. 1-3, are by way of lemmas. First, if <I>CD</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	5999 the semi-diameter parallel to the tangent at <I>P</I> to a central
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6000 conic, and if the tangent meet the axis <I>AA</I>′ in <I>T</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6001 <MATH><I>PT</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=<I>NT</I>:<I>CN</I></MATH>. (VII. 4.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6002 Draw <I>AE, TF</I> at right angles to <I>CA</I> to meet <I>CP</I>, and let <I>AE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6003 meet <I>PT</I> in <I>O.</I> Then, if <I>p</I>′ be the parameter of the ordinates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6004 to <I>CP</I>, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6005 <MATH>1/2<I>p</I>′:<I>PT</I>=<I>OP</I>:<I>PE</I> (I. 49, 50.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6006 =<I>PT</I>:<I>PF</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6007 or <MATH>1/2<I>p</I>′.<I>PF</I>=<I>PT</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6008 <p>Therefore <MATH><I>PT</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=1/2<I>p</I>′.<I>PF</I>:1/2<I>p</I>′.<I>CP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6009 =<I>PF</I>:<I>CP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6010 =<I>NT</I>:<I>CN</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6011 <pb n=170><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6012 <p>Secondly, Apollonius proves that, if <I>PN</I> be a principal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6013 ordinate in a parabola, <I>p</I> the principal parameter, <I>p</I>′ the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6014 parameter of the ordinates to the diameter through <I>P</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6015 <MATH><I>p</I>′=<I>p</I>+4<I>AN</I></MATH> (VII. 5); this is proved by means of the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6016 property as VII. 4, namely <MATH>1/2<I>p</I>′:<I>PT</I>=<I>OP</I>:<I>PE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6017 <p>Much use is made in the remainder of the Book of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6018 points <I>Q</I> and <I>M</I>, where <I>AQ</I> is drawn parallel to the conjugate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6019 diameter <I>CD</I> to meet the curve in <I>Q</I>, and <I>M</I> is the foot of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6020 the principal ordinate at <I>Q</I>; since the diameter <I>CP</I> bisects
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6021 both <I>AA</I>′ and <I>QA</I>, it follows that <I>A</I>′<I>Q</I> is parallel to <I>CP.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6022 Many ratios between functions of <I>PP</I>′, <I>DD</I>′ are expressed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6023 terms of <I>AM, A</I>′<I>M, MH, MH</I>′, <I>AH, A</I>′<I>H</I>, &c. The first pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6024 positions of the Book proper (VII. 6, 7) prove, for instance,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6025 that <MATH><I>PP</I>′<SUP>2</SUP>:<I>DD</I>′<SUP>2</SUP>=<I>MH</I>′:<I>MH</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6026 <p>For <MATH><I>PT</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=<I>NT</I>:<I>CN</I>=<I>AM</I>:<I>A</I>′<I>M</I></MATH>, by similar triangles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6027 <p>Also <MATH><I>CP</I><SUP>2</SUP>:<I>PT</I><SUP>2</SUP>=<I>A</I>′<I>Q</I><SUP>2</SUP>:<I>AQ</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6028 <p>Therefore, <I>ex aequali</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6029 <MATH><I>CP</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=(<I>AM</I>:<I>A</I>′<I>M</I>)X(<I>A</I>′<I>Q</I><SUP>2</SUP>:<I>AQ</I><SUP>2</SUP>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6030 =(<I>AM</I>:<I>A</I>′<I>M</I>)X(<I>A</I>′<I>Q</I><SUP>2</SUP>:<I>A</I>′<I>M.MH</I>′)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6031 X(<I>A</I>′<I>M.MH</I>′:<I>AM.MH</I>)X(<I>AM.MH</I>:<I>AQ</I><SUP>2</SUP>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6032 =(<I>AM</I>:<I>A</I>′<I>M</I>)X(<I>AA</I>′:<I>AH</I>′)X(<I>A</I>′<I>M</I>:<I>AM</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6033 X(<I>MH</I>′:<I>MH</I>)X(<I>A</I>′<I>H</I>:<I>AA</I>′)</MATH>, by aid of VII. 2, 3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6034 <p>Therefore <MATH><I>PP</I>′<SUP>2</SUP>:<I>DD</I>′<SUP>2</SUP>=<I>MH</I>′:<I>MH</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6035 <p>Next (VII. 8, 9, 10, 11) the following relations are proved,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6036 namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6037 <p>(1) <MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′±<I>DD</I>′)<SUP>2</SUP>=<I>A</I>′<I>H.MH</I>′:{<I>MH</I>′±√(<I>MH.MH</I>′)}<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6038 <p>(2) <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′.<I>DD</I>′=<I>A</I>′<I>H</I>:√(<I>MH.MH</I>′)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6039 <p>(3) <MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP>)=<I>A</I>′<I>H</I>:<I>MH</I>±<I>MH</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6040 <p>The steps by which these results are obtained are as follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6041 <p>First, <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>=<I>A</I>′<I>H</I>:<I>MH</I>′ (<G>a</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6042 =<I>A</I>′<I>H.MH</I>′:<I>MH</I>′<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6043 <p>(This is proved thus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6044 <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>=<I>CA</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6045 =<I>CN.CT</I>:<I>CP</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6046 =<I>A</I>′<I>M.A</I>′<I>A</I>:<I>A</I>′<I>Q</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6047 <pb n=171><head>THE <I>CONICS</I>, BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6048 <p>But <MATH><I>A</I>′<I>Q</I><SUP>2</SUP>:<I>A</I>′<I>M.MH</I>′=<I>AA</I>′:<I>AH</I>′ (VII. 2, 3)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6049 =<I>AA</I>′:<I>A</I>′<I>H</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6050 =<I>A</I>′<I>M.AA</I>′:<I>A</I>′<I>M.A</I>′<I>H</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6051 so that, alternately,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6052 <MATH><I>A</I>′<I>M.AA</I>′:<I>A</I>′<I>Q</I><SUP>2</SUP>=<I>A</I>′<I>M.A</I>′<I>H</I>:<I>A</I>′<I>M.MH</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6053 =<I>A</I>′<I>H</I>:<I>MH</I>′</MATH>.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6054 <p>Next, <MATH><I>PP</I>′<SUP>2</SUP>:<I>DD</I>′<SUP>2</SUP>=<I>MH</I>′:<I>MH</I>, as above, (<G>b</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6055 =<I>MH</I>′<SUP>2</SUP>:<I>MH.MH</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6056 whence <MATH><I>PP</I>′:<I>DD</I>′=<I>MH</I>′:√(<I>MH.MH</I>′)</MATH>, (<G>g</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6057 and <MATH><I>PP</I>′<SUP>2</SUP>:(<I>PP</I>′±<I>DD</I>′)<SUP>2</SUP>=<I>MH</I>′<SUP>2</SUP>:{<I>MH</I>′±√(<I>MH.MH</I>′)}<SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6058 <p>(1) above follows from this relation and (<G>a</G>) <I>ex aequali</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6059 <p>(2) follows from (<G>a</G>) and (<G>g</G>) <I>ex aequali</I>, and (3) from (<G>a</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6060 and (<G>b</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6061 <p>We now obtain immediately the important proposition that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6062 <MATH><I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP></MATH> is constant, whatever be the position of <I>P</I> on an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6063 ellipse or hyperbola (the upper sign referring to the ellipse),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6064 and is equal to <MATH><I>AA</I>′<SUP>2</SUP>±<I>BB</I>′<SUP>2</SUP></MATH> (VII. 12, 13, 29, 30).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6065 <p>For <MATH><I>AA</I>′<SUP>2</SUP>:<I>BB</I>′<SUP>2</SUP>=<I>AA</I>′:<I>p</I>=<I>A</I>′<I>H</I>:<I>AH</I>=<I>A</I>′<I>H</I>:<I>A</I>′<I>H</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6066 by construction;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6067 therefore <MATH><I>AA</I>′<SUP>2</SUP>:<I>AA</I>′<SUP>2</SUP>±<I>BB</I>′<SUP>2</SUP>=<I>A</I>′<I>H</I>:<I>HH</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6068 also, from (<G>a</G>) above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6069 <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>=<I>A</I>′<I>H</I>:<I>MH</I>′</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6070 and, by means of (<G>b</G>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6071 <MATH><I>PP</I>′<SUP>2</SUP>:(<I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP>)=<I>MH</I>′:<I>MH</I>′±<I>MH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6072 =<I>MH</I>′:<I>HH</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6073 <p><I>Ex aequali</I>, from the last two relations, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6074 <MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP>)=<I>A</I>′<I>H</I>:<I>HH</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6075 =<I>AA</I>′<SUP>2</SUP>:<I>AA</I>′<SUP>2</SUP>±<I>BB</I>′<SUP>2</SUP></MATH>, from above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6076 whence <MATH><I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP>=<I>AA</I>′<SUP>2</SUP>±<I>BB</I>′<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6077 <pb n=172><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6078 <p>A number of other ratios are expressed in terms of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6079 straight lines terminating at <I>A, A</I>′, <I>H, H</I>′, <I>M, M</I>′ as follows
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6080 (VII. 14-20).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6081 <p>In the ellipse <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>-<I>DD</I>′<SUP>2</SUP>=<I>A</I>′<I>H</I>:2<I>CM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6082 and in the hyperbola or ellipse (if <I>p</I> be the parameter of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6083 ordinates to <I>PP</I>′)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6084 <MATH><I>AA</I>′<SUP>2</SUP>:<I>p</I><SUP>2</SUP>=<I>A</I>′<I>H.MH</I>′:<I>MH</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6085 <MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′±<I>p</I>)<SUP>2</SUP>=<I>A</I>′<I>H.MH</I>′:(<I>MH</I>±<I>MH</I>′)<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6086 <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′.<I>p</I>=<I>A</I>′<I>H</I>:<I>MH</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6087 and <MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′<SUP>2</SUP>±<I>p</I><SUP>2</SUP>)=<I>A</I>′<I>H.MH</I>′:(<I>MH</I>′<SUP>2</SUP>±<I>MH</I><SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6088 <p>Apollonius is now in a position, by means of all these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6089 relations, resting on the use of the auxiliary points <I>H, H</I>′, <I>M</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6090 to compare different functions of any conjugate diameters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6091 with the same functions of the axes, and to show how the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6092 former vary (by way of increase or diminution) as <I>P</I> moves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6093 away from <I>A.</I> The following is a list of the functions com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6094 pared, where for brevity I shall use <I>a, b</I> to represent <I>AA</I>′, <I>BB</I>′;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6095 <I>a</I>′, <I>b</I>′ to represent <I>PP</I>′, <I>DD</I>′; and <I>p, p</I>′ to represent the para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6096 meters of the ordinates to <I>AA</I>′, <I>PP</I>′ respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6097 <p>In a hyperbola, according as <I>a</I> > or < <I>b, a</I>′ > or < <I>b</I>′, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6098 ratio <I>a</I>′:<I>b</I>′ decreases or increases as <I>P</I> moves from <I>A</I> on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6099 either side; also, if <MATH><I>a</I>=<I>b</I></MATH>, <MATH><I>a</I>′=<I>b</I>′</MATH> (VII. 21-3); in an ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6100 <MATH><I>a</I>:<I>b</I> > <I>a</I>′:<I>b</I>′</MATH>, and the latter ratio diminishes as <I>P</I> moves from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6101 <I>A</I> to <I>B</I> (VII. 24).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6102 <p>In a hyperbola or ellipse <MATH><I>a</I>+<I>b</I> < <I>a</I>′+<I>b</I>′</MATH>, and <MATH><I>a</I>′+<I>b</I>′</MATH> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6103 hyperbola increases continually as <I>P</I> moves farther from <I>A</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6104 but in the ellipse increases till <I>a</I>′, <I>b</I>′ take the position of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6105 equal conjugate diameters when it is a <I>maximum</I> (VII.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6106 25, 26).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6107 <p>In a hyperbola in which <I>a, b</I> are unequal, or in an ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6108 <MATH><I>a</I>-<I>b</I> > <I>a</I>′-<I>b</I>′</MATH>, and <MATH><I>a</I>′-<I>b</I>′</MATH> diminishes as <I>P</I> moves away from <I>A</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6109 in the hyperbola continually, and in the ellipse till <I>a</I>′, <I>b</I>′ are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6110 the equal conjugate diameters (VII. 27).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6111 <p><MATH><I>ab</I> < <I>a</I>′<I>b</I>′</MATH>, and <I>a</I>′<I>b</I>′ increases as <I>P</I> moves away from <I>A</I>, in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6112 hyperbola continually, and in the ellipse till <I>a</I>′, <I>b</I>′ coincide with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6113 the equal conjugate diameters (VII. 28).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6114 <p>VII. 31 is the important proposition that, if <I>PP</I>′, <I>DD</I>′ are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6115 <pb n=173><head>THE <I>CONICS</I>, BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6116 conjugate diameters in an ellipse or conjugate hyperbolas, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6117 if the tangents at their extremities form the parallelogram
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6118 <I>LL</I>′<I>MM</I>′, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6119 the parallelogram <MATH><I>LL</I>′<I>MM</I>′=rect.<I>AA</I>′.<I>BB</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6120 <p>The proof is interesting. Let the tangents at <I>P, D</I> respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6121 tively meet the major or transverse axis in <I>T, T</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6122 <p>Now (by VII. 4) <MATH><I>PT</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=<I>NT</I>:<I>CN</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6123 therefore <MATH>2&utri;<I>CPT</I>:2&utri;<I>T</I>′<I>DC</I>=<I>NT</I>:<I>CN</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6124 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6125 <p>But <MATH>2&utri;<I>CPT</I>:(<I>CL</I>)=<I>PT</I>:<I>CD</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6126 =<I>CP</I>:<I>DT</I>′, by similar triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6127 =(<I>CL</I>):2&utri;<I>T</I>′<I>DC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6128 <p>That is, (<I>CL</I>) is a mean proportional between 2&utri;<I>CPT</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6129 2&utri;<I>T</I>′<I>DC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6130 <p>Therefore, since <MATH>√(<I>NT.CN</I>)</MATH> is a mean proportional between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6131 <I>NT</I> and <I>CN</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6132 <pb n=174><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6133 <MATH>2&utri;<I>CPT</I>:(<I>CL</I>)=√(<I>CN.NT</I>):<I>CN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6134 =<I>PN.CA</I>/<I>CB</I>:<I>CN</I> (I. 37, 39)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6135 =<I>PN.CT</I>:<I>CT.CN.CB</I>/<I>CA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6136 =2&utri;<I>CPT</I>:<I>CA.CB</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6137 therefore <MATH>(<I>CL</I>)=<I>CA.CB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6138 <p>The remaining propositions of the Book trace the variations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6139 of different functions of the conjugate diameters, distinguishing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6140 the maximum values, &c. The functions treated are the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6141 following:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6142 <p><I>p</I>′, the parameter of the ordinates to <I>PP</I>′ in the hyperbola,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6143 according as <I>AA</I>′ is (1) not less than <I>p</I>, the parameter corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6144 sponding to <I>AA</I>′, (2) less than <I>p</I> but not less than 1/2<I>p</I>, (3) less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6145 than 1/2<I>p</I> (VII. 33-5).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6146 <p><MATH><I>PP</I>′-<I>p</I>′</MATH>, as compared with <MATH><I>AA</I>′-<I>p</I></MATH> in the hyperbola (VII. 36)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6147 or the ellipse (VII. 37).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6148 <p><MATH><I>PP</I>′+<I>p</I>′</MATH> ”” <MATH><I>AA</I>′+<I>p</I></MATH> in the hyperbola (VII.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6149 38-40) or the ellipse (VII. 41).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6150 <p><MATH><I>PP</I>′.<I>p</I>′</MATH> ”” <MATH><I>AA</I>′.<I>p</I></MATH> in the hyperbola (VII. 42)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6151 or the ellipse (VII. 43).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6152 <p><MATH><I>PP</I>′<SUP>2</SUP>+<I>p</I>′<SUP>2</SUP></MATH> ”” <MATH><I>AA</I>′<SUP>2</SUP>+<I>p</I><SUP>2</SUP></MATH> in the hyperbola, accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6153 ing as (1) <I>AA</I>′ is not less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6154 <I>p</I>, or (2) <MATH><I>AA</I>′ < <I>p</I></MATH>, but <I>AA</I>′<SUP>2</SUP> not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6155 less than <MATH>1/2(<I>AA</I>′-<I>p</I>)<SUP>2</SUP></MATH>, or (3)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6156 <MATH><I>AA</I>′<SUP>2</SUP> < 1/2(<I>AA</I>′-<I>p</I>)<SUP>2</SUP></MATH> (VII. 44-6).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6157 <p><MATH><I>PP</I>′<SUP>2</SUP>+<I>p</I>′<SUP>2</SUP></MATH> ”” <MATH><I>AA</I>′<SUP>2</SUP>+<I>p</I><SUP>2</SUP></MATH> in the ellipse, according
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6158 as <I>AA</I>′<SUP>2</SUP> is not greater, or is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6159 greater, than <MATH>(<I>AA</I>′+<I>p</I>)<SUP>2</SUP></MATH> (VII.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6160 47, 48).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6161 <p><MATH><I>PP</I>′<SUP>2</SUP>-<I>p</I>′<SUP>2</SUP></MATH> ”” <MATH><I>AA</I>′<SUP>2</SUP>-<I>p</I><SUP>2</SUP></MATH> in the hyperbola, accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6162 ing as <I>AA</I>′ > or < <I>p</I> (VII.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6163 49, 50).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6164 <p><MATH><I>PP</I>′<SUP>2</SUP>-<I>p</I>′<SUP>2</SUP></MATH> ”” <MATH><I>AA</I>′<SUP>2</SUP>-<I>p</I><SUP>2</SUP></MATH> or <MATH><I>BB</I>′<SUP>2</SUP>-<I>p</I><SUB>b</SUB><SUP>2</SUP></MATH> in the ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6165 according as <I>PP</I>′ > or < <I>p</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6166 (VII. 51).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6167 <pb n=175><head>THE <I>CONICS</I>, BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6168 <p>As we have said, Book VIII is lost. The nature of its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6169 contents can only be conjectured from Apollonius's own
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6170 remark that it contained determinate conic problems for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6171 which Book VII was useful, particularly in determining
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6172 limits of possibility. Unfortunately, the lemmas of Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6173 do not enable us to form any clearer idea. But it is probable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6174 enough that the Book contained a number of problems having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6175 for their object the finding of conjugate diameters in a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6176 conic such that certain functions of their lengths have given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6177 values. It was on this assumption that Halley attempted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6178 a restoration of the Book.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6179 <p>If it be thought that the above account of the <I>Conics</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6180 disproportionately long for a work of this kind, it must be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6181 remembered that the treatise is a great classic which deserves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6182 to be more known than it is. What militates against its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6183 being read in its original form is the great extent of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6184 exposition (it contains 387 separate propositions), due partly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6185 to the Greek habit of proving particular cases of a general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6186 proposition separately from the proposition itself, but more to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6187 the cumbrousness of the enunciations of complicated proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6188 tions in general terms (without the help of letters to denote
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6189 particular points) and to the elaborateness of the Euclidean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6190 form, to which Apollonius adheres throughout.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6191 <C>Other works by Apollonius.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6192 <p>Pappus mentions and gives a short indication of the con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6193 tents of six other works of Apollonius which formed part of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6194 <I>Treasury of Analysis.</I><note>Pappus, vii, pp. 640-8, 660-72.</note> Three of these should be mentioned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6195 in close connexion with the <I>Conics.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6196 <C><I>(a) On the Cutting-off of a Ratio (<G>lo/gou a)potomh/</G>)</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6197 two Books.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6198 <p>This work alone of the six mentioned has survived, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6199 that only in the Arabic; it was published in a Latin trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6200 lation by Edmund Halley in 1706. It deals with the general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6201 problem, ‘<I>Given two straight lines, parallel to one another or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6202 intersecting, and a fixed point on each line, to draw through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6203 <pb n=176><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6204 a given point a straight line which shall cut off segments from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6205 each line (measured from the fixed points) bearing a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6206 ratio to one another.</I>’ Thus, let <I>A, B</I> be fixed points on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6207 two given straight lines <I>AC, BK</I>, and let <I>O</I> be the given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6208 point. It is required to draw through <I>O</I> a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6209 cutting the given straight lines in points <I>M, N</I> respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6210 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6211 such that <I>AM</I> is to <I>BN</I> in a given ratio. The two Books of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6212 the treatise discussed the various possible cases of this pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6213 blem which arise according to the relative positions of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6214 given straight lines and points, and also the necessary condi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6215 tions and limits of possibility in cases where a solution is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6216 always possible. The first Book begins by supposing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6217 given lines to be parallel, and discusses the different cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6218 which arise; Apollonius then passes to the cases in which the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6219 straight lines intersect, but one of the given points, <I>A</I> or <I>B</I>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6220 at the intersection of the two lines. Book II proceeds to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6221 general case shown in the above figure, and first proves that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6222 the general case can be reduced to the case in Book I where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6223 one of the given points, <I>A</I> or <I>B</I>, is at the intersection of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6224 two lines. The reduction is easy. For join <I>OB</I> meeting <I>AC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6225 in <I>B</I>′, and draw <I>B</I>′<I>N</I>′ parallel to <I>BN</I> to meet <I>OM</I> in <I>N</I>′. Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6226 the ratio <MATH><I>B</I>′<I>N</I>′:<I>BN</I></MATH>, being equal to the ratio <MATH><I>OB</I>′:<I>OB</I></MATH>, is con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6227 stant. Since, therefore, <MATH><I>BN</I>:<I>AM</I></MATH> is a given ratio, the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6228 <MATH><I>B</I>′<I>N</I>′:<I>AM</I></MATH> is also given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6229 <p>Apollonius proceeds in all cases by the orthodox method of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6230 analysis and synthesis. Suppose the problem solved and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6231 <I>OMN</I> drawn through <I>O</I> in such a way that <MATH><I>B</I>′<I>N</I>′:<I>AM</I></MATH> is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6232 given ratio =<G>l</G>, say.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6233 <pb n=177><head><I>ON THE CUTTING-OFF OF A RATIO</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6234 <p>Draw <I>OC</I> parallel to <I>BN</I> or <I>B′N′</I> to meet <I>AM</I> in <I>C.</I> Take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6235 <I>D</I> on <I>AM</I> such that <MATH><I>OC</I>:<I>AD</I> = <G>l</G> = <I>B′N′</I>:<I>AM</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6236 <p>Then <MATH><I>AM</I>:<I>AD</I> = <I>B′N′</I>:<I>OC</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6237 <MATH>= <I>B′M</I>:<I>CM</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6238 therefore <MATH><I>MD</I>:<I>AD</I> = <I>B′C</I>:<I>CM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6239 or <MATH><I>CM.MD</I> = <I>AD.B′C</I></MATH>, a given rectangle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6240 <p>Hence the problem is reduced to one of <I>applying to CD a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6241 rectangle</I> (<I>CM.MD</I>) <I>equal to a given rectangle</I> (<I>AD.B′C</I>) <I>but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6242 falling short by a square figure.</I> In the case as drawn, whatever
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6243 be the value of <G>l</G>, the solution is always possible because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6244 the given rectangle <I>AD.CB′</I> is always less than <I>CA.AD,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6245 therefore always less than 1/4<I>CD</I><SUP>2</SUP>; one of the positions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6246 <I>M</I> falls between <I>A</I> and <I>D</I> because <MATH><I>CM.MD</I><<I>CA.AD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6247 <p>The proposition III. 41 of the <I>Conics</I> about the intercepts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6248 made on two tangents to a parabola by a third tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6249 (pp. 155-6 above) suggests an obvious application of our problem.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6250 We had, with the notation of that proposition,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6251 <MATH><I>Pr</I>:<I>rq</I> = <I>rQ</I>:<I>Qp</I> = <I>qp</I>:<I>pR</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6252 Suppose that the two tangents <I>qP, qR</I> are given as fixed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6253 tangents with their points of contact <I>P, R.</I> Then we can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6254 draw another tangent if we can draw a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6255 intersecting <I>qP, qR</I> in such a way that <MATH><I>Pr</I>:<I>rq</I> = <I>qp</I>:<I>pR</I></MATH> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6256 <MATH><I>Pq</I>:<I>qr</I> = <I>qR</I>:<I>pR</I></MATH>, i.e. <MATH><I>qr</I>:<I>pR</I> = <I>Pq</I>:<I>qR</I></MATH> (a constant ratio);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6257 i.e. we have to draw a straight line such that the intercept by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6258 it on <I>qP</I> measured from <I>q</I> has a given ratio to the intercept
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6259 by it on <I>qR</I> measured from <I>R.</I> This is a particular case of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6260 our problem to which, as a matter of fact, Apollonius devotes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6261 special attention. In the annexed figure the letters have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6262 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6263 same meaning as before, and <I>N′M</I> has to be drawn through <I>O</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6264 such that <MATH><I>B′N′</I>:<I>AM</I> = <G>l</G></MATH>. In this case there are limits to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6265 <pb n=178><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6266 the value of <G>l</G> in order that the solution may be possible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6267 Apollonius begins by stating the limiting case, saying that we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6268 obtain a solution in a special manner in the case where <I>M</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6269 the middle point of <I>CD,</I> so that the rectangle <I>CM.MD</I> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6270 <I>CB′.AD</I> has its maximum value.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6271 <p>The corresponding limiting value of <G>l</G> is determined by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6272 finding the corresponding position of <I>D</I> or <I>M.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6273 <p>We have <MATH><I>B′C</I>:<I>MD</I> = <I>CM</I>:<I>AD</I></MATH>, as before,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6274 <MATH>= <I>B′M</I>:<I>MA</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6275 whence, since <MATH><I>MD</I> = <I>CM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6276 <MATH><I>B′C</I>:<I>B′M</I> = <I>CM</I>:<I>MA</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6277 <MATH>= <I>B′M</I>:<I>B′A</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6278 so that <MATH><I>B′M</I><SUP>2</SUP> = <I>B′C.B′A</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6279 <p>Thus <I>M</I> is found and therefore <I>D</I> also.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6280 <p>According, therefore, as <G>l</G> is less or greater than the particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6281 value of <I>OC</I>:<I>AD</I> thus determined, Apollonius finds no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6282 solution or two solutions.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6283 <p>Further, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6284 <MATH><I>AD</I> = <I>B′A</I> + <I>B′C</I>-(<I>B′D</I> + <I>B′C</I>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6285 <MATH>= <I>B′A</I> + <I>B′C</I>-2<I>B′M</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6286 <MATH>= <I>B′A</I> + <I>B′C</I>-2√(<I>B′A. B′C</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6287 <p>If then we refer the various points to a system of coordinates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6288 in which <I>B′A, B′N′</I> are the axes of <I>x</I> and <I>y,</I> and if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6289 we denote <I>O</I> by (<I>x, y</I>) and the length <I>B′A</I> by <I>h</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6290 <MATH><G>l</G> = <I>OC</I>/<I>AD</I> = <I>y</I>/(<I>h</I> + <I>x</I> - 2√(<I>hx</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6291 <p>If we suppose Apollonius to have used these results for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6292 parabola, he cannot have failed to observe that the limiting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6293 case described is that in which <I>O</I> is on the parabola, while
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6294 <I>N′OM</I> is the tangent at <I>O</I>; for, as above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6295 <MATH><I>B′M</I>:<I>B′A</I> = <I>B′C</I>:<I>B′M</I> = <I>N′O</I>:<I>N′M</I></MATH>, by parallels,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6296 so that <I>B′A, N′M</I> are divided at <I>M, O</I> respectively in the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6297 proportion.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6298 <pb n=179><head><I>ON THE CUTTING-OFF OF A RATIO</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6299 <p>Further, if we put for <G>l</G> the ratio between the lengths of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6300 two fixed tangents, then if <I>h, k</I> be those lengths,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6301 <MATH><I>k</I>/<I>h</I> = <I>y</I>/(<I>h</I>+<I>x</I>-2√(<I>hx</I>)</MATH>,........
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6302 which can easily be reduced to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6303 <MATH>(<I>x</I>/<I>h</I>)<SUP>1/2</SUP> + (<I>y</I>/<I>k</I>)<SUP>1/2</SUP>=1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6304 the equation of the parabola referred to the two fixed tangents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6305 as axes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6306 <C>(<G>b</G>) <I>On the cutting-off of an area</I> (<G>xwri/ou a)potomh/</G>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6307 two Books.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6308 <p>This work, also in two Books, dealt with a similar problem,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6309 with the difference that the intercepts on the given straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6310 lines measured from the given points are required, not to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6311 have a given ratio, but to contain a given rectangle. Halley
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6312 included an attempted restoration of this work in his edition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6313 of the <I>De sectione rationis.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6314 <p>The general case can here again be reduced to the more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6315 special one in which one of the fixed points is at the inter-section
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6316 of the two given straight lines. Using the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6317 figure as before, but with <I>D</I> taking the position shown by (<I>D</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6318 in the figure, we take that point such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6319 <MATH><I>OC.AD</I> = the given rectangle</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6320 <p>We have then to draw <I>ON′M</I> through <I>O</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6321 <MATH><I>B′N′.AM</I> = <I>OC.AD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6322 or <MATH><I>B′N′</I>:<I>OC</I> = <I>AD</I>:<I>AM</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6323 <p>But, by parallels, <MATH><I>B′N′</I>:<I>OC</I> = <I>B′M</I>:<I>CM</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6324 therefore <MATH><I>AM</I>:<I>CM</I> = <I>AD</I>:<I>B′M</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6325 <MATH>= <I>MD</I>:<I>B′C</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6326 so that <MATH><I>B′M.MD</I> = <I>AD.B′C</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6327 <p>Hence, as before, the problem is reduced to an application
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6328 of a rectangle in the well-known manner. The complete
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6329 <pb n=180><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6330 treatment of this problem in all its particular cases with their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6331 <G>diorismoi/</G> could present no difficulty to Apollonius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6332 <p>If the two straight lines are parallel, the solution of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6333 problem gives a means of drawing any number of tangents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6334 to an ellipse when two parallel tangents, their points of contact,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6335 and the length of the parallel semi-diameter are given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6336 (see <I>Conics,</I> III. 42). In the case of the hyperbola (III. 43)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6337 the intercepts made by any tangent on the asymptotes contain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6338 a constant rectangle. Accordingly the drawing of tangents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6339 depends upon the particular case of our problem in which both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6340 fixed points are the intersection of the two fixed lines.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6341 <C>(<G>g</G>) <I>On determinate section</I> (<G>diwrisme/nh tomh/</G>), two Books.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6342 <p>The general problem here is, Given four points <I>A, B, C, D</I> on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6343 a straight line, to determine another point <I>P</I> on the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6344 straight line such that the ratio <I>AP.CP</I>:<I>BP.DP</I> has a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6345 given value. It is clear from Pappus's account<note>Pappus, vii, pp. 642-4.</note> of the contents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6346 of this work, and from his extensive collection of lemmas to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6347 the different propositions in it, that the question was very
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6348 exhaustively discussed. To determine <I>P</I> by means of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6349 equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6350 <MATH><I>AP.CP</I> = <G>l</G>.<I>BP.DP</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6351 where <I>A, B, C, D</I>, <G>l</G> are given, is in itself an easy matter since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6352 the problem can at once be put into the form of a quadratic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6353 equation, and the Greeks would have no difficulty in reducing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6354 it to the usual <I>application of areas.</I> If, however (as we may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6355 fairly suppose), it was intended for application in further
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6356 investigations, the complete, discussion of it would naturally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6357 include not only the finding of a solution, but also the determination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6358 of the limits of possibility and the number of possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6359 solutions for different positions of the point-pairs <I>A, C</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6360 <I>B, D,</I> for the cases in which the points in either pair coincide,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6361 or in which one of the points is infinitely distant, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6362 This agrees with what we find in Pappus, who makes it clear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6363 that, though we do not meet with any express mention of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6364 <I>series</I> of point-pairs determined by the equation for different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6365 values of <G>l</G>, yet the treatise contained what amounts to a com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6366 <pb n=181><head><I>ON DETERMINATE SECTION</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6367 plete <I>Theory of Involution.</I> Pappus says that the separate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6368 cases were dealt with in which the given ratio was that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6369 either (1) the square of one abscissa measured from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6370 required point or (2) the rectangle contained by two such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6371 abscissae to any one of the following: (1) the square of one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6372 abscissa, (2) the rectangle contained by one abscissa and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6373 another separate line of given length independent of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6374 position of the required point, (3) the rectangle contained by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6375 two abscissae. We learn also that maxima and minima were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6376 investigated. From the lemmas, too, we may draw other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6377 conclusions, e.g.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6378 <p>(1) that, in the case where <G>l</G> = 1, or <MATH><I>AP.CP</I> = <I>BP.DP</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6379 Apollonius used the relation <MATH><I>BP</I>:<I>DP</I> = <I>AB.BC</I>:<I>AD.DC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6380 <p>(2) that Apollonius probably obtained a double point <I>E</I> of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6381 involution determined by the point-pairs <I>A, C</I> and <I>B, D</I> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6382 means of the relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6383 <MATH><I>AB.BC</I>:<I>AD.DC</I> = <I>BE</I><SUP>2</SUP>:<I>DE</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6384 <p>A possible application of the problem was the determination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6385 of the points of intersection of the given straight line with a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6386 conic determined as a four-line locus, since <I>A, B, C, D</I> are in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6387 fact the points of intersection of the given straight line with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6388 the four lines to which the locus has reference.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6389 <C>(<G>d</G>) <I>On Contacts</I> or <I>Tangencies</I> (<G>e)pafai/</G>), two Books.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6390 <p>Pappus again comprehends in one enunciation the varieties
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6391 of problems dealt with in the treatise, which we may repro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6392 duce as follows: <I>Given three things, each of which may be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6393 either a point, a straight line or a circle, to draw a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6394 which shall pass through each of the given points</I> (<I>so far as it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6395 is points that are given</I>) <I>and touch the straight lines or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6396 circles.</I><note>Pappus, vii, p. 644, 25-8.</note> The possibilities as regards the different data are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6397 ten. We may have any one of the following: (1) three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6398 points, (2) three straight lines, (3) two points and a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6399 line, (4) two straight lines and a point, (5) two points and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6400 a circle, (6) two circles and a point, (7) two straight lines and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6401 <pb n=182><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6402 a circle, (8) two circles and a straight line, (9) a point, a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6403 and a straight line, (10) three circles. Of these varieties the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6404 first two are treated in Eucl. IV; Book I of Apollonius's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6405 treatise treated of (3), (4), (5), (6), (8), (9), while (7), the case of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6406 two straight lines and a circle, and (10), that of the three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6407 circles, occupied the whole of Book II.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6408 <p>The last problem (10), where the data are three circles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6409 has exercised the ingenuity of many distinguished geometers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6410 including Vieta and Newton. Vieta (1540-1603) set the pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6411 blem to Adrianus Romanus (van Roomen, 1561-1615) who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6412 solved it by means of a hyperbola. Vieta was not satisfied
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6413 with this, and rejoined with his <I>Apollonius Gallus</I> (1600) in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6414 which he solved the problem by plane methods. A solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6415 of the same kind is given by Newton in his <I>Arithmetica
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6416 Universalis</I> (Prob. xlvii), while an equivalent problem is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6417 solved by means of two hyperbolas in the <I>Principia</I>, Lemma
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6418 xvi. The problem is quite capable of a ‘plane’ solution, and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6419 as a matter of fact, it is not difficult to restore the actual
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6420 solution of Apollonius (which of course used the ‘plane’ method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6421 depending on the straight line and circle only), by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6422 the lemmas given by Pappus. Three things are necessary to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6423 the solution. (1) A proposition, used by Pappus elsewhere<note>Pappus, iv, pp. 194-6.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6424 and easily proved, that, if two circles touch internally or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6425 externally, any straight line through the point of contact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6426 divides the circles into segments respectively similar. (2) The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6427 proposition that, given three circles, their six centres of similitude
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6428 (external and internal) lie three by three on four straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6429 lines. This proposition, though not proved in Pappus, was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6430 certainly known to the ancient geometers; it is even possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6431 that Pappus omitted to prove it because it was actually proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6432 by Apollonius in his treatise. (3) An auxiliary problem solved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6433 by Pappus and enunciated by him as follows.<note><I>Ib.</I> vii, p. 848.</note> Given a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6434 <I>ABC,</I> and given three points <I>D, E, F</I> in a straight line, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6435 inflect (the broken line) <I>DAE</I> (to the circle) so as to make <I>BC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6436 in a straight line with <I>CF</I>; in other words, to inscribe in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6437 circle a triangle the sides of which, when produced, pass
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6438 respectively through three given points lying in a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6439 line. This problem is interesting as a typical example of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6440 ancient analysis followed by synthesis. Suppose the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6441 <pb n=183><head><I>ON CONTACTS OR TANGENCIES</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6442 solved, i.e. suppose <I>DA, EA</I> drawn to the circle cutting it in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6443 points <I>B, C</I> such that <I>BC</I> produced passes through <I>F</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6444 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6445 <p>Draw <I>BG</I> parallel to <I>DF</I>; join <I>GC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6446 and produce it to meet <I>DE</I> in <I>H.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6447 <p>Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6448 <MATH>&angle; <I>BAC</I> = &angle; <I>BGC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6449 = &angle; <I>CHF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6450 = supplement of &angle;<I>CHD</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6451 therefore <I>A, D, H, C</I> lie on a circle, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6452 <MATH><I>DE.EH</I> = <I>AE.EC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6453 <p>Now <I>AE.EC</I> is given, being equal to the square on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6454 tangent from <I>E</I> to the circle; and <I>DE</I> is given; therefore <I>HE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6455 is given, and therefore the point <I>H.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6456 <p>But <I>F</I> is also given; therefore the problem is reduced to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6457 drawing <I>HC, FC</I> to meet the circle in such a way that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6458 <I>HC, FC</I> produced meet the circle again in <I>G, B,</I> the straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6459 line <I>BG</I> is parallel to <I>HF</I>: a problem which Pappus has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6460 previously solved.<note>Pappus, vii, pp. 830-2.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6461 <p>Suppose this done, and draw <I>BK</I> the tangent at <I>B</I> meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6462 <I>HF</I> in <I>K</I>. Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6463 <MATH><I>&angle; KBC</I> = <I>&angle;BGC</I>, in the alternate segment,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6464 = <I>&angle;CHF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6465 <p>Also the angle <I>CFK</I> is common to the two triangles <I>KBF,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6466 CHF</I>; therefore the triangles are similar, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6467 <MATH><I>CF</I>:<I>FH</I> = <I>KF</I>:<I>FB</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6468 or <MATH><I>HF.FK</I> = <I>BF.FC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6469 <p>Now <I>BF.FC</I> is given, and so is <I>HF</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6470 therefore <I>FK</I> is given, and therefore <I>K</I> is given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6471 <p>The synthesis is as follows. Take a point <I>H</I> on <I>DE</I> such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6472 that <I>DE.EH</I> is equal to the square on the tangent from <I>E</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6473 the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6474 <p>Next take <I>K</I> on <I>HF</I> such that <I>HF.FK</I> = the square on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6475 tangent from <I>F</I> to the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6476 <p>Draw the tangent to the circle from <I>K,</I> and let <I>B</I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6477 point of contact. Join <I>BF</I> meeting the circle in <I>C</I>, and join
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6478 <pb n=184><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6479 <I>HC</I> meeting the circle again in <I>G.</I> It is then easy to prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6480 that <I>BG</I> is parallel to <I>DF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6481 <p>Now join <I>EC</I>, and produce it to meet the circle again at <I>A</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6482 join <I>AB.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6483 <p>We have only to prove that <I>AB, BD</I> are in one straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6484 <p>Since <I>DE.EH</I> = <I>AE.EC,</I> the points <I>A, D, H, C</I> are concyclic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6485 <p>Now the angle <I>CHF,</I> which is the supplement of the angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6486 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6487 <I>CHD,</I> is equal to the angle <I>BGC,</I> and therefore to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6488 angle <I>BAC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6489 <p>Therefore the angle <I>BAC</I> is equal to the supplement of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6490 angle <I>DHC,</I> so that the angle <I>BAC</I> is equal to the angle <I>DAC,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6491 and <I>AB, BD</I> are in a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6492 <p>The problem of Apollonius is now easy. We will take the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6493 case in which the required circle touches all the three given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6494 circles externally as shown in the figure. Let the radii of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6495 <pb n=185><head><I>ON CONTACTS OR TANGENCIES</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6496 given circles be <I>a, b, c</I> and their centres <I>A, B, C.</I> Let <I>D, E, F</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6497 be the external centres of similitude so that <I>BD</I>:<I>DC</I>=<I>b</I>:<I>c</I>, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6498 <p>Suppose the problem solved, and let <I>P, Q, R</I> be the points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6499 of contact. Let <I>PQ</I> produced meet the circles with centres
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6500 <I>A, B</I> again in <I>K, L.</I> Then, by the proposition (1) above, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6501 segments <I>KGP, QHL</I> are both similar to the segment <I>PYQ</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6502 therefore they are similar to one another. It follows that <I>PQ</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6503 produced beyond <I>L</I> passes through <I>F.</I> Similarly <I>QR, PR</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6504 produced pass respectively through <I>D, E.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6505 <p>Let <I>PE, QD</I> meet the circle with centre <I>C</I> again in <I>M, N.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6506 Then, the segments <I>PQR, RNM</I> being similar, the angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6507 <I>PQR, RNM</I> are equal, and therefore <I>MN</I> is parallel to <I>PQ.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6508 Produce <I>NM</I> to meet <I>EF</I> in <I>V.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6509 <p>Then <MATH><I>EV</I>:<I>EF</I> = <I>EM</I>:<I>EP</I> = <I>EC</I>:<I>EA</I> = <I>c</I>:<I>a</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6510 therefore the point <I>V</I> is given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6511 <p>Accordingly the problem reduces itself to this: Given three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6512 points <I>V, E, D</I> in a straight line, it is required to draw <I>DR, ER</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6513 to a point <I>R</I> on the circle with centre <I>C</I> so that, if <I>DR, ER</I> meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6514 the circle again in <I>N, M, NM</I> produced shall pass through <I>V.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6515 This is the problem of Pappus just solved.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6516 <p>Thus <I>R</I> is found, and <I>DR, ER</I> produced meet the circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6517 with centres <I>B</I> and <I>A</I> in the other required points <I>Q, P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6518 respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6519 <C>(<G>e</G>) <I>Plane loci,</I> two Books.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6520 <p>Pappus gives a pretty full account of the contents of this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6521 work, which has sufficed to enable restorations of it to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6522 be made by three distinguished geometers, Fermat, van
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6523 Schooten, and (most completely) by Robert Simson. Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6524 prefaces his account by a classification of loci on two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6525 different plans. Under the first classification loci are of three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6526 kinds: (1) <G>e)fektikoi/</G>, <I>holding-in</I> or <I>fixed</I>; in this case the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6527 locus of a point is a point, of a line a line, and of a solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6528 a solid, where presumably the line or solid can only move on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6529 itself so that it does not change its position: (2) <G>diexo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6530 dikoi/</G>, <I>passing-along</I>: this is the ordinary sense of a locus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6531 where the locus of a point is a line, and of a line a solid:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6532 (3) <G>a)nastrofikoi/</G>, <I>moving backwards and forwards,</I> as it were,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6533 in which sense a plane may be the locus of a point and a solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6534 <pb n=186><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6535 of a line.<note>Pappus, vii, pp. 660. 18-662. 5.</note> The second classification is the familiar division into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6536 <I>plane, solid,</I> and <I>linear</I> loci, <I>plane</I> loci being straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6537 and circles only, <I>solid</I> loci conic sections only, and <I>linear</I> loci
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6538 those which are not straight lines nor circles nor any of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6539 conic sections. The loci dealt with in our treatise are accordingly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6540 all straight lines or circles. The proof of the propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6541 is of course enormously facilitated by the use of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6542 Cartesian coordinates, and many of the loci are really the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6543 geometrical equivalent of fundamental theorems in analytical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6544 or algebraical geometry. Pappus begins with a composite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6545 enunciation, including a number of propositions, in these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6546 terms, which, though apparently confused, are not difficult
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6547 to follow out:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6548 <p>‘If two straight lines be drawn, from one given point or from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6549 two, which are (<I>a</I>) in a straight line or (<I>b</I>) parallel or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6550 (<I>c</I>) include a given angle, and either (<G>a</G>) bear a given ratio to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6551 one another or (<G>b</G>) contain a given rectangle, then, if the locus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6552 of the extremity of one of the lines is a plane locus given in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6553 position, the locus of the extremity of the other will also be a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6554 plane locus given in position, which will sometimes be of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6555 same kind as the former, sometimes of the other kind, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6556 will sometimes be similarly situated with reference to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6557 straight line, and sometimes contrarily, according to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6558 particular differences in the suppositions.’<note><I>Ib.</I> vii, pp. 662. 25-664. 7.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6559 <p>(The words ‘with reference to <I>the straight line</I>’ are obscure, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6560 the straight line is presumably some obvious straight line in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6561 each figure, e.g., when there are two given points, the straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6562 line joining them.) After quoting three obvious loci ‘added
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6563 by Charmandrus’, Pappus gives three loci which, though containing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6564 an unnecessary restriction in the third case, amount
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6565 to the statement that any equation of the first degree between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6566 coordinates inclined at fixed angles to (<I>a</I>) two axes perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6567 or oblique, (<I>b</I>) to any number of axes, represents a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6568 straight line. The enunciations (5-7) are as follows.<note><I>Ib.,</I> pp. 664. 20-666. 6.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6569 <p>5. ‘If, when a straight line is given in magnitude and is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6570 moved so as always to be parallel to a certain straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6571 given in position, one of the extremities (of the moving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6572 straight line), lies on a straight line given in position, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6573 <pb n=187><head><I>PLANE LOCI</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6574 other extremity will also lie on a straight line given in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6575 position.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6576 <p>(That is, <MATH><I>x</I> = <I>a</I> or <I>y</I> = <I>b</I></MATH> in Cartesian coordinates represents a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6577 straight line.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6578 <p>6. ‘If from any point straight lines be drawn to meet at given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6579 angles two straight lines either parallel or intersecting, and if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6580 the straight lines so drawn have a given ratio to one another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6581 or if the sum of one of them and a line to which the other has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6582 a given ratio be given (in length), then the point will lie on a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6583 straight line given in position.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6584 <p>(This includes the equivalent of saying that, if <I>x, y</I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6585 coordinates of the point, each of the equations <MATH><I>x</I> = <I>my,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6586 x</I> + <I>my</I> = <I>c</I></MATH> represents a straight line.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6587 <p>7. ‘If any number of straight lines be given in position, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6588 straight lines be drawn from a point to meet them at given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6589 angles, and if the straight lines so drawn be such that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6590 rectangle contained by one of them and a given straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6591 added to the rectangle contained by another of them and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6592 (another) given straight line is equal to the rectangle contained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6593 by a third and a (third) given straight line, and similarly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6594 with the others, the point will lie on a straight line given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6595 in position.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6596 <p>(Here we have trilinear or multilinear coordinates proportional
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6597 to the distances of the variable point from each of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6598 three or more fixed lines. When there are three fixed lines,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6599 the statement is that <MATH><I>ax</I> + <I>by</I> = <I>cz</I></MATH> represents a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6600 The precise meaning of the words ‘and similarly with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6601 the others’ or ‘of the others’—<G>kai\ tw=n loipw=n o(moi/ws</G>—is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6602 uncertain; the words seem to imply that, when there were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6603 more than three rectangles <I>ax, by, cz</I> ..., two of them were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6604 taken to be equal to the sum of all the others; but it is quite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6605 possible that Pappus meant that any linear equation between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6606 these rectangles represented a straight line. Precisely how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6607 far Apollonius went in generality we are not in a position to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6608 judge.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6609 <p>The last enunciation (8) of Pappus referring to Book I
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6610 states that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6611 <p>‘If from any point (two) straight lines be drawn to meet (two)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6612 parallel straight lines given in position at given angles, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6613 <pb n=188><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6614 cut off from the parallels straight lines measured from given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6615 points on them such that (<I>a</I>) they have a given ratio or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6616 (<I>b</I>) they contain a given rectangle or (<I>c</I>) the sum or difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6617 of figures of given species described on them respectively is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6618 equal to a given area, the point will lie on a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6619 given in position.’<note>Pappus, vii, p. 666. 7-13.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6620 <p>The contents of Book II are equally interesting. Some of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6621 the enunciations shall for brevity be given by means of letters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6622 instead of in general terms. If from two given points <I>A, B</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6623 two straight lines be ‘inflected’ (<G>klasqw=sin</G>) to a point <I>P,</I> then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6624 (1), if <I>AP</I><SUP>2</SUP> <01> <I>BP</I><SUP>2</SUP> is given, the locus of <I>P</I> is a straight line;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6625 (2) if <I>AP, BP</I> are in a given ratio, the locus is a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6626 or a circle [this is the proposition quoted by Eutocius in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6627 commentary on the <I>Conics,</I> but already known to Aristotle];
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6628 (4) if <I>AP</I><SUP>2</SUP> is ‘greater by a given area than in a given ratio’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6629 to <I>BP</I><SUP>2</SUP>, i.e. if <I>AP</I><SUP>2</SUP> = <I>a</I><SUP>2</SUP> + <I>m.BP</I><SUP>2</SUP>, the locus is a circle given in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6630 position. An interesting proposition is (5) that, ‘If from any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6631 number of given points whatever straight lines be inflected to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6632 one point, and the figures (given in species) described on all of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6633 them be together equal to a given area, the point will lie on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6634 a circumference (circle) given in position’; that is to say, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6635 <G>a</G>. <I>AP</I><SUP>2</SUP> + <G>b</G>.<I>BP</I><SUP>2</SUP> + <G>g</G>.<I>CP</I><SUP>2</SUP> + ... = a given area (where <G>a, b, g</G> ...
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6636 are constants), the locus of <I>P</I> is a circle. (3) states that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6637 <I>AN</I> be a fixed straight line and <I>A</I> a fixed point on it, and if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6638 <I>AP</I> be any straight line drawn to a point <I>P</I> such that, if <I>PN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6639 is perpendicular to <I>AN, AP</I><SUP>2</SUP> = <I>a.AN</I> or <I>a.BN,</I> where <I>a</I> is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6640 given length and <I>B</I> is another fixed point on <I>AN,</I> then the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6641 locus of <I>P</I> is a circle given in position; this is equivalent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6642 to the fact that, if <I>A</I> be the origin, <I>AN</I> the axis of <I>x,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6643 <MATH><I>x</I> = <I>AN, y</I> = <I>PN</I></MATH> be the coordinates of <I>P,</I> the locus <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>ax</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6644 or <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>a</I>(<I>x</I>-<I>b</I>)</MATH> is a circle. (6) is somewhat obscurely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6645 enunciated: ‘If from two given points straight lines be inflected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6646 (to a point), and from the point (of concourse) a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6647 line be drawn parallel to a straight line given in position and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6648 cutting off from another straight line given in position an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6649 intercept measured from a given point on it, and if the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6650 figures (given in species) described on the two inflected lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6651 be equal to the rectangle contained by a given straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6652 and the intercept, the point at which the straight lines are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6653 <pb n=189><head><I>PLANE LOCI</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6654 inflected lies on a circle given in position.’ The meaning
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6655 seems to be this: Given two fixed points <I>A, B,</I> a length <I>a,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6656 a straight line <I>OX</I> with a point <I>O</I> fixed upon it, and a direction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6657 represented, say, by any straight line <I>OZ</I> through <I>O,</I> then,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6658 if <I>AP, BP</I> be drawn to <I>P,</I> and <I>PM</I> parallel to <I>OZ</I> meets <I>OX</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6659 in <I>M,</I> the locus of <I>P</I> will be a circle given in position if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6660 <MATH><G>a</G>.<I>AP</I><SUP>2</SUP> + <G>b</G>.<I>BP</I><SUP>2</SUP> = <G>a</G>.<I>OM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6661 where <G>a, b</G> are constants. The last two loci are again
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6662 obscurely expressed, but the sense is this: (7) If <I>PQ</I> be any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6663 chord of a circle passing through a fixed internal point <I>O,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6664 <I>R</I> be an external point on <I>PQ</I> produced such that either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6665 <MATH>(<I>a</I>) <I>OR</I><SUP>2</SUP> = <I>PR.RQ</I> or (<I>b</I>) <I>OR</I><SUP>2</SUP> + <I>PO.OQ</I> = <I>PR.RQ</I></MATH>, the locus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6666 of <I>R</I> is a straight line given in position. (8) is the reciprocal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6667 of this: Given the fixed point <I>O,</I> the straight line which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6668 the locus of <I>R,</I> and also the relation (<I>a</I>) or (<I>b</I>), the locus of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6669 <I>P, Q</I> is a circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6670 <C>(<G>z</G>) <G>*neu/seis</G> (<I>Vergings</I> or <I>Inclinations</I>), two Books.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6671 <p>As we have seen, the problem in a <G>neu=sis</G> is to place
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6672 between two straight lines, a straight line and a curve, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6673 two curves, a straight line of given length in such a way
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6674 that it <I>verges</I> towards a fixed point, i.e. it will, if produced,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6675 pass through a fixed point. Pappus observes that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6676 when we come to particular cases, the problem will be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6677 ‘plane’, ‘solid’ or ‘linear’, according to the nature of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6678 particular hypotheses; but a selection had been made from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6679 the class which could be solved by plane methods, i.e. by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6680 means of the straight line and circle, the object being to give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6681 those which were more generally useful in geometry. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6682 following were the cases thus selected and proved.<note>Pappus, vii, pp. 670-2.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6683 <p>I. Given (<I>a</I>) a semicircle and a straight line at right angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6684 to the base, or (<I>b</I>) two semicircles with their bases in a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6685 line, to insert a straight line of given length verging to an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6686 angle of the semicircle [or of one of the semicircles].
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6687 <p>II. Given a rhombus with one side produced, to insert
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6688 a straight line of given length in the external angle so that it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6689 verges to the opposite angle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6690 <pb n=190><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6691 <p>III. Given a circle, to insert a chord of given length verging
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6692 to a given point.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6693 <p>In Book I of Apollonius's work there were four cases of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6694 I (<I>a</I>), two cases of III, and two of II; the second Book contained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6695 ten cases of I (<I>b</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6696 <p>Restorations were attempted by Marino Ghetaldi (<I>Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6697 redivivus,</I> Venice, 1607, and <I>Apollonius redivivus . . . Liber
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6698 secundus,</I> Venice, 1613), Alexander Anderson (in a <I>Supplementum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6699 Apollonii redivivi,</I> 1612), and Samuel Horsley
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6700 (Oxford, 1770); the last is much the most complete.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6701 <p>In the case of the rhombus (II) the construction of Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6702 can be restored with certainty. It depends on a lemma given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6703 by Pappus, which is as follows: Given a rhombus <I>AD</I> with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6704 diagonal <I>BC</I> produced to <I>E,</I> if <I>F</I> be taken on <I>BC</I> such that <I>EF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6705 is a mean proportional between <I>BE</I> and <I>EC,</I> and if a circle be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6706 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6707 described with <I>E</I> as centre and <I>EF</I> as radius cutting <I>CD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6708 in <I>K</I> and <I>AC</I> produced in <I>H,</I> then shall <I>B, K, H</I> be in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6709 straight line.<note>Pappus, vii, pp. 778-80.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6710 <p>Let the circle cut <I>AC</I> in <I>L,</I> join <I>LK</I> meeting <I>BC</I> in <I>M,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6711 join <I>HE, LE, KE.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6712 <p>Since now <I>CL, CK</I> are equally inclined to the diameter of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6713 the circle, <I>CL</I> = <I>CK.</I> Also <I>EL</I> = <I>EK,</I> and it follows that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6714 triangles <I>ECK, ECL</I> are equal in all respects, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6715 <MATH>&angle; <I>CKE</I> = &angle; <I>CLE</I> = &angle; <I>CHE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6716 <p>By hypothesis, <MATH><I>EB</I>:<I>EF</I> = <I>EF</I>:<I>EC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6717 or <MATH><I>EB</I>:<I>EK</I> = <I>EK</I>:<I>EC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6718 <pb n=191><head><G>negsei*s</G> (<I>VERGINGS OR INCLINATIONS</I>)</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6719 <p>Therefore the triangles <I>BEK, KEC,</I> which have the angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6720 <I>BEK</I> common, are similar, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6721 <MATH>&angle; <I>CBK</I> = &angle; <I>CKE</I> = &angle; <I>CHE</I> (from above)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6722 <p>But <MATH>&angle; <I>HCE</I> = &angle; <I>ACB</I> = &angle; <I>BCK</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6723 <p>Therefore in the triangles <I>CBK, CHE</I> two angles are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6724 respectively equal, so that &angle; <I>CEH</I> = &angle; <I>CKB</I> also.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6725 <p>But since &angle; <I>CKE</I> = &angle; <I>CHE</I> (from above), <I>K, C, E, H</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6726 concyclic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6727 <p>Hence <MATH>&angle; <I>CEH</I> + &angle; <I>CKH</I> = (two right angles)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6728 therefore, since <MATH>&angle; <I>CEH</I> = &angle; <I>CKB</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6729 <MATH>&angle; <I>CKB</I> + &angle; <I>CKH</I> = (two right angles)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6730 and <I>BKH</I> is a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6731 <p>It is certain, from the nature of this lemma, that Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6732 made his construction by drawing the circle shown in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6733 figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6734 <p>He would no doubt arrive at it by analysis somewhat as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6735 follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6736 <p>Suppose the problem solved, and <I>HK</I> inserted as required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6737 (= <I>k</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6738 <p>Bisect <I>HK</I> in <I>N,</I> and draw <I>NE</I> at right angles to <I>KH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6739 meeting <I>BC</I> produced in <I>E.</I> Draw <I>KM</I> perpendicular to <I>BC,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6740 and produce it to meet <I>AC</I> in <I>L.</I> Then, by the property of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6741 the rhombus, <I>LM</I> = <I>MK,</I> and, since <I>KN</I> = <I>NH</I> also, <I>MN</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6742 parallel to <I>LH.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6743 <p>Now, since the angles at <I>M, N</I> are right, <I>M, K, N, E</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6744 concyclic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6745 <p>Therefore &angle; <I>CEK</I> = &angle; <I>MNK</I> = &angle; <I>CHK,</I> so that <I>C, K, H, E</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6746 are concyclic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6747 <p>Therefore &angle; <I>BCD</I> = supplement of <I>KCE</I> = &angle; <I>EHK</I> = &angle; <I>EKH,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6748 and the triangles <I>EKH, DCB</I> are similar.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6749 <p>Lastly,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6750 &angle; <I>EBK</I> = &angle; <I>EKH</I> - &angle; <I>CEK</I> = &angle; <I>EHK</I> - &angle; <I>CHK</I> = &angle; <I>EHC</I> = &angle; <I>EKC</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6751 therefore the triangles <I>EBK, EKC</I> are similar, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6752 <MATH><I>BE</I>:<I>EK</I> = <I>EK</I>:<I>EC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6753 or <MATH><I>BE.EC</I> = <I>EK</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6754 <pb n=192><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6755 <p>But, by similar triangles <I>EKH, DCB</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6756 <MATH><I>EK</I>:<I>KH</I> = <I>DC</I>:<I>CB</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6757 and, since the ratio <I>DC</I>:<I>CB,</I> as well as <I>KH,</I> is given, <I>EK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6758 is given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6759 <p>The construction then is as follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6760 <p>If <I>k</I> be the given length, take a straight line <I>p</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6761 <MATH><I>p</I>:<I>k</I>=<I>AB</I>:<I>BC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6762 apply to <I>BC</I> a rectangle <I>BE.EC</I> equal to <I>p</I><SUP>2</SUP> and exceeding by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6763 a square; then with <I>E</I> as centre and radius equal to <I>p</I> describe a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6764 circle cutting <I>AC</I> produced in <I>H</I> and <I>CD</I> in <I>K.</I> <I>HK</I> is then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6765 equal to <I>k</I> and, by Pappus's lemma, verges towards <I>B.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6766 <p>Pappus adds an interesting solution of the same problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6767 with reference to a square instead of a rhombus; the solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6768 is by one Heraclitus and depends on a lemma which Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6769 also gives.<note>Pappus, vii, pp. 780-4.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6770 <p>We hear of yet other lost works by Apollonius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6771 <p>(<G>h</G>) A <I>Comparison of the dodecahedron with the icosahedron.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6772 This is mentioned by Hypsicles in the preface to the so-called
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6773 Book XIV of Euclid. Like the <I>Conics,</I> it appeared in two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6774 editions, the second of which contained the proposition that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6775 if there be a dodecahedron and an icosahedron inscribed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6776 one and the same sphere, the surfaces of the solids are in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6777 same ratio as their volumes; this was established by showing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6778 that the perpendiculars from the centre of the sphere to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6779 a pentagonal face of the dodecahedron and to a triangular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6780 face of the icosahedron are equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6781 <p>(<G>q</G>) Marinus on Euclid's <I>Data</I> speaks of a <I>General Treatise</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6782 (<G>kaqo/lou pragmatei/a</G>) in which Apollonius used the word
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6783 <I>assigned</I> (<G>tetagme/non</G>) as a comprehensive term to describe the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6784 <I>datum</I> in general. It would appear that this work must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6785 have dealt with the fundamental principles of mathematics,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6786 definitions, axioms, &c., and that to it must be referred the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6787 various remarks on such subjects attributed to Apollonius by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6788 Proclus, the elucidation of the notion of a line, the definition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6789 <pb n=193><head>OTHER LOST WORKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6790 of plane and solid angles, and his attempts to prove the axioms;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6791 it must also have included the three definitions (13-15) in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6792 Euclid's <I>Data</I> which, according to a scholium, were due to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6793 Apollonius and must therefore have been interpolated (they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6794 are definitions of <G>kathgme/nh, a)nhgme/nh</G>, and the elliptical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6795 phrase <G>para\ qe/sei</G>, which means ‘parallel to a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6796 given in position’). Probably the same work also contained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6797 Apollonius's alternative constructions for the problems of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6798 Eucl. I. 10, 11 and 23 given by Proclus. Pappus speaks
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6799 of a mention by Apollonius ‘before his own elements’ of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6800 class of locus called <G>e)fektiko/s</G>, and it may be that the treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6801 now in question is referred to rather than the <I>Plane Loci</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6802 itself.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6803 <p>(<G>i</G>) The work <I>On the Cochlias</I> was on the cylindrical helix.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6804 It included the theoretical generation of the curve on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6805 surface of the cylinder, and the proof that the curve is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6806 <I>homoeomeric</I> or uniform, i.e. such that any part will fit upon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6807 or coincide with any other.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6808 <p>(<G>k</G>) A work on <I>Unordered Irrationals</I> is mentioned by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6809 Proclus, and a scholium on Eucl. X. 1 extracted from Pappus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6810 commentary remarks that ‘Euclid did not deal with all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6811 rationals and irrationals, but only with the simplest kinds by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6812 the combination of which an infinite number of irrationals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6813 are formed, of which latter Apollonius also gave some’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6814 To a like effect is a passage of the fragment of Pappus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6815 commentary on Eucl. X discovered in an Arabic translation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6816 by Woepcke: ‘it was Apollonius who, besides the <I>ordered</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6817 irrational magnitudes, showed the existence of the <I>unordered,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6818 and by accurate methods set forth a great number of them’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6819 The hints given by the author of the commentary seem to imply
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6820 that Apollonius's extensions of the theory of irrationals took
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6821 two directions, (1) generalizing the <I>medial</I> straight line of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6822 Euclid, on the basis that, between two lines commensurable in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6823 square (only), we may take not only one sole medial line but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6824 three or four, and so on <I>ad infinitum,</I> since we can take,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6825 between any two given straight lines, as many lines as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6826 we please in continued proportion, (2) forming compound
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6827 irrationals by the addition and subtraction of more than two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6828 terms of the sort composing the <I>binomials, apotomes,</I> &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6829 <pb n=194><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6830 <p>(<G>l</G>) <I>On the burning-mirror</I> (<G>peri\ tou= puri/ou</G>) is the title of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6831 another work of Apollonius mentioned by the author of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6832 <I>Fragmentum mathematicum Bobiense,</I> which is attributed by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6833 Heiberg to Anthemius but is more likely (judging by its survivals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6834 of antiquated terminology) to belong to a much earlier
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6835 date. The fragment shows that Apollonius discussed the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6836 spherical form of mirror among others. Moreover, the extant
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6837 fragment by Anthemius himself (on burning mirrors) proves the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6838 property of mirrors of parabolic section, using the properties of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6839 the parabola (<I>a</I>) that the tangent at any point makes equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6840 angles with the axis and with the focal distance of the point,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6841 and (<I>b</I>) that the distance of any point on the curve from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6842 focus is equal to its distance from a certain straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6843 (our ‘directrix’); and we can well believe that the parabolic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6844 form of mirror was also considered in Apollonius's work, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6845 that he was fully aware of the focal properties of the parabola,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6846 notwithstanding the omission from the <I>Conics</I> of all mention
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6847 of the focus of a parabola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6848 <p>(<G>m</G>) In a work called <G>w)kuto/kion</G> (‘quick-delivery’) Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6849 is said to have found an approximation to the value of <G>p</G> ‘by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6850 a different calculation (from that of Archimedes), bringing it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6851 within closer limits’.<note>v. Eutocius on Archimedes, <I>Measurement of a Circle.</I></note> Whatever these closer limits may have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6852 been, they were considered to be less suitable for practical use
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6853 than those of Archimedes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6854 <p>It is a moot question whether Apollonius's system of arithmetical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6855 notation (by tetrads) for expressing large numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6856 and performing the usual arithmetical operations with them,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6857 as described by Pappus, was included in this same work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6858 Heiberg thinks it probable, but there does not seem to be any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6859 necessary reason why the notation for large numbers, classifying
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6860 them into myriads, double myriads, triple myriads, &c.,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6861 i.e. according to powers of 10,000, need have been connected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6862 with the calculation of the value of <G>p</G>, unless indeed the numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6863 used in the calculation were so large as to require the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6864 tetradic system for the handling of them.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6865 <p>We have seen that Apollonius is credited with a solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6866 of the problem of the two mean proportionals (vol. i,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6867 pp. 262-3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6868 <pb n=195><head>OTHER LOST WORKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6869 <C><I>Astronomy.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6870 <p>We are told by Ptolemaeus Chennus<note><I>apud Photium,</I> Cod. cxc, p. 151 b 18, ed. Bekker.</note> that Apollonius was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6871 famed for his astronomy, and was called <G>e</G> (Epsilon) because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6872 the form of that letter is associated with that of the moon, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6873 which his accurate researches principally related. Hippolytus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6874 says he made the distance of the moon's circle from the surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6875 of the earth to be 500 myriads of stades.<note>Hippol. <I>Refut.</I> iv. 8, p. 66, ed. Duncker.</note> This figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6876 can hardly be right, for, the diameter of the earth being,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6877 according to Eratosthenes's evaluation, about eight myriads of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6878 stades, this would make the distance of the moon from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6879 earth about 125 times the earth's radius. This is an unlikely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6880 figure, seeing that Aristarchus had given limits for the ratios
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6881 between the distance of the moon and its diameter, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6882 between the diameters of the moon and the earth, which lead
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6883 to about 19 as the ratio of the moon's distance to the earth's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6884 radius. Tannery suggests that perhaps Hippolytus made a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6885 mistake in copying from his source and took the figure of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6886 5,000,000 stades to be the length of the radius instead of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6887 <I>diameter</I> of the moon's orbit.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6888 <p>But we have better evidence of the achievements of Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6889 in astronomy. In Ptolemy's <I>Syntaxis</I><note>Ptolemy, <I>Syntaxis,</I> xii. 1.</note> he appears as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6890 an authority upon the hypotheses of epicycles and eccentrics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6891 designed to account for the apparent motions of the planets.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6892 The propositions of Apollonius quoted by Ptolemy contain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6893 exact statements of the alternative hypotheses, and from this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6894 fact it was at one time concluded that Apollonius invented
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6895 the two hypotheses. This, however, is not the case. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6896 hypothesis of epicycles was already involved, though with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6897 restricted application, in the theory of Heraclides of Pontus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6898 that the two inferior planets, Mercury and Venus, revolve in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6899 circles like satellites round the sun, while the sun itself
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6900 revolves in a circle round the earth; that is, the two planets
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6901 describe epicycles about the material sun as moving centre.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6902 In order to explain the motions of the superior planets by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6903 means of epicycles it was necessary to conceive of an epicycle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6904 about a point as moving centre which is not a material but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6905 a mathematical point. It was some time before this extension
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6906 of the theory of epicycles took place, and in the meantime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6907 <pb n=196><head>APOLLONIUS OF PERGA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6908 another hypothesis, that of eccentrics, was invented to account
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6909 for the movements of the superior planets only. We are at this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6910 stage when we come to Apollonius. His enunciations show
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6911 that he understood the theory of epicycles in all its generality,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6912 but he states specifically that the theory of eccentrics can only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6913 be applied to the three planets which can be at any distance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6914 from the sun. The reason why he says that the eccentric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6915 hypothesis will not serve for the inferior planets is that, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6916 order to make it serve, we should have to suppose the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6917 described by the centre of the eccentric circle to be greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6918 than the eccentric circle itself. (Even this generalization was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6919 made later, at or before the time of Hipparchus.) Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6920 further says in his enunciation about the eccentric that ‘the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6921 centre of the eccentric circle moves about the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6922 zodiac in the direct order of the signs and <I>at a speed equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6923 that of the sun,</I> while the star moves on the eccentric about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6924 its centre in the inverse order of the signs and at a speed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6925 equal to the anomaly’. It is clear from this that the theory
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6926 of eccentrics was invented for the specific purpose of explaining
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6927 the movements of Mars, Jupiter, and Saturn about the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6928 sun and for that purpose alone. This explanation, combined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6929 with the use of epicycles about the sun as centre to account
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6930 for the motions of Venus and Mercury, amounted to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6931 system of Tycho Brahe; that system was therefore anticipated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6932 by some one intermediate in date between Heraclides and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6933 Apollonius and probably nearer to the latter, or it may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6934 have been Apollonius himself who took this important step.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6935 If it was, then Apollonius, coming after Aristarchus of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6936 Samos, would be exactly the Tycho Brahe of the Copernicus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6937 of antiquity. The actual propositions quoted by Ptolemy as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6938 proved by Apollonius among others show mathematically at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6939 what points, under each of the two hypotheses, the apparent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6940 forward motion changes into apparent retrogradation and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6941 vice versa, or the planet appears to be <I>stationary.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6942 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6943 <C>XV
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6944 THE SUCCESSORS OF THE GREAT GEOMETERS</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6945 <p>WITH Archimedes and Apollonius Greek geometry reached
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6946 its culminating point. There remained details to be filled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6947 in, and no doubt in a work such as, for instance, the <I>Conics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6948 geometers of the requisite calibre could have found proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6949 tions containing the germ of theories which were capable of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6950 independent development. But, speaking generally, the fur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6951 ther progress of geometry on general lines was practically
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6952 barred by the restrictions of method and form which were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6953 inseparable from the classical Greek geometry. True, it was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6954 open to geometers to discover and investigate curves of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6955 higher order than conics, such as spirals, conchoids, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6956 like. But the Greeks could not get very far even on these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6957 lines in the absence of some system of coordinates and without
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6958 freer means of manipulation such as are afforded by modern
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6959 algebra, in contrast to the geometrical algebra, which could
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6960 only deal with equations connecting lines, areas, and volumes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6961 but involving no higher dimensions than three, except in so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6962 far as the use of proportions allowed a very partial exemp-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6963 tion from this limitation. The theoretical methods available
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6964 enabled quadratic, cubic and bi-quadratic equations or their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6965 equivalents to be solved. But all the solutions were <I>geometri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6966 cal</I>; in other words, quantities could only be represented by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6967 lines, areas and volumes, or ratios between them. There was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6968 nothing corresponding to operations with general algebraical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6969 quantities irrespective of what they represented. There were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6970 no <I>symbols</I> for such quantities. In particular, the irrational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6971 was discovered in the form of incommensurable <I>lines</I>; hence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6972 irrationals came to be represented by straight lines as they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6973 are in Euclid, Book X, and the Greeks had no other way of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6974 representing them. It followed that a product of two irra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6975 tionals could only be represented by a <I>rectangle</I>, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6976 Even when Diophantus came to use a symbol for an unknown
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6977 <pb n=198><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6978 quantity, it was only an abbreviation for the word <G>a)riqmo/s</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6979 with the meaning of ‘an undetermined multitude of units’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6980 not a general quantity. The restriction then of the algebra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6981 employed by geometers to the geometrical form of algebra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6982 operated as an insuperable obstacle to any really new depar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6983 ture in theoretical geometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6984 <p>It might be thought that there was room for further exten-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6985 sions in the region of solid geometry. But the fundamental
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6986 principles of solid geometry had also been laid down in Euclid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6987 Books XI-XIII; the theoretical geometry of the sphere had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6988 been fully treated in the ancient <I>sphaeric</I>; and any further
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6989 application of solid geometry, or of loci in three dimensions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6990 was hampered by the same restrictions of method which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6991 hindered the further progress of plane geometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6992 <p>Theoretical geometry being thus practically at the end of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6993 its resources, it was natural that mathematicians, seeking for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6994 an opening, should turn to the <I>applications</I> of geometry. One
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6995 obvious branch remaining to be worked out was the geometry
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6996 of measurement, or <I>mensuration</I> in its widest sense, which of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6997 course had to wait on pure theory and to be based on its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6998 results. One species of mensuration was immediately required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	6999 for astronomy, namely the measurement of triangles, especially
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7000 spherical triangles; in other words, trigonometry plane and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7001 spherical. Another species of mensuration was that in which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7002 an example had already been set by Archimedes, namely the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7003 measurement of areas and volumes of different shapes, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7004 arithmetical approximations to their true values in cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7005 where they involved surds or the ratio (<G>p</G>) between the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7006 circumference of a circle and its diameter; the object of such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7007 mensuration was largely practical. Of these two kinds of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7008 mensuration, the first (trigonometry) is represented by Hip-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7009 parchus, Menelaus and Ptolemy; the second by Heron of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7010 Alexandria. These mathematicians will be dealt with in later
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7011 chapters; this chapter will be devoted to the successors of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7012 great geometers who worked on the same lines as the latter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7013 <p>Unfortunately we have only very meagre information as to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7014 what these geometers actually accomplished in keeping up the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7015 tradition. No geometrical works by them have come down
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7016 to us in their entirety, and we are dependent on isolated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7017 extracts or scraps of information furnished by commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7018 <pb n=199><head>NICOMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7019 tators, and especially by Pappus and Eutocius. Some of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7020 these are very interesting, and it is evident from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7021 extracts from the works of such writers as Diocles and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7022 Dionysodorus that, for some time after Archimedes and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7023 Apollonius, mathematicians had a thorough grasp of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7024 contents of the works of the great geometers, and were able
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7025 to use the principles and methods laid down therein with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7026 ease and skill.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7027 <p>Two geometers properly belonging to this chapter have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7028 already been dealt with. The first is NICOMEDES, the inventor
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7029 of the conchoid, who was about intermediate in date between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7030 Eratosthenes and Apollonius. The conchoid has already been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7031 described above (vol. i, pp. 238-40). It gave a general method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7032 of solving any <G>neu=sis</G> where one of the lines which cut off an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7033 intercept of given length on the line verging to a given point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7034 is a straight line; and it was used both for the finding of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7035 mean proportionals and for the trisection of any angle, these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7036 problems being alike reducible to a <G>neu=sis</G> of this kind. How
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7037 far Nicomedes discussed the properties of the curve in itself
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7038 is uncertain; we only know from Pappus that he proved two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7039 properties, (1) that the so-called ‘ruler’ in the instrument for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7040 constructing the curve is an asymptote, (2) that any straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7041 line drawn in the space between the ‘ruler’ or asymptote and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7042 the conchoid must, if produced, be cut by the conchoid.<note>Pappus, iv, p. 244. 21-8.</note> The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7043 equation of the curve referred to polar coordinates is, as we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7044 have seen, <MATH><I>r</I> = <I>a</I> + <I>b</I> sec <G>q</G></MATH>. According to Eutocius, Nicomedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7045 prided himself inordinately on his discovery of this curve,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7046 contrasting it with Eratosthenes's mechanism for finding any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7047 number of mean proportionals, to which he objected formally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7048 and at length on the ground that it was impracticable and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7049 entirely outside the spirit of geometry.<note>Eutoc. on Archimedes, <I>On the Sphere and Cylinder</I>, Archimedes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7050 vol. iii, p. 98.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7051 <p>Nicomedes is associated by Pappus with Dinostratus, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7052 brother of Menaechmus, and others as having applied to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7053 squaring of the circle the curve invented by Hippias and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7054 known as the <I>quadratrix</I>,<note>Pappus, iv, pp. 250. 33-252. 4. Cf. vol. i, p. 225 sq.</note> which was originally intended for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7055 the purpose of trisecting any angle. These facts are all that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7056 we know of Nicomedes's achievements.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7057 <pb n=200><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7058 <p>The second name is that of DIOCLES. We have already
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7059 (vol. i, pp. 264-6) seen him as the discoverer of the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7060 known as the <I>cissoid</I>, which he used to solve the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7061 of the two mean proportionals, and also (pp. 47-9 above)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7062 as the author of a method of solving the equivalent of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7063 a certain cubic equation by means of the intersection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7064 of an ellipse and a hyperbola. We are indebted for our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7065 information on both these subjects to Eutocius,<note>Eutocius, <I>loc. cit.</I>, p. 66. 8 sq., p. 160. 3 sq.</note> who tells
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7066 us that the fragments which he quotes came from Diocles's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7067 work <G>peri\ purei/wn</G>, <I>On burning-mirrors.</I> The connexion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7068 the two things with the subject of this treatise is not obvious,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7069 and we may perhaps infer that it was a work of considerable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7070 scope. What exactly were the forms of the burning-mirrors
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7071 discussed in the treatise it is not possible to say, but it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7072 probably safe to assume that among them were concave
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7073 mirrors in the forms (1) of a sphere, (2) of a paraboloid, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7074 (3) of the surface described by the revolution of an ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7075 about its major axis. The author of the <I>Fragmentum mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7076 maticum Bobiense</I> says that Apollonius in his book <I>On the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7077 burning-mirror</I> discussed the case of the concave spherical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7078 mirror, showing about what point ignition would take place;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7079 and it is certain that Apollonius was aware that an ellipse has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7080 the property of reflecting all rays through one focus to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7081 other focus. Nor is it likely that the corresponding property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7082 of a parabola with reference to rays parallel to the axis was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7083 unknown to Apollonius. Diocles therefore, writing a century
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7084 or more later than Apollonius, could hardly have failed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7085 deal with all three cases. True, Anthemius (died about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7086 A.D. 534) in his fragment on burning-mirrors says that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7087 ancients, while mentioning the usual burning-mirrors and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7088 saying that such figures are conic sections, omitted to specify
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7089 which conic sections, and how produced, and gave no geo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7090 metrical proofs of their properties. But if the properties
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7091 were commonly known and quoted, it is obvious that they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7092 must have been proved by the ancients, and the explanation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7093 of Anthemius's remark is presumably that the original works
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7094 in which they were proved (e.g. those of Apollonius and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7095 Diocles) were already lost when he wrote. There appears to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7096 be no trace of Diocles's work left either in Greek or Arabic,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7097 <pb n=201><head>DIOCLES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7098 unless we have a fragment from it in the <I>Fragmentum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7099 mathematicum Bobiense.</I> But Moslem writers regarded Diocles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7100 as the discoverer of the parabolic burning-mirror; ‘the ancients’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7101 says al Sing&amacr;r&imacr; (Sach&amacr;w&imacr;, An⋅&amacr;r&imacr;), ‘made mirrors of plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7102 surfaces. Some made them concave (i.e. spherical) until
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7103 Diocles (Di&umacr;klis) showed and proved that, if the surface of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7104 these mirrors has its curvature in the form of a parabola, they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7105 then have the greatest power and burn most strongly. There
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7106 is a work on this subject composed by Ibn al-Haitham.’ This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7107 work survives in Arabic and in Latin translations, and is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7108 reproduced by Heiberg and Wiedemann<note><I>Bibliotheca mathematica</I>, x<SUB>3</SUB>, 1910, pp. 201-37.</note>; it does not, how-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7109 ever, mention the name of Diocles, but only those of Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7110 medes and Anthemius. Ibn al-Haitham says that famous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7111 men like Archimedes and Anthemius had used mirrors made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7112 up of a number of spherical rings; afterwards, he adds, they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7113 considered the form of curves which would reflect rays to one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7114 point, and found that the concave surface of a paraboloid of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7115 revolution has this property. It is curious to find Ibn al-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7116 Haitham saying that the ancients had not set out the proofs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7117 sufficiently, nor the method by which they discovered them,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7118 words which almost exactly recall those of Anthemius himself.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7119 Nevertheless the whole course of Ibn al-Haitham's proofs is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7120 on the Greek model, Apollonius being actually quoted by name
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7121 in the proof of the main property of the parabola required,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7122 namely that the tangent at any point of the curve makes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7123 equal angles with the focal distance of the point and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7124 straight line drawn through it parallel to the axis. A proof
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7125 of the property actually survives in the Greek <I>Fragmentum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7126 mathematicum Bobiense</I>, which evidently came from some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7127 treatise on the parabolic burning-mirror; but Ibn al-Haitham
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7128 does not seem to have had even this fragment at his disposal,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7129 since his proof takes a different course, distinguishing three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7130 different cases, reducing the property by analysis to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7131 known property <I>AN</I> = <I>AT</I>, and then working out the syn-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7132 thesis. The proof in the <I>Fragmentum</I> is worth giving. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7133 substantially as follows, beginning with the preliminary lemma
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7134 that, if <I>PT</I>, the tangent at any point <I>P</I>, meets the axis at <I>T</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7135 and if <I>AS</I> be measured along the axis from the vertex <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7136 equal to 1/4<I>AL</I>, where <I>AL</I> is the parameter, then <I>SP</I> = <I>ST.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7137 <pb n=202><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7138 <p>Let <I>PN</I> be the ordinate from <I>P</I>; draw <I>AY</I> at right angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7139 to the axis meeting <I>PT</I> in <I>Y</I>, and join <I>SY.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7140 <p>Now <MATH><I>PN</I><SUP>2</SUP> = <I>AL.AN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7141 = 4<I>AS.AN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7142 = 4<I>AS.AT</I> (since <I>AN</I> = <I>AT</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7143 <p>But <MATH><I>PN</I> = 2<I>AY</I> (since <I>AN</I> = <I>AT</I>);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7144 therefore <I>AY</I><SUP>2</SUP> = <I>TA.AS</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7145 and the angle <I>TYS</I> is right.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7146 <p>The triangles <I>SYT, SYP</I> being right-angled, and <I>TY</I> being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7147 equal to <I>YP</I>, it follows that <I>SP</I> = <I>ST.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7148 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7149 <p>With the same figure, let <I>BP</I> be a ray parallel to <I>AN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7150 impinging on the curve at <I>P.</I> It is required to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7151 the angles of incidence and reflection (to <I>S</I>) are equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7152 <p>We have <I>SP</I> = <I>ST</I>, so that ‘the angles at the points <I>T, P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7153 are equal. So’, says the author, ‘are the angles <I>TPA, KPR</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7154 [the angles between the tangent and the <I>curve</I> on each side of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7155 the point of contact]. Let the difference between the angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7156 be taken; therefore the angles <I>SPA, RPB</I> which remain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7157 [again ‘mixed’ angles] are equal. Similarly we shall show
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7158 that all the lines drawn parallel to <I>AS</I> will be reflected at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7159 equal angles to the point <I>S.</I>’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7160 <p>The author then proceeds: ‘Thus burning-mirrors con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7161 structed with the surface of impact (in the form) of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7162 <I>section of a right-angled cone</I> may easily, in the manner
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7163 <pb n=203><head>DIOCLES. PERSEUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7164 above shown, be proved to bring about ignition at the point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7165 indicated.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7166 <p>Heiberg held that the style of this fragment is Byzantine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7167 and that it is probably by Anthemius. Cantor conjectured
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7168 that here we might, after all, have an extract from Diocles's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7169 work. Heiberg's supposition seems to me untenable because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7170 of the author's use (1) of the ancient terms ‘section of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7171 a right-angled cone’ for parabola and ‘diameter’ for axis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7172 (to say nothing of the use of the parameter, of which there is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7173 no word in the genuine fragment of Anthemius), and (2) of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7174 the mixed ‘angles of contact’. Nor does it seem likely that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7175 even Diocles, living a century after Apollonius, would have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7176 spoken of the ‘section of a right-angled cone’ instead of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7177 parabola, or used the ‘mixed’ angle of which there is only the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7178 merest survival in Euclid. The assumption of the equality
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7179 of the two angles made by the curve with the tangent on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7180 both sides of the point of contact reminds us of Aristotle's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7181 assumption of the equality of the angles ‘<I>of</I> a segment’ of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7182 a circle as prior to the truth proved in Eucl. I. 5. I am
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7183 inclined, therefore, to date the fragment much earlier even
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7184 than Diocles. Zeuthen suggested that the property of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7185 paraboloidal mirror may have been discovered by Archimedes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7186 who, according to a Greek tradition, wrote <I>Catoptrica.</I> This,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7187 however, does not receive any confirmation in Ibn al-Haitham
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7188 or in Anthemius, and we can only say that the fragment at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7189 least goes back to an original which was probably not later
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7190 than Apollonius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7191 <p>PERSEUS is only known, from allusions to him in Proclus,<note>Proclus on Eucl. I, pp. 111. 23-112. 8, 356. 12. Cf. vol. i, p. 226.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7192 as the discoverer and investigator of the <I>spiric sections.</I> They
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7193 are classed by Proclus among curves obtained by cutting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7194 solids, and in this respect they are associated with the conic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7195 sections. We may safely infer that they were discovered
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7196 after the conic sections, and only after the theory of conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7197 had been considerably developed. This was already the case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7198 in Euclid's time, and it is probable, therefore, that Perseus was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7199 not earlier than Euclid. On the other hand, by that time
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7200 the investigation of conics had brought the exponents of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7201 subject such fame that it would be natural for mathematicians
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7202 to see whether there was not an opportunity for winning a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7203 <pb n=204><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7204 like renown as discoverers of other curves to be obtained by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7205 cutting well-known solid figures other than the cone and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7206 cylinder. A particular case of one such solid figure, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7207 <G>spei=ra</G>, had already been employed by Archytas, and the more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7208 general form of it would not unnaturally be thought of as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7209 likely to give sections worthy of investigation. Since Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7210 is Proclus's authority, Perseus may have lived at any date from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7211 Euclid's time to (say) 75 B.C., but the most probable supposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7212 tion seems to be that he came before Apollonius and near to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7213 Euclid in date.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7214 <p>The <I>spire</I> in one of its forms is what we call a <I>tore</I>, or an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7215 anchor-ring. It is generated by the revolution of a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7216 about a straight line in its plane in such a way that the plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7217 of the circle always passes through the axis of revolution. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7218 takes three forms according as the axis of revolution is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7219 (<I>a</I>) altogether outside the circle, when the spire is <I>open</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7220 (<G>diexh/s</G>), (<I>b</I>) a tangent to the circle, when the surface is <I>con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7221 tinuous</I> (<G>sunexh/s</G>), or (<I>c</I>) a chord of the circle, when it is <I>inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7222 laced</I> (<G>e)mpeplegme/nh</G>), or <I>crossing-itself</I> (<G>e)palla/ttousa</G>); an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7223 alternative name for the surface was <G>kri/kos</G>, a <I>ring.</I> Perseus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7224 celebrated his discovery in an epigram to the effect that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7225 ‘Perseus on his discovery of three lines (curves) upon five
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7226 sections gave thanks to the gods therefor’.<note>Proclus on Eucl. I, p. 112. 2.</note> There is some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7227 doubt about the meaning of ‘three lines <I>upon</I> five sections’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7228 (<G>trei=s gramma\s e)pi\ pe/nte tomai=s</G>). We gather from Proclus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7229 account of three sections distinguished by Perseus that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7230 plane of section was always parallel to the axis of revolution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7231 or perpendicular to the plane which cuts the tore symmetri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7232 cally like the division in a split-ring. It is difficult to inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7233 pret the phrase if it means three curves made by five different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7234 sections. Proclus indeed implies that the three curves were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7235 sections of the three kinds of tore respectively (the open, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7236 closed, and the interlaced), but this is evidently a slip.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7237 Tannery interprets the phrase as meaning ‘three curves <I>in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7238 addition to</I> five sections’.<note>See Tannery, <I>Mémoires scientifiques</I>, II, pp. 24-8.</note> Of these the five sections belong
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7239 to the open tore, in which the distance (<I>c</I>) of the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7240 generating circle from the axis of revolution is greater than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7241 the radius (<I>a</I>) of the generating circle. If <I>d</I> be the perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7242 <pb n=205><head>PERSEUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7243 dicular distance of the plane of section from the axis of rota-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7244 tion, we can distinguish the following cases:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7245 <p>(1) <MATH><I>c</I> + <I>a</I> > <I>d</I> > <I>c</I></MATH>. Here the curve is an oval.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7246 <p>(2) <MATH><I>d</I> = <I>c</I></MATH>: transition from case <B>(1)</B> to the next case.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7247 <p>(3) <MATH><I>c</I> > <I>d</I> > <I>c</I> - <I>a</I></MATH>. The curve is now a closed curve narrowest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7248 in the middle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7249 <p>(4) <MATH><I>d</I> = <I>c</I> - <I>a</I></MATH>. In this case the curve is the <I>hippopede</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7250 (horse-fetter), a curve in the shape of the figure of 8. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7251 lemniscate of Bernoulli is a particular case of this curve, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7252 namely in which <I>c</I> = 2<I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7253 <p>(5) <MATH><I>c</I> - <I>a</I> > <I>d</I> > 0</MATH>. In this case the section consists of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7254 ovals symmetrical with one another.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7255 <p>The three curves specified by Proclus are those correspond-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7256 ing to (1), (3) and (4).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7257 <p>When the tore is ‘continuous’ or closed, <I>c</I> = <I>a</I>, and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7258 sections corresponding to (1), (2) and (3) only; (4) reduces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7259 two circles touching one another.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7260 <p>But Tannery finds in the third, the interlaced, form of tore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7261 three new sections corresponding to (1) (2) (3), each with an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7262 oval in the middle. This would make three curves in addi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7263 tion to the five sections, or eight curves in all. We cannot be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7264 certain that this is the true explanation of the phrase in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7265 epigram; but it seems to be the best suggestion that has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7266 been made.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7267 <p>According to Proclus, Perseus worked out the property of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7268 his curves, as Nicomedes did that of the conchoid, Hippias
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7269 that of the <I>quadratrix</I>, and Apollonius those of the three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7270 conic sections. That is, Perseus must have given, in some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7271 form, the equivalent of the Cartesian equation by which we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7272 can represent the different curves in question. If we refer the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7273 tore to three axes of coordinates at right angles to one another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7274 with the centre of the tore as origin, the axis of <I>y</I> being taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7275 to be the axis of revolution, and those of <I>z, x</I> being perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7276 dicular to it in the plane bisecting the tore (making it a split-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7277 ring), the equation of the tore is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7278 <MATH>(<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP> - <I>a</I><SUP>2</SUP>)<SUP>2</SUP> = 4<I>c</I><SUP>2</SUP> (<I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7279 <pb n=206><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7280 where <I>c, a</I> have the same meaning as above. The different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7281 sections parallel to the axis of revolution are obtained by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7282 giving (say) <I>z</I> any value between 0 and <I>c</I> + <I>a.</I> For the value
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7283 <I>z</I> = <I>a</I> the curve is the oval of Cassini which has the property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7284 that, if <I>r, r</I>′ be the distances of any point on the curve from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7285 two fixed points as poles, <I>rr</I>′ = const. For, if <I>z</I> = <I>a</I>, the equa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7286 tion becomes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7287 <MATH>(<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>)<SUP>2</SUP> = 4<I>c</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + 4<I>c</I><SUP>2</SUP><I>a</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7288 or {&horbar;(<I>c</I> - <I>x</I><SUP>2</SUP>) + <I>y</I><SUP>2</SUP>} {&horbar;(<I>c</I> + <I>x</I><SUP>2</SUP>) + <I>y</I><SUP>2</SUP>} = 4<I>c</I><SUP>2</SUP><I>a</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7289 and this is equivalent to <I>rr</I>′ = ± 2<I>ca</I> if <I>x, y</I> are the coordinates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7290 of any point on the curve referred to <I>Ox, Oy</I> as axes, where <I>O</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7291 is the middle point of the line (2<I>c</I> in length) joining the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7292 poles, and <I>Ox</I> lies along that line in either direction, while <I>Oy</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7293 is perpendicular to it. Whether Perseus discussed this case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7294 and arrived at the property in relation to the two poles is of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7295 course quite uncertain.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7296 <C>Isoperimetric figures.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7297 <p>The subject of isoperimetric figures, that is to say, the com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7298 parison of the areas of figures having different shapes but the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7299 same perimeter, was one which would naturally appeal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7300 early Greek mathematicians. We gather from Proclus's notes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7301 on Eucl. I. 36, 37 that those theorems, proving that parallelo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7302 grams or triangles on the same or equal bases and between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7303 the same parallels are equal in area, appeared to the ordinary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7304 person paradoxical because they meant that, by moving the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7305 side opposite to the base in the parallelogram, or the vertex
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7306 of the triangle, to the right or left as far as we please, we may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7307 increase the perimeter of the figure to any extent while keep-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7308 ing the same area. Thus the perimeter in parallelograms or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7309 triangles is in itself no criterion as to their area. Misconcep-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7310 tion on this subject was rife among non-mathematicians.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7311 Proclus tells us of describers of countries who inferred
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7312 the size of cities from their perimeters; he mentions also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7313 certain members of communistic societies in his own time who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7314 cheated their fellow-members by giving them land of greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7315 perimeter but less area than the plots which they took
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7316 <pb n=207><head>ISOPERIMETRIC FIGURES. ZENODORUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7317 themselves, so that, while they got a reputation for greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7318 honesty, they in fact took more than their share of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7319 produce.<note>Proclus on Eucl. I, p. 403. 5 sq.</note> Several remarks by ancient authors show the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7320 prevalence of the same misconception. Thucydides estimates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7321 the size of Sicily according to the time required for circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7322 navigating it.<note>Thuc. vi. 1.</note> About 130 B.C. Polybius observed that there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7323 were people who could not understand that camps of the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7324 periphery might have different capacities.<note>Polybius, ix. 21.</note> Quintilian has a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7325 similar remark, and Cantor thinks he may have had in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7326 mind the calculations of Pliny, who compares the size of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7327 different parts of the earth by adding their lengths to their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7328 breadths.<note>Pliny, <I>Hist. nat</I>, vi. 208.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7329 <p>ZENODORUS wrote, at some date between (say) 200 B.C. and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7330 A.D 90, a treatise <G>peri\ i)some/trwn sxhma/twn</G>, <I>On isometric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7331 figures.</I> A number of propositions from it are preserved in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7332 the commentary of Theon of Alexandria on Book I of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7333 Ptolemy's <I>Syntaxis</I>; and they are reproduced in Latin in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7334 third volume of Hultsch's edition of Pappus, for the purpose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7335 of comparison with Pappus's own exposition of the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7336 propositions at the beginning of his Book V, where he appears
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7337 to have followed Zenodorus pretty closely while making some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7338 changes in detail.<note>Pappus, v, p. 308 sq.</note> From the closeness with which the style
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7339 of Zenodorus follows that of Euclid and Archimedes we may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7340 judge that his date was not much later than that of Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7341 medes, whom he mentions as the author of the proposition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7342 (<I>Measurement of a Circle</I>, Prop. 1) that the area of a circle is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7343 half that of the rectangle contained by the perimeter of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7344 circle and its radius. The important propositions proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7345 Zenodorus and Pappus include the following: (1) <I>Of all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7346 regular polygons of equal perimeter, that is the greatest in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7347 area which has the most angles.</I> (2) <I>A circle is greater than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7348 any regular polygon of equal contour.</I> (3) <I>Of all polygons of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7349 the same number of sides and equal perimeter the equilateral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7350 and equiangular polygon is the greatest in area.</I> Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7351 added the further proposition that <I>Of all segments of a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7352 having the same circumference the semicircle is the greatest in</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7353 <pb n=208><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7354 <I>area.</I> Zenodorus's treatise was not confined to propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7355 about plane figures, but gave also the theorem that <I>Of all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7356 solid figures the surfaces of which are equal, the sphere is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7357 greatest in solid content.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7358 <p>We will briefly indicate Zenodorus's method of proof. To
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7359 begin with (1); let <I>ABC, DEF</I> be equilateral and equiangular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7360 polygons of the same perimeter, <I>DEF</I> having more angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7361 than <I>ABC.</I> Let <I>G, H</I> be the centres of the circumscribing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7362 circles, <I>GK, HL</I> the perpendiculars from <I>G, H</I> to the sides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7363 <I>AB, DE</I>, so that <I>K, L</I> bisect those sides.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7364 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7365 <p>Since the perimeters are equal, <I>AB</I> > <I>DE</I>, and <I>AK</I> > <I>DL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7366 Make <I>KM</I> equal to <I>DL</I> and join <I>GM.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7367 <p>Since <I>AB</I> is the same fraction of the perimeter that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7368 angle <I>AGB</I> is of four right angles, and <I>DE</I> is the same fraction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7369 of the same perimeter that the angle <I>DHE</I> is of four right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7370 angles, it follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7371 <MATH><I>AB</I> : <I>DE</I> = &angle; <I>AGB</I> : &angle; <I>DHE</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7372 that is, <I>AK</I> : <I>MK</I> = &angle; <I>AGK</I> : &angle; <I>DHL</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7373 <p>But <MATH><I>AK</I> : <I>MK</I> > &angle; <I>AGK</I> : &angle; <I>MGK</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7374 (this is easily proved in a lemma following by the usual
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7375 method of drawing an arc of a circle with <I>G</I> as centre and <I>GM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7376 as radius cutting <I>GA</I> and <I>GK</I> produced. The proposition is of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7377 course equivalent to tan <G>a</G> / tan <G>b</G> > <G>a</G> / <G>b</G>, where 1/2<G>p</G> > <G>a</G> > <G>b</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7378 <p>Therefore <MATH>&angle; <I>MGK</I> > &angle; <I>DHL</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7379 and consequently &angle; <I>GMK</I> < &angle; <I>HDL.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7380 <p>Make the angle <I>NMK</I> equal to the angle <I>HDL</I>, so that <I>MN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7381 meets <I>KG</I> produced in <I>N.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7382 <pb n=209><head>ZENODORUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7383 <p>The triangles <I>NMK, HDL</I> are now equal in all respects, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7384 <I>NK</I> is equal to <I>HL</I>, so that <I>GK</I> < <I>HL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7385 <p>But the area of the polygon <I>ABC</I> is half the rectangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7386 contained by <I>GK</I> and the perimeter, while the area of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7387 polygon <I>DEF</I> is half the rectangle contained by <I>HL</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7388 the same perimeter. Therefore the area of the polygon <I>DEF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7389 is the greater.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7390 <p>(2) The proof that a circle is greater than any regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7391 polygon with the same perimeter is deduced immediately from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7392 Archimedes's proposition that the area of a circle is equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7393 to the right-angled triangle with perpendicular side equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7394 the radius and base equal to the perimeter of the circle;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7395 Zenodorus inserts a proof <I>in extenso</I> of Archimedes's pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7396 position, with preliminary lemma. The perpendicular from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7397 the centre of the circle circumscribing the polygon is easily
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7398 proved to be less than the radius of the given circle with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7399 perimeter equal to that of the polygon; whence the proposition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7400 follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7401 <p>(3) The proof of this proposition depends on some pre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7402 liminary lemmas. The first proves that, if there be two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7403 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7404 triangles on the same base and with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7405 same perimeter, one being isosceles and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7406 the other scalene, the isosceles triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7407 has the greater area. (Given the scalene
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7408 triangle <I>BDC</I> on the base <I>BC</I>, it is easy to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7409 draw on <I>BC</I> as base the isosceles triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7410 having the same perimeter. We have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7411 only to take <I>BH</I> equal to 1/2(<I>BD</I> + <I>DC</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7412 bisect <I>BC</I> at <I>E</I>, and erect at <I>E</I> the per-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7413 pendicular <I>AE</I> such that <MATH><I>AE</I><SUP>2</SUP> = <I>BH</I><SUP>2</SUP> - <I>BE</I><SUP>2</SUP></MATH>.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7414 <p>Produce <I>BA</I> to <I>F</I> so that <I>BA</I> = <I>AF</I>, and join <I>AD, DF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7415 <p>Then <MATH><I>BD</I> + <I>DF</I> > <I>BF</I></MATH>, i.e. <I>BA</I> + <I>AC</I>, i.e. <I>BD</I> + <I>DC</I>, by hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7416 thesis; therefore <I>DF</I> > <I>DC</I>, whence in the triangles <I>FAD</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7417 <I>CAD</I> the angle <I>FAD</I> > the angle <I>CAD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7418 <p>Therefore <MATH>&angle; <I>FAD</I> > 1/2&angle; <I>FAC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7419 > &angle; <I>BCA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7420 <p>Make the angle <I>FAG</I> equal to the angle <I>BCA</I> or <I>ABC</I>, so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7421 that <I>AG</I> is parallel to <I>BC</I>; let <I>BD</I> produced meet <I>AG</I> in <I>G</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7422 and join <I>GC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7423 <pb n=210><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7424 <p>Then <MATH>&utri; <I>ABC</I> = &utri; <I>GBC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7425 > &utri; <I>DBC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7426 <p>The second lemma is to the effect that, given two isosceles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7427 triangles not similar to one another, if we construct on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7428 same bases two triangles <I>similar to one another</I> such that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7429 sum of their perimeters is equal to the sum of the perimeters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7430 of the first two triangles, then the sum of the areas of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7431 similar triangles is greater than the sum of the areas of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7432 the non-similar triangles. (The easy construction of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7433 similar triangles is given in a separate lemma.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7434 <p>Let the bases of the isosceles triangles, <I>EB, BC</I>, be placed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7435 one straight line, <I>BC</I> being greater than <I>EB.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7436 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7437 <p>Let <I>ABC, DEB</I> be the similar isosceles triangles, and <I>FBC,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7438 GEB</I> the non-similar, the triangles being such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7439 <MATH><I>BA</I> + <I>AC</I> + <I>ED</I> + <I>DB</I> = <I>BF</I> + <I>FC</I> + <I>EG</I> + <I>GB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7440 <p>Produce <I>AF, GD</I> to meet the bases in <I>K, L.</I> Then clearly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7441 <I>AK, GL</I> bisect <I>BC, EB</I> at right angles at <I>K, L.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7442 <p>Produce <I>GL</I> to <I>H</I>, making <I>LH</I> equal to <I>GL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7443 <p>Join <I>HB</I> and produce it to <I>N</I>; join <I>HF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7444 <p>Now, since the triangles <I>ABC, DEB</I> are similar, the angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7445 <I>ABC</I> is equal to the angle <I>DEB</I> or <I>DBE.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7446 <p>Therefore <MATH>&angle; <I>NBC</I> (= &angle; <I>HBE</I> = &angle; <I>GBE</I>) > &angle; <I>DBE</I> or &angle; <I>ABC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7447 therefore the angle <I>ABH</I>, and <I>a fortiori</I> the angle <I>FBH</I>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7448 less than two right angles, and <I>HF</I> meets <I>BK</I> in some point <I>M.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7449 <pb n=211><head>ZENODORUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7450 <p>Now, by hypothesis, <MATH><I>DB</I> + <I>BA</I> = <I>GB</I> + <I>BF</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7451 therefore <I>DB</I> + <I>BA</I> = <I>HB</I> + <I>BF</I> > <I>HF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7452 <p>By an easy lemma, since the triangles <I>DEB, ABC</I> are similar,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7453 <MATH>(<I>DB</I> + <I>BA</I>)<SUP>2</SUP> = (<I>DL</I> + <I>AK</I>)<SUP>2</SUP> + (<I>BL</I> + <I>BK</I>)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7454 = (<I>DL</I> + <I>AK</I>)<SUP>2</SUP> + <I>LK</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7455 <p>Therefore <MATH>(<I>DL</I> + <I>AK</I>)<SUP>2</SUP> + <I>LK</I><SUP>2</SUP> > <I>HF</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7456 > (<I>GL</I> + <I>FK</I>)<SUP>2</SUP> + <I>LK</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7457 whence <I>DL</I> + <I>AK</I> > <I>GL</I> + <I>FK</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7458 and it follows that <I>AF</I> > <I>GD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7459 <p>But <I>BK</I> > <I>BL</I>; therefore <I>AF.BK</I> > <I>GD.BL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7460 <p>Hence the ‘hollow-angled (figure)’ (<G>koilogw/nion</G>) <I>ABFC</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7461 greater than the hollow-angled (figure) <I>GEDB.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7462 <p>Adding &utri; <I>DEB</I> + &utri; <I>BFC</I> to each, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7463 <MATH>&utri; <I>DEB</I> + &utri; <I>ABC</I> > &utri; <I>GEB</I> + &utri; <I>FBC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7464 <p>The above is the only case taken by Zenodorus. The proof
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7465 still holds if <I>EB</I> = <I>BC</I>, so that <I>BK</I> = <I>BL.</I> But it fails in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7466 case in which <I>EB</I> > <I>BC</I> and the vertex <I>G</I> of the triangle <I>EB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7467 belonging to the non-similar pair is still above <I>D</I> and not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7468 below it (as <I>F</I> is below <I>A</I> in the preceding case). This was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7469 no doubt the reason why Pappus gave a proof intended to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7470 apply to all the cases without distinction. This proof is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7471 same as the above proof by Zenodorus up to the point where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7472 it is proved that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7473 <MATH><I>DL</I> + <I>AK</I> > <I>GL</I> + <I>FK</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7474 but there diverges. Unfortunately the text is bad, and gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7475 no sufficient indication of the course of the proof; but it would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7476 seem that Pappus used the relations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7477 <MATH><I>DL</I> : <I>GL</I> = &utri; <I>DEB</I> : &utri; <I>GEB</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7478 <I>AK</I> : <I>FK</I> = &utri; <I>ABC</I> : &utri; <I>FBC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7479 and <I>AK</I><SUP>2</SUP> : <I>DL</I><SUP>2</SUP> = &utri; <I>ABC</I> : &utri; <I>DEB</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7480 combined of course with the fact that <MATH><I>GB</I> + <I>BF</I> = <I>DB</I> + <I>BA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7481 in order to prove the proposition that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7482 according as <MATH><I>DL</I> + <I>AK</I> > or < <I>GL</I> + <I>FK</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7483 &utri; <I>DEB</I> + &utri; <I>ABC</I> > or < &utri; <I>GEB</I> + &utri; <I>FBC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7484 <pb n=212><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7485 <p>The proof of his proposition, whatever it was, Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7486 indicates that he will give later; but in the text as we have it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7487 the promise is not fulfilled.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7488 <p>Then follows the proof that the maximum polygon of given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7489 perimeter is both equilateral and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7490 equiangular.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7491 <p>(1) It is equilateral.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7492 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7493 <p>For, if not, let two sides of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7494 maximum polygon, as <I>AB, BC</I>, be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7495 unequal. Join <I>AC</I>, and on <I>AC</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7496 base draw the isosceles triangle <I>AFC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7497 such that <I>AF</I> + <I>FC</I> = <I>AB</I> + <I>BC.</I> The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7498 area of the triangle <I>AFC</I> is then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7499 greater than the area of the triangle <I>ABC</I>, and the area of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7500 the whole polygon has been increased by the construc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7501 tion: which is impossible, as by hypothesis the area is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7502 maximum.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7503 <p>Similarly it can be proved that no other side is unequal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7504 to any other.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7505 <p>(2) It is also equiangular.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7506 <p>For, if possible, let the maximum polygon <I>ABCDE</I> (which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7507 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7508 we have proved to be equilateral)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7509 have the angle at <I>B</I> greater than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7510 the angle at <I>D.</I> Then <I>BAC, DEC</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7511 non-similar isosceles triangles. On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7512 <I>AC, CE</I> as bases describe the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7513 isosceles triangles <I>FAC, GEC</I> similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7514 to one another which have the sum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7515 of their perimeters equal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7516 sum of the perimeters of <I>BAC,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7517 DEC</I>. Then the sum of the areas of the two similar isosceles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7518 triangles is greater than the sum of the areas of the triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7519 <I>BAC, DEC</I>; the area of the polygon is therefore increased,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7520 which is contrary to the hypothesis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7521 <p>Hence no two angles of the polygon can be unequal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7522 <p>The maximum polygon of given perimeter is therefore both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7523 equilateral and equiangular.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7524 <p>Dealing with the sphere in relation to other solids having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7525 <pb n=213><head>ZENODORUS. HYPSICLES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7526 their surfaces equal to that of the sphere, Zenodorus confined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7527 himself to proving (1) that the sphere is greater if the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7528 solid with surface equal to that of the sphere is a solid formed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7529 by the revolution of a regular polygon about a diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7530 bisecting it as in Archimedes, <I>On the Sphere and Cylinder</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7531 Book I, and (2) that the sphere is greater than any of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7532 the regular solids having its surface equal to that of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7533 sphere.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7534 <p>Pappus's treatment of the subject is more complete in that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7535 he proves that the sphere is greater than the cone or cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7536 the surface of which is equal to that of the sphere, and further
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7537 that of the five regular solids which have the same surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7538 that which has more faces is the greater.<note>Pappus, v, Props. 19, 38-56.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7539 <p>HYPSICLES (second half of second century B.C.) has already
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7540 been mentioned (vol. i, pp. 419-20) as the author of the con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7541 tinuation of the <I>Elements</I> known as Book XIV. He is quoted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7542 by Diophantus as having given a definition of a polygonal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7543 number as follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7544 <p>‘If there are as many numbers as we please beginning from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7545 1 and increasing by the same common difference, then, when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7546 the common difference is 1, the sum of all the numbers is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7547 a triangular number; when 2, a square; when 3, a pentagonal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7548 number [and so on]. And the number of angles is called
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7549 after the number which exceeds the common difference by 2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7550 and the side after the number of terms including 1.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7551 <p>This definition amounts to saying that the <I>n</I>th <I>a</I>-gonal num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7552 ber (1 counting as the first) is <MATH>1/2<I>n</I> {2 + (<I>n</I>-1) (<I>a</I>-2)}</MATH>. If, as is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7553 probable, Hypsicles wrote a treatise on polygonal numbers, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7554 has not survived. On the other hand, the <G>*)anaforiko/s</G> (<I>Ascen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7555 siones</I>) known by his name has survived in Greek as well as in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7556 Arabic, and has been edited with translation.<note>Manitius, <I>Des Hypsikles Schrift Anaphorikos</I>, Dresden, Lehmannsche Buchdruckerei, 1888.</note> True, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7557 treatise (if it really be by Hypsicles, and not a clumsy effort
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7558 by a beginner working from an original by Hypsicles)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7559 does no credit to its author; but it is in some respects
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7560 interesting, and in particular because it is the first Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7561 <pb n=214><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7562 work in which we find the division of the ecliptic circle into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7563 360 ‘parts’ or degrees. The author says, after the preliminary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7564 propositions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7565 <p>‘The circle of the zodiac having been divided into 360 equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7566 circumferences (arcs), let each of the latter be called a <I>degree
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7567 in space</I> (<G>moi=ra topikh/</G>, ‘local’ or ‘spatial part’). And simi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7568 larly, supposing that the time in which the zodiac circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7569 returns to any position it has left is divided into 360 equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7570 times, let each of these be called a <I>degree in time</I> (<G>moi=ra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7571 xronikh/</G>).’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7572 <p>From the word <G>kalei/sqw</G> (‘let it be called’) we may perhaps
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7573 infer that the terms were new in Greece. This brings us to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7574 the question of the origin of the division (1) of the circle of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7575 the zodiac, (2) of the circle in general, into 360 parts. On this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7576 question innumerable suggestions have been made. With
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7577 reference to (1) it was suggested as long ago as 1788 (by For-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7578 maleoni) that the division was meant to correspond to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7579 number of days in the year. Another suggestion is that it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7580 would early be discovered that, in the case of any circle the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7581 inscribed hexagon dividing the circumference into six parts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7582 has each of its sides equal to the radius, and that this would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7583 naturally lead to the circle being regularly divided into six
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7584 parts; after this, the very ancient sexagesimal system would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7585 naturally come into operation and each of the parts would be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7586 divided into 60 subdivisions, giving 360 of these for the whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7587 circle. Again, there is an explanation which is not even
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7588 geometrical, namely that in the Babylonian numeral system,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7589 which combined the use of 6 and 10 as bases, the numbers 6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7590 60, 360, 3600 were fundamental round numbers, and these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7591 numbers were transferred from arithmetic to the heavens.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7592 The obvious objection to the first of these explanations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7593 (referring the 360 to the number of days in the solar year) is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7594 that the Babylonians were well acquainted, as far back as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7595 monuments go, with 365.2 as the number of days in the year.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7596 A variant of the hexagon-theory is the suggestion that a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7597 <I>natural</I> angle to be discovered, and to serve as a measure of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7598 others, is the angle of an equilateral triangle, found by draw-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7599 ing a star * like a six-spoked wheel without any circle. If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7600 the base of a sundial was so divided into six angles, it would be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7601 <pb n=215><head>HYPSICLES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7602 natural to divide each of the sixth parts into either 10 or 60
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7603 parts; the former division would account for the attested
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7604 division of the day into 60 hours, while the latter division on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7605 the sexagesimal system would give the 360 time-degrees (each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7606 of 4 minutes) making up the day of 24 hours. The purely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7607 arithmetical explanation is defective in that the series of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7608 numbers for which the Babylonians had special names is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7609 60, 360, 3600 but 60 (Soss), <I>600</I> (Ner), and 3600 or 60<SUP>2</SUP> (Sar).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7610 On the whole, after all that has been said, I know of no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7611 better suggestion than that of Tannery.<note>Tannery, ‘La coudée astronomique et les anciennes divisions du cercle’ (<I>Mémoires scientifiques</I>, ii, pp. 256-68).</note> It is certain that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7612 both the division of the ecliptic into 360 degrees and that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7613 the <G>nuxqh/meron</G> into 360 time-degrees were adopted by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7614 Greeks from Babylon. Now the earliest division of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7615 ecliptic was into 12 parts, the signs, and the question is, how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7616 were the signs subdivided? Tannery observes that, accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7617 ing to the cuneiform inscriptions, as well as the testimony of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7618 Greek authors, the sign was divided into parts one of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7619 (<I>dargatu</I>) was double of the other (<I>murran</I>), the former being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7620 1/30th, the other (called <I>stadium</I> by Manilius) 1/60th, of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7621 sign; the former division would give 360 parts, the latter 720
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7622 parts for the whole circle. The latter division was more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7623 natural, in view of the long-established system of sexagesimal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7624 fractions; it also had the advantage of corresponding toler-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7625 ably closely to the apparent diameter of the sun in comparison
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7626 with the circumference of the sun's apparent circle. But, on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7627 the other hand, the double fraction, the 1/30th, was contained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7628 in the circle of the zodiac approximately the same number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7629 times as there are days in the year, and consequently corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7630 sponded nearly to the distance described by the sun along the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7631 zodiac in one day. It would seem that this advantage was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7632 sufficient to turn the scale in favour of diyiding each sign of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7633 the zodiac into 30 parts, giving 360 parts for the whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7634 circle. While the Chaldaeans thus divided the ecliptic into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7635 360 parts, it does not appear that they applied the same divi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7636 sion to the equator or any other circle. They measured angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7637 in general by <I>ells</I>, an ell representing 2°, so that the complete
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7638 circle contained 180, not 360, parts, which they called ells.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7639 The explanation may perhaps be that the Chaldaeans divided
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7640 <pb n=216><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7641 the <I>diameter</I> of the circle into 60 ells in accordance with their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7642 usual sexagesimal division, and then came to divide the cir-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7643 cumference into 180 such ells on the ground that the circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7644 ference is roughly three times the diameter. The measure-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7645 ment in <I>ells</I> and <I>dactyli</I> (of which there were 24 to the ell)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7646 survives in Hipparchus (<I>On the Phaenomena of Eudoxus and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7647 Aratus</I>), and some measurements in terms of the same units
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7648 are given by Ptolemy. It was Hipparchus who first divided
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7649 the circle in general into 360 parts or degrees, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7650 introduction of this division coincides with his invention of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7651 trigonometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7652 <p>The contents of Hypsicles's tract need not detain us long.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7653 The problem is: If we know the ratio which the length of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7654 longest day bears to the length of the shortest day at any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7655 given place, to find how many time-degrees it takes any given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7656 sign to rise; and, after this has been found, the author finds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7657 what length of time it takes any given degree in any sign to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7658 rise, i.e. the interval between the rising of one degree-point on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7659 the ecliptic and that of the next following. It is explained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7660 that the longest day is the time during which one half of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7661 zodiac (Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius) rises,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7662 and the shortest day the time during which the other half
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7663 (Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini) rises.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7664 Now at Alexandria the longest day is to the shortest as 7
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7665 to 5; the longest therefore contains 210 ‘time-degrees’, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7666 shortest 150. The two quadrants Cancer-Virgo and Libra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7667 Sagittarius take the same time to rise, namely 105 time-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7668 degrees, and the two quadrants Capricornus-Pisces and Aries-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7669 Gemini each take the same time, namely 75 time-degrees.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7670 It is further assumed that the times taken by Virgo, Leo,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7671 Cancer, Gemini, Taurus, Aries are in descending arithmetical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7672 progression, while the times taken by Libra, Scorpio, Sagit-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7673 tarius, Capricornus, Aquarius, Pisces continue the same de-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7674 scending arithmetical series. The following lemmas are used
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7675 and proved:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7676 <p>I. If <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB><I>n</I></SUB>, <I>a</I><SUB><I>n</I>+1</SUB>, <I>a</I><SUB><I>n</I>+2</SUB> ... <I>a</I><SUB>2<I>n</I></SUB> is a descending arithmeti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7677 cal progression of 2<I>n</I> terms with <G>d</G> (= <I>a</I><SUB>1</SUB> - <I>a</I><SUB>2</SUB> = <I>a</I><SUB>2</SUB> - <I>a</I><SUB>3</SUB> = ...)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7678 as common difference,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7679 <MATH><I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> + ... + <I>a</I><SUB><I>n</I></SUB> - (<I>a</I><SUB><I>n</I>+1</SUB> + <I>a</I><SUB><I>n</I>+2</SUB> + ... + <I>a</I><SUB>2<I>n</I></SUB>) = <I>n</I><SUP>2</SUP><G>d</G></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7680 <pb n=217><head>HYPSICLES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7681 <p>II. If <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB><I>n</I></SUB> ... <I>a</I><SUB>2<I>n</I>-1</SUB> is a descending arithmetical pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7682 gression of 2<SUB><I>n</I>-1</SUB> terms with <G>d</G> as common difference and <I>a</I><SUB><I>n</I></SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7683 is the middle term, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7684 <MATH><I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> + ... + <I>a</I><SUB>2<I>n</I>-1</SUB> = (2<I>n</I> - 1)<I>a</I><SUB><I>n</I></SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7685 <p>III. If <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB><I>n</I></SUB>, <I>a</I><SUB><I>n</I>+1</SUB> ... <I>a</I><SUB>2<I>n</I></SUB> is a descending arithmetical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7686 progression of 2<I>n</I> terms, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7687 <MATH><I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> + ... + <I>a</I><SUB>2<I>n</I></SUB> = <I>n</I>(<I>a</I><SUB>1</SUB> + <I>a</I><SUB>2<I>n</I></SUB>) = <I>n</I>(<I>a</I><SUB>2</SUB> + <I>a</I><SUB>2<I>n</I>-1</SUB>) = ...
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7688 = <I>n</I>(<I>a</I><SUB><I>n</I></SUB> + <I>a</I><SUB><I>n</I>+1</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7689 <p>Now let <I>A, B, C</I> be the descending series the sum of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7690 is 105, and <I>D, E, F</I> the next three terms in the same series
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7691 the sum of which is 75, the common difference being <G>d</G>; we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7692 then have, by (I),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7693 <MATH><I>A</I> + <I>B</I> + <I>C</I> - (<I>D</I> + <I>E</I> + <I>F</I>) = 9<G>d</G>, or 30 = 9<G>d</G></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7694 and <G>d</G> = 3 1/3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7695 <p>Next, by (II), <MATH><I>A</I> + <I>B</I> + <I>C</I> = 3<I>B</I></MATH>, or 3<I>B</I> = 105, and <I>B</I> = 35;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7696 therefore <I>A, B, C, D, E, F</I> are equal to 38 1/3, 35, 31 2/3, 28 1/3, 25,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7697 21 2/3 time-degrees respectively, which the author of the tract
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7698 expresses in time-degrees and minutes as 38<SUP><I>t</I></SUP> 20′, 35<SUP><I>t</I></SUP>, 31<SUP><I>t</I></SUP> 40′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7699 28<SUP><I>t</I></SUP> 20′, 25<SUP><I>t</I></SUP>, 21<SUP><I>t</I></SUP> 40′. We have now to carry through the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7700 procedure for each degree in each sign. If the difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7701 between the times taken to rise by one sign and the next
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7702 is 3<SUP><I>t</I></SUP> 20′, what is the difference for each of the 30 degrees in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7703 the sign? We have here 30 terms followed by 30 other terms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7704 of the same descending arithmetical progression, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7705 formula (I) gives <MATH>3<SUP><I>t</I></SUP> . 20′ = (30)<SUP>2</SUP><I>d</I></MATH>, where <I>d</I> is the common
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7706 difference; therefore <MATH><I>d</I> = 1/900 X 3<SUP><I>t</I></SUP> . 20′ = 0<SUP><I>t</I></SUP>0′13″20‴</MATH>. Lastly,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7707 take the sign corresponding to 21<SUP><I>t</I></SUP> 40′. This is the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7708 a descending arithmetical progression of 30 terms <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB>30</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7709 with common difference 0<SUP><I>t</I></SUP>0′13″20‴. Therefore, by (III),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7710 <MATH>21<SUP><I>t</I></SUP> 40′ = 15 (<I>a</I><SUB>1</SUB> + <I>a</I><SUB>30</SUB>)</MATH>, whence <MATH><I>a</I><SUB>1</SUB> + <I>a</I><SUB>30</SUB> = 1<SUP><I>t</I></SUP> 26′ 40″</MATH>. Now,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7711 since there are 30 terms <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB>30</SUB>, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7712 <MATH><I>a</I><SUB>1</SUB> - <I>a</I><SUB>30</SUB> = 29<I>d</I> = 0<SUP><I>t</I></SUP> 6′ 26″ 40‴</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7713 It follows that <I>a</I><SUB>30</SUB> = 0<SUP><I>t</I></SUP> 40′ 6″ 40‴ and <I>a</I><SUB>1</SUB> = 0<SUP><I>t</I></SUP> 46′ 33″ 20‴,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7714 <pb n=218><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7715 and from these and the common difference 0<SUP><I>t</I></SUP> 0′ 13″ 20‴ all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7716 the times corresponding to all the degrees in the circle can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7717 found.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7718 <p>The procedure was probably, as Tannery thinks, taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7719 direct from the Babylonians, who would no doubt use it for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7720 the purpose of enabling the time to be determined at any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7721 hour of the night. Another view is that the object was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7722 astrological rather than astronomical (Manitius). In either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7723 case the method was exceedingly rough, and the assumed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7724 increases and decreases in the times of the risings of the signs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7725 in arithmetical progression are not in accordance with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7726 facts. The book could only have been written before the in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7727 vention of trigonometry by Hipparchus, for the problem of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7728 finding the times of rising of the signs is really one of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7729 spherical trigonometry, and these times were actually cal-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7730 culated by Hipparchus and Ptolemy by means of tables of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7731 chords.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7732 <p>DIONYSODORUS is known in the first place as the author of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7733 a solution of the cubic equation subsidiary to the problem of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7734 Archimedes, <I>On the Sphere and Cylinder</I>, II. 4, To cut a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7735 sphere by a plane so that the volumes of the segments have to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7736 one another a given ratio (see above, p. 46). Up to recently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7737 this Dionysodorus was supposed to be Dionysodorus of Amisene
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7738 in Pontus, whom Suidas describes as ‘a mathematician worthy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7739 of mention in the field of education’. But we now learn from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7740 a fragment of the Herculaneum Roll, No. 1044, that ‘Philonides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7741 was a pupil, first of Eudemus, and afterwards of Dionysodorus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7742 the son of Dionysodorus the Caunian’. Now Eudemus is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7743 evidently Eudemus of Pergamum to whom Apollonius dedi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7744 cated the first two Books of his <I>Conics</I>, and Apollonius actually
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7745 asks him to show Book II to Philonides. In another frag-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7746 ment Philonides is said to have published some lectures by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7747 Dionysodorus. Hence our Dionysodorus may be Dionysodorus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7748 of Caunus and a contemporary of Apollonius, or very little
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7749 later.<note>W. Schmidt in <I>Bibliotheca mathematica</I>, iv<SUB>3</SUB>, pp. 321-5.</note> A Dionysodorus is also mentioned by Heron<note>Heron, <I>Metrica</I>, ii. 13, p. 128. 3.</note> as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7750 author of a tract <I>On the Spire</I> (or tore) in which he proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7751 that, if <I>d</I> be the diameter of the revolving circle which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7752 <pb n=219><head>DIONYSODORUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7753 generates the tore, and <I>c</I> the distance of its centre from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7754 axis of revolution,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7755 <MATH>(volume of tore):<G>p</G><I>c</I><SUP>2</SUP> . <I>d</I> = 1/4<G>p</G><I>d</I><SUP>2</SUP>:1/2<I>cd</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7756 that is, <MATH>(volume of tore) = 1/2<G>p</G><SUP>2</SUP> . <I>cd</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7757 which is of course the product of the area of the generating
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7758 circle and the length of the path of its centre of gravity. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7759 form in which the result is stated, namely that the tore is to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7760 the cylinder with height <I>d</I> and radius <I>c</I> as the generating
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7761 circle of the tore is to half the parallelogram <I>cd</I>, indicates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7762 quite clearly that Dionysodorus proved his result by the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7763 procedure as that employed by Archimedes in the <I>Method</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7764 in the book <I>On Conoids and Spheroids</I>; and indeed the proof
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7765 on Archimedean lines is not difficult.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7766 <p>Before passing to the mathematicians who are identified
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7767 with the discovery and development of trigonometry, it will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7768 be convenient, I think, to dispose of two more mathematicians
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7769 belonging to the last century B.C., although this involves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7770 a slight departure from chronological order; I mean Posidonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7771 and Geminus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7772 <p>POSIDONIUS, a Stoic, the teacher of Cicero, is known as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7773 Posidonius of Apamea (where he was born) or of Rhodes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7774 (where he taught); his date may be taken as approximately
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7775 135-51 B.C. In pure mathematics he is mainly quoted as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7776 author of certain definitions, or for views on technical terms,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7777 e.g. ‘theorem’ and ‘problem’, and subjects belonging to ele-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7778 mentary geometry. More important were his contributions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7779 to mathematical geography and astronomy. He gave his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7780 great work on geography the title <I>On the Ocean</I>, using the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7781 word which had always had such a fascination for the Greeks;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7782 its contents are known to us through the copious quotations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7783 from it in Strabo; it dealt with physical as well as mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7784 matical geography, the zones, the tides and their connexion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7785 with the moon, ethnography and all sorts of observations made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7786 during extensive travels. His astronomical book bore the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7787 title <I>Meteorologica</I> or <G>peri\ metew/rwn</G>, and, while Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7788 wrote a commentary on or exposition of this work, we may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7789 assign to it a number of views quoted from Posidonius in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7790 <pb n=220><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7791 Cleomedes's work <I>De motu circulari corporum caelestium.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7792 Posidonius also wrote a separate tract on the size of the sun.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7793 <p>The two things which are sufficiently important to deserve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7794 mention here are (1) Posidonius's measurement of the circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7795 ference of the earth, (2) his hypothesis as to the distance and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7796 size of the sun.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7797 <p>(1) He estimated the circumference of the earth in this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7798 way. He assumed (according to Cleomedes<note>Cleomedes, <I>De motu circulari</I>, i. 10, pp. 92-4.</note>) that, whereas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7799 the star Canopus, invisible in Greece, was just seen to graze the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7800 horizon at Rhodes, rising and setting again immediately, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7801 meridian altitude of the same star at Alexandria was ‘a fourth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7802 part of a sign, that is, one forty-eighth part of the zodiac
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7803 circle’ (= 7 1/2°); and he observed that the distance between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7804 the two places (supposed to lie on the same meridian) ‘was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7805 considered to be 5,000 stades’. The circumference of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7806 earth was thus made out to be 240,000 stades. Unfortunately
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7807 the estimate of the difference of latitude, 7 1/2°, was very far
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7808 from correct, the true difference being 5 1/4° only; moreover
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7809 the estimate of 5,000 stades for the distance was incorrect,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7810 being only the maximum estimate put upon it by mariners,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7811 while some put it at 4,000 and Eratosthenes, by observations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7812 of the shadows of gnomons, found it to be 3,750 stades only.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7813 Strabo, on the other hand, says that Posidonius favoured ‘the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7814 latest of the measurements which gave the smallest dimen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7815 sions to the earth, namely about 180,000 stades’.<note>Strabo, ii. c. 95.</note> This is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7816 evidently 48 times 3,750, so that Posidonius combined Erato-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7817 sthenes's figure of 3,750 stades with the incorrect estimate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7818 of 7 1/2° for the difference of latitude, although Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7819 presumably obtained the figure of 3,750 stades from his own
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7820 estimate (250,000 or 252,000) of the circumference of the earth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7821 combined with an estimate of the difference of latitude which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7822 was about 5 2/5° and therefore near the truth.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7823 <p>(2) Cleomedes<note>Cleomedes, <I>op. cit.</I> ii. 1, pp. 144-6, p. 98. 1-5.</note> tells us that Posidonius supposed the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7824 in which the sun apparently moves round the earth to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7825 10,000 times the size of a circular section of the earth through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7826 its centre, and that with this assumption he combined the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7827 <pb n=221><head>POSIDONIUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7828 statement of Eratosthenes (based apparently upon hearsay)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7829 that at Syene, which is under the summer tropic, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7830 throughout a circle round it of 300 stades in diameter, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7831 upright gnomon throws no shadow at noon. It follows from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7832 this that the diameter of the sun occupies a portion of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7833 sun's circle 3,000,000 stades in length; in other words, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7834 diameter of the sun is 3,000,000 stades. The assumption that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7835 the sun's circle is 10,000 times as large as a great circle of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7836 earth was presumably taken from Archimedes, who had proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7837 in the <I>Sand-reckoner</I> that the diameter of the sun's orbit is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7838 <I>less</I> than 10,000 times that of the earth; Posidonius in fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7839 took the maximum value to be the true value; but his esti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7840 mate of the sun's size is far nearer the truth than the estimates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7841 of Aristarchus, Hipparchus, and Ptolemy. Expressed in terms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7842 of the mean diameter of the earth, the estimates of these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7843 astronomers give for the diameter of the sun the figures 6 3/4,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7844 12 1/3, and 5 1/2 respectively; Posidonius's estimate gives 39 1/4, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7845 true figure being 108.9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7846 <p>In elementary geometry Posidonius is credited by Proclus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7847 with certain definitions. He defined ‘figure’ as ‘confining
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7848 limit’ (<G>pe/ras sugklei=on</G>)<note>Proclus on Eucl. I, p. 143. 8.</note> and ‘parallels’ as ‘those lines which,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7849 being in one plane, neither converge nor diverge, but have all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7850 the perpendiculars equal which are drawn from the points of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7851 one line to the other’.<note><I>Ib.</I>, p. 176. 6-10.</note> (Both these definitions are included
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7852 in the <I>Definitions</I> of Heron.) He also distinguished seven
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7853 species of quadrilaterals, and had views on the distinction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7854 between <I>theorem</I> and <I>problem.</I> Another indication of his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7855 interest in the fundamentals of elementary geometry is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7856 fact<note><I>Ib.</I>, pp. 199. 14-200. 3.</note> that he wrote a separate work in refutation of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7857 Epicurean Zeno of Sidon, who had objected to the very begin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7858 nings of the <I>Elements</I> on the ground that they contained un-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7859 proved assumptions. Thus, said Zeno, even Eucl.I. 1 requires it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7860 to be admitted that ‘two straight lines cannot have a common
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7861 segment’; and, as regards the ‘proof’ of this fact deduced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7862 from the bisection of a circle by its diameter, he would object
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7863 that it has to be assumed that two arcs of circles cannot have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7864 a common part. Zeno argued generally that, even if we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7865 admit the fundamental principles of geometry, the deductions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7866 <pb n=222><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7867 from them cannot be proved without the admission of some-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7868 thing else as well which has not been included in the said
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7869 principles, and he intended by means of these criticisms to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7870 destroy the whole of geometry.<note>Proclus on Eucl. I, pp. 214. 18-215. 13, p. 216. 10-19, p. 217. 10-23.</note> We can understand, there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7871 fore, that the tract of Posidonius was a serious work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7872 <p>A definition of the centre of gravity by one ‘Posidonius a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7873 Stoic’ is quoted in Heron's <I>Mechanics</I>, but, as the writer goes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7874 on to say that Archimedes introduced a further distinction, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7875 may fairly assume that the Posidonius in question is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7876 Posidonius of Rhodes, but another, perhaps Posidonius of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7877 Alexandria, a pupil of Zeno of Cittium in the third cen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7878 tury B.C.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7879 <p>We now come to GEMINUS, a very important authority on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7880 many questions belonging to the history of mathematics, as is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7881 shown by the numerous quotations from him in Proclus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7882 <I>Commentary on Euclid, Book I.</I> His date and birthplace are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7883 uncertain, and the discussions on the subject now form a whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7884 literature for which reference must be made to Manitius's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7885 edition of the so-called <I>Gemini elementa astronomiae</I> (Teubner,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7886 1898) and the article ‘Geminus’ in Pauly-Wissowa's <I>Real-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7887 Encyclopädie.</I> The doubts begin with his name. Petau, who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7888 included the treatise mentioned in his <I>Uranologion</I> (Paris,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7889 1630), took it to be the Latin Gem&icaron;nus. Manitius, the latest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7890 editor, satisfied himself that it was Gem&imacr;nus, a Greek name,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7891 judging from the fact that it consistently appears with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7892 properispomenon accent in Greek (<G>*gemi=nos</G>), while it is also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7893 found in inscriptions with the spelling <G>*gemei=nos</G>; Manitius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7894 suggests the derivation from <G>gem</G>, as <G>*)ergi=vos</G> from <G>e)rg</G>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7895 <G>*)alexi=nos</G> from <G>a)lex</G>; he compares also the unmistakably
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7896 Greek names <G>)ikti=nos, krati=nos</G>. Now, however, we are told
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7897 (by Tittel) that the name is, after all, the Latin Gém&ibreve;nus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7898 that <G>*gemi=nos</G> came to be so written through false analogy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7899 with <G>alexi=nos</G>, &c., and that <G>ge[m]ei=nos</G>, if the reading is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7900 correct, is also wrongly formed on the model of <G>*antwnei=nos,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7901 *agrippei/na</G>. The occurrence of a Latin name in a centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7902 of Greek culture need not surprise us, since Romans settled in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7903 such centres in large numbers during the last century B.C.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7904 Geminus, however, in spite of his name, was thoroughly Greek.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7905 <pb n=223><head>GEMINUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7906 An upper limit for his date is furnished by the fact that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7907 wrote a commentary on or exposition of Posidonius's work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7908 <G>peri\ metew/rwn</G>; on the other hand, Alexander Aphrodisiensis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7909 (about A.D. 210) quotes an important passage from an ‘epitome’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7910 of this <G>e)xh/ghsis</G> by Geminus. The view most generally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7911 accepted is that he was a Stoic philosopher, born probably
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7912 in the island of Rhodes, and a pupil of Posidonius, and that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7913 he wrote about 73-67 B.C.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7914 <p>Of Geminus's works that which has most interest for us
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7915 is a comprehensive work on mathematics. Proclus, though
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7916 he makes great use of it, does not mention its title, unless
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7917 indeed, in the passage where, after quoting from Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7918 a classification of lines which never meet, he says ‘these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7919 remarks I have selected from the <G>filokali/a</G> of Geminus’,<note>Proclus on Eucl. I, p. 177. 24.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7920 the word <G>filokali/a</G> is a title or an alternative title. Pappus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7921 however, quotes a work of Geminus ‘on the classification of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7922 the mathematics’ (<G>e)n tw=\| peri\ th=s tw=n maqhma/twn ta/xews</G>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7923 while Eutocius quotes from ‘the sixth book of the doctrine of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7924 the mathematics’ (<G>e)n tw=\| e(/ktw\| th=s tw=n maqhma/twn qewri/as</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7925 The former title corresponds well enough to the long extract
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7926 on the division of the mathematical sciences into arithmetic,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7927 geometry, mechanics, astronomy, optics, geodesy, canonic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7928 (musical harmony) and logistic which Proclus gives in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7929 first prologue, and also to the fragments contained in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7930 <I>Anonymi variae collectiones</I> published by Hultsch in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7931 edition of Heron; but it does not suit most of the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7932 passages borrowed by Proclus. The correct title was most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7933 probably that given by Eutocius, <I>The Doctrine</I>, or <I>Theory,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7934 of the Mathematics</I>; and Pappus probably refers to one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7935 particular section of the work, say the first Book. If the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7936 sixth Book treated of conics, as we may conclude from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7937 Eutocius's reference, there must have been more Books to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7938 follow; for Proclus has preserved us details about higher
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7939 curves, which must have come later. If again Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7940 finished his work and wrote with the same fullness about the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7941 other branches of mathematics as he did about geometry,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7942 there must have been a considerable number of Books
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7943 altogether. It seems to have been designed to give a com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7944 plete view of the whole science of mathematics, and in fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7945 <pb n=224><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7946 to have been a sort of encyclopaedia of the subject. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7947 quotations of Proclus from Geminus's work do not stand
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7948 alone; we have other collections of extracts, some more and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7949 some less extensive, and showing varieties of tradition accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7950 ing to the channel through which they came down. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7951 scholia to Euclid's <I>Elements</I>, Book I, contain a considerable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7952 part of the commentary on the Definitions of Book I, and are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7953 valuable in that they give Geminus pure and simple, whereas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7954 Proclus includes extracts from other authors. Extracts from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7955 Geminus of considerable length are included in the Arabic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7956 commentary by an-Nair&imacr;z&imacr; (about A.D. 900) who got them
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7957 through the medium of Greek commentaries on Euclid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7958 especially that of Simplicius. It does not appear to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7959 doubted any longer that ‘Aganis’ in an-Nair&imacr;z&imacr; is really
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7960 Geminus; this is inferred from the close agreement between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7961 an-Nair&imacr;z&imacr;'s quotations from ‘Aganis’ and the correspond-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7962 ing passages in Proclus; the difficulty caused by the fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7963 that Simplicius calls Agains ‘socius noster’ is met by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7964 suggestion that the particular word <I>socius</I> is either the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7965 result of the double translation from the Greek or means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7966 nothing more, in the mouth of Simplicius, than ‘colleague’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7967 in the sense of a worker in the same field, or ‘authority’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7968 A few extracts again are included in the <I>Anonymi variae
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7969 collectiones</I> in Hultsch's <I>Heron.</I> Nos. 5-14 give definitions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7970 geometry, logistic, geodesy and their subject-matter, remarks
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7971 on bodies as continuous magnitudes, the three dimensions as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7972 ‘principles’ of geometry, the purpose of geometry, and lastly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7973 on optics, with its subdivisions, optics proper, <I>Catoptrica</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7974 <G>skhnografikh/</G>, scene-painting (a sort of perspective), with some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7975 fundamental principles of optics, e.g. that all light travels
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7976 along straight lines (which are broken in the cases of reflection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7977 and refraction), and the division between optics and natural
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7978 philosophy (the theory of light), it being the province of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7979 latter to investigate (what is a matter of indifference to optics)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7980 whether (1) visual rays issue from the eye, (2) images proceed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7981 from the object and impinge on the eye, or (3) the intervening
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7982 air is aligned or compacted with the beam-like breath or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7983 emanation from the eye.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7984 <p>Nos. 80-6 again in the same collection give the Peripatetic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7985 explanation of the name mathematics, adding that the term
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7986 <pb n=225><head>GEMINUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7987 was applied by the early Pythagoreans more particularly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7988 to geometry and arithmetic, sciences which deal with the pure,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7989 the eternal and the unchangeable, but was extended by later
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7990 writers to cover what we call ‘mixed’ or applied mathematics,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7991 which, though theoretical, has to do with sensible objects, e.g.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7992 astronomy and optics. Other extracts from Geminus are found
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7993 in extant manuscripts in connexion with Damianus's treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7994 on optics (published by R. Schöne, Berlin, 1897). The defini-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7995 tions of logistic and geometry also appear, but with decided
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7996 differences, in the scholia to Plato's <I>Charmides</I> 165 E. Lastly,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7997 isolated extracts appear in Eutocius, (1) a remark reproduced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7998 in the commentary on Archimedes's <I>Plane Equilibriums</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	7999 the effect that Archimedes in that work gave the name of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8000 postulates to what are really axioms, (2) the statement that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8001 before Apollonius's time the conics were produced by cutting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8002 different cones (right-angled, acute-angled, and obtuse-angled)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8003 by sections perpendicular in each case to a generator.<note>Eutocius, <I>Comm. on Apollonius's Conics, ad init.</I></note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8004 <p>The object of Geminus's work was evidently the examina-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8005 tion of the first principles, the logical building up of mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8006 matics on the basis of those admitted principles, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8007 defence of the whole structure against the criticisms of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8008 the enemies of the science, the Epicureans and Sceptics, some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8009 of whom questioned the unproved principles, and others the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8010 logical validity of the deductions from them. Thus in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8011 geometry Geminus dealt first with the principles or hypotheses
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8012 (<G>a)rxai/, u(poqe/seis</G>) and then with the logical deductions, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8013 theorems and problems (<G>ta\ meta\ ta\s a)rxa/s</G>). The distinction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8014 is between the things which must be taken for granted but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8015 are incapable of proof and the things which must not be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8016 assumed but are matter for demonstration. The principles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8017 consisting of definitions, postulates, and axioms, Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8018 subjected them severally to a critical examination from this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8019 point of view, distinguishing carefully between postulates and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8020 axioms, and discussing the legitimacy or otherwise of those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8021 formulated by Euclid in each class. In his notes on the defini-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8022 tions Geminus treated them historically, giving the various
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8023 alternative definitions which had been suggested for each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8024 fundamental concept such as ‘line’, ‘surface’, ‘figure’, ‘body’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8025 ‘angle’, &c., and frequently adding instructive classifications
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8026 <pb n=226><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8027 of the different species of the thing defined. Thus in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8028 case of ‘lines’ (which include curves) he distinguishes, first,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8029 the composite (e.g. a broken line forming an angle) and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8030 incomposite. The incomposite are subdivided into those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8031 ‘forming a figure’ (<G>sxhmatopoiou=sai</G>) or determinate (e.g.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8032 circle, ellipse, cissoid) and those not forming a figure, inde-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8033 terminate and extending without limit (e.g. straight line,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8034 parabola, hyperbola, conchoid). In a second classification
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8035 incomposite lines are divided into (1) ‘simple’, namely the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8036 and straight line, the one ‘making a figure’, the other extend-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8037 ing without limit, and (2) ‘mixed’. ‘Mixed’ lines again are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8038 divided into (<I>a</I>) ‘lines in planes’, one kind being a line meet-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8039 ing itself (e.g. the cissoid) and another a line extending
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8040 without limit, and (<I>b</I>) ‘lines on solids’, subdivided into lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8041 formed by <I>sections</I> (e.g. conic sections, <I>spiric</I> curves) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8042 ‘lines <I>round</I> solids’ (e.g. a helix round a cylinder, sphere, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8043 cone, the first of which is uniform, homoeomeric, alike in all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8044 its parts, while the others are non-uniform). Geminus gave
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8045 a corresponding division of surfaces into simple and mixed,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8046 the former being plane surfaces and spheres, while examples
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8047 of the latter are the tore or anchor-ring (though formed by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8048 the revolution of a circle about an axis) and the conicoids of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8049 revolution (the right-angled conoid, the obtuse-angled conoid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8050 and the two spheroids, formed by the revolution of a para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8051 bola, a hyperbola, and an ellipse respectively about their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8052 axes). He observes that, while there are three <I>homoeomeric</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8053 or uniform ‘lines’ (the straight line, the circle, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8054 cylindrical helix), there are only two homoeomeric surfaces,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8055 the plane and the sphere. Other classifications are those of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8056 ‘angles’ (according to the nature of the two lines or curves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8057 which form them) and of figures and plane figures.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8058 <p>When Proclus gives definitions, &c., by Posidonius, it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8059 evident that he obtained them from Geminus's work. Such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8060 are Posidonius's definitions of ‘figure’ and ‘parallels’, and his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8061 division of quadrilaterals into seven kinds. We may assume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8062 further that, even where Geminus did not mention the name
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8063 of Posidonius, he was, at all events so far as the philosophy of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8064 mathematics was concerned, expressing views which were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8065 mainly those of his master.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8066 <pb n=227><head>GEMINUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8067 <C><I>Attempt to prove the Parallel-Postulate.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8068 <p>Geminus devoted much attention to the distinction between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8069 postulates and axioms, giving the views of earlier philoso-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8070 phers and mathematicians (Aristotle, Archimedes, Euclid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8071 Apollonius, the Stoics) on the subject as well as his own. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8072 was important in view of the attacks of the Epicureans and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8073 Sceptics on mathematics, for (as Geminus says) it is as futile
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8074 to attempt to prove the indemonstrable (as Apollonius did
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8075 when he tried to prove the axioms) as it is incorrect to assume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8076 what really requires proof, ‘as Euclid did in the fourth postu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8077 late [that all right angles are equal] and in the fifth postulate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8078 [the parallel-postulate]’.<note>Proclus on Eucl. I, pp. 178-82. 4; 183. 14-184. 10.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8079 <p>The fifth postulate was the special stumbling-block.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8080 Geminus observed that the converse is actually proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8081 Euclid in I. 17; also that it is conclusively proved that an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8082 angle equal to a right angle is not necessarily itself a right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8083 angle (e.g. the ‘angle’ between the circumferences of two semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8084 circles on two equal straight lines with a common extremity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8085 and at right angles to one another); we cannot therefore admit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8086 that the converses are incapable of demonstration.<note><I>Ib.</I>, pp. 183. 26-184. 5.</note> And
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8087 <p>‘we have learned from the very pioneers of this science not to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8088 have regard to mere plausible imaginings when it is a ques-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8089 tion of the reasonings to be included in our geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8090 doctrine. As Aristotle says, it is as justifiable to ask scien-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8091 tific proofs from a rhetorician as to accept mere plausibilities
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8092 from a geometer ... So in this case (that of the parallel-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8093 postulate) the fact that, when the right angles are lessened, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8094 straight lines converge is true and necessary; but the state-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8095 ment that, since they converge more and more as they are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8096 produced, they will sometime meet is plausible but not neces-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8097 sary, in the absence of some argument showing that this is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8098 true in the case of straight lines. For the fact that some lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8099 exist which approach indefinitely but yet remain non-secant
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8100 (<G>a)su/mptwtoi</G>), although it seems improbable and paradoxical,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8101 is nevertheless true and fully ascertained with reference to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8102 other species of lines [the hyperbola and its asymptote and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8103 the conchoid and its asymptote, as Geminus says elsewhere].
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8104 May not then the same thing be possible in the case of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8105 <pb n=228><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8106 straight lines which happens in the case of the lines referred
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8107 to? Indeed, until the statement in the postulate is clinched
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8108 by proof, the facts shown in the case of the other lines may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8109 direct our imagination the opposite way. And, though the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8110 controversial arguments against the meeting of the straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8111 lines should contain much that is surprising, is there not all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8112 the more reason why we should expel from our body of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8113 doctrine this merely plausible and unreasoned (hypothesis)?
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8114 It is clear from this that we must seek a proof of the present
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8115 theorem, and that it is alien to the special character of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8116 postulates.’<note>Proclus on Eucl. I, pp. 192. 5-193. 3.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8117 <p>Much of this might have been written by a modern
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8118 geometer. Geminus's attempted remedy was to substitute
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8119 a definition of parallels like that of Posidonius, based on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8120 notion of <I>equidistance.</I> An-Nair&imacr;z&imacr; gives the definition as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8121 follows: ‘Parallel straight lines are straight lines situated in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8122 the same plane and such that the distance between them, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8123 they are produced without limit in both directions at the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8124 time, is everywhere the same’, to which Geminus adds the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8125 statement that the said distance is the shortest straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8126 that can be drawn between them. Starting from this,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8127 Geminus proved to his own satisfaction the propositions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8128 Euclid regarding parallels and finally the parallel-postulate.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8129 He first gave the propositions (1) that the ‘distance’ between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8130 the two lines as defined is perpendicular to both, and (2) that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8131 if a straight line is perpendicular to each of two straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8132 and meets both, the two straight lines are parallel, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8133 ‘distance’ is the intercept on the perpendicular (proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8134 <I>reductio ad absurdum</I>). Next come (3) Euclid's propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8135 I. 27, 28 that, if two lines are parallel, the alternate angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8136 made by any transversal are equal, &c. (easily proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8137 drawing the two equal ‘distances’ through the points of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8138 intersection with the transversal), and (4) Eucl. I. 29, the con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8139 verse of I. 28, which is proved by <I>reductio ad absurdum</I>, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8140 means of (2) and (3). Geminus still needs Eucl. I. 30, 31
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8141 (about parallels) and I. 33, 34 (the first two propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8142 relating to parallelograms) for his final proof of the postulate,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8143 which is to the following effect.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8144 <p>Let <I>AB, CD</I> be two straight lines met by the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8145 <pb n=229><head>GEMINUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8146 <I>EF</I>, and let the interior angles <I>BEF, EFD</I> be together less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8147 than two right angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8148 <p>Take any point <I>H</I> on <I>FD</I> and draw <I>HK</I> parallel to <I>AB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8149 meeting <I>EF</I> in <I>K.</I> Then, if we bisect <I>EF</I> at <I>L, LF</I> at <I>M, MF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8150 at <I>N</I>, and so on, we shall at last have a length, as <I>FN</I>, less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8151 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8152 than <I>FK.</I> Draw <I>FG, NOP</I> parallel to <I>AB.</I> Produce <I>FO</I> to <I>Q</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8153 and let <I>FQ</I> be the same multiple of <I>FO</I> that <I>FE</I> is of <I>FN</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8154 then shall <I>AB, CD</I> meet in <I>Q.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8155 <p>Let <I>S</I> be the middle point of <I>FQ</I> and <I>R</I> the middle point of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8156 <I>FS.</I> Draw through <I>R, S, Q</I> respectively the straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8157 <I>RPG, STU, QV</I> parallel to <I>EF.</I> Join <I>MR, LS</I> and produce
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8158 them to <I>T, V.</I> Produce <I>FG</I> to <I>U.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8159 <p>Then, in the triangles <I>FON, ROP</I>, two angles are equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8160 respectively, the vertically opposite angles <I>FON, ROP</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8161 the alternate angles <I>NFO, PRO</I>; and <I>FO</I> = <I>OR</I>; therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8162 <I>RP</I> = <I>FN.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8163 <p>And <I>FN, PG</I> in the parallelogram <I>FNPG</I> are equal; there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8164 fore <I>RG</I> = 2<I>FN</I> = <I>FM</I> (whence <I>MR</I> is parallel to <I>FG</I> or <I>AB</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8165 <p>Similarly we prove that <I>SU</I> = 2<I>FM</I> = <I>FL</I>, and <I>LS</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8166 parallel to <I>FG</I> or <I>AB.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8167 <p>Lastly, by the triangles <I>FLS, QVS</I>, in which the sides <I>FS</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8168 <I>SQ</I> are equal and two angles are respectively equal, <I>QV</I> = <I>FL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8169 <p>Therefore <I>QV</I> = <I>LE.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8170 <p>Since then <I>EL, QV</I> are equal and parallel, so are <I>EQ, LV</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8171 and (says Geminus) it follows that <I>AB</I> passes through <I>Q.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8172 <pb n=230><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8173 <p>What follows is actually that both <I>EQ</I> and <I>AB</I> (or <I>EB</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8174 are parallel to <I>LV</I>, and Geminus assumes that <I>EQ, AB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8175 are coincident (in other words, that through a given point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8176 only one parallel can be drawn to a given straight line, an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8177 assumption known as Playfair's Axiom, though it is actually
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8178 stated in Proclus on Eucl. I. 31).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8179 <p>The proof therefore, apparently ingenious as it is, breaks
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8180 down. Indeed the method is unsound from the beginning,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8181 since (as Saccheri pointed out), before even the definition of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8182 parallels by Geminus can be used, it has to be <I>proved</I> that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8183 ‘the geometrical locus of points equidistant from a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8184 line is a straight line’, and this cannot be proved without a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8185 postulate. But the attempt is interesting as the first which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8186 has come down to us, although there must have been many
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8187 others by geometers earlier than Geminus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8188 <p>Coming now to the things which follow from the principles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8189 (<G>ta\ meta\ ta\s a)rxa/s</G>), we gather from Proclus that Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8190 carefully discussed such generalities as the nature of <I>elements</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8191 the different views which had been held of the distinction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8192 between theorems and problems, the nature and significance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8193 of <G>diorismoi/</G> (conditions and limits of possibility), the meaning
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8194 of ‘porism’ in the sense in which Euclid used the word in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8195 <I>Porisms</I> as distinct from its other meaning of ‘corollary’, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8196 different sorts of problems and theorems, the two varieties of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8197 converses (complete and partial), <I>topical</I> or <I>locus</I>-theorems,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8198 with the classification of loci. He discussed also philosophical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8199 questions, e.g. the question whether a line is made up of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8200 indivisible parts (<G>e)x a)merw=n</G>), which came up in connexion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8201 with Eucl. I. 10 (the bisection of a straight line).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8202 <p>The book was evidently not less exhaustive as regards
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8203 higher geometry. Not only did Geminus mention the <I>spiric</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8204 curves, conchoids and cissoids in his classification of curves;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8205 he showed how they were obtained, and gave proofs, presum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8206 ably of their principal properties. Similarly he gave the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8207 proof that there are three homoeomeric or uniform lines or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8208 curves, the straight line, the circle and the cylindrical helix.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8209 The proof of ‘uniformity’ (the property that any portion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8210 the line or curve will coincide with any other portion of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8211 same length) was preceded by a proof that, if two straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8212 lines be drawn from any point to meet a uniform line or curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8213 <pb n=231><head>GEMINUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8214 and make equal angles with it, the straight lines are equal.<note>Proclus on Eucl. I, pp. 112. 22-113. 3, p. 251. 3-11.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8215 As Apollonius wrote on the cylindrical helix and proved the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8216 fact of its uniformity, we may fairly assume that Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8217 was here drawing upon Apollonius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8218 <p>Enough has been said to show how invaluable a source of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8219 information Geminus's work furnished to Proclus and all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8220 writers on the history of mathematics who had access to it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8221 <p>In astronomy we know that Geminus wrote an <G>e)xh/ghsis</G> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8222 Posidonius's work, the <I>Meteorologica</I> or <G>peri\ metew/rwn</G>. This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8223 is the source of the famous extract made from Geminus by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8224 Alexander Aphrodisiensis, and reproduced by Simplicius in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8225 his commentary on the <I>Physics</I> of Aristotle,<note>Simpl. <I>in Phys.</I>, pp. 291-2, ed. Diels.</note> on which Schia-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8226 parelli relied in his attempt to show that it was Heraclides of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8227 Pontus, not Aristarchus of Samos, who first put forward the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8228 heliocentric hypothesis. The extract is on the distinction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8229 between physical and astronomical inquiry as applied to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8230 heavens. It is the business of the physicist to consider the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8231 substance of the heaven and stars, their force and quality,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8232 their coming into being and decay, and he is in a position to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8233 prove the facts about their size, shape, and arrangement;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8234 astronomy, on the other hand, ignores the physical side,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8235 proving the arrangement of the heavenly bodies by considera-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8236 tions based on the view that the heaven is a real <G>ko/smos</G>, and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8237 when it tells us of the shapes, sizes and distances of the earth,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8238 sun and moon, of eclipses and conjunctions, and of the quality
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8239 and extent of the movements of the heavenly bodies, it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8240 connected with the mathematical investigation of quantity,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8241 size, form, or shape, and uses arithmetic and geometry to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8242 prove its conclusions. Astronomy deals, not with causes, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8243 with facts; hence it often proceeds by hypotheses, stating
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8244 certain expedients by which the phenomena may be saved.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8245 For example, why do the sun, the moon and the planets
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8246 appear to move irregularly? To explain the observed facts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8247 we may assume, for instance, that the orbits are eccentric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8248 circles or that the stars describe epicycles on a carrying
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8249 circle; and then we have to go farther and examine other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8250 ways in which it is possible for the phenomena to be brought
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8251 about. ‘<I>Hence we actually find a certain person</I> [Heraclides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8252 <pb n=232><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8253 of Pontus] <I>coming forward and saying that, even on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8254 assumption that the earth moves in a certain way, while
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8255 the sun is in a certain way at rest, the apparent irregularity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8256 with reference to the sun may be saved.</I>’ Philological con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8257 siderations as well as the other notices which we possess
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8258 about Heraclides make it practically certain that ‘Heraclides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8259 of Pontus’ is an interpolation and that Geminus said <G>tis</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8260 simply, ‘a certain person’, without any name, though he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8261 doubtless meant Aristarchus of Samos.<note>Cf. <I>Aristarchus of Samos</I>, pp. 275-83.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8262 <p>Simplicius says that Alexander quoted this extract from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8263 the <I>epitome</I> of the <G>e)xh/ghsis</G> by Geminus. As the original
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8264 work was apparently made the subject of an abridgement, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8265 gather that it must have been of considerable scope. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8266 a question whether <G>e)xh/ghsis</G> means ‘commentary’ or ‘ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8267 position’; I am inclined to think that the latter interpretation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8268 is the correct one, and that Geminus reproduced Posidonius's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8269 work in its entirety with elucidations and comments; this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8270 seems to me to be suggested by the words added by Simplicius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8271 immediately after the extract ‘this is the account given by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8272 Geminus, <I>or Posidonius in Geminus</I>, of the difference between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8273 physics and astronomy’, which seems to imply that Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8274 in our passage reproduced Posidonius textually.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8275 <p>‘<I>Introduction to the Phaenomena’ attributed to Geminus.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8276 <p>There remains the treatise, purporting to be by Geminus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8277 which has come down to us under the title <G>*gemi/nou ei)sagwgh\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8278 ei)s ta\ *faino/mena</G>.<note>Edited by Manitius (Teubner, 1898).</note> What, if any, is the relation of this work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8279 to the exposition of Posidonius's <I>Meteorologica</I> or the epitome
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8280 of it just mentioned? One view is that the original <I>Isagoge</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8281 of Geminus and the <G>e)xh/ghsis</G> of Posidonius were one and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8282 same work, though the <I>Isagoge</I> as we have it is not by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8283 Geminus, but by an unknown compiler. The objections to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8284 this are, first, that it does not contain the extract given by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8285 Simplicius, which would have come in usefully at the begin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8286 ning of an Introduction to Astronomy, nor the other extract
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8287 given by Alexander from Geminus and relating to the rainbow
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8288 which seems likewise to have come from the <G>e)xh/ghsis</G><note>Alex. Aphr. on Aristotle's <I>Meteorologica</I>, iii. 4, 9 (Ideler, ii, p. 128; p. 152. 10, Hayduck).</note>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8289 <pb n=233><head>GEMINUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8290 secondly, that it does not anywhere mention the name of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8291 Posidonius (not, perhaps, an insuperable objection); and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8292 thirdly, that there are views expressed in it which are not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8293 those held by Posidonius but contrary to them. Again, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8294 writer knows how to give a sound judgement as between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8295 divergent views, writes in good style on the whole, and can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8296 hardly have been the mere compiler of extracts from Posi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8297 donius which the view in question assumes him to be. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8298 seems in any case safer to assume that the <I>Isagoge</I> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8299 <G>e)xh/ghsis</G> were separate works. At the same time, the <I>Isagoge</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8300 as we have it, contains errors which we cannot attribute to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8301 Geminus. The choice, therefore, seems to lie between two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8302 alternatives: either the book is by Geminus in the main, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8303 has in the course of centuries suffered deterioration by inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8304 polations, mistakes of copyists, and so on, or it is a compilation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8305 of extracts from an original <I>Isagoge</I> by Geminus with foreign
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8306 and inferior elements introduced either by the compiler him-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8307 self or by other prentice hands. The result is a tolerable ele-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8308 mentary treatise suitable for teaching purposes and containing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8309 the most important doctrines of Greek astronomy represented
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8310 from the standpoint of Hipparchus. Chapter 1 treats of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8311 zodiac, the solar year, the irregularity of the sun's motion,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8312 which is explained by the eccentric position of the sun's orbit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8313 relatively to the zodiac, the order and the periods of revolution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8314 of the planets and the moon. In &sc; 23 we are told that all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8315 the fixed stars do not lie on one spherical surface, but some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8316 are farther away than others—a doctrine due to the Stoics.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8317 Chapter 2, again, treats of the twelve signs of the zodiac,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8318 chapter 3 of the constellations, chapter 4 of the axis of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8319 the universe and the poles, chapter 5 of the circles on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8320 sphere (the equator and the parallel circles, arctic, summer-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8321 tropical, winter-tropical, antarctic, the colure-circles, the zodiac
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8322 or ecliptic, the horizon, the meridian, and the Milky Way),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8323 chapter 6 of Day and Night, their relative lengths in different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8324 latitudes, their lengthening and shortening, chapter 7 of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8325 the times which the twelve signs take to rise. Chapter 8
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8326 is a clear, interesting and valuable chapter on the calendar,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8327 the length of months and years and the various cycles, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8328 octaëteris, the 16-years and 160-years cycles, the 19-years
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8329 cycle of Euctemon (and Meton), and the cycle of Callippus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8330 <pb n=234><head>SUCCESSORS OF THE GREAT GEOMETERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8331 (76 years). Chapter 9 deals with the moon's phases, chapters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8332 10, 11 with eclipses of the sun and moon, chapter 12 with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8333 problem of accounting for the motions of the sun, moon and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8334 planets, chapter 13 with Risings and Settings and the various
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8335 technical terms connected therewith, chapter 14 with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8336 circles described by the fixed stars, chapters 15 and 16 with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8337 mathematical and physical geography, the zones, &c. (Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8338 follows Eratosthenes's evaluation of the circumference of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8339 earth, not that of Posidonius). Chapter 17, on weather indica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8340 tions, denies the popular theory that changes of atmospheric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8341 conditions depend on the rising and setting of certain stars,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8342 and states that the predictions of weather (<G>e)pishmasi/ai</G>) in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8343 calendars (<G>paraph/gmata</G>) are only derived from experience
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8344 and observation, and have no scientific value. Chapter 18 is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8345 on the <G>e)xeligmo/s</G>, the shortest period which contains an integral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8346 number of synodic months, of days, and of anomalistic revolu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8347 tions of the moon; this period is three times the Chaldaean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8348 period of 223 lunations used for predicting eclipses. The end
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8349 of the chapter deals with the maximum, mean, and minimum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8350 daily motion of the moon. The chapter as a whole does not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8351 correspond to the rest of the book; it deals with more difficult
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8352 matters, and is thought by Manitius to be a fragment only of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8353 a discussion to which the compiler did not feel himself equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8354 At the end of the work is a calendar (<I>Parapegma</I>) giving the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8355 number of days taken by the sun to traverse each sign of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8356 the zodiac, the risings and settings of various stars and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8357 weather indications noted by various astronomers, Democritus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8358 Eudoxus, Dositheus, Euctemon, Meton, Callippus; this calendar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8359 is unconnected with the rest of the book and the contents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8360 are in several respects inconsistent with it, especially the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8361 division of the year into quarters which follows Callippus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8362 rather than Hipparchus. Hence it has been, since Boeckh's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8363 time, generally considered not to be the work of Geminus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8364 Tittel, however, suggests that it is not impossible that Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8365 may have reproduced an older <I>Parapegma</I> of Callippus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8366 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8367 <C>XVI
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8368 SOME HANDBOOKS</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8369 <p>THE description of the handbook on the elements of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8370 astronomy entitled the <I>Introduction to the Phaenomena</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8371 attributed to Geminus might properly have been reserved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8372 for this chapter. It was, however, convenient to deal with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8373 Geminus in close connexion with Posidonius; for Geminus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8374 wrote an exposition of Posidonius's <I>Meteorologica</I> related to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8375 original work in such a way that Simplicius, in quoting a long
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8376 passage from an epitome of this work, could attribute the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8377 passage to either Geminus or ‘Posidonius in Geminus’; and it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8378 is evident that, in other subjects too, Geminus drew from, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8379 was influenced by, Posidonius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8380 <p>The small work <I>De motu circulari corporum caelestium</I> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8381 CLEOMEDES (<G>*kleomh/dous kuklikh\ qewri/a</G>) in two Books is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8382 production of a much less competent person, but is much more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8383 largely based on Posidonius. This is proved by several refer-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8384 ences to Posidonius by name, but it is specially true of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8385 very long first chapter of Book II (nearly half of the Book)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8386 which seems for the most part to be copied bodily from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8387 Posidonius, in accordance with the author's remark at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8388 end of Book I that, in giving the refutation of the Epicurean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8389 assertion that the sun is just as large as it looks, namely one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8390 foot in diameter, he will give so much as suffices for such an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8391 introduction of the particular arguments used by ‘certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8392 authors who have written whole treatises on this one topic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8393 (i. e. the size of the sun), among whom is Posidonius’. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8394 interest of the book then lies mainly in what is quoted from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8395 Posidonius; its mathematical interest is almost <I>nil.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8396 <p>The date of Cleomedes is not certainly ascertained, but, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8397 he mentions no author later than Posidonius, it is permissible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8398 to suppose, with Hultsch, that he wrote about the middle of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8399 <pb n=236><head>SOME HANDBOOKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8400 the first century B. C. As he seems to know nothing of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8401 works of Ptolemy, he can hardly, in any case, have lived
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8402 later than the beginning of the second century A. D.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8403 <p>Book I begins with a chapter the object of which is to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8404 prove that the universe, which has the shape of a sphere,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8405 is limited and surrounded by void extending without limit in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8406 all directions, and to refute objections to this view. Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8407 follow chapters on the five parallel circles in the heaven and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8408 the zones, habitable and uninhabitable (chap. 2); on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8409 motion of the fixed stars and the independent (<G>proairetikai/</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8410 movements of the planets including the sun and moon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8411 (chap. 3); on the zodiac and the effect of the sun's motion in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8412 it (chap. 4); on the inclination of the axis of the universe and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8413 its effects on the lengths of days and nights at different places
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8414 (chap. 5); on the inequality in the rate of increase in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8415 lengths of the days and nights according to the time of year,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8416 the different lengths of the seasons due to the motion of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8417 sun in an eccentric circle, the difference between a day-and-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8418 night and an exact revolution of the universe owing to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8419 separate motion of the sun (chap. 6); on the habitable regions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8420 of the globe including Britain and the ‘island of Thule’, said
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8421 to have been visited by Pytheas, where, when the sun is in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8422 Cancer and visible, the day is a month long; and so on (chap. 7).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8423 Chap. 8 purports to prove that the universe is a sphere by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8424 proving first that the earth is a sphere, and then that the air
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8425 about it, and the ether about that, must necessarily make up
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8426 larger spheres. The earth is proved to be a sphere by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8427 method of exclusion; it is assumed that the only possibilities
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8428 are that it is (<I>a</I>) flat and plane, or (<I>b</I>) hollow and deep, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8429 (<I>c</I>) square, or (<I>d</I>) pyramidal, or (<I>e</I>) spherical, and, the first four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8430 hypotheses being successively disposed of, only the fifth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8431 remains. Chap. 9 maintains that the earth is in the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8432 the universe; chap. 10, on the size of the earth, contains the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8433 interesting reproduction of the details of the measurements of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8434 the earth by Posidonius and Eratosthenes respectively which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8435 have been given above in their proper places (p.220, pp.106-7);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8436 chap. 11 argues that the earth is in the relation of a point to,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8437 i. e. is negligible in size in comparison with, the universe or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8438 even the sun's circle, but not the moon's circle (cf. p. 3 above).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8439 <p>Book II, chap. 1, is evidently the <I>pièce de résistance</I>, con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8440 <pb n=237><head>CLEOMEDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8441 sisting of an elaborate refutation of Epicurus and his followers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8442 who held that the sun is just as large as it <I>looks</I>, and further
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8443 asserted (according to Cleomedes) that the stars are lit up as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8444 they rise and extinguished as they set. The chapter seems to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8445 be almost wholly taken from Posidonius; it ends with some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8446 pages of merely vulgar abuse, comparing Epicurus with Ther-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8447 sites, with more of the same sort. The value of the chapter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8448 lies in certain historical traditions mentioned in it, and in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8449 account of Posidonius's speculation as to the size and distance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8450 of the sun, which does, as a matter of fact, give results much
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8451 nearer the truth than those obtained by Aristarchus, Hippar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8452 chus, and Ptolemy. Cleomedes observes (1) that by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8453 water-clocks it is found that the apparent diameter of the sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8454 is 1/750th of the sun's circle, and that this method of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8455 measuring it is said to have been first invented by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8456 Egyptians; (2) that Hipparchus is said to have found that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8457 the sun is 1,050 times the size of the earth, though, as regards
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8458 this, we have the better authority of Adrastus (in Theon of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8459 Smyrna) and of Chalcidius, according to whom Hipparchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8460 made the sun nearly 1,880 times the size of the earth (both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8461 figures refer of course to the solid content). We have already
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8462 described Posidonius's method of arriving at the size and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8463 distance of the sun (pp. 220-1). After he has given this, Cleo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8464 medes, apparently deserting his guide, adds a calculation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8465 his own relating to the sizes and distances of the moon and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8466 the sun which shows how little he was capable of any scien-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8467 tific inquiry.<note>He says (pp. 146. 17-148. 27) that in an eclipse the breadth of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8468 earth's shadow is stated to be two moon-breadths; hence, he says, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8469 seems credible (<G>piqano/n</G>) that the earth is twice the size of the moon (this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8470 practically assumes that the breadth of the earth's shadow is equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8471 the diameter of the earth, or that the cone of the earth's shadow is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8472 a cylinder!). Since then the circumference of the earth, according to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8473 Eratosthenes, is 250,000 stades, and its diameter therefore ‘more than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8474 80,000’ (he evidently takes <G>p</G>=3), the diameter of the moon will be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8475 40,000 stades. Now, the moon's circle being 750 times the moon's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8476 diameter, the radius of the moon's circle, i.e. the distance of the moon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8477 from the earth, will be 1/6th of this (i.e. <G>p</G>=3) or 125 moon-diameters;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8478 therefore the moon's distance is 5,000,000 stades (which is much too
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8479 great). Again, since the moon traverses its orbit 13 times to the sun's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8480 once, he assumes that the sun's orbit is 13 times as large as the moon's,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8481 and consequently that the diameter of the sun is 13 times that of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8482 moon, or 520,000 stades and its distance 13 times 5,000,000 or 65,000,000
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8483 stades!</note> Chap. 2 purports to prove that the sun is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8484 <pb n=238><head>SOME HANDBOOKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8485 larger than the earth; and the remaining chapters deal with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8486 the size of the moon and the stars (chap. 3), the illumination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8487 of the moon by the sun (chap. 4), the phases of the moon and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8488 its conjunctions with the sun (chap. 5), the eclipses of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8489 moon (chap. 6), the maximum deviation in latitude of the five
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8490 planets (given as 5° for Venus, 4° for Mercury, 2 1/2° for Mars
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8491 and Jupiter, 1° for Saturn), the maximum elongations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8492 Mercury and Venus from the sun (20° and 50° respectively),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8493 and the synodic periods of the planets (Mercury 116 days,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8494 Venus 584 days, Mars 780 days, Jupiter 398 days, Saturn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8495 378 days) (chap. 7).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8496 <p>There is only one other item of sufficient interest to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8497 mentioned here. In Book II, chap. 6, Cleomedes mentions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8498 that there were stories of extraordinary eclipses which ‘the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8499 more ancient of the mathematicians had vainly tried to ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8500 plain’; the supposed ‘paradoxical’ case was that in which,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8501 while the sun seems to be still above the horizon, the <I>eclipsed</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8502 moon rises in the east. The passage has been cited above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8503 (vol. i, pp. 6-7), where I have also shown that Cleomedes him-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8504 self gives the true explanation of the phenomenon, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8505 that it is due to atmospheric refraction.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8506 <p>The first and second centuries of the Christian era saw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8507 a continuation of the work of writing manuals or introduc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8508 tions to the different mathematical subjects. About A. D. 100
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8509 came NICOMACHUS, who wrote an <I>Introduction to Arithmetic</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8510 and an <I>Introduction to Harmony</I>; if we may judge by a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8511 remark of his own,<note>Nicom. <I>Arith.</I> ii. 6. 1.</note> he would appear to have written an intro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8512 duction to geometry also. The <I>Arithmetical Introduction</I> has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8513 been sufficiently described above (vol. i, pp. 97-112).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8514 <p>There is yet another handbook which needs to be mentioned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8515 separately, although we have had occasion to quote from it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8516 several times already. This is the book by THEON OF SMYRNA
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8517 which goes by the title <I>Expositio rerum mathematicarum ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8518 legendum Platonem utilium.</I> There are two main divisions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8519 of this work, contained in two Venice manuscripts respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8520 tively. The first was edited by Bullialdus (Paris, 1644), the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8521 second by T. H. Martin (Paris, 1849); the whole has been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8522 <pb n=239><head>THEON OF SMYRNA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8523 edited by E. Hiller (Teubner, 1878) and finally, with a French
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8524 translation, by J. Dupuis (Paris, 1892).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8525 <p>Theon's date is approximately fixed by two considerations.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8526 He is clearly the person whom Theon of Alexandria called
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8527 ‘the old Theon’, <G>to\n palaio\n *qe/wna</G>,<note>Theon of Alexandria, <I>Comm. on Ptolemy's Syntaxis</I>, Basel edition,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8528 pp. 390, 395, 396.</note> and there is no reason
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8529 to doubt that he is the ‘Theon the mathematician’ (<G>o( maqh-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8530 matiko/s</G>) who is credited by Ptolemy with four observations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8531 of the planets Mercury and Venus made in A.D. 127, 129, 130
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8532 and 132.<note>Ptolemy, <I>Syntaxis</I>, ix. 9, x. 1, 2.</note> The latest writers whom Theon himself mentions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8533 are Thrasyllus, who lived in the reign of Tiberius, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8534 Adrastus the Peripatetic, who belongs to the middle of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8535 second century A.D. Theon's work itself is a curious medley,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8536 valuable, not intrinsically, but for the numerous historical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8537 notices which it contains. The title, which claims that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8538 book contains things useful for the study of Plato, must not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8539 be taken too seriously. It was no doubt an elementary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8540 <I>introduction</I> or vade-mecum for students of philosophy, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8541 there is little in it which has special reference to the mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8542 matical questions raised in Plato. The connexion consists
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8543 mostly in the long proem quoting the views of Plato on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8544 paramount importance of mathematics in the training of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8545 the philosopher, and the mutual relation of the five different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8546 branches, arithmetic, geometry, stereometry, astronomy and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8547 music. The want of care shown by Theon in the quotations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8548 from particular dialogues of Plato prepares us for the patch-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8549 work character of the whole book.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8550 <p>In the first chapter he promises to give the mathematical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8551 theorems most necessary for the student of Plato to know,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8552 in arithmetic, music, and geometry, with its application to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8553 stereometry and astronomy.<note>Theon of Smyrna, ed. Hiller, p. 1. 10-17.</note> But the promise is by no means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8554 kept as regards geometry and stereometry: indeed, in a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8555 later passage Theon seems to excuse himself from including
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8556 theoretical geometry in his plan, on the ground that all those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8557 who are likely to read his work or the writings of Plato may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8558 be assumed to have gone through an elementary course of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8559 theoretical geometry.<note><I>Ib.</I>, p. 16. 17-20.</note> But he writes at length on figured
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8560 <pb n=240><head>SOME HANDBOOKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8561 numbers, plane and solid, which are of course analogous to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8562 the corresponding geometrical figures, and he may have con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8563 sidered that he was in this way sufficiently fulfilling his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8564 promise with regard to geometry and stereometry. Certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8565 geometrical definitions, of point, line, straight line, the three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8566 dimensions, rectilinear plane and solid figures, especially
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8567 parallelograms and parallelepipedal figures including cubes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8568 <I>plinthides</I> (square bricks) and <G>doki/des</G> (beams), and <I>scalene</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8569 figures with sides unequal every way (=<G>bwmi/skoi</G> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8570 classification of solid numbers), are dragged in later (chaps.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8571 53-5 of the section on music)<note>Theon of Smyrna, ed. Hiller, pp. 111-13.</note> in the middle of the discussion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8572 of proportions and means; if this passage is not an inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8573 polation, it confirms the supposition that Theon included in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8574 his work only this limited amount of geometry and stereo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8575 metry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8576 <p>Section I is on Arithmetic in the same sense as Nicomachus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8577 <I>Introduction.</I> At the beginning Theon observes that arith-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8578 metic will be followed by music. Of music in its three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8579 aspects, music in instruments (<G>e)n o)rga/nois</G>), music in numbers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8580 i.e. musical intervals expressed in numbers or pure theoretical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8581 music, and the music or harmony in the universe, the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8582 kind (instrumental music) is not exactly essential, but the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8583 two must be discussed immediately after arithmetic.<note><I>Ib.</I>, pp. 16. 24-17. 11.</note> The con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8584 tents of the arithmetical section have been sufficiently indicated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8585 in the chapter on Pythagorean arithmetic (vol. i, pp. 112-13);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8586 it deals with the classification of numbers, odd, even, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8587 their subdivisions, prime numbers, composite numbers with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8588 equal or unequal factors, plane numbers subdivided into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8589 square, oblong, triangular and polygonal numbers, with their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8590 respective ‘gnomons’ and their properties as the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8591 successive terms of arithmetical progressions beginning with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8592 1 as the first term, circular and spherical numbers, solid num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8593 bers with three factors, pyramidal numbers and truncated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8594 pyramidal numbers, perfect numbers with their correlatives,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8595 the over-perfect and the deficient; this is practically what
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8596 we find in Nicomachus. But the special value of Theon's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8597 exposition lies in the fact that it contains an account of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8598 famous ‘side-’ and ‘diameter-’ numbers of the Pythagoreans.<note><I>Ib.</I>, pp. 42. 10-45. 9. Cf. vol. i, pp. 91-3.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8599 <pb n=241><head>THEON OF SMYRNA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8600 <p>In the Section on Music Theon says he will first speak of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8601 the two kinds of music, the audible or instrumental, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8602 intelligible or theoretical subsisting in numbers, after which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8603 he promises to deal lastly with ratio as predicable of mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8604 matical entities in general and the ratio constituting the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8605 harmony in the universe, ‘not scrupling to set out once again
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8606 the things discovered by our predecessors, just as we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8607 given the things handed down in former times by the Pytha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8608 goreans, with a view to making them better known, without
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8609 ourselves claiming to have discovered any of them’.<note>Theon of Smyrna, ed. Hiller, pp. 46. 20-47. 14.</note> Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8610 follows a discussion of audible music, the intervals which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8611 give harmonies, &c., including substantial quotations from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8612 Thrasyllus and Adrastus, and references to views of Aris-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8613 toxenus, Hippasus, Archytas, Eudoxus and Plato. With
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8614 chap. 17 (p. 72) begins the account of the ‘harmony in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8615 numbers’, which turns into a general discussion of ratios,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8616 proportions and means, with more quotations from Plato,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8617 Eratosthenes and Thrasyllus, followed by Thrasyllus's <I>divisio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8618 canonis</I>, chaps. 35, 36 (pp. 87-93). After a promise to apply
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8619 the latter division to the sphere of the universe, Theon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8620 purports to return to the subject of proportion and means.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8621 This, however, does not occur till chap. 50 (p. 106), the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8622 intervening chapters being taken up with a discussion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8623 the <G>deka/s</G> and <G>tetraktu/s</G> (with eleven applications of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8624 latter) and the mystic or curious properties of the numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8625 from 2 to 10; here we have a part of the <I>theologumena</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8626 arithmetic. The discussion of proportions and the different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8627 kinds of means after Eratosthenes and Adrastus is again
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8628 interrupted by the insertion of the geometrical definitions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8629 already referred to (chaps. 53-5, pp. 111-13), after which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8630 Theon resumes the question of means for ‘more precise’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8631 treatment.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8632 <p>The Section on Astronomy begins on p. 120 of Hiller's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8633 edition. Here again Theon is mainly dependent upon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8634 Adrastus, from whom he makes long quotations. Thus, on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8635 the sphericity of the earth, he says that for the neces-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8636 sary conspectus of the arguments it will be sufficient to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8637 refer to the grounds stated summarily by Adrastus. In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8638 explaining (p. 124) that the unevennesses in the surface of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8639 <pb n=242><head>SOME HANDBOOKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8640 the earth, represented e.g. by mountains, are negligible in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8641 comparison with the size of the whole, he quotes Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8642 and Dicaearchus as claiming to have discovered that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8643 perpendicular height of the highest mountain above the normal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8644 level of the land is no more than 10 stades; and to obtain the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8645 diameter of the earth he uses Eratosthenes's figure of approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8646 mately 252,000 stades for the circumference of the earth,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8647 which, with the Archimedean value of 22/7 for <G>p</G>, gives a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8648 diameter of about 80,182 stades. The principal astronomical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8649 circles in the heaven are next described (chaps. 5-12, pp.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8650 129-35); then (chap. 12) the assumed maximum deviations in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8651 latitude are given, that of the sun being put at 1°, that of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8652 moon and Venus at 12°, and those of the planets Mercury,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8653 Mars, Jupiter and Saturn at 8°, 5°, 5° and 3° respectively; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8654 obliquity of the ecliptic is given as the side of a regular polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8655 of 15 sides described in a circle, i.e. as 24° (chap. 23, p. 151).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8656 Next the order of the orbits of the sun, moon and planets is ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8657 plained (the system is of course geocentric); we are told (p.138)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8658 that ‘some of the Pythagoreans’ made the order (reckoning
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8659 outwards from the earth) to be moon, Mercury, Venus, sun,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8660 Mars, Jupiter, Saturn, whereas (p. 142) Eratosthenes put the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8661 sun next to the moon, and the mathematicians, agreeing with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8662 Eratosthenes in this, differed only in the order in which they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8663 placed Venus and Mercury after the sun, some putting Mercury
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8664 next and some Venus (p. 143). The order adopted by ‘some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8665 of the Pythagoreans’ is the Chaldaean order, which was not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8666 followed by any Greek before Diogenes of Babylon (second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8667 century B.C.); ‘some of the Pythagoreans’ are therefore the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8668 later Pythagoreans (of whom Nicomachus was one); the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8669 order, moon, sun, Venus, Mercury, Mars, Jupiter, Saturn, was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8670 that of Plato and the early Pythagoreans. In chap. 15
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8671 (p. 138 sq.) Theon quotes verses of Alexander ‘the Aetolian’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8672 (not really the ‘Aetolian’, but Alexander of Ephesus, a con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8673 temporary of Cicero, or possibly Alexander of Miletus, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8674 Chalcidius calls him) assigning to each of the planets (includ-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8675 ing the earth, though stationary) with the sun and moon and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8676 the sphere of the fixed stars one note, the intervals between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8677 the notes being so arranged as to bring the nine into an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8678 octave, whereas with Eratosthenes and Plato the earth was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8679 excluded, and the eight notes of the octachord were assigned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8680 <pb n=243><head>THEON OF SMYRNA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8681 to the seven heavenly bodies and the sphere of the fixed stars.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8682 The whole of this passage (chaps. 15 to 16, pp. 138-47) is no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8683 doubt intended as the promised account of the ‘harmony in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8684 the universe’, although at the very end of the work Theon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8685 implies that this has still to be explained on the basis of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8686 Thrasyllus's exposition combined with what he has already
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8687 given himself.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8688 <p>The next chapters deal with the forward movements, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8689 stationary points, and the retrogradations, as they respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8690 appear to us, of the five planets, and the ‘saving of the pheno-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8691 mena’ by the alternative hypotheses of eccentric circles and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8692 epicycles (chaps. 17-30, pp. 147-78). These hypotheses are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8693 explained, and the identity of the motion produced by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8694 two is shown by Adrastus in the case of the sun (chaps. 26, 27,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8695 pp. 166-72). The proof is introduced with the interesting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8696 remark that ‘Hipparchus says it is worthy of investigation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8697 by mathematicians why, on two hypotheses so different from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8698 one another, that of eccentric circles and that of concentric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8699 circles with epicycles, the same results appear to follow’. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8700 is not to be supposed that the proof of the identity could be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8701 other than easy to a mathematician like Hipparchus; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8702 remark perhaps merely suggests that the two hypotheses were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8703 discovered quite independently, and it was not till later that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8704 the effect was discovered to be the same, when of course the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8705 fact would seem to be curious and a mathematical proof would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8706 immediately be sought. Another passage (p. 188) says that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8707 Hipparchus preferred the hypothesis of the epicycle, as being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8708 his own. If this means that Hipparchus claimed to have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8709 discovered the epicycle-hypothesis, it must be a misapprehen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8710 sion; for Apollonius already understood the theory of epi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8711 cycles in all its generality. According to Theon, the epicycle-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8712 hypothesis is more ‘according to nature’; but it was presum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8713 ably preferred because it was applicable to all the planets,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8714 whereas the eccentric-hypothesis, when originally suggested,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8715 applied only to the three superior planets; in order to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8716 it apply to the inferior planets it is necessary to suppose the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8717 circle described by the centre of the eccentric to be greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8718 than the eccentric circle itself, which extension of the hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8719 thesis, though known to Hipparchus, does not seem to have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8720 occurred to Apollonius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8721 <pb n=244><head>SOME HANDBOOKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8722 <p>We next have (chap. 31, p. 178) an allusion to the systems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8723 of Eudoxus, Callippus and Aristotle, and a description
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8724 (p. 180 sq.) of a system in which the ‘carrying’ spheres
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8725 (called ‘hollow’) have between them ‘solid spheres which by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8726 their own motion will roll (<G>a)neli/xousi</G>) the carrying spheres in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8727 the opposite direction, being in contact with them’. These
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8728 ‘solid’ spheres (which carry the planet fixed at a point on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8729 their surface) act in practically the same way as epicycles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8730 In connexion with this description Theon (i.e. Adrastus)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8731 speaks (chap. 33, pp. 186-7) of two alternative hypotheses in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8732 which, by comparison with Chalcidius,<note>Chalcidius, <I>Comm. on Timaeus</I>, c. 110. Cf. <I>Aristarchus of Samos</I>, pp. 256-8.</note> we recognize (after
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8733 eliminating epicycles erroneously imported into both systems)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8734 the hypotheses of Plato and Heraclides respectively. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8735 this passage which enables us to conclude for certain that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8736 Heraclides made Venus and Mercury revolve in circles about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8737 the sun, like satellites, while the sun in its turn revolves in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8738 a circle about the earth as centre. Theon (p. 187) gives the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8739 maximum arcs separating Mercury and Venus respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8740 from the sun as 20° and 50°, these figures being the same as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8741 those given by Cleomedes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8742 <p>The last chapters (chaps. 37-40), quoted from Adrastus, deal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8743 with conjunctions, transits, occultations and eclipses. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8744 book concludes with a considerable extract from Dercyllides,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8745 a Platonist with Pythagorean leanings, who wrote (before the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8746 time of Tiberius and perhaps even before Varro) a book on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8747 Plato's philosophy. It is here (p. 198. 14) that we have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8748 passage so often quoted from Eudemus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8749 <p>‘Eudemus relates in his Astronomy that it was Oenopides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8750 who first discovered the girdling of the zodiac and the revolu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8751 tion (or cycle) of the Great Year, that Thales was the first to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8752 discover the eclipse of the sun and the fact that the sun's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8753 period with respect to the solstices is not always the same,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8754 that Anaximander discovered that the earth is (suspended) on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8755 high and lies (substituting <G>kei=tai</G> for the reading of the manu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8756 scripts, <G>kinei=tai</G>, moves) about the centre of the universe, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8757 that Anaximenes said that the moon has its light from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8758 sun and (explained) how its eclipses come about’ (Anaxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8759 menes is here apparently a mistake for Anaxagoras).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8760 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8761 <C>XVII
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8762 TRIGONOMETRY: HIPPARCHUS, MENELAUS,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8763 PTOLEMY</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8764 <p>WE have seen that <I>Sphaeric</I>, the geometry of the sphere,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8765 was very early studied, because it was required so soon as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8766 astronomy became mathematical; with the Pythagoreans the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8767 word <I>Sphaeric</I>, applied to one of the subjects of the quadrivium,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8768 actually meant astronomy. The subject was so far advanced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8769 before Euclid's time that there was in existence a regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8770 textbook containing the principal propositions about great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8771 and small circles on the sphere, from which both Autolycus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8772 and Euclid quoted the propositions as generally known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8773 These propositions, with others of purely astronomical in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8774 terest, were collected afterwards in a work entitled <I>Sphaerica</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8775 in three Books, by THEODOSIUS.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8776 <p>Suidas has a notice, <I>s. v.</I> <G>*qeodo/sios</G>, which evidently con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8777 fuses the author of the <I>Sphaerica</I> with another Theodosius,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8778 a Sceptic philosopher, since it calls him ‘Theodosius, a philoso-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8779 pher’, and attributes to him, besides the mathematical works,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8780 ‘Sceptic chapters’ and a commentary on the chapters of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8781 Theudas. Now the commentator on Theudas must have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8782 belonged, at the earliest, to the second half of the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8783 century A.D., whereas our Theodosius was earlier than Mene-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8784 laus (<I>fl.</I> about A. D. 100), who quotes him by name. The next
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8785 notice by Suidas is of yet another Theodosius, a poet, who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8786 came from Tripolis. Hence it was at one time supposed that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8787 our Theodosius was of Tripolis. But Vitruvius<note><I>De architectura</I> ix. 9.</note> mentions a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8788 Theodosius who invented a sundial ‘for any climate’; and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8789 Strabo, in speaking of certain Bithynians distinguished in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8790 their particular sciences, refers to ‘Hipparchus, Theodosius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8791 and his sons, mathematicians’<note>Strabo, xii. 4, 9, p. 566.</note>. We conclude that our Theo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8792 <pb n=246><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8793 dosius was of Bithynia and not later in date than Vitruvius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8794 (say 20 B.C.); but the order in which Strabo gives the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8795 names makes it not unlikely that he was contemporary with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8796 Hipparchus, while the character of his <I>Sphaerica</I> suggests a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8797 date even earlier rather than later.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8798 <C>Works by Theodosius.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8799 <p>Two other works of Theodosius besides the <I>Sphaerica</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8800 namely <I>On habitations</I> and <I>On Days and Nights</I>, seem to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8801 have been included in the ‘Little Astronomy’ (<G>mikro\s a)stro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8802 nomou/menos</G>, <I>sc.</I> <G>to/pos</G>). These two treatises need not detain us
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8803 long. They are extant in Greek (in the great MS. Vaticanus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8804 Graecus 204 and others), but the Greek text has not appar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8805 ently yet been published. In the first, <I>On habitations</I>, in 12
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8806 propositions, Theodosius explains the different phenomena due
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8807 to the daily rotation of the earth, and the particular portions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8808 of the whole system which are visible to inhabitants of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8809 different zones. In the second, <I>On Days and Nights</I>, contain-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8810 ing 13 and 19 propositions in the two Books respectively,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8811 Theodosius considers the arc of the ecliptic described by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8812 sun each day, with a view to determining the conditions to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8813 satisfied in order that the solstice may occur in the meridian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8814 at a given place, and in order that the day and the night may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8815 really be equal at the equinoxes; he shows also that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8816 variations in the day and night must recur exactly after
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8817 a certain time, if the length of the solar year is commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8818 surable with that of the day, while on the contrary assump-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8819 tion they will not recur so exactly.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8820 <p>In addition to the works bearing on astronomy, Theodosius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8821 is said<note>Suidas, <I>loc. cit.</I></note> to have written a commentary, now lost, on the <G>e)fo/dion</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8822 or <I>Method</I> of Archimedes (see above, pp. 27-34).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8823 <C>Contents of the <I>Sphaerica.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8824 <p>We come now to the <I>Sphaerica</I>, which deserves a short
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8825 description from the point of view of this chapter. A text-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8826 book on the geometry of the sphere was wanted as a supple-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8827 ment to the <I>Elements</I> of Euclid. In the <I>Elements</I> themselves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8828 <pb n=247><head>THEODOSIUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8829 (Books XII and XIII) Euclid included no general properties
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8830 of the sphere except the theorem proved in XII. 16-18, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8831 the volumes of two spheres are in the triplicate ratio of their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8832 diameters; apart from this, the sphere is only introduced in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8833 the propositions about the regular solids, where it is proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8834 that they are severally inscribable in a sphere, and it was doubt-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8835 less with a view to his proofs of this property in each case that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8836 he gave a new definition of a sphere as the figure described by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8837 the revolution of a semicircle about its diameter, instead of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8838 the more usual definition (after the manner of the definition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8839 of a circle) as the locus of all points (in space instead of in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8840 a plane) which are equidistant from a fixed point (the centre).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8841 No doubt the exclusion of the geometry of the sphere from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8842 the <I>Elements</I> was due to the fact that it was regarded as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8843 belonging to astronomy rather than pure geometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8844 <p>Theodosius defines the sphere as ‘a solid figure contained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8845 by one surface such that all the straight lines falling upon it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8846 from one point among those lying within the figure are equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8847 to one another’, which is exactly Euclid's definition of a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8848 with ‘solid’ inserted before ‘figure’ and ‘surface’ substituted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8849 for ‘line’. The early part of the work is then generally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8850 developed on the lines of Euclid's Book III on the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8851 Any plane section of a sphere is a circle (Prop. 1). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8852 straight line from the centre of the sphere to the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8853 a circular section is perpendicular to the plane of that section
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8854 (1, Por. 2; cf. 7, 23); thus a plane section serves for finding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8855 the centre of the sphere just as a chord does for finding that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8856 of a circle (Prop. 2). The propositions about tangent planes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8857 (3-5) and the relation between the sizes of circular sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8858 and their distances from the centre (5, 6) correspond to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8859 Euclid III. 16-19 and 15; as the small circle corresponds to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8860 any chord, the great circle (‘greatest circle’ in Greek) corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8861 sponds to the diameter. The poles of a circular section
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8862 correspond to the extremities of the diameter bisecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8863 a chord of a circle at right angles (Props. 8-10). Great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8864 circles bisecting one another (Props. 11-12) correspond to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8865 chords which bisect one another (diameters), and great circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8866 bisecting small circles at right angles and passing through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8867 their poles (Props. 13-15) correspond to diameters bisecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8868 chords at right angles. The distance of any point of a great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8869 <pb n=248><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8870 circle from its pole is equal to the side of a square inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8871 in the great circle and conversely (Props. 16, 17). Next come
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8872 certain problems: To find a straight line equal to the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8873 of any circular section or of the sphere itself (Props. 18, 19);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8874 to draw the great circle through any two given points on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8875 the surface (Prop. 20); to find the pole of any given circu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8876 lar section (Prop. 21). Prop. 22 applies Eucl. III. 3 to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8877 sphere.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8878 <p>Book II begins with a definition of circles on a sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8879 which touch one another; this happens ‘when the common
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8880 section of the planes (of the circles) touches both circles’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8881 Another series of propositions follows, corresponding again
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8882 to propositions in Eucl., Book III, for the circle. Parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8883 circular sections have the same poles, and conversely (Props.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8884 1, 2). Props. 3-5 relate to circles on the sphere touching
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8885 one another and therefore having their poles on a great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8886 circle which also passes through the point of contact (cf.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8887 Eucl. III. 11, [12] about circles touching one another). If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8888 a great circle touches a small circle, it also touches another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8889 small circle equal and parallel to it (Props. 6, 7), and if a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8890 great circle be obliquely inclined to another circular section,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8891 it touches each of two equal circles parallel to that section
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8892 (Prop. 8). If two circles on a sphere cut one another, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8893 great circle drawn through their poles bisects the intercepted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8894 segments of the circles (Prop. 9). If there are any number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8895 parallel circles on a sphere, and any number of great circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8896 drawn through their poles, the arcs of the parallel circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8897 intercepted between any two of the great circles are similar,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8898 and the arcs of the great circles intercepted between any two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8899 of the parallel circles are equal (Prop. 10).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8900 <p>The last proposition forms a sort of transition to the portion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8901 of the treatise (II. 11-23 and Book III) which contains pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8902 positions of purely astronomical interest, though expressed as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8903 propositions in pure geometry without any specific reference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8904 to the various circles in the heavenly sphere. The proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8905 tions are long and complicated, and it would neither be easy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8906 nor worth while to attempt an enumeration. They deal with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8907 circles or parts of circles (arcs intercepted on one circle by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8908 series of other circles and the like). We have no difficulty in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8909 recognizing particular circles which come into many proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8910 <pb n=249><head>THEODOSIUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8911 tions. A particular small circle is the circle which is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8912 limit of the stars which do not set, as seen by an observer at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8913 a particular place on the earth's surface; the pole of this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8914 circle is the pole in the heaven. A great circle which touches
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8915 this circle and is obliquely inclined to the ‘parallel circles’ is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8916 circle of the horizon; the parallel circles of course represent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8917 the apparent motion of the fixed stars in the diurnal rotation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8918 and have the pole of the heaven as pole. A second great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8919 circle obliquely inclined to the parallel circles is of course the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8920 circle of the zodiac or ecliptic. The greatest of the ‘parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8921 circles’ is naturally the equator. All that need be said of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8922 various propositions (except two which will be mentioned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8923 separately) is that the sort of result proved is like that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8924 Props. 12 and 13 of Euclid's <I>Phaenomena</I> to the effect that in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8925 the half of the zodiac circle beginning with Cancer (or Capri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8926 cornus) equal arcs set (or rise) in unequal times; those which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8927 are nearer the tropic circle take a longer time, those further
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8928 from it a shorter; those which take the shortest time are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8929 those adjacent to the equinoctial points; those which are equi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8930 distant from the equator rise and set in equal times. In like
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8931 manner Theodosius (III. 8) in effect takes equal and con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8932 tiguous arcs of the ecliptic all on one side of the equator,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8933 draws through their extremities great circles touching the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8934 circumpolar ‘parallel’ circle, and proves that the correspond-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8935 ing arcs of the equator intercepted between the latter great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8936 circles are unequal and that, of the said arcs, that correspond-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8937 ing to the arc of the ecliptic which is nearer the tropic circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8938 is the greater. The successive great circles touching the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8939 circumpolar circle are of course successive positions of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8940 horizon as the earth revolves about its axis, that is to say,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8941 the same length of arc on the ecliptic takes a longer or shorter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8942 time to rise according as it is nearer to or farther from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8943 tropic, in other words, farther from or nearer to the equinoctial
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8944 points.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8945 <p>It is, however, obvious that investigations of this kind,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8946 which only prove that certain arcs are greater than others,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8947 and do not give the actual numerical ratios between them, are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8948 useless for any practical purpose such as that of telling the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8949 hour of the night by the stars, which was one of the funda-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8950 mental problems in Greek astronomy; and in order to find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8951 <pb n=250><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8952 the required numerical ratios a new method had to be invented,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8953 namely trigonometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8954 <C><I>No actual trigonometry in Theodosius.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8955 <p>It is perhaps hardly correct to say that spherical triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8956 are nowhere referred to in Theodosius, for in III. 3 the con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8957 gruence-theorem for spherical triangles corresponding to Eucl.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8958 I. 4 is practically proved; but there is nothing in the book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8959 that can be called trigonometrical. The nearest approach is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8960 in III. 11, 12, where ratios between certain straight lines are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8961 compared with ratios between arcs. <I>ACc</I> (Prop. 11) is a great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8962 circle through the poles <I>A, A′</I>; <I>CDc, C′D</I> are two other great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8963 circles, both of which are at right angles to the plane of <I>AC′c</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8964 but <I>CDc</I> is perpendicular to <I>AA′</I>, while <I>C′D</I> is inclined to it at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8965 an acute angle. Let any other great circle <I>AB′BA′</I> through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8966 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8967 <I>AA′</I> cut <I>CD</I> in any point <I>B</I> between <I>C</I> and <I>D</I>, and <I>C′D</I> in <I>B′.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8968 Let the ‘parallel’ circle <I>EB′e</I> be drawn through <I>B′</I>, and let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8969 <I>C′c′</I> be the diameter of the ‘parallel’ circle touching the great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8970 circle <I>C′D.</I> Let <I>L, K</I> be the centres of the ‘parallel’ circles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8971 and let <I>R</I>, <G>r</G> be the radii of the ‘parallel’ circles <I>CDc, C′c′</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8972 respectively. It is required to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8973 <MATH>2<I>R</I>:2<G>r</G> > (arc <I>CB</I>):(arc <I>C′B′</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8974 <p>Let <I>C′O, Ee</I> meet in <I>N</I>, and join <I>NB′.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8975 <p>Then <I>B′N</I>, being the intersection of two planes perpendicu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8976 lar to the plane of <I>AC′CA′</I>, is perpendicular to that plane and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8977 therefore to both <I>Ee</I> and <I>C′O.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8978 <pb n=251><head>THEODOSIUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8979 <p>Now, the triangle <I>NLO</I> being right-angled at <I>L, NO</I> > <I>NL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8980 <p>Measure <I>NT</I> along <I>NO</I> equal to <I>NL</I>, and join <I>TB′.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8981 <p>Then in the triangles <I>B′NT, B′NL</I> two sides <I>B′N, NT</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8982 equal to two sides <I>B′N, NL</I>, and the included angles (both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8983 being right) are equal; therefore the triangles are equal in all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8984 respects, and &angle; <I>NLB′</I> = &angle; <I>NTB′.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8985 <p>Now <MATH>2<I>R</I>:2<G>r</G> = <I>OC′</I>:<I>C′K</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8986 = <I>ON</I>:<I>NL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8987 = <I>ON</I>:<I>NT</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8988 [=tan <I>NTB′</I>:tan<I>NOB′</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8989 > &angle;<I>NTB′</I>:&angle;<I>NOB′</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8990 > &angle;<I>NLB′</I>:&angle;<I>NOB′</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8991 > &angle;<I>COB</I>:&angle;<I>NOB′</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8992 > (arc <I>BC</I>):(arc <I>B′C′</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8993 <p>If <I>a′, b′, c′</I> are the sides of the spherical triangle <I>AB′C′</I>, this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8994 result is equivalent (since the angle <I>COB</I> subtended by the arc
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8995 <I>CB</I> is equal to <I>A</I>) to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8996 <MATH>1:sin <I>b′</I> = tan <I>A</I>:tan <I>a′</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8997 <MATH>> <I>a</I>:<I>a′</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8998 where <I>a</I> = <I>BC</I>, the side opposite <I>A</I> in the triangle <I>ABC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	8999 <p>The proof is based on the fact (proved in Euclid's <I>Optics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9000 and assumed as known by Aristarchus of Samos and Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9001 medes) that, if <G>a, b</G> are angles such that 1/2<G>p</G> > <G>a</G> > <G>b</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9002 tan <G>a</G>/tan <G>b</G> > <G>a</G>/<G>b</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9003 <p>While, therefore, Theodosius proves the equivalent of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9004 formula, applicable in the solution of a spherical triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9005 right-angled at <I>C</I>, that tan <I>a</I> = sin <I>b</I> tan <I>A</I>, he is unable, for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9006 want of trigonometry, to find the actual value of <I>a</I>/<I>a′</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9007 can only find a limit for it. He is exactly in the same position
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9008 as Aristarchus, who can only approximate to the values of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9009 trigonometrical ratios which he needs, e.g. sin 1°, cos 1°, sin 3°,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9010 by bringing them within upper and lower limits with the aid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9011 of the inequalities
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9012 <MATH>tan<G>a</G>/tan<G>b</G> > <G>a</G>/<G>b</G> > sin<G>a</G>/sin<G>b</G></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9013 where 1/2 <G>p</G> > <G>a</G> > <G>b</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9014 <pb n=252><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9015 <p>We may contrast with this proposition of Theodosius the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9016 corresponding proposition in Menelaus's <I>Sphaerica</I> (III. 15)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9017 dealing with the more general case in which <I>C′</I>, instead of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9018 being the tropical point on the ecliptic, is, like <I>B′</I>, any point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9019 between the tropical point and <I>D.</I> If <I>R</I>, <G>r</G> have the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9020 meaning as above and <I>r</I><SUB>1</SUB>, <I>r</I><SUB>2</SUB> are the radii of the parallel circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9021 through <I>B′</I> and the new <I>C′</I>, Menelaus proves that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9022 <MATH>sin<I>a</I>/sin<I>a′</I> = <I>R</I><G>r</G>/<I>r</I><SUB>1</SUB><I>r</I><SUB>2</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9023 which, of course, with the aid of Tables, gives the means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9024 of finding the actual values of <I>a</I> or <I>a′</I> when the other elements
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9025 are given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9026 <p>The proposition III. 12 of Theodosius proves a result similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9027 to that of III. 11 for the case where the great circles <I>AB′B</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9028 <I>AC′C</I>, instead of being great circles through the poles, are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9029 great circles touching ‘the circle of the always-visible stars’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9030 i.e. different positions of the horizon, and the points <I>C′, B′</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9031 any points on the arc of the oblique circle between the tropical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9032 and the equinoctial points; in this case, with the same notation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9033 <MATH>4<I>R</I>:2<G>r</G> > (arc <I>BC</I>):(arc <I>B′C′</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9034 <p>It is evident that Theodosius was simply a laborious com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9035 piler, and that there was practically nothing original in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9036 work. It has been proved, by means of propositions quoted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9037 <I>verbatim</I> or assumed as known by Autolycus in his <I>Moving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9038 Sphere</I> and by Euclid in his <I>Phaenomena</I>, that the following
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9039 propositions in Theodosius are pre-Euclidean, I. 1, 6 a, 7, 8, 11,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9040 12, 13, 15, 20; II. 1, 2, 3, 5, 8, 9, 10 a, 13, 15, 17, 18, 19, 20, 22;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9041 III. 1 b, 2, 3, 7, 8, those shown in thick type being quoted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9042 word for word.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9043 <C>The beginnings of trigonometry.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9044 <p>But this is not all. In Menelaus's <I>Sphaerica</I>, III. 15, there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9045 is a reference to the proposition (III. 11) of Theodosius proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9046 above, and in Gherard of Cremona's translation from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9047 Arabic, as well as in Halley's translation from the Hebrew
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9048 of Jacob b. Machir, there is an addition to the effect that this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9049 proposition was used by Apollonius in a book the title of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9050 which is given in the two translations in the alternative
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9051 <pb n=253><head>BEGINNINGS OF TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9052 forms ‘<I>liber aggregativus</I>’ and ‘liber de principiis universa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9053 libus’. Each of these expressions may well mean the work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9054 of Apollonius which Marinus refers to as the ‘General
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9055 Treatise’ (<G>h( kaqo/lou pragmatei/a</G>). There is no apparent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9056 reason to doubt that the remark in question was really
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9057 contained in Menelaus's original work; and, even if it is an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9058 Arabian interpolation, it is not likely to have been made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9059 without some definite authority. If then Apollonius was the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9060 discoverer of the proposition, the fact affords some ground for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9061 thinking that the beginnings of trigonometry go as far back,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9062 at least, as Apollonius. Tannery<note>Tannery, <I>Recherches sur l'hist. de l'astronomie ancienne</I>, p. 64.</note> indeed suggested that not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9063 only Apollonius but Archimedes before him may have com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9064 piled a ‘table of chords’, or at least shown the way to such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9065 a compilation, Archimedes in the work of which we possess
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9066 only a fragment in the <I>Measurement of a Circle</I>, and Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9067 in the <G>w)kuto/kion</G>, where he gave an approximation to the value
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9068 of <G>p</G> closer than that obtained by Archimedes; Tannery
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9069 compares the Indian Table of Sines in the <I>S&umacr;rya-Siddh&amacr;nta</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9070 where the angles go by 24ths of a right angle (1/24th = 3°45′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9071 2/24ths = 7°30′, &c.), as possibly showing Greek influence.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9072 This is, however, in the region of conjecture; the first person
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9073 to make systematic use of trigonometry is, so far as we know,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9074 Hipparchus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9075 <p>HIPPARCHUS, the greatest astronomer of antiquity, was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9076 born at Nicaea in Bithynia. The period of his activity is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9077 indicated by references in Ptolemy to observations made by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9078 him the limits of which are from 161 B.C. to 126 B.C. Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9079 further says that from Hipparchus's time to the beginning of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9080 the reign of Antoninus Pius (A.D. 138) was 265 years.<note>Ptolemy, <I>Syntaxis</I>, vii. 2 (vol. ii, p. 15).</note> The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9081 best and most important observations made by Hipparchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9082 were made at Rhodes, though an observation of the vernal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9083 equinox at Alexandria on March 24, 146 B.C., recorded by him
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9084 may have been his own. His main contributions to theoretical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9085 and practical astronomy can here only be indicated in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9086 briefest manner.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9087 <pb n=254><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9088 <C>The work of Hipparchus.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9089 <C><I>Discovery of precession.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9090 <p>1. The greatest is perhaps his discovery of the precession
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9091 of the equinoxes. Hipparchus found that the bright star
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9092 Spica was, at the time of his observation of it, 6° distant
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9093 from the autumnal equinoctial point, whereas he deduced from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9094 observations recorded by Timocharis that Timocharis had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9095 made the distance 8°. Consequently the motion had amounted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9096 to 2° in the period between Timocharis's observations, made in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9097 283 or 295 B.C., and 129/8 B.C., a period, that is, of 154 or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9098 166 years; this gives about 46.8″ or 43.4″ a year, as compared
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9099 with the true value of 50.3757″.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9100 <C><I>Calculation of mean lunar month.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9101 <p>2. The same discovery is presupposed in his work <I>On the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9102 length of the Year</I>, in which, by comparing an observation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9103 of the summer solstice by Aristarchus in 281/0 B.C. with his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9104 own in 136/5 B.C., he found that after 145 years (the interval
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9105 between the two dates) the summer solstice occurred half
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9106 a day-and-night earlier than it should on the assumption of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9107 exactly 365 1/4 days to the year; hence he concluded that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9108 <I>tropical</I> year contained about 1/300th of a day-and-night less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9109 than 365 1/4 days. This agrees very nearly with Censorinus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9110 statement that Hipparchus's cycle was 304 years, four times
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9111 the 76 years of Callippus, but with 111,035 days in it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9112 instead of 111,036 (= 27,759 x 4). Counting in the 304 years
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9113 12 x 304 + 112 (intercalary) months, or 3,760 months in all,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9114 Hipparchus made the mean lunar month 29 days 12 hrs.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9115 44 min. 2 1/2 sec., which is less than a second out in comparison
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9116 with the present accepted figure of 29.53059 days!
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9117 <p>3. Hipparchus attempted a new determination of the sun's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9118 motion by means of exact equinoctial and solstitial obser-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9119 vations; he reckoned the eccentricity of the sun's course
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9120 and fixed the apogee at the point 5°30′ of <I>Gemini.</I> More
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9121 remarkable still was his investigation of the moon's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9122 course. He determined the eccentricity and the inclination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9123 of the orbit to the ecliptic, and by means of records of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9124 observations of eclipses determined the moon's period with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9125 extraordinary accuracy (as remarked above). We now learn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9126 <pb n=255><head>HIPPARCHUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9127 that the lengths of the mean synodic, the sidereal, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9128 anomalistic and the draconitic month obtained by Hipparchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9129 agree exactly with Babylonian cuneiform tables of date not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9130 later than Hipparchus, and it is clear that Hipparchus was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9131 in full possession of all the results established by Babylonian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9132 astronomy.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9133 <C><I>Improved estimates of sizes and distances of sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9134 and moon.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9135 <p>4. Hipparchus improved on Aristarchus's calculations of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9136 sizes and distances of the sun and moon, determining the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9137 apparent diameters more exactly and noting the changes in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9138 them; he made the mean distance of the sun 1,245<I>D</I>, the mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9139 distance of the moon 33 2/3<I>D</I>, the diameters of the sun and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9140 moon 12 1/3<I>D</I> and 1/3<I>D</I> respectively, where <I>D</I> is the mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9141 diameter of the earth.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9142 <C><I>Epicycles and eccentrics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9143 <p>5. Hipparchus, in investigating the motions of the sun, moon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9144 and planets, proceeded on the alternative hypotheses of epi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9145 cycles and eccentrics; he did not invent these hypotheses,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9146 which were already fully understood and discussed by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9147 Apollonius. While the motions of the sun and moon could
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9148 with difficulty be accounted for by the simple epicycle and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9149 eccentric hypotheses, Hipparchus found that for the planets it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9150 was necessary to combine the two, i.e. to superadd epicycles to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9151 motion in eccentric circles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9152 <C><I>Catalogue of stars.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9153 <p>6. He compiled a catalogue of fixed stars including 850 or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9154 more such stars; apparently he was the first to state their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9155 positions in terms of coordinates in relation to the ecliptic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9156 (latitude and longitude), and his table distinguished the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9157 apparent sizes of the stars. His work was continued by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9158 Ptolemy, who produced a catalogue of 1,022 stars which,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9159 owing to an error in his solar tables affecting all his longi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9160 tudes, has by many erroneously been supposed to be a mere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9161 reproduction of Hipparchus's catalogue. That Ptolemy took
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9162 many observations himself seems certain.<note>See two papers by Dr. J. L. E. Dreyer in the <I>Monthly Notices of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9163 Royal Astronomical Society</I>, 1917, pp. 528-39, and 1918, pp. 343-9.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9164 <pb n=256><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9165 <C><I>Improved Instruments.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9166 <p>7. He made great improvements in the instruments used for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9167 observations. Among those which he used were an improved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9168 dioptra, a ‘meridian-instrument’ designed for observations in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9169 the meridian only, and a universal instrument (<G>a)strola/bon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9170 o)/rganon</G>) for more general use. He also made a globe on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9171 which he showed the positions of the fixed stars as determined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9172 by him; it appears that he showed a larger number of stars
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9173 on his globe than in his catalogue.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9174 <C><I>Geography.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9175 <p>In geography Hipparchus wrote a criticism of Eratosthenes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9176 in great part unfair. He checked Eratosthenes's data by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9177 means of a sort of triangulation; he insisted on the necessity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9178 of applying astronomy to geography, of fixing the position of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9179 places by latitude and longitude, and of determining longitudes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9180 by observations of lunar eclipses.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9181 <p>Outside the domain of astronomy and geography, Hipparchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9182 wrote a book <I>On things borne down by their weight</I> from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9183 which Simplicius (on Aristotle's <I>De caelo</I>, p. 264 sq.) quotes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9184 two propositions. It is possible, however, that even in this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9185 work Hipparchus may have applied his doctrine to the case of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9186 the heavenly bodies.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9187 <p>In pure mathematics he is said to have considered a problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9188 in permutations and combinations, the problem of finding the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9189 number of different possible combinations of 10 axioms or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9190 assumptions, which he made to be 103,049 (<I>v.l.</I> 101,049)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9191 or 310,952 according as the axioms were affirmed or denied<note>Plutarch, <I>Quaest. Conviv.</I> viii. 9. 3, 732 F, <I>De Stoicorum repugn.</I> 29.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9192 1047 D.</note>:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9193 it seems impossible to make anything of these figures. When
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9194 the <I>Fihrist</I> attributes to him works ‘On the art of algebra,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9195 known by the title of the Rules’ and ‘On the division of num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9196 bers’, we have no confirmation: Suter suspects some confusion,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9197 in view of the fact that the article immediately following in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9198 the <I>Fihrist</I> is on Diophantus, who also ‘wrote on the art of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9199 algebra’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9200 <pb n=257><head>HIPPARCHUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9201 <C>First systematic use of Trigonometry.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9202 <p>We come now to what is the most important from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9203 point of view of this work, Hipparchus's share in the develop-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9204 ment of trigonometry. Even if he did not invent it,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9205 Hipparchus is the first person of whose systematic use of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9206 trigonometry we have documentary evidence. (1) Theon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9207 of Alexandria says on the <I>Syntaxis</I> of Ptolemy, aà propos of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9208 Ptolemy's Table of Chords in a circle (equivalent to sines),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9209 that Hipparchus, too, wrote a treatise in twelve books on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9210 straight lines (i.e. chords) in a circle, while another in six
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9211 books was written by Menelaus.<note>Theon, <I>Comm. on Syntaxis</I>, p. 110, ed. Halma.</note> In the <I>Syntaxis</I> I. 10
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9212 Ptolemy gives the necessary explanations as to the notation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9213 used in his Table. The circumference of the circle is divided
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9214 into 360 parts or degrees; the diameter is also divided into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9215 120 parts, and one of such parts is the unit of length in terms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9216 of which the length of each chord is expressed; each part,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9217 whether of the circumference or diameter, is divided into 60
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9218 parts, each of these again into 60, and so on, according to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9219 system of sexagesimal fractions. Ptolemy then sets out the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9220 minimum number of propositions in plane geometry upon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9221 which the calculation of the chords in the Table is based (<G>dia\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9222 th=s e)k tw=n grammw=n meqodikh=s au)tw=n susta/sews</G>). The pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9223 positions are famous, and it cannot be doubted that Hippar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9224 chus used a set of propositions of the same kind, though his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9225 exposition probably ran to much greater length. As Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9226 definitely set himself to give the necessary propositions in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9227 shortest form possible, it will be better to give them under
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9228 Ptolemy rather than here. (2) Pappus, in speaking of Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9229 propositions about the inequality of the times which equal arcs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9230 of the zodiac take to rise, observes that ‘Hipparchus in his book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9231 <I>On the rising of the twelve signs of the zodiac</I> shows <I>by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9232 of numerical calculations</I> (<G>di' a)riqmw=n</G>) that equal arcs of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9233 semicircle beginning with Cancer which set in times having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9234 a certain relation to one another do not everywhere show the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9235 same relation between the times in which they rise’,<note>Pappus, vi, p. 600. 9-13.</note> and so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9236 on. We have seen that Euclid, Autolycus, and even Theo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9237 dosius could only prove that the said times are greater or less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9238 <pb n=258><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9239 in relation to one another; they could not calculate the actual
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9240 times. As Hipparchus proved corresponding propositions by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9241 means of <I>numbers</I>, we can only conclude that he used proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9242 tions in spherical trigonometry, calculating arcs from others
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9243 which are given, by means of tables. (3) In the only work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9244 of his which survives, the <I>Commentary on the Phaenomena
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9245 of Eudoxus and Aratus</I> (an early work anterior to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9246 discovery of the precession of the equinoxes), Hipparchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9247 states that (presumably in the latitude of Rhodes) a star which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9248 lies 27 1/3° north of the equator describes above the horizon an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9249 arc containing 3 minutes less than 15/24ths of the whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9250 circle<note>Ed. Manitius, pp. 148-50.</note>; then, after some more inferences, he says, ‘For each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9251 of the aforesaid facts is proved <I>by means of lines</I> (<G>dia\ tw=n
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9252 grammw=n</G>) in the general treatises on these matters compiled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9253 by me’. In other places<note><I>Ib.</I>, pp. 128. 5, 148. 20.</note> of the <I>Commentary</I> he alludes to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9254 a work <I>On simultaneous risings</I> (<G>ta\ peri\ tw=n sunanatolw=n</G>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9255 and in II.4. 2 he says he will state summarily, about each of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9256 the fixed stars, along with what sign of the zodiac it rises and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9257 sets and from which degree to which degree of each sign it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9258 rises or sets in the regions about Greece or wherever the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9259 longest day is 14 1/2 equinoctial hours, adding that he has given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9260 special proofs in another work designed so that it is possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9261 in practically every place in the inhabited earth to follow
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9262 the differences between the concurrent risings and settings.<note><I>Ib.</I>, pp. 182. 19-184. 5.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9263 Where Hipparchus speaks of proofs ‘by means of lines’, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9264 does not mean a merely graphical method, by construction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9265 only, but theoretical determination by geometry, followed by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9266 calculation, just as Ptolemy uses the expression <G>e)k tw=n gram-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9267 mw=n</G> of his calculation of chords and the expressions <G>sfairikai\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9268 dei/xeis</G> and <G>grammikai\ dei/xeis</G> of the fundamental proposition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9269 in spherical trigonometry (Menelaus's theorem applied to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9270 sphere) and its various applications to particular cases. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9271 is significant that in the <I>Syntaxis</I> VIII. 5, where Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9272 applies the proposition to the very problem of finding the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9273 times of concurrent rising, culmination and setting of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9274 fixed stars, he says that the times can be obtained ‘by lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9275 only’ (<G>dia\ mo/nwn tw=n grammw=n</G>).<note><I>Syntaxis</I>, vol. ii, p. 193.</note> Hence we may be certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9276 that, in the other books of his own to which Hipparchus refers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9277 <pb n=259><head>HIPPARCHUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9278 in his <I>Commentary</I>, he used the formulae of spherical trigono-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9279 metry to get his results. In the particular case where it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9280 required to find the time in which a star of 27 1/3° northern
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9281 declination describes, in the latitude of Rhodes, the portion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9282 its arc above the horizon, Hipparchus must have used the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9283 equivalent of the formula in the solution of a right-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9284 spherical triangle, tan <I>b</I> = cos <I>A</I> tan <I>c</I>, where <I>C</I> is the right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9285 angle. Whether, like Ptolemy, Hipparchus obtained the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9286 formulae, such as this one, which he used from different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9287 applications of the one general theorem (Menelaus's theorem)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9288 it is not possible to say. There was of course no difficulty
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9289 in calculating the tangent or other trigonometrical function
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9290 of an angle if only a table of sines was given; for Hippar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9291 chus and Ptolemy were both aware of the fact expressed by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9292 sin<SUP>2</SUP> <G>a</G> + cos<SUP>2</SUP> <G>a</G> = 1 or, as they would have written it,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9293 <MATH>(crd. 2<G>a</G>)<SUP>2</SUP> + {crd. (180° - 2<G>a</G>)}<SUP>2</SUP> = 4<I>r</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9294 where (crd. 2<G>a</G>) means the chord subtending an arc 2<G>a</G>, and <G>g</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9295 is the radius, of the circle of reference.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9296 <C>Table of Chords.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9297 <p>We have no details of Hipparchus's Table of Chords suffi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9298 cient to enable us to compare it with Ptolemy's, which goes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9299 by half-degrees, beginning with angles of 1/2°, 1°, 1 1/2°, and so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9300 on. But Heron<note>Heron, <I>Metrica</I>, i. 22, 24, pp. 58. 19 and 62. 17.</note> in his <I>Metrica</I> says that ‘it is proved in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9301 books about chords in a circle’ that, if <I>a</I><SUB>9</SUB> and <I>a</I><SUB>11</SUB> are the sides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9302 of a regular enneagon (9-sided figure) and hendecagon (11-sided
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9303 figure) inscribed in a circle of diameter <I>d</I>, then (1) <I>a</I><SUB>9</SUB> = 1/3<I>d</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9304 (2) <I>a</I><SUB>11</SUB> = 7/25<I>d</I> very nearly, which means that sin 20° was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9305 taken as equal to 0.3333 . . . (Ptolemy's table makes it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9306 1/60(20 + 31/60 + 16 1/2/60<SUP>2</SUP>), so that the first approximation is 1/3), and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9307 sin 1/11.180° or sin 16° 21′ 49″ was made equal to 0.28 (this cor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9308 responds to the chord subtending an angle of 32° 43′ 38″, nearly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9309 half-way between 32 1/2° and 33°, and the mean between the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9310 chords subtending the latter angles gives 1/60(16 + 54/60 + 55/60<SUP>2</SUP>) as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9311 the required sine, while 1/60(16 9/10) = 169/600, which only differs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9312 <pb n=260><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9313 by 1/600 from 168/600 or 7/25, Heron's figure). There is little doubt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9314 that it is to Hipparchus's work that Heron refers, though the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9315 author is not mentioned.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9316 <p>While for our knowledge of Hipparchus's trigonometry we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9317 have to rely for the most part upon what we can infer from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9318 Ptolemy, we fortunately possess an original source of infor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9319 mation about Greek trigonometry in its highest development
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9320 in the <I>Sphaerica</I> of Menelaus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9321 <p>The date of MENELAUS of Alexandria is roughly indi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9322 cated by the fact that Ptolemy quotes an observation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9323 his made in the first year of Trajan's reign (A.D. 98). He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9324 was therefore a contemporary of Plutarch, who in fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9325 represents him as being present at the dialogue <I>De facie in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9326 orbe lunae</I>, where (chap. 17) Lucius apologizes to Menelaus ‘the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9327 mathematician’ for questioning the fundamental proposition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9328 in optics that the angles of incidence and reflection are equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9329 <p>He wrote a variety of treatises other than the <I>Sphaerica.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9330 We have seen that Theon mentions his work on <I>Chords in a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9331 Circle</I> in six Books. Pappus says that he wrote a treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9332 (<G>pragmatei/a</G>) on the setting (or perhaps only rising) of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9333 different arcs of the zodiac.<note>Pappus, vi, pp. 600-2.</note> Proclus quotes an alternative
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9334 proof by him of Eucl. I. 25, which is direct instead of by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9335 <I>reductio ad absurdum</I>,<note>Proclus on Eucl. I, pp. 345. 14-346. 11.</note> and he would seem to have avoided
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9336 the latter kind of proof throughout. Again, Pappus, speaking
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9337 of the many complicated curves ‘discovered by Demetrius of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9338 Alexandria (in his “Linear considerations”) and by Philon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9339 of Tyana as the result of interweaving plectoids and other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9340 surfaces of all kinds’, says that one curve in particular was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9341 investigated by Menelaus and called by him ‘paradoxical’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9342 (<G>para/doxos</G>)<note>Pappus, iv, p. 270. 25.</note>; the nature of this curve can only be conjectured
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9343 (see below).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9344 <p>But Arabian tradition refers to other works by Menelaus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9345 (1) <I>Elements of Geometry</I>, edited by Th&amacr;bit b. Qurra, in three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9346 Books, (2) a Book on triangles, and (3) a work the title of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9347 which is translated by Wenrich <I>de cognitione quantitatis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9348 discretac corporum permixtorum.</I> Light is thrown on this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9349 last title by one al-Ch&amacr;zin&imacr; who (about A.D. 1121) wrote a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9350 <pb n=261><head>MENELAUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9351 treatise about the hydrostatic balance, i.e. about the deter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9352 mination of the specific gravity of homogeneous or mixed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9353 bodies, in the course of which he mentions Archimedes and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9354 Menelaus (among others) as authorities on the subject; hence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9355 the treatise (3) must have been a book on hydrostatics dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9356 cussing such problems as that of the crown solved by Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9357 medes. The alternative proof of Eucl. I. 25 quoted by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9358 Proclus might have come either from the <I>Elements of Geometry</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9359 or the Book on triangles. With regard to the geometry, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9360 ‘liber trium fratrum’ (written by three sons of M&umacr;s&amacr; b. Sh&amacr;kir
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9361 in the ninth century) says that it contained a solution of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9362 duplication of the cube, which is none other than that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9363 Archytas. The solution of Archytas having employed the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9364 intersection of a tore and a cylinder (with a cone as well),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9365 there would, on the assumption that Menelaus reproduced the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9366 solution, be a certain appropriateness in the suggestion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9367 Tannery<note>Tannery, <I>Mémoires scientifiques</I>, ii, p. 17.</note> that the curve which Menelaus called the <G>para/doxos
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9368 grammh/</G> was in reality the curve of double curvature, known
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9369 by the name of Viviani, which is the intersection of a sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9370 with a cylinder touching it internally and having for its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9371 diameter the radius of the sphere. This curve is a particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9372 case of Eudoxus's <I>hippopede</I>, and it has the property that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9373 portion left outside the curve of the surface of the hemisphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9374 on which it lies is equal to the square on the diameter of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9375 sphere; the fact of the said area being squareable would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9376 justify the application of the word <G>para/doxos</G> to the curve,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9377 and the quadrature itself would not probably be beyond the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9378 powers of the Greek mathematicians, as witness Pappus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9379 determination of the area cut off between a complete turn of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9380 a certain spiral on a sphere and the great circle touching it at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9381 the origin.<note>Pappus, iv, pp. 264-8.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9382 <C>The <I>Sphaerica</I> of Menelaus.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9383 <p>This treatise in three Books is fortunately preserved in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9384 the Arabic, and although the extant versions differ con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9385 siderably in form, the substance is beyond doubt genuine;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9386 the original translator was apparently Ish&amacr;q b. Hunain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9387 (died A. D. 910). There have been two editions, (1) a Latin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9388 <pb n=262><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9389 translation by Maurolycus (Messina, 1558) and (2) Halley's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9390 edition (Oxford, 1758). The former is unserviceable because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9391 Maurolycus's manuscript was very imperfect, and, besides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9392 trying to correct and restore the propositions, he added
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9393 several of his own. Halley seems to have made a free
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9394 translation of the Hebrew version of the work by Jacob b.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9395 Machir (about 1273), although he consulted Arabic manuscripts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9396 to some extent, following them, e.g., in dividing the work into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9397 three Books instead of two. But an earlier version direct
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9398 from the Arabic is available in manuscripts of the thirteenth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9399 to fifteenth centuries at Paris and elsewhere; this version is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9400 without doubt that made by the famous translator Gherard
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9401 of Cremona (1114-87). With the help of Halley's edition,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9402 Gherard's translation, and a Leyden manuscript (930) of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9403 the redaction of the work by Ab&umacr;-Nasr-Mans&umacr;r made in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9404 A.D. 1007-8, Björnbo has succeeded in presenting an adequate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9405 reproduction of the contents of the <I>Sphaerica.</I><note>Björnbo, <I>Studien über Menelaos' Sphärik</I> (Abhandlungen zur Gesch. d.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9406 math. Wissenschaften, Heft xiv. 1902).</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9407 <C>Book I.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9408 <p>In this Book for the first time we have the conception and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9409 definition of a <I>spherical triangle.</I> Menelaus does not trouble
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9410 to give the usual definitions of points and circles related to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9411 the sphere, e.g. pole, great circle, small circle, but begins with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9412 that of a spherical triangle as ‘the area included by arcs of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9413 great circles on the surface of a sphere’, subject to the restric-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9414 tion (Def. 2) that each of the sides or legs of the triangle is an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9415 arc less than a semicircle. The angles of the triangle are the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9416 angles contained by the arcs of great circles on the sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9417 (Def. 3), and one such angle is equal to or greater than another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9418 according as the planes containing the arcs forming the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9419 angle are inclined at the same angle as, or a greater angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9420 than, the planes of the arcs forming the other (Defs. 4, 5).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9421 The angle is a right angle if the planes of the arcs are at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9422 angles (Def. 6). Pappus tells us that Menelaus in his <I>Sphaerica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9423 calls the figure in question (the spherical triangle) a ‘three-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9424 side’ (<G>tri/pleuron</G>)<note>Pappus, vi, p. 476. 16.</note>; the word <I>triangle</I> (<G>tri/gwnon</G>) was of course
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9425 <pb n=263><head>MENELAUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9426 already appropriated for the plane triangle. We should gather
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9427 from this, as well as from the restriction of the definitions to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9428 the spherical triangle and its parts, that the discussion of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9429 spherical triangle as such was probably new; and if the pre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9430 face in the Arabic version addressed to a prince and beginning
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9431 with the words, ‘O prince! I have discovered an excellent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9432 method of proof . . .’ is genuine, we have confirmatory evidence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9433 in the writer's own claim.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9434 <p>Menelaus's object, so far as Book I is concerned, seems to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9435 have been to give the main propositions about spherical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9436 triangles corresponding to Euclid's propositions about plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9437 triangles. At the same time he does not restrict himself to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9438 Euclid's methods of proof even where they could be adapted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9439 to the case of the sphere; he avoids the form of proof by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9440 <I>reductio ad absurdum</I>, but, subject to this, he prefers the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9441 easiest proofs. In some respects his treatment is more com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9442 plete than Euclid's treatment of the analogous plane cases.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9443 In the congruence-theorems, for example, we have I. 4a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9444 corresponding to Eucl. I. 4, I. 4b to Eucl. I. 8, I. 14, 16 to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9445 Eucl. I. 26a, b; but Menelaus includes (I. 13) what we know
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9446 as the ‘ambiguous case’, which is enunciated on the lines of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9447 Eucl. VI. 7. I. 12 is a particular case of I. 16. Menelaus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9448 includes also the further case which has no analogue in plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9449 triangles, that in which the three angles of one triangle are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9450 severally equal to the three angles of the other (I. 17). He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9451 makes, moreover, no distinction between the congruent and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9452 the symmetrical, regarding both as covered by congruent. I. 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9453 is a problem, to construct a spherical angle equal to a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9454 spherical angle, introduced only as a lemma because required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9455 in later propositions. I. 2, 3 are the propositions about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9456 isosceles triangles corresponding to Eucl. I. 5, 6; Eucl. I. 18, 19
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9457 (greater side opposite greater angle and vice versa) have their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9458 analogues in I. 7, 9, and Eucl. I. 24, 25 (two sides respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9459 equal and included angle, or third side, in one triangle greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9460 than included angle, or third side, in the other) in I. 8. I. 5
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9461 (two sides of a triangle together greater than the third) corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9462 sponds to Eucl. I. 20. There is yet a further group of proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9463 tions comparing parts of spherical triangles, I. 6, 18, 19, where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9464 I. 6 (corresponding to Eucl. I. 21) is deduced from I. 5, just as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9465 the first part of Eucl. I. 21 is deduced from Eucl. I. 20.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9466 <pb n=264><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9467 <p>Eucl. I. 16, 32 are not true of spherical triangles, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9468 Menelaus has therefore the corresponding but different pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9469 positions. I. 10 proves that, with the usual notation <I>a, b, c,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9470 A, B, C,</I> for the sides and opposite angles of a spherical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9471 triangle, the exterior angle at <I>C,</I> or <MATH>180°-<I>C,</I> < = or > <I>A</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9472 according as <MATH><I>c</I>+<I>a</I> > = or < 180°</MATH>, and vice versa. The proof
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9473 of this and the next proposition shall be given as specimens.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9474 <p>In the triangle <I>ABC</I> suppose that <MATH><I>c</I>+<I>a</I> > = or < 180°</MATH>; let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9475 <I>D</I> be the pole opposite to <I>A.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9476 <p>Then, according as <MATH><I>c</I>+<I>a</I> > = or < 180°, <I>BC</I> > = or < <I>BD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9477 (since <I>AD</I> = 180°)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9478 and therefore <MATH>&angle;<I>D</I> > = or < &angle;<I>BCD</I> (= 180°-<I>C</I>)</MATH>, [I. 9]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9479 i.e. <MATH>(since &angle;<I>D</I> = &angle;<I>A</I>) 180°-<I>C</I> < = or > <I>A</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9480 <p>Menelaus takes the converse for granted.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9481 <p>As a consequence of this, I. 11 proves that <MATH><I>A</I>+<I>B</I>+<I>C</I> > 180°</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9482 <p>Take the same triangle <I>ABC,</I> with the pole <I>D</I> opposite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9483 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9484 to <I>A,</I> and from <I>B</I> draw the great circle <I>BE</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9485 <MATH>&angle;<I>DBE</I> = &angle;<I>BDE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9486 <p>Then <MATH><I>CE</I>+<I>EB</I> = <I>CD</I> < 180°</MATH>, so that, by the preceding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9487 proposition, the exterior angle <I>ACB</I> to the triangle <I>BCE</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9488 greater than &angle;<I>CBE,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9489 i.e. <MATH><I>C</I> > &angle;<I>CBE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9490 <p>Add <I>A</I> or <I>D</I> (= &angle;<I>EBD</I>) to the unequals;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9491 therefore <MATH><I>C</I>+<I>A</I> > &angle;<I>CBD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9492 whence <MATH><I>A</I>+<I>B</I>+<I>C</I> > &angle;<I>CBD</I>+<I>B</I> or 180°</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9493 <p>After two lemmas I. 21, 22 we have some propositions introducing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9494 <I>M, N, P</I> the middle points of <I>a, b, c</I> respectively. I. 23
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9495 proves, e.g., that the arc <I>MN</I> of a great circle > 1/2<I>c,</I> and I. 20
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9496 that <MATH><I>AM</I> < = or > 1/2<I>a</I></MATH> according as <MATH><I>A</I> > = or < (<I>B</I>+<I>C</I>)</MATH>. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9497 last group of propositions, 26-35, relate to the figure formed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9498 <pb n=265><head>MENELAUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9499 by the triangle <I>ABC</I> with great circles drawn through <I>B</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9500 meet <I>AC</I> (between <I>A</I> and <I>C</I>) in <I>D, E</I> respectively, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9501 case where <I>D</I> and <I>E</I> coincide, and they prove different results
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9502 arising from different relations between <I>a</I> and <I>c</I> (<I>a</I> > <I>c</I>), com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9503 bined with the equality of <I>AD</I> and <I>EC</I> (or <I>DC</I>), of the angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9504 <I>ABD</I> and <I>EBC</I> (or <I>DBC</I>), or of <I>a</I>+<I>c</I> and <I>BD</I>+<I>BE</I> (or 2 <I>BD</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9505 respectively, according as <MATH><I>a</I>+<I>c</I> < = or > 180°</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9506 <p>Book II has practically no interest for us. The object of it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9507 is to establish certain propositions, of astronomical interest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9508 only, which are nothing more than generalizations or exten-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9509 sions of propositions in Theodosius's <I>Sphaerica,</I> Book III.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9510 Thus Theodosius III. 5, 6, 9 are included in Menelaus II. 10,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9511 Theodosius III. 7-8 in Menelaus II. 12, while Menelaus II. 11
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9512 is an extension of Theodosius III. 13. The proofs are quite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9513 different from those of Theodosius, which are generally very
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9514 long-winded.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9515 <C>Book III. Trigonometry.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9516 <p>It will have been noticed that, while Book I of Menelaus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9517 gives the geometry of the spherical triangle, neither Book I
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9518 nor Book II contains any trigonometry. This is reserved for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9519 Book III. As I shall throughout express the various results
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9520 obtained in terms of the trigonometrical ratios, sine, cosine,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9521 tangent, it is necessary to explain once for all that the Greeks
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9522 did not use this terminology, but, instead of sines, they used
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9523 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9524 the chords subtended by arcs of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9525 circle. In the accompanying figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9526 let the arc <I>AD</I> of a circle subtend an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9527 angle <G>a</G> at the centre <I>O.</I> Draw <I>AM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9528 perpendicular to <I>OD,</I> and produce it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9529 to meet the circle again in <I>A</I>′. Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9530 sin <MATH><G>a</G> = <I>AM/AO</I></MATH>, and <I>AM</I> is 1/2<I>AA</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9531 or half the chord subtended by an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9532 angle 2<G>a</G> at the centre, which may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9533 shortly be denoted by 1/2(crd. 2<G>a</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9534 Since Ptolemy expresses the chords as so many 120th parts of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9535 the diameter of the circle, while <MATH><I>AM/AO</I> = <I>AA</I>′/2<I>AO</I></MATH>, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9536 follows that sin <G>a</G> and 1/2(crd. 2<G>a</G>) are equivalent. Cos <G>a</G> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9537 of course sin (90°-<G>a</G>) and is therefore equivalent to 1/2 crd.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9538 (180°-2<G>a</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9539 <pb n=266><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9540 <C>(<G>a</G>) <I>‘Menelaus's theorem’ for the sphere.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9541 <p>The first proposition of Book III is the famous ‘Menelaus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9542 theorem’ with reference to a spherical triangle and any trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9543 versal (great circle) cutting the sides of a triangle, produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9544 if necessary. Menelaus does not, however, use a spherical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9545 triangle in his enunciation, but enunciates the proposition in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9546 terms of intersecting great circles. ‘Between two arcs <I>ADB,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9547 AEC</I> of great circles are two other arcs of great circles <I>DFC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9548 and <I>BFE</I> which intersect them and also intersect each other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9549 in <I>F.</I> All the arcs are less than a semicircle. It is required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9550 to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9551 <MATH>sin <I>CE</I>/sin <I>EA</I> = sin <I>CF</I>/sin <I>FD</I>.sin <I>DB</I>/sin <I>BA</I></MATH>.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9552 <p>It appears that Menelaus gave three or four cases, sufficient
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9553 to prove the theorem completely. The proof depends on two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9554 simple propositions which Menelaus assumes without proof;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9555 the proof of them is given by Ptolemy.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9556 <p>(1) In the figure on the last page, if <I>OD</I> be a radius cutting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9557 a chord <I>AB</I> in <I>C,</I> then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9558 <MATH><I>AC</I>:<I>CB</I> = sin <I>AD</I>:sin <I>DB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9559 <p>For draw <I>AM, BN</I> perpendicular to <I>OD.</I> Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9560 <MATH><I>AC</I>:<I>CB</I> = <I>AM</I>:<I>BN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9561 = 1/2 (crd. 2<I>AD</I>):1/2 (crd. 2<I>DB</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9562 = sin <I>AD</I>:sin <I>DB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9563 <p>(2) If <I>AB</I> meet the radius <I>OC</I> produced in <I>T,</I> then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9564 <MATH><I>AT</I>:<I>BT</I> = sin <I>AC</I>:sin <I>BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9565 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9566 <pb n=267><head>MENELAUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9567 <p>For, if <I>AM, BN</I> are perpendicular to <I>OC,</I> we have, as before,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9568 <MATH><I>AT</I>:<I>TB</I> = <I>AM</I>:<I>BN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9569 = 1/2 (crd. 2 <I>AC</I>):1/2(crd. 2<I>BC</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9570 = sin <I>AC</I>:sin <I>BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9571 <p>Now let the arcs of great circles <I>ADB, AEC</I> be cut by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9572 arcs of great circles <I>DFC, BFE</I> which themselves meet in <I>F.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9573 <p>Let <I>G</I> be the centre of the sphere and join <I>GB, GF, GE, AD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9574 <p>Then the straight lines <I>AD, GB,</I> being in one plane, are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9575 either parallel or not parallel. If they are not parallel, they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9576 will meet either in the direction of <I>D, B</I> or of <I>A, G.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9577 <p>Let <I>AD, GB</I> meet in <I>T.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9578 <p>Draw the straight lines <I>AKC, DLC</I> meeting <I>GE, GF</I> in <I>K, L</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9579 respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9580 <p>Then <I>K, L, T</I> must lie on a straight line, namely the straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9581 line which is the section of the planes determined by the arc
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9582 <I>EFB</I> and by the triangle <I>ACD.</I><note>So Ptolemy. In other words, since the straight lines <I>GB, GE, GF,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9583 which are in one plane, respectively intersect the straight lines <I>AD, AC,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9584 CD</I> which are also in one plane, the points of intersection <I>T, K, L</I> are in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9585 both planes, and therefore lie on the straight line in which the planes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9586 intersect.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9587 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9588 <p>Thus we have two straight lines <I>AC, AT</I> cut by the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9589 straight lines <I>CD, TK</I> which themselves intersect in <I>L.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9590 <p>Therefore, by Menelaus's proposition in plane geometry,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9591 <MATH><I>CK</I>/<I>KA</I> = <I>CL</I>/<I>LD</I>.<I>DT</I>/<I>TA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9592 <pb n=268><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9593 <p>But, by the propositions proved above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9594 <MATH><I>CK</I>/<I>KA</I> = sin <I>CE</I>/sin <I>EA</I>, <I>CL</I>/<I>LD</I> = sin <I>CF</I>/sin <I>FD</I></MATH>, and <MATH><I>DT</I>/<I>TA</I> = sin <I>DB</I>/sin <I>BA</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9595 therefore, by substitution, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9596 <MATH>sin <I>CE</I>/sin <I>EA</I> = sin <I>CF</I>/sin <I>FD</I>.sin <I>DB</I>/sin <I>BA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9597 <p>Menelaus apparently also gave the proof for the cases in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9598 which <I>AD, GB</I> meet towards <I>A, G,</I> and in which <I>AD, GB</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9599 parallel respectively, and also proved that in like manner, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9600 the above figure,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9601 <MATH>sin <I>CA</I>/sin <I>AE</I> = sin <I>CD</I>/sin <I>DF</I>.sin <I>FB</I>/sin <I>BE</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9602 (the triangle cut by the transversal being here <I>CFE</I> instead of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9603 <I>ADC</I>). Ptolemy<note>Ptolemy, <I>Syntaxis,</I> i. 13, vol. i, p. 76.</note> gives the proof of the above case only, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9604 dismisses the last-mentioned result with a ‘similarly’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9605 <C>(<G>b</G>) <I>Deductions from Menelaus's Theorem.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9606 <p>III. 2 proves, by means of I. 14, 10 and III. 1, that, if <I>ABC,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9607 A</I>′<I>B</I>′<I>C</I>′ be two spherical triangles in which <I>A</I> = <I>A</I>′, and <I>C, C</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9608 are either equal or supplementary, <MATH>sin <I>c</I>/sin <I>a</I> = sin <I>c</I>′/sin <I>a</I>′</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9609 and conversely. The particular case in which <I>C, C</I>′ are right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9610 angles gives what was afterwards known as the ‘regula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9611 quattuor quantitatum’ and was fundamental in Arabian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9612 trigonometry.<note>See Braunmühl, <I>Gesch. der Trig.</I> i, pp. 17, 47, 58-60, 127-9.</note> A similar association attaches to the result of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9613 III. 3, which is the so-called ‘tangent’ or ‘shadow-rule’ of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9614 Arabs. If <I>ABC, A</I>′<I>B</I>′<I>C</I>′ be triangles right-angled at <I>A, A</I>′, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9615 <I>C, C</I>′ are equal and both either > or < 90°, and if <I>P, P</I>′ be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9616 the poles of <I>AC, A</I>′<I>C</I>′, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9617 <MATH>sin <I>AB</I>/sin <I>AC</I> = sin <I>A</I>′<I>B</I>′/sin <I>A</I>′<I>C</I>′.sin <I>BP</I>/sin <I>B</I>′<I>P</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9618 <p>Apply the triangles so that <I>C</I>′ falls on <I>C, C</I>′<I>B</I>′ on <I>CB</I> as <I>CE,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9619 and <I>C</I>′<I>A</I>′ on <I>CA</I> as <I>CD;</I> then the result follows directly from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9620 III. 1. Since <MATH>sin <I>BP</I> = cos <I>AB</I></MATH>, and <MATH>sin <I>B</I>′<I>P</I>′ = cos <I>A</I>′<I>B</I>′</MATH>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9621 result becomes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9622 <MATH>sin <I>CA</I>/sin <I>C</I>′<I>A</I>′ = tan <I>AB</I>/tan <I>A</I>′<I>B</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9623 which is the ‘tangent-rule’ of the Arabs.<note>Cf. Braunmühl, <I>op. cit.</I> i, pp. 17-18, 58, 67-9, &c.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9624 <pb n=269><head>MENELAUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9625 <p>It follows at once (Prop. 4) that, if <I>AM, A</I>′<I>M</I>′ are great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9626 circles drawn perpendicular to the bases <I>BC, B</I>′<I>C</I>′ of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9627 spherical triangles <I>ABC, A</I>′<I>B</I>′<I>C</I>′ in which <I>B</I> = <I>B</I>′, <I>C</I> = <I>C</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9628 <MATH>sin <I>BM</I>/sin <I>B</I>′<I>M</I>′ = sin <I>MC</I>/sin <I>M</I>′<I>C</I>′ (since both are equal to tan <I>AM</I>/tan <I>A</I>′<I>M</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9629 <p>III. 5 proves that, if there are two spherical triangles <I>ABC,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9630 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9631 <I>A</I>′<I>B</I>′<I>C</I>′ right-angled at <I>A, A</I>′ and such that <I>C</I> = <I>C</I>′, while <I>b</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9632 and <I>b</I>′ are less than 90°,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9633 <MATH>sin (<I>a</I>+<I>b</I>)/sin (<I>a</I>-<I>b</I>) = sin (<I>a</I>′+<I>b</I>′)/sin (<I>a</I>′-<I>b</I>′)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9634 from which we may deduce<note>Braunmühl, <I>op. cit.</I> i, p. 18; Björnbo, p. 96.</note> the formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9635 <MATH>sin (<I>a</I>+<I>b</I>)/sin (<I>a</I>-<I>b</I>) = (1+cos <I>C</I>)/(1-cos <I>C</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9636 which is equivalent to tan <I>b</I> = tan <I>a</I> cos <I>C.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9637 <C>(<G>g</G>) <I>Anharmonic property of four great circles through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9638 one point.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9639 <p>But more important than the above result is the fact that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9640 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9641 the proof assumes as known the anhar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9642 monic property of four great circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9643 drawn from a point on a sphere in rela-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9644 tion to any great circle intersecting them
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9645 all, viz. that, if <I>ABCD, A</I>′<I>B</I>′<I>C</I>′<I>D</I>′ be two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9646 transversals,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9647 <MATH>sin <I>AD</I>/sin <I>DC</I>.sin <I>BC</I>/sin <I>AB</I> = sin <I>A</I>′<I>D</I>′/sin <I>D</I>′<I>C</I>′.sin <I>B</I>′<I>C</I>′/sin <I>A</I>′<I>B</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9648 <pb n=270><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9649 <p>It follows that this proposition was known before Mene-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9650 laus's time. It is most easily proved by means of ‘Menelaus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9651 Theorem’, III. 1, or alternatively it may be deduced for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9652 sphere from the corresponding proposition in plane geometry,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9653 just as Menelaus's theorem is transferred by him from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9654 plane to the sphere in III. 1. We may therefore fairly con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9655 clude that both the anharmonic property and Menelaus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9656 theorem with reference to the sphere were already included
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9657 in some earlier text-book; and, as Ptolemy, who built so much
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9658 upon Hipparchus, deduces many of the trigonometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9659 formulae which he uses from the one theorem (III. 1) of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9660 Menelaus, it seems probable enough that both theorems were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9661 known to Hipparchus. The corresponding plane theorems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9662 appear in Pappus among his lemmas to Euclid's <I>Porisms,</I><note>Pappus, vii, pp. 870-2, 874.</note> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9663 there is therefore every probability that they were assumed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9664 by Euclid as known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9665 <C>(<G>d</G>) <I>Propositions anulogous to Eucl. VI. 3.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9666 <p>Two theorems following, III. 6, 8, have their analogy in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9667 Eucl. VI. 3. In III. 6 the vertical angle <I>A</I> of a spherical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9668 triangle is bisected by an arc of a great circle meeting <I>BC</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9669 <I>D,</I> and it is proved that <MATH>sin <I>BD</I>/sin <I>DC</I> = sin <I>BA</I>/sin <I>AC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9670 in III. 8 we have the vertical angle bisected both internally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9671 and externally by arcs of great circles meeting <I>BC</I> in <I>D</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9672 <I>E,</I> and the proposition proves the harmonic property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9673 <MATH>sin <I>BE</I>/sin <I>EC</I> = sin <I>BD</I>/sin <I>DC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9674 <p>III. 7 is to the effect that, if arcs of great circles be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9675 through <I>B</I> to meet the opposite side <I>AC</I> of a spherical triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9676 in <I>D, E</I> so that <MATH>&angle;<I>ABD</I> = &angle;<I>EBC</I></MATH>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9677 <MATH>(sin <I>EA</I>.sin <I>AD</I>)/(sin <I>DC</I>.sin <I>CE</I>) = sin<SUP>2</SUP> <I>AB</I>/sin<SUP>2</SUP> <I>BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9678 As this is analogous to plane propositions given by Pappus as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9679 lemmas to different works included in the <I>Treasury of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9680 Analysis,</I> it is clear that these works were familiar to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9681 Menelaus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9682 <pb n=271><head>MENELAUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9683 <p>III. 9 and III. 10 show, for a spherical triangle, that (1) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9684 great circles bisecting the three angles, (2) the great circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9685 through the angular points meeting the opposite sides at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9686 right angles meet in a point.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9687 <p>The remaining propositions, III. 11-15, return to the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9688 sort of astronomical problem as those dealt with in Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9689 <I>Phaenomena,</I> Theodosius's <I>Sphaerica</I> and Book II of Mene-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9690 laus's own work. Props. 11-14 amount to theorems in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9691 spherical trigonometry such as the following.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9692 <p>Given arcs <G>a</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB>, <G>a</G><SUB>3</SUB>, <G>a</G><SUB>4</SUB>, <G>b</G><SUB>1</SUB>, <G>b</G><SUB>2</SUB>, <G>b</G><SUB>3</SUB>, <G>b</G><SUB>4</SUB>, such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9693 <MATH>90°≥<G>a</G><SUB>1</SUB> > <G>a</G><SUB>2</SUB> > <G>a</G><SUB>3</SUB> > <G>a</G><SUB>4</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9694 90° > <G>b</G><SUB>1</SUB> > <G>b</G><SUB>2</SUB> > <G>b</G><SUB>3</SUB> > <G>b</G><SUB>4</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9695 and also <MATH><G>a</G><SUB>1</SUB> > <G>b</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB> > <G>b</G><SUB>2</SUB>, <G>a</G><SUB>3</SUB> > <G>b</G><SUB>3</SUB>, <G>a</G><SUB>4</SUB> > <G>b</G><SUB>4</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9696 <p>(1) If <MATH>sin <G>a</G><SUB>1</SUB>:sin <G>a</G><SUB>2</SUB>:sin <G>a</G><SUB>3</SUB>:sin <G>a</G><SUB>4</SUB> = sin <G>b</G><SUB>1</SUB>:sin <G>b</G><SUB>2</SUB>:sin <G>b</G><SUB>3</SUB>:sin <G>b</G><SUB>4</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9697 then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>3</SUB>-<G>a</G><SUB>4</SUB>)>(<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>3</SUB>-<G>b</G><SUB>4</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9698 <p>(2) If <MATH>sin (<G>a</G><SUB>1</SUB>+<G>b</G><SUB>1</SUB>)/sin (<G>a</G><SUB>1</SUB>-<G>b</G><SUB>1</SUB>) = sin (<G>a</G><SUB>2</SUB>+<G>b</G><SUB>2</SUB>)/sin (<G>a</G><SUB>2</SUB>-<G>b</G><SUB>2</SUB>) = sin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9699 (<G>a</G><SUB>3</SUB>+<G>b</G><SUB>3</SUB>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9700 = sin (<G>a</G><SUB>4</SUB>+<G>b</G><SUB>4</SUB>)/sin (<G>a</G><SUB>4</SUB>-<G>b</G><SUB>4</SUB>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9701 then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>3</SUB>-<G>a</G><SUB>4</SUB>) > (<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>3</SUB>-<G>b</G><SUB>4</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9702 <p>(3) If <MATH>sin (<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/sin (<G>a</G><SUB>3</SUB>-<G>a</G><SUB>4</SUB>) < sin (<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/sin (<G>b</G><SUB>3</SUB>-<G>b</G><SUB>4</SUB>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9703 then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>3</SUB>-<G>a</G><SUB>4</SUB>) < (<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>3</SUB>-<G>b</G><SUB>4</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9704 <p>Again, given three series of three arcs such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9705 <MATH><G>a</G><SUB>1</SUB> > <G>a</G><SUB>2</SUB> > <G>a</G><SUB>3</SUB>, <G>b</G><SUB>1</SUB> > <G>b</G><SUB>2</SUB> > <G>b</G><SUB>3</SUB>, 90°><G>g</G><SUB>1</SUB> > <G>g</G><SUB>2</SUB> > <G>g</G><SUB>3</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9706 and <MATH>sin (<G>a</G><SUB>1</SUB>-<G>g</G><SUB>1</SUB>):sin (<G>a</G><SUB>2</SUB>-<G>g</G><SUB>2</SUB>):sin (<G>a</G><SUB>3</SUB>-<G>g</G><SUB>3</SUB>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9707 = sin (<G>b</G><SUB>1</SUB>-<G>g</G><SUB>1</SUB>):sin (<G>b</G><SUB>2</SUB>-<G>g</G><SUB>2</SUB>):sin (<G>b</G><SUB>3</SUB>-<G>g</G><SUB>3</SUB>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9708 = sin <G>g</G><SUB>1</SUB>:sin <G>g</G><SUB>2</SUB>:sin <G>g</G><SUB>3</SUB></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9709 <pb n=272><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9710 <p>(1) If <MATH><G>a</G><SUB>1</SUB> > <G>b</G><SUB>1</SUB> > 2<G>g</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB> > <G>b</G><SUB>2</SUB> > 2<G>g</G><SUB>2</SUB>, <G>a</G><SUB>3</SUB> > <G>b</G><SUB>3</SUB> > 2<G>g</G><SUB>3</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9711 then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>2</SUB>-<G>a</G><SUB>3</SUB>)>(<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>2</SUB>-<G>b</G><SUB>3</SUB>)</MATH>; and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9712 <p>(2) If <MATH><G>b</G><SUB>1</SUB> < <G>a</G><SUB>1</SUB> < <G>g</G><SUB>1</SUB>, <G>b</G><SUB>2</SUB> < <G>a</G><SUB>2</SUB> < <G>g</G><SUB>2</SUB>, <G>b</G><SUB>3</SUB> < <G>a</G><SUB>3</SUB> < <G>g</G><SUB>3</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9713 then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>2</SUB>-<G>a</G><SUB>3</SUB>)<(<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>2</SUB>-<G>b</G><SUB>3</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9714 <p>III. 15, the last proposition, is in four parts. The first part
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9715 is the proposition corresponding to Theodosius III. 11 above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9716 alluded to. Let <I>BA, BC</I> be two quadrants of great circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9717 (in which we easily recognize the equator and the ecliptic),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9718 <I>P</I> the pole of the former, <I>PA</I><SUB>1</SUB>, <I>PA</I><SUB>3</SUB> quadrants of great circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9719 meeting the other quadrants in <I>A</I><SUB>1</SUB>, <I>A</I><SUB>3</SUB> and <I>C</I><SUB>1</SUB>, <I>C</I><SUB>3</SUB> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9720 Let <I>R</I> be the radius of the sphere, <I>r, r</I><SUB>1</SUB>, <I>r</I><SUB>3</SUB> the radii of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9721 ‘parallel circles’ (with pole <I>P</I>) through <I>C, C</I><SUB>1</SUB>, <I>C</I><SUB>3</SUB> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9722 <p>Then shall <MATH>sin <I>A</I><SUB>1</SUB><I>A</I><SUB>3</SUB>/sin <I>C</I><SUB>1</SUB><I>C</I><SUB>3</SUB> = <I>Rr</I>/<I>r</I><SUB>1</SUB><I>r</I><SUB>3</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9723 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9724 <p>In the triangles <I>PCC</I><SUB>3</SUB>, <I>BA</I><SUB>3</SUB><I>C</I><SUB>3</SUB> the angles at <I>C, A</I><SUB>3</SUB> are right,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9725 and the angles at <I>C</I><SUB>3</SUB> equal; therefore (III. 2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9726 <MATH>sin <I>PC</I>/sin <I>PC</I><SUB>3</SUB> = sin <I>BA</I><SUB>3</SUB>/sin <I>BC</I><SUB>3</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9727 <pb n=273><head>MENELAUS'S <I>SPHAERICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9728 <p>But, by III. 1 applied to the triangle <I>BC</I><SUB>1</SUB><I>A</I><SUB>1</SUB> cut by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9729 transversal <I>PC</I><SUB>3</SUB><I>A</I><SUB>3</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9730 <MATH>sin <I>A</I><SUB>1</SUB><I>A</I><SUB>3</SUB>/sin <I>BA</I><SUB>3</SUB> = sin <I>C</I><SUB>1</SUB><I>C</I><SUB>3</SUB>/sin <I>BC</I><SUB>3</SUB>.sin <I>PA</I><SUB>1</SUB>/sin <I>PC</I><SUB>1</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9731 or <MATH>sin <I>A</I><SUB>1</SUB><I>A</I><SUB>3</SUB>/sin <I>C</I><SUB>1</SUB><I>C</I><SUB>3</SUB> = sin <I>PA</I><SUB>1</SUB>/sin <I>PC</I><SUB>1</SUB>.sin <I>BA</I><SUB>3</SUB>/sin <I>BC</I><SUB>3</SUB> = sin <I>PA</I><SUB>1</SUB>/sin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9732 <I>PC</I><SUB>1</SUB>.sin <I>PC</I>/sin <I>PC</I><SUB>3</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9733 from above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9734 = <I>Rr</I>/<I>r</I><SUB>1</SUB><I>r</I><SUB>3</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9735 <p>Part 2 of the proposition proves that, if <I>PC</I><SUB>2</SUB><I>A</I><SUB>2</SUB> be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9736 such that <MATH>sin<SUP>2</SUP> <I>PC</I><SUB>2</SUB> = sin <I>PA</I><SUB>2</SUB>.sin <I>PC,</I> or <I>r</I><SUB>2</SUB><SUP>2</SUP> = <I>Rr</I></MATH> (where <I>r</I><SUB>2</SUB> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9737 the radius of the parallel circle through <I>C</I><SUB>2</SUB>), <I>BC</I><SUB>2</SUB>-<I>BA</I><SUB>2</SUB> is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9738 maximum, while Parts 3, 4 discuss the limits to the value of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9739 the ratio between the arcs <I>A</I><SUB>1</SUB><I>A</I><SUB>3</SUB> and <I>C</I><SUB>1</SUB><I>C</I><SUB>3</SUB>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9740 <p>Nothing is known of the life of CLAUDIUS PTOLEMY except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9741 that he was of Alexandria, made observations between the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9742 years A.D. 125 and 141 or perhaps 151, and therefore presum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9743 ably wrote his great work about the middle of the reign of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9744 Antoninus Pius (A.D. 138-61). A tradition handed down by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9745 the Byzantine scholar Theodorus Meliteniota (about 1361)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9746 states that he was born, not at Alexandria, but at Ptolemais
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9747 <G>h( *(ermei/ou</G>. Arabian traditions, going back probably to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9748 &Hdot;unain b. Is&hdot;&amacr;q, say that he lived to the age of 78, and give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9749 a number of personal details to which too much weight must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9750 not be attached.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9751 <C>The <G>*maqhmatikh\ du/ntaxis</G> (Arab. <I>Almagest</I>).</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9752 <p>Ptolemy's great work, the definitive achievement of Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9753 astronomy, bore the title <G>maqhmatikh=s sunta/xews bibli/a ig</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9754 the <I>Mathematical Collection</I> in thirteen Books. By the time
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9755 of the commentators who distinguished the lesser treatises on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9756 astronomy forming an introduction to Ptolemy's work as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9757 <G>mikro\s a)stronomou/menos (to/pos)</G>, the ‘Little Astronomy’, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9758 book came to be called the ‘Great Collection’, <G>mega/lh su/n-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9759 taxis</G>. Later still the Arabs, combining the article Al with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9760 <pb n=274><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9761 the superlative <G>me/gistos</G>, made up a word Al-majisti, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9762 became <I>Almagest;</I> and it has been known by this name ever
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9763 since. The complicated character of the system expounded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9764 by Ptolemy is no doubt responsible for the fact that it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9765 speedily became the subject of elaborate commentaries.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9766 <C>Commentaries on the <I>Syntaxis.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9767 <p>Pappus<note>Pappus, viii, p. 1106. 13.</note> cites a passage from his own commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9768 Book I of the <I>Mathematica,</I> which evidently means Ptolemy's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9769 work. Part of Pappus's commentary on Book V, as well as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9770 his commentary on Book VI, are actually extant in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9771 original. Theon of Alexandria, who wrote a commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9772 the <I>Syntaxis</I> in eleven Books, incorporated as much as was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9773 available of Pappus's commentary on Book V with full
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9774 acknowledgement, though not in Pappus's exact words. In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9775 his commentary on Book VI Theon made much more partial
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9776 quotations from Pappus; indeed the greater part of the com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9777 mentary on this Book is Theon's own or taken from other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9778 sources. Pappus's commentaries are called <I>scholia,</I> Theon's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9779 <G>u(pomnh/mata</G>. Passages in Pappus's commentary on Book V
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9780 allude to ‘the scholia preceding this one’ (in the plural), and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9781 in particular to the scholium on Book IV. It is therefore all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9782 but certain that he wrote on all the Books from I to VI at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9783 least. The text of the eleven Books of Theon's commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9784 was published at Basel by Joachim Camerarius in 1538, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9785 it is rare and, owing to the way in which it is printed, with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9786 insufficient punctuation marks, gaps in places, and any number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9787 of misprints, almost unusable; accordingly little attention has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9788 so far been paid to it except as regards the first two Books,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9789 which were included, in a more readable form and with a Latin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9790 translation, by Halma in his edition of Ptolemy.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9791 <C>Translations and editions.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9792 <p>The <I>Syntaxis</I> was translated into Arabic, first (we are told)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9793 by translators unnamed at the instance of Ya&hdot;y&amacr; b. Kh&amacr;lid b.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9794 Barmak, then by al-&Hdot;ajj&amacr;j, the translator of Euclid (about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9795 786-835), and again by the famous translator Is&hdot;&amacr;q b. &Hdot;unain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9796 (d. 910), whose translation, as improved by Th&amacr;bit b. Quarra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9797 <pb n=275><head>PTOLEMY'S <I>SYNTAXIS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9798 (died 901), is extant in part, as well as the version by Nas&imacr;rad-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9799 d&imacr;n a&tdot;-&Tdot;&umacr;s&imacr; (1201-74).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9800 <p>The first edition to be published was the Latin translation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9801 made by Gherard of Cremona from the Arabic, which was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9802 finished in 1175 but was not published till 1515, when it was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9803 brought out, without the author's name, by Peter Liechten-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9804 stein at Venice. A translation from the Greek had been made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9805 about 1160 by an unknown writer for a certain Henricus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9806 Aristippus, Archdeacon of Catania, who, having been sent by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9807 William I, King of Sicily, on a mission to the Byzantine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9808 Emperor Manuel I. Comnenus in 1158, brought back with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9809 him a Greek manuscript of the <I>Syntaxis</I> as a present; this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9810 translation, however, exists only in manuscripts in the Vatican
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9811 and at Florence. The first Latin translation from the Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9812 to be published was that made by Georgius ‘of Trebizond’ for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9813 Pope Nicolas V in 1451; this was revised and published by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9814 Lucas Gauricus at Venice in 1528. The <I>editio princeps</I> of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9815 Greek text was brought out by Grynaeus at Basel in 1538.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9816 The next complete edition was that of Halma published
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9817 1813-16, which is now rare. All the more welcome, there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9818 fore, is the definitive Greek text of the astronomical works
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9819 of Ptolemy edited by Heiberg (1899-1907), to which is now
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9820 added, so far as the <I>Syntaxis</I> is concerned, a most valuable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9821 supplement in the German translation (with notes) by Manitius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9822 (Teubner, 1912-13).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9823 <C>Summary of Contents.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9824 <p>The <I>Syntaxis</I> is most valuable for the reason that it con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9825 tains very full particulars of observations and investigations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9826 by Hipparchus, as well as of the earlier observations recorded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9827 by him, e.g. that of a lunar eclipse in 721 B.C. Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9828 based himself very largely upon Hipparchus, e.g. in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9829 preparation of a Table of Chords (equivalent to sines), the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9830 theory of eccentrics and epicycles, &c.; and it is questionable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9831 whether he himself contributed anything of great value except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9832 a definite theory of the motion of the five planets, for which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9833 Hipparchus had only collected material in the shape of obser-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9834 vations made by his predecessors and himself. A very short
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9835 indication of the subjects of the different Books is all that can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9836 <pb n=276><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9837 be given here. Book I: Indispensable preliminaries to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9838 study of the Ptolemaic system, general explanations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9839 the different motions of the heavenly bodies in relation to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9840 the earth as centre, propositions required for the preparation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9841 of Tables of Chords, the Table itself, some propositions in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9842 spherical geometry leading to trigonometrical calculations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9843 the relations of arcs of the equator, ecliptic, horizon and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9844 meridian, a ‘Table of Obliquity’, for calculating declinations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9845 for each degree-point on the ecliptic, and finally a method of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9846 finding the right ascensions for arcs of the ecliptic equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9847 one-third of a sign or 10°. Book II: The same subject con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9848 tinued, i.e. problems on the sphere, with special reference to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9849 the differences between various latitudes, the length of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9850 longest day at any degree of latitude, and the like. Book III:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9851 On the length of the year and the motion of the sun on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9852 eccentric and epicycle hypotheses. Book IV: The length of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9853 months and the theory of the moon. Book V: The construc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9854 tion of the astrolabe, and the theory of the moon continued,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9855 the diameters of the sun, the moon and the earth's shadow,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9856 the distance of the sun and the dimensions of the sun, moon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9857 and earth. Book VI: Conjunctions and oppositions of sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9858 and moon, solar and lunar eclipses and their periods. Books
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9859 VII and VIII are about the fixed stars and the precession of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9860 the equinoxes, and Books IX-XIII are devoted to the move-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9861 ments of the planets.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9862 <C>Trigonometry in Ptolemy.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9863 <p>What interests the historian of mathematics is the trigono-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9864 metry in Ptolemy. It is evident that no part of the trigono-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9865 metry, or of the matter preliminary to it, in Ptolemy was new.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9866 What he did was to abstract from earlier treatises, and to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9867 condense into the smallest possible space, the minimum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9868 propositions necessary to establish the methods and formulae
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9869 used. Thus at the beginning of the preliminaries to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9870 Table of Chords in Book I he says:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9871 <p>‘We will first show how we can establish a systematic and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9872 speedy method of obtaining the lengths of the chords based on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9873 the uniform use of the smallest possible number of proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9874 tions, so that we may not only have the lengths of the chords
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9875 <pb n=277><head>PTOLEMY'S <I>SYNTAXIS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9876 set out correctly, but may be in possession of a ready proof of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9877 our method of obtaining them based on geometrical con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9878 siderations.’<note>Ptolemy, <I>Syntaxis,</I> i. 10, pp. 31 2.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9879 <p>He explains that he will use the division (1) of the circle into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9880 360 equal parts or degrees and (2) of the diameter into 120
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9881 equal parts, and will express fractions of these parts on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9882 sexagesimal system. Then come the geometrical propositions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9883 as follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9884 <C>(<G>a</G>) <I>Lemma for finding</I> sin 18° <I>and</I> sin 36°.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9885 <p>To find the side of a pentagon and decagon inscribed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9886 a circle or, in other words, the chords subtending arcs of 72°
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9887 and 36° respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9888 <p>Let <I>AB</I> be the diameter of a circle, <I>O</I> the centre, <I>OC</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9889 radius perpendicular to <I>AB.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9890 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9891 <p>Bisect <I>OB</I> at <I>D,</I> join <I>DC,</I> and measure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9892 <I>DE</I> along <I>DA</I> equal to <I>DC.</I> Join <I>EC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9893 <p>Then shall <I>OE</I> be the side of the in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9894 scribed regular decagon, and <I>EC</I> the side
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9895 of the inscribed regular pentagon.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9896 <p>For, since <I>OB</I> is bisected at <I>D,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9897 <MATH><I>BE.EO</I>+<I>OD</I><SUP>2</SUP> = <I>DE</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9898 = <I>DC</I><SUP>2</SUP> = <I>DO</I><SUP>2</SUP>+<I>OC</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9899 <p>Therefore <MATH><I>BE.EO</I> = <I>OC</I><SUP>2</SUP> = <I>OB</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9900 and <I>BE</I> is divided in extreme and mean ratio.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9901 <p>But (Eucl. XIII. 9) the sides of the regular hexagon and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9902 regular decagon inscribed in a circle when placed in a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9903 line with one another form a straight line divided in extreme
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9904 and mean ratio at the point of division.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9905 <p>Therefore, <I>BO</I> being the side of the hexagon, <I>EO</I> is the side
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9906 of the decagon.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9907 <p>Also (by Eucl. XIII. 10)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9908 <MATH>(side of pentagon)<SUP>2</SUP> = (side of hexagon)<SUP>2</SUP>+(side of decagon)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9909 = <I>CO</I><SUP>2</SUP>+<I>OE</I><SUP>2</SUP> = <I>EC</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9910 therefore <I>EC</I> is the side of the regular pentagon inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9911 in the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9912 <pb n=278><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9913 <p>The construction in fact easily leads to the results
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9914 <MATH><I>EO</I> = 1/2<I>a</I>(√5-1), <I>EC</I> = 1/2<I>a</I>√(10-2√5)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9915 where <I>a</I> is the radius of the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9916 <p>Ptolemy does not however use these radicals, but calculates
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9917 the lengths in terms of ‘parts’ of the diameter thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9918 <MATH><I>DO</I> = 30</MATH>, and <MATH><I>DO</I><SUP>2</SUP> = 900</MATH>; <MATH><I>OC</I> = 60</MATH> and <MATH><I>OC</I><SUP>2</SUP> = 3600</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9919 therefore <MATH><I>DE</I><SUP>2</SUP> = <I>DC</I><SUP>2</SUP> = 4500</MATH>, and <MATH><I>DE</I> = 67<SUP><I>p</I></SUP>4′55″</MATH> nearly;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9920 therefore side of decagon or <MATH>(crd. 36°) = <I>DE</I>-<I>DO</I> = 37<SUP><I>p</I></SUP>4′55″</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9921 <p>Again <MATH><I>OE</I><SUP>2</SUP> = (37<SUP><I>p</I></SUP>4′55″)<SUP>2</SUP> = 1375.4′15″</MATH>, and <MATH><I>OC</I><SUP>2</SUP> = 3600</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9922 therefore <MATH><I>CE</I><SUP>2</SUP> = 4975.4′15″</MATH>, and <MATH><I>CE</I> = 70<SUP><I>p</I></SUP>32′3″</MATH> nearly,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9923 i.e. side of pentagon or <MATH>(crd. 72°) = 70<SUP><I>p</I></SUP>32′3″</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9924 <p>The method of extracting the square root is explained by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9925 Theon in connexion with the first of these cases, √4500 (see
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9926 above, vol. i, pp. 61-3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9927 <p>The chords which are the sides of other regular inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9928 figures, the hexagon, the square and the equilateral triangle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9929 are next given, namely,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9930 <MATH>crd. 60° = 60<SUP><I>p</I></SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9931 crd. 90° = √(2.60<SUP>2</SUP>) = √(7200) = 84<SUP><I>p</I></SUP>51′10″,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9932 crd. 120° = √(3.60<SUP>2</SUP>) = √(10800) = 103<SUP><I>p</I></SUP>55′23″</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9933 <C>(<G>b</G>) <I>Equivalent of</I> <MATH>sin<SUP>2</SUP> <G>q</G>+cos<SUP>2</SUP> <G>q</G> = 1</MATH>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9934 <p>It is next observed that, if <I>x</I> be any arc,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9935 <MATH>(crd. <I>x</I>)<SUP>2</SUP>+{crd. (180°-<I>x</I>)}<SUP>2</SUP> = (diam.)<SUP>2</SUP> = 120<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9936 a formula which is of course equivalent to <MATH>sin<SUP>2</SUP> <G>q</G>+cos<SUP>2</SUP> <G>q</G> = 1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9937 <p>We can therefore, from crd. 72°, derive crd. 108°, from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9938 crd. 36°, crd. 144°, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9939 <C>(<G>g</G>) ‘<I>Ptolemy's theorem</I>’, <I>giving the equivalent of</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9940 <MATH>sin (<G>q</G>-<G>f</G>) = sin <G>q</G> cos <G>f</G>-cos <G>q</G> sin <G>f</G></MATH>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9941 <p>The next step is to find a formula which will give us
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9942 crd. (<G>a</G>-<G>b</G>) when crd. <G>a</G> and crd. <G>b</G> are given. (This for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9943 instance enables us to find crd. 12° from crd. 72° and crd. 60°.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9944 <pb n=279><head>PTOLEMY'S <I>SYNTAXIS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9945 <p>The proposition giving the required formula depends upon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9946 a lemma, which is the famous ‘Ptolemy's theorem’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9947 <p>Given a quadrilateral <I>ABCD</I> inscribed in a circle, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9948 diagonals being <I>AC, BD,</I> to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9949 <MATH><I>AC.BD</I> = <I>AB.DC</I>+<I>AD.BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9950 <p>The proof is well known. Draw <I>BE</I> so that the angle <I>ABE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9951 is equal to the angle <I>DBC,</I> and let <I>BE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9952 meet <I>AC</I> in <I>E.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9953 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9954 <p>Then the triangles <I>ABE, DBC</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9955 equiangular, and therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9956 <MATH><I>AB</I>:<I>AE</I> = <I>BD</I>:<I>DC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9957 or <MATH><I>AB.DC</I> = <I>AE.BD</I></MATH>. (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9958 <p>Again, to each of the equal angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9959 <I>ABE, DBC</I> add the angle <I>EBD;</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9960 then the angle <I>ABD</I> is equal to the angle <I>EBC,</I> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9961 triangles <I>ABD, EBC</I> are equiangular;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9962 therefore <MATH><I>BC</I>:<I>CE</I> = <I>BD</I>:<I>DA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9963 or <MATH><I>AD.BC</I> = <I>CE.BD</I></MATH>. (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9964 <p>By adding (1) and (2), we obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9965 <MATH><I>AB.DC</I>+<I>AD.BC</I> = <I>AC.BD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9966 <p>Now let <I>AB, AC</I> be two arcs terminating at <I>A,</I> the extremity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9967 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9968 of the diameter <I>AD</I> of a circle, and let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9969 <I>AC</I> (= <G>a</G>) be greater than <I>AB</I> (= <G>b</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9970 Suppose that (crd. <I>AC</I>) and (crd. <I>AB</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9971 are given: it is required to find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9972 (crd. <I>BC</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9973 <p>Join <I>BD, CD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9974 <p>Then, by the above theorem,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9975 <MATH><I>AC.BD</I> = <I>BC.AD</I>+<I>AB.CD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9976 <p>Now <I>AB, AC</I> are given; therefore <MATH><I>BD</I> = crd. (180°-<I>AB</I>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9977 and <MATH><I>CD</I> = crd. (180°-<I>AC</I>)</MATH> are known. And <I>AD</I> is known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9978 Hence the remaining chord <I>BC</I> (crd. <I>BC</I>) is known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9979 <pb n=280><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9980 <p>The equation in fact gives the formula,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9981 <MATH>{crd. (<G>a</G>-<G>b</G>)}.(crd. 180°) = (crd. <G>a</G>).{crd. (180°-<G>b</G>)}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9982 -(crd. <G>b</G>).{crd. (180°-<G>a</G>)}</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9983 which is, of course, equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9984 <MATH>sin (<G>q</G>-<G>f</G>) = sin <G>q</G> cos <G>f</G>-cos <G>q</G> sin <G>f</G>, where <G>a</G> = 2<G>q</G>, <G>b</G> = 2<G>f</G></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9985 <p>By means of this formula Ptolemy obtained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9986 <MATH>crd. 12° = crd. (72°-60°) = 12<SUP><I>p</I></SUP>32′36″</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9987 <C>(<G>d</G>) <I>Equivalent of</I> <MATH>sin<SUP>2</SUP> 1/2<G>q</G> = 1/2 (1-cos <G>q</G>)</MATH>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9988 <p>But, in order to get the chords of smaller angles still, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9989 want a formula for finding the chord of half an arc when the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9990 chord of the arc is given. This is the subject of Ptolemy's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9991 next proposition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9992 <p>Let <I>BC</I> be an arc of a circle with diameter <I>AC,</I> and let the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9993 arc <I>BC</I> be bisected at <I>D.</I> Given (crd. <I>BC</I>), it is required to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9994 find (crd. <I>DC</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9995 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9996 <p>Draw <I>DF</I> perpendicular to <I>AC,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9997 and join <I>AB, AD, BD, DC.</I> Measure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9998 <I>AE</I> along <I>AC</I> equal to <I>AB,</I> and join
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	9999 <I>DE.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10000 <p>Then shall <I>FC</I> be equal to <I>EF,</I> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10001 <I>FC</I> shall be half the difference be-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10002 tween <I>AC</I> and <I>AB.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10003 <p>For the triangles <I>ABD, AED</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10004 equal in all respects, since two sides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10005 of the one are equal to two sides of the other and the included
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10006 angles <I>BAD, EAD,</I> standing on equal arcs, are equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10007 <p>Therefore <MATH><I>ED</I> = <I>BD</I> = <I>DC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10008 and the right-angled triangles <I>DEF, DCF</I> are equal in all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10009 respects, whence <MATH><I>EF</I> = <I>FC</I></MATH>, or <MATH><I>CF</I> = 1/2(<I>AC</I>-<I>AB</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10010 <p>Now <MATH><I>AC.CF</I> = <I>CD</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10011 whence <MATH>(crd. <I>CD</I>)<SUP>2</SUP> = 1/2 <I>AC</I> (<I>AC</I>-<I>AB</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10012 = 1/2 (crd. 180°).{(crd. 180°)-(crd.&horbar;(180°-<I>BC</I>))}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10013 <p>This is, of course, equivalent to the formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10014 <MATH>sin<SUP>2</SUP> 1/2<G>q</G> = 1/2(1-cos <G>q</G>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10015 <pb n=281><head>PTOLEMY'S <I>SYNTAXIS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10016 <p>By successively applying this formula, Ptolemy obtained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10017 (crd. 6°), (crd. 3°) and finally <MATH>(crd. 1 1/2°) = 1<SUP><I>p</I></SUP>34′15″</MATH> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10018 <MATH>(crd. 3/4°) = 0<SUP><I>p</I></SUP>47′8″</MATH>. But we want a table going by half-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10019 degrees, and hence two more things are necessary; we have to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10020 get a value for (crd. 1°) lying between (crd. 1 1/2°) and (crd. 3/4°),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10021 and we have to obtain an <I>addition</I> formula enabling us when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10022 (crd. <G>a</G>) is given to find {crd. (<G>a</G>+1/2°)}, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10023 <C>(<G>e</G>) <I>Equivalent of</I> <MATH>cos (<G>q</G>+<G>f</G>) = cos <G>q</G> cos <G>f</G>-sin <G>q</G> sin <G>f</G></MATH>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10024 <p>To find the addition formula. Suppose <I>AD</I> is the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10025 of a circle, and <I>AB, BC</I> two arcs. Given (crd. <I>AB</I>) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10026 (crd. <I>BC</I>), to find (crd. <I>AC</I>). Draw the diameter <I>BOE,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10027 join <I>CE, CD, DE, BD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10028 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10029 <p>Now, (crd. <I>AB</I>) being known,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10030 (crd. <I>BD</I>) is known, and therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10031 also (crd. <I>DE</I>), which is equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10032 (crd. <I>AB</I>); and, (crd. <I>BC</I>) being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10033 known, (crd. <I>CE</I>) is known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10034 <p>And, by Ptolemy's theorem,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10035 <MATH><I>BD.CE</I> = <I>BC.DE</I>+<I>BE.CD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10036 <p>The diameter <I>BE</I> and all the chords in this equation except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10037 <I>CD</I> being given, we can find <I>CD</I> or crd. (180°-<I>AC</I>). We have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10038 in fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10039 <MATH>(crd. 180°).{crd. (180°-<I>AC</I>)}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10040 = {crd. (180°-<I>AB</I>)}.{crd. (180°-<I>BC</I>)}-(crd. <I>AB</I>).(crd. <I>BC</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10041 thus crd. (180°-<I>AC</I>) and therefore (crd. <I>AC</I>) is known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10042 <p>If <MATH><I>AB</I> = 2<G>q</G>, <I>BC</I> = 2<G>f</G></MATH>, the result is equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10043 <MATH>cos (<G>q</G>+<G>f</G>) = cos <G>q</G> cos <G>f</G>-sin <G>q</G> sin <G>f</G></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10044 <C>(<G>z</G>) <I>Method of interpolation based on formula</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10045 <MATH>sin <G>a</G>/sin <G>b</G> < <G>a</G>/<G>b</G> (<I>where</I> 1/2<G>p</G> > <G>a</G> > <G>b</G>)</MATH>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10046 <p>Lastly we have to find (crd. 1°), having given (crd. 1 1/2°) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10047 (crd. 3/4°).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10048 <p>Ptolemy uses an ingenious method of <I>interpolation</I> based on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10049 a proposition already assumed as known by Aristarchus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10050 <p>If <I>AB, BC</I> be unequal chords in a circle, <I>BC</I> being the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10051 <pb n=282><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10052 greater, then shall the ratio of <I>CB</I> to <I>BA</I> be less than the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10053 ratio of the arc <I>CB</I> to the arc <I>BA.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10054 <p>Let <I>BD</I> bisect the angle <I>ABC,</I> meeting <I>AC</I> in <I>E</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10055 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10056 the circumference in <I>D.</I> The arcs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10057 <I>AD, DC</I> are then equal, and so are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10058 the chords <I>AD, DC.</I> Also <MATH><I>CE</I> > <I>EA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10059 (since <I>CB</I>:<I>BA</I> = <I>CE</I>:<I>EA</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10060 <p>Draw <I>DF</I> perpendicular to <I>AC;</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10061 then <I>AD</I> > <I>DE</I> > <I>DF,</I> so that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10062 circle with centre <I>D</I> and radius <I>DE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10063 will meet <I>DA</I> in <I>G</I> and <I>DF</I> produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10064 in <I>H.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10065 <p>Now <MATH><I>FE</I>:<I>EA</I> = &utri;<I>FED</I>:&utri;<I>AED</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10066 <(sector <I>HED</I>):(sector <I>GED</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10067 <&angle;<I>FDE</I>:&angle;<I>EDA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10068 <I>Componendo,</I> <MATH><I>FA</I>:<I>AE</I><&angle;<I>FDA</I>:&angle;<I>ADE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10069 <p>Doubling the antecedents, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10070 <MATH><I>CA</I>:<I>AE</I><&angle;<I>CDA</I>:&angle;<I>ADE</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10071 and, <I>separando,</I> <MATH><I>CE</I>:<I>EA</I><&angle;<I>CDE</I>:&angle;<I>EDA</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10072 therefore <MATH>(since <I>CB</I>:<I>BA</I> = <I>CE</I>:<I>EA</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10073 <I>CB</I>:<I>BA</I><&angle;<I>CDB</I>:&angle;<I>BDA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10074 <(arc <I>CB</I>):(arc <I>BA</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10075 i.e. <MATH>(crd. <I>CB</I>):(crd. <I>BA</I>)<(arc <I>CB</I>):(arc <I>BA</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10076 [This is of course equivalent to <MATH>sin <G>a</G>:sin <G>b</G> < <G>a</G>:<G>b</G></MATH>, where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10077 <MATH>1/2<G>p</G> > <G>a</G> > <G>b</G></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10078 <p>It follows (1) that <MATH>(crd. 1°):(crd. 3/4°)<1:3/4</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10079 and (2) that <MATH>(crd. 1 1/2°):(crd. 1°)<1 1/2:1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10080 <p>That is, <MATH>4/3.(crd. 3/4°)>(crd. 1°)>2/3.(crd. 1 1/2°)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10081 <p>But <MATH>(crd. 3/4°) = 0<SUP><I>p</I></SUP>47′8″</MATH>, so that <MATH>4/3(crd. 3/4°) = 1<SUP><I>p</I></SUP>2′50″</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10082 nearly (actually 1<SUP><I>p</I></SUP>2′50 2/3″);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10083 and <MATH>(crd. 1 1/2°) = 1<SUP><I>p</I></SUP>34′15″</MATH>, so that <MATH>2/3(crd. 1 1/2°) = 1<SUP><I>p</I></SUP>2′50″</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10084 <p>Since, then, (crd. 1°) is both less and greater than a length
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10085 which only differs inappreciably from 1<SUP><I>p</I></SUP>2′50″, we may say
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10086 that <MATH>(crd. 1°) = 1<SUP><I>p</I></SUP>2′50″</MATH> as nearly as possible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10087 <pb n=283><head>PTOLEMY'S <I>SYNTAXIS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10088 <C>(<G>h</G>) <I>Table of Chords.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10089 <p>From this Ptolemy deduces that (crd. 1/2°) is very nearly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10090 0<SUP>p</SUP> 31′ 25″, and by the aid of the above propositions he is in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10091 a position to complete his Table of Chords for arcs subtending
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10092 angles increasing from 1/2° to 180° by steps of 1/2°; in other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10093 words, a Table of Sines for angles from 1/4° to 90° by steps
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10094 of 1/4°.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10095 <C>(<G>q</G>) <I>Further use of proportional increase.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10096 <p>Ptolemy carries further the principle of proportional in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10097 crease as a method of finding approximately the chords of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10098 arcs containing an odd number of minutes between 0′ and 30′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10099 Opposite each chord in the Table he enters in a third column
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10100 1/30th of the excess of that chord over the one before, i.e. the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10101 chord of the arc containing 30′ less than the chord in question.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10102 For example (crd. 2 1/2°) is stated in the second column of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10103 Table as 2<SUP>p</SUP> 37′ 4″. The excess of (crd. 2 1/2°) over (crd. 2°) in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10104 Table is 0<SUP>p</SUP> 31′ 24″; 1/30th of this is 0<SUP>p</SUP> 1′ 2″ 48‴, which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10105 therefore the amount entered in the third column opposite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10106 (crd. 2 1/2°). Accordingly, if we want (crd. 2° 25′), we take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10107 (crd. 2°) or 2<SUP>p</SUP> 5′ 40″ and add 25 times 0<SUP>p</SUP> 1′ 2″ 48‴; or we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10108 take (crd. 2 1/2°) or 2<SUP>p</SUP> 37′ 4″ and subtract 5 times 0<SUP>p</SUP> 1′ 2″ 48‴.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10109 Ptolemy adds that if, by using the approximation for 1° and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10110 1/2°, we gradually accumulate an error, we can check the calcu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10111 lation by comparing the chord with that of other related arcs,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10112 e.g. the double, or the supplement (the difference between the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10113 arc and the semicircle).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10114 <p>Some particular results obtained from the Table may be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10115 mentioned. Since <MATH>(crd. 1°) = 1<SUP>p</SUP> 2′ 50″</MATH>, the whole circumference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10116 <MATH>= 360 (1<SUP>p</SUP> 2′ 50″)</MATH>, nearly, and, the length of the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10117 being 120<SUP>p</SUP>, the value of <G>p</G> is <MATH>3 (1 + 2/60 + 50/3600) = 3 + 8/60 + 30/3600</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10118 which is the value used later by Ptolemy and is equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10119 3.14166... Again, <MATH>√3 = 2 sin 60°</MATH> and, 2 (crd. 120°) being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10120 equal to 2 (103<SUP>p</SUP> 55′ 23″), we have <MATH>√3 = 2/120 (103 + 55/60 + 23/3600)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10121 = 1 + 43/60 + 55/60<SUP>2</SUP> + 23/60<SUP>3</SUP> = 1.7320509</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10122 which is correct to 6 places of decimals. Speaking generally,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10123 <pb n=284><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10124 the sines obtained from Ptolemy's Table are correct to 5
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10125 places.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10126 <C>(<G>i</G>) <I>Plane trigonometry in effect used.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10127 <p>There are other cases in Ptolemy in which plane trigono-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10128 metry is in effect used, e.g. in the determination of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10129 eccentricity of the sun's orbit.<note>Ptolemy, <I>Syntaxis</I>, iii. 4, vol. i, pp. 234-7.</note> Suppose that <I>ACBD</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10130 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10131 the eccentric circle with centre <I>O</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10132 and <I>AB, CD</I> are chords at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10133 angles through <I>E</I>, the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10134 earth. To find <I>OE.</I> The are <I>BC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10135 is known (= <G>a</G>, say) as also the arc
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10136 <I>CA</I> (= <G>b</G>). If <I>BF</I> be the chord
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10137 parallel to <I>CD</I>, and <I>CG</I> the chord
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10138 parallel to <I>AB</I>, and if <I>N, P</I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10139 middle points of the arcs <I>BF, GC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10140 Ptolemy finds (1) the arc <I>BF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10141 <MATH>(= <G>a</G> + <G>b</G> - 180°)</MATH>, then the chord <I>BF</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10142 crd. <MATH>(<G>a</G> + <G>b</G> - 180°)</MATH>, then the half of it, (2) the arc <MATH><I>GC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10143 = arc (<G>a</G> + <G>b</G> - 2<G>b</G>)</MATH> or arc (<G>a</G> - <G>b</G>), then the chord <I>GC</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10144 lastly half of it. He then adds the squares on the half-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10145 chords, i.e. he obtains
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10146 <MATH><I>OE</I><SUP>2</SUP> = 1/4{crd. (<G>a</G> + <G>b</G> - 180)}<SUP>2</SUP> + 1/4 {crd. (<G>a</G> - <G>b</G>)}<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10147 that is, <MATH><I>OE</I><SUP>2</SUP>/r<SUP>2</SUP> = cos<SUP>2</SUP> 1/2 (<G>a</G> + <G>b</G>) + sin<SUP>2</SUP>1/2(<G>a</G> - <G>b</G>).</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10148 He proceeds to obtain the angle <I>OEC</I> from its sine <I>OR / OE</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10149 which he expresses as a chord of double the angle in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10150 circle on <I>OE</I> as diameter in relation to that diameter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10151 <C>Spherical trigonometry: formulae in solution of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10152 spherical triangles.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10153 <p>In spherical trigonometry, as already stated, Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10154 obtains everything that he wants by using the one funda-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10155 mental proposition known as ‘Menelaus's theorem’ applied
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10156 to the sphere (Menelaus III. 1), of which he gives a proof
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10157 following that given by Menelaus of the first case taken in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10158 his proposition. Where Ptolemy has occasion for other pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10159 positions of Menelaus's <I>Sphaerica</I>, e.g. III. 2 and 3, he does
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10160 <pb n=285><head>PTOLEMY'S <I>SYNTAXIS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10161 not quote those propositions, as he might have done, but proves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10162 them afresh by means of Menelaus's theorem.<note><I>Syntaxis</I>, vol. i, p. 169 and pp. 126-7 respectively.</note> The appli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10163 cation of the theorem in other cases gives in effect the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10164 following different formulae belonging to the solution of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10165 a spherical triangle <I>ABC</I> right-angled at <I>C</I>, viz.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10166 <MATH>sin <I>a</I> = sin <I>c</I> sin <I>A</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10167 <MATH>tan <I>a</I> = sin <I>b</I> tan <I>A</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10168 <MATH>cos <I>c</I> = cos <I>a</I> cos <I>b</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10169 <MATH>tan <I>b</I> = tan <I>c</I> cos <I>A.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10170 <p>One illustration of Ptolemy's procedure will be sufficient.<note><I>Ib.</I>, vol. i, pp. 121-2.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10171 Let <I>HAH′</I> be the horizon, <I>PEZH</I> the meridian circle, <I>EE′</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10172 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10173 the equator, <I>ZZ′</I> the ecliptic, <I>F</I> an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10174 equinoctial point. Let <I>EE′, ZZ′</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10175 cut the horizon in <I>A, B</I>. Let <I>P</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10176 the pole, and let the great circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10177 through <I>P, B</I> cut the equator at <I>C.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10178 Now let it be required to find the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10179 time which the arc <I>FB</I> of the ecliptic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10180 takes to rise; this time will be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10181 measured by the arc <I>FA</I> of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10182 equator. (Ptolemy has previously found the length of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10183 arcs <I>BC</I>, the declination, and <I>FC</I>, the right ascension, of <I>B</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10184 I. 14, 16.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10185 <p>By Menelaus's theorem applied to the arcs <I>AE′, E′P</I> cut by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10186 the arcs <I>AH′, PC</I> which also intersect one another in <I>B</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10187 <MATH>(crd. 2 <I>PH′</I>)/(crd. 2 <I>H′E′</I>) = (crd. 2 <I>PB</I>/crd. 2 <I>BC</I>) . (crd. 2 <I>CA</I>/crd. 2 <I>AE′</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10188 that is, <MATH>sin <I>PH′</I>/sin <I>H′E′</I> = (sin <I>PB</I>/sin <I>BC</I>) . (sin <I>CA</I>/sin <I>AE′</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10189 <p>Now <MATH>sin <I>PH′</I> = cos <I>H′E′</I>, sin <I>PB</I> = cos <I>BC</I></MATH>, and <MATH>sin <I>AE′</I> = 1</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10190 therefore <MATH>cot <I>H′E′</I> = cot <I>BC</I> . sin <I>CA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10191 in other words, in the triangle <I>ABC</I> right-angled at <I>C</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10192 <MATH>cot <I>A</I> = cot <I>a</I> sin <I>b</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10193 or <MATH>tan <I>a</I> = sin <I>b</I> tan <I>A</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10194 <pb n=286><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10195 <p>Thus <I>AC</I> is found, and therefore <I>FC-AC</I> or <I>FA.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10196 <p>The lengths of <I>BC, FC</I> are found in I. 14, 16 by the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10197 method, the four intersecting great circles used in the figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10198 being in that case the equator <I>EE′</I>, the ecliptic <I>ZZ′</I>, the great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10199 circle <I>PBCP′</I> through the poles, and the great circle <I>PKLP′</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10200 passing through the poles of both the ecliptic and the equator.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10201 In this case the two arcs <I>PL, AE′</I> are cut by the intersecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10202 great circles <I>PC, FK</I>, and Menelaus's theorem gives (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10203 <MATH>sin <I>PL</I>/sin <I>KL</I> = (sin <I>CP</I>/sin <I>BC</I>) . (sin <I>BF</I>/sin <I>FK</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10204 <p>But <MATH>sin <I>PL</I> = 1, sin <I>KL</I> = sin <I>BFC</I>, sin <I>CP</I> = 1, sin <I>FK</I> = 1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10205 and it follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10206 <MATH>sin <I>BC</I> = sin <I>BF</I> sin <I>BFC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10207 corresponding to the formula for a triangle right-angled at <I>C</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10208 <MATH>sin <I>a</I> = sin <I>c</I> sin <I>A</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10209 <p>(2) We have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10210 <MATH>sin <I>PK</I>/sin <I>KL</I> = (sin <I>PB</I>/sin <I>BC</I>) . (sin <I>CF</I>/sin <I>FL</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10211 and <MATH>sin <I>PK</I> = cos <I>KL</I> = cos <I>BFC</I>, sin <I>PB</I> = cos <I>BC</I>, sin <I>FL</I> = 1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10212 so that <MATH>tan <I>BC</I> = sin <I>CF</I> tan <I>BFC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10213 corresponding to the formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10214 <MATH>tan <I>a</I> = sin <I>b</I> tan <I>A</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10215 <p>While, therefore, Ptolemy's method implicitly gives the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10216 formulae for the solution of right-angled triangles above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10217 quoted, he does not speak of right-angled triangles at all, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10218 only of arcs of intersecting great circles. The advantage
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10219 from his point of view is that he works in sines and cosines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10220 only, avoiding tangents as such, and therefore he requires
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10221 tables of only one trigonometrical ratio, namely the sine (or,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10222 as he has it, the chord of the double arc).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10223 <C>The <I>Analemma</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10224 <p>Two other works of Ptolemy should be mentioned here.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10225 The first is the <I>Analemma.</I> The object of this is to explain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10226 a method of representing on one plane the different points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10227 <pb n=287><head>THE <I>ANALEMMA</I> OF PTOLEMY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10228 and arcs of the heavenly sphere by means of <I>orthogonal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10229 projection</I> upon three planes mutually at right angles, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10230 meridian, the horizon, and the ‘prime vertical’. The definite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10231 problem attacked is that of showing the position of the sun at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10232 any given time of the day, and the use of the method and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10233 of the instruments described in the book by Ptolemy was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10234 connected with the construction of sundials, as we learn from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10235 Vitruvius.<note>Vitruvius, <I>De architect.</I> ix. 4.</note> There was another <G>a)na/lhmma</G> besides that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10236 Ptolemy; the author of it was Diodorus of Alexandria, a con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10237 temporary of Caesar and Cicero (‘Diodorus, famed among the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10238 makers of gnomons, tell me the time!’ says the Anthology<note><I>Anth. Palat.</I> xiv. 139.</note>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10239 and Pappus wrote a commentary upon it in which, as he tells
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10240 us,<note>Pappus, iv, p. 246. 1.</note> he used the conchoid in order to trisect an angle, a problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10241 evidently required in the <I>Analemma</I> in order to divide any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10242 arc of a circle into six equal parts (hours). The word
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10243 <G>a)na/lhmma</G> evidently means ‘taking up’ (‘Aufnahme’) in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10244 sense of ‘making a graphic representation’ of something, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10245 this case the representation on a plane of parts of the heavenly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10246 sphere. Only a few fragments remain of the Greek text of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10247 the <I>Analemma</I> of Ptolemy; these are contained in a palimpsest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10248 (Ambros. Gr. L. 99 sup., now 491) attributed to the seventh
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10249 century but probably earlier. Besides this, we have a trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10250 lation by William of Moerbeke from an Arabic version.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10251 This Latin translation was edited with a valuable commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10252 by the indefatigable Commandinus (Rome, 1562); but it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10253 now available in William of Moerbeke's own words, Heiberg
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10254 having edited it from Cod. Vaticanus Ottobon. lat. 1850 of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10255 thirteenth century (written in William's own hand), and in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10256 cluded it with the Greek fragments (so far as they exist) in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10257 parallel columns in vol. ii of Ptolemy's works (Teubner, 1907).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10258 <p>The figure is referred to three fixed planes (1) the meridian,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10259 (2) the horizon, (3) the prime vertical; these planes are the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10260 planes of the three circles <I>APZB, ACB, ZQC</I> respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10261 shown in the diagram below. Three other great circles are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10262 used, one of which, the equator with pole <I>P</I>, is fixed; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10263 other two are movable and were called by special names;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10264 the first is the circle represented by any position of the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10265 of the horizon as it revolves round <I>COC′</I> as diameter (<I>CSM</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10266 <pb n=288><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10267 the diagram is one position of it, coinciding with the equator),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10268 and it was called <G>e(kth/moros ku/klos</G> (‘the circle in six parts’)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10269 because the highest point of it above the horizon corresponds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10270 to the lapse of six hours; the second, called the <I>hour-circle</I>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10271 the circle represented by any position, as <I>BSQA</I>, of the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10272 of the horizon as it revolves round <I>BA</I> as axis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10273 <p>The problem is, as above stated, to find the position of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10274 sun at a given hour of the day. In order to illustrate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10275 the method, it is sufficient, with A. v. Braunmühl,<note>Braunmühl, <I>Gesch. der Trigonometrie</I>, i, pp. 12, 13.</note> to take the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10276 simplest case where the sun is on the equator, i.e. at one of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10277 the equinoctial points, so that the <I>hectemoron</I> circle coincides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10278 with the equator.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10279 <p>Let <I>S</I> be the position of the sun, lying on the equator <I>MSC,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10280 P</I> the pole, <I>MZA</I> the meridian, <I>BCA</I> the horizon, <I>BSQA</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10281 <I>hour-circle</I>, and let the vertical great circle <I>ZSV</I> be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10282 through <I>S</I>, and the vertical great circle <I>ZQC</I> through <I>Z</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10283 zenith and <I>C</I> the east-point.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10284 <p>We are given the arc <MATH><I>SC</I> = 90° - <I>t</I></MATH>, where <I>t</I> is the hour-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10285 angle, and the arc <MATH><I>MB</I> = 90° - <G>f</G></MATH>, where <G>f</G> is the elevation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10286 the pole; and we have to find the arcs <I>SV</I> (the sun's altitude),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10287 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10288 <I>VC</I>, the ‘ascensional difference’, <I>SQ</I> and <I>QC.</I> Ptolemy, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10289 fact, practically determines the position of <I>S</I> in terms of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10290 certain spherical coordinates.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10291 <p>Draw the perpendiculars, <I>SF</I> to the plane of the meridian,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10292 <I>SH</I> to that of the horizon, and <I>SE</I> to the plane of the prime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10293 <pb n=289><head>THE <I>ANALEMMA</I> OF PTOLEMY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10294 vertical; and draw <I>FG</I> perpendicular to <I>BA</I>, and <I>ET</I> to <I>OZ.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10295 Join <I>HG</I>, and we have <MATH><I>FG</I> = <I>SH</I>, <I>GH</I> = <I>FS</I> = <I>ET</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10296 <p>We now represent <I>SF</I> in a separate figure (for clearness'
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10297 sake, as Ptolemy uses only one figure), where <I>B′Z′A′</I> corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10298 sponds to <I>BZA, P′</I> to <I>P</I> and <I>O′M′</I> to <I>OM.</I> Set off the arc
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10299 <I>P′S′</I> equal to <I>CS</I> (= 90° - <I>t</I>), and draw <I>S′F′</I> perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10300 to <I>O′M′.</I> Then <MATH><I>S′M′</I> = <I>SM</I></MATH>, and <MATH><I>S′F′</I> = <I>SF</I></MATH>; it is as if in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10301 original figure we had turned the quadrant <I>MSC</I> round <I>MO</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10302 till it coincided with the meridian circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10303 <p>In the two figures draw <I>IFK, I′F′K′</I> parallel to <I>BA, B′A′</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10304 and <I>LFG, L′F′G′</I> parallel to <I>OZ, O′Z′.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10305 <p>Then (1) arc <MATH><I>ZI</I> = arc <I>ZS</I> = arc (90° - <I>SV</I>)</MATH>, because if we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10306 turn the quadrant <I>ZSV</I> about <I>ZO</I> till it coincides with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10307 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10308 meridian, <I>S</I> falls on <I>I</I>, and <I>V</I> on <I>B.</I> It follows that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10309 required arc <I>SV</I> = arc <I>B′I′</I> in the second figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10310 <p>(2) To find the arc <I>VC</I>, set off <I>G′X</I> (in the second figure)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10311 along <I>G′F′</I> equal to <I>FS</I> or <I>F′S′</I>, and draw <I>O′X</I> through to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10312 meet the circle in <I>X′.</I> Then arc <MATH><I>Z′X′</I> = arc <I>VC</I></MATH>; for it is as if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10313 we had turned the quadrant <I>BVC</I> about <I>BO</I> till it coincided
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10314 with the meridian, when (since <MATH><I>G′X</I> = <I>FS</I> = <I>GH</I></MATH>) <I>H</I> would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10315 coincide with <I>X</I> and <I>V</I> with <I>X′.</I> Therefore <I>BV</I> is also equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10316 to <I>B′X′.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10317 <p>(3) To find <I>QC</I> or <I>ZQ</I>, set off along <I>T′F′</I> in the second figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10318 <I>T′Y</I> equal to <I>F′S′</I>, and draw <I>O′Y</I> through to <I>Y′</I> on the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10319 <p>Then arc <I>B′Y′</I> = arc <I>QC</I>; for it is as if we turned the prime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10320 vertical <I>ZQC</I> about <I>ZO</I> till it coincided with the meridian,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10321 when (since <I>T′Y</I> = <I>S′F′</I> = <I>TE</I>) <I>E</I> would fall on <I>Y</I>, the radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10322 <I>OEQ</I> on <I>O′YY′</I> and <I>Q</I> on <I>Y′.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10323 <p>(4) Lastly, arc <MATH><I>BS</I> = arc <I>BL</I> = arc <I>B′L′</I></MATH>, because <I>S, L</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10324 <pb n=290><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10325 both in the plane <I>LSHG</I> at right angles to the meridian;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10326 therefore arc <I>SQ</I> = arc <I>L′Z′.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10327 <p>Hence all four arcs <I>SV, VC, QC, QS</I> are represented in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10328 auxiliary figure in one plane.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10329 <p>So far the procedure amounts to a method of <I>graphically</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10330 constructing the arcs required as parts of an auxiliary circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10331 in one plane. But Ptolemy makes it clear that practical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10332 calculation followed on the basis of the figure.<note>See Zeuthen in <I>Bibliotheca mathematica</I>, i<SUB>3</SUB>, 1900, pp. 23-7.</note> The lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10333 used in the construction are <MATH><I>SF</I> = sin <I>t</I></MATH> (where the radius = 1),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10334 <MATH><I>FT</I> = <I>OF</I> sin <G>f</G>, <I>FG</I> = <I>OF</I> sin (90° - <G>f</G>)</MATH>, and this was fully
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10335 realized by Ptolemy. Thus he shows how to calculate the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10336 arc <I>SZ</I>, the zenith distance (= <I>d</I>, say) or its complement <I>SV</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10337 the height of the sun (= <I>h</I>, say), in the following way. He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10338 says in effect: Since <I>G</I> is known, and <MATH>&angle;<I>F′O′G′</I> = 90° - <G>f</G></MATH>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10339 ratios <I>O′F′</I>:<I>F′T′</I> and <I>O′F′</I>:<I>O′T′</I> are known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10340 <p>[In fact <MATH><I>O′F′</I>/<I>O′T′</I> = <I>D</I>/crd. (180° - 2<G>f</G>)</MATH>, where <I>D</I> is the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10341 of the sphere.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10342 <p>Next, since the arc <I>MS</I> or <I>M′S′</I> is known [= <I>t</I>], and there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10343 fore the arc <I>P′S′</I> [= 90° - <I>t</I>], the ratio of <I>O′F′</I> to <I>D</I> is known
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10344 [in fact <MATH><I>O′F′</I> / <I>D</I> = {crd. (180 - 2<I>t</I>)} / 2 <I>D.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10345 <p>It follows from these two results that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10346 <MATH><I>O</I>′<I>T</I>′ = crd. (180° - 2<I>t</I>)/2<I>D</I>.crd. (180° - 2<G>f</G>)].</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10347 <p>Lastly, the arc <I>SV</I> (= <I>h</I>) being equal to <I>B′I′</I>, the angle <I>h</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10348 equal to the angle <I>O′I′T′</I> in the triangle <I>I′O′T′.</I> And in this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10349 triangle <I>O′I′</I>, the radius, is known, while <I>O′T′</I> has been found;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10350 and we have therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10351 <MATH><I>O′T′</I>/<I>O′I′</I> = crd. (2<I>h</I>)/<I>D</I> = (crd. (180° - 2<I>t</I>)/<I>D</I>).(crd. (180° - 2<G>f</G>)/<I>D</I>)</MATH>, from above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10352 <p>[In other words, <MATH>sin <I>h</I> = cos<I>t</I>cos <G>f</G></MATH>; or, if <MATH><I>u</I> = <I>SC</I> = 90° - <I>t</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10353 sin <I>h</I> = sin <I>u</I> cos <G>f</G></MATH>, the formula for finding sin <I>h</I> in the right-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10354 angled spherical triangle <I>SVC.</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10355 <p>For the azimuth <MATH><G>w</G> (arc <I>BV</I> = arc <I>B′X′</I>)</MATH>, the figure gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10356 <MATH>tan <G>w</G> = <I>XG′</I>/<I>G′O′</I> = <I>S′F′</I>/<I>F′T′</I> = (<I>S′F′</I>/<I>O′F′</I>) . (<I>O′F′</I>/<I>F′T′</I>) = tan <I>t</I> . (1/sin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10357 <G>f</G>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10358 <pb n=291><head>THE <I>ANALEMMA</I> OF PTOLEMY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10359 or tan <I>VC</I> = tan <I>SC</I> cos <I>SCV</I> in the right-angled spherical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10360 triangle <I>SVC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10361 <p>Thirdly,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10362 <MATH>tan <I>QZ</I> = tan <I>Z′Y′</I> = <I>S′F′</I>/<I>O′T′</I> = (<I>S′F′</I>/<I>O′F′</I>) . (<I>O′F′</I>/<I>O′T′</I>) = tan<I>t</I> . (1/cos<G>f</G>);</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10363 that is, <MATH>tan <I>QZ</I>/tan <I>SM</I> = sin <I>BZ</I>/sin <I>BM</I></MATH>, which is Menelaus, <I>Sphaerica</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10364 III. 3, applied to the right-angled spherical triangles <I>ZBQ</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10365 <I>MBS</I> with the angle <I>B</I> common.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10366 <p>Zeuthen points out that later in the same treatise Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10367 finds the arc 2<G>a</G> described above the horizon by a star of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10368 given declination <G>d</G>′, by a procedure equivalent to the formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10369 <MATH>cos <G>a</G> = tan <G>d</G>′ tan <G>f</G></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10370 and this is the same formula which, as we have seen,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10371 Hipparchus must in effect have used in his <I>Commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10372 the Phaenomena of Eudoxus and Aratus.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10373 <p>Lastly, with regard to the calculations of the height <I>h</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10374 the azimuth <G>w</G> in the general case where the sun's declination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10375 is <G>d</G>′, Zeuthen has shown that they may be expressed by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10376 formulae
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10377 <MATH>sin <I>h</I> = (cos <G>d</G>′ cos <I>t</I> - sin <G>d</G>′ tan <G>f</G>) cos <G>f</G></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10378 and <MATH>tan <G>w</G> = cos <G>d</G>′ sin <I>t</I>/(sin <G>d</G>′/cos <G>f</G> + (cos <G>d</G>′ cos <I>t</I> - sin <G>d</G>′ tan <G>f</G>) sin <G>f</G>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10379 or <MATH>cos <G>d</G>′ sin <I>t</I>/(sin <G>d</G>′ cos <G>f</G> + cos <G>d</G>′ cos <I>t</I> sin <G>f</G>).</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10380 <p>The statement therefore of A. v. Braunmühl<note>Braunmühl, i, pp. 13, 14, 38-41.</note> that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10381 Indians were the first to utilize the method of projection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10382 contained in the <I>Analemma</I> for actual trigonometrical calcu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10383 lations with the help of the Table of Chords or Sines requires
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10384 modification in so far as the Greeks at all events showed the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10385 way to such use of the figure. Whether the practical applica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10386 tion of the method of the <I>Analemma</I> for what is equivalent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10387 to the solution of spherical triangles goes back as far as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10388 Hipparchus is not certain; but it is quite likely that it does,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10389 <pb n=292><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10390 seeing that Diodorus wrote his <I>Analemma</I> in the next cen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10391 tury. The other alternative source for Hipparchus's spherical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10392 trigonometry is the Menelaus-theorem applied to the sphere,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10393 on which alone Ptolemy, as we have seen, relies in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10394 <I>Syntaxis.</I> In any case the Table of Chords or Sines was in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10395 full use in Hipparchus's works, for it is presupposed by either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10396 method.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10397 <C>The <I>Planisphaerium</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10398 <p>With the <I>Analemma</I> of Ptolemy is associated another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10399 work of somewhat similar content, the <I>Planisphaerium.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10400 This again has only survived in a Latin translation from an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10401 Arabic version made by one Maslama b. Ahmad al-Majriti, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10402 Cordova (born probably at Madrid, died 1007/8); the transla-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10403 tion is now found to be, not by Rudolph of Bruges, but by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10404 ‘Hermannus Secundus’, whose pupil Rudolph was; it was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10405 first published at Basel in 1536, and again edited, with com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10406 mentary, by Commandinus (Venice, 1558). It has been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10407 re-edited from the manuscripts by Heiberg in vol. ii. of his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10408 text of Ptolemy. The book is an explanation of the system
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10409 of projection known as <I>stereographic</I>, by which points on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10410 heavenly sphere are represented on the plane of the equator
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10411 by projection from one point, a pole; Ptolemy naturally takes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10412 the south pole as centre of projection, as it is the northern
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10413 hemisphere which he is concerned to represent on a plane.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10414 Ptolemy is aware that the projections of all circles on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10415 sphere (great circles—other than those through the poles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10416 which project into straight lines—and small circles either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10417 parallel or not parallel to the equator) are likewise circles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10418 It is curious, however, that he does not give any general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10419 proof of the fact, but is content to prove it of particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10420 circles, such as the ecliptic, the horizon, &c. This is remark-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10421 able, because it is easy to show that, if a cone be described
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10422 with the pole as vertex and passing through any circle on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10423 sphere, i.e. a circular cone, in general oblique, with that circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10424 as base, the section of the cone by the plane of the equator
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10425 satisfies the criterion found for the ‘subcontrary sections’ by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10426 Apollonius at the beginning of his <I>Conics</I>, and is therefore a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10427 circle. The fact that the method of stereographic projection is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10428 so easily connected with the property of subcontrary sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10429 <pb n=293><head>THE <I>PLANISPHAERIUM</I> OF PTOLEMY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10430 of oblique circular cones has led to the conjecture that Apollo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10431 nius was the discoverer of the method. But Ptolemy makes no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10432 mention of Apollonius, and all that we know is that Synesius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10433 of Cyrene (a pupil of Hypatia, and born about A.D. 365-370)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10434 attributes the discovery of the method and its application to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10435 Hipparchus; it is curious that he does not mention Ptolemy's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10436 treatise on the subject, but speaks of himself alone as having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10437 perfected the theory. While Ptolemy is fully aware that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10438 circles on the sphere become circles in the projection, he says
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10439 nothing about the other characteristic of this method of pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10440 jection, namely that the angles on the sphere are represented
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10441 by equal angles on the projection.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10442 <p>We must content ourselves with the shortest allusion to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10443 other works of Ptolemy. There are, in the first place, other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10444 minor astronomical works as follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10445 <p>(1) <G>*fa/seis a)planw=n a)ste/rwn</G> of which only Book II sur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10446 vives, (2) <G>*(gpoqe/seis tw=n planwme/nwn</G> in two Books, the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10447 of which is extant in Greek, the second in Arabic only, (3) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10448 inscription in Canobus, (4) <G>*proxei/rwn kano/nwn dia/tasis kai\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10449 yhfofori/a</G>. All these are included in Heiberg's edition,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10450 vol. ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10451 <C>The <I>Optics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10452 <p>Ptolemy wrote an <I>Optics</I> in five Books, which was trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10453 lated from an Arabic version into Latin in the twelfth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10454 century by a certain Admiral Eugenius Siculus<note>See G. Govi, <I>L'ottica di Claudio Tolomeo di Eugenio Ammiraglio di
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10455 Sicilia</I>, . . . Torino, 1884; and particulars in G. Loria. <I>Le scienze esatte
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10456 nell' antica Grecia</I>, pp. 570, 571.</note>; Book I,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10457 however, and the end of Book V are wanting. Books I, II
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10458 were physical, and dealt with generalities; in Book III
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10459 Ptolemy takes up the theory of mirrors, Book IV deals with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10460 concave and composite mirrors, and Book V with refraction.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10461 The theoretical portion would suggest that the author was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10462 not very proficient in geometry. Many questions are solved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10463 incorrectly owing to the assumption of a principle which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10464 clearly false, namely that ‘the image of a point on a mirror is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10465 at the point of concurrence of two lines, one of which is drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10466 from the luminous point to the centre of curvature of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10467 mirror, while the other is the line from the eye to the point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10468 <pb n=294><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10469 on the mirror where the reflection takes place’; Ptolemy uses
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10470 the principle to solve various special cases of the following
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10471 problem (depending in general on a biquadratic equation and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10472 now known as the problem of Alhazen), ‘Given a reflecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10473 surface, the position of a luminous point, and the position
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10474 of a point through which the reflected ray is required to pass,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10475 to find the point on the mirror where the reflection will take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10476 place.’ Book V is the most interesting, because it seems to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10477 be the first attempt at a theory of refraction. It contains
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10478 many details of experiments with different media, air, glass,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10479 and water, and gives tables of angles of refraction (<I>r</I>) corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10480 sponding to different angles of incidence (<I>i</I>); these are calcu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10481 lated on the supposition that <I>r</I> and <I>i</I> are connected by an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10482 equation of the following form,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10483 <MATH><I>r</I> = <I>ai</I> - <I>bi</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10484 where <I>a, b</I> are constants, which is worth noting as the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10485 recorded attempt to state a law of refraction.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10486 <p>The discovery of Ptolemy's <I>Optics</I> in the Arabic at once
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10487 made it clear that the work <I>De speculis</I> formerly attributed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10488 to Ptolemy is not his, and it is now practically certain that it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10489 is, at least in substance, by Heron. This is established partly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10490 by internal evidence, e.g. the style and certain expressions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10491 recalling others which are found in the same author's <I>Auto-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10492 mata</I> and <I>Dioptra</I>, and partly by a quotation by Damianus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10493 (<I>On hypotheses in Optics</I>, chap. 14) of a proposition proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10494 ‘the mechanician Heron in his own <I>Catoptrica</I>’, which appears
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10495 in the work in question, but is not found in Ptolemy's <I>Optics</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10496 or in Euclid's. The proposition in question is to the effect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10497 that of all broken straight lines from the eye to the mirror
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10498 and from that again to the object, that particular broken line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10499 is shortest in which the two parts make equal angles with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10500 surface of the mirror; the inference is that, as nature does
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10501 nothing in vain, we must assume that, in reflection from a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10502 mirror, the ray takes the shortest course, i.e. the angles of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10503 incidence and reflection are equal. Except for the notice in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10504 Damianus and a fragment in Olympiodorus<note>Olympiodorus on Aristotle, <I>Meteor.</I> iii. 2, ed. Ideler, ii, p. 96, ed.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10505 Stüve, pp. 212. 5-213. 20.</note> containing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10506 proof of the proposition, nothing remains of the Greek text;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10507 <pb n=295><head>THE <I>OPTICS</I> OF PTOLEMY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10508 but the translation into Latin (now included in the Teubner
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10509 edition of Heron, ii, 1900, pp. 316-64), which was made by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10510 William of Moerbeke in 1269, was evidently made from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10511 Greek and not from the Arabic, as is shown by Graecisms in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10512 the translation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10513 <C>A mechanical work, <G>*peri\ r(opw=n</G>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10514 <p>There are allusions in Simplicius<note>Simplicius on Arist. <I>De caelo</I>, p. 710. 14, Heib. (Ptolemy, ed. Heib.,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10515 vol. ii, p. 263).</note> and elsewhere to a book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10516 by Ptolemy of mechanical content, <G>peri\ r(opw=n</G>, on balancings
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10517 or turnings of the scale, in which Ptolemy maintained as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10518 against Aristotle that air or water (e.g.) in their own ‘place’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10519 have no weight, and, when they are in their own ‘place’, either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10520 remain at rest or rotate simply, the tendency to go up or to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10521 fall down being due to the desire of things which are not in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10522 their own places to move to them. Ptolemy went so far as to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10523 maintain that a bottle full of air was not only not heavier
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10524 than the same bottle empty (as Aristotle held), but actually
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10525 lighter when inflated than when empty. The same work is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10526 apparently meant by the ‘book on the elements’ mentioned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10527 by Simplicius.<note><I>Ib.</I>, p. 20. 10 sq.</note> Suidas attributes to Ptolemy three Books of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10528 <I>Mechanica.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10529 <p>Simplicius<note><I>Ib.</I>, p. 9. 21 sq., (Ptolemy, ed. Heib., vol. ii, p. 265).</note> also mentions a single book, <G>peri\ diasta/sews</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10530 ‘<I>On dimension</I>’, i.e. dimensions, in which Ptolemy tried to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10531 show that the possible number of dimensions is limited to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10532 three.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10533 <C>Attempt to prove the Parallel-Postulate.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10534 <p>Nor should we omit to notice Ptolemy's attempt to prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10535 the Parallel-Postulate. Ptolemy devoted a tract to this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10536 subject, and Proclus<note>Proclus on Eucl. I, pp. 362. 14 sq., 365. 7-367. 27 (Ptolemy, ed. Heib.,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10537 vol. ii, pp. 266-70).</note> has given us the essentials of the argu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10538 ment used. Ptolemy gives, first, a proof of Eucl. I. 28, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10539 then an attempted proof of I. 29, from which he deduces
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10540 Postulate 5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10541 <pb n=296><head>TRIGONOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10542 <p>I. To prove I. 28, Ptolemy takes two straight lines <I>AB, CD</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10543 and a transversal <I>EFGH.</I> We have to prove that, if the sum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10544 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10545 of the angles <I>BFG, FGD</I> is equal to two right angles, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10546 straight lines <I>AB, CD</I> are parallel, i.e. non-secant.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10547 <p>Since <I>AFG</I> is the supplement of <I>BFG</I>, and <I>FGC</I> of <I>FGD</I>, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10548 follows that the sum of the angles <I>AFG, FGC</I> is also equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10549 two right angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10550 <p>Now suppose, if possible, that <I>FB, GD</I>, making the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10551 the angles <I>BFG, FGD</I> equal to two right angles, meet at <I>K</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10552 then similarly <I>FA, GC</I> making the sum of the angles <I>AFG,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10553 FGC</I> equal to two right angles must also meet, say at <I>L.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10554 <p>[Ptolemy would have done better to point out that not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10555 only are the two sums equal but the angles themselves are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10556 equal in pairs, i.e. <I>AFG</I> to <I>FGD</I> and <I>FGC</I> to <I>BFG</I>, and we can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10557 therefore take the triangle <I>KFG</I> and apply it to <I>FG</I> on the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10558 side so that the sides <I>FK, GK</I> may lie along <I>GC, FA</I> respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10559 tively, in which case <I>GC, FA</I> will meet at the point where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10560 <I>K</I> falls.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10561 <p>Consequently the straight lines <I>LABK, LCDK</I> enclose a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10562 space: which is impossible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10563 <p>It follows that <I>AB, CD</I> cannot meet in either direction;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10564 they are therefore parallel.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10565 <p>II. To prove I. 29, Ptolemy takes two parallel lines <I>AB,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10566 CD</I> and the transversal <I>FG</I>, and argues thus. It is required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10567 to prove that <MATH>&angle;<I>AFG</I> + &angle;<I>CGF</I> =</MATH> two right angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10568 <p>For, if the sum is not equal to two right angles, it must be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10569 either (1) greater or (2) less.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10570 <p>(1) If it is greater, the sum of the angles on the other side,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10571 <I>BFG, FGD</I>, which are the supplements of the first pair of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10572 angles, must be <I>less</I> than two right angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10573 <p>But <I>AF, CG</I> are no more parallel than <I>FB, GD, so that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10574 FG makes one pair of angles AFG, FGC together greater than</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10575 <pb n=297><head>PTOLEMY ON THE PARALLEL-POSTULATE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10576 <I>two right angles, it must also make the other pair BFG, FGD
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10577 together greater than two right angles.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10578 <p>But the latter pair of angles were proved less than two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10579 right angles: which is impossible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10580 <p>Therefore the sum of the angles <I>AFG, FGC</I> cannot be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10581 <I>greater</I> than two right angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10582 <p>(2) Similarly we can show that the sum of the two angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10583 <I>AFG, FGC</I> cannot be <I>less</I> than two right angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10584 <p>Therefore <MATH>&angle;<I>AFG</I> + &angle;<I>CGF</I> =</MATH> two right angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10585 <p>[The fallacy here lies in the inference which I have marked
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10586 by italics. When Ptolemy says that <I>AF, CG</I> are no more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10587 parallel than <I>FB, GD</I>, he is in effect assuming that <I>through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10588 any one point only one parallel can be drawn to a given straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10589 line</I>, which is an equivalent for the very Postulate he is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10590 endeavouring to prove. The alternative Postulate is known
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10591 as ‘Playfair's axiom’, but it is of ancient origin, since it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10592 distinctly enunciated in Proclus's note on Eucl. I. 31.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10593 <p>III. Post. 5 is now deduced, thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10594 <p>Suppose that the straight lines making with a transversal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10595 angles the sum of which is less than two right angles do not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10596 meet on the side on which those angles are.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10597 <p>Then, <I>a fortiori</I>, they will not meet on the other side on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10598 which are the angles the sum of which is <I>greater</I> than two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10599 right angles. [This is enforced by a supplementary proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10600 tion showing that, if the lines met on that side, Eucl. I. 16
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10601 would be contradicted.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10602 <p>Hence the straight lines cannot meet in either direction:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10603 they are therefore <I>parallel.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10604 <p>But in that case the angles made with the transversal are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10605 <I>equal</I> to two right angles: which contradicts the assumption.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10606 <p>Therefore the straight lines will meet.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10607 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10608 <C>XVIII
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10609 MENSURATION: HERON OF ALEXANDRIA
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10610 Controversies as to Heron's date.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10611 <p>THE vexed question of Heron's date has perhaps called
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10612 forth as much discussion as any doubtful point in the history
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10613 of mathematics. In the early stages of the controversy much
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10614 was made of the supposed relation of Heron to Ctesibius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10615 The <I>Belopoeïca</I> of Heron has, in the best manuscript, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10616 heading <G>(/hrwnos kthsibi/ou *belopoii+ka/</G>, and from this, coupled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10617 with an expression used by an anonymous Byzantine writer
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10618 of the tenth century, <G>o( )askrhno\s kthsi/bios o( tou= *)alexandre/ws
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10619 *(/hrwnos kaqhghth/s</G>, ‘Ctesibius of Ascra, the teacher of Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10620 of Alexandria’, it was inferred that Heron was a pupil of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10621 Ctesibius. The question then was, when did Ctesibius live?
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10622 Martin took him to be a certain barber of that name who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10623 lived in the time of Ptolemy Euergetes II, that is, Ptolemy VII,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10624 called Physcon (died 117 B.C.), and who is said to have made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10625 an improved water-organ<note>Athenaeus, <I>Deipno-Soph.</I> iv. c. 75, p. 174 b-e: cf. Vitruvius, x. 9, 13.</note>; Martin therefore placed Heron at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10626 the beginning of the first century (say 126-50) B.C. But
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10627 Philon of Byzantium, who repeatedly mentions Ctesibius by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10628 name, says that the first mechanicians (<G>texni=tai</G>) had the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10629 great advantage of being under kings who loved fame and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10630 supported the arts.<note>Philon, <I>Mechan. Synt.,</I> p. 50. 38, ed. Schöne.</note> This description applies much better
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10631 to Ptolemy II Philadelphus (285-247) and Ptolemy III Euer-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10632 getes I (247-222). It is more probable, therefore, that Ctesibius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10633 was the mechanician Ctesibius who is mentioned by Athenaeus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10634 as having made an elegant drinking-horn in the time of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10635 Ptolemy Philadelphus<note>Athenaeus, xi. c. 97, p. 497 b-e.</note>; a pupil then of Ctesibius would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10636 probably belong to the end of the third and the beginning of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10637 the second century B.C. But in truth we cannot safely con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10638 clude that Heron was an immediate pupil of Ctesibius. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10639 Byzantine writer probably only inferred this from the title
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10640 <pb n=299><head>CONTROVERSIES AS TO HERON'S DATE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10641 above quoted; the title, however, in itself need not imply
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10642 more than that Heron's work was a new edition of a similar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10643 work by Ctesibius, and the <G>*kthsibi/ou</G> may even have been added
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10644 by some well-read editor who knew both works and desired to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10645 indicate that the greater part of the contents of Heron's work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10646 was due to Ctesibius. One manuscript has <G>(/hrwnos )alexan-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10647 dre/ws *belopoii+ka/</G>, which corresponds to the titles of the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10648 works of Heron and is therefore more likely to be genuine.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10649 <p>The discovery of the Greek text of the <I>Metrica</I> by R. Schöne
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10650 in 1896 made it possible to fix with certainty an upper limit.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10651 In that work there are a number of allusions to Archimedes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10652 three references to the <G>xwri/ou a)potomh/</G> of Apollonius, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10653 two to ‘the (books) about straight lines (chords) in a circle’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10654 (<G>de/deiktai de\ e)n toi=s peri\ tw=n e)n ku/klw\| eu)qeiw=n</G>). Now, although
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10655 the first beginnings of trigonometry may go back as far as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10656 Apollonius, we know of no work giving an actual Table of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10657 Chords earlier than that of Hipparchus. We get, therefore,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10658 at once the date 150 B.C. or thereabouts as the <I>terminus post
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10659 quem.</I> A <I>terminus ante quem</I> is furnished by the date of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10660 composition of Pappus's <I>Collection;</I> for Pappus alludes to, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10661 draws upon, the works of Heron. As Pappus was writing in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10662 the reign of Diocletian (A.D. 284-305), it follows that Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10663 could not be much later than, say, A.D. 250. In speaking of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10664 the solutions by ‘the old geometers’ (<G>oi( palaioi\ gewme/trai</G>) of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10665 the problem of finding the two mean proportionals, Pappus may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10666 seem at first sight to include Heron along with Eratosthenes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10667 Nicomedes and Philon in that designation, and it has been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10668 argued, on this basis, that Heron lived long before Pappus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10669 But a close examination of the passage<note>Pappus, iii, pp. 54-6.</note> shows that this is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10670 by no means necessary. The relevant words are as follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10671 <p>‘The ancient geometers were not able to solve the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10672 of the two straight lines [the problem of finding two mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10673 proportionals to them] by ordinary geometrical methods, since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10674 the problem is by nature “solid” . . . but by attacking it with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10675 mechanical means they managed, in a wonderful way, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10676 reduce the question to a practical and convenient construction,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10677 as may be seen in the <I>Mesolabon</I> of Eratosthenes and in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10678 mechanics of Philon and Heron . . . Nicomedes also solved it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10679 by means of the cochloid curve, with which he also trisected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10680 an angle.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10681 <pb n=300><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10682 <p>Pappus goes on to say that he will give four solutions, one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10683 of which is his own; the first, second, and third he describes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10684 as those of Eratosthenes, Nicomedes and Heron. But in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10685 earlier sentence he mentions Philon along with Heron, and we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10686 know from Eutocius that Heron's solution is practically the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10687 same as Philon's. Hence we may conclude that by the third
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10688 solution Pappus really meant Philon's, and that he only men-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10689 tioned Heron's <I>Mechanics</I> because it was a convenient place in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10690 which to find the same solution.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10691 <p>Another argument has been based on the fact that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10692 extracts from Heron's <I>Mechanics</I> given at the end of Pappus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10693 Book VIII, as we have it, are introduced by the author with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10694 a complaint that the copies of Heron's works in which he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10695 found them were in many respects corrupt, having lost both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10696 beginning and end.<note>Pappus, viii, p. 1116. 4-7.</note> But the extracts appear to have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10697 added, not by Pappus, but by some later writer, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10698 argument accordingly falls to the ground.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10699 <p>The limits of date being then, say, 150 B.C. to A.D. 250, our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10700 only course is to try to define, as well as possible, the relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10701 in time between Heron and the other mathematicians who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10702 come, roughly, within the same limits. This method has led
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10703 one of the most recent writers on the subject (Tittel<note>Art. ‘Heron von Alexandreia’ in Pauly-Wissowa's <I>Real-Encyclopädie
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10704 der class. Altertumswissenschaft,</I> vol. 8. 1, 1912.</note>) to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10705 place Heron not much later than 100 B.C., while another,<note>I. Hammer-Jensen in <I>Hermes,</I> vol. 48, 1913, pp. 224-35.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10706 relying almost entirely on a comparison between passages in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10707 Ptolemy and Heron, arrives at the very different conclusion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10708 that Heron was later than Ptolemy and belonged in fact to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10709 the second century A.D.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10710 <p>In view of the difference between these results, it will be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10711 convenient to summarize the evidence relied on to establish
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10712 the earlier date, and to consider how far it is or is not con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10713 clusive against the later. We begin with the relation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10714 Heron to Philon. Philon is supposed to come not more than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10715 a generation later than Ctesibius, because it would appear that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10716 machines for throwing projectiles constructed by Ctesibius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10717 and Philon respectively were both available at one time for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10718 inspection by experts on the subject<note>Philon, <I>Mech. Synt.</I> iv, pp. 68. 1, 72. 36.</note>; it is inferred that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10719 <pb n=301><head>CONTROVERSIES AS TO HERON'S DATE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10720 Philon's date cannot be later than the end of the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10721 century B.C. (If Ctesibius flourished before 247 B.C. the argu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10722 ment would apparently suggest rather the beginning than the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10723 end of the second century.) Next, Heron is supposed to have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10724 been a younger contemporary of Philon, the grounds being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10725 the following. (1) Heron mentions a ‘stationary-automaton’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10726 representation by Philon of the Nauplius-story,<note>Heron, <I>Autom.,</I> pp. 404. 11-408. 9.</note> and this is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10727 identified by Tittel with a representation of the same story by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10728 some contemporary of Heron's (<G>oi( kaq' h(ma=s</G><note><I>Ib.,</I> p. 412. 13.</note>). But a careful
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10729 perusal of the whole passage seems to me rather to suggest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10730 that the latter representation was not Philon's, and that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10731 Philon was included by Heron among the ‘ancient’ auto-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10732 maton-makers, and not among his contemporaries.<note>The relevant remarks of Heron are as follows. (1) He says that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10733 has found no arrangements of ‘stationary automata’ better or more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10734 instructive than those described by Philon of Byzantium (p. 404. 11).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10735 As an instance he mentions Philon's setting of the Nauplius-story, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10736 which he found everything good except two things (<I>a</I>) the mechanism
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10737 for the appearance of Athene, which was too difficult (<G>e)rgwde/steron</G>), and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10738 (<I>b</I>) the absence of an incident promised by Philon in his description,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10739 namely the falling of a thunderbolt on Ajax with a sound of thunder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10740 accompanying it (pp. 404. 15-408. 9). This latter incident Heron could
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10741 not find anywhere in Philon, though he had consulted a great number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10742 of copies of his work. He continues (p. 408. 9-13) that we are not to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10743 suppose that he is running down Philon or charging him with not being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10744 capable of carrying out what he promised. On the contrary, the omission
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10745 was probably due to a slip of memory, for it is easy enough to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10746 stage-thunder (he proceeds to show how to do it). But the rest of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10747 Philon's arrangements seemed to him satisfactory, and this, he says, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10748 why he has not ignored Philon's work: ‘for I think that my readers will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10749 get the most benefit if they are shown, first what has been well said by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10750 the ancients and then, separately from this, what the ancients overlooked
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10751 or what in their work needed improvement’ (pp. 408.22-410.6). (2) The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10752 next chapter (pp. 410. 7-412. 2) explains generally the sort of thing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10753 automaton-picture has to show, and Heron says he will give one example
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10754 which he regards as the best. Then (3), after drawing a contrast between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10755 the simpler pictures made by ‘the ancients’, which involved three different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10756 movements only, and the contemporary (<G>oi( kaq' h(ma=s</G>) representations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10757 interesting stories by means of more numerous and varied movements
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10758 (p. 412. 3-15), he proceeds to describe a setting of the Nauplius-story.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10759 This is the representation which Tittel identifies with Philon's. But it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10760 is to be observed that the description includes that of the episode of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10761 thunderbolt striking Ajax (c. 30, pp. 448. 1-452. 7) which Heron expressly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10762 says that Philon omitted. Further, the mechanism for the appearance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10763 of Athene described in c. 29 is clearly not. Philon's ‘more difficult’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10764 arrangement, but the simpler device described (pp. 404. 18-408. 5) as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10765 possible and preferable to Philon's (cf. Heron, vol. i, ed. Schmidt, pp.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10766 lxviii-lxix).</note> (2) Another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10767 argument adduced to show that Philon was contemporary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10768 <pb n=302><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10769 with Heron is the fact that Philon has some criticisms of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10770 details of construction of projectile-throwers which are found
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10771 in Heron, whence it is inferred that Philon had Heron's work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10772 specifically in view. But if Heron's <G>*belopoii+ka/</G> was based on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10773 the work of Ctesibius, it is equally possible that Philon may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10774 be referring to Ctesibius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10775 <p>A difficulty in the way of the earlier date is the relation in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10776 which Heron stands to Posidonius. In Heron's <I>Mechanics,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10777 i. 24, there is a definition of ‘centre of gravity’ which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10778 attributed by Heron to ‘Posidonius a Stoic’. But this can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10779 hardly be Posidonius of Apamea, Cicero's teacher, because the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10780 next sentence in Heron, stating a distinction drawn by Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10781 medes in connexion with this definition, seems to imply that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10782 the Posidonius referred to lived before Archimedes. But the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10783 <I>Definitions</I> of Heron do contain definitions of geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10784 notions which are put down by Proclus to Posidonius of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10785 Apamea or Rhodes, and, in particular, definitions of ‘figure’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10786 and of ‘parallels’. Now Posidonius lived from 135 to 51 B.C.,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10787 and the supporters of the earlier date for Heron can only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10788 suggest that either Posidonius was not the first to give these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10789 definitions, or alternatively, if he was, and if they were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10790 included in Heron's <I>Definitions</I> by Heron himself and not by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10791 some later editor, all that this obliges us to admit is that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10792 Heron cannot have lived before the first century B.C.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10793 <p>Again, if Heron lived at the beginning of the first cen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10794 tury B.C., it is remarkable that he is nowhere mentioned by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10795 Vitruvius. The <I>De architectura</I> was apparently brought out
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10796 in 14 B.C. and in the preface to Book VII Vitruvius gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10797 a list of authorities on <I>machinationes</I> from whom he made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10798 extracts. The list contains twelve names and has every
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10799 appearance of being scrupulously complete; but, while it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10800 includes Archytas (second), Archimedes (third), Ctesibius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10801 (fourth), and Philon of Byzantium (sixth), it does not men-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10802 tion Heron. Nor is it possible to establish interdependence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10803 between Heron and Vitruvius; the differences seem, on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10804 whole, to be more numerous than the resemblances. A few of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10805 the differences may be mentioned. Vitruvius uses 3 as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10806 value of <G>p</G>, whereas Heron always uses the Archimedean value
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10807 3 1/7. Both writers make extracts from the Aristotelian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10808 <G>*mhxanika\ problh/mata</G>, but their selections are different. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10809 <pb n=303><head>CONTROVERSIES AS TO HERON'S DATE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10810 machines used by the two for the same purpose frequently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10811 differ in details; e.g. in Vitruvius's hodometer a pebble drops
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10812 into a box at the end of each Roman mile,<note>Vitruvius, x. 14.</note> while in Heron's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10813 the distance completed is marked by a pointer.<note>Heron, <I>Dioptra,</I> c. 34.</note> It is indeed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10814 pointed out that the water-organ of Heron is in many respects
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10815 more primitive than that of Vitruvius; but, as the instru-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10816 ments are altogether different, this can scarcely be said to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10817 prove anything.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10818 <p>On the other hand, there are points of contact between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10819 certain propositions of Heron and of the Roman <I>agrimen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10820 sores.</I> Columella, about A.D. 62, gave certain measurements of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10821 plane figures which agree with the formulae used by Heron,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10822 notably those for the equilateral triangle, the regular hexagon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10823 (in this case not only the formula but the actual figures agree
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10824 with Heron's) and the segment of a circle which is less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10825 a semicircle, the formula in the last case being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10826 <MATH>1/2(<I>s</I>+<I>h</I>)<I>h</I>+1/14(1/2<I>s</I>)<SUP>2</SUP>,</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10827 where <I>s</I> is the chord and <I>h</I> the height of the segment. Here
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10828 there might seem to be dependence, one way or the other;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10829 but the possibility is not excluded that the two writers may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10830 merely have drawn from a common source; for Heron, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10831 giving the formula for the area of the segment of a circle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10832 states that it was the formula used by ‘the more accurate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10833 investigators’ (<G>oi( a)kribe/steron e)zhthko/tes</G>).<note>Heron, <I>Metrica,</I> i. 31, p. 74. 21.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10834 <p>We have, lastly, to consider the relation between Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10835 and Heron. If Heron lived about 100 B.C., he was 200 years
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10836 earlier than Ptolemy (A.D. 100-178). The argument used to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10837 prove that Ptolemy came some time after Heron is based on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10838 a passage of Proclus where Ptolemy is said to have remarked
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10839 on the untrustworthiness of the method in vogue among the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10840 ‘more ancient’ writers of measuring the apparent diameter of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10841 the sun by means of water-clocks.<note>Proclus, <I>Hypotyposis,</I> pp. 120. 9-15, 124. 7-26.</note> Hipparchus, says Pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10842 clus, used his dioptra for the purpose, and Ptolemy followed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10843 him. Proclus proceeds:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10844 <p>‘Let us then set out here not only the observations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10845 the ancients but also the construction of the dioptra of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10846 <pb n=304><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10847 Hipparchus. And first we will show how we can measure an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10848 interval of time by means of the regular efflux of water,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10849 a procedure which was explained by Heron the mechanician
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10850 in his treatise on water-clocks.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10851 <p>Theon of Alexandria has a passage to a similar effect.<note>Theon, <I>Comm. on the Syntaxis,</I> Basel, 1538, pp. 261 sq. (quoted in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10852 Proclus, <I>Hypotyposis,</I> ed. Manitius, pp. 309-11).</note> He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10853 first says that the most ancient mathematicians contrived
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10854 a vessel which would let water flow out uniformly through a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10855 small aperture at the bottom, and then adds at the end, almost
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10856 in the same words as Proclus uses, that Heron showed how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10857 this is managed in the first book of his work on water-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10858 clocks. Theon's account is from Pappus's Commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10859 the <I>Syntaxis,</I> and this is also Proclus's source, as is shown by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10860 the fact that Proclus gives a drawing of the water-clock
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10861 which appears to have been lost in Theon's transcription from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10862 Pappus, but which Pappus must have reproduced from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10863 work of Heron. Tittel infers that Heron must have ranked
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10864 as one of the ‘more ancient’ writers as compared with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10865 Ptolemy. But this again does not seem to be a necessary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10866 inference. No doubt Heron's work was a convenient place to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10867 refer to for a description of a water-clock, but it does not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10868 necessarily follow that Ptolemy was referring to Heron's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10869 clock rather than some earlier form of the same instrument.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10870 <p>An entirely different conclusion from that of Tittel is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10871 reached in the article ‘Ptolemaios and Heron’ already alluded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10872 to.<note>Hammer-Jensen, <I>op. cit.</I></note> The arguments are shortly these. (1) Ptolemy says in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10873 his <I>Geography</I> (c. 3) that his predecessors had only been able
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10874 to measure the distance between two places (as an are of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10875 great circle on the earth's circumference) in the case where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10876 the two places are on the same meridian. He claims that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10877 himself invented a way of doing this even in the case where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10878 the two places are neither on the same meridian nor on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10879 same parallel circle, provided that the heights of the pole at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10880 the two places respectively, and the angle between the great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10881 circle passing through both and the meridian circle through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10882 one of the places, are known. Now Heron in his <I>Dioptra</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10883 deals with the problem of measuring the distance between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10884 two places by means of the dioptra, and takes as an example
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10885 <pb n=305><head>CONTROVERSIES AS TO HERON'S DATE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10886 the distance between Rome and Alexandria.<note>Heron, <I>Dioptra,</I> c. 35 (vol. iii, pp. 302-6).</note> Unfortunately
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10887 the text is in places corrupt and deficient, so that the method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10888 cannot be reconstructed in detail. But it involved the obser-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10889 vation of the same lunar eclipse at Rome and Alexandria
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10890 respectively and the drawing of the <I>analemma</I> for Rome.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10891 That is to say, the mathematical method which Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10892 claims to have invented is spoken of by Heron as a thing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10893 generally known to experts and not more remarkable than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10894 other technical matters dealt with in the same book. Conse-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10895 quently Heron must have been later than Ptolemy. (It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10896 right to add that some hold that the chapter of the <I>Dioptra</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10897 in question is not germane to the subject of the treatise, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10898 was probably not written by Heron but interpolated by some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10899 later editor; if this is so, the argument based upon it falls to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10900 the ground.) (2) The dioptra described in Heron's work is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10901 fine and accurate instrument, very much better than anything
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10902 Ptolemy had at his disposal. If Ptolemy had been aware of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10903 its existence, it is highly unlikely that he would have taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10904 the trouble to make his separate and imperfect ‘parallactic’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10905 instrument, since it could easily have been grafted on to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10906 Heron's dioptra. Not only, therefore, must Heron have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10907 later than Ptolemy but, seeing that the technique of instru-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10908 ment-making had made such strides in the interval, he must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10909 have been considerably later. (3) In his work <G>peri\ r(opw=n</G><note>Simplicius on <I>De caelo,</I> p. 710. 14, Heib. (Ptolemy, vol. ii, p. 263).</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10910 Ptolemy, as we have seen, disputed the view of Aristotle that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10911 air has weight even when surrounded by air. Aristotle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10912 satisfied himself experimentally that a vessel full of air is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10913 heavier than the same vessel empty; Ptolemy, also by ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10914 periment, convinced himself that the former is actually the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10915 lighter. Ptolemy then extended his argument to water, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10916 held that water with water round it has no weight, and that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10917 the diver, however deep he dives, does not feel the weight of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10918 the water above him. Heron<note>Heron, <I>Pneumatica,</I> i. Pref. (vol. i, p. 22. 14 sq.).</note> asserts that water has no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10919 appreciable weight and has no appreciable power of com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10920 pressing the air in a vessel inverted and forced down into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10921 the water. In confirmation of this he cites the case of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10922 diver, who is not prevented from breathing when far below
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10923 <pb n=306><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10924 the surface. He then inquires what is the reason why the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10925 diver is not oppressed though he has an unlimited weight of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10926 water on his back. He accepts, therefore, the view of Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10927 as to the fact, however strange this may seem. But he is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10928 satisfied with the explanation given: ‘Some say’, he goes on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10929 ‘it is because water in itself is uniformly heavy (<G>i)sobare\s au)to\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10930 kaq' au(to/</G>)’—this seems to be equivalent to Ptolemy's dictum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10931 that water in water has no weight—‘but they give no ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10932 planation whatever why divers...’ He himself attempts an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10933 explanation based on Archimedes. It is suggested, therefore,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10934 that Heron's criticism is directed specifically against Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10935 and no one else. (4) It is suggested that the Dionysius to whom
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10936 Heron dedicated his <I>Definitions</I> is a certain Dionysius who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10937 was <I>praefectus urbi</I> at Rome in A.D. 301. The grounds are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10938 these (<I>a</I>) Heron addresses Dionysius as <G>*dionu/sie lampro/tate</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10939 where <G>lampro/tatos</G> obviously corresponds to the Latin <I>clarissi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10940 mus,</I> a title which in the third century and under Diocletian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10941 was not yet in common use. Further, this Dionysius was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10942 <I>curator aquarum</I> and <I>curator operum publicorum,</I> so that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10943 was the sort of person who would have to do with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10944 engineers, architects and craftsmen for whom Heron wrote.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10945 Lastly, he is mentioned in an inscription commemorating an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10946 improvement of water supply and dedicated ‘to Tiberinus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10947 father of all waters, and to the ancient inventors of marvel-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10948 lous constructions’ (<I>repertoribus admirabilium fabricarum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10949 priscis viris</I>), an expression which is not found in any other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10950 inscription, but which recalls the sort of tribute that Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10951 frequently pays to his predecessors. This identification of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10952 two persons named Dionysius is an ingenious conjecture, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10953 the evidence is not such as to make it anything more.<note>Dionysius was of course a very common name. Diophantus dedicated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10954 his <I>Arithmetica</I> to a person of this name (<G>timiw/tate/ moi *dionu/sie</G>), whom he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10955 praised for his ambition to learn the solutions of arithmetical problems.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10956 This Dionysius must have lived in the second half of the third century
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10957 A.D., and if Heron also belonged to this time, is it not possible that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10958 Heron's Dionysius was the same person?</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10959 <p>The result of the whole investigation just summarized is to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10960 place Heron in the third century A.D., and perhaps little, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10961 anything, earlier than Pappus. Heiberg accepts this conclu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10962 sion,<note>Heron, vol. v, p. ix.</note> which may therefore, I suppose, be said to hold the field
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10963 for the present.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10964 <pb n=307><head>CONTROVERSIES AS TO HERON'S DATE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10965 <p>Heron was known as <G>o( *)alexandreu/s</G> (e.g. by Pappus) or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10966 <G>o( mhxaniko/s</G> (<I>mechanicus</I>), to distinguish him from other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10967 persons of the same name; Proclus and Damianus use the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10968 latter title, while Pappus also speaks of <G>oi( peri\ to\n *(/hrwna
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10969 mhxanikoi/</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10970 <C>Character of works.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10971 <p>Heron was an almost encyclopaedic writer on mathematical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10972 and physical subjects. Practical utility rather than theoreti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10973 cal completeness was the object aimed at; his environment in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10974 Egypt no doubt accounts largely for this. His <I>Metrica</I> begins
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10975 with the old legend of the traditional origin of geometry in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10976 Egypt, and in the <I>Dioptra</I> we find one of the very problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10977 which geometry was intended to solve, namely that of re-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10978 establishing boundaries of lands when the flooding of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10979 Nile had destroyed the land-marks: ‘When the boundaries
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10980 of an area have become obliterated to such an extent that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10981 only two or three marks remain, in addition to a plan of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10982 area, to supply afresh the remaining marks.’<note>Heron, <I>Dioptra,</I> c. 25, p. 268. 17-19.</note> Heron makes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10983 little or no claim to originality; he often quotes authorities,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10984 but, in accordance with Greek practice, he more frequently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10985 omits to do so, evidently without any idea of misleading any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10986 one; only when he has made what is in his opinion any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10987 slight improvement on the methods of his predecessors does
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10988 he trouble to mention the fact, a habit which clearly indi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10989 cates that, except in these cases, he is simply giving the best
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10990 traditional methods in the form which seemed to him easiest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10991 of comprehension and application. The <I>Metrica</I> seems to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10992 richest in definite references to the discoveries of prede-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10993 cessors; the names mentioned are Archimedes; Dionysodorus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10994 Eudoxus, Plato; in the <I>Dioptra</I> Eratosthenes is quoted, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10995 in the introduction to the <I>Catoptrica</I> Plato and Aristotle are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10996 mentioned.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10997 <p>The practical utility of Heron's manuals being so great, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10998 was natural that they should have great vogue, and equally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	10999 natural that the most popular of them at any rate should be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11000 re-edited, altered and added to by later writers; this was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11001 inevitable with books which, like the <I>Elements</I> of Euclid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11002 were in regular use in Greek, Byzantine, Roman, and Arabian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11003 <pb n=308><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11004 education for centuries. The geometrical or mensurational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11005 books in particular gave scope for expansion by multiplication
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11006 of examples, so that it is difficult to disentangle the genuine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11007 Heron from the rest of the collections which have come down
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11008 to us under his name. Hultsch's considered criterion is as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11009 follows: ‘The Heron texts which have come down to our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11010 time are authentic in so far as they bear the author's name
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11011 and have kept the original design and form of Heron's works,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11012 but are unauthentic in so far as, being constantly in use for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11013 practical purposes, they were repeatedly re-edited and, in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11014 course of re-editing, were rewritten with a view to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11015 particular needs of the time.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11016 <C>List of Treatises.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11017 <p>Such of the works of Heron as have survived have reached
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11018 us in very different ways. Those which have come down in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11019 the Greek are:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11020 <p>I. The <I>Metrica,</I> first discovered in 1896 in a manuscript
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11021 of the eleventh (or twelfth) century at Constantinople by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11022 R. Schöne and edited by his son, H. Schöne (<I>Heronis Opera,</I> iii,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11023 Teubner, 1903).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11024 <p>II. <I>On the Dioptra,</I> edited in an Italian version by Venturi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11025 in 1814; the Greek text was first brought out by A. J. H.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11026 Vincent<note><I>Notices et extraits des manuscrits de la Bibliothèque impériale,</I> xix, pt. 2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11027 pp. 157-337.</note> in 1858, and the critical edition of it by H. Schöne is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11028 included in the Teubner vol. iii just mentioned.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11029 <p>III. The <I>Pneumatica,</I> in two Books, which appeared first in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11030 a Latin translation by Commandinus, published after his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11031 death in 1575; the Greek text was first edited by Thévenot
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11032 in <I>Veterum mathematicorum opera Graece et Latine edita</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11033 (Paris, 1693), and is now available in <I>Heronis Opera,</I> i (Teub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11034 ner, 1899), by W. Schmidt.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11035 <p>IV. <I>On the art of constructing automata</I> (<G>peri\ au)tomato-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11036 poihtikh=s</G>), or <I>The automaton-theatre,</I> first edited in an Italian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11037 translation by B. Baldi in 1589; the Greek text was included
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11038 in Thévenot's <I>Vet. math.,</I> and now forms part of <I>Heronis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11039 Opera,</I> vol. i, by W. Schmidt.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11040 <p>V. <I>Belopoeïca</I> (on the construction of engines of war), edited
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11041 <pb n=309><head>LIST OF TREATISES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11042 by B. Baldi (Augsburg, 1616), Thévenot (<I>Vet. math.</I>), Köchly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11043 and Rüstow (1853) and by Wescher (<I>Poliorcétique des Grecs,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11044 1867, the first critical edition).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11045 <p>VI. The <I>Cheirobalistra</I> (<G>*(/hrwnos xeiroballi/stras kataskeuh\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11046 kai\ summetri/a</G> (?)), edited by V. Prou, <I>Notices et extraits,</I> xxvi. 2
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11047 (Paris, 1877).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11048 <p>VII. The geometrical works, <I>Definitiones, Geometria, Geo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11049 daesia, Stereometrica</I> I and II, <I>Mensurae, Liber Geeponicus,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11050 edited by Hultsch with <I>Variae collectiones</I> (<I>Heronis Alexan-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11051 drini geometrioorum et stereometricorum reliquiae,</I> 1864).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11052 This edition will now be replaced by that of Heiberg in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11053 Teubner collection (vols. iv, v), which contains much addi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11054 tional matter from the Constantinople manuscript referred to,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11055 but omits the <I>Liber Geeponicus</I> (except a few extracts) and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11056 <I>Geodaesia</I> (which contains only a few extracts from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11057 <I>Geometry</I> of Heron).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11058 <p>Only fragments survive of the Greek text of the <I>Mechanics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11059 in three Books, which, however, is extant in the Arabic (now
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11060 edited, with German translation, in <I>Heronis Opera,</I> vol. ii,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11061 by L. Nix and W. Schmidt, Teubner, 1901).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11062 <p>A smaller separate mechanical treatise, the <G>*baroulko/s</G>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11063 quoted by Pappus.<note>Pappus, viii, p. 1060. 5.</note> The object of it was ‘to move a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11064 weight by means of a given force’, and the machine consisted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11065 of an arrangement of interacting toothed wheels with different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11066 diameters.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11067 <p>At the end of the <I>Dioptra</I> is a description of a <I>hodometer</I> for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11068 measuring distances traversed by a wheeled vehicle, a kind of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11069 taxameter, likewise made of a combination of toothed wheels.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11070 <p>A work on <I>Water-clocks</I> (<G>peri\ u(dri/wn w(roskopei/wn</G>) is men-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11071 tioned in the <I>Pneumatica</I> as having contained four Books,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11072 and is also alluded to by Pappus.<note><I>Ib.,</I> p. 1026. 1.</note> Fragments are preserved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11073 in Proclus (<I>Hypotyposis,</I> chap. 4) and in Pappus's commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11074 on Book V of Ptolemy's <I>Syntaxis</I> reproduced by Theon.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11075 <p>Of Heron's <I>Commentary on Euclid's Elements</I> only very
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11076 meagre fragments survive in Greek (Proclus), but a large
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11077 number of extracts are fortunately preserved in the Arabic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11078 commentary of an-Nair&imacr;z&imacr;, edited (1) in the Latin version of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11079 Gherard of Cremona by Curtze (Teubner, 1899); and (2) by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11080 <pb n=310><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11081 Besthorn and Heiberg (<I>Codex Leidensis</I> 399. 1, five parts of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11082 which had appeared up to 1910). The commentary extended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11083 as far as <I>Elem.</I> VIII. 27 at least.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11084 <p>The <I>Catoptrica,</I> as above remarked under Ptolemy, exists in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11085 a Latin translation from the Greek, presumed to be by William
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11086 of Moerbeke, and is included in vol. ii of <I>Heronis Opera,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11087 edited, with introduction, by W. Schmidt.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11088 <p>Nothing is known of the <I>Camarica</I> (‘on vaultings’) men-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11089 tioned by Eutocius (on Archimedes, <I>Sphere and Cylinder</I>), the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11090 <I>Zygia</I> (balancings) associated by Pappus with the <I>Automata,</I><note>Pappus, viii, p. 1024. 28.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11091 or of a work on the use of the astrolabe mentioned in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11092 <I>Fihrist.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11093 <p>We are in this work concerned with the treatises of mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11094 matical content, and therefore can leave out of account such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11095 works as the <I>Pneumatica,</I> the <I>Automata,</I> and the <I>Belopoeïca.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11096 The <I>Pneumatica</I> and <I>Automata</I> have, however, an interest to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11097 the historian of physics in so far as they employ the force of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11098 compressed air, water, or steam. In the <I>Pneumatica</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11099 reader will find such things as siphons, ‘Heron's fountain’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11100 ‘penny-in-the-slot’ machines, a fire-engine, a water-organ, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11101 many arrangements employing the force of steam.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11102 <C>Geometry.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11103 <C>(<I>a</I>) <I>Commentary on Euclid's Elements.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11104 <p>In giving an account of the geometry and mensuration
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11105 (or geodesy) of Heron it will be well, I think, to begin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11106 with what relates to the <I>elements,</I> and first the Commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11107 tary on Euclid's <I>Elements,</I> of which we possess a number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11108 of extracts in an-Nair&imacr;z&imacr; and Proclus, enabling us to form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11109 a general idea of the character of the work. Speaking
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11110 generally, Heron's comments do not appear to have contained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11111 much that can be called important. They may be classified
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11112 as follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11113 <p>(1) A few general notes, e.g. that Heron would not admit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11114 more than three axioms.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11115 <p>(2) Distinctions of a number of particular <I>cases</I> of Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11116 propositions according as the figure is drawn in one way
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11117 or another.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11118 <pb n=311><head>GEOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11119 <p>Of this class are the different cases of I. 35, 36, III. 7, 8
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11120 (where the chords to be compared are drawn on different sides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11121 of the diameter instead of on the same side), III. 12 (which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11122 not Euclid's at all but Heron's own, adding the case of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11123 external to that of internal contact in III. 11, VI. 19 (where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11124 the triangle in which an additional line is drawn is taken to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11125 be the <I>smaller</I> of the two), VII. 19 (where the particular case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11126 is given of <I>three</I> numbers in continued proportion instead of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11127 four proportionals).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11128 <p>(3) Alternative proofs.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11129 <p>It appears to be Heron who first introduced the easy but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11130 uninstructive semi-algebraical method of proving the proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11131 tions II. 2-10 which is now so popular. On this method the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11132 propositions are proved ‘without figures’ as consequences of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11133 II. 1 corresponding to the algebraical formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11134 <MATH><I>a</I>(<I>b</I>+<I>c</I>+<I>d</I>+...)=<I>ab</I>+<I>ac</I>+<I>ad</I>+...</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11135 <p>Heron explains that it is not possible to prove II. 1 without
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11136 drawing a number of lines (i.e. without actually drawing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11137 rectangles), but that the following propositions up to II. 10
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11138 can be proved by merely drawing one line. He distinguishes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11139 two varieties of the method, one by <I>dissolutio,</I> the other by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11140 <I>compositio,</I> by which he seems to mean <I>splitting-up</I> of rect-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11141 angles and squares and <I>combination</I> of them into others.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11142 But in his proofs he sometimes combines the two varieties.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11143 <p>Alternative proofs are given (<I>a</I>) of some propositions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11144 Book III, namely III. 25 (placed after III. 30 and starting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11145 from the <I>are</I> instead of the chord), III. 10 (proved by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11146 of III. 9), III. 13 (a proof preceded by a lemma to the effect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11147 that a straight line cannot meet a circle in more than two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11148 points).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11149 <p>A class of alternative proof is (<I>b</I>) that which is intended to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11150 meet a particular objection (<G>e)/nstasis</G>) which had been or might
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11151 be raised to Euclid's constructions. Thus in certain cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11152 Heron avoids <I>producing</I> a certain straight line, where Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11153 produces it, the object being to meet the objection of one who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11154 should deny our right to assume that there is <I>any space
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11155 available.</I> Of this class are his proofs of I. 11, 20 and his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11156 note on I. 16. Similarly in I. 48 he supposes the right-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11157 <pb n=312><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11158 triangle which is constructed to be constructed on the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11159 side of the common side as the given triangle is.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11160 <p>A third class (<I>c</I>) is that which avoids <I>reductio ad absurdum,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11161 e.g. a direct proof of I. 19 (for which he requires and gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11162 a preliminary lemma) and of I. 25.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11163 <p>(4) Heron supplies certain <I>converses</I> of Euclid's propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11164 e.g. of II. 12, 13 and VIII. 27.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11165 <p>(5) A few additions to, and extensions of, Euclid's propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11166 are also found. Some are unimportant, e.g. the construction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11167 of isosceles and scalene triangles in a note on I. 1 and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11168 construction of <I>two</I> tangents in III. 17. The most important
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11169 extension is that of III. 20 to the case where the angle at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11170 circumference is greater than a right angle, which gives an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11171 easy way of proving the theorem of III. 22. Interesting also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11172 are the notes on I. 37 (on I. 24 in Proclus), where Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11173 proves that two triangles with two sides of the one equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11174 to two sides of the other and with the included angles <I>supple-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11175 mentary</I> are equal in area, and compares the areas where the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11176 sum of the included angles (one being supposed greater than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11177 the other) is less or greater than two right angles, and on I. 47,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11178 where there is a proof (depending on preliminary lemmas) of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11179 the fact that, in the figure of Euclid's proposition (see next
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11180 page), the straight lines <I>AL, BG, CE</I> meet in a point. This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11181 last proof is worth giving. First come the lemmas.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11182 <p>(1) If in a triangle <I>ABC</I> a straight line <I>DE</I> be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11183 parallel to the base <I>BC</I> cutting the sides <I>AB, AC</I> or those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11184 sides produced in <I>D, E,</I> and if <I>F</I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11185 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11186 middle point of <I>BC,</I> then the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11187 <I>AF</I> (produced if necessary) will also bisect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11188 <I>DE.</I> (<I>HK</I> is drawn through <I>A</I> parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11189 <I>DE,</I> and <I>HDL, KEM</I> through <I>D, E</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11190 to <I>AF</I> meeting the base in <I>L, M</I> respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11191 tively. Then the triangles <I>ABF, AFC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11192 between the same parallels are equal. So are the triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11193 <I>DBF, EFC.</I> Therefore the differences, the triangles <I>ADF,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11194 AEF,</I> are equal and so therefore are the parallelograms <I>HF,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11195 KF.</I> Therefore <MATH><I>LF</I>=<I>FM,</I></MATH> or <MATH><I>DG</I>=<I>GE.</I></MATH>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11196 <p>(2) is the converse of Eucl. I. 43. If a parallelogram is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11197 <pb n=313><head>GEOMETRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11198 cut into four others <I>ADGE, DF, FGCB, CE</I>, so that <I>DF, CE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11199 are equal, the common vertex <I>G</I> will lie on the diagonal <I>AB.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11200 <p>Heron produces <I>AG</I> to meet <I>CF</I> in <I>H,</I> and then proves that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11201 <I>AHB</I> is a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11202 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11203 <p>Since <I>DF, CE</I> are equal, so are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11204 the triangles <I>DGF, ECG.</I> Adding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11205 the triangle <I>GCF,</I> we have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11206 triangles <I>ECF, DCF</I> equal, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11207 <I>DE, CF</I> are parallel.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11208 <p>But (by I. 34, 29, 26) the tri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11209 angles <I>AKE, GKD</I> are congruent,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11210 so that <MATH><I>EK</I>=<I>KD</I></MATH>; and by lemma (1) it follows that <MATH><I>CH</I>=<I>HF.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11211 <p>Now, in the triangles <I>FHB, CHG,</I> two sides (<I>BF, FH</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11212 <I>GC, CH</I>) and the included angles are equal; therefore the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11213 triangles are congruent, and the angles <I>BHF, GHC</I> are equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11214 <p>Add to each the angle <I>GHF,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11215 <MATH>&angle;<I>BHF</I>+&angle;<I>FHG</I>=&angle;<I>CHG</I>+&angle;<I>GHF</I>=two right angles.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11216 <p>To prove his substantive proposition Heron draws <I>AKL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11217 perpendicular to <I>BC,</I> and joins <I>EC</I> meeting <I>AK</I> in <I>M.</I> Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11218 we have only to prove that <I>BMG</I> is a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11219 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11220 <p>Complete the parallelogram <I>FAHO,</I> and draw the diagonals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11221 <I>OA, FH</I> meeting in <I>Y.</I> Through <I>M</I> draw <I>PQ, SR</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11222 respectively to <I>BA, AC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11223 <pb n=314><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11224 <p>Now the triangles <I>FAH</I>, <I>BAC</I> are equal in all respects;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11225 therefore <MATH>&angle;<I>HFA</I>=&angle;<I>ABC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11226 =&angle;<I>CAK</I></MATH> (since <I>AK</I> is at right angles to <I>BC</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11227 <p>But, the diagonals of the rectangle <I>FH</I> cutting one another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11228 in <I>Y</I>, we have <I>FY</I>=<I>YA</I> and &angle;<I>HFA</I>=&angle;<I>OAF</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11229 therefore &angle;<I>OAF</I>=&angle;<I>CAK</I>, and <I>OA</I> is in a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11230 with <I>AKL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11231 <p>Therefore, <I>OM</I> being the diagonal of <I>SQ</I>, <I>SA</I>=<I>AQ</I>, and, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11232 we add <I>AM</I> to each, <I>FM</I>=<I>MH.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11233 <p>Also, since <I>EC</I> is the diagonal of <I>FN</I>, <I>FM</I>=<I>MN.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11234 <p>Therefore the parallelograms <I>MH</I>, <I>MN</I> are equal; and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11235 hence, by the preceding lemma, <I>BMG</I> is a straight line. Q.E.D.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11236 <C>(<G>b</G>) The <I>Definitions.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11237 <p>The elaborate collection of <I>Definitions</I> is dedicated to one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11238 Dionysius in a preface to the following effect:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11239 <p>‘In setting out for you a sketch, in the shortest possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11240 form, of the technical terms premised in the elements of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11241 geometry, I shall take as my point of departure, and shall
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11242 base my whole arrangement upon, the teaching of Euclid, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11243 author of the elements of theoretical geometry; for by this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11244 means I think that I shall give you a good general under-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11245 standing not only of Euclid's doctrine but of many other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11246 works in the domain of geometry. I shall begin then with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11247 the <I>point.</I>’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11248 <p>He then proceeds to the definitions of the point, the line,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11249 the different sorts of lines, straight, circular, ‘curved’ and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11250 ‘spiral-shaped’ (the Archimedean spiral and the cylindrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11251 helix), Defs. 1-7; surfaces, plane and not plane, solid body,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11252 Defs. 8-11; angles and their different kinds, plane, solid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11253 rectilinear and not rectilinear, right, acute and obtuse angles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11254 Defs. 12-22; figure, boundaries of figure, varieties of figure,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11255 plane, solid, composite (of homogeneous or non-homogeneous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11256 parts) and incomposite, Defs. 23-6. The incomposite plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11257 figure is the circle, and definitions follow of its parts, segments
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11258 (which are composite of non-homogeneous parts), the semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11259 circle, the <G>a(yi/s</G> (less than a semicircle), and the segment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11260 greater than a semicircle, angles in segments, the sector,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11261 <pb n=315><head>THE <I>DEFINITIONS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11262 ‘concave’ and ‘convex’, lune, garland (these last two are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11263 composite of homogeneous parts) and <I>axe</I> (<G>pe/lekus</G>), bounded by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11264 four circular arcs, two concave and two convex, Defs. 27-38.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11265 Rectilineal figures follow, the various kinds of triangles and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11266 of quadrilaterals, the gnomon in a parallelogram, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11267 gnomon in the more general sense of the figure which added
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11268 to a given figure makes the whole into a similar figure,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11269 polygons, the parts of figures (side, diagonal, height of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11270 triangle), perpendicular, parallels, the three figures which will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11271 fill up the space round a point, Defs. 39-73. Solid figures are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11272 next classified according to the surfaces bounding them, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11273 lines on surfaces are divided into (1) simple and circular,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11274 (2) mixed, like the conic and spiric curves, Defs. 74, 75. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11275 sphere is then defined, with its parts, and stated to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11276 the figure which, of all figures having the same surface, is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11277 greatest in content, Defs. 76-82. Next the cone, its different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11278 species and its parts are taken up, with the distinction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11279 between the three conics, the section of the acute-angled cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11280 (‘by some also called <I>ellipse</I>’) and the sections of the right-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11281 angled and obtuse-angled cones (also called <I>parabola</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11282 <I>hyperbola</I>), Defs. 83-94; the cylinder, a section in general,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11283 the <I>spire</I> or <I>tore</I> in its three varieties, open, continuous (or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11284 just closed) and ‘crossing-itself’, which respectively have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11285 sections possessing special properties, ‘square rings’ which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11286 are cut out of cylinders (i.e. presumably rings the cross-section
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11287 of which through the centre is two squares), and various other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11288 figures cut out of spheres or mixed surfaces, Defs. 95-7;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11289 rectilineal solid figures, pyramids, the five regular solids, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11290 semi-regular solids of Archimedes two of which (each with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11291 fourteen faces) were known to Plato, Defs. 98-104; prisms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11292 of different kinds, parallelepipeds, with the special varieties,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11293 the cube, the <I>beam</I>, <G>doko/s</G> (length longer than breadth and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11294 depth, which may be equal), the <I>brick</I>, <G>plinqi/s</G> (length less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11295 than breadth and depth), the <G>sfhni/skos</G> or <G>bwmi/skos</G> with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11296 length, breadth and depth unequal, Defs. 105-14.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11297 <p>Lastly come definitions of relations, equality of lines, sur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11298 faces, and solids respectively, similarity of figures, ‘reciprocal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11299 figures’, Defs. 115-18; indefinite increase in magnitude,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11300 parts (which must be homogeneous with the wholes, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11301 e.g. the horn-like angle is not a part or submultiple of a right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11302 <pb n=316><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11303 or any angle), multiples, Defs. 119-21; proportion in magni-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11304 tudes, what magnitudes can have a ratio to one another,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11305 magnitudes in the same ratio or magnitudes in proportion,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11306 definition of greater ratio, Defs. 122-5; transformation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11307 ratios (<I>componendo, separando, convertendo, alternando, in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11308 vertendo</I> and <I>ex aequali</I>), Defs. 126-7; commensurable and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11309 incommensurable magnitudes and straight lines, Defs. 128,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11310 129. There follow two tables of measures, Defs. 130-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11311 <p>The <I>Definitions</I> are very valuable from the point of view of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11312 the historian of mathematics, for they give the different alter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11313 native definitions of the fundamental conceptions; thus we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11314 find the Archimedean ‘definition’ of a straight line, other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11315 definitions which we know from Proclus to be due to Apol-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11316 lonius, others from Posidonius, and so on. No doubt the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11317 collection may have been recast by some editor or editors
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11318 after Heron's time, but it seems, at least in substance, to go
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11319 back to Heron or earlier still. So far as it contains original
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11320 definitions of Posidonius, it cannot have been compiled earlier
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11321 than the first century B.C.; but its content seems to belong in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11322 the main to the period before the Christian era. Heiberg
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11323 adds to his edition of the <I>Definitions</I> extracts from Heron's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11324 Geometry, postulates and axioms from Euclid, extracts from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11325 Geminus on the classification of mathematics, the principles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11326 of geometry, &c., extracts from Proclus or some early collec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11327 tion of scholia on Euclid, and extracts from Anatolius and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11328 Theon of Smyrna, which followed the actual definitions in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11329 manuscripts. These various additions were apparently collected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11330 by some Byzantine editor, perhaps of the eleventh century.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11331 <C>Mensuration.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11332 <C>The <I>Metrica, Geometrica, Stereometrica, Geodaesia,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11333 Mensurae.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11334 <p>We now come to the mensuration of Heron. Of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11335 different works under this head the <I>Metrica</I> is the most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11336 important from our point of view because it seems, more than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11337 any of the others, to have preserved its original form. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11338 also more fundamental in that it gives the theoretical basis of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11339 the formulae used, and is not a mere application of rules to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11340 particular examples. It is also more akin to theory in that it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11341 <pb n=317><head>MENSURATION</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11342 does not use concrete measures, but simple numbers or units
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11343 which may then in particular cases be taken to be feet, cubits,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11344 or any other unit of measurement. Up to 1896, when a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11345 manuscript of it was discovered by R. Schöne at Constanti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11346 nople, it was only known by an allusion to it in Eutocius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11347 (on Archimedes's <I>Measurement of a Circle</I>), who states that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11348 the way to obtain an approximation to the square root of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11349 a non-square number is shown by Heron in his <I>Metrica</I>, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11350 well as by Pappus, Theon, and others who had commented on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11351 the <I>Syntaxis</I> of Ptolemy.<note>Archimedes, vol. iii, p. 232. 13-17.</note> Tannery<note>Tannery, <I>Mémoires scientifiques</I>, ii, 1912, pp. 447-54.</note> had already in 1894
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11352 discovered a fragment of Heron's <I>Metrica</I> giving the particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11353 rule in a Paris manuscript of the thirteenth century contain-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11354 ing Prolegomena to the <I>Syntaxis</I> compiled presumably from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11355 the commentaries of Pappus and Theon. Another interesting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11356 difference between the <I>Metrica</I> and the other works is that in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11357 the former the Greek way of writing fractions (which is our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11358 method) largely preponderates, the Egyptian form (which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11359 expresses a fraction as the sum of diminishing submultiples)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11360 being used comparatively rarely, whereas the reverse is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11361 case in the other works.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11362 <p>In view of the greater authority of the <I>Metrica</I>, we shall
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11363 take it as the basis of our account of the mensuration, while
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11364 keeping the other works in view. It is desirable at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11365 outset to compare broadly the contents of the various collec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11366 tions. Book I of the <I>Metrica</I> contains the mensuration of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11367 squares, rectangles and triangles (chaps. 1-9), parallel-trapezia,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11368 rhombi, rhomboids and quadrilaterals with one angle right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11369 (10-16), regular polygons from the equilateral triangle to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11370 regular dodecagon (17-25), a ring between two concentric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11371 circles (26), segments of circles (27-33), an ellipse (34), a para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11372 bolic segment (35), the surfaces of a cylinder (36), an isosceles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11373 cone (37), a sphere (38) and a segment of a sphere (39).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11374 Book II gives the mensuration of certain solids, the solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11375 content of a cone (chap. 1), a cylinder (2), rectilinear solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11376 figures, a parallelepiped, a prism, a pyramid and a frustum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11377 &c. (3-8), a frustum of a cone (9, 10), a sphere and a segment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11378 of a sphere (11, 12), a <I>spire</I> or <I>tore</I> (13), the section of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11379 cylinder measured in Archimedes's <I>Method</I> (14), and the solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11380 <pb n=318><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11381 formed by the intersection of two cylinders with axes at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11382 angles inscribed in a cube, also measured in the <I>Method</I> (15),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11383 the five regular solids (16-19). Book III deals with the divi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11384 sion of figures into parts having given ratios to one another,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11385 first plane figures (1-19), then solids, a pyramid, a cone and a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11386 frustum, a sphere (20-3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11387 <p>The <I>Geometria</I> or <I>Geometrumena</I> is a collection based upon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11388 Heron, but not his work in its present form. The addition of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11389 a theorem due to Patricius<note><I>Geometrica</I>, 21 26 (vol. iv, p. 386. 23).</note> and a reference to him in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11390 <I>Stereometrica</I> (I. 22) suggest that Patricius edited both works,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11391 but the date of Patricius is uncertain. Tannery identifies
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11392 him with a mathematical professor of the tenth century,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11393 Nicephorus Patricius; if this is correct, he would be contem-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11394 porary with the Byzantine writer (erroneously called Heron)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11395 who is known to have edited genuine works of Heron, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11396 indeed Patricius and the anonymous Byzantine might be one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11397 and the same person. The mensuration in the <I>Geometry</I> has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11398 reference almost entirely to the same figures as those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11399 measured in Book I of the <I>Metrica</I>, the difference being that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11400 in the <I>Geometry</I> (1) the rules are not explained but merely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11401 applied to examples, (2) a large number of numerical illustra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11402 tions are given for each figure, (3) the Egyptian way of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11403 writing fractions as the sum of submultiples is followed,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11404 (4) lengths and areas are given in terms of particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11405 measures, and the calculations are lengthened by a consider-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11406 able amount of conversion from one measure into another.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11407 The first chapters (1-4) are of the nature of a general intro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11408 duction, including certain definitions and ending with a table
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11409 of measures. Chaps. 5-99, Hultsch (=5-20, 14, Heib.), though
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11410 for the most part corresponding in content to <I>Metrica</I> I,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11411 seem to have been based on a different collection, because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11412 chaps. 100-3 and 105 (=21, 1-25, 22, 3-24, Heib.) are clearly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11413 modelled on the <I>Metrica</I>, and 101 is headed ‘A definition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11414 (really ‘measurement’) of a circle in another book of Heron’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11415 Heiberg transfers to the <I>Geometrica</I> a considerable amount of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11416 the content of the so-called <I>Liber Geeponicus</I>, a badly ordered
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11417 collection consisting to a large extent of extracts from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11418 other works. Thus it begins with 41 definitions identical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11419 with the same number of the <I>Definitiones.</I> Some sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11420 <pb n=319><head>MENSURATION</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11421 Heiberg puts side by side with corresponding sections of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11422 <I>Geometrica</I> in parallel columns; others he inserts in suitable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11423 places; sections 78. 79 contain two important problems in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11424 indeterminate analysis (=<I>Geom.</I> 24, 1-2, Heib.). Heiberg
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11425 adds, from the Constantinople manuscript containing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11426 <I>Metrica</I>, eleven more sections (chap. 24, 3-13) containing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11427 indeterminate problems, and other sections (chap. 24, 14-30 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11428 37-51) giving the mensuration, mainly, of figures inscribed in or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11429 circumscribed to others, e.g. squares or circles in triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11430 circles in squares, circles about triangles, and lastly of circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11431 and segments of circles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11432 <p>The <I>Stereometrica</I> I has at the beginning the title <G>*ei)sa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11433 gwgai\ tw=n stereometroume/nwn *(/hrwnos</G> but, like the <I>Geometrica</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11434 seems to have been edited by Patricius. Chaps. 1-40 give the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11435 mensuration of the geometrical solid figures, the sphere, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11436 cone, the frustum of a cone, the obelisk with circular base,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11437 the cylinder, the ‘pillar’, the cube, the <G>sfhni/skos</G> (also called
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11438 <G>o)/nux</G>), the <G>mei/ouron proeskarifeume/non</G>, pyramids, and frusta.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11439 Some portions of this section of the book go back to Heron;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11440 thus in the measurement of the sphere chap. 1=<I>Metrica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11441 II. 11, and both here and elsewhere the ordinary form of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11442 fractions appears. Chaps. 41-54 measure the contents of cer-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11443 tain buildings or other constructions, e.g. a theatre, an amphi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11444 theatre, a swimming-bath, a well, a ship, a wine-butt, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11445 the like.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11446 <p>The second collection, <I>Stereometrica</I> II, appears to be of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11447 Byzantine origin and contains similar matter to <I>Stereometrica</I> I,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11448 parts of which are here repeated. Chap. 31 (27, Heib.) gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11449 the problem of Thales, to find the height of a pillar or a tree
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11450 by the measurement of shadows; the last sections measure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11451 various pyramids, a prism, a <G>bwmi/skos</G> (little altar).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11452 <p>The <I>Geodaesia</I> is not an independent work, but only con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11453 tains extracts from the <I>Geometry</I>; thus chaps. 1-16=<I>Geom.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11454 5-31, Hultsch (=5, 2-12, 32, Heib.); chaps. 17-19 give the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11455 methods of finding, in any scalene triangle the sides of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11456 are given, the segments of the base made by the perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11457 from the vertex, and of finding the area direct by the well-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11458 known ‘formula of Heron’; i.e. we have here the equivalent of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11459 <I>Metrica</I> I. 5-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11460 <p>Lastly, the <G>metrh/seis</G>, or <I>Mensurae</I>, was attributed to Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11461 <pb n=320><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11462 in an Archimedes manuscript of the ninth century, but can-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11463 not in its present form be due to Heron, although portions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11464 it have points of contact with the genuine works. Sects. 2-27
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11465 measure all sorts of objects, e.g. stones of different shapes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11466 a pillar, a tower, a theatre, a ship, a vault, a hippodrome; but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11467 sects. 28-35 measure geometrical figures, a circle and segments
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11468 of a circle (cf. <I>Metrica</I> I), and sects. 36-48 on spheres, segments
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11469 of spheres, pyramids, cones and frusta are closely connected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11470 with <I>Stereom.</I> I and <I>Metrica</I> II; sects. 49-59, giving the men-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11471 suration of receptacles and plane figures of various shapes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11472 seem to have a different origin. We can now take up the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11473 <C>Contents of the <I>Metrica.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11474 <C>Book I. Measurement of Areas.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11475 <p>The preface records the tradition that the first geometry
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11476 arose out of the practical necessity of measuring and dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11477 tributing land (whence the name ‘geometry’), after which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11478 extension to three dimensions became necessary in order to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11479 measure solid bodies. Heron then mentions Eudoxus and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11480 Archimedes as pioneers in the discovery of difficult measure-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11481 ments, Eudoxus having been the first to prove that a cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11482 is three times the cone on the same base and of equal height,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11483 and that circles are to one another as the squares on their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11484 diameters, while Archimedes first proved that the surface of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11485 a sphere is equal to four times the area of a great circle in it,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11486 and the volume two-thirds of the cylinder circumscribing it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11487 <C>(<I>a</I>) <I>Area of scalene triangle.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11488 <p>After the easy cases of the rectangle, the right-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11489 triangle and the isosceles triangle, Heron gives two methods
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11490 of finding the area of a scalene triangle (acute-angled or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11491 obtuse-angled) when the lengths of the three sides are given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11492 <p>The first method is based on Eucl. II. 12 and 13. If <I>a</I>, <I>b</I>, <I>c</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11493 be the sides of the triangle opposite to the angles <I>A</I>, <I>B</I>, <I>C</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11494 respectively, Heron observes (chap. 4) that any angle, e.g. <I>C</I>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11495 acute, right or obtuse according as <MATH><I>c</I><SUP>2</SUP><=or><I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP></MATH>, and this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11496 is the criterion determining which of the two propositions is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11497 applicable. The method is directed to determining, first the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11498 segments into which any side is divided by the perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11499 <pb n=321><head>AREA OF SCALENE TRIANGLE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11500 from the opposite vertex, and thence the length of the per-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11501 pendicular itself. We have, in the cases of the triangle acute-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11502 angled at <I>C</I> and the triangle obtuse-angled at <I>C</I> respectively,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11503 <MATH><I>c</I><SUP>2</SUP>=<I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>&mnplus;2<I>a</I>.<I>CD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11504 or <MATH><I>CD</I>={(<I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>)-<I>c</I><SUP>2</SUP>}/2<I>a</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11505 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11506 whence <MATH><I>AD</I><SUP>2</SUP>(=<I>b</I><SUP>2</SUP>-<I>CD</I><SUP>2</SUP>)</MATH> is found, so that we know the area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11507 <MATH>(=1/2<I>a</I>.<I>AD</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11508 <p>In the cases given in <I>Metrica</I> I. 5, 6 the sides are (14, 15, 13)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11509 and (11, 13, 20) respectively, and <I>AD</I> is found to be rational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11510 (=12). But of course both <I>CD</I> (or <I>BD</I>) and <I>AD</I> may be surds,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11511 in which case Heron gives approximate values. Cf. <I>Geom.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11512 53, 54, Hultsch (15, 1-4, Heib.), where we have a triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11513 in which <MATH><I>a</I>=8, <I>b</I>=4, <I>c</I>=6</MATH>, so that <MATH><I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>=44</MATH> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11514 <MATH><I>CD</I>=44/16=2 1/2 1/4</MATH>. Thus <MATH><I>AD</I><SUP>2</SUP>=16-(2 1/2 1/4)<SUP>2</SUP>=16-7 1/2 1/(16)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11515 =8 1/4 1/8 1/(16)</MATH>, and <MATH><I>AD</I>=√(8 1/4 1/8 1/(16))=2 2/3 1/4</MATH> approximately, whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11516 the area <MATH>=4x2 2/3 1/4=11 2/3</MATH>. Heron then observes that we get
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11517 a nearer result still if we multiply <I>AD</I><SUP>2</SUP> by (1/2<I>a</I>)<SUP>2</SUP> before
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11518 extracting the square root, for the area is then <MATH>√(16x8 1/4 1/8 1/(16))</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11519 or √(135), which is very nearly 11 1/2 1/(14) 1/(21) or 11 (13)/(21).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11520 <p>So in <I>Metrica</I> I. 9, where the triangle is 10, 8, 12 (10 being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11521 the base), Heron finds the perpendicular to be √(63), but he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11522 obtains the area as <MATH>√(1/4<I>AD</I><SUP>2</SUP>.<I>BC</I><SUP>2</SUP>)</MATH>, or √(1575), while observing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11523 that we <I>can</I>, of course, take the approximation to √(63), or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11524 7 1/2 1/4 1/8 1/(16), and multiply it by half 10, obtaining 39 1/2 1/8 1/(16) as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11525 the area.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11526 <p><I>Proof of the formula</I> <MATH>&utri;=√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11527 <p>The second method is that known as the ‘formula of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11528 Heron’, namely, in our notation, <MATH>&utri;=√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11529 The proof of the formula is given in <I>Metrica</I> I. 8 and also in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11530 <pb n=322><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11531 chap. 30 of the <I>Dioptra</I>; but it is now known (from Arabian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11532 sources) that the proposition is due to Archimedes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11533 <p>Let the sides of the triangle <I>ABC</I> be given in length.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11534 <p>Inscribe the circle <I>DEF</I>, and let <I>O</I> be the centre.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11535 <p>Join <I>AO</I>, <I>BO</I>, <I>CO</I>, <I>DO</I>, <I>EO</I>, <I>FO.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11536 <p>Then <MATH><I>BC</I>.<I>OD</I>=2&utri;<I>BOC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11537 <I>CA</I>.<I>OE</I>=2&utri;<I>COA</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11538 <I>AB</I>.<I>OF</I>=2&utri;<I>AOB</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11539 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11540 whence, by addition,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11541 <MATH><I>p</I>.<I>OD</I>=2&utri;<I>ABC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11542 where <I>p</I> is the perimeter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11543 <p>Produce <I>CB</I> to <I>H</I>, so that <I>BH</I>=<I>AF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11544 <p>Then, since <MATH><I>AE</I>=<I>AF</I>, <I>BF</I>=<I>BD</I></MATH>, and <I>CE</I>=<I>CD</I>, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11545 <MATH><I>CH</I>=1/2<I>p</I>=<I>s.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11546 <p>Therefore <MATH><I>CH</I>.<I>OD</I>=&utri;<I>ABC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11547 <p>But <I>CH</I>.<I>OD</I> is the ‘side’ of the product <I>CH</I><SUP>2</SUP>.<I>OD</I><SUP>2</SUP>, i.e.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11548 <MATH>√(<I>CH</I><SUP>2</SUP>.<I>OD</I><SUP>2</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11549 so that <MATH>(&utri;<I>ABC</I>)<SUP>2</SUP>=<I>CH</I><SUP>2</SUP>.<I>OD</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11550 <pb n=323><head>PROOF OF THE ‘FORMULA OF HERON’</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11551 <p>Draw <I>OL</I> at right angles to <I>OC</I> cutting <I>BC</I> in <I>K</I>, and <I>BL</I> at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11552 right angles to <I>BC</I> meeting <I>OL</I> in <I>L.</I> Join <I>CL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11553 <p>Then, since each of the angles <I>COL</I>, <I>CBL</I> is right, <I>COBL</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11554 a quadrilateral in a circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11555 <p>Therefore <MATH>&angle;<I>COB</I>+&angle;<I>CLB</I>=2<I>R.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11556 <p>But <MATH>&angle;<I>COB</I>+&angle;<I>AOF</I>=2<I>R</I></MATH>, because <I>AO</I>, <I>BO</I>, <I>CO</I> bisect the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11557 angles round <I>O</I>, and the angles <I>COB</I>, <I>AOF</I> are together equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11558 to the angles <I>AOC</I>, <I>BOF</I>, while the sum of all four angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11559 is equal to 4<I>R.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11560 <p>Consequently <MATH>&angle;<I>AOF</I>=&angle;<I>CLB.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11561 <p>Therefore the right-angled triangles <I>AOF</I>, <I>CLB</I> are similar;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11562 therefore <MATH><I>BC</I>:<I>BL</I>=<I>AF</I>:<I>FO</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11563 =<I>BH</I>:<I>OD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11564 and, alternately, <MATH><I>CB</I>:<I>BH</I>=<I>BL</I>:<I>OD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11565 =<I>BK</I>:<I>KD</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11566 whence, <I>componendo</I>, <MATH><I>CH</I>:<I>HB</I>=<I>BD</I>:<I>DK.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11567 <p>It follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11568 <MATH><I>CH</I><SUP>2</SUP>:<I>CH</I>.<I>HB</I>=<I>BD</I>.<I>DC</I>:<I>CD</I>.<I>DK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11569 =<I>BD</I>.<I>DC</I>:<I>OD</I><SUP>2</SUP></MATH>, since the angle <I>COK</I> is right.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11570 <p>Therefore <MATH>(&utri;<I>ABC</I>)<SUP>2</SUP>=<I>CH</I><SUP>2</SUP>.<I>OD</I><SUP>2</SUP></MATH> (from above)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11571 <MATH>=<I>CH</I>.<I>HB</I>.<I>BD</I>.<I>DC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11572 =<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11573 <C>(<G>b</G>) <I>Method of approximating to the square root of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11574 a non-square number.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11575 <p>It is à propos of the triangle 7, 8, 9 that Heron gives the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11576 important statement of his method of approximating to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11577 value of a surd, which before the discovery of the passage
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11578 of the <I>Metrica</I> had been a subject of unlimited conjecture
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11579 as bearing on the question how Archimedes obtained his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11580 approximations to √(3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11581 <p>In this case <MATH><I>s</I>=12, <I>s</I>-<I>a</I>=5, <I>s</I>-<I>b</I>=4, <I>s</I>-<I>c</I>=3</MATH>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11582 <MATH>&utri;=√(12.5.4.3)=√(720)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11583 <pb n=324><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11584 <p>‘Since’, says Heron,<note><I>Metrica</I>, i. 8, pp. 18, 22-20. 5.</note> ‘720 has not its side rational, we can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11585 obtain its side within a very small difference as follows. Since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11586 the next succeeding square number is 729, which has 27 for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11587 its side, divide 720 by 27. This gives 26 2/3. Add 27 to this,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11588 making 53 2/3, and take half of this or 26 1/2 1/3. The side of 720
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11589 will therefore be very nearly 26 1/2 1/3. In fact, if we multiply
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11590 26 1/2 1/3 by itself, the product is 720 1/36, so that the difference (in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11591 the square) is 1/36.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11592 <p>‘If we desire to make the difference still smaller than 1/36, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11593 shall take 720 1/36 instead of 729 [or rather we should take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11594 26 1/2 1/3 instead of 27], and by proceeding in the same way we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11595 shall find that the resulting difference is much less than 1/36.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11596 <p>In other words, if we have a non-square number <I>A</I>, and <I>a</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11597 is the nearest square number to it, so that <MATH><I>A</I>=<I>a</I><SUP>2</SUP>±<I>b</I></MATH>, then we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11598 have, as the first approximation to √(<I>A</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11599 <MATH><G>a</G><SUB>1</SUB>=1/2(<I>a</I>+<I>A</I>/<I>a</I>)</MATH>; (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11600 for a second approximation we take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11601 <MATH><G>a</G><SUB>2</SUB>=1/2(<G>a</G><SUB>1</SUB>+<I>A</I>/(<G>a</G><SUB>1</SUB>))</MATH>, (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11602 and so on.<note>The method indicated by Heron was known to Barlaam and Nicolas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11603 Rhabdas in the fourteenth century. The equivalent of it was used by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11604 Luca Paciuolo (fifteenth-sixteenth century), and it was known to the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11605 Italian algebraists of the sixteenth century. Thus Luca Paciuolo gave
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11606 2 1/2, 2 9/(20) and 2 (881)/(1960) as successive approximations to √(6). He obtained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11607 the first as <MATH>2+2/(2.2)</MATH>, the second as <MATH>2 1/2-((2 1/2)<SUP>2</SUP>-6)/(2.2 1/2)</MATH>, and the third as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11608 <MATH>(49)/(20)-(((49)/(20))<SUP>2</SUP>-6)/(2.(49)/(20))</MATH>. The above rule gives <MATH>1/2(2+6/2)=2 1/2, 1/2 (5/2+2/5)=2 9/(20),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11609 1/2((49)/(20)+(120)/(49))=2 (881)/(1960)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11610 <p>The formula of Heron was again put forward, in modern times, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11611 Buzengeiger as a means of accounting for the Archimedean approxima-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11612 tion to √(3), apparently without knowing its previous history. Bertrand
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11613 also stated it in a treatise on arithmetic (1851). The method, too, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11614 which Oppermann and Alexeieff sought to account for Archimedes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11615 approximations is in reality the same. The latter method depends on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11616 the formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11617 <MATH>1/2 (<G>a</G>+<G>b</G>):√(<G>ab</G>)=√(<G>ab</G>):(2<G>ab</G>)/(<G>a</G>+<G>b</G>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11618 Alexeieff separated <I>A</I> into two factors <I>a</I><SUB>0</SUB>, <I>b</I><SUB>0</SUB>, and pointed out that if, say,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11619 <MATH><I>a</I><SUB>0</SUB>>√(<I>A</I>)><I>b</I><SUB>0</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11620 then, <MATH>1/2 (<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)>√(<I>A</I>)>(2<I>A</I>)/(<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)</MATH> or <MATH>(2<I>a</I><SUB>0</SUB><I>b</I><SUB>0</SUB>)/(<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11621 and again, if <MATH>1/2(<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)=<I>a</I><SUB>1</SUB>, 2<I>A</I>/(<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)=<I>b</I><SUB>1</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11622 1/2(<I>a</I><SUB>1</SUB>+<I>b</I><SUB>1</SUB>)>√(<I>A</I>)>(2<I>A</I>)/(<I>a</I><SUB>1</SUB>+<I>b</I><SUB>1</SUB>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11623 and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11624 <p>Now suppose that, in Heron's formulae, we put <MATH><I>a</I>=<I>X</I><SUB>0</SUB>, <I>A</I>/<I>a</I>=<I>x</I><SUB>0</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11625 <G>a</G><SUB>1</SUB>=<I>X</I><SUB>1</SUB>, <I>A</I>/<G>a</G><SUB>1</SUB>=<I>x</I><SUB>1</SUB></MATH>, and so on. We then have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11626 <MATH><I>X</I><SUB>1</SUB>=1/2(<I>a</I>+<I>A</I>/<I>a</I>)=1/2(<I>X</I><SUB>0</SUB>+<I>x</I><SUB>0</SUB>), <I>x</I><SUB>1</SUB>=<I>A</I>/<I>X</I><SUB>1</SUB>=<I>A</I>/(1/2(<I>X</I><SUB>0</SUB>+<I>x</I><SUB>0</SUB>))</MATH> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11627 <MATH>(2<I>X</I><SUB>0</SUB><I>x</I><SUB>0</SUB>)/(<I>X</I><SUB>0</SUB>+<I>x</I><SUB>0</SUB>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11628 that is, <I>X</I><SUB>1</SUB>, <I>x</I><SUB>1</SUB> are, respectively, the arithmetic and harmonic means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11629 between <I>X</I><SUB>0</SUB>, <I>x</I><SUB>0</SUB>; <I>X</I><SUB>2</SUB>, <I>x</I><SUB>2</SUB> are the arithmetic and harmonic means between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11630 <I>X</I><SUB>1</SUB>, <I>x</I><SUB>1</SUB>, and so on, exactly as in Alexeieff's formulae.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11631 <p>Let us now try to apply the method to Archimedes's case, √(3), and we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11632 shall see to what extent it serves to give what we want. Suppose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11633 we begin with 3>√(3)>1. We then have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11634 <MATH>1/2 (3+1)>√(3)>3/1/2(3+1)</MATH>, or <MATH>2>√(3)>3/2</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11635 and from this we derive successively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11636 <MATH>7/4>√(3)>(12)/7, (97)/(56)>√(3)>(168)/(97), (18817)/(10864)>√(3)>(32592)/(18817)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11637 But, if we start from 5/3, obtained by the formula <MATH><I>a</I>+<I>b</I>/(2<I>a</I>+1)<√(<I>a</I><SUP>2</SUP>+<I>b</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11638 we obtain the following approximations by excess,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11639 <MATH>1/2(5/3+9/5)=(26)/(15), 1/2((26)/(15)+(45)/(26))=(1351)/(780)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11640 The second process then gives one of Archimedes's results, (1351)/(780), but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11641 neither of the two processes gives the other, (265)/(153), directly. The latter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11642 can, however, be obtained by using the formula that, if <MATH><I>a</I>/<I>b</I><<I>c</I>/<I>d</I></MATH>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11643 <MATH><I>a</I>/<I>b</I><(<I>ma</I>+<I>nc</I>)/(<I>mb</I>+<I>nd</I>)<<I>c</I>/<I>d</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11644 <p>For we can obtain (265)/(153) from (97)/(56) and (168)/(97) thus: <MATH>(97+168)/(56+97)=(265)/(153)</MATH>, or from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11645 (97)/(56) and 7/4 thus: <MATH>(11.97-7)/(11.56-4)=(1060)/(612)=(265)/(153)</MATH>; and so on. Or again (1351)/(780) can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11646 be obtained from (18817)/(10864) and (97)/(56) thus: <MATH>(18817+97)/(10864+56)=(18914)/(10920)=(1351)/(780)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11647 <p>The advantage of the method is that, as compared with that of con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11648 tinued fractions, it is a very rapid way of arriving at a close approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11649 mation. Günther has shown that the (<I>m</I>+1)th approximation obtained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11650 by Heron's formula is the 2<SUP><I>m</I></SUP>th obtained by continued fractions. (‘Die
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11651 quadratischen Irrationalitäten der Alten und deren Entwickelungs-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11652 methoden in <I>Abhandlungen zur Gesch. d. Math.</I> iv. 1882, pp. 83-6.)</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11653 <pb n=325><head>APPROXIMATIONS TO SURDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11654 <p>Substituting in (1) the value <I>a</I><SUP>2</SUP>±<I>b</I> for <I>A</I>, we obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11655 <MATH><G>a</G><SUB>1</SUB>=<I>a</I>±<I>b</I>/(2<I>a</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11656 <p>Heron does not seem to have used this formula with a nega-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11657 tive sign, unless in <I>Stereom.</I> I. 33 (34, Hultsch), where √(63)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11658 <pb n=326><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11659 is given as approximately 8-1/(16). In <I>Metrica</I> I. 9, as we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11660 have seen, √(63) is given as 7 1/2 1/4 1/8 1/(16), which was doubtless
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11661 obtained from the formula (1) as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11662 <MATH>1/2 (8+(63)/8)=1/2 (8+7 7/8)=7 1/2 1/4 1/8 1/(16)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11663 <p>The above seems to be the only <I>classical</I> rule which has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11664 been handed down for finding second and further approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11665 mations to the value of a surd. But, although Heron thus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11666 shows how to obtain a second approximation, namely by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11667 formula (2), he does not seem to make any direct use of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11668 this method himself, and consequently the question how the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11669 approximations closer than the first which are to be found in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11670 his works were obtained still remains an open one.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11671 <C>(<G>g</G>) <I>Quadrilaterals.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11672 <p>It is unnecessary to give in detail the methods of measuring
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11673 the areas of quadrilaterals (chaps. 11-16). Heron deals with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11674 the following kinds, the parallel-trapezium (isosceles or non-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11675 isosceles), the rhombus and rhomboid, and the quadrilateral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11676 which has one angle right and in which the four sides have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11677 given lengths. Heron points out that in the rhombus or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11678 rhomboid, and in the general case of the quadrilateral, it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11679 necessary to know a diagonal as well as the four sides. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11680 mensuration in all the cases reduces to that of the rectangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11681 and triangle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11682 <C>(<G>d</G>) <I>The regular polygons with 3, 4, 5, 6, 7, 8, 9, 10, 11,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11683 or 12 sides.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11684 <p>Beginning with the <I>equilateral triangle</I> (chap. 17), Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11685 proves that, if <I>a</I> be the side and <I>p</I> the perpendicular from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11686 a vertex on the opposite side, <MATH><I>a</I><SUP>2</SUP>:<I>p</I><SUP>2</SUP>=4:3</MATH>, whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11687 <MATH><I>a</I><SUP>4</SUP>:<I>p</I><SUP>2</SUP><I>a</I><SUP>2</SUP>=4:3=16:12</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11688 so that <MATH><I>a</I><SUP>4</SUP>:(&utri;<I>ABC</I>)<SUP>2</SUP>=16:3</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11689 and <MATH>(&utri;<I>ABC</I>)<SUP>2</SUP>=3/(16)<I>a</I><SUP>4</SUP></MATH>. In the particular case taken <I>a</I>=10
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11690 and <MATH>&utri;<SUP>2</SUP>=1875</MATH>, whence <MATH>&utri;=43 1/3</MATH> nearly.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11691 <p>Another method is to use an approximate value for √(3) in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11692 the formula √(3).<I>a</I><SUP>2</SUP>/4. This is what is done in the <I>Geometrica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11693 14 (10, Heib.), where we are told that the area is <MATH>(1/3+1/(10))<I>a</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11694 <pb n=327><head>THE REGULAR POLYGONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11695 now <MATH>1/3+1/(10)=(13)/(30)=1/4((26)/(15))</MATH>, so that the approximation used by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11696 Heron for √(3) is here (26)/(15). For the side 10, the method gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11697 the same result as above, for <MATH>(13)/(30).100=43 1/3</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11698 <p>The regular <I>pentagon</I> is next taken (chap. 18). Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11699 premises the following lemma.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11700 <p>Let <I>ABC</I> be a right-angled triangle, with the angle <I>A</I> equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11701 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11702 to 2/5<I>R.</I> Produce <I>AC</I> to <I>O</I> so that <I>CO</I>=<I>AC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11703 If now <I>AO</I> is divided in extreme and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11704 mean ratio, <I>AB</I> is equal to the greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11705 segment. (For produce <I>AB</I> to <I>D</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11706 <I>AD</I>=<I>AO</I>, and join <I>BO</I>, <I>DO.</I> Then, since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11707 <I>ADO</I> is isosceles and the angle at <MATH><I>A</I>=2/5<I>R</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11708 &angle;<I>ADO</I>=&angle;<I>AOD</I>=4/5<I>R</I></MATH>, and, from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11709 equality of the triangles <MATH><I>ABC</I>, <I>OBC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11710 &angle;<I>AOB</I>=&angle;<I>BAO</I>=2/5<I>R</I></MATH>. It follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11711 the triangle <I>ADO</I> is the isosceles triangle of Eucl. IV. 10, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11712 <I>AD</I> is divided in extreme and mean ratio in <I>B.</I>) Therefore,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11713 says Heron, <MATH>(<I>BA</I>+<I>AC</I>)<SUP>2</SUP>=5<I>AC</I><SUP>2</SUP></MATH>. [This is Eucl. XIII. 1.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11714 <p>Now, since <MATH>&angle;<I>BOC</I>=2/5<I>R</I></MATH>, if <I>BC</I> be produced to <I>E</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11715 <I>CE</I>=<I>BC</I>, <I>BE</I> subtends at <I>O</I> an angle equal to 4/5<I>R</I>, and there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11716 fore <I>BE</I> is the side of a regular pentagon inscribed in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11717 circle with <I>O</I> as centre and <I>OB</I> as radius. (This circle also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11718 passes through <I>D</I>, and <I>BD</I> is the side of a regular decagon in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11719 the same circle). If now <MATH><I>BO</I>=<I>AB</I>=<I>r</I>, <I>OC</I>=<I>p</I>, <I>BE</I>=<I>a</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11720 we have from above, <MATH>(<I>r</I>+<I>p</I>)<SUP>2</SUP>=5<I>p</I><SUP>2</SUP></MATH>, whence, since √(5) is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11721 approximately 9/4, we obtain approximately <MATH><I>r</I>=5/4<I>p</I></MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11722 <MATH>1/2<I>a</I>=3/4<I>p</I></MATH>, so that <MATH><I>p</I>=2/3<I>a</I></MATH> Hence <MATH>1/2<I>pa</I>=1/3<I>a</I><SUP>2</SUP></MATH>, and the area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11723 of the pentagon=5/3<I>a</I><SUP>2</SUP>. Heron adds that, if we take a closer
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11724 approximation to √(5) than 9/4, we shall obtain the area still
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11725 more exactly. In the <I>Geometry</I><note><I>Geom.</I> 102 (21, 14, Heib.).</note> the formula is given as (12)/7<I>a</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11726 <p>The regular <I>hexagon</I> (chap. 19) is simply 6 times the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11727 equilateral triangle with the same side. If &utri; be the area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11728 of the equilateral triangle with side <I>a</I>, Heron has proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11729 that <MATH>&utri;<SUP>2</SUP>=3/(16)<I>a</I><SUP>4</SUP></MATH> (<I>Metrica</I> I. 17), hence <MATH>(hexagon)<SUP>2</SUP>=(27)/4<I>a</I><SUP>4</SUP></MATH>. If,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11730 e.g. <MATH><I>a</I>=10, (hexagon)<SUP>2</SUP>=67500</MATH>, and (hexagon)=259 nearly.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11731 In the <I>Geometry</I><note><I>Ib.</I> 102 (21, 16, 17, Heib.).</note> the formula is given as (13)/5<I>a</I><SUP>2</SUP>, while ‘another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11732 book’ is quoted as giving <MATH>6(1/3+1/(10))<I>a</I><SUP>2</SUP></MATH>; it is added that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11733 latter formula, obtained from the area of the triangle, <MATH>(1/3+1/(10))<I>a</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11734 represents the more accurate procedure, and is fully set out by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11735 <pb n=328><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11736 Heron. As a matter of fact, however, <MATH>6(1/3+1/(10))=(13)/5</MATH> exactly,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11737 and only the <I>Metrica</I> gives the more accurate calculation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11738 <p>The regular <I>heptagon.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11739 <p>Heron assumes (chap. 20) that, if <I>a</I> be the side and <I>r</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11740 radius of the circumscribing circle, <I>a</I>=7/8<I>r</I>, being approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11741 mately equal to the perpendicular from the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11742 circle to the side of the regular hexagon inscribed in it (for 7/8
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11743 is the approximate value of 1/2√(3)). This theorem is quoted by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11744 Jordanus Nemorarius (d. 1237) as an ‘Indian rule’; he pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11745 bably obtained it from Ab&ubreve;'l Waf&amacr; (940-98). The <I>Metrica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11746 shows that it is of Greek origin, and, if Archimedes really
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11747 wrote a book on the heptagon in a circle, it may be due to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11748 him. If then <I>p</I> is the perpendicular from the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11749 circle on the side (<I>a</I>) of the inscribed heptagon, <MATH><I>r</I>/(1/2<I>a</I>)=8/3 1/2</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11750 or 16/7, whence <MATH><I>p</I><SUP>2</SUP>/(1/2<I>a</I>)<SUP>2</SUP>=(207)/(49)</MATH>, and <I>p</I>/1/2<I>a</I>=(approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11751 mately) 14 1/3/7 or 43/21. Consequently the area of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11752 heptagon <MATH>=7.1/2<I>pa</I>=7.(43)/(84)<I>a</I><SUP>2</SUP>=(43)/(12)<I>a</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11753 <p>The regular <I>octagon, decagon</I> and <I>dodecagon.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11754 <p>In these cases (chaps. 21, 23, 25) Heron finds <I>p</I> by drawing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11755 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11756 the perpendicular <I>OC</I> from <I>O</I>, the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11757 circumscribed circle, on a side <I>AB</I>, and then making
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11758 the angle <I>OAD</I> equal to the angle <I>AOD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11759 <p>For the octagon,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11760 <MATH>&angle;<I>ADC</I>=1/2<I>R</I></MATH>, and <MATH><I>p</I>=1/2<I>a</I>(1+√(2))=1/2<I>a</I>(1+(17)/(12))</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11761 or 1/2<I>a</I>.(29)/(12) approximately.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11762 <p>For the decagon,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11763 <MATH>&angle;<I>ADC</I>=2/5<I>R</I></MATH>, and <MATH><I>AD</I>:<I>DC</I>=5:4</MATH> nearly (see preceding page);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11764 hence <MATH><I>AD</I>:<I>AC</I>=5:3</MATH>, and <MATH><I>p</I>=1/2<I>a</I>(5/3+4/3)=3/2<I>a.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11765 <p>For the dodecagon,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11766 <MATH>&angle;<I>ADC</I>=1/3<I>R</I></MATH>, and <MATH><I>p</I>=1/2<I>a</I>(2+√(3))=1/2<I>a</I>(2+7/4)=(15)/8<I>a</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11767 approximately.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11768 <p>Accordingly <MATH><I>A</I><SUB>8</SUB>=(29)/6<I>a</I><SUP>2</SUP>, <I>A</I><SUB>10</SUB>=(15)/2<I>a</I><SUP>2</SUP>, <I>A</I><SUB>12</SUB>=(45)/4<I>a</I><SUP>2</SUP></MATH>, where <I>a</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11769 the side in each case.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11770 <p>The regular <I>enneagon</I> and <I>hendecagon.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11771 <p>In these cases (chaps. 22, 24) the Table of Chords (i.e.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11772 <pb n=329><head>THE REGULAR POLYGONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11773 presumably Hipparchus's Table) is appealed to. If <I>AB</I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11774 side (<I>a</I>) of an enneagon or hendecagon inscribed in a circle, <I>AC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11775 the diameter through <I>A</I>, we are told that the Table of Chords
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11776 gives 1/3 and 7/(25) as the respective approximate values of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11777 ratio <I>AB</I>/<I>AC.</I> The angles subtended at the centre <I>O</I> by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11778 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11779 side <I>AB</I> are 40° and 32 8/(11)° respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11780 tively, and Ptolemy's Table gives,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11781 as the chords subtended by angles of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11782 40° and 33° respectively, 41<SUP><I>p</I></SUP>2′33″
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11783 and 34<SUP><I>p</I></SUP>4′55″ (expressed in 120th
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11784 parts of the diameter); Heron's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11785 figures correspond to 40<SUP><I>p</I></SUP> and 33<SUP><I>p</I></SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11786 36′ respectively. For the <I>enneagon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11787 AC</I><SUP>2</SUP>=9<I>AB</I><SUP>2</SUP>, whence <I>BC</I><SUP>2</SUP>=8<I>AB</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11788 or approximately (289)/(36)<I>AB</I><SUP>2</SUP>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11789 <I>BC</I>=(17)/6<I>a</I>; therefore <MATH>(area of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11790 enneagon)=9/2.&utri;<I>ABC</I>=(51)/8<I>a</I><SUP>2</SUP></MATH>. For
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11791 the <MATH><I>hendecagon AC</I><SUP>2</SUP>=(625)/(49)<I>AB</I><SUP>2</SUP></MATH> and <MATH><I>BC</I><SUP>2</SUP>=(576)/(49)<I>AB</I><SUP>2</SUP></MATH>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11792 <I>BC</I>=(24)/7<I>a</I>, and area of hendecagon <MATH>=(11)/2.&utri;<I>ABC</I>=(66)/7<I>a</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11793 <p>An ancient formula for the ratio between the side of any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11794 regular polygon and the diameter of the circumscribing circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11795 is preserved in <I>Geëpon.</I> 147 sq. (= Pseudo-Dioph. 23-41),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11796 namely <MATH><I>d</I><SUB><I>n</I></SUB>=<I>n</I>(<I>a</I><SUB><I>n</I></SUB>)/3</MATH>. Now the ratio <I>na</I><SUB><I>n</I></SUB>/<I>d</I><SUB><I>n</I></SUB> tends to <G>p</G> as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11797 number (<I>n</I>) of sides increases, and the formula indicates a time
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11798 when <G>p</G> was generally taken as = 3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11799 <C>(<G>e</G>) <I>The Circle.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11800 <p>Coming to the circle (<I>Metrica</I> I. 26) Heron uses Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11801 medes's value for <G>p</G>, namely (22)/7, making the circumference of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11802 a circle (44)/7<I>r</I> and the area (11)/(14)<I>d</I><SUP>2</SUP>, where <I>r</I> is the radius and <I>d</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11803 diameter. It is here that he gives the more exact limits
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11804 for <G>p</G> which he says that Archimedes found in his work <I>On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11805 Plinthides and Cylinders</I>, but which are not convenient for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11806 calculations. The limits. as we have seen, are given in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11807 text as <MATH>(211875)/(67441)<<G>p</G><(197888)/(62351)</MATH>, and with Tannery's alteration to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11808 <MATH>(211872)/(67441)<<G>p</G><(195882)/(62351)</MATH> are quite satisfactory.<note>See vol. i, pp. 232-3.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11809 <pb n=330><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11810 <C>(<G>z</G>) <I>Segment of a circle.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11811 <p>According to Heron (<I>Metrica</I> I. 30) the ancients measured
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11812 the area of a segment rather inaccurately, taking the area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11813 to be <MATH>1/2(<I>b</I>+<I>h</I>)<I>h</I></MATH>, where <I>b</I> is the base and <I>h</I> the height. He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11814 conjectures that it arose from taking <G>p</G>=3, because, if we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11815 apply the formula to the semicircle, the area becomes 1/2.3<I>r</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11816 where <I>r</I> is the radius. Those, he says (chap. 31), who have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11817 investigated the area more accurately have added 1/(14) (1/2<I>b</I>)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11818 to the above formula, making it <MATH>1/2(<I>b</I>+<I>h</I>)<I>h</I>+1/(14)(1/2<I>b</I>)<SUP>2</SUP></MATH>, and this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11819 seems to correspond to the value 3 1/7 for <G>p</G>, since, when applied
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11820 to the semicircle, the formula gives <MATH>1/2(3<I>r</I><SUP>2</SUP>+1/7<I>r</I><SUP>2</SUP>)</MATH>. He adds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11821 that this formula should only be applied to segments of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11822 a circle less than a semicircle, and not even to all of these, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11823 only in cases where <I>b</I> is not greater than 3<I>h.</I> Suppose e.g.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11824 that <I>b</I>=60, <I>h</I>=1; in that case even <MATH>1/(14)(1/2<I>b</I>)<SUP>2</SUP>=1/(14).900=64 2/7</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11825 which is greater even than the parallelogram with 60, 1 as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11826 sides, which again is greater than the segment. Where there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11827 fore <I>b</I>>3<I>h</I>, he adopts another procedure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11828 <p>This is exactly modelled on Archimedes's quadrature of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11829 a segment of a parabola. Heron proves (<I>Metrica</I> I. 27-29, 32)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11830 that, if <I>ADB</I> be a segment of a circle, and <I>D</I> the middle point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11831 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11832 of the arc, and if the arcs <I>AD</I>, <I>DB</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11833 similarly bisected at <I>E</I>, <I>F</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11834 <MATH>&utri;<I>ADB</I><4(&utri;<I>AED</I>+&utri;<I>DFB</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11835 <p>Similarly, if the same construction be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11836 made for the segments <I>AED</I>, <I>BFD</I>, each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11837 of them is less than 4 times the sum of the two small triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11838 in the segments left over. It follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11839 (area of segmt. <MATH><I>ADB</I>)>&utri;<I>ADB</I>{1+1/4+(1/4)<SUP>2</SUP>+...}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11840 >4/3&utri;<I>ADB.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11841 ‘If therefore we measure the triangle, and add one-third of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11842 it, we shall obtain the area of the segment as nearly as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11843 possible.’ That is, for segments in which <I>b</I>>3<I>h</I>, Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11844 takes the area to be equal to that of the parabolic segment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11845 with the same base and height, or 2/3<I>bh.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11846 <p>In addition to these three formulae for <I>S</I>, the area of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11847 a segment, there are yet others, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11848 <MATH><I>S</I>=1/2(<I>b</I>+<I>h</I>)<I>h</I>(1+1/(21))</MATH>, <I>Mensurae</I> 29,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11849 <MATH><I>S</I>=1/2(<I>b</I>+<I>h</I>)<I>h</I>(1+1/(16))</MATH>, ” 31.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11850 <pb n=331><head>SEGMENT OF A CIRCLE</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11851 The first of these formulae is applied to a segment greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11852 than a semicircle, the second to a segment less than a semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11853 circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11854 <p>In the <I>Metrica</I> the area of a segment greater than a semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11855 circle is obtained by subtracting the area of the complementary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11856 segment from the area of the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11857 <p>From the <I>Geometrica</I><note>Cf. <I>Geom.</I>, 94, 95 (19. 2, 4, Heib.), 97. 4 (20. 7, Heib.).</note> we find that the circumference of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11858 segment less than a semicircle was taken to be <MATH>√(<I>b</I><SUP>2</SUP>+4<I>h</I><SUP>2</SUP>)+1/4<I>h</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11859 or alternatively <MATH>√(<I>b</I><SUP>2</SUP>+4<I>h</I><SUP>2</SUP>)+{√(<I>b</I><SUP>2</SUP>+4<I>h</I><SUP>2</SUP>)-<I>b</I>}<I>h</I>/<I>b.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11860 <C>(<G>h</G>) <I>Ellipse, parabolic segment, surface of cylinder, right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11861 cone, sphere and segment of sphere.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11862 <p>After the area of an ellipse (<I>Metrica</I> I. 34) and of a parabolic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11863 segment (chap. 35), Heron gives the surface of a cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11864 (chap. 36) and a right cone (chap. 37); in both cases he unrolls
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11865 the surface on a plane so that the surface becomes that of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11866 parallelogram in the one case and a sector of a circle in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11867 other. For the surface of a sphere (chap. 38) and a segment of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11868 it (chap. 39) he simply uses Archimedes's results.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11869 <p>Book I ends with a hint how to measure irregular figures,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11870 plane or not. If the figure is plane and bounded by an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11871 irregular curve, neighbouring points are taken on the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11872 such that, if they are joined in order, the contour of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11873 polygon so formed is not much different from the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11874 itself, and the polygon is then measured by dividing it into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11875 triangles. If the surface of an irregular solid figure is to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11876 found, you wrap round it pieces of very thin paper or cloth,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11877 enough to cover it, and you then spread out the paper or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11878 cloth and measure that.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11879 <C>Book II. Measurement of volumes.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11880 <p>The preface to Book II is interesting as showing how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11881 vague the traditions about Archimedes had already become.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11882 <p>‘After the measurement of surfaces, rectilinear or not, it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11883 proper to proceed to the solid bodies, the surfaces of which we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11884 have already measured in the preceding book, surfaces plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11885 and spherical, conical and cylindrical, and irregular surfaces
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11886 as well. The methods of dealing with these solids are, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11887 <pb n=332><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11888 view of their surprising character, referred to Archimedes by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11889 certain writers who give the traditional account of their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11890 origin. But whether they belong to Archimedes or another,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11891 it is necessary to give a sketch of these methods as well.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11892 <p>The Book begins with generalities about figures all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11893 sections of which parallel to the base are equal to the base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11894 and similarly situated, while the centres of the sections are on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11895 a straight line through the centre of the base, which may be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11896 either obliquely inclined or perpendicular to the base; whether
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11897 the said straight line (‘the axis’) is or is not perpendicular to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11898 the base, the volume is equal to the product of the area of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11899 base and the <I>perpendicular</I> height of the top of the figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11900 from the base. The term ‘height’ is thenceforward restricted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11901 to the length of the perpendicular from the top of the figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11902 on the base.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11903 <C>(<I>a</I>) <I>Cone, cylinder, parallelepiped</I> (<I>prism</I>), <I>pyramid, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11904 frustum.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11905 <p>II. 1-7 deal with a cone, a cylinder, a ‘parallelepiped’ (the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11906 base of which is not restricted to the parallelogram but is in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11907 the illustration given a regular hexagon, so that the figure is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11908 more properly a prism with polygonal bases), a triangular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11909 prism, a pyramid with base of any form, a frustum of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11910 triangular pyramid; the figures are in general <I>oblique.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11911 <C>(<G>b</G>) <I>Wedge-shaped solid</I> (<G>bwmi/skos</G> <I>or</I> <G>sfhni/skos</G>).</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11912 <p>II. 8 is a case which is perhaps worth giving. It is that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11913 a rectilineal solid, the base of which is a rectangle <I>ABCD</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11914 has opposite to it another rectangle <I>EFGH</I>, the sides of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11915 are respectively parallel but not necessarily proportional to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11916 those of <I>ABCD.</I> Take <I>AK</I> equal to <I>EF</I>, and <I>BL</I> equal to <I>FG.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11917 Bisect <I>BK</I>, <I>CL</I> in <I>V</I>, <I>W</I>, and draw <I>KRPU</I>, <I>VQOM</I> parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11918 <I>AD</I>, and <I>LQRN</I>, <I>WOPT</I> parallel to <I>AB.</I> Join <I>FK</I>, <I>GR</I>, <I>LG</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11919 <I>GU</I>, <I>HN.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11920 <p>Then the solid is divided into (1) the parallelepiped with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11921 <I>AR</I>, <I>EG</I> as opposite faces, (2) the prism with <I>KL</I> as base and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11922 <I>FG</I> as the opposite edge, (3) the prism with <I>NU</I> as base and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11923 <I>GH</I> as opposite edge, and (4) the pyramid with <I>RLCU</I> as base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11924 and <I>G</I> as vertex. Let <I>h</I> be the ‘height’ of the figure. Now
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11925 <pb n=333><head>MEASUREMENT OF SOLIDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11926 the parallelepiped (1) is on <I>AR</I> as base and has height <I>h</I>; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11927 prism (2) is equal to a parallelepiped on <I>KQ</I> as base and with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11928 height <I>h</I>; the prism (3) is equal to a parallelepiped with <I>NP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11929 as base and height <I>h</I>; and finally the pyramid (4) is equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11930 a parallelepiped of height <I>h</I> and one-third of <I>RC</I> as base.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11931 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11932 <p>Therefore the whole solid is equal to one parallelepiped
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11933 with height <I>h</I> and base equal to <MATH>(<I>AR</I>+<I>KQ</I>+<I>NP</I>+<I>RO</I>+1/3<I>RO</I>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11934 or <I>AO</I>+1/3<I>RO.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11935 <p>Now, if <MATH><I>AB</I>=<I>a</I>, <I>BC</I>=<I>b</I>, <I>EF</I>=<I>c</I>, <I>FG</I>=<I>d</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11936 <I>AV</I>=1/2(<I>a</I>+<I>c</I>), <I>AT</I>=1/2(<I>b</I>+<I>d</I>), <I>RQ</I>=1/2(<I>a</I>-<I>c</I>), <I>RP</I>=1/2(<I>b</I>-<I>d</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11937 <p>Therefore volume of solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11938 <MATH>={1/4(<I>a</I>+<I>c</I>)(<I>b</I>+<I>d</I>)+1/(12)(<I>a</I>-<I>c</I>)(<I>b</I>-<I>d</I>)}<I>h.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11939 <p>The solid in question is evidently the true <G>bwmi/skos</G> (‘little
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11940 altar’), for the formula is used to calculate the content of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11941 a <G>bwmi/skos</G> in <I>Stereom.</I> II. 40 (68, Heib.) It is also, I think,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11942 the <G>sfhni/skos</G> (‘little wedge’), a measurement of which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11943 given in <I>Stereom.</I> I. 26 (25, Heib.) It is true that the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11944 term of the first factor <MATH>1/(12)(<I>a</I>-<I>c</I>)(<I>b</I>-<I>d</I>)</MATH> is there neglected,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11945 perhaps because in the case taken <MATH>(<I>a</I>=7, <I>b</I>=6, <I>c</I>=5, <I>d</I>=4)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11946 this term (=1/3) is small compared with the other (=30). A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11947 particular <G>sfhni/skos</G>, in which either <I>c</I>=<I>a</I> or <I>d</I>=<I>b</I>, was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11948 called <G>o)/nux</G>; the second term in the factor of the content
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11949 vanishes in this case, and, if e.g. <I>c</I>=<I>a</I>, the content is <MATH>1/2(<I>b</I>+<I>d</I>)<I>ah.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11950 Another <G>bwmi/skos</G> is measured in <I>Stereom.</I> I. 35 (34, Heib.),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11951 where the solid is inaccurately called ‘a pyramid oblong
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11952 (<G>e(teromh/khs</G>) and truncated (<G>ko/louros</G>) or half-perfect’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11953 <pb n=334><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11954 <p>The method is the same <I>mutatis mutandis</I> as that used in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11955 II. 6 for the frustum of a pyramid with any triangle for base,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11956 and it is applied in II. 9 to the case of a frustum of a pyramid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11957 with a square base, the formula for which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11958 <MATH>[{1/2(<I>a</I>+<I>a</I>′)}<SUP>2</SUP>+1/3{1/2(<I>a</I>-<I>a</I>′)}<SUP>2</SUP>]<I>h</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11959 where <I>a, a</I>′ are the sides of the larger and smaller bases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11960 respectively, and <I>h</I> the height; the expression is of course
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11961 easily reduced to <MATH>1/3<I>h</I>(<I>a</I><SUP>2</SUP>+<I>aa</I>′+<I>a</I>′<SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11962 <C>(<G>g</G>) <I>Frustum of cone, sphere, and segment thereof</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11963 <p>A <I>frustum of a cone</I> is next measured in two ways, (1) by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11964 comparison with the corresponding frustum of the circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11965 scribing pyramid with square base, (2) directly as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11966 difference between two cones (chaps. 9, 10). The volume of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11967 the frustum of the cone is to that of the frustum of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11968 circumscribing pyramid as the area of the base of the cone to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11969 that of the base of the pyramid; i.e. the volume of the frus-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11970 tum of the cone is 1/4<G>p</G>, or 11/14, times the above expression for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11971 the frustum of the pyramid with <I>a</I><SUP>2</SUP>, <I>a</I>′<SUP>2</SUP> as bases, and it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11972 reduces to <MATH>1/12<G>p</G><I>h</I> (<I>a</I><SUP>2</SUP>+<I>aa</I>′+<I>a</I>′<SUP>2</SUP>)</MATH>, where <I>a, a</I>′ are the <I>diameters</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11973 of the two bases. For the <I>sphere</I> (chap. 11) Heron uses
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11974 Archimedes's proposition that the circumscribing cylinder is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11975 1 1/2 times the sphere, whence the volume of the sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11976 =2/3.<I>d</I>.11/14<I>d</I><SUP>2</SUP> or 11/21<I>d</I><SUP>3</SUP>; for a <I>segment of a sphere</I> (chap. 12) he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11977 likewise uses Archimedes's result (<I>On the Sphere and Cylinder</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11978 II. 4).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11979 <C>(<G>d</G>) <I>Anchor-ring or tore</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11980 <p>The anchor-ring or <I>tore</I> is next measured (chap. 13) by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11981 means of a proposition which Heron quotes from Dionyso-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11982 dorus, and which is to the effect that, if <I>a</I> be the radius of either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11983 circular section of the <I>tore</I> through the axis of revolution, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11984 <I>c</I> the distance of its centre from that axis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11985 <MATH><G>p</G><I>a</I><SUP>2</SUP>:<I>ac</I>=(volume of tore):<G>p</G><I>c</I><SUP>2</SUP>.2<I>a</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11986 <MATH>[whence volume of tore=2<G>p</G><SUP>2</SUP><I>ca</I><SUP>2</SUP>]</MATH>. In the particular case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11987 taken <MATH><I>a</I>=6, <I>c</I>=14</MATH>, and Heron obtains, from the proportion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11988 113 1/7:84=<I>V</I>:7392, <I>V</I>=9956 4/7. But he shows that he is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11989 aware that the volume is the product of the area of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11990 <pb n=335><head>MEASUREMENT OF SOLIDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11991 describing circle and the length of the path of its centre.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11992 For, he says, since 14 is a radius (of the path of the centre),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11993 28 is its diameter and 88 its circumference. ‘If then the tore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11994 be straightened out and made into a cylinder, it will have 88
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11995 for its length, and the diameter of the base of the cylinder is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11996 12; so that the solid content of the cylinder is, as we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11997 seen, <MATH>9956 4/7’(=88.11/14.144)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11998 <C>(<G>e</G>) <I>The two special solids of Archimedes's ‘Method’</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	11999 <p>Chaps. 14, 15 give the measurement of the two remarkable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12000 solids of Archimedes's <I>Method</I>, following Archimedes's results.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12001 <C>(<G>z</G>) <I>The five regular solids</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12002 <p>In chaps. 16-18 Heron measures the content of the five
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12003 regular solids after the cube. He has of course in each case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12004 to find the perpendicular from the centre of the circumscrib-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12005 ing sphere on any face. Let <I>p</I> be this perpendicular, <I>a</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12006 edge of the solid, <I>r</I> the radius of the circle circumscribing any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12007 face. Then (1) for the <I>tetrahedron</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12008 <MATH><I>a</I><SUP>2</SUP>=3<I>r</I><SUP>2</SUP>,<I>p</I><SUP>2</SUP>=<I>a</I><SUP>2</SUP>-1/3<I>a</I><SUP>2</SUP>=2/3<I>a</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12009 (2) In the case of the <I>octahedron</I>, which is the sum of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12010 equal pyramids on a square base, the content is one-third
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12011 of that base multiplied by the diagonal of the figure,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12012 i.e. 1/3.<I>a</I><SUP>2</SUP>.√2<I>a</I> or 1/3√2.<I>a</I><SUP>3</SUP>; in the case taken <MATH><I>a</I>=7</MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12013 Heron takes 10 as an approximation to √(2.7<SUP>2</SUP>) or √98, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12014 result being 1/3.10.49 or 163 1/3. (3) In the case of the <I>icosa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12015 hedron</I> Heron merely says that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12016 <MATH><I>p</I>:<I>a</I>=93:127 (the real value of the ratio is 1/2√(7+3√5)/6)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12017 (4) In the case of the <I>dodecahedron</I>, Heron says that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12018 <MATH><I>p</I>:<I>a</I>=9:8 (the true value is 1/2√(25+11√5)/10</MATH>, and, if √5 is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12019 put equal to 9/4, Heron's ratio is readily obtained).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12020 <p>Book II ends with an allusion to the method attributed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12021 Archimedes for measuring the contents of irregular bodies by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12022 immersing them in water and measuring the amount of fluid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12023 displaced.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12024 <pb n=336><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12025 <C>Book III. Divisions of figures.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12026 <p>This book has much in common with Euclid's book <I>On divi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12027 sions</I> (<I>of figures</I>), the problem being to divide various figures,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12028 plane or solid, by a straight line or plane into parts having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12029 a given ratio. In III. 1-3 a triangle is divided into two parts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12030 in a given ratio by a straight line (1) passing through a vertex,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12031 (2) parallel to a side, (3) through any point on a side.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12032 III. 4 is worth description: ‘Given a triangle <I>ABC</I>, to cut
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12033 out of it a triangle <I>DEF</I> (where <I>D, E, F</I> are points on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12034 sides respectively) given in magnitude and such that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12035 triangles <I>AEF, BFD, CED</I> may be equal in area.’ Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12036 <I>assumes</I> that, if <I>D, E, F</I> divide the sides so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12037 <MATH><I>AF</I>:<I>FB</I>=<I>BD</I>:<I>DC</I>=<I>CE</I>:<I>EA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12038 the latter three triangles are equal in area.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12039 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12040 <p>He then has to find the value of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12041 each of the three ratios which will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12042 result in the triangle <I>DEF</I> having a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12043 given area.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12044 <p>Join <I>AD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12045 <p>Since <MATH><I>BD</I>:<I>CD</I>=<I>CE</I>:<I>EA</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12046 <I>BC</I>:<I>CD</I>=<I>CA</I>:<I>AE</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12047 and <MATH>&utri;<I>ABC</I>:&utri;<I>ADC</I>=&utri;<I>ADC</I>:&utri;<I>ADE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12048 <p>Also <MATH>&utri;<I>ABC</I>:&utri;<I>ABD</I>=&utri;<I>ADC</I>:&utri;<I>EDC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12049 <p>But (since the area of the triangle <I>DEF</I> is given) &utri;<I>EDC</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12050 given, as well as &utri;<I>ABC.</I> Therefore &utri;<I>ABD</I>x&utri;<I>ADC</I> is given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12051 <p>Therefore, if <I>AH</I> be perpendicular to <I>BC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12052 <MATH><I>AH</I><SUP>2</SUP>.<I>BD.DC</I></MATH> is given;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12053 therefore <I>BD.DC</I> is given, and, since <I>BC</I> is given, <I>D</I> is given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12054 in position (we have to apply to <I>BC</I> a rectangle equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12055 <I>BD.DC</I> and falling short by a square).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12056 <p>As an example Heron takes <MATH><I>AB</I>=13, <I>BC</I>=14, <I>CA</I>=15,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12057 &utri;<I>DEF</I>=24. &utri;<I>ABC</I></MATH> is then 84, and <MATH><I>AH</I>=12</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12058 <p>Thus <MATH>&utri;<I>EDC</I>=20</MATH>, and <MATH><I>AH</I><SUP>2</SUP>.<I>BD.DC</I>=4.84.20=6720</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12059 therefore <MATH><I>BD.DC</I>=6720/144 or 46 2/3</MATH> (the text omits the 2/3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12060 <p>Therefore, says Heron, <MATH><I>BD</I>=8</MATH> approximately. For 8 we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12061 <pb n=337><head>DIVISIONS OF FIGURES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12062 should apparently have 8 1/2, since <I>DC</I> is immediately stated to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12063 be 5 1/2 (not 6). That is, in solving the equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12064 <MATH><I>x</I><SUP>2</SUP>-14<I>x</I>+46 2/3=0</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12065 which gives <MATH><I>x</I>=7±√(2 1/3)</MATH>, Heron apparently substituted 2 1/4 or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12066 9/4 for 2 1/3, thereby obtaining 1 1/2 as an approximation to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12067 surd.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12068 <p>(The lemma assumed in this proposition is easily proved.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12069 Let <I>m</I>:<I>n</I> be the ratio <MATH><I>AF</I>:<I>FB</I>=<I>BD</I>:<I>DC</I>=<I>CE</I>:<I>EA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12070 <p>Then <MATH><I>AF</I>=<I>mc</I>/(<I>m</I>+<I>n</I>), <I>FB</I>=<I>nc</I>/(<I>m</I>+<I>n</I>), <I>CE</I>=<I>mb</I>/(<I>m</I>+<I>n</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12071 <I>EA</I>=<I>nb</I>/(<I>m</I>+<I>n</I>), &c</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12072 Hence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12073 <MATH>&utri;<I>AFE</I>/&utri;<I>ABC</I>=(<I>mn</I>)/((<I>m</I>+<I>n</I>)<SUP>2</SUP>)=&utri;<I>BDF</I>/&utri;<I>ABC</I>=&utri;<I>CDE</I>/&utri;<I>ABC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12074 and the triangles <I>AFE, BDF, CDE</I> are equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12075 <p>Pappus<note>Pappus, viii, pp. 1034-8. Cf. pp. 430-2 <I>post.</I></note> has the proposition that the triangles <I>ABC, DEF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12076 have the same centre of gravity.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12077 <p>Heron next shows how to divide a parallel-trapezium into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12078 two parts in a given ratio by a straight line (1) through the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12079 point of intersection of the non-parallel sides, (2) through a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12080 given point on one of the parallel sides, (3) parallel to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12081 parallel sides, (4) through a point on one of the non-parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12082 sides (III. 5-8). III. 9 shows how to divide the area of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12083 circle into parts which have a given ratio by means of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12084 inner circle with the same centre. For the problems begin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12085 ning with III. 10 Heron says that numerical calculation alone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12086 no longer suffices, but geometrical methods must be applied.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12087 Three problems are reduced to problems solved by Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12088 in his treatise <I>On cutting off an area.</I> The first of these is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12089 III. 10, to cut off from the angle of a triangle a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12090 proportion of the triangle by a straight line through a point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12091 on the opposite side produced. III. 11, 12, 13 show how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12092 to cut any quadrilateral into parts in a given ratio by a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12093 straight line through a point (1) on a side (<I>a</I>) dividing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12094 side in the given ratio, (<I>b</I>) not so dividing it, (2) not on any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12095 side, (<I>a</I>) in the case where the quadrilateral is a trapezium,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12096 i.e. has two sides parallel, (<I>b</I>) in the case where it is not; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12097 last case (<I>b</I>) is reduced (like III. 10) to the ‘cutting-off of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12098 <pb n=338><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12099 area’. These propositions are ingenious and interesting.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12100 III. 11 shall be given as a specimen.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12101 <p>Given any quadrilateral <I>ABCD</I> and a point <I>E</I> on the side
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12102 <I>AD</I>, to draw through <I>E</I> a straight line <I>EF</I> which shall cut
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12103 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12104 the quadrilateral into two parts in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12105 the ratio of <I>AE</I> to <I>ED.</I> (We omit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12106 the analysis.) Draw <I>CG</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12107 to <I>DA</I> to meet <I>AB</I> produced in <I>G.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12108 <p>Join <I>BE</I>, and draw <I>GH</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12109 to <I>BE</I> meeting <I>BC</I> in <I>H.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12110 <p>Join <I>CE, EH, EG.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12111 <p>Then <MATH>&utri;<I>GBE</I>=&utri;<I>HBE</I></MATH> and, adding &utri;<I>ABE</I> to each, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12112 <MATH>&utri;<I>AGE</I>=(quadrilateral <I>ABHE</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12113 <p>Therefore <MATH>(quadr. <I>ABHE</I>):&utri;<I>CED</I>=&utri;<I>GAE</I>:&utri;<I>CED</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12114 =<I>AE</I>:<I>ED</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12115 <p>But (quadr. <I>ABHE</I>) and &utri;<I>CED</I> are parts of the quadri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12116 lateral, and they leave over only the triangle <I>EHC.</I> We have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12117 therefore only to divide &utri;<I>EHC</I> in the same ratio <I>AE:ED</I> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12118 the straight line <I>EF.</I> This is done by dividing <I>HC</I> at <I>F</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12119 the ratio <I>AE</I>:<I>ED</I> and joining <I>EF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12120 <p>The next proposition (III. 12) is easily reduced to this.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12121 <p>If <I>AE</I>:<I>ED</I> is not equal to the given ratio, let <I>F</I> divide <I>AD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12122 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12123 in the given ratio, and through <I>F</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12124 draw <I>FG</I> dividing the quadri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12125 lateral in the given ratio (III. 11).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12126 <p>Join <I>EG</I>, and draw <I>FH</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12127 to <I>EG.</I> Let <I>FH</I> meet <I>BC</I> in <I>H</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12128 and join <I>EH.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12129 <p>Then is <I>EH</I> the required straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12130 line through <I>E</I> dividing the quad-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12131 rilateral in the given ratio.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12132 <p>For <MATH>&utri;<I>FGE</I>=&utri;<I>HGE</I></MATH>. Add to each (quadr. <I>GEDC</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12133 <p>Therefore <MATH>(quadr. <I>CGFD</I>)=(quadr. <I>CHED</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12134 <p>Therefore <I>EH</I> divides the quadrilateral in the given ratio,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12135 just as <I>FG</I> does.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12136 <p>The case (III. 13) where <I>E</I> is not on a side of the quadri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12137 lateral [(2) above] takes two different forms according as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12138 <pb n=339><head>DIVISIONS OF FIGURES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12139 two opposite sides which the required straight line cuts are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12140 (<I>a</I>) parallel or (<I>b</I>) not parallel. In the first case (<I>a</I>) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12141 problem reduces to drawing a straight line through <I>E</I> inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12142 secting the parallel sides in points <I>F, G</I> such that <I>BF</I>+<I>AG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12143 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12144 is equal to a given length. In the second case (<I>b</I>) where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12145 <I>BC, AD</I> are not parallel Heron supposes them to meet in <I>H.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12146 The angle at <I>H</I> is then given, and the area <I>ABH.</I> It is then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12147 a question of cutting off from a triangle with vertex <I>H</I> a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12148 triangle <I>HFG</I> of given area by a straight line drawn from <I>E</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12149 which is again a problem in Apollonius's <I>Cutting-off of an</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12150 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12151 <I>area.</I> The auxiliary problem in case (<I>a</I>) is easily solved in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12152 III. 16. Measure <I>AH</I> equal to the given length. Join <I>BH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12153 and bisect it at <I>M.</I> Then <I>EM</I> meets <I>BC, AD</I> in points such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12154 that <MATH><I>BF</I>+<I>AG</I>=the given length</MATH>. For, by congruent triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12155 <MATH><I>BF</I>=<I>GH</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12156 <p>The same problems are solved for the case of any polygon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12157 in III. 14, 15. A sphere is then divided (III. 17) into segments
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12158 such that their surfaces are in a given ratio, by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12159 Archimedes, <I>On the Sphere and Cylinder</I>, II. 3, just as, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12160 III. 23, Prop. 4 of the same Book is used to divide a sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12161 into segments having their volumes in a given ratio.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12162 <p>III. 18 is interesting because it recalls an ingenious pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12163 position in Euclid's book <I>On Divisions.</I> Heron's problem is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12164 ‘To divide a given circle into three equal parts by two straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12165 <pb n=340><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12166 lines’, and he observes that, ‘as the problem is clearly not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12167 rational, we shall, for practical convenience, make the division,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12168 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12169 as exactly as possible, in the follow-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12170 ing way.’ <I>AB</I> is the side of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12171 equilateral triangle inscribed in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12172 circle. Let <I>CD</I> be the parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12173 diameter, <I>O</I> the centre of the circle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12174 and join <I>AO, BO, AD, DB.</I> Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12175 shall the segment <I>ABD</I> be very
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12176 nearly one-third of the circle. For,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12177 since <I>AB</I> is the side of an equi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12178 lateral triangle in the circle, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12179 sector <I>OAEB</I> is one-third of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12180 circle. And the triangle <I>AOB</I> forming part of the sector
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12181 is equal to the triangle <I>ADB</I>; therefore the segment <I>AEB
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12182 plus</I> the triangle <I>ABD</I> is equal to one-third of the circle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12183 and the segment <I>ABD</I> only differs from this by the small
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12184 segment on <I>BD</I> as base, which may be neglected. Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12185 proposition is to cut off one-third (or any fraction) of a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12186 between two parallel chords (see vol. i, pp. 429-30).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12187 <p>III. 19 finds a point <I>D</I> within any triangle <I>ABC</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12188 the triangles <I>DBC, DCA, DAB</I> are all equal; and then Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12189 passes to the division of solid figures.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12190 <p>The solid figures divided in a given ratio (besides the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12191 sphere) are the pyramid with base of any form (III. 20),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12192 the cone (III. 21) and the frustum of a cone (III. 22), the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12193 cutting planes being parallel to the base in each case. These
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12194 problems involve the extraction of the cube root of a number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12195 which is in general not an exact cube, and the point of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12196 interest is Heron's method of approximating to the cube root
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12197 in such a case. Take the case of the cone, and suppose that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12198 the portion to be cut off at the top is to the rest of the cone as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12199 <I>m</I> to <I>n.</I> We have to find the ratio in which the height or the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12200 edge is cut by the plane parallel to the base which cuts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12201 the cone in the given ratio. The volume of a cone being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12202 1/3<G>p</G><I>c</I><SUP>2</SUP><I>h</I>, where <I>c</I> is the radius of the base and <I>h</I> the height,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12203 we have to find the height of the cone the volume of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12204 is <I>m</I>/(<I>m</I>+<I>n</I>).1/3<G>p</G><I>c</I><SUP>2</SUP><I>h</I>, and, as the height <I>h</I>′ is to the radius <I>c</I>′ of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12205 its base as <I>h</I> is to <I>c</I>, we have simply to find <I>h</I>′ where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12206 <pb n=341><head>DIVISIONS OF FIGURES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12207 <MATH><I>h</I>′<SUP>3</SUP>/<I>h</I><SUP>3</SUP>=<I>m</I>/(<I>m</I>+<I>n</I>)</MATH>. Or, if we take the edges <I>e, e</I>′ instead
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12208 of the heights, <MATH><I>e</I>′<SUP>3</SUP>/<I>e</I><SUP>3</SUP>=<I>m</I>/(<I>m</I>+<I>n</I>)</MATH>. In the case taken by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12209 Heron <MATH><I>m</I>:<I>n</I>=4:1</MATH>, and <MATH><I>e</I>=5</MATH>. Consequently <MATH><I>e</I>′<SUP>3</SUP>=4/5.5<SUP>3</SUP>=100</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12210 Therefore, says Heron, <MATH><I>e</I>′=4 9/14</MATH> approximately, and in III. 20
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12211 he shows how this is arrived at.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12212 <C><I>Approximation to the cube root of a non-cube number.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12213 <p>‘Take the nearest cube numbers to 100 both above and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12214 below; these are 125 and 64.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12215 <p>Then 125-100=25,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12216 and 100-64=36.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12217 <p>Multiply 5 into 36; this gives 180. Add 100, making 280.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12218 <Divide 180 by 280>; this gives 9/14. Add this to the side of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12219 the smaller cube: this gives 4 9/14. This is as nearly as possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12220 the cube root (“cubic side”) of 100 units.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12221 <p>We have to conjecture Heron's formula from this example.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12222 Generally, if <MATH><I>a</I><SUP>3</SUP><<I>A</I><(<I>a</I>+1)<SUP>3</SUP></MATH>, suppose that <MATH><I>A</I>-<I>a</I><SUP>3</SUP>=<I>d</I><SUB>1</SUB></MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12223 <MATH>(<I>a</I>+1)<SUP>3</SUP>-<I>A</I>=<I>d</I><SUB>2</SUB></MATH>. The best suggestion that has been made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12224 is Wertheim's,<note><I>Zeitschr. f. Math. u. Physik</I>, xliv, 1899, hist.-litt. Abt., pp. 1-3.</note> namely that Heron's formula for the approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12225 mate cube root was <I>a</I>+((<I>a</I>+1)<I>d</I><SUB>1</SUB>)/((<I>a</I>+1)<I>d</I><SUB>1</SUB>+<I>ad</I><SUB>2</SUB>). The 5 multiplied
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12226 into the 36 might indeed have been the square root of 25 or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12227 √<I>d</I><SUB>2</SUB>, and the 100 added to the 180 in the denominator of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12228 fraction might have been the original number 100 (<I>A</I>) and not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12229 4.25 or <I>ad</I><SUB>2</SUB>, but Wertheim's conjecture is the more satisfactory
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12230 because it can be evolved out of quite elementary considera-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12231 tions. This is shown by G. Enestr&odblac;m as follows.<note><I>Bibliotheca Mathematica</I>, viii<SUB>3</SUB>, 1907-8, pp. 412-13.</note> Using the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12232 same notation, Enestr&odblac;m further supposes that <I>x</I> is the exact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12233 value of √<SUP>3</SUP><I>A</I> and that <MATH>(<I>x</I>-<I>a</I>)<SUP>3</SUP>=<G>d</G><SUB>1</SUB>, (<I>a</I>+1-<I>x</I>)<SUP>3</SUP>=<G>d</G><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12234 <p>Thus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12235 <MATH><G>d</G><SUB>1</SUB>=<I>x</I><SUP>3</SUP>-3<I>x</I><SUP>2</SUP><I>a</I>+3<I>xa</I><SUP>2</SUP>-<I>a</I><SUP>3</SUP></MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12236 <MATH>3<I>ax</I>(<I>x</I>-<I>a</I>)=<I>x</I><SUP>3</SUP>-<I>a</I><SUP>3</SUP>-<G>d</G><SUB>1</SUB>=<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12237 <p>Similarly from <MATH><G>d</G><SUB>2</SUB>=(<I>a</I>+1-<I>x</I>)<SUP>3</SUP></MATH> we derive
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12238 <MATH>3(<I>a</I>+1)<I>x</I>(<I>a</I>+1-<I>x</I>)=(<I>a</I>+1)<SUP>3</SUP>-<I>x</I><SUP>3</SUP>-<G>d</G><SUB>2</SUB>=<I>d</I><SUB>2</SUB>-<G>d</G><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12239 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12240 <MATH>(<I>d</I><SUB>2</SUB>-<G>d</G><SUB>2</SUB>)/(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>)=(3(<I>a</I>+1)<I>x</I>(<I>a</I>+1-<I>x</I>))/(3<I>ax</I>(<I>x</I>-<I>a</I>))=((<I>a</I>+1){1-(<I>x</I>-<I>a</I>)})/(<I>a</I>(<I>x</I>-<I>a</I>))
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12241 =(<I>a</I>+1)/<I>a</I>(<I>x</I>-<I>a</I>)-(<I>a</I>+1)/<I>a</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12242 <pb n=342><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12243 and, solving for <I>x</I>-<I>a</I>, we obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12244 <MATH><I>x</I>-<I>a</I>=((<I>a</I>+1)(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>))/((<I>a</I>+1)(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>)+<I>a</I>(<I>d</I><SUB>2</SUB>-<G>d</G><SUB>2</SUB>))</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12245 or <MATH>√<SUP>3</SUP><I>A</I>=<I>a</I>+((<I>a</I>+1)(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>))/((<I>a</I>+1)(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>)+<I>a</I>(<I>d</I><SUB>2</SUB>-<G>d</G><SUB>2</SUB>))</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12246 <p>Since <G>d</G><SUB>1</SUB>, <G>d</G><SUB>2</SUB> are in any case the cubes of fractions, we may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12247 neglect them for a first approximation, and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12248 <MATH>√<SUP>3</SUP><I>A</I>=<I>a</I>+((<I>a</I>+1)<I>d</I><SUB>1</SUB>)/((<I>a</I>+1)<I>d</I><SUB>1</SUB>+<I>ad</I><SUB>2</SUB>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12249 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12250 <p>III. 22, which shows how to cut a frustum of a cone in a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12251 ratio by a section parallel to the bases, shall end our account
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12252 of the <I>Metrica.</I> I shall give the general formulae on the left
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12253 and Heron's case on the right. Let <I>ABED</I> be the frustum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12254 let the <I>diameters</I> of the bases be <I>a, a</I>′, and the height <I>h.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12255 Complete the cone, and let the height of <I>CDE</I> be <I>x.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12256 <p>Suppose that the frustum has to be cut by a plane <I>FG</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12257 such a way that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12258 <MATH>(frustum <I>DG</I>):(frustum <I>FB</I>)=<I>m</I>:<I>n</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12259 <p>In the case taken by Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12260 <MATH><I>a</I>=28, <I>a</I>′=21, <I>h</I>=12, <I>m</I>=4, <I>n</I>=1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12261 <p>Draw <I>DH</I> perpendicular to <I>AB.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12262 <pb n=343><head>DIVISIONS OF FIGURES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12263 <p>Since <MATH>(<I>DG</I>):(<I>FB</I>)=<I>m</I>:<I>n</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12264 (<I>DB</I>):(<I>DG</I>)=(<I>m</I>+<I>n</I>):<I>m</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12265 <p>Now
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12266 <MATH>(<I>DB</I>)=1/12<G>p</G><I>h</I>(<I>a</I><SUP>2</SUP>+<I>aa</I>′+<I>a</I>′<SUP>2</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12267 and <MATH>(<I>DG</I>)=<I>m</I>/(<I>m</I>+<I>n</I>)(<I>DB</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12268 <p>Let <I>y</I> be the height (<I>CM</I>) of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12269 cone <I>CFG.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12270 <p>Then <MATH><I>DH</I>:<I>AH</I>=<I>CK</I>:<I>KA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12271 or <MATH><I>h</I>:1/2(<I>a</I>-<I>a</I>′)=(<I>x</I>+<I>h</I>):1/2<I>a</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12272 whence <I>x</I> is known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12273 <p><MATH>Cone <I>CDE</I>=1/12<G>p</G><I>a</I>′<SUP>2</SUP><I>x</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12274 cone <I>CFG</I>=(<I>CDE</I>)+<I>m</I>/(<I>m</I>+<I>n</I>)(<I>DB</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12275 cone <I>CAB</I>=(<I>CDE</I>)+(<I>DB</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12276 <p>Now, says Heron,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12277 <MATH>((<I>CAB</I>)+(<I>CDE</I>))/(<I>CFG</I>)=((<I>x</I>+<I>h</I>)<SUP>3</SUP>+<I>x</I><SUP>3</SUP>)/<I>y</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12278 <p>[He might have said simply
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12279 <MATH>(<I>CDE</I>):(<I>CFG</I>)=<I>x</I><SUP>3</SUP>:<I>y</I><SUP>3</SUP></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12280 <p>This gives <I>y</I> or <I>CM</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12281 whence <I>LM</I> is known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12282 <p>Now <MATH><I>AD</I><SUP>2</SUP>=<I>AH</I><SUP>2</SUP>+<I>DH</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12283 ={1/2(<I>a</I>-<I>a</I>′)}<SUP>2</SUP>+<I>h</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12284 so that <I>AD</I> is known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12285 <p>Therefore <MATH><I>DF</I>=(<I>y</I>-<I>x</I>)/<I>h</I>.<I>AD</I></MATH> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12286 known.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12287 <MATH>(<I>DG</I>):(<I>FB</I>)=4:1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12288 (<I>DB</I>):(<I>DG</I>)=5:4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12289 (<I>DB</I>)=5698,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12290 (<I>DG</I>)=4558 2/5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12291 <I>x</I>+<I>h</I>=(14.12)/3 1/2=48</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12292 and <MATH><I>x</I>=48-12=36.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12293 (cone <I>CDE</I>)=4158,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12294 (cone <I>CFG</I>)=4158+4558 2/5=8716 2/5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12295 (cone <I>CAB</I>)=4158+5698=9856.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12296 <I>y</I><SUP>3</SUP>=8716 2/5/(9856+4158).(48<SUP>3</SUP>+36<SUP>3</SUP>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12297 =8716 2/5.157248/14014=97805</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12298 whence <MATH><I>y</I>=46</MATH> approximately.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12299 <p>Therefore <MATH><I>LM</I>=<I>y</I>-<I>x</I>=10.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12300 <I>AD</I><SUP>2</SUP>=(3 1/2)<SUP>2</SUP>+12<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12301 =156 1/4</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12302 and <MATH><I>AD</I>=12 1/2</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12303 <p>Therefore <MATH><I>DF</I>=10/12.12 1/2
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12304 =10 5/12</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12305 <pb n=344><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12306 <C><I>Quadratic equations solved in Heron.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12307 <p>We have already met with one such equation (in <I>Metrica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12308 III. 4), namely <MATH><I>x</I><SUP>2</SUP>-14<I>x</I>+46 2/3=0</MATH>, the result only <MATH>(<I>x</I>=8 1/2)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12309 being given. There are others in the <I>Geometrica</I> where the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12310 process of solution is shown.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12311 <p>(1) <I>Geometrica</I> 24, 3 (Heib.). ‘Given a square such that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12312 sum of its area and perimeter is 896 feet: to separate the area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12313 from the perimenter’: i.e. <MATH><I>x</I><SUP>2</SUP>+4<I>x</I>=896</MATH>. Heron takes half of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12314 4 and adds its square, completing the square on the left side.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12315 <p>(2) <I>Geometrica</I> 21, 9 and 24, 46 (Heib.) give one and the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12316 equation, <I>Geom.</I> 24, 47 another like it. ‘Given the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12317 the diameter, perimeter and area of a circle, to find each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12318 of them.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12319 <p>The two equations are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12320 <MATH>11/14<I>d</I><SUP>2</SUP>+29/7<I>d</I>=212</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12321 and <MATH>11/14<I>d</I><SUP>2</SUP>+29/7<I>d</I>=67 1/2</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12322 <p>Our usual method is to begin by dividing by 11/14 throughout,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12323 so as to leave <I>d</I><SUP>2</SUP> as the first term. Heron's is to <I>multiply</I> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12324 such a number as will leave a square as the first term. In this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12325 case he multiplies by 154, giving <MATH>11<SUP>2</SUP><I>d</I><SUP>2</SUP>+58.11<I>d</I>=212.154
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12326 or 67 1/2.154</MATH> as the case may be. Completing the square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12327 he obtains <MATH>(11<I>d</I>+29)<SUP>2</SUP>=32648+841 or 10395+841</MATH>. Thus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12328 <MATH>11<I>d</I>+29=√(33489) or √(11236)</MATH>, that is, 183 or 106.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12329 Thus <MATH>11<I>d</I>=154 or 77</MATH>, and <MATH><I>d</I>=14 or 7</MATH>, as the case may be.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12330 <C>Indeterminate problems in the <I>Geometrica.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12331 <p>Some very interesting indeterminate problems are now
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12332 included by Heiberg in the <I>Geometrica.</I><note><I>Heronis Alexandrini opera</I>, vol. iv, p. 414. 28 sq.</note> Two of them (chap.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12333 24, 1-2) were included in the <I>Ge&edblac;ponicus</I> in Hultsch's edition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12334 (sections 78, 79); the rest are new, having been found in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12335 Constantinople manuscript from which Sch&odblac;ne edited the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12336 <I>Metrica.</I> As, however, these problems, to whatever period
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12337 they belong, are more akin to algebra than to mensuration,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12338 they will be more properly described in a later chapter on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12339 Algebra.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12340 <pb n=345><head>THE <I>DIOPTRA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12341 <C>The <I>Dioptra</I> (<G>peri\ dio/ptras</G>).</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12342 <p>This treatise begins with a careful description of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12343 <I>dioptra</I>, an instrument which served with the ancients for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12344 the same purpose as a theodolite with us (chaps. 1-5). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12345 problems with which the treatise goes on to deal are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12346 (<I>a</I>) problems of ‘heights and distances’, (<I>b</I>) engineering pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12347 blems, (<I>c</I>) problems of mensuration, to which is added
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12348 (chap. 34) a description of a ‘hodometer’, or taxameter, con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12349 sisting of an arrangement of toothed wheels and endless
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12350 screws on the same axes working on the teeth of the next
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12351 wheels respectively. The book ends with the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12352 (chap. 37), ‘With a given force to move a given weight by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12353 means of interacting toothed wheels’, which really belongs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12354 to mechanics, and was apparently added, like some other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12355 problems (e.g. 31, ‘to measure the outflow of, i.e. the volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12356 of water issuing from, a spring’), in order to make the book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12357 more comprehensive. The essential problems dealt with are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12358 such as the following. To determine the difference of level
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12359 between two given points (6), to draw a straight line connect-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12360 ing two points the one of which is not visible from the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12361 (7), to measure the least breadth of a river (9), the distance of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12362 two inaccessible points (10), the height of an inaccessible point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12363 (12), to determine the difference between the heights of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12364 inaccessible points and the position of the straight line joining
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12365 them (13), the depth of a ditch (14); to bore a tunnel through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12366 a mountain going straight from one mouth to the other (15), to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12367 sink a shaft through a mountain perpendicularly to a canal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12368 flowing underneath (16); given a subterranean canal of any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12369 form, to find on the ground above a point from which a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12370 vertical shaft must be sunk in order to reach a given point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12371 on the canal (for the purpose e.g. of removing an obstruction)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12372 (20); to construct a harbour on the model of a given segment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12373 of a circle, given the ends (17), to construct a vault so that it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12374 may have a spherical surface modelled on a given segment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12375 (18). The mensuration problems include the following: to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12376 measure an irregular area, which is done by inscribing a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12377 rectilineal figure and then drawing perpendiculars to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12378 sides at intervals to meet the contour (23), or by drawing one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12379 straight line across the area and erecting perpendiculars from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12380 <pb n=346><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12381 that to meet the contour on both sides (24); given that all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12382 the boundary stones of a certain area have disappeared except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12383 two or three, but that the plan of the area is forthcoming,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12384 to determine the position of the lost boundary stones (25).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12385 Chaps. 26-8 remind us of the <I>Metrica</I>: to divide a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12386 area into given parts by straight lines drawn from one point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12387 (26); to measure a given area without entering it, whether
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12388 because it is thickly covered with trees, obstructed by houses,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12389 or entry is forbidden! (27); chaps. <MATH>28-30=<I>Metrica</I> III. 7,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12390 III. 1, and I. 7</MATH>, the last of these three propositions being the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12391 proof of the ‘formula of Heron’ for the area of a triangle in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12392 terms of the sides. Chap. 35 shows how to find the distance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12393 between Rome and Alexandria along a great circle of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12394 earth by means of the observation of the same eclipse at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12395 the two places, the <I>analemma</I> for Rome, and a concave hemi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12396 sphere constructed for Alexandria to show the position of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12397 sun at the time of the said eclipse. It is here mentioned that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12398 the estimate by Eratosthenes of the earth's circumference in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12399 his book <I>On the Measurement of the Earth</I> was the most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12400 accurate that had been made up to date.<note>Heron, vol. iii, p. 302. 13-17.</note> Some hold that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12401 the chapter, like some others which have no particular con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12402 nexion with the real subject of the <I>Dioptra</I> (e.g. chaps. 31, 34,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12403 37-8) were probably inserted by a later editor, ‘in order to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12404 make the treatise as complete as possible’.<note><I>Ib.</I>, p. 302. 9.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12405 <C>The <I>Mechanics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12406 <p>It is evident that the <I>Mechanics</I>, as preserved in the Arabic,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12407 is far from having kept its original form, especially in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12408 Book I. It begins with an account of the arrangement of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12409 toothed wheels designed to solve the problem of moving a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12410 given weight by a given force; this account is the same as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12411 that given at the end of the Greek text of the <I>Dioptra</I>, and it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12412 is clearly the same description as that which Pappus<note>Pappus, viii, p. 1060 sq.</note> found in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12413 the work of Heron entitled <G>*baroulko/s</G> (‘weight-lifter’) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12414 himself reproduced with a ratio of force to weight altered
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12415 from 5:1000 to 4:160 and with a ratio of 2:1 substituted for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12416 5:1 in the diameters of successive wheels. It would appear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12417 that the chapter from the <G>*baroulko/s</G> was inserted in place of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12418 <pb n=347><head>THE <I>MECHANICS</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12419 the first chapter or chapters of the real <I>Mechanics</I> which had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12420 been lost. The treatise would doubtless begin with generalities
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12421 introductory to mechanics such as we find in the (much
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12422 interpolated) beginning of Pappus, Book VIII. It must then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12423 apparently have dealt with the properties of circles, cylinders,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12424 and spheres with reference to their importance in mechanics;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12425 for in Book II. 21 Heron says that the circle is of all figures
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12426 the most movable and most easily moved, the same thing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12427 applying also to the cylinder and sphere, and he adds in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12428 support of this a reference to a proof ‘in the preceding Book’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12429 This reference may be to I. 21, but at the end of that chapter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12430 he says that ‘cylinders, even when heavy, if placed on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12431 ground so that they touch it in one line only, are easily
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12432 moved, and the same is true of spheres also, a matter <I>which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12433 we have already discussed’</I>; the discussion may have come
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12434 earlier in the Book, in a chapter now lost.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12435 <p>The treatise, beginning with chap. 2 after the passage
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12436 interpolated from the <G>*baroulko/s</G>, is curiously disconnected.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12437 Chaps. 2-7 discuss the motion of circles or wheels, equal or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12438 unequal, moving on different axes (e.g. interacting toothed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12439 wheels), or fixed on the same axis, much after the fashion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12440 the Aristotelian <I>Mechanical problems.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12441 <C><I>Aristotle's Wheel</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12442 <p>In particular (chap. 7) Heron attempts to explain the puzzle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12443 of the ‘Wheel of Aristotle’, which remained a puzzle up to quite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12444 modern times, and gave rise to the proverb, ‘rotam Aristotelis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12445 magis torquere, quo magis torqueretur’.<note>See Van Capelle, <I>Aristotelis quaestiones mechanicae</I>, 1812, p. 263 sq.</note> ‘The question is’, says
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12446 the Aristotelian problem 24, ‘why does the greater circle roll an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12447 equal distance with the lesser circle when they are placed about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12448 the same centre, whereas, when they roll separately, as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12449 size of one is to the size of the other, so are the straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12450 traversed by them to one another?’<note>Arist. <I>Mechanica</I>, 855 a 28.</note> Let <I>AC, BD</I> be quadrants
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12451 of circles with centre <I>O</I> bounded by the same radii, and draw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12452 tangents <I>AE, BF</I> at <I>A</I> and <I>B.</I> In the first case suppose the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12453 circle <I>BD</I> to roll along <I>BF</I> till <I>D</I> takes the position <I>H</I>; then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12454 the radius <I>ODC</I> will be at right angles to <I>AE</I>, and <I>C</I> will be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12455 at <I>G</I>, a point such that <I>AG</I> is equal to <I>BH.</I> In the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12456 <pb n=348><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12457 case suppose the circle <I>AC</I> to roll along <I>AE</I> till <I>ODC</I> takes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12458 the position <I>O</I>′<I>FE</I>; then <I>D</I> will be at <I>F</I> where <I>AE</I>=<I>BF</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12459 And similarly if a whole revolution is performed and <I>OBA</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12460 again perpendicular to <I>AE.</I> Contrary, therefore, to the prin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12461 ciple that the greater circle moves quicker than the smaller on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12462 the same axis, it would appear that the movement of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12463 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12464 smaller in this case is as quick as that of the greater, since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12465 <MATH><I>BH</I>=<I>AG</I></MATH>, and <MATH><I>BF</I>=<I>AE</I></MATH>. Heron's explanation is that, e.g.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12466 in the case where the larger circle rolls on <I>AE</I>, the lesser
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12467 circle maintains the same speed as the greater because it has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12468 <I>two</I> motions; for if we regard the smaller circle as merely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12469 fastened to the larger, and not rolling at all, its centre <I>O</I> will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12470 move to <I>O</I>′ traversing a distance <I>OO</I>′ equal to <I>AE</I> and <I>BF</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12471 hence the greater circle will take the lesser with it over an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12472 equal distance, the rolling of the lesser circle having no effect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12473 upon this.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12474 <C><I>The parallelogram of velocities</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12475 <p>Heron next proves the parallelogram of velocities (chap. 8);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12476 he takes the case of a rectangle, but the proof is applicable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12477 generally.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12478 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12479 <p>The way it is put is this. A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12480 point moves with uniform velocity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12481 along a straight line <I>AB</I>, from <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12482 to <I>B</I>, while at the same time <I>AB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12483 moves with uniform velocity always
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12484 parallel to itself with its extremity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12485 <I>A</I> describing the straight line <I>AC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12486 Suppose that, when the point arrives at <I>B</I>, the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12487 <pb n=349><head>THE PARALLELOGRAM OF VELOCITIES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12488 reaches the position <I>CD.</I> Let <I>EF</I> be any intermediate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12489 position of <I>AB</I>, and <I>G</I> the position at the same instant
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12490 of the moving point on it. Then clearly <MATH><I>AE</I>:<I>AC</I>=<I>EG</I>:<I>EF</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12491 therefore <MATH><I>AE</I>:<I>EG</I>=<I>AC</I>:<I>EF</I>=<I>AC</I>:<I>CD</I></MATH>, and it follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12492 <I>G</I> lies on the diagonal <I>AD</I>, which is therefore the actual path
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12493 of the moving point.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12494 <p>Chaps. 9-19 contain a digression on the construction of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12495 plane and solid figures similar to given figures but greater or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12496 less in a given ratio. Heron observes that the case of plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12497 figures involves the finding of a mean proportional between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12498 two straight lines, and the case of solid figures the finding of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12499 <I>two</I> mean proportionals; in chap. 11 he gives his solution of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12500 the latter problem, which is preserved in Pappus and Eutocius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12501 as well, and has already been given above (vol. i, pp. 262-3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12502 <p>The end of chap. 19 contains, quite inconsequently, the con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12503 struction of a toothed wheel to move on an endless screw,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12504 after which chap. 20 makes a fresh start with some observa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12505 tions on weights in equilibrium on a horizontal plane but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12506 tending to fall when the plane is inclined, and on the ready
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12507 mobility of objects of cylindrical form which touch the plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12508 in one line only.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12509 <C><I>Motion on an inclined plane</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12510 <p>When a weight is hanging freely by a rope over a pulley,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12511 no force applied to the other end of the rope less than the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12512 weight itself will keep it up, but, if the weight is placed on an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12513 inclined plane, and both the plane and the portion of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12514 weight in contact with it are smooth, the case is different.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12515 Suppose, e.g., that a weight in the form of a cylinder is placed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12516 on an inclined plane so that the line in which they touch is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12517 horizontal; then the force required to be applied to a rope
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12518 parallel to the line of greatest slope in the plane in order to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12519 keep the weight in equilibrium is less than the weight. For
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12520 the vertical plane passing through the line of contact between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12521 the cylinder and the plane divides the cylinder into two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12522 unequal parts, that on the downward side of the plane being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12523 the greater, so that the cylinder will tend to roll down; but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12524 the force required to support the cylinder is the ‘equivalent’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12525 not of the weight of the whole cylinder, but of the difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12526 <pb n=350><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12527 between the two portions into which the vertical plane cuts it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12528 (chap. 23).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12529 <C><I>On the centre of gravity</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12530 <p>This brings Heron to the centre of gravity (chap. 24). Here
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12531 a definition by Posidonius, a Stoic, of the ‘centre of gravity’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12532 or ‘centre of inclination’ is given, namely ‘a point such that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12533 if the body is hung up at it, the body is divided into two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12534 equal parts’ (he should obviously have said ‘divided <I>by any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12535 vertical plane through the point of suspension</I> into two equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12536 parts’). But, Heron says, Archimedes distinguished between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12537 the ‘centre of gravity’ and the ‘point of suspension’, defining
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12538 the latter as a point on the body such that, if the body is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12539 hung up at it, all the parts of the body remain in equilibrium
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12540 and do not oscillate or incline in any direction. ‘“Bodies”, said
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12541 Archimedes, “may rest (without inclining one way or another)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12542 with either a line, or only one point, in the body fixed”.’ The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12543 ‘centre of inclination’, says Heron, ‘is one single point in any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12544 particular body to which all the vertical lines through the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12545 points of suspension converge.’ Comparing Simplicius's quo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12546 tation of a definition by Archimedes in his <G>*kentrobarika/</G>, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12547 the effect that the centre of gravity is a certain point in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12548 body such that, if the body is hung up by a string attached to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12549 that point, it will remain in its position without inclining in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12550 any direction,<note>Simplicius on <I>De caelo</I>, p. 543. 31-4, Heib.</note> we see that Heron directly used a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12551 treatise of Archimedes. So evidently did Pappus, who has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12552 a similar definition. Pappus also speaks of a body supported
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12553 at a point by a vertical stick: if, he says, the body is in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12554 equilibrium, the line of the stick produced upwards must pass
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12555 through the centre of gravity.<note>Pappus, viii, p. 1032. 5-24.</note> Similarly Heron says that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12556 the same principles apply when the body is supported as when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12557 it is suspended. Taking up next (chaps. 25-31) the question
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12558 of ‘supports’, he considers cases of a heavy beam or a wall
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12559 supported on a number of pillars, equidistant or not, even
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12560 or not even in number, and projecting or not projecting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12561 beyond one or both of the extreme pillars, and finds how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12562 much of the weight is supported on each pillar. He says
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12563 that Archimedes laid down the principles in his ‘Book on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12564 <pb n=351><head>ON THE CENTRE OF GRAVITY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12565 Supports’. As, however, the principles are the same whether
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12566 the body is supported or hung up, it does not follow that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12567 this was a different work from that known as <G>peri\ zugw=n</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12568 Chaps. 32-3, which are on the principles of the lever or of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12569 weighing, end with an explanation amounting to the fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12570 that ‘greater circles overpower smaller when their movement
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12571 is about the same centre’, a proposition which Pappus says
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12572 that Archimedes proved in his work <G>peri\ zugw=n</G>.<note>Pappus, viii, p. 1068. 20-3.</note> In chap. 32,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12573 too, Heron gives as his authority a proof given by Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12574 in the same work. With I. 33 may be compared II. 7,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12575 where Heron returns to the same subject of the greater and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12576 lesser circles moving about the same centre and states the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12577 fact that weights reciprocally proportional to their radii are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12578 in equilibrium when suspended from opposite ends of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12579 horizontal diameters, observing that Archimedes proved the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12580 proposition in his work ‘On the equalization of inclination’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12581 (presumably <G>i)sorropi/ai</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12582 <C>Book II. The five mechanical powers.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12583 <p>Heron deals with the wheel and axle, the lever, the pulley,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12584 the wedge and the screw, and with combinations of these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12585 powers. The description of the powers comes first, chaps. 1-6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12586 and then, after II. 7, the proposition above referred to, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12587 theory of the several powers based upon it (chaps. 8-20).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12588 Applications to specific cases follow. Thus it is shown how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12589 to move a weight of 1000 talents by means of a force of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12590 5 talents, first by the system of wheels described in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12591 <G>*baroulko/s</G>, next by a system of pulleys, and thirdly by a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12592 combination of levers (chaps. 21-5). It is possible to combine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12593 the different powers (other than the wedge) to produce the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12594 same result (chap. 29). The wedge and screw are discussed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12595 with reference to their angles (chaps. 30-1), and chap. 32 refers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12596 to the effect of friction.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12597 <C><I>Mechanics in daily life; queries and answers</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12598 <p>After a prefatory chapter (33), a number of queries resem-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12599 bling the Aristotelian problems are stated and answered
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12600 (chap. 34), e.g. ‘Why do waggons with two wheels carry
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12601 a weight more easily than those with four wheels?’, ‘Why
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12602 <pb n=352><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12603 do great weights fall to the ground in a shorter time than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12604 lighter ones?’, ‘Why does a stick break sooner when one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12605 puts one's knee against it in the middle?’, ‘Why do people
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12606 use pincers rather than the hand to draw a tooth?’, ‘Why
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12607 is it easy to move weights which are suspended?’, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12608 ‘Why is it the more difficult to move such weights the farther
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12609 the hand is away from them, right up to the point of suspension
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12610 or a point near it?’, ‘Why are great ships turned by a rudder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12611 although it is so small?’, ‘Why do arrows penetrate armour
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12612 or metal plates but fail to penetrate cloth spread out?’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12613 <C><I>Problems on the centre of gravity, &c.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12614 <p>II. 35, 36, 37 show how to find the centre of gravity of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12615 a triangle, a quadrilateral and a pentagon respectively. Then,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12616 assuming that a triangle of uniform thickness is supported by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12617 a prop at each angle, Heron finds what weight is supported
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12618 by each prop, (<I>a</I>) when the props support the triangle only,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12619 (<I>b</I>) when they support the triangle plus a given weight placed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12620 at any point on it (chaps. 38, 39). Lastly, if known weights
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12621 are put on the triangle at each angle, he finds the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12622 gravity of the system (chap. 40); the problem is then extended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12623 to the case of any polygon (chap. 41).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12624 <p>Book III deals with the practical construction of engines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12625 for all sorts of purposes, machines employing pulleys with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12626 one, two, or more supports for lifting weights, oil-presses, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12627 <C>The <I>Catoptrica.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12628 <p>This work need not detain us long. Several of the theoretical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12629 propositions which it contains are the same as propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12630 in the so-called <I>Catoptrica</I> of Euclid, which, as we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12631 seen, was in all probability the work of Theon of Alexandria
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12632 and therefore much later in date. In addition to theoretical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12633 propositions, it contains problems the purpose of which is to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12634 construct mirrors or combinations of mirrors of such shape
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12635 as will reflect objects in a particular way, e.g. to make the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12636 right side appear as the right in the picture (instead of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12637 reverse), to enable a person to see his back or to appear in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12638 the mirror head downwards, with face distorted, with three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12639 eyes or two noses, and so forth. Concave and convex
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12640 <pb n=353><head>THE <I>CATOPTRICA</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12641 cylindrical mirrors play a part in these arrangements. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12642 whole theory of course ultimately depends on the main pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12643 positions 4 and 5 that the angles of incidence and reflection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12644 are equal whether the mirror is plane or circular.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12645 <C><I>Heron's proof of equality of angles of incidence and reflection</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12646 <p>Let <I>AB</I> be a plane mirror, <I>C</I> the eye, <I>D</I> the object seen.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12647 The argument rests on the fact that nature ‘does nothing in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12648 vain’. Thus light travels in a straight line, that is, by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12649 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12650 quickest road. Therefore, even
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12651 when the ray is a line broken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12652 at a point by reflection, it must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12653 mark the shortest broken line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12654 of the kind connecting the eye
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12655 and the object. Now, says
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12656 Heron, I maintain that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12657 shortest of the broken lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12658 (broken at the mirror) which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12659 connect <I>C</I> and <I>D</I> is the line, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12660 <I>CAD</I>, the parts of which make equal angles with the mirror.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12661 Join <I>DA</I> and produce it to meet in <I>F</I> the perpendicular from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12662 <I>C</I> to <I>AB.</I> Let <I>B</I> be any point on the mirror other than <I>A</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12663 and join <I>FB, BD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12664 <p>Now <MATH>&angle;<I>EAF</I>=&angle;<I>BAD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12665 =&angle;<I>CAE</I></MATH>, by hypothesis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12666 <p>Therefore the triangles <I>AEF, AEC</I>, having two angles equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12667 and <I>AE</I> common, are equal in all respects.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12668 <p>Therefore <MATH><I>CA</I>=<I>AF</I></MATH>, and <MATH><I>CA</I>+<I>AD</I>=<I>DF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12669 <p>Since <MATH><I>FE</I>=<I>EC</I></MATH>, and <I>BE</I> is perpendicular to <MATH><I>FC, BF</I>=<I>BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12670 <p>Therefore <MATH><I>CB</I>+<I>BD</I>=<I>FB</I>+<I>BD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12671 > <I>FD</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12672 i.e. > <I>CA</I>+<I>AD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12673 <p>The proposition was of course known to Archimedes. We
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12674 gather from a scholium to the Pseudo-Euclidean <I>Catoptrica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12675 that he proved it in a different way, namely by <I>reductio ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12676 absurdum</I>, thus: Denote the angles <I>CAE, DAB</I> by <G>a, b</G> re-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12677 spectively. Then, <G>a</G> is >= or <<G>b</G>. Suppose <G>a>b</G>. Then,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12678 <pb n=354><head>HERON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12679 reversing the ray so that the eye is at <I>D</I> instead of <I>C</I>, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12680 object at <I>C</I> instead of <I>D</I>, we must have <G>b>a</G>. But <G>b</G> was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12681 less than <G>a</G>, which is impossible. (Similarly it can be proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12682 that <G>a</G> is not less than <G>b</G>.) Therefore <MATH><G>a=b</G></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12683 <p>In the Pseudo-Euclidean <I>Catoptrica</I> the proposition is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12684 practically assumed; for the third assumption or postulate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12685 at the beginning states in effect that, in the above figure, if <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12686 be the point of incidence, <MATH><I>CE</I>:<I>EA</I>=<I>DH</I>:<I>HA</I></MATH> (where <I>DH</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12687 perpendicular to <I>AB</I>). It follows instantaneously (Prop. 1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12688 that &angle;<I>CAE</I>=&angle;<I>DAH</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12689 <p>If the mirror is the convex side of a circle, the same result
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12690 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12691 follows <I>a fortiori.</I> Let <I>CA, AD</I> meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12692 the arc at equal angles, and <I>CB, BD</I> at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12693 unequal angles. Let <I>AE</I> be the tan-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12694 gent at <I>A</I>, and complete the figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12695 Then, says Heron, (the angles <I>GAC,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12696 BAD</I> being by hypothesis equal), if we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12697 subtract the equal angles <I>GAE, BAF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12698 from the equal angles <I>GAC, BAD</I> (both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12699 pairs of angles being ‘mixed’, be it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12700 observed), we have &angle;<I>EAC</I>=&angle;<I>FAD.</I> Therefore <MATH><I>CA</I>+<I>AD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12701 <<I>CF</I>+<I>FD</I></MATH> and <I>a fortiori</I> <<I>CB</I>+<I>BD</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12702 <p>The problems solved (though the text is so corrupt in places
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12703 that little can be made of it) were such as the following:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12704 11, To construct a right-handed mirror (i.e. a mirror which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12705 makes the right side right and the left side left instead of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12706 the opposite); 12, to construct the mirror called <I>polytheoron</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12707 (‘with many images’); 16, to construct a mirror inside the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12708 window of a house, so that you can see in it (while inside
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12709 the room) everything that passes in the street; 18, to arrange
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12710 mirrors in a given place so that a person who approaches
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12711 cannot actually see either himself or any one else but can see
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12712 any image desired (a ‘ghost-seer’).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12713 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12714 <C>XIX
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12715 PAPPUS OF ALEXANDRIA</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12716 <p>WE have seen that the Golden Age of Greek geometry
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12717 ended with the time of Apollonius of Perga. But the influence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12718 of Euclid, Archimedes and Apollonius continued, and for some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12719 time there was a succession of quite competent mathematicians
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12720 who, although not originating anything of capital importance,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12721 kept up the tradition. Besides those who were known for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12722 particular investigations, e.g. of new curves or surfaces, there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12723 were such men as Geminus who, it cannot be doubted, were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12724 thoroughly familiar with the great classics. Geminus, as we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12725 have seen, wrote a comprehensive work of almost encyclopaedic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12726 character on the classification and content of mathematics,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12727 including the history of the development of each subject.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12728 But the beginning of the Christian era sees quite a different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12729 state of things. Except in sphaeric and astronomy (Menelaus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12730 and Ptolemy), production was limited to elementary text-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12731 books of decidedly feeble quality. In the meantime it would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12732 seem that the study of higher geometry languished or was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12733 completely in abeyance, until Pappus arose to revive interest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12734 in the subject. From the way in which he thinks it necessary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12735 to describe the contents of the classical works belonging to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12736 the <I>Treasury of Analysis</I>, for example, one would suppose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12737 that by his time many of them were, if not lost, completely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12738 forgotten, and that the great task which he set himself was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12739 the re-establishment of geometry on its former high plane of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12740 achievement. Presumably such interest as he was able to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12741 arouse soon flickered out, but for us his work has an in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12742 estimable value as constituting, after the works of the great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12743 mathematicians which have actually survived, the most im-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12744 portant of all our sources.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12745 <pb n=356><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12746 <C>Date of Pappus.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12747 <p>Pappus lived at the end of the third century A.D. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12748 authority for this date is a marginal note in a Leyden manu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12749 script of chronological tables by Theon of Alexandria, where,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12750 opposite to the name of Diocletian, a scholium says, ‘In his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12751 time Pappus wrote’. Diocletian reigned from 284 to 305,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12752 and this must therefore be the period of Pappus's literary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12753 activity. It is true that Suidas makes him a contemporary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12754 of Theon of Alexandria, adding that they both lived under
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12755 Theodosius I (379-395). But Suidas was evidently not well
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12756 acquainted with the works of Pappus; though he mentions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12757 a description of the earth by him and a commentary on four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12758 Books of Ptolemy's <I>Syntaxis</I>, he has no word about his greatest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12759 work, the <I>Synagoge.</I> As Theon also wrote a commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12760 Ptolemy and incorporated a great deal of the commentary of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12761 Pappus, it is probable that Suidas had Theon's commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12762 before him and from the association of the two names wrongly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12763 inferred that they were contemporaries.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12764 <C>Works (commentaries) other than the <I>Collection.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12765 <p>Besides the <I>Synagoge</I>, which is the main subject of this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12766 chapter, Pappus wrote several commentaries, now lost except for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12767 fragments which have survived in Greek or Arabic. One was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12768 a commentary on the <I>Elements</I> of Euclid. This must presum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12769 ably have been pretty complete, for, while Proclus (on Eucl. I)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12770 quotes certain things from Pappus which may be assumed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12771 have come in the notes on Book I, fragments of his commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12772 tary on Book X actually survive in the Arabic (see above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12773 vol. i, pp. 154-5, 209), and again Eutocius in his note on Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12774 medes, <I>On the Sphere and Cylinder</I>, I. 13, says that Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12775 explained in his commentary on the <I>Elements</I> how to inscribe
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12776 in a circle a polygon similar to a polygon inscribed in another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12777 circle, which problem would no doubt be solved by Pappus, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12778 it is by a scholiast, in a note on XII. 1. Some of the references
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12779 by Proclus deserve passing mention. (1) Pappus said that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12780 the converse of Post. 4 (equality of all right angles) is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12781 true, i.e. it is not true that all angles equal to a right angle are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12782 themselves right, since the ‘angle’ between the conterminous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12783 arcs of two semicircles which are equal and have their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12784 <pb n=357><head>WORKS OTHER THAN THE <I>COLLECTION</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12785 diameters at right angles and terminating at one point is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12786 equal to, but is not, a right angle.<note>Proclus on Eucl. I, pp. 189-90.</note> (2) Pappus said that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12787 in addition to the genuine axioms of Euclid, there were others
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12788 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12789 on record about unequals added to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12790 equals and equals added to unequals.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12791 Others given by Pappus are (says
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12792 Proclus) involved by the definitions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12793 e.g. that ‘all parts of the plane and of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12794 the straight line coincide with one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12795 another’, that ‘a point divides a line,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12796 a line a surface, and a surface a solid’, and that ‘the infinite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12797 is (obtained) in magnitudes both by addition and diminution’.<note><I>Ib.</I>, pp. 197. 6-198. 15.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12798 (3) Pappus gave a pretty proof of Eucl. I. 5, which modern
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12799 editors have spoiled when introducing it into text-books. If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12800 <I>AB, AC</I> are the equal sides in an isosceles triangle, Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12801 compares the triangles <I>ABC</I> and <I>ACB</I> (i.e. as if he were com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12802 paring the triangle <I>ABC</I> seen from the front with the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12803 triangle seen from the back), and shows that they satisfy the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12804 conditions of I. 4, so that they are equal in all respects, whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12805 the result follows.<note><I>Ib.</I>, pp. 249. 20-250. 12.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12806 <p>Marinus at the end of his commentary on Euclid's <I>Data</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12807 refers to a commentary by Pappus on that book.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12808 <p>Pappus's commentary on Ptolemy's <I>Syntaxis</I> has already
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12809 been mentioned (p. 274); it seems to have extended to six
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12810 Books, if not to the whole of Ptolemy's work. The <I>Fihrist</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12811 says that he also wrote a commentary on Ptolemy's <I>Plani-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12812 sphaerium</I>, which was translated into Arabic by Th&amacr;bit b.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12813 Qurra. Pappus himself alludes to his own commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12814 the <I>Analemma</I> of Diodorus, in the course of which he used the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12815 conchoid of Nicomedes for the purpose of trisecting an angle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12816 <p>We come now to Pappus's great work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12817 <C>The <I>Synagoge</I> or <I>Collection.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12818 <C>(<G>a</G>) <I>Character of the work; wide range.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12819 <p>Obviously written with the object of reviving the classical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12820 Greek geometry, it covers practically the whole field. It is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12821 <pb n=358><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12822 however, a handbook or guide to Greek geometry rather than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12823 an encyclopaedia; it was intended, that is, to be read with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12824 original works (where still extant) rather than to enable them
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12825 to be dispensed with. Thus in the case of the treatises
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12826 included in the <I>Treasury of Analysis</I> there is a general intro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12827 duction, followed by a general account of the contents, with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12828 lemmas, &c., designed to facilitate the reading of the treatises
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12829 themselves. On the other hand, where the history of a subject
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12830 is given, e.g. that of the problem of the duplication of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12831 cube or the finding of the two mean proportionals, the various
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12832 solutions themselves are reproduced, presumably because they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12833 were not easily accessible, but had to be collected from various
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12834 sources. Even when it is some accessible classic which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12835 being described, the opportunity is taken to give alternative
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12836 methods, or to make improvements in proofs, extensions, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12837 so on. Without pretending to great originality, the whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12838 work shows, on the part of the author, a thorough grasp of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12839 all the subjects treated, independence of judgment, mastery
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12840 of technique; the style is terse and clear; in short, Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12841 stands out as an accomplished and versatile mathematician,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12842 a worthy representative of the classical Greek geometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12843 <C>(<G>b</G>) <I>List of authors mentioned.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12844 <p>The immense range of the <I>Collection</I> can be gathered from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12845 a mere enumeration of the names of the various mathematicians
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12846 quoted or referred to in the course of it. The greatest of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12847 them, Euclid, Archimedes and Apollonius, are of course con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12848 tinually cited, others are mentioned for some particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12849 achievement, and in a few cases the mention of a name by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12850 Pappus is the whole of the information we possess about the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12851 person mentioned. In giving the list of the names occurring
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12852 in the book, it will, I think, be convenient and may economize
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12853 future references if I note in brackets the particular occasion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12854 of the reference to the writers who are mentioned for one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12855 achievement or as the authors of a particular book or investi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12856 gation. The list in alphabetical order is: Apollonius of Perga,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12857 Archimedes, Aristaeus the elder (author of a treatise in five
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12858 Books on the Elements of Conics or of ‘five Books on Solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12859 Loci connected with the conics’), Aristarchus of Samos (<I>On the</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12860 <pb n=359><head>THE <I>COLLECTION</I></head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12861 <I>sizes and distances of the sun and moon</I>), Autolycus (<I>On the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12862 moving sphere</I>), Carpus of Antioch (who is quoted as having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12863 said that Archimedes wrote only one mechanical book, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12864 on sphere-making, since he held the mechanical appliances
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12865 which made him famous to be nevertheless unworthy of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12866 written description: Carpus himself, who was known as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12867 <I>mechanicus</I>, applied geometry to other arts of this practical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12868 kind), Charmandrus (who added three simple and obvious loci
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12869 to those which formed the beginning of the <I>Plane Loci</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12870 Apollonius), Conon of Samos, the friend of Archimedes (cited
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12871 as the propounder of a theorem about the spiral in a plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12872 which Archimedes proved: this would, however, seem to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12873 a mistake, as Archimedes says at the beginning of his treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12874 that he sent certain theorems, without proofs, to Conon, who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12875 would certainly have proved them had he lived), Demetrius of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12876 Alexandria (mentioned as the author of a work called ‘Linear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12877 considerations’, <G>grammikai\ e)pista/seis</G>, i.e. considerations on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12878 curves, as to which nothing more is known), Dinostratus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12879 the brother of Menaechmus (cited, with Nicomedes, as having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12880 used the curve of Hippias, to which they gave the name of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12881 <I>quadratrix</I>, <G>tetragwni/zousa</G>, for the squaring of the circle),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12882 Diodorus (mentioned as the author of an <I>Analemma</I>), Erato-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12883 sthenes (whose <I>mean-finder</I>, an appliance for finding two or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12884 any number of geometric means, is described, and who is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12885 further mentioned as the author of two Books ‘On means’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12886 and of a work entitled ‘Loci with reference to means’),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12887 Erycinus (from whose <I>Paradoxa</I> are quoted various problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12888 seeming at first sight to be inconsistent with Eucl. I. 21, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12889 being shown that straight lines can be drawn from two points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12890 on the base of a triangle to a point within the triangle which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12891 are together greater than the other two sides, provided that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12892 points in the base may be points other than the extremities),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12893 Euclid, Geminus the mathematician (from whom is cited a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12894 remark on Archimedes contained in his book ‘On the classifica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12895 tion of the mathematical sciences’, see above, p. 223), Heraclitus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12896 (from whom Pappus quotes an elegant solution of a <G>neu=sis</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12897 with reference to a square), Hermodorus (Pappus's son, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12898 whom he dedicated Books VII, VIII of his <I>Collection</I>), Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12899 of Alexandria (whose mechanical works are extensively quoted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12900 from), Hierius the philosopher (a contemporary of Pappus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12901 <pb n=360><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12902 who is mentioned as having asked Pappus's opinion on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12903 attempted solution by ‘plane’ methods of the problem of the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12904 means, which actually gives a method of approximating to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12905 a solution<note>See vol. i, pp. 268-70.</note>), Hipparchus (quoted as practically adopting three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12906 of the hypotheses of Aristarchus of Samos), Megethion (to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12907 whom Pappus dedicated Book V of his <I>Collection</I>), Menelaus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12908 of Alexandria (quoted as the author of <I>Sphaerica</I> and as having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12909 applied the name <G>para/doxos</G> to a certain curve), Nicomachus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12910 (on three means additional to the first three), Nicomedes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12911 Pandrosion (to whom Book III of the <I>Collection</I> is dedicated),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12912 Pericles (editor of Euclid's <I>Data</I>), Philon of Byzantium (men-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12913 tioned along with Heron), Philon of Tyana (mentioned as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12914 discoverer of certain complicated curves derived from the inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12915 weaving of plectoid and other surfaces), Plato (with reference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12916 to the five regular solids), Ptolemy, Theodosius (author of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12917 <I>Sphaerica</I> and <I>On Days and Nights</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12918 <C>(<G>g</G>) <I>Translations and editions.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12919 <p>The first published edition of the <I>Collection</I> was the Latin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12920 translation by Commandinus (Venice 1589, but dated at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12921 end ‘Pisauri apud Hieronymum Concordiam 1588’; reissued
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12922 with only the title-page changed ‘Pisauri . . . 1602’). Up to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12923 1876 portions only of the Greek text had appeared, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12924 Books VII, VIII in Greek and German, by C. J. Gerhardt, 1871,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12925 chaps. 33-105 of Book V, by Eisenmann, Paris 1824, chaps.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12926 45-52 of Book IV in <I>Iosephi Torelli Veronensis Geometrica</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12927 1769, the remains of Book II, by John Wallis (in <I>Opera
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12928 mathematica</I>, III, Oxford 1699); in addition, the restorers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12929 of works of Euclid and Apollonius from the indications
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12930 furnished by Pappus give extracts from the Greek text
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12931 relating to the particular works, Bretonde Champ on Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12932 <I>Porisms</I>, Halley in his edition of the <I>Conics</I> of Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12933 (1710) and in his translation from the Arabic and restoration
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12934 respectively of the <I>De sectione rationis</I> and <I>De sectione spatii</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12935 of Apollonius (1706), Camerer on Apollonius's <I>Tactiones</I> (1795),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12936 Simson and Horsley in their restorations of Apollonius's <I>Plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12937 Loci</I> and <I>Inclinationes</I> published in the years 1749 and 1770
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12938 respectively. In the years 1876-8 appeared the only com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12939 <pb n=361><head>THE <I>COLLECTION.</I> BOOKS I, II, III</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12940 plete Greek text, with apparatus, Latin translation, com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12941 mentary, appendices and indices, by Friedrich Hultsch; this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12942 great edition is one of the first monuments of the revived
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12943 study of the history of Greek mathematics in the last half
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12944 of the nineteenth century, and has properly formed the model
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12945 for other definitive editions of the Greek text of the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12946 classical Greek mathematicians, e.g. the editions of Euclid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12947 Archimedes, Apollonius, &c., by Heiberg and others. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12948 Greek index in this edition of Pappus deserves special mention
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12949 because it largely serves as a dictionary of mathematical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12950 terms used not only in Pappus but by the Greek mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12951 maticians generally.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12952 <C>(<G>d</G>) <I>Summary of contents.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12953 <p>At the beginning of the work, Book I and the first 13 pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12954 positions (out of 26) of Book II are missing. The first 13
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12955 propositions of Book II evidently, like the rest of the Book,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12956 dealt with Apollonius's method of working with very large
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12957 numbers expressed in successive powers of the myriad, 10000.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12958 This system has already been described (vol. i, pp. 40, 54-7).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12959 The work of Apollonius seems to have contained 26 proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12960 tions (25 leading up to, and the 26th containing, the final
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12961 continued multiplication).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12962 <p>Book III consists of four sections. Section (1) is a sort of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12963 history of the problem of <I>finding two mean proportionals, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12964 continued proportion, between two given struight lines.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12965 <p>It begins with some general remarks about the distinction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12966 between theorems and problems. Pappus observes that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12967 whereas the ancients called them all alike by one name, some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12968 regarding them all as problems and others as theorems, a clear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12969 distinction was drawn by those who favoured more exact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12970 terminology. According to the latter a problem is that in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12971 which it is proposed to <I>do</I> or <I>construct</I> something, a theorem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12972 that in which, given certain hypotheses, we investigate that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12973 which follows from and is necessarily implied by them.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12974 Therefore he who propounds a theorem, no matter how he has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12975 become aware of the fact which is a necessary consequence of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12976 the premisses, must state, as the object of inquiry, the right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12977 result and no other. On the other hand, he who propounds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12978 <pb n=362><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12979 a problem may bid us do something which is in fact im-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12980 possible, and that without necessarily laying himself open
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12981 to blame or criticism. For it is part of the solver's duty
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12982 to determine the conditions under which the problem is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12983 possible or impossible, and, ‘if possible, when, how, and in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12984 how many ways it is possible’. When, however, a man pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12985 fesses to know mathematics and yet commits some elementary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12986 blunder, he cannot escape censure. Pappus gives, as an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12987 example, the case of an unnamed person ‘who was thought to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12988 be a great geometer’ but who showed ignorance in that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12989 claimed to know how to solve the problem of the two mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12990 proportionals by ‘plane’ methods (i.e. by using the straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12991 line and circle only). He then reproduces the argument of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12992 the anonymous person, for the purpose of showing that it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12993 does not solve the problem as its author claims. We have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12994 seen (vol. i, pp. 269-70) how the method, though not actually
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12995 solving the problem, does furnish a series of successive approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12996 mations to the real solution. Pappus adds a few simple
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12997 lemmas assumed in the exposition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12998 <p>Next comes the passage<note>Pappus, iii, p. 54. 7-22.</note>, already referred to, on the dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	12999 tinction drawn by the ancients between (1) <I>plane</I> problems or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13000 problems which can be solved by means of the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13001 and circle, (2) <I>solid</I> problems, or those which require for their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13002 solution one or more conic sections, (3) <I>linear</I> problems, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13003 those which necessitate recourse to higher curves still, curves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13004 with a more complicated and indeed a forced or unnatural
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13005 origin (<G>bebiasme/nhn</G>) such as spirals, quadratrices, cochloids
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13006 and cissoids, which have many surprising properties of their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13007 own. The problem of the two mean proportionals, being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13008 a <I>solid</I> problem, required for its solution either conics or some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13009 equivalent, and, as conics could not be constructed by purely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13010 geometrical means, various mechanical devices were invented
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13011 such as that of Eratosthenes (the <I>mean-finder</I>), those described
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13012 in the <I>Mechanics</I> of Philon and Heron, and that of Nicomedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13013 (who used the ‘cochloidal’ curve). Pappus proceeds to give the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13014 solutions of Eratosthenes, Nicomedes and Heron, and then adds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13015 a fourth which he claims as his own, but which is practically
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13016 the same as that attributed by Eutocius to Sporus. All these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13017 solutions have been given above (vol. i, pp. 258-64, 266-8).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13018 <pb n=363><head>THE <I>COLLECTION.</I> BOOK III</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13019 <C>Section (2). <I>The theory of means.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13020 <p>Next follows a section (pp. 69-105) on the theory of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13021 different kinds of <I>means.</I> The discussion takes its origin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13022 from the statement of the ‘second problem’, which was that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13023 of ‘exhibiting the three means’ (i.e. the arithmetic, geometric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13024 and harmonic) ‘in a semicircle’. Pappus first gives a con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13025 struction by which another geometer (<G>a)/llos tis</G>) claimed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13026 have solved this problem, but he does not seem to have under-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13027 stood it, and returns to the same problem later (pp. 80-2).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13028 <p>In the meantime he begins with the definitions of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13029 three means and then shows how, given any two of three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13030 terms <I>a, b, c</I> in arithmetical, geometrical or harmonical pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13031 gression, the third can be found. The definition of the mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13032 (<I>b</I>) of three terms <I>a, b, c</I> in harmonic progression being that it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13033 satisfies the relation <MATH><I>a:c</I>=<I>a--b:b--c</I>,</MATH> Pappus gives alternative
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13034 definitions for the arithmetic and geometric means in corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13035 sponding form, namely for the arithmetic mean <MATH><I>a:a</I>=<I>a--b:b--c</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13036 and for the geometric <MATH><I>a:b</I>=<I>a--b:b--c.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13037 <p>The construction for the harmonic mean is perhaps worth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13038 giving. Let <I>AB, BG</I> be two given straight lines. At <I>A</I> draw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13039 <I>DAE</I> perpendicular to <I>AB</I>, and make <I>DA, AE</I> equal. Join
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13040 <I>DB, BE.</I> From <I>G</I> draw <I>GF</I> at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13041 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13042 angles to <I>AB</I> meeting <I>DB</I> in <I>F.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13043 Join <I>EF</I> meeting <I>AB</I> in <I>C.</I> Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13044 <I>BC</I> is the required harmonic mean.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13045 <p>For
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13046 <MATH><I>AB:BG</I>=<I>DA:FG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13047 =<I>EA:FG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13048 =<I>AC:CG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13049 =<I>(AB--BC):(BC--BG).</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13050 <p>Similarly, by means of a like figure, we can find <I>BG</I> when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13051 <I>AB, BC</I> are given, and <I>AB</I> when <I>BC, BG</I> are given (in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13052 the latter case the perpendicular <I>DE</I> is drawn through <I>G</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13053 instead of <I>A</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13054 <p>Then follows a proposition that, if the three means and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13055 several extremes are represented in one set of lines, there must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13056 be five of them at least, and, after a set of five such lines have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13057 been found in the smallest possible integers, Pappus passes to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13058 <pb n=364><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13059 the problem of representing the three means with the respective
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13060 extremes by <I>six</I> lines drawn in a semicircle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13061 <p>Given a semicircle on the diameter <I>AC</I>, and <I>B</I> any point on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13062 the diameter, draw <I>BD</I> at right angles to <I>AC.</I> Let the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13063 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13064 at <I>D</I> meet <I>AC</I> produced in <I>G</I>, and measure <I>DH</I> along the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13065 tangent equal to <I>DG.</I> Join <I>HB</I> meeting the radius <I>OD</I> in <I>K.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13066 Let <I>BF</I> be perpendicular to <I>OD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13067 <p>Then, exactly as above, it is shown that <I>OK</I> is a harmonic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13068 mean between <I>OF</I> and <I>OD.</I> Also <I>BD</I> is the geometric mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13069 between <I>AB, BC</I>, while <I>OC</I> (=<I>OD</I>) is the arithmetic mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13070 between <I>AB, BC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13071 <p>Therefore the <I>six</I> lines <I>DO</I> (=<I>OC</I>), <I>OK, OF, AB, BC, BD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13072 supply the three means with the respective extremes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13073 <p>But Pappus seems to have failed to observe that the ‘certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13074 other geometer’, who has the same figure excluding the dotted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13075 lines, supplied the same in <I>five</I> lines. For he said that <I>DF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13076 is ‘a harmonic mean’. It is in fact the harmonic mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13077 between <I>AB, BC</I>, as is easily seen thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13078 <p>Since <I>ODB</I> is a right-angled triangle, and <I>BF</I> perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13079 to <I>OD</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13080 <MATH><I>DF:BD</I>=<I>BD:DO</I>,</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13081 or <MATH><I>DF.DO</I>=<I>BD</I><SUP>2</SUP>=<I>AB.BC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13082 <p>But <MATH><I>DO</I>=1/2(<I>AB</I>+<I>BC</I>);</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13083 therefore <MATH><I>DF.</I>(<I>AB</I>+<I>BC</I>)=2<I>AB.BC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13084 <p>Therefore <MATH><I>AB.(DF--BC)</I>=<I>BC.(AB--DF)</I>,</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13085 that is, <MATH><I>AB:BC</I>=<I>(AB--DF):(DF--BC)</I>,</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13086 and <I>DF</I> is the harmonic mean between <I>AB, BC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13087 <p>Consequently the <I>five</I> lines <I>DO</I> (=<I>OC</I>), <I>DF, AB, BC, BD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13088 exhibit all the three means with the extremes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13089 <pb n=365><head>THE <I>COLLECTION.</I> BOOK III</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13090 <p>Pappus does not seem to have seen this, for he observes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13091 that the geometer in question, though saying that <I>DF</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13092 a harmonic mean, does not say how it is a harmonic mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13093 or between what straight liues.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13094 <p>In the next chapters (pp. 84-104) Pappus, following Nico-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13095 machus and others, defines seven more means, three of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13096 were ancient and the last four more modern, and shows how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13097 we can form all ten means as linear functions of <G>a, b, g</G>, where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13098 <G>a, b, g</G> are in geometrical progression. The exposition has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13099 already been described (vol. i, pp. 86-9).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13100 <C>Section (3). <I>The ‘Paradoxes’ of Erycinus.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13101 <p>The third section of Book III (pp. 104-30) contains a series
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13102 of propositions, all of the same sort, which are curious rather
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13103 than geometrically important. They appear to have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13104 taken direct from a collection of <I>Paradoxes</I> by one Erycinus.<note>Pappus, iii, p. 106. 5-9.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13105 The first set of these propositions (Props. 28-34) are connected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13106 with Eucl. I. 21, which says that, if from the extremities
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13107 of the base of any triangle two straight lines be drawn meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13108 at any point within the triangle, the straight lines are together
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13109 less than the two sides of the triangle other than the base,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13110 but contain a greater angle. It is pointed out that, if the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13111 straight lines are allowed to be drawn from points in the base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13112 other than the extremities, their sum may be greater than the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13113 other two sides of the triangle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13114 <p>The first case taken is that of a right-angled triangle <I>ABC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13115 right-angled at <I>B.</I> Draw <I>AD</I> to any point <I>D</I> on <I>BC.</I> Measure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13116 on it <I>DE</I> equal to <I>AB</I>, bisect <I>AE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13117 in <I>F</I>, and join <I>FC.</I> Then shall
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13118 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13119 <MATH><I>DF</I>+<I>FC</I> be><I>BA</I>+<I>AC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13120 <p>For <MATH><I>EF</I>+<I>FC</I>=<I>AF</I>+<I>FC</I>><I>AC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13121 <p>Add <I>DE</I> and <I>AB</I> respectively,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13122 and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13123 <MATH><I>DF</I>+<I>FC</I>><I>BA</I>+<I>AC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13124 <p>More elaborate propositions are next proved, such as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13125 following.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13126 <p>1. In any triangle, except an equilateral triangle or an isosceles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13127 <pb n=366><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13128 triangle with base less than one of the other sides, it is possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13129 to construct on the base and within the triangle two straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13130 lines meeting at a point, the sum of which is <I>equal</I> to the sum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13131 of the other two sides of the triangle (Props. 29, 30).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13132 <p>2. In any triangle in which it is possible to construct two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13133 straight lines from the base to one internal point the sum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13134 of which is equal to the sum of the two sides of the triangle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13135 it is also possible to construct two other such straight lines the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13136 sum of which is <I>greater</I> than that sum (Prop. 31).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13137 <p>3. Under the same conditions, if the base is greater than either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13138 of the other two sides, two straight lines can be so constructed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13139 from the base to an internal point which are <I>respectively</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13140 greater than the other two sides of the triangle; and the lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13141 may be constructed so as to be respectively <I>equal</I> to the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13142 sides, if one of those two sides is less than the other and each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13143 of them is less than the base (Props. 32, 33).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13144 <p>4. The lines may be so constructed that their sum will bear to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13145 the sum of the two sides of the triangle any ratio less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13146 2:1 (Prop. 34).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13147 <p>As examples of the proofs, we will take the case of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13148 scalene triangle, and prove the first and Part 1 of the third of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13149 the above propositions for such a triangle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13150 <p>In the triangle <I>ABC</I> with base <I>BC</I> let <I>AB</I> be greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13151 than <I>AC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13152 <p>Take <I>D</I> on <I>BA</I> such that <MATH><I>BD</I>=1/2 (<I>BA</I>+<I>AC</I>).</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13153 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13154 <p>On <I>DA</I> between <I>D</I> and <I>A</I> take any point <I>E</I>, and draw <I>EF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13155 parallel to <I>BC.</I> Let <I>G</I> be any point on <I>EF</I>; draw <I>GH</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13156 to <I>AB</I> and join <I>GC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13157 <pb n=367><head>THE <I>COLLECTION.</I> BOOK III</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13158 <p>Now <MATH><I>EA</I>+<I>AC</I>><I>EF</I>+<I>FC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13159 ><I>EG</I>+<I>GC</I> and ><I>GC</MATH>, a fortiori.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13160 <p>Produce <I>GC</I> to <I>K</I> so that <MATH><I>GK</I>=<I>EA</I>+<I>AC</I>,</MATH> and with <I>G</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13161 centre and <I>GK</I> as radius describe a circle. This circle will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13162 meet <I>HC</I> and <I>HG</I>, because <MATH><I>GH</I>=<I>EB</I>><I>BD</I> or <I>DA</I>+<I>AC</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13163 ><I>GK</MATH>, a fortiori.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13164 <p>Then <MATH><I>HG</I>+<I>GL</I>=<I>BE</I>+<I>EA</I>+<I>AC</I>=<I>BA</I>+<I>AC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13165 <p>To obtain two straight lines <I>HG</I>′, <I>G</I>′<I>L</I> such that <MATH><I>HG</I>′+<I>G</I>′<I>L</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13166 ><I>BA</I>+<I>AC</I>,</MATH> we have only to choose <I>G</I>′ so that <I>HG</I>′, <I>G</I>′<I>L</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13167 enclose the straight lines <I>HG, GL</I> completely.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13168 <p>Next suppose that, given a triangle <I>ABC</I> in which <I>BC</I>><I>BA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13169 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13170 ><I>AC</I>, we are required to draw from two points on <I>BC</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13171 an internal point two straight lines greater <I>respectively</I> than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13172 <I>BA, AC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13173 <p>With <I>B</I> as centre and <I>BA</I> as radius describe the arc <I>AEF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13174 Take any point <I>E</I> on it, and any point <I>D</I> on <I>BE</I> produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13175 but within the triangle. Join <I>DC</I>, and produce it to <I>G</I> so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13176 that <MATH><I>DG</I>=<I>AC.</I></MATH> Then with <I>D</I> as centre and <I>DG</I> as radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13177 describe a circle. This will meet both <I>BC</I> and <I>BD</I> because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13178 <MATH><I>BA</I>><I>AC</I>,</MATH> and <I>a fortiori</I> <MATH><I>DB</I>><I>DG.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13179 <p>Then, if <I>L</I> be any point on <I>BH</I>, it is clear that <I>BD, DL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13180 are two straight lines satisfying the conditions.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13181 <p>A point <I>L</I>′ on <I>BH</I> can be found such that <I>DL</I>′ is <I>equal</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13182 to <I>AB</I> by marking off <I>DN</I> on <I>DB</I> equal to <I>AB</I> and drawing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13183 with <I>D</I> as centre and <I>DN</I> as radius a circle meeting <I>BH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13184 in <I>L</I>′. Also, if <I>DH</I> be joined, <MATH><I>DH</I>=<I>AC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13185 <p>Propositions follow (35-9) having a similar relation to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13186 Postulate in Archimedes, <I>On the Sphere and Cylinder</I>, I,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13187 about conterminous broken lines one of which wholly encloses
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13188 <pb n=368><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13189 the other, i.e. it is shown that broken lines, consisting of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13190 several straight lines, can be drawn with two points on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13191 base of a triangle or parallelogram as extremities, and of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13192 greater total length than the remaining two sides of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13193 triangle or three sides of the parallelogram.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13194 <p>Props. 40-2 show that triangles or parallelograms can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13195 constructed with sides respectively greater than those of a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13196 triangle or parallelogram but having a less area.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13197 <C>Section (4). <I>The inscribing of the five regular solids
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13198 in a sphere.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13199 <p>The fourth section of Book III (pp. 132-62) solves the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13200 problems of inscribing each of the five regular solids in a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13201 given sphere. After some preliminary lemmas (Props. 43-53),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13202 Pappus attacks the substantive problems (Props. 54-8), using
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13203 the method of analysis followed by synthesis in the case of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13204 each solid.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13205 <p>(<I>a</I>) In order to inscribe a regular pyramid or tetrahedron in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13206 the sphere, he finds two circular sections equal and parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13207 to one another, each of which contains one of two opposite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13208 edges as its diameter. If <I>d</I> be the diameter of the sphere, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13209 parallel circular sections have <I>d</I>′ as diameter, where <MATH><I>d</I><SUP>2</SUP>=3/2<I>d</I>′<SUP>2</SUP>.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13210 <p>(<I>b</I>) In the case of the cube Pappus again finds two parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13211 circular sections with diameter <I>d</I>′ such that <MATH><I>d</I><SUP>2</SUP>=3/2<I>d</I>′<SUP>2</SUP>;</MATH> a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13212 inscribed in one of these circles is one face of the cube and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13213 the square with sides parallel to those of the first square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13214 inscribed in the second circle is the opposite face.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13215 <p>(<I>c</I>) In the case of the octahedron the same two parallel circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13216 sections with diameter <I>d</I>′ such that <MATH><I>d</I><SUP>2</SUP>=3/2<I>d</I>′<SUP>2</SUP></MATH> are used; an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13217 equilateral triangle inscribed in one circle is one face, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13218 opposite face is an equilateral triangle inscribed in the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13219 circle but placed in exactly the opposite way.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13220 <p>(<I>d</I>) In the case of the icosahedron Pappus finds four parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13221 circular sections each passing through three of the vertices of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13222 the icosahedron; two of these are small circles circumscribing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13223 two opposite triangular faces respectively, and the other two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13224 circles are between these two circles, parallel to them, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13225 equal to one another. The pairs of circles are determined in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13226 <pb n=369><head>THE <I>COLLECTION.</I> BOOKS III, IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13227 this way. If <I>d</I> be the diameter of the sphere, set out two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13228 straight lines <I>x, y</I> such that <I>d, x, y</I> are in the ratio of the sides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13229 of the regular pentagon, hexagon and decagon respectively
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13230 described in one and the same circle. The smaller pair of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13231 circles have <I>r</I> as radius where <MATH><I>r</I><SUP>2</SUP>=1/3<I>y</I><SUP>2</SUP>,</MATH> and the larger pair
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13232 have <I>r</I>′ as radius where <MATH><I>r</I>′<SUP>2</SUP>=1/3<I>x</I><SUP>2</SUP>.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13233 <p>(<I>e</I>) In the case of the dodecahedron the <I>same</I> four parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13234 circular sections are drawn as in the case of the icosahedron.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13235 Inscribed pentagons set the opposite way are inscribed in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13236 two smaller circles; these pentagons form opposite faces.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13237 Regular pentagons inscribed in the larger circles with vertices
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13238 at the proper points (and again set the opposite way) determine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13239 ten more vertices of the inscribed dodecahedron.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13240 <p>The constructions are quite different from those in Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13241 XIII. 13, 15, 14, 16, 17 respectively, where the problem is first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13242 to construct the particular regular solid and then to ‘com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13243 prehend it in a sphere’, i.e. to determine the circumscribing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13244 sphere in each case. I have set out Pappus's propositions in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13245 detail elsewhere.<note><I>Vide</I> notes to Euclid's propositions in <I>The Thirteen Books of Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13246 Elements</I>, pp. 473, 480, 477, 489-91, 501-3.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13247 <C>Book IV.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13248 <p>At the beginning of Book IV the title and preface are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13249 missing, and the first section of the Book begins immediately
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13250 with an enunciation. The first section (pp. 176-208) contains
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13251 Propositions 1-12 which, with the exception of Props. 8-10,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13252 seem to be isolated propositions given for their own sakes and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13253 not connected by any general plan.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13254 <C>Section (1). <I>Extension of the theorem of Pythagoras.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13255 <p>The first proposition is of great interest, being the generaliza-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13256 tion of Eucl. I. 47, as Pappus himself calls it, which is by this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13257 time pretty widely known to mathematicians. The enunciation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13258 is as follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13259 <p>‘If <I>ABC</I> be a triangle and on <I>AB, AC</I> any parallelograms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13260 whatever be described, as <I>ABDE, ACFG</I>, and if <I>DE, FG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13261 produced meet in <I>H</I> and <I>HA</I> be joined, then the parallelo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13262 grams <I>ABDE, ACFG</I> are together equal to the parallelogram
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13263 <pb n=370><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13264 contained by <I>BC, HA</I> in an angle which is equal to the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13265 the angles <I>ABC, DHA.</I>’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13266 <p>Produce <I>HA</I> to meet <I>BC</I> in <I>K</I>, draw <I>BL, CM</I> parallel to <I>KH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13267 meeting <I>DE</I> in <I>L</I> and <I>FG</I> in <I>M</I>, and join <I>LNM.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13268 <p>Then <I>BLHA</I> is a parallelogram, and <I>HA</I> is equal and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13269 parallel to <I>BL.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13270 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13271 <p>Similarly <I>HA, CM</I> are equal and parallel; therefore <I>BL, CM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13272 are equal and parallel.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13273 <p>Therefore <I>BLMC</I> is a parallelogram; and its angle <I>LBK</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13274 equal to the sum of the angles <I>ABC, DHA.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13275 <p>Now <MATH><I>&rect; ABDE</I>=<I>&rect; BLHA</I>,</MATH> in the same parallels,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13276 <MATH>=<I>&rect; BLNK</I>,</MATH> for the same reason.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13277 <p>Similarly <MATH><I>&rect; ACFG</I>=<I>&rect; ACMH</I>=<I>&rect; NKCM.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13278 <p>Therefore, by addition, <MATH><I>&rect; ABDE</I>+<I>&rect; ACFG</I>=<I>&rect; BLMC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13279 <p>It has been observed (by Professor Cook Wilson<note><I>Mathematical Gazette</I>, vii, p. 107 (May 1913).</note>) that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13280 parallelograms on <I>AB, AC</I> need not necessarily be erected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13281 <I>outwards</I> from <I>AB, AC.</I> If one of them, e.g. that on <I>AC</I>, be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13282 drawn inwards, as in the second figure above, and Pappus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13283 construction be made, we have a similar result with a negative
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13284 sign, namely,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13285 <MATH><I>&rect; BLMC</I>=<I>&rect; BLNK -- &rect; CMNK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13286 =<I>&rect; ABDE -- &rect; ACFG.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13287 <p>Again, if both <I>ABDE</I> and <I>ACFG</I> were drawn inwards, their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13288 sum would be equal to <I>BLMC</I> drawn <I>outwards.</I> Generally, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13289 the areas of the parallelograms described outwards are regarded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13290 as of opposite sign to those of parallelograms drawn inwards,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13291 <pb n=371><head>THE <I>COLLECTION.</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13292 we may say that the algebraic sum of the three parallelograms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13293 is equal to zero.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13294 <p>Though Pappus only takes one case, as was the Greek habit,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13295 I see no reason to doubt that he was aware of the results
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13296 in the other possible cases.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13297 <p>Props. 2, 3 are noteworthy in that they use the method and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13298 phraseology of Eucl. X, proving that a certain line in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13299 figure is the irrational called <I>minor</I> (see Eucl. X. 76), and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13300 a certain line in another figure is ‘the excess by which the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13301 <I>binomial</I> exceeds the <I>straight line which produces with a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13302 rational area a medial whole’</I> (Eucl. X. 77). The propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13303 4-7 and 11-12 are quite interesting as geometrical exercises,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13304 but their bearing is not obvious: Props. 4 and 12 are remark-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13305 able in that they are cases of analysis followed by synthesis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13306 applied to the proof of <I>theorems.</I> Props. 8-10 belong to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13307 subject of <I>tangencies</I>, being the sort of propositions that would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13308 come as particular cases in a book such as that of Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13309 <I>On Contacts</I>; Prop. 8 shows that, if there are two equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13310 circles and a given point outside both, the diameter of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13311 circle passing through the point and touching both circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13312 is ‘given’; the proof is in many places obscure and assumes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13313 lemmas of the same kind as those given later à propos of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13314 Apollonius's treatise; Prop. 10 purports to show how, given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13315 three unequal circles touching one another two and two, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13316 find the diameter of the circle including them and touching
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13317 all three.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13318 <C>Section (2). <I>On circles inscribed in the</I> <G>a)/bhlos</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13319 (<I>‘shoemaker's knife’</I>).</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13320 <p>The next section (pp. 208-32), directed towards the demon-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13321 stration of a theorem about the relative sizes of successive
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13322 circles inscribed in the <G>a)/rbhlos</G> (shoemaker's knife), is ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13323 tremely interesting and clever, and I wish that I had space
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13324 to reproduce it completely. The <G>a)/rbhlos</G>, which we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13325 already met with in Archimedes's ‘Book of Lemmas’, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13326 formed thus. <I>BC</I> is the diameter of a semicircle <I>BGC</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13327 <I>BC</I> is divided into two parts (in general unequal) at <I>D</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13328 semicircles are described on <I>BD, DC</I> as diameters on the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13329 side of <I>BC</I> as <I>BGC</I> is; the figure included between the three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13330 semicircles is the <G>a)/rbhlos</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13331 <pb n=372><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13332 <p>There is, says Pappus, on record an ancient proposition to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13333 the following effect. Let successive circles be inscribed in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13334 <G>a)/rbhlos</G> touching the semicircles and one another as shown
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13335 in the figure on p. 376, their centres being <I>A, P, O</I> .... Then, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13336 <I>p</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB> ... be the perpendiculars from the centres <I>A, P, O</I> ...
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13337 on <I>BC</I> and <I>d</I><SUB>1</SUB>, <I>d</I><SUB>2</SUB>, <I>d</I><SUB>3</SUB> ... the diameters of the corresponding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13338 circles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13339 <MATH><I>p</I><SUB>1</SUB>=<I>d</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>=2<I>d</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB>=3<I>d</I><SUB>3</SUB>....</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13340 <p>He begins by some lemmas, the course of which I shall
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13341 reproduce as shortly as I can.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13342 <p>I. If (Fig. 1) two circles with centres <I>A, C</I> of which the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13343 former is the greater touch externally at <I>B,</I> and another circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13344 with centre <I>G</I> touches the two circles at <I>K, L</I> respectively,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13345 then <I>KL</I> produced cuts the circle <I>BL</I> again in <I>D</I> and meets
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13346 <I>AC</I> produced in a point <I>E</I> such that <MATH><I>AB</I>:<I>BC</I>=<I>AE</I>:<I>EC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13347 This is easily proved, because the circular segments <I>DL, LK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13348 are similar, and <I>CD</I> is parallel to <I>AG.</I> Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13349 <MATH><I>AB</I>:<I>BC</I>=<I>AK</I>:<I>CD</I>=<I>AE</I>:<I>EC.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13350 <p>Also <MATH><I>KE.EL</I>=<I>EB</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13351 <p>For <MATH><I>AE</I>:<I>EC</I>=<I>AB</I>:<I>BC</I>=<I>AB</I>:<I>CF</I>=(<I>AE</I>-<I>AB</I>):(<I>EC</I>-<I>CF</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13352 =<I>BE</I>:<I>EF.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13353 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13354 <CAP>FIG 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13355 <p>But <MATH><I>AE</I>:<I>EC</I>=<I>KE</I>:<I>ED</I></MATH>; therefore <MATH><I>KE</I>:<I>ED</I>=<I>BE</I>:<I>EF.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13356 <p>Therefore <MATH><I>KE.EL</I>:<I>EL.ED</I>=<I>BE</I><SUP>2</SUP>:<I>BE.EF.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13357 <p>And <MATH><I>EL.ED</I>=<I>BE.EF</I></MATH>; therefore <MATH><I>KE.EL</I>=<I>EB</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13358 <pb n=373><head>THE <I>COLLECTION.</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13359 <p>II. Let (Fig. 2) <I>BC, BD,</I> being in one straight line, be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13360 diameters of two semicircles <I>BGC, BED,</I> and let any circle as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13361 <I>FGH</I> touch both semicircles, <I>A</I> being the centre of the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13362 Let <I>M</I> be the foot of the perpendicular from <I>A</I> on <I>BC, r</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13363 radius of the circle <I>FGH.</I> There are two cases according
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13364 as <I>BD</I> lies along <I>BC</I> or <I>B</I> lies between <I>D</I> and <I>C;</I> i.e. in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13365 first case the two semicircles are the outer and one of the inner
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13366 semicircles of the <G>a)/rbhlos</G>, while in the second case they are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13367 the two inner semicircles; in the latter case the circle <I>FGH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13368 may either include the two semicircles or be entirely external
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13369 to them. Now, says Pappus, it is to be proved that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13370 in case (1) <MATH><I>BM</I>:<I>r</I>=(<I>BC</I>+<I>BD</I>):(<I>BC</I>-<I>BD</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13371 and in case (2) <MATH><I>BM</I>:<I>r</I>=(<I>BC</I>-<I>BD</I>):(<I>BC</I>+<I>BD</I>).</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13372 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13373 <CAP>FIG 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13374 <p>We will confine ourselves to the first case, represented in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13375 the figure (Fig. 2).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13376 <p>Draw through <I>A</I> the diameter <I>HF</I> parallel to <I>BC.</I> Then,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13377 since the circles <I>BGC, HGF</I> touch at <I>G,</I> and <I>BC, HF</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13378 parallel diameters, <I>GHB, GFC</I> are both straight lines.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13379 <p>Let <I>E</I> be the point of contact of the circles <I>FGH</I> and <I>BED;</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13380 then, similarly, <I>BEF, HED</I> are straight lines.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13381 <p>Let <I>HK, FL</I> be drawn perpendicular to <I>BC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13382 <p>By the similar triangles <I>BGC, BKH</I> we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13383 <MATH><I>BC</I>:<I>BG</I>=<I>BH</I>:<I>BK,</I></MATH> or <MATH><I>CB.BK</I>=<I>GB.BH</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13384 and by the similar triangles <I>BLF, BED</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13385 <MATH><I>BF</I>:<I>BL</I>=<I>BD</I>:<I>BE,</I></MATH> or <MATH><I>DB.BL</I>=<I>FB.BE.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13386 <pb n=374><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13387 <p>But <MATH><I>GB.BH</I>=<I>FB.BE</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13388 therefore <MATH><I>CB.BK</I>=<I>DB.BL</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13389 or <MATH><I>BC</I>:<I>BD</I>=<I>BL</I>:<I>BK.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13390 <p>Therefore <MATH>(<I>BC</I>+<I>BD</I>):(<I>BC</I>-<I>BD</I>)=(<I>BL</I>+<I>BK</I>):(<I>BL</I>-<I>BK</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13391 =2<I>BM</I>:<I>KL.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13392 <p>And <MATH><I>KL</I>=<I>HF</I>=2<I>r</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13393 therefore <MATH><I>BM</I>:<I>r</I>=(<I>BC</I>+<I>BD</I>):(<I>BC</I>-<I>BD</I>).</MATH> (<I>a</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13394 <p>It is next proved that <MATH><I>BK.LC</I>=<I>AM</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13395 <p>For, by similar triangles <I>BKH, FLC,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13396 <MATH><I>BK</I>:<I>KH</I>=<I>FL</I>:<I>LC,</I></MATH> or <MATH><I>BK.LC</I>=<I>KH.FL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13397 =<I>AM</I><SUP>2</SUP></MATH>. (<I>b</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13398 <p>Lastly, since <MATH><I>BC</I>:<I>BD</I>=<I>BL</I>:<I>BK,</I></MATH> from above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13399 <MATH><I>BC</I>:<I>CD</I>=<I>BL</I>:<I>KL,</I></MATH> or <MATH><I>BL.CD</I>=<I>BC.KL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13400 =<I>BC.</I>2<I>r.</I></MATH> (<I>c</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13401 <p>Also <MATH><I>BD</I>:<I>CD</I>=<I>BK</I>:<I>KL,</I></MATH> or <MATH><I>BK.CD</I>=<I>BD.KL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13402 =<I>BD.</I>2<I>r.</I></MATH> (<I>d</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13403 <p>III. We now (Fig. 3) take any two circles touching the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13404 semicircles <I>BGC, BED</I> and one another. Let their centres be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13405 <I>A</I> and <I>P, H</I> their point of contact, <I>d, d</I>′ their diameters respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13406 tively. Then, if <I>AM, PN</I> are drawn perpendicular to <I>BC,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13407 Pappus proves that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13408 <MATH>(<I>AM</I>+<I>d</I>):<I>d</I>=<I>PN</I>:<I>d</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13409 <p>Draw <I>BF</I> perpendicular to <I>BC</I> and therefore touching the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13410 semicircles <I>BGC, BED</I> at <I>B.</I> Join <I>AP,</I> and produce it to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13411 meet <I>BF</I> in <I>F.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13412 <p>Now, by II. (<I>a</I>) above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13413 <MATH>(<I>BC</I>+<I>BD</I>):(<I>BC</I>-<I>BD</I>)=<I>BM</I>:<I>AH,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13414 and for the same reason =<I>BN</I>:<I>PH;</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13415 it follows that <MATH><I>AH</I>:<I>PH</I>=<I>BM</I>:<I>BN</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13416 =<I>FA</I>:<I>FP.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13417 <pb n=375><head>THE <I>COLLECTION,</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13418 <p>Therefore (Lemma I), if the two circles touch the semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13419 circle <I>BED</I> in <I>R, E</I> respectively, <I>FRE</I> is a straight line and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13420 <MATH><I>EF.FR</I>=<I>FH</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13421 <p>But <MATH><I>EF.FR</I>=<I>FB</I><SUP>2</SUP></MATH>; therefore <MATH><I>FH</I>=<I>FB.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13422 <p>If now <I>BH</I> meets <I>PN</I> in <I>O</I> and <I>MA</I> produced in <I>S,</I> we have,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13423 by similar triangles, <MATH><I>FH</I>:<I>FB</I>=<I>PH</I>:<I>PO</I>=<I>AH</I>:<I>AS,</I></MATH> whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13424 <MATH><I>PH</I>=<I>PO</I></MATH> and <MATH><I>SA</I>=<I>AH,</I></MATH> so that <I>O, S</I> are the intersections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13425 of <I>PN, AM</I> with the respective circles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13426 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13427 <CAP>FIG 3.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13428 <p>Join <I>BP,</I> and produce it to meet <I>MA</I> in <I>K.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13429 <p>Now <MATH><I>BM</I>:<I>BN</I>=<I>FA</I>:<I>FP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13430 =<I>AH</I>:<I>PH,</I> from above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13431 =<I>AS</I>:<I>PO.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13432 <p>And <MATH><I>BM</I>:<I>BN</I>=<I>BK</I>:<I>BP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13433 =<I>KS</I>:<I>PO.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13434 <p>Therefore <MATH><I>KS</I>=<I>AS,</I></MATH> and <MATH><I>KA</I>=<I>d,</I></MATH> the diameter of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13435 circle <I>EHG.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13436 <p>Lastly, <MATH><I>MK</I>:<I>KS</I>=<I>PN</I>:<I>PO,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13437 that is, <MATH>(<I>AM</I>+<I>d</I>):1/2<I>d</I>=<I>PN</I>:1/2<I>d</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13438 or <MATH>(<I>AM</I>+<I>d</I>):<I>d</I>=<I>PN</I>:<I>d</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13439 <pb n=376><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13440 <p>IV. We now come to the substantive theorem.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13441 <p>Let <I>FGH</I> be the circle touching all three semicircles (Fig. 4).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13442 We have then, as in Lemma II,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13443 <MATH><I>BC.BK</I>=<I>BD.BL,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13444 and for the same reason (regarding <I>FGH</I> as touching the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13445 semicircles <I>BGC, DUC</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13446 <MATH><I>BC.CL</I>=<I>CD.CK.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13447 <p>From the first relation we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13448 <MATH><I>BC</I>:<I>BD</I>=<I>BL</I>:<I>BK,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13449 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13450 <CAP>FIG 4.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13451 whence <MATH><I>DC</I>:<I>BD</I>=<I>KL</I>:<I>BK,</I></MATH> and inversely <MATH><I>BD</I>:<I>DC</I>=<I>BK</I>:<I>KL,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13452 while, from the second relation, <MATH><I>BC</I>:<I>CD</I>=<I>CK</I>:<I>CL,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13453 whence <MATH><I>BD</I>:<I>DC</I>=<I>KL</I>:<I>CL.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13454 <p>Consequently <MATH><I>BK</I>:<I>KL</I>=<I>KL</I>:<I>CL,</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13455 or <MATH><I>BK.LC</I>=<I>KL</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13456 <p>But we saw in Lemma II (<I>b</I>) that <MATH><I>BK.LC</I>=<I>AM</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13457 <p>Therefore <MATH><I>KL</I>=<I>AM,</I></MATH> or <MATH><I>p</I><SUB>1</SUB>=<I>d</I><SUB>1</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13458 <p>For the second circle Lemma III gives us
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13459 <MATH>(<I>p</I><SUB>1</SUB>+<I>d</I><SUB>1</SUB>):<I>d</I><SUB>1</SUB>=<I>p</I><SUB>2</SUB>:<I>d</I><SUB>2</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13460 whence, since <MATH><I>p</I><SUB>1</SUB>=<I>d</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>=2<I>d</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13461 <p>For the third circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13462 <MATH>(<I>p</I><SUB>2</SUB>+<I>d</I><SUB>2</SUB>):<I>d</I><SUB>2</SUB>=<I>p</I><SUB>3</SUB>:<I>d</I><SUB>3</SUB></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13463 whence <MATH><I>p</I><SUB>3</SUB>=3<I>d</I><SUB>3</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13464 <p>And so on <I>ad infinitum.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13465 <pb n=377><head>THE <I>COLLECTION.</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13466 <p>The same proposition holds when the successive circles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13467 instead of being placed between the large and one of the small
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13468 semicircles, come down between the two small semicircles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13469 <p>Pappus next deals with special cases (1) where the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13470 smaller semicircles become straight lines perpendicular to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13471 diameter of the other semicircle at its extremities, (2) where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13472 we replace one of the smaller semicircles by a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13473 through <I>D</I> at right angles to <I>BC,</I> and lastly (3) where instead
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13474 of the semicircle <I>DUC</I> we simply have the straight line <I>DC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13475 and make the first circle touch it and the two other semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13476 circles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13477 <p>Pappus's propositions of course include as particular cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13478 the partial propositions of the same kind included in the ‘Book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13479 of Lemmas’ attributed to Archimedes (Props. 5, 6); cf. p. 102.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13480 <C>Sections (3) and (4). <I>Methods of squaring the circle, and of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13481 trisecting (or dividing in any ratio) any given angle.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13482 <p>The last sections of Book IV (pp. 234-302) are mainly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13483 devoted to the solutions of the problems (1) of squaring or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13484 rectifying the circle and (2) of trisecting any given angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13485 or dividing it into two parts in any ratio. To this end Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13486 gives a short account of certain curves which were used for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13487 the purpose.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13488 <C>(<G>a</G>) <I>The Archimedean spiral.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13489 <p>He begins with the spiral of Archimedes, proving some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13490 of the fundamental properties. His method of finding the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13491 area included (1) between the first turn and the initial line,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13492 (2) between any radius vector on the first turn and the curve,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13493 is worth giving because it differs from the method of Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13494 medes. It is the area of the whole first turn which Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13495 works out in detail. We will take the area up to the radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13496 vector <I>OB,</I> say.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13497 <p>With centre <I>O</I> and radius <I>OB</I> draw the circle <I>A</I>′<I>BCD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13498 <p>Let <I>BC</I> be a certain fraction, say 1/<I>n</I>th, of the arc <I>BCDA</I>′,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13499 and <I>CD</I> the same fraction, <I>OC, OD</I> meeting the spiral in <I>F, E</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13500 respectively. Let <I>KS, SV</I> be the same fraction of a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13501 line <I>KR,</I> the side of a square <I>KNLR.</I> Draw <I>ST, VW</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13502 to <I>KN</I> meeting the diagonal <I>KL</I> of the square in <I>U, Q</I> respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13503 tively, and draw <I>MU, PQ</I> parallel to <I>KR.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13504 <pb n=378><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13505 <p>With <I>O</I> as centre and <I>OE, OF</I> as radii draw arcs of circles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13506 meeting <I>OF, OB</I> in <I>H, G</I> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13507 <p>For brevity we will now denote a cylinder in which <I>r</I> is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13508 radius of the base and <I>h</I> the height by (cyl. <I>r, h</I>) and the cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13509 with the same base and height by (cone <I>r, h</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13510 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13511 <p>By the property of the spiral,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13512 <MATH><I>OB</I>:<I>BG</I>=(arc <I>A</I>′<I>DCB</I>):(arc <I>CB</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13513 =<I>RK</I>:<I>KS</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13514 =<I>NK</I>:<I>KM</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13515 whence <MATH><I>OB</I>:<I>OG</I>=<I>NK</I>:<I>NM.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13516 <p>Now
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13517 <MATH>(sector <I>OBC</I>):(sector <I>OGF</I>)=<I>OB</I><SUP>2</SUP>:<I>OG</I><SUP>2</SUP>=<I>NK</I><SUP>2</SUP>:<I>MN</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13518 =(cyl. <I>KN, NT</I>):(cyl. <I>MN, NT</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13519 <p>Similarly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13520 <MATH>(sector <I>OCD</I>):(sector <I>OEH</I>)=(cyl. <I>ST, TW</I>):(cyl. <I>PT, TW</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13521 and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13522 <p>The sectors <I>OBC, OCD</I> ... form the sector <I>OA</I>′<I>DB,</I> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13523 sectors <I>OFG, OEH</I> ... form a figure inscribed to the spiral.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13524 In like manner the cylinders (<I>KN, TN</I>), (<I>ST, TW</I>) ... form the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13525 cylinder (<I>KN, NL</I>), while the cylinders (<I>MN, NT</I>), (<I>PT, TW</I>) ...
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13526 form a figure inscribed to the cone (<I>KN, NL</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13527 <p>Consequently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13528 <MATH>(sector <I>OA</I>′<I>DB</I>):(fig. inscr. in spiral)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13529 =(cyl. <I>KN, NL</I>):(fig. inscr. in cone <I>KN, NL</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13530 <pb n=379><head>THE <I>COLLECTION.</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13531 <p>We have a similar proportion connecting a figure circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13532 scribed to the spiral and a figure circumscribed to the cone.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13533 <p>By increasing <I>n</I> the inscribed and circumscribed figures can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13534 be compressed together, and by the usual method of exhaustion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13535 we have ultimately
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13536 <MATH>(sector <I>OA</I>′<I>DB</I>):(area of spiral)=(cyl. <I>KN, NL</I>):(cone <I>KN, NL</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13537 =3:1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13538 or <MATH>(area of spiral cut off by <I>OB</I>)=1/3(sector <I>OA</I>′<I>DB</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13539 <p>The ratio of the sector <I>OA</I>′<I>DB</I> to the complete circle is that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13540 of the angle which the radius vector describes in passing from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13541 the position <I>OA</I> to the position <I>OB</I> to four right angles, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13542 is, by the property of the spiral, <I>r</I>:<I>a,</I> where <MATH><I>r</I>=<I>OB,</I> <I>a</I>=<I>OA.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13543 <p>Therefore <MATH>(area of spiral cut off by <I>OB</I>)=1/3<I>r/a</I>.<G>p</G><I>r</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13544 <p>Similarly the area of the spiral cut off by any other radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13545 vector <MATH><I>r</I>′=1/3<I>r</I>′/<I>a</I>.<G>p</G><I>r</I>′<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13546 <p>Therefore (as Pappus proves in his next proposition) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13547 first area is to the second as <I>r</I><SUP>3</SUP> to <I>r</I>′<SUP>3</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13548 <p>Considering the areas cut off by the radii vectores at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13549 points where the revolving line has passed through angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13550 of 1/2<G>p</G>, <G>p</G>, 3/2<G>p</G> and 2<G>p</G> respectively, we see that the areas are in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13551 the ratio of (1/4)<SUP>3</SUP>, (1/2)<SUP>3</SUP>, (3/4)<SUP>3</SUP>, 1 or 1, 8, 27, 64, so that the areas of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13552 the spiral included in the four quadrants are in the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13553 of 1, 7, 19, 37 (Prop. 22).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13554 <C>(<G>b</G>) <I>The conchoid of Nicomedes.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13555 <p>The conchoid of Nicomedes is next described (chaps. 26-7),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13556 and it is shown (chaps. 28, 29) how it can be used to find two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13557 geometric means between two straight lines, and consequently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13558 to find a cube having a given ratio to a given cube (see vol. i,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13559 pp. 260-2 and pp. 238-40, where I have also mentioned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13560 Pappus's remark that the conchoid which he describes is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13561 <I>first</I> conchoid, while there also exist a <I>second,</I> a <I>third</I> and a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13562 <I>fourth</I> which are of use for other theorems).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13563 <C>(<G>g</G>) <I>The quadratrix.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13564 <p>The <I>quadratrix</I> is taken next (chaps. 30-2), with Sporus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13565 criticism questioning the construction as involving a <I>petitio</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13566 <pb n=380><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13567 <I>principii.</I> Its use for squaring the circle is attributed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13568 Dinostratus and Nicomedes. The whole substance of this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13569 subsection is given above (vol. i, pp. 226-30).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13570 <C><I>Two constructions for the quadratrix by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13571 ‘surface-loci’.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13572 <p>In the next chapters (chaps. 33, 34, Props. 28, 29) Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13573 gives two alternative ways of producing the <I>quadratrix</I> ‘by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13574 means of surface-loci’, for which he claims the merit that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13575 they are geometrical rather than ‘too mechanical’ as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13576 traditional method (of Hippias) was.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13577 <p>(1) The first method uses a cylindrical helix thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13578 <p>Let <I>ABC</I> be a quadrant of a circle with centre <I>B,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13579 let <I>BD</I> be any radius. Suppose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13580 that <I>EF,</I> drawn from a point <I>E</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13581 on the radius <I>BD</I> perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13582 to <I>BC,</I> is (for all such radii) in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13583 a given ratio to the arc <I>DC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13584 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13585 <p>‘I say’, says Pappus, ‘that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13586 locus of <I>E</I> is a certain curve.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13587 <p>Suppose a right cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13588 erected from the quadrant and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13589 a cylindrical helix <I>CGH</I> drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13590 upon its surface. Let <I>DH</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13591 the generator of this cylinder through <I>D,</I> meeting the helix
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13592 in <I>H.</I> Draw <I>BL, EI</I> at right angles to the plane of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13593 quadrant, and draw <I>HIL</I> parallel to <I>BD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13594 <p>Now, by the property of the helix, <MATH><I>EI</I>(=<I>DH</I>)</MATH> is to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13595 arc <I>CD</I> in a given ratio. Also <MATH><I>EF</I>:(arc <I>CD</I>)=a given ratio.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13596 <p>Therefore the ratio <I>EF</I>:<I>EI</I> is given. <I>A</I>nd since <I>EF, EI</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13597 given in position, <I>FI</I> is given in position. But <I>FI</I> is perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13598 dicular to <I>BC.</I> Therefore <I>FI</I> is in a plane given in position,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13599 and so therefore is <I>I.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13600 <p>But <I>I</I> is also on a certain surface described by the line <I>LH,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13601 which moves always parallel to the plane <I>ABC,</I> with one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13602 extremity <I>L</I> on <I>BL</I> and the other extremity <I>H</I> on the helix.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13603 Therefore <I>I</I> lies on the intersection of this surface with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13604 plane through <I>FI.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13605 <pb n=381><head>THE <I>COLLECTION.</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13606 <p>Hence <I>I</I> lies on a certain curve. Therefore <I>E,</I> its projection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13607 on the plane <I>ABC,</I> also lies on a curve.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13608 <p>In the particular case where the given ratio of <I>EF</I> to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13609 arc <I>CD</I> is equal to the ratio of <I>BA</I> to the arc <I>CA,</I> the locus of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13610 <I>E</I> is a <I>quadratrix.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13611 <p>[The surface described by the straight line <I>LH</I> is a <I>plectoid.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13612 The shape of it is perhaps best realized as a <I>continuous</I> spiral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13613 staircase, i.e. a spiral staircase with infinitely small steps.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13614 The <I>quadratrix</I> is thus produced as the orthogonal projection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13615 of the curve in which the plectoid is intersected by a plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13616 through <I>BC</I> inclined at a given angle to the plane <I>ABC.</I> It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13617 not difficult to verify the result analytically.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13618 <p>(2) The second method uses a right cylinder the base of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13619 is an Archimedean spiral.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13620 <p>Let <I>ABC</I> be a quadrant of a circle, as before, and <I>EF,</I> per-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13621 pendicular at <I>F</I> to <I>BC,</I> a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13622 line of such length that <I>EF</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13623 to the arc <I>DC</I> as <I>AB</I> is to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13624 arc <I>ADC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13625 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13626 <p>Let a point on <I>AB</I> move uni-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13627 formly from <I>A</I> to <I>B</I> while, in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13628 same time, <I>AB</I> itself revolves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13629 uniformly about <I>B</I> from the position <I>BA</I> to the position <I>BC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13630 The point thus describes the spiral <I>AGB.</I> If the spiral cuts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13631 <I>BD</I> in <I>G,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13632 <MATH><I>BA</I>:<I>BG</I>=(arc <I>ADC</I>):(arc <I>DC</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13633 or <MATH><I>BG</I>:(arc <I>DC</I>)=<I>BA</I>:(arc <I>ADC</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13634 <p>Therefore <MATH><I>BG</I>=<I>EF.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13635 <p>Draw <I>GK</I> at right angles to the plane <I>ABC</I> and equal to <I>BG.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13636 Then <I>GK,</I> and therefore <I>K,</I> lies on a right cylinder with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13637 spiral as base.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13638 <p>But <I>BK</I> also lies on a conical surface with vertex <I>B</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13639 its generators all make an angle of 1/4<G>p</G> with the plane <I>ABC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13640 <p>Consequently <I>K</I> lies on the intersection of two surfaces,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13641 and therefore on a curve.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13642 <p>Through <I>K</I> draw <I>LKI</I> parallel to <I>BD,</I> and let <I>BL, EI</I> be at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13643 right angles to the plane <I>ABC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13644 <p>Then <I>LKI,</I> moving always parallel to the plane <I>ABC,</I> with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13645 one extremity on <I>BL</I> and passing through <I>K</I> on a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13646 <pb n=382><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13647 curve, describes a certain plectoid, which therefore contains the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13648 point <I>I.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13649 <p>Also <MATH><I>IE</I>=<I>EF</I></MATH>, <I>IF</I> is perpendicular to <I>BC,</I> and hence <I>IF,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13650 therefore <I>I,</I> lies on a fixed plane through <I>BC</I> inclined to <I>ABC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13651 at an angle of 1/4<G>p</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13652 <p>Therefore <I>I,</I> lying on the intersection of the plectoid and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13653 said plane, lies on a certain curve. So therefore does the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13654 projection of <I>I</I> on <I>ABC,</I> i.e. the point <I>E.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13655 <p>The locus of <I>E</I> is clearly the <I>quadratrix.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13656 <p>[This result can also be verified analytically.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13657 <C>(<G>d</G>) <I>Digression: a spiral on a sphere.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13658 <p>Prop. 30 (chap. 35) is a digression on the subject of a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13659 spiral described on a sphere, suggested by the discussion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13660 a spiral in a plane.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13661 <p>Take a hemisphere bounded by the great circle <I>KLM,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13662 with <I>H</I> as pole. Suppose that the quadrant of a great circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13663 <I>HNK</I> revolves uniformly about the radius <I>HO</I> so that <I>K</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13664 describes the circle <I>KLM</I> and returns to its original position
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13665 at <I>K,</I> and suppose that a point moves uniformly at the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13666 time from <I>H</I> to <I>K</I> at such speed that the point arrives at <I>K</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13667 at the same time that <I>HK</I> resumes its original position. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13668 point will thus describe a spiral on the surface of the sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13669 between the points <I>H</I> and <I>K</I> as shown in the figure.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13670 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13671 <p>Pappus then sets himself to prove that the portion of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13672 surface of the sphere cut off towards the pole between the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13673 spiral and the arc <I>HNK</I> is to the surface of the hemisphere in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13674 <pb n=383><head>THE <I>COLLECTION.</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13675 a certain ratio shown in the second figure where <I>ABC</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13676 a quadrant of a circle equal to a great circle in the sphere,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13677 namely the ratio of the segment <I>ABC</I> to the sector <I>DABC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13678 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13679 <p>Draw the tangent <I>CF</I> to the quadrant at <I>C.</I> With <I>C</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13680 centre and radius <I>CA</I> draw the circle <I>AEF</I> meeting <I>CF</I> in <I>F.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13681 <p>Then the sector <I>CAF</I> is equal to the sector <I>ADC</I> (since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13682 <MATH><I>CA</I><SUP>2</SUP>=2<I>AD</I><SUP>2</SUP></MATH>, while <MATH>&angle;<I>ACF</I>=1/2&angle;<I>ADC</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13683 <p>It is required, therefore, to prove that, if <I>S</I> be the area cut
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13684 off by the spiral as above described,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13685 <MATH><I>S</I>:(surface of hemisphere)=(segmt. <I>ABC</I>):(sector <I>CAF</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13686 <p>Let <I>KL</I> be a (small) fraction, say 1/<I>n</I>th, of the circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13687 ference of the circle <I>KLM,</I> and let <I>HPL</I> be the quadrant of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13688 great circle through <I>H, L</I> meeting the spiral in <I>P.</I> Then, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13689 the property of the spiral,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13690 <MATH>(arc <I>HP</I>):(arc <I>HL</I>)=(arc <I>KL</I>):(circumf. of <I>KLM</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13691 =1:<I>n.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13692 <p>Let the small circle <I>NPQ</I> passing through <I>P</I> be described
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13693 about the pole <I>H.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13694 <p>Next let <I>FE</I> be the same fraction, 1/<I>n</I>th, of the arc <I>FA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13695 that <I>KL</I> is of the circumference of the circle <I>KLM,</I> and join <I>EC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13696 meeting the arc <I>ABC</I> in <I>B.</I> With <I>C</I> as centre and <I>CB</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13697 radius describe the arc <I>BG</I> meeting <I>CF</I> in <I>G.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13698 <p>Then the arc <I>CB</I> is the same fraction, 1/<I>n</I>th, of the arc
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13699 <I>CBA</I> that the arc <I>FE</I> is of <I>FA</I> (for it is easily seen that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13700 <MATH>&angle;<I>FCE</I>=1/2&angle;<I>BDC,</I></MATH> while <MATH>&angle;<I>FCA</I>=1/2&angle;<I>CDA</I>)</MATH>. Therefore, since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13701 <MATH>(arc <I>CBA</I>)=(arc <I>HPL</I>), (arc <I>CB</I>)=(arc <I>HP</I>)</MATH>, and chord <MATH><I>CB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13702 =chord <I>HP.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13703 <pb n=384><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13704 <p>Now <MATH>(sector <I>HPN</I> on sphere):(sector <I>HKL</I> on sphere)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13705 =(chord <I>HP</I>)<SUP>2</SUP>:(chord <I>HL</I>)<SUP>2</SUP></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13706 (a consequence of Archimedes, <I>On Sphere and Cylinder,</I> I. 42).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13707 <p>And <MATH><I>HP</I><SUP>2</SUP>:<I>HL</I><SUP>2</SUP>=<I>CB</I><SUP>2</SUP>:<I>CA</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13708 =<I>CB</I><SUP>2</SUP>:<I>CE</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13709 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13710 <MATH>(sector <I>HPN</I>):(sector <I>HKL</I>)=(sector <I>CBG</I>):(sector <I>CEF</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13711 <p>Similarly, if the arc <I>LL</I>′ be taken equal to the arc <I>KL</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13712 the great circle through <I>H, L</I>′ cuts the spiral in <I>P</I>′, and a small
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13713 circle described about <I>H</I> and through <I>P</I>′ meets the arc <I>HPL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13714 in <I>p;</I> and if likewise the arc <I>BB</I>′ is made equal to the arc <I>BC,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13715 and <I>CB</I>′ is produced to meet <I>AF</I> in <I>E</I>′, while again a circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13716 arc with <I>C</I> as centre and <I>CB</I>′ as radius meets <I>CE</I> in <I>b,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13717 <MATH>(sector <I>HP</I>′<I>p</I> on sphere):(sector <I>HLL</I>′ on sphere)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13718 =(sector <I>CB</I>′<I>b</I>):(sector <I>CE</I>′<I>E</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13719 <p>And so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13720 <p>Ultimately then we shall get a figure consisting of sectors
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13721 on the sphere circumscribed about the area <I>S</I> of the spiral and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13722 a figure consisting of sectors of circles circumscribed about the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13723 segment <I>CBA;</I> and in like manner we shall have inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13724 figures in each case similarly made up.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13725 <p>The method of exhaustion will then give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13726 <MATH><I>S</I>:(surface of hemisphere)=(segmt. <I>ABC</I>):(sector <I>CAF</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13727 =(segmt. <I>ABC</I>):(sector <I>DAC</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13728 <p>[We may, as an illustration, give the analytical equivalent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13729 of this proposition. If <G>r, w</G> be the spherical coordinates of <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13730 with reference to <I>H</I> as pole and the arc <I>HNK</I> as polar axis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13731 the equation of Pappus's curve is obviously <MATH><G>w</G>=4<G>r</G>.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13732 <p>If now the radius of the sphere is taken as unity, we have as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13733 the element of area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13734 <MATH><I>dA</I>=<I>d</I><G>w</G>(1-cos<G>r</G>)=4<I>d</I><G>r</G>(1-cos<G>r</G>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13735 <p>Therefore <MATH><I>A</I>=∫<SUP>1/2<G>p</G></SUP><SUB>0</SUB>4<I>d</I><G>r</G>(1-cos<G>r</G>)=2<G>p</G>-4</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13736 <pb n=385><head>THE <I>COLLECTION.</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13737 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13738 <MATH><I>A</I>/(surface of hemisphere)=(2<G>p</G>-4)/(2<G>p</G>)=(1/4<G>p</G>-1/2)/(1/4<G>p</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13739 =(segment <I>ABC</I>)/(sector <I>DABC</I>).]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13740 <p>The second part of the last section of Book IV (chaps. 36-41,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13741 pp. 270-302) is mainly concerned with the problem of tri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13742 secting any given angle or dividing it into parts in any given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13743 ratio. Pappus begins with another account of the distinction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13744 between <I>plane, solid</I> and <I>linear</I> problems (cf. Book III, chaps.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13745 20-2) according as they require for their solution (1) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13746 straight line and circle only, (2) conics or their equivalent,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13747 (3) higher curves still, ‘which have a more complicated and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13748 forced (or unnatural) origin, being produced from more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13749 irregular surfaces and involved motions. Such are the curves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13750 which are discovered in the so-called <I>loci on surfaces,</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13751 well as others more complicated still and many in number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13752 discovered by Demetrius of Alexandria in his <I>Linear con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13753 siderations</I> and by Philon of Tyana by means of the inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13754 lacing of plectoids and other surfaces of all sorts, all of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13755 curves possess many remarkable properties peculiar to them.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13756 Some of these curves have been thought by the more recent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13757 writers to be worthy of considerable discussion; one of them is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13758 that which also received from Menelaus the name of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13759 <I>paradoxical</I> curve. Others of the same class are spirals,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13760 quadratrices, cochloids and cissoids.’ He adds the often-quoted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13761 reflection on the error committed by geometers when they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13762 solve a problem by means of an ‘inappropriate class’ (of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13763 curve or its equivalent), illustrating this by the use in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13764 Apollonius, Book V, of a rectangular hyperbola for finding the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13765 feet of normals to a <I>parabola</I> passing through one point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13766 (where a circle would serve the purpose), and by the assump-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13767 tion by Archimedes of a <I>solid</I> <G>neu=sis</G> in his book <I>On Spirals</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13768 (see above, pp. 65-8).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13769 <C><I>Trisection (or division in any ratio) of any angle.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13770 <p>The method of trisecting any angle based on a certain <G>neu=sis</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13771 is next described, with the solution of the <G>neu=sis</G> itself by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13772 <pb n=386><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13773 means of a hyperbola which has to be constructed from certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13774 data, namely the asymptotes and a certain point through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13775 which the curve must pass (this easy construction is given in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13776 Prop. 33, chap. 41-2). Then the problem is directly solved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13777 (chaps. 43, 44) by means of a hyperbola in two ways prac-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13778 tically equivalent, the hyperbola being determined in the one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13779 case by the ordinary Apollonian property, but in the other by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13780 means of the <I>focus-directrix</I> property. Solutions follow of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13781 the problem of dividing any angle in a given ratio by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13782 (1) of the <I>quadratrix,</I> (2) of the spiral of Archimedes (chaps.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13783 45, 46). All these solutions have been sufficiently described
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13784 above (vol. i, pp. 235-7, 241-3, 225-7).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13785 <p>Some problems follow (chaps. 47-51) depending on these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13786 results, namely those of constructing an isosceles triangle in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13787 which either of the base angles has a given ratio to the vertical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13788 angle (Prop. 37), inscribing in a circle a regular polygon of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13789 any number of sides (Prop. 38), drawing a circle the circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13790 ference of which shall be equal to a given straight line (Prop.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13791 39), constructing on a given straight line <I>AB</I> a segment of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13792 a circle such that the arc of the segment may have a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13793 ratio to the base (Prop. 40), and constructing an angle incom-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13794 mensurable with a given angle (Prop. 41).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13795 <C>Section (5). <I>Solution of the</I> <G>neu=sis</G> <I>of Archimedes, ‘On Spirals’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13796 Prop. 8, by means of conics.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13797 <p>Book IV concludes with the solution of the <G>neu=sis</G> which,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13798 according to Pappus, Archimedes unnecessarily assumed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13799 <I>On Spirals,</I> Prop. 8. Archimedes's assumption is this. Given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13800 a circle, a chord (<I>BC</I>) in it less than the diameter, and a point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13801 <I>A</I> on the circle the perpendicular from which to <I>BC</I> cuts <I>BC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13802 in a point <I>D</I> such that <I>BD</I>><I>DC</I> and meets the circle again
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13803 in <I>E,</I> it is possible to draw through <I>A</I> a straight line <I>ARP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13804 cutting <I>BC</I> in <I>R</I> and the circle in <I>P</I> in such a way that <I>RP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13805 shall be equal to <I>DE</I> (or, in the phraseology of <G>neu/seis</G>, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13806 place between the straight line <I>BC</I> and the circumference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13807 of the circle a straight line equal to <I>DE</I> and <I>verging</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13808 towards <I>A</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13809 <p>Pappus makes the problem rather more general by not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13810 requiring <I>PR</I> to be equal to <I>DE,</I> but making it of any given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13811 <pb n=387><head>THE <I>COLLECTION.</I> BOOK IV</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13812 length (consistent with a real solution). The problem is best
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13813 exhibited by means of analytical geometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13814 <p>If <MATH><I>BD</I>=<I>a,</I> <I>DC</I>=<I>b,</I> <I>AD</I>=<I>c</I></MATH> (so that <MATH><I>DE</I>=<I>ab/c</I></MATH>), we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13815 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13816 to find the point <I>R</I> on <I>BC</I> such that <I>AR</I> produced solves the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13817 problem by making <I>PR</I> equal to <I>k,</I> say.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13818 <p>Let <MATH><I>DR</I>=<I>x.</I></MATH> Then, since <MATH><I>BR.RC</I>=<I>PR.RA,</I></MATH> we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13819 <MATH>(<I>a</I>-<I>x</I>)(<I>b</I>+<I>x</I>)=<I>k</I>√(<I>c</I><SUP>2</SUP>+<I>x</I><SUP>2</SUP>).</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13820 <p>An obvious expedient is to put <I>y</I> for √(<I>c</I><SUP>2</SUP>+<I>x</I><SUP>2</SUP>, when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13821 we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13822 <MATH>(<I>a</I>-<I>x</I>)(<I>b</I>+<I>x</I>)=<I>ky,</I></MATH> (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13823 and <MATH><I>y</I><SUP>2</SUP>=<I>c</I><SUP>2</SUP>+<I>x</I><SUP>2</SUP></MATH>. (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13824 <p>These equations represent a parabola and a hyperbola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13825 respectively, and Pappus does in fact solve the problem by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13826 means of the intersection of a parabola and a hyperbola; one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13827 of his preliminary lemmas is, however, again a little more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13828 general. In the above figure <I>y</I> is represented by <I>RQ.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13829 <p>The first lemma of Pappus (Prop. 42, p. 298) states that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13830 from a given point <I>A</I> any straight line be drawn meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13831 a straight line <I>BC</I> given in position in <I>R,</I> and if <I>RQ</I> be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13832 at right angles to <I>BC</I> and of length bearing a given ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13833 to <I>AR,</I> the locus of <I>Q</I> is a <I>hyperbola.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13834 <p>For draw <I>AD</I> perpendicular to <I>BC</I> and produce it to <I>A</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13835 so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13836 <MATH><I>QR</I>:<I>RA</I>=<I>A</I>′<I>D</I>:<I>DA</I>=the given ratio</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13837 <pb n=388><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13838 <p>Measure <I>DA</I>″ along <I>DA</I> equal to <I>DA</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13839 <p>Then, if <I>QN</I> be perpendicular to <I>AD,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13840 <MATH>(<I>AR</I><SUP>2</SUP>-<I>AD</I><SUP>2</SUP>):(<I>QR</I><SUP>2</SUP>-<I>A</I>′<I>D</I><SUP>2</SUP>)=(const.)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13841 that is, <MATH><I>QN</I><SUP>2</SUP>:<I>A</I>′<I>N.A</I>″<I>N</I>=(const.)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13842 and the locus of <I>Q</I> is a hyperbola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13843 <p>The equation of the hyperbola is clearly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13844 <MATH><I>x</I><SUP>2</SUP>=<G>m</G>(<I>y</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13845 where <G>m</G> is a constant. In the particular case taken by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13846 Archimedes <MATH><I>QR</I>=<I>RA,</I></MATH> or <MATH><G>m</G>=1</MATH>, and the hyperbola becomes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13847 the rectangular hyperbola (2) above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13848 <p>The second lemma (Prop. 43, p. 300) proves that, if <I>BC</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13849 given in length, and <I>Q</I> is such a point that, when <I>QR</I> is drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13850 perpendicular to <I>BC</I>, <MATH><I>BR.RC</I>=<I>k.QR,</I></MATH> where <I>k</I> is a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13851 length, the locus of <I>Q</I> is a <I>parabola.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13852 <p>Let <I>O</I> be the middle point of <I>BC,</I> and let <I>OK</I> be drawn at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13853 right angles to <I>BC</I> and of length such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13854 <MATH><I>OC</I><SUP>2</SUP>=<I>k.KO.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13855 <p>Let <I>QN</I>′ be drawn perpendicular to <I>OK.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13856 <p>Then <MATH><I>QN</I>′<SUP>2</SUP>=<I>OR</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13857 =<I>OC</I><SUP>2</SUP>-<I>BR.RC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13858 =<I>k.</I>(<I>KO</I>-<I>QR</I>), by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13859 =<I>k.KN</I>′.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13860 <p>Therefore the locus of <I>Q</I> is a parabola.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13861 <p>The equation of the parabola referred to <I>DB, DE</I> as axes of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13862 <I>x</I> and <I>y</I> is obviously
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13863 <MATH>{1/2(<I>a</I>-<I>b</I>)-<I>x</I>}<SUP>2</SUP>=<I>k</I>{(<I>a</I>+<I>b</I>)<SUP>2</SUP>/4<I>k</I>-<I>y</I>}</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13864 which easily reduces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13865 <MATH>(<I>a</I>-<I>x</I>)(<I>b</I>+<I>x</I>)=<I>ky,</I></MATH> as above (1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13866 <p>In Archimedes's particular case <MATH><I>k</I>=<I>ab/c.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13867 <p>To solve the problem then we have only to draw the para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13868 bola and hyperbola in question, and their intersection then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13869 gives <I>Q,</I> whence <I>R,</I> and therefore <I>ARP,</I> is determined.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13870 <pb n=389>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13871 <head>THE <I>COLLECTION.</I> BOOKS IV, V</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13872 <C>Book V. Preface on the Sagacity of Bees.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13873 <p>It is characteristic of the great Greek mathematicians that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13874 whenever they were free from the restraint of the technical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13875 language of mathematics, as when for instance they had occa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13876 sion to write a preface, they were able to write in language of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13877 the highest literary quality, comparable with that of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13878 philosophers, historians, and poets. We have only to recall
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13879 the introductions to Archimedes's treatises and the prefaces
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13880 to the different Books of Apollonius's <I>Conics.</I> Heron, though
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13881 severely practical, is no exception when he has any general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13882 explanation, historical or other, to give. We have now to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13883 note a like case in Pappus, namely the preface to Book V of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13884 the <I>Collection.</I> The editor, Hultsch, draws attention to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13885 elegance and purity of the language and the careful writing;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13886 the latter is illustrated by the studied avoidance of hiatus.<note>Pappus, vol. iii, p. 1233.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13887 The subject is one which a writer of taste and imagination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13888 would naturally find attractive, namely the practical intelli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13889 gence shown by bees in selecting the hexagonal form for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13890 cells in the honeycomb. Pappus does not disappoint us; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13891 passage is as attractive as the subject, and deserves to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13892 reproduced.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13893 <p>‘It is of course to men that God has given the best and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13894 most perfect notion of wisdom in general and of mathematical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13895 science in particular, but a partial share in these things he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13896 allotted to some of the unreasoning animals as well. To men,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13897 as being endowed with reason, he vouchsafed that they should
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13898 do everything in the light of reason and demonstration, but to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13899 the other animals, while denying them reason, he granted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13900 that each of them should, by virtue of a certain natural
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13901 instinct, obtain just so much as is needful to support life.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13902 This instinct may be observed to exist in very many other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13903 species of living creatures, but most of all in bees. In the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13904 place their orderliness and their submission to the queens who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13905 rule in their state are truly admirable, but much more admirable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13906 still is their emulation, the cleanliness they observe in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13907 gathering of honey, and the forethought and housewifely care
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13908 they devote to its custody. Presumably because they know
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13909 themselves to be entrusted with the task of bringing from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13910 the gods to the accomplished portion of mankind a share of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13911 <pb n=390>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13912 <head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13913 ambrosia in this form, they do not think it proper to pour it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13914 carelessly on ground or wood or any other ugly and irregular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13915 material; but, first collecting the sweets of the most beautiful
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13916 flowers which grow on the earth, they make from them, for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13917 the reception of the honey, the vessels which we call honey-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13918 combs, (with cells) all equal, similar and contiguous to one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13919 another, and hexagonal in form. And that they have con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13920 trived this by virtue of a certain geometrical forethought we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13921 may infer in this way. They would necessarily think that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13922 the figures must be such as to be contiguous to one another,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13923 that is to say, to have their sides common, in order that no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13924 foreign matter could enter the interstices between them and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13925 so defile the purity of their produce. Now only three recti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13926 lineal figures would satisfy the condition, I mean regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13927 figures which are equilateral and equiangular; for the bees
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13928 would have none of the figures which are not uniform. . . .
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13929 There being then three figures capable by themselves of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13930 exactly filling up the space about the same point, the bees by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13931 reason of their instinctive wisdom chose for the construction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13932 of the honeycomb the figure which has the most angles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13933 because they conceived that it would contain more honey than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13934 either of the two others.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13935 <p>‘Bees, then, know just this fact which is of service to them-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13936 selves, that the hexagon is greater than the square and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13937 triangle and will hold more honey for the same expenditure of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13938 material used in constructing the different figures. We, how-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13939 ever, claiming as we do a greater share in wisdom than bees,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13940 will investigate a problem of still wider extent, namely that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13941 of all equilateral and equiangular plane figures having an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13942 equal perimeter, that which has the greater number of angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13943 is always greater, and the greatest plane figure of all those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13944 which have a perimeter equal to that of the polygons is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13945 circle.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13946 <p>Book V then is devoted to what we may call <I>isoperimetry</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13947 including in the term not only the comparison of the areas of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13948 different plane figures with the same perimeter, but that of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13949 contents of different solid figures with equal surfaces.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13950 <C>Section (1). <I>Isoperimetry after Zenodorus.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13951 <p>The first section of the Book relating to plane figures
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13952 (chaps. 1-10, pp. 308-34) evidently followed very closely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13953 the exposition of Zenodorus <G>peri\ i)some/trwn sxhma/twn</G> (see
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13954 pp. 207-13, above); but before passing to solid figures Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13955 inserts the proposition that <I>of all circular segments having</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13956 <pb n=391>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13957 <head>THE <I>COLLECTION.</I> BOOK V</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13958 <I>the same circumference the semicircle is the greatest</I>, with some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13959 preliminary lemmas which deserve notice (chaps. 15, 16).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13960 <p>(1) <I>ABC</I> is a triangle right-angled at <I>B.</I> With <I>C</I> as centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13961 and radius <I>CA</I> describe the arc
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13962 <I>AD</I> cutting <I>CB</I> produced in <I>D.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13963 To prove that (<I>R</I> denoting a right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13964 angle)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13965 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13966 <MATH>(sector <I>CAD</I>):(area <I>ABD</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13967 ><I>R</I>:&angle;<I>BCA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13968 <p>Draw <I>AF</I> at right angles to <I>CA</I> meeting <I>CD</I> produced in <I>F</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13969 and draw <I>BH</I> perpendicular to <I>AF.</I> With <I>A</I> as centre and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13970 <I>AB</I> as radius describe the arc <I>GBE.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13971 <p>Now <MATH>(area <I>EBF</I>):(area <I>EBH</I>)>(area <I>EBF</I>):(sector <I>ABE</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13972 and, <I>componendo</I>, <MATH>&utri;<I>FBH</I>:(<I>EBH</I>)>&utri;<I>ABF</I>:(<I>ABE</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13973 <p>But (by an easy lemma which has just preceded)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13974 <MATH>&utri;<I>FBH</I>:(<I>EBH</I>)=&utri;<I>ABF</I>:(<I>ABD</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13975 whence <MATH>&utri;<I>ABF</I>:(<I>ABD</I>)>&utri;<I>ABF</I>:(<I>ABE</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13976 and <MATH>(<I>ABE</I>)>(<I>ABD</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13977 <p>Therefore <MATH>(<I>ABE</I>):(<I>ABG</I>)>(<I>ABD</I>):(<I>ABG</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13978 >(<I>ABD</I>):&utri;<I>ABC</I></MATH>, <I>a fortiori.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13979 <p>Therefore <MATH>&angle;<I>BAF</I>:&angle;<I>BAC</I>>(<I>ABD</I>):&utri;<I>ABC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13980 whence, inversely, <MATH>&utri;<I>ABC</I>:(<I>ABD</I>)>&angle;<I>BAC</I>:&angle;<I>BAF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13981 and, <I>componendo</I>, <MATH>(sector <I>ACD</I>):(<I>ABD</I>)><I>R</I>:&angle;<I>BCA</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13982 <p>[If <G>a</G> be the circular measure of &angle;<I>BCA</I>, this gives (if <MATH><I>AC</I>=<I>b</I></MATH>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13983 <MATH>1/2 <G>a</G><I>b</I><SUP>2</SUP>:(1/2 <G>a</G><I>b</I><SUP>2</SUP>-1/2 sin<G>a</G>cos<G>a</G>.<I>b</I><SUP>2</SUP>)>1/2 <G>p</G>:<G>a</G></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13984 or <MATH>2<G>a</G>:(2<G>a</G>-sin2<G>a</G>)><G>p</G>:2<G>a</G></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13985 that is, <MATH><G>q</G>/(<G>q</G>-sin<G>q</G>)><G>p</G>/<G>q</G></MATH>, where <MATH>0<<G>q</G><<G>p</G></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13986 <p>(2) <I>ABC</I> is again a triangle right-angled at <I>B.</I> With <I>C</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13987 centre and <I>CA</I> as radius draw a circle <I>AD</I> meeting <I>BC</I> pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13988 duced in <I>D.</I> To prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13989 <MATH>(sector <I>CAD</I>):(area <I>ABD</I>)><I>R</I>:&angle;<I>ACD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13990 <pb n=392>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13991 <head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13992 <p>Draw <I>AE</I> at right angles to <I>AC.</I> With <I>A</I> as centre and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13993 <I>AC</I> as radius describe the circle <I>FCE</I> meeting <I>AB</I> produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13994 in <I>F</I> and <I>AE</I> in <I>E.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13995 <p>Then, since <MATH>&angle;<I>ACD</I>>&angle;<I>CAE</I>, (sector <I>ACD</I>)>(sector <I>ACE</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13996 <p>Therefore <MATH>(<I>ACD</I>):&utri;<I>ABC</I>>(<I>ACE</I>):&utri;<I>ABC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13997 >(<I>ACE</I>):(<I>ACF</I>)</MATH>, <I>a fortiori</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13998 <MATH>>&angle;<I>EAC</I>:&angle;<I>CAB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	13999 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14000 Inversely,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14001 <MATH>&utri;<I>ABC</I>:(<I>ACD</I>)<&angle;<I>CAB</I>:&angle;<I>EAC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14002 and, <I>componendo</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14003 <MATH>(<I>ABD</I>):(<I>ACD</I>)<&angle;<I>EAB</I>:&angle;<I>EAC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14004 <p>Inversely, <MATH>(<I>ACD</I>):(<I>ABD</I>)>&angle;<I>EAC</I>:&angle;<I>EAB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14005 ><I>R</I>:&angle;<I>ACD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14006 <p>We come now to the application of these lemmas to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14007 proposition comparing the area of a semicircle with that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14008 other segments of equal circumference (chaps. 17, 18).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14009 <C><I>A semicircle is the greatest of all segments of circles which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14010 have the same circumference.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14011 <p>Let <I>ABC</I> be a semicircle with centre <I>G</I>, and <I>DEF</I> another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14012 segment of a circle such that the circumference <I>DEF</I> is equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14013 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14014 to the circumference <I>ABC.</I> I say that the area of <I>ABC</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14015 greater than the area of <I>DEF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14016 <p>Let <I>H</I> be the centre of the circle <I>DEF.</I> Draw <I>EHK, BG</I> at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14017 right angles to <I>DF, AC</I> respectively. Join <I>DH</I>, and draw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14018 <I>LHM</I> parallel to <I>DF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14019 <pb n=393>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14020 <head>THE <I>COLLECTION.</I> BOOK V</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14021 <p>Then <MATH><I>LH</I>:<I>AG</I>=(arc <I>LE</I>):(arc <I>AB</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14022 =(arc <I>LE</I>):(arc <I>DE</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14023 =(sector <I>LHE</I>):(sector <I>DHE</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14024 <p>Also <MATH><I>LH</I><SUP>2</SUP>:<I>AG</I><SUP>2</SUP>=(sector <I>LHE</I>):(sector <I>AGB</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14025 <p>Therefore the sector <I>LHE</I> is to the sector <I>AGB</I> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14026 ratio duplicate of that which the sector <I>LHE</I> has to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14027 sector <I>DHE.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14028 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14029 <MATH>(sector <I>LHE</I>):(sector <I>DHE</I>)=(sector <I>DHE</I>):(sector <I>AGB</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14030 <p>Now (1) in the case of the segment less than a semicircle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14031 and (2) in the case of the segment greater than a semicircle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14032 <MATH>(sector <I>EDH</I>):(<I>EDK</I>)><I>R</I>:&angle;<I>DHE</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14033 by the lemmas (1) and (2) respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14034 <p>That is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14035 <MATH>(sector <I>EDH</I>):(<I>EDK</I>)>&angle;<I>LHE</I>:&angle;<I>DHE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14036 >(sector <I>LHE</I>):(sector <I>DHE</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14037 >(sector <I>EDH</I>):(sector <I>AGB</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14038 from above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14039 <p>Therefore the half segment <I>EDK</I> is less than the half
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14040 semicircle <I>AGB</I>, whence the semicircle <I>ABC</I> is greater than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14041 the segment <I>DEF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14042 <p>We have already described the content of Zenodorus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14043 treatise (pp. 207-13, above) to which, so far as plane figures
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14044 are concerned, Pappus added nothing except the above pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14045 position relating to segments of circles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14046 <C>Section (2). <I>Comparison of volumes of solids having their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14047 surfaces equal. Case of the sphere.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14048 <p>The portion of Book V dealing with solid figures begins
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14049 (p. 350. 20) with the statement that the philosophers who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14050 considered that the creator gave the universe the form of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14051 sphere because that was the most beautiful of all shapes also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14052 asserted that the sphere is the greatest of all solid figures
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14053 <pb n=394>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14054 <head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14055 which have their surfaces equal; this, however, they had not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14056 proved, nor could it be proved without a long investigation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14057 Pappus himself does not attempt to prove that the sphere is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14058 greater than <I>all</I> solids with the same surface, but only that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14059 the sphere is greater than any of the five regular solids having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14060 the same surface (chap. 19) and also greater than either a cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14061 or a cylinder of equal surface (chap. 20).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14062 <C>Section (3). <I>Digression on the semi-regular solids
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14063 of Archimedes.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14064 <p>He begins (chap. 19) with an account of the thirteen <I>semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14065 regular</I> solids discovered by Archimedes, which are contained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14066 by polygons all equilateral and all equiangular but not all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14067 similar (see pp. 98-101, above), and he shows how to determine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14068 the number of solid angles and the number of edges which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14069 they have respectively; he then gives them the go-by for his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14070 present purpose because they are not completely regular; still
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14071 less does he compare the sphere with any irregular solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14072 having an equal surface.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14073 <C><I>The sphere is greater than any of the regular solids which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14074 has its surface equal to that of the sphere.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14075 <p>The proof that the sphere is greater than any of the regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14076 solids with surface equal to that of the sphere is the same as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14077 that given by Zenodorus. Let <I>P</I> be any one of the regular solids,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14078 <I>S</I> the sphere with surface equal to that of <I>P.</I> To prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14079 <MATH><I>S</I>><I>P</I></MATH>. Inscribe in the solid a sphere <I>s</I>, and suppose that <I>r</I> is its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14080 radius. Then the surface of <I>P</I> is greater than the surface of <I>s</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14081 and accordingly, if <I>R</I> is the radius of <I>S</I>, <MATH><I>R</I>><I>r</I></MATH>. But the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14082 volume of <I>S</I> is equal to the cone with base equal to the surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14083 of <I>S</I>, and therefore of <I>P</I>, and height equal to <I>R</I>; and the volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14084 of <I>P</I> is equal to the cone with base equal to the surface of <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14085 and height equal to <I>r.</I> Therefore, since <MATH><I>R</I>><I>r</I></MATH>, volume of <I>S</I>>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14086 volume of <I>P.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14087 <C>Section (4). <I>Propositions on the lines of Archimedes,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14088 ‘On the Sphere and Cylinder’.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14089 <p>For the fact that the volume of a sphere is equal to the cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14090 with base equal to the surface, and height equal to the radius,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14091 <pb n=395>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14092 <head>THE <I>COLLECTION.</I> BOOK V</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14093 of the sphere, Pappus quotes Archimedes, <I>On the Sphere and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14094 Cylinder</I>, but thinks proper to add a series of propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14095 (chaps. 20-43, pp. 362-410) on much the same lines as those of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14096 Archimedes and leading to the same results as Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14097 obtains for the surface of a segment of a sphere and of the whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14098 sphere (Prop. 28), and for the volume of a sphere (Prop. 35).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14099 Prop. 36 (chap. 42) shows how to divide a sphere into two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14100 segments such that their surfaces are in a given ratio and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14101 Prop. 37 (chap. 43) proves that the volume as well as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14102 surface of the cylinder circumscribing a sphere is 1 1/2 times
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14103 that of the sphere itself.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14104 <p>Among the lemmatic propositions in this section of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14105 Book Props. 21, 22 may be mentioned. Prop. 21 proves that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14106 if <I>C, E</I> be two points on the tangent at <I>H</I> to a semicircle such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14107 that <MATH><I>CH</I>=<I>HE</I></MATH>, and if <I>CD, EF</I> be drawn perpendicular to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14108 diameter <I>AB</I>, then <MATH>(<I>CD</I>+<I>EF</I>)<I>CE</I>=<I>AB.DF</I></MATH>; Prop. 22 proves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14109 a like result where <I>C, E</I> are points on the semicircle, <I>CD, EF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14110 are as before perpendicular to <I>AB</I>, and <I>EH</I> is the chord of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14111 the circle subtending the arc which with <I>CE</I> makes up a semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14112 circle; in this case <MATH>(<I>CD</I>+<I>EF</I>)<I>CE</I>=<I>EH.DF</I></MATH>. Both results
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14113 are easily seen to be the equivalent of the trigonometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14114 formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14115 <MATH>sin(<I>x</I>+<I>y</I>)+sin(<I>x</I>-<I>y</I>)=2sin<I>x</I>cos<I>y</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14116 or, if certain different angles be taken as <I>x, y</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14117 <MATH>(sin<I>x</I>+sin<I>y</I>)/(cos<I>y</I>-cos<I>x</I>)=cot1/2(<I>x</I>-<I>y</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14118 <C>Section (5). <I>Of regular solids with surfaces equal, that is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14119 greater which has more faces.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14120 <p>Returning to the main problem of the Book, Pappus shows
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14121 that, of the five regular solid figures assumed to have their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14122 surfaces equal, that is greater which has the more faces, so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14123 that the pyramid, the cube, the octahedron, the dodecahedron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14124 and the icosahedron of equal surface are, as regards solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14125 content, in ascending order of magnitude (Props. 38-56).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14126 Pappus indicates (p. 410. 27) that ‘some of the ancients’ had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14127 worked out the proofs of these propositions by the analytical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14128 method; for himself, he will give a method of his own by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14129 <pb n=396>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14130 <head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14131 synthetical deduction, for which he claims that it is clearer
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14132 and shorter. We have first propositions (with auxiliary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14133 lemmas) about the perpendiculars from the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14134 circumscribing sphere to a face of (<I>a</I>) the octahedron, (<I>b</I>) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14135 icosahedron (Props. 39, 43), then the proposition that, if a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14136 dodecahedron and an icosahedron be inscribed in the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14137 sphere, the same small circle in the sphere circumscribes both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14138 the pentagon of the dodecahedron and the triangle of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14139 icosahedron (Prop. 48); this last is the proposition proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14140 Hypsicles in the so-called ‘Book XIV of Euclid’, Prop. 2, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14141 Pappus gives two methods of proof, the second of which (chap.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14142 56) corresponds to that of Hypsicles. Prop. 49 proves that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14143 twelve of the regular pentagons inscribed in a circle are together
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14144 greater than twenty of the equilateral triangles inscribed in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14145 the same circle. The final propositions proving that the cube
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14146 is greater than the pyramid with the same surface, the octa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14147 hedron greater than the cube, and so on, are Props. 52-6
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14148 (chaps. 60-4). Of Pappus's auxiliary propositions, Prop. 41
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14149 is practically contained in Hypsicles's Prop. 1, and Prop. 44
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14150 in Hypsicles's last lemma; but otherwise the exposition is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14151 different.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14152 <C>Book VI.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14153 <p>On the contents of Book VI we can be brief. It is mainly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14154 astronomical, dealing with the treatises included in the so-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14155 called <I>Little Astronomy</I>, that is, the smaller astronomical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14156 treatises which were studied as an introduction to the great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14157 <I>Syntaxis</I> of Ptolemy. The preface says that many of those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14158 who taught the <I>Treasury of Astronomy</I>, through a careless
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14159 understanding of the propositions, added some things as being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14160 necessary and omitted others as unnecessary. Pappus mentions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14161 at this point an incorrect addition to Theodosius, <I>Sphaerica</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14162 III. 6, an omission from Euclid's <I>Phaenomena</I>, Prop. 2, an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14163 inaccurate representation of Theodosius, <I>On Days and Nights</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14164 Prop. 4, and the omission later of certain other things as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14165 being unnecessary. His object is to put these mistakes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14166 right. Allusions are also found in the Book to Menelaus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14167 <I>Sphaerica</I>, e.g. the statement (p. 476. 16) that Menelaus in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14168 his <I>Sphaerica</I> called a spherical triangle <G>tri/pleurov</G>, <I>three-side.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14169 <pb n=397>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14170 <head>THE <I>COLLECTION.</I> BOOKS V, VI</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14171 The <I>Sphaerica</I> of Theodosius is dealt with at some length
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14172 (chaps. 1-26, Props. 1-27), and so are the theorems of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14173 Autolycus <I>On the moving Sphere</I> (chaps. 27-9), Theodosius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14174 <I>On Days and Nights</I> (chaps. 30-6, Props. 29-38), Aristarchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14175 <I>On the sizes and distances of the Sun and Moon</I> (chaps. 37-40,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14176 including a proposition, Prop. 39 with two lemmas, which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14177 corrupt at the end and is not really proved), Euclid's <I>Optics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14178 (chaps. 41-52, Props. 42-54), and Euclid's <I>Phaenomena</I> (chaps.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14179 53-60, Props. 55-61).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14180 <C><I>Problem arising out of Euclid's ‘Optics’.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14181 <p>There is little in the Book of general mathematical interest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14182 except the following propositions which occur in the section on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14183 Euclid's <I>Optics.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14184 <p>Two propositions are fundamental in solid geometry,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14185 namely:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14186 <p>(<I>a</I>) If from a point <I>A</I> above a plane <I>AB</I> be drawn perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14187 dicular to the plane, and if from <I>B</I> a straight line <I>BD</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14188 drawn perpendicular to any straight line <I>EF</I> in the plane,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14189 then will <I>AD</I> also be perpendicular to <I>EF</I> (Prop. 43).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14190 <p>(<I>b</I>) If from a point <I>A</I> above a plane <I>AB</I> be drawn to the plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14191 but not at right angles to it, and <I>AM</I> be drawn perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14192 to the plane (i.e. if <I>BM</I> be the orthogonal projection of <I>BA</I> on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14193 the plane), the angle <I>ABM</I> is the least of all the angles which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14194 <I>AB</I> makes with any straight lines through <I>B</I>, as <I>BP</I>, in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14195 plane; the angle <I>ABP</I> increases as <I>BP</I> moves away from <I>BM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14196 on either side; and, given any straight line <I>BP</I> making
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14197 a certain angle with <I>BA</I>, only one other straight line in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14198 plane will make the same angle with <I>BA</I>, namely a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14199 line <I>BP</I>′ on the other side of <I>BM</I> making the same angle with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14200 it that <I>BP</I> does (Prop. 44).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14201 <p>These are the first of a series of lemmas leading up to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14202 main problem, the investigation of the apparent form of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14203 a circle as seen from a point outside its plane. In Prop. 50
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14204 (=Euclid, <I>Optics</I>, 34) Pappus proves the fact that all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14205 diameters of the circle will appear equal if the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14206 drawn from the point representing the eye to the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14207 the circle is either (<I>a</I>) at right angles to the plane of the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14208 or (<I>b</I>), if not at right angles to the plane of the circle, is equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14209 <pb n=398>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14210 <head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14211 in length to the radius of the circle. In all other cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14212 (Prop. 51=Eucl. <I>Optics</I>, 35) the diameters will appear unequal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14213 Pappus's other propositions carry farther Euclid's remark
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14214 that the circle seen under these conditions will appear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14215 deformed or distorted (<G>parespasme/nos</G>), proving (Prop. 53,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14216 pp. 588-92) that the apparent form will be an ellipse with its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14217 centre not, ‘as some think’, at the centre of the circle but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14218 at another point in it, determined in this way. Given a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14219 <I>ABDE</I> with centre <I>O</I>, let the eye be at a point <I>F</I> above the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14220 plane of the circle such that <I>FO</I> is neither perpendicular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14221 to that plane nor equal to the radius of the circle. Draw <I>FG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14222 perpendicular to the plane of the circle and let <I>ADG</I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14223 diameter through <I>G</I>. Join <I>AF, DF,</I> and bisect the angle <I>AFD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14224 by the straight line <I>FC</I> meeting <I>AD</I> in <I>C</I>. Through <I>C</I> draw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14225 <I>BE</I> perpendicular to <I>AD</I>, and let the tangents at <I>B, E</I> meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14226 <I>AG</I> produced in <I>K</I>. Then Pappus proves that <I>C</I> (not <I>O</I>) is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14227 centre of the apparent ellipse, that <I>AD, BE</I> are its major and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14228 minor axes respectively, that the ordinates to <I>AD</I> are parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14229 to <I>BE</I> both really and apparently, and that the ordinates to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14230 <I>BE</I> will pass through <I>K</I> but will appear to be parallel to <I>AD.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14231 Thus in the figure, <I>C</I> being the centre of the apparent ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14232 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14233 it is proved that, if <I>LCM</I> is any straight line through <I>C, LC</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14234 apparently equal to <I>CM</I> (it is practically assumed--a proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14235 tion proved later in Book VII, Prop. 156--that, if <I>LK</I> meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14236 the circle again in <I>P</I>, and if <I>PM</I> be drawn perpendicular to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14237 <I>AD</I> to meet the circle again in <I>M, LM</I> passes through <I>C</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14238 <pb n=399>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14239 <head>THE <I>COLLECTION.</I> BOOKS VI, VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14240 The test of apparent equality is of course that the two straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14241 lines should subtend equal angles at <I>F.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14242 <p>The main points in the proof are these. The plane through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14243 <I>CF, CK</I> is perpendicular to the planes <I>BFE, PFM</I> and <I>LFR</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14244 hence <I>CF</I> is perpendicular to <I>BE, QF</I> to <I>PM</I> and <I>HF</I> to <I>LR</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14245 whence <I>BC</I> and <I>CE</I> subtend equal angles at <I>F</I>: so do <I>LH, HR</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14246 and <I>PQ, QM</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14247 <p>Since <I>FC</I> bisects the angle <I>AFD</I>, and <MATH><I>AC</I>:<I>CD</I>=<I>AK</I>:<I>KD</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14248 (by the polar property), &angle;<I>CFK</I> is a right angle. And <I>CF</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14249 the intersection of two planes at right angles, namely <I>AFK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14250 and <I>BFE</I>, in the former of which <I>FK</I> lies; therefore <I>KF</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14251 perpendicular to the plane <I>BFE</I>, and therefore to <I>FN</I>. Since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14252 therefore (by the polar property) <MATH><I>LN</I>:<I>NP</I>=<I>LK</I>:<I>KP</I></MATH>, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14253 follows that the angle <I>LFP</I> is bisected by <I>FN</I>; hence <I>LN, NP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14254 are apparently equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14255 <p>Again <MATH><I>LC</I>:<I>CM</I>=<I>LN</I>:<I>NP</I>=<I>LF</I>:<I>FP</I>=<I>LF</I>:<I>FM</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14256 <p>Therefore the angles <I>LFC, CFM</I> are equal, and <I>LC, CM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14257 are apparently equal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14258 <p>Lastly <MATH><I>LR</I>:<I>PM</I>=<I>LK</I>:<I>KP</I>=<I>LN</I>:<I>NP</I>=<I>LF</I>:<I>FP</I></MATH>; therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14259 the isosceles triangles <I>FLR, FPM</I> are equiangular; there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14260 fore the angles <I>PFM, LFR</I>, and consequently <I>PFQ, LFH</I>, are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14261 equal. Hence <I>LP, RM</I> will appear to be parallel to <I>AD</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14262 <p>We have, based on this proposition, an easy method of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14263 solving Pappus's final problem (Prop. 54). ‘Given a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14264 <I>ABDE</I> and any point within it, to find outside the plane of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14265 the circle a point from which the circle will have the appear-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14266 ance of an ellipse with centre <I>C.</I>’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14267 <p>We have only to produce the diameter <I>AD</I> through <I>C</I> to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14268 pole <I>K</I> of the chord <I>BE</I> perpendicular to <I>AD</I> and then, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14269 the plane through <I>AK</I> perpendicular to the plane of the circle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14270 to describe a semicircle on <I>CK</I> as diameter. Any point <I>F</I> on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14271 this semicircle satisfies the condition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14272 <C>Book VII. <I>On the ‘Treasury of Analysis’</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14273 <p>Book VII is of much greater importance, since it gives an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14274 account of the books forming what was called the <I>Treasury of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14275 Analysis</I> (<G>a)naluo/menos to/pos</G>) and, as regards those of the books
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14276 which are now lost, Pappus's account, with the hints derivable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14277 from the large collection of lemmas supplied by him to each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14278 <pb n=400>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14279 <head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14280 book, practically constitutes our only source of information.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14281 The Book begins (p. 634) with a definition of <I>analysis</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14282 <I>synthesis</I> which, as being the most elaborate Greek utterance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14283 on the subject, deserves to be quoted in full.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14284 <p>‘The so-called <G>*)analuo/menos</G> is, to put it shortly, a special
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14285 body of doctrine provided for the use of those who, after
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14286 finishing the ordinary Elements, are desirous of acquiring the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14287 power of solving problems which may be set them involving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14288 (the construction of) lines, and it is useful for this alone. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14289 the work of three men, Euclid the author of the Elements,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14290 Apollonius of Perga and Aristaeus the elder, and proceeds by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14291 way of analysis and synthesis.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14292 <C><I>Definition of Analysis and Synthesis.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14293 <p>‘<I>Analysis</I>, then, takes that which is sought as if it were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14294 admitted and passes from it through its successive conse-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14295 quences to something which is admitted as the result of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14296 synthesis: for in analysis we assume that which is sought
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14297 as if it were already done (<G>lelono/s</G>), and we inquire what it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14298 from which this results, and again what is the antecedent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14299 cause of the latter, and so on, until by so retracing our steps
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14300 we come upon something already known or belonging to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14301 class of first principles, and such a method we call analysis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14302 as being solution backwards (<G>a)na/palin lu/sin</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14303 <p>‘But in <I>synthesis</I>, reversing the process, we take as already
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14304 done that which was last arrived at in the analysis and, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14305 arranging in their natural order as consequences what before
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14306 were antecedents, and successively connecting them one with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14307 another, we arrive finally at the construction of what was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14308 sought; and this we call synthesis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14309 <p>‘Now analysis is of two kinds, the one directed to searching
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14310 for the truth and called <I>theoretical</I>, the other directed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14311 finding what we are told to find and called <I>problematical.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14312 (1) In the <I>theoretical</I> kind we assume what is sought as if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14313 it were existent and true, after which we pass through its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14314 successive consequences, as if they too were true and established
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14315 by virtue of our hypothesis, to something admitted: then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14316 (<I>a</I>), if that something admitted is true, that which is sought
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14317 will also be true and the proof will correspond in the reverse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14318 order to the analysis, but (<I>b</I>), if we come upon something
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14319 admittedly false, that which is sought will also be false.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14320 (2) In the <I>problematical</I> kind we assume that which is pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14321 pounded as if it were known, after which we pass through its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14322 <pb n=401>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14323 <head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14324 successive consequences, taking them as true, up to something
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14325 admitted: if then (<I>a</I>) what is admitted is possible and obtain-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14326 able, that is, what mathematicians call <I>given</I>, what was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14327 originally proposed will also be possible, and the proof will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14328 again correspond in the reverse order to the analysis, but if (<I>b</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14329 we come upon something admittedly impossible, the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14330 will also be impossible.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14331 <p>This statement could hardly be improved upon except that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14332 it ought to be added that each step in the chain of inference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14333 in the analysis must be <I>unconditionally convertible</I>; that is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14334 when in the analysis we say that, if <I>A</I> is true, <I>B</I> is true,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14335 we must be sure that each statement is a necessary conse-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14336 quence of the other, so that the truth of <I>A</I> equally follows
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14337 from the truth of <I>B.</I> This, however, is almost implied by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14338 Pappus when he says that we inquire, not what it is (namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14339 <I>B</I>) which follows from <I>A</I>, but what it is (<I>B</I>) from which <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14340 follows, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14341 <C><I>List of works in the ‘Treasury of Analysis’.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14342 <p>Pappus adds a list, in order, of the books forming the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14343 <G>*)analuo/menos</G>, namely:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14344 <p>‘Euclid's <I>Data</I>, one Book, Apollonius's <I>Cutting-off of a ratio</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14345 two Books, <I>Cutting-off of an area</I>, two Books, <I>Determinate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14346 Section</I>, two Books, <I>Contacts</I>, two Books, Euclid's <I>Porisms</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14347 three Books, Apollonius's <I>Inclinations</I> or <I>Vergings</I> (<G>neu/seis</G>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14348 two Books, the same author's <I>Plane Loci</I>, two Books, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14349 <I>Conics</I>, eight Books, Aristaeus's <I>Solid Loci</I>, five Books, Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14350 <I>Surface-Loci</I>, two Books, Eratosthenes's <I>On means</I>, two Books.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14351 There are in all thirty-three Books, the contents of which up
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14352 to the <I>Conics</I> of Apollonius I have set out for your considera-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14353 tion, including not only the number of the propositions, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14354 <I>diorismi</I> and the cases dealt with in each Book, but also the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14355 lemmas which are required; indeed I have not, to the best
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14356 of my belief, omitted any question arising in the study of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14357 Books in question.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14358 <C><I>Description of the treatises.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14359 <p>Then follows the short description of the contents of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14360 various Books down to Apollonius's <I>Conics</I>; no account is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14361 given of Aristaeus's <I>Solid Loci</I>, Euclid's <I>Surface-Loci</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14362 <pb n=402>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14363 <head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14364 Eratosthenes's <I>On means</I>, nor are there any lemmas to these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14365 works except two on the <I>Surface-Loci</I> at the end of the Book.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14366 <p>The contents of the various works, including those of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14367 lost treatises so far as they can be gathered from Pappus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14368 have been described in the chapters devoted to their authors,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14369 and need not be further referred to here, except for an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14370 <I>addendum</I> to the account of Apollonius's <I>Conics</I> which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14371 remarkable. Pappus has been speaking of the ‘locus with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14372 respect to three or four lines’ (which is a conic), and proceeds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14373 to say (p. 678. 26) that we may in like manner have loci with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14374 reference to five or six or even more lines; these had not up
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14375 to his time become generally known, though the synthesis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14376 of one of them, not by any means the most obvious, had been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14377 worked out and its utility shown. Suppose that there are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14378 five or six lines, and that <I>p</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB>, <I>p</I><SUB>4</SUB>, <I>p</I><SUB>5</SUB> or <I>p</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB>, <I>p</I><SUB>4</SUB>, <I>p</I><SUB>5</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14379 <I>p</I><SUB>6</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14380 are the lengths of straight lines drawn from a point to meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14381 the five or six at given angles, then, if in the first case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14382 <MATH><I>p</I><SUB>1</SUB><I>p</I><SUB>2</SUB><I>p</I><SUB>3</SUB>=<G>l</G><I>p</I><SUB>4</SUB><I>p</I><SUB>5</SUB><I>a</I></MATH> (where <G>l</G> is a constant ratio and <I>a</I> a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14383 length), and in the second case <MATH><I>p</I><SUB>1</SUB><I>p</I><SUB>2</SUB><I>p</I><SUB>3</SUB>=<G>l</G><I>p</I><SUB>4</SUB><I>p</I><SUB>5</SUB><I>p</I><SUB>6</SUB></MATH>, the locus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14384 of the point is in each case a certain curve given in position.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14385 The relation could not be expressed in the same form if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14386 there were more lines than six, because there are only three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14387 dimensions in geometry, although certain recent writers had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14388 allowed themselves to speak of a rectangle multiplied by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14389 a square or a rectangle without giving any intelligible idea of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14390 what they meant by such a thing (is Pappus here alluding to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14391 Heron's proof of the formula for the area of a triangle in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14392 terms of its sides given on pp. 322-3, above?). But the system
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14393 of compounded ratios enables it to be expressed for any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14394 number of lines thus, <MATH>(<I>p</I><SUB>1</SUB>/<I>p</I><SUB>2</SUB>).(<I>p</I><SUB>3</SUB>/<I>p</I><SUB>4</SUB>)....(<I>p<SUB>n</SUB>/a</I>)</MATH> (or <MATH><I>p</I><SUB><I>n</I>-1</SUB>/<I>p<SUB>n</SUB></I>))=<G>l</G></MATH>. Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14395 proceeds in language not very clear (p. 680. 30); but the gist
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14396 seems to be that the investigation of these curves had not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14397 attracted men of light and leading, as, for instance, the old
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14398 geometers and the best writers. Yet there were other impor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14399 tant discoveries still remaining to be made. For himself, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14400 noticed that every one in his day was occupied with the elements,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14401 the first principles and the natural origin of the subject-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14402 matter of investigation; ashamed to pursue such topics, he had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14403 himself proved propositions of much more importance and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14404 <pb n=403>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14405 <head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14406 utility. In justification of this statement and ‘in order that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14407 he may not appear empty-handed when leaving the subject’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14408 he will present his readers with the following.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14409 <C>(<I>Anticipation of Guldin's Theorem.</I>)</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14410 <p>The enunciations are not very clearly worded, but there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14411 is no doubt as to the sense.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14412 <p>‘<I>Figures generated by a complete revolution of a plane figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14413 about an axis are in a ratio compounded (1) of the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14414 of the areas of the figures, and (2) of the ratio of the straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14415 lines similarly drawn to (i.e. drawn to meet at the same angles)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14416 the axes of rotation from the respective centres of gravity.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14417 Figures generated by incomplete revolutions are in the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14418 compounded (1) of the ratio of the areas of the figures and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14419 (2) of the ratio of the arcs described by the centres of gravity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14420 of the respective figures, the latter ratio being itself compounded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14421 (a) of the ratio of the straight lines similarly drawn (from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14422 the respective centres of gravity to the axes of rotation) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14423 (b) of the ratio of the angles contained (i.e. described) about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14424 the axes of revolution by the extremities of the said straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14425 lines (i.e. the centres of gravity).</I>’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14426 <p>Here, obviously, we have the essence of the celebrated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14427 theorem commonly attributed to P. Guldin (1577-1643),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14428 ‘quantitas rotunda in viam rotationis ducta producit Pote-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14429 statem Rotundam uno grado altiorem Potestate sive Quantitate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14430 Rotata’.<note><I>Centrobaryca</I>, Lib. ii, chap. viii, Prop. 3. Viennae 1641.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14431 <p>Pappus adds that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14432 <p>‘these propositions, which are practically one, include any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14433 number of theorems of all sorts about curves, surfaces, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14434 solids, all of which are proved at once by one demonstration,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14435 and include propositions both old and new, and in particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14436 those proved in the twelfth Book of these Elements.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14437 <p>Hultsch attributes the whole passage (pp. 680. 30-682. 20)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14438 to an interpolator, I do not know for what reason; but it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14439 seems to me that the propositions are quite beyond what
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14440 could be expected from an interpolator, indeed I know of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14441 no Greek mathematician from Pappus's day onward except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14442 Pappus himself who was capable of discovering such a pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14443 position.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14444 <pb n=404><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14445 <p>If the passage is genuine, it seems to indicate, what is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14446 elsewhere confirmed, that the <I>Collection</I> originally contained,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14447 or was intended to contain, twelve Books.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14448 <C><I>Lemmas to the different treatises.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14449 <p>After the description of the treatises forming the <I>Treasury
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14450 of Analysis</I> come the collections of lemmas given by Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14451 to assist the student of each of the books (except Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14452 <I>Data</I>) down to Apollonius's <I>Conics</I>, with two isolated lemmas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14453 to the <I>Surface-Loci</I> of Euclid. It is difficult to give any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14454 summary or any general idea of these lemmas, because they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14455 are very numerous, extremely various, and often quite diffi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14456 cult, requiring first-rate ability and full command of all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14457 resources of pure geometry. Their number is also greatly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14458 increased by the addition of alternative proofs, often requiring
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14459 lemmas of their own, and by the separate formulation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14460 particular cases where by the use of algebra and conventions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14461 with regard to sign we can make one proposition cover all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14462 cases. The style is admirably terse, often so condensed as to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14463 make the argument difficult to follow without some little
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14464 filling-out; the hand is that of a master throughout. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14465 only misfortune is that, the books elucidated being lost (except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14466 the <I>Conics</I> and the <I>Cutting-off of a ratio</I> of Apollonius), it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14467 difficult, often impossible, to see the connexion of the lemmas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14468 with one another and the problems of the book to which they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14469 relate. In the circumstances, all that I can hope to do is to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14470 indicate the types of propositions included in the lemmas and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14471 by way of illustration, now and then to give a proof where it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14472 is sufficiently out of the common.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14473 <p>(<I>a</I>) Pappus begins with Lemmas to the <I>Sectio rationis</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14474 <I>Sectio spatii</I> of Apollonius (Props. 1-21, pp. 684-704). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14475 first two show how to divide a straight line in a given ratio,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14476 and how, given the first, second and fourth terms of a pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14477 portion between straight lines, to find the third term. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14478 next section (Props. 3-12 and 16) shows how to manipulate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14479 relations between greater and less ratios by transforming
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14480 them, e.g. <I>componendo, convertendo</I>, &c., in the same way
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14481 as Euclid transforms <I>equal</I> ratios in Book V; Prop. 16 proves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14482 that, according as <I>a</I>:<I>b</I> > or < <I>c</I>:<I>d, ad</I> > or < <I>bc.</I> Props.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14483 <pb n=405><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14484 17-20 deal with three straight lines <I>a, b, c</I> in geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14485 progression, showing how to mark on a straight line containing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14486 <I>a, b, c</I> as segments (including the whole among ‘segments’),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14487 lengths equal to <MATH><I>a</I>+<I>c</I>±2√(<I>ac</I>)</MATH>; the lengths are of course equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14488 to <MATH><I>a</I>+<I>c</I>±2<I>b</I></MATH> respectively. These lemmas are preliminary to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14489 the problem (Prop. 21), Given two straight lines <I>AB, BC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14490 (<I>C</I>lying between <I>A</I> and <I>B</I>), to find a point <I>D</I> on <I>BA</I> produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14491 such that <MATH><I>BD</I>:<I>DA</I>=<I>CD</I>:(<I>AB</I>+<I>BC</I>-2√(<I>AB.BC</I>))</MATH>. This is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14492 of course, equivalent to the quadratic equation <MATH>(<I>a</I>+<I>x</I>):<I>x</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14493 =(<I>a</I>-<I>c</I>+<I>x</I>):(<I>a</I>+<I>c</I>-2√(<I>ac</I>))</MATH>, and, after marking off <I>AE</I> along
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14494 <I>AD</I> equal to the fourth term of this proportion, Pappus solves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14495 the equation in the usual way by application of areas.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14496 <C>(<G>b</G>) <I>Lemmas to the</I> ‘<I>Determinate Section</I>’ <I>of Apollonius.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14497 <p>The next set of Lemmas (Props. 22-64, pp. 704-70) belongs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14498 to the <I>Determinate Section</I> of Apollonius. As we have seen
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14499 (pp. 180-1, above), this work seems to have amounted to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14500 a <I>Theory of Involution.</I> Whether the application of certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14501 of Pappus's lemmas corresponded to the conjecture of Zeuthen
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14502 or not, we have at all events in this set of lemmas some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14503 remarkable applications of ‘geometrical algebra’. They may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14504 be divided into groups as follows
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14505 <p>I. Props. 22, 25, 29.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14506 <p>If in the figure <MATH><I>AD.DC</I>=<I>BD.DE</I></MATH>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14507 <MATH><I>BD</I>:<I>DE</I>=<I>AB.BC</I>:<I>AE.EC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14508 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14509 <p>The proofs by proportions are not difficult. Prop. 29 is an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14510 alternative proof by means of Prop. 26 (see below). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14511 algebraic equivalent may be expressed thus: if <MATH><I>ax</I>=<I>by</I></MATH>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14512 <MATH><I>b</I>/<I>y</I>=((<I>a</I>+<I>b</I>)(<I>b</I>+<I>x</I>))/((<I>a</I>+<I>y</I>)(<I>x</I>+<I>y</I>))</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14513 <p>II. Props. 30, 32, 34.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14514 <p>If in the same figure <MATH><I>AD.DE</I>=<I>BD.DC</I></MATH>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14515 <MATH><I>BD</I>:<I>DC</I>=(<I>AB.BE</I>):(<I>EC.CA</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14516 <pb n=406><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14517 <p>Props. 32, 34 are alternative proofs based on other lemmas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14518 (Props. 31, 33 respectively). The algebraic equivalent may be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14519 stated thus: if <MATH><I>ax</I>=<I>by</I>, then <I>b</I>/<I>y</I>=((<I>a</I>+<I>b</I>)(<I>b</I>-<I>x</I>))/((<I>x</I>+<I>y</I>)(<I>a</I>-<I>y</I>))</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14520 <p>III. Props. 35, 36.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14521 <p>If <MATH><I>AB.BE</I>=<I>CB.BD</I>, then <I>AB</I>:<I>BE</I>=(<I>DA.AC</I>):(<I>CE.ED</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14522 and <MATH><I>CB</I>:<I>BD</I>=(<I>AC.CE</I>):(<I>AD.DE</I>)</MATH>, results equivalent to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14523 following: if <MATH><I>ax</I>=<I>by</I></MATH>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14524 <MATH><I>a</I>/<I>x</I>=((<I>a</I>-<I>y</I>)(<I>a</I>-<I>b</I>))/((<I>b</I>-<I>x</I>)(<I>y</I>-<I>x</I>)) and <I>b</I>/<I>y</I>=((<I>a</I>-<I>b</I>)(<I>b</I>-<I>x</I>))/((<I>a</I>-<I>y</I>)(<I>y</I>-<I>x</I>))</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14525 <p>IV. Props. 23, 24, 31, 57, 58.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14526 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14527 <p>If <MATH><I>AB</I>=<I>CD</I></MATH>, and <I>E</I> is any point in <I>CD</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14528 <MATH><I>AB.CD</I>=<I>AE.ED</I>+<I>BE.EC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14529 and similar formulae hold for other positions of <I>E.</I> If <I>E</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14530 between <I>B</I> and <I>C</I>, <MATH><I>AC.CD</I>=<I>AE.ED</I>-<I>BE.EC</I></MATH>; and if <I>E</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14531 is on <I>AD</I> produced, <MATH><I>BE.EC</I>=<I>AE.ED</I>+<I>BD.DC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14532 <p>V. A small group of propositions relate to a triangle <I>ABC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14533 with two straight lines <I>AD, AE</I> drawn from the vertex <I>A</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14534 points on the base <I>BC</I> in accordance with one or other of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14535 conditions (<I>a</I>) that the angles <I>BAC, DAE</I> are supplementary,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14536 (<I>b</I>) that the angles <I>BAE, DAC</I> are both right angles or, as we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14537 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14538 may add from Book VI, Prop. 12, (<I>c</I>) that the angles <I>BAD,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14539 EAC</I> are equal. The theorems are:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14540 In case <MATH>(<I>a</I>) <I>BC.CD</I>:<I>BE.ED</I>=<I>CA</I><SUP>2</SUP>:<I>AE</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14541 ” <MATH>(<I>b</I>) <I>BC.CE</I>:<I>BD.DE</I>=<I>CA</I><SUP>2</SUP>:<I>AD</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14542 ” <MATH>(<I>c</I>) <I>DC.CE</I>:<I>EB.BD</I>=<I>AC</I><SUP>2</SUP>:<I>AB</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14543 <pb n=407><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14544 Two proofs are given of the first theorem. We will give the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14545 first (Prop. 26) because it is a case of <I>theoretical analysis</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14546 followed by <I>synthesis</I>. Describe a circle about <I>ABD</I>: produce
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14547 <I>EA, CA</I> to meet the circle again in <I>F, G</I>, and join <I>BF, FG</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14548 <p>Substituting <I>GC.CA</I> for <I>BC.CD</I> and <I>FE.EA</I> for <I>BE.ED</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14549 we have to inquire whether <MATH><I>GC.CA</I>:<I>CA</I><SUP>2</SUP>=<I>FE.EA</I>:<I>AE</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14550 i.e. whether <MATH><I>GC</I>:<I>CA</I>=<I>FE</I>:<I>EA</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14551 i.e. whether <MATH><I>GA</I>:<I>AC</I>=<I>FA</I>:<I>AE</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14552 i.e. whether the triangles <I>GAF, CAE</I> are similar or, in other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14553 words, whether <I>GF</I> is parallel to <I>BC</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14554 <p>But <I>GF is</I> parallel to <I>BC</I>, because, the angles <I>BAC, DAE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14555 being supplementary, <MATH>&angle;<I>DAE</I>=&angle;<I>GAB</I>=&angle;<I>GFB</I></MATH>, while at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14556 same time <MATH>&angle;<I>DAE</I>=suppt. of &angle;<I>FAD</I>=&angle;<I>FBD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14557 <p>The synthesis is obvious.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14558 <p>An alternative proof (Prop. 27) dispenses with the circle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14559 and only requires <I>EKH</I> to be drawn parallel to <I>CA</I> to meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14560 <I>AB, AD</I> in <I>H, K</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14561 <p>Similarly (Prop. 28) for case (<I>b</I>) it is only necessary to draw
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14562 <I>FG</I> through <I>D</I> parallel to <I>AC</I> meeting <I>BA</I> in <I>F</I> and <I>AE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14563 produced in <I>G</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14564 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14565 <p>Then, <MATH>&angle;<I>FAG</I>, &angle;<I>ADF</I> (=&angle;<I>DAC</I>)</MATH> being both right angles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14566 <MATH><I>FD.DG</I>=<I>DA</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14567 <p>Therefore <MATH><I>CA</I><SUP>2</SUP>:<I>AD</I><SUP>2</SUP>=<I>CA</I><SUP>2</SUP>:<I>FD.DG</I>=(<I>CA</I>:<I>FD</I>).(<I>CA</I>:<I>DG</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14568 =(<I>BC</I>:<I>BD</I>).(<I>CE</I>:<I>DE</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14569 =<I>BC.CE</I>:<I>BD.DE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14570 <p>In case (<I>c</I>) a circle is circumscribed to <I>ADE</I> cutting <I>AB</I> in <I>F</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14571 and <I>AC</I> in <I>G</I>. Then, since <MATH>&angle;<I>FAD</I>=&angle;<I>GAE</I></MATH>, the arcs <I>DF, EG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14572 are equal and therefore <I>FG</I> is parallel to <I>DE</I>. The proof is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14573 like that of case (<I>a</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14574 <pb n=408><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14575 VI. Props. 37, 38.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14576 <p>If <MATH><I>AB</I>:<I>BC</I>=<I>AD</I><SUP>2</SUP>:<I>DC</I><SUP>2</SUP></MATH>, whether <I>AB</I> be greater or less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14577 than <I>AD</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14578 <MATH><I>AB.BC</I>=<I>BD</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14579 <p>[<I>E</I> in the figure is a point such that <MATH><I>ED</I>=<I>CD</I></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14580 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14581 <p>The algebraical equivalent is: If <MATH><I>a</I>/<I>c</I>=(<I>a</I>±<I>b</I>)<SUP>2</SUP>/(<I>b</I>±<I>c</I>)<SUP>2</SUP>, then <I>ac</I>=<I>b</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14582 <p>These lemmas are subsidiary to the next (Props. 39, 40),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14583 being used in the first proofs of them.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14584 <p>Props. 39, 40 prove the following:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14585 <p>If <I>ACDEB</I> be a straight line, and if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14586 <MATH><I>BA.AE</I>:<I>BD.DE</I>=<I>AC</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14587 then <I>AB.BD</I>:<I>AE.ED</I>=<I>BC</I><SUP>2</SUP>:<I>CE</I><SUP>2</SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14588 if, again, <I>AC.CB</I>:<I>AE.EB</I>=<I>CD</I><SUP>2</SUP>:<I>DE</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14589 then <I>EA.AC</I>:<I>CB.BE</I>=<I>AD</I><SUP>2</SUP>:<I>DB</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14590 <p>If <MATH><I>AB</I>=<I>a</I>, <I>BC</I>=<I>b</I>, <I>BD</I>=<I>c</I>, <I>BE</I>=<I>d</I></MATH>, the algebraic equiva-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14591 lents are the following.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14592 <p>If <MATH>(<I>a</I>(<I>a</I>-<I>d</I>))/(<I>c</I>(<I>c</I>-<I>d</I>))=(<I>a</I>-<I>b</I>)<SUP>2</SUP>/(<I>b</I>-<I>c</I>)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14593 then <I>ac</I>/((<I>a</I>-<I>d</I>)(<I>c</I>-<I>d</I>))=<I>b</I><SUP>2</SUP>/(<I>b</I>-<I>d</I>)<SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14594 and if <MATH>((<I>a</I>-<I>b</I>)<I>b</I>)/((<I>a</I>-<I>d</I>)<I>d</I>)=(<I>b</I>-<I>c</I>)<SUP>2</SUP>/(<I>c</I>-<I>d</I>)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14595 then ((<I>a</I>-<I>d</I>)(<I>a</I>-<I>b</I>))/<I>bd</I>=(<I>a</I>-<I>c</I>)<SUP>2</SUP>/<I>e</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14596 <p>VII. Props. 41, 42, 43.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14597 <p>If <MATH><I>AD.DC</I>=<I>BD.DE</I></MATH>, suppose that in Figures (1) and (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14598 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14599 <MATH><I>k</I>=<I>AE</I>+<I>CB</I></MATH>, and in Figure (3) <MATH><I>k</I>=<I>AE</I>-<I>BC</I></MATH>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14600 <MATH><I>k.AD</I>=<I>BA.AE</I>, <I>k.CD</I>=<I>BC.CE</I>, <I>k.BD</I>=<I>AB.BC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14601 <I>k.DE</I>=<I>AE.EC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14602 <pb n=409><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14603 <p>The algebraical equivalents for Figures (1) and (2) re-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14604 spectively may be written (if <MATH><I>a</I>=<I>AD</I>, <I>b</I>=<I>DC</I>, <I>c</I>=<I>BD</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14605 <I>d</I>=<I>DE</I></MATH>):
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14606 <p>If <MATH><I>ab</I>=<I>cd</I>, then (<I>a</I>±<I>d</I>+<I>c</I>±<I>b</I>) <I>a</I>=(<I>a</I>+<I>c</I>)(<I>a</I>±<I>d</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14607 (<I>a</I>±<I>d</I>+<I>c</I>±<I>b</I>) <I>b</I>=(<I>c</I>±<I>b</I>)(<I>b</I>+<I>d</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14608 (<I>a</I>±<I>d</I>+<I>c</I>±<I>b</I>) <I>c</I>=(<I>c</I>+<I>a</I>)(<I>c</I>±<I>b</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14609 (<I>a</I>±<I>d</I>+<I>c</I>±<I>b</I>) <I>d</I>=(<I>a</I>±<I>d</I>)(<I>d</I>+<I>b</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14610 <p>Figure (3) gives other varieties of sign. Troubles about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14611 sign can be avoided by measuring all lengths in one direction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14612 from an origin <I>O</I> outside the line. Thus, if <I>OA</I>=<I>a</I>, <I>OB</I>=<I>b</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14613 &c., the proposition may be as follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14614 <p>If <MATH>(<I>d</I>-<I>a</I>)(<I>d</I>-<I>c</I>)=(<I>b</I>-<I>d</I>)(<I>e</I>-<I>d</I>) and <I>k</I>=<I>e</I>-<I>a</I>+<I>b</I>-<I>c</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14615 then <MATH><I>k</I>(<I>d</I>-<I>a</I>)=(<I>b</I>-<I>a</I>)(<I>e</I>-<I>a</I>), <I>k</I>(<I>d</I>-<I>c</I>)=(<I>b</I>-<I>c</I>)(<I>e</I>-<I>c</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14616 <I>k</I>(<I>b</I>-<I>d</I>)=(<I>b</I>-<I>a</I>)(<I>b</I>-<I>c</I>) and <I>k</I>(<I>e</I>-<I>d</I>)=(<I>e</I>-<I>a</I>)(<I>e</I>-<I>c</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14617 <p>VIII. Props. 45-56.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14618 <p>More generally, if <MATH><I>AD.DC</I>=<I>BD.DE</I></MATH> and <MATH><I>k</I>=<I>AE</I>±<I>BC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14619 then, if <I>F</I> be any point on the line, we have, according to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14620 position of <I>F</I> in relation to <I>A, B, C, D, E</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14621 <MATH>±<I>AF.FC</I>±<I>EF.FB</I>=<I>k.DF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14622 <p>Algebraically, if <MATH><I>OA</I>=<I>a</I>, <I>OB</I>=<I>b...OF</I>=<I>x</I></MATH>, the equivalent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14623 is: If <MATH>(<I>d</I>-<I>a</I>)(<I>d</I>-<I>c</I>)=(<I>b</I>-<I>d</I>)(<I>e</I>-<I>d</I>), and <I>k</I>=(<I>e</I>-<I>a</I>)+(<I>b</I>-<I>c</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14624 then <MATH>(<I>x</I>-<I>a</I>)(<I>x</I>-<I>c</I>)+(<I>x</I>-<I>e</I>)(<I>b</I>-<I>x</I>)=<I>k</I>(<I>x</I>-<I>d</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14625 <p>By making <MATH><I>x</I>=<I>a, b, c, e</I></MATH> successively in this equation, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14626 obtain the results of Props. 41-3 above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14627 <p>IX. Props. 59-64.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14628 <p>In this group Props. 59, 60, 63 are lemmas required for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14629 remarkable propositions (61, 62, 64) in which Pappus investi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14630 gates ‘singular and minimum’ values of the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14631 <MATH><I>AP.PD</I>:<I>BP.PC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14632 where (<I>A, D</I>), (<I>B, C</I>) are point-pairs on a straight line and <I>P</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14633 is another point on the straight line. He finds, not only when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14634 the ratio has the ‘singular and minimum (or maximum)’ value,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14635 <pb n=410><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14636 but also what the value is, for three different positions of <I>P</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14637 relation to the four given points.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14638 <p>I will give, as an illustration, the first case, on account of its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14639 elegance. It depends on the following <I>Lemma. AEB</I> being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14640 a semicircle on <I>AB</I> as diameter, <I>C, D</I> any two points on <I>AB</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14641 and <I>CE, DF</I> being perpendicular to <I>AB</I>, let <I>EF</I> be joined and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14642 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14643 produced, and let <I>BG</I> be drawn perpendicular to <I>EG</I>. To
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14644 prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14645 <MATH><I>CB.BD</I>=<I>BG</I><SUP>2</SUP></MATH>, (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14646 <MATH><I>AC.DB</I>=<I>FG</I><SUP>2</SUP></MATH>, (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14647 <MATH><I>AD.BC</I>=<I>EG</I><SUP>2</SUP></MATH>. (3)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14648 <p>Join <I>GC, GD, FB, EB, AF</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14649 <p>(1) Since the angles at <I>G, D</I> are right, <I>F, G, B, D</I> are concyclic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14650 Similarly <I>E, G, B, C</I> are concyclic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14651 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14652 <MATH>&angle;<I>BGD</I>=&angle;<I>BFD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14653 =&angle;<I>FAB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14654 =&angle;<I>FEB</I>, in the same segment of the semicircle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14655 =&angle;<I>GCB</I></MATH>, in the same segment of the circle <I>EGBC</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14656 <p>And the triangles <I>GCB, DGB</I> also have the angle <I>CBG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14657 common; therefore they are similar, and <MATH><I>CB</I>:<I>BG</I>=<I>BG</I>:<I>BD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14658 or <MATH><I>CB.BD</I>=<I>BG</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14659 <p>(2) We have <MATH><I>AB.BD</I>=<I>BF</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14660 therefore, by subtraction, <MATH><I>AC.DB</I>=<I>BF</I><SUP>2</SUP>-<I>BG</I><SUP>2</SUP>=<I>FG</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14661 <p>(3) Similarly <MATH><I>AB.BC</I>=<I>BE</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14662 therefore, by subtraction, from the same result (1),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14663 <MATH><I>AD.BC</I>=<I>BE</I><SUP>2</SUP>-<I>BG</I><SUP>2</SUP>=<I>EG</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14664 <p>Thus the lemma gives an extremely elegant construction for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14665 squares equal to each of the three rectangles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14666 <pb n=411><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14667 <p>Now suppose (<I>A, D</I>), (<I>B, C</I>) to be two point-pairs on a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14668 straight line, and let <I>P</I>, another point on it, be determined by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14669 the relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14670 <MATH><I>AB.BD</I>:<I>AC.CD</I>=<I>BP</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14671 then, says Pappus, the ratio <I>AP.PD</I>:<I>BP.PC</I> is singular and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14672 a minimum, and is equal to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14673 <MATH><I>AD</I><SUP>2</SUP>:(√(<I>AC.BD</I>)-√(<I>AB.CD</I>))<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14674 <p>On <I>AD</I> as diameter draw a circle, and draw <I>BF, CG</I> perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14675 dicular to <I>AD</I> on opposite sides.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14676 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14677 <p>Then, by hypothesis, <MATH><I>AB.BD</I>:<I>AC.CD</I>=<I>BP</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14678 therefore <MATH><I>BF</I><SUP>2</SUP>:<I>CG</I><SUP>2</SUP>=<I>BP</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14679 or <MATH><I>BF</I>:<I>CG</I>=<I>BP</I>:<I>CP</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14680 whence the triangles <I>FBP, GCP</I> are similar and therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14681 equiangular, so that <I>FPG</I> is a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14682 <p>Produce <I>GC</I> to meet the circle in <I>H</I>, join <I>FH</I>, and draw <I>DK</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14683 perpendicular to <I>FH</I> produced. Draw the diameter <I>FL</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14684 join <I>LH</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14685 <p>Now, by the lemma, <MATH><I>FK</I><SUP>2</SUP>=<I>AC.BD</I>, and <I>HK</I><SUP>2</SUP>=<I>AB.CD</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14686 therefore <MATH><I>FH</I>=<I>FK</I>-<I>HK</I>=√(<I>AC.BD</I>)-√(<I>AB.CD</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14687 <p>Since, in the triangles <I>FHL, PCG</I>, the angles at <I>H, C</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14688 right and <MATH>&angle;<I>FLH</I>=&angle;<I>PGC</I></MATH>, the triangles are similar, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14689 <MATH><I>GP</I>:<I>PC</I>=<I>FL</I>:<I>FH</I>=<I>AD</I>:<I>FH</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14690 =<I>AD</I>:{√(<I>AC.BD</I>)-√(<I>AB.CD</I>)}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14691 But <MATH><I>GP</I>:<I>PC</I>=<I>FP</I>:<I>PB</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14692 therefore <MATH><I>GP</I><SUP>2</SUP>:<I>PC</I><SUP>2</SUP>=<I>FP.PG</I>:<I>BP.PC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14693 =<I>AP.PD</I>:<I>BP.PC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14694 <pb n=412><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14695 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14696 <MATH><I>AP.PD</I>:<I>BP.PC</I>=<I>AD</I><SUP>2</SUP>:{√(<I>AC.BD</I>)-√(<I>AB.CD</I>)}<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14697 <p>The proofs of Props. 62 and 64 are different, the former
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14698 being long and involved. The results are:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14699 <p>Prop. 62. If <I>P</I> is between <I>C</I> and <I>D</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14700 <MATH><I>AD.DB</I>:<I>AC.CB</I>=<I>DP</I><SUP>2</SUP>:<I>PC</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14701 then the ratio <MATH><I>AP.PB</I>:<I>CP.PD</I></MATH> is singular and a minimum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14702 and is equal to <MATH>{√(<I>AC.BD</I>)+√(<I>AD.BC</I>)}<SUP>2</SUP>:<I>DC</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14703 <p>Prop. 64. If <I>P</I> is on <I>AD</I> produced, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14704 <MATH><I>AB.BD</I>:<I>AC.CD</I>=<I>BP</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14705 then the ratio <MATH><I>AP.PD</I>:<I>BP.PC</I></MATH> is singular and a maximum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14706 and is equal to <MATH><I>AD</I><SUP>2</SUP>:{√(<I>AC.BD</I>)+√(<I>AB.CD</I>)}<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14707 <C>(<G>g</G>) <I>Lemmas on the</I> <G>*neu/seis</G> <I>of Apollonius</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14708 <p>After a few easy propositions (e.g. the equivalent of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14709 proposition that, if <MATH><I>ax</I>+<I>x</I><SUP>2</SUP>=<I>by</I>+<I>y</I><SUP>2</SUP></MATH>, then, according as <I>a</I> >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14710 or < <I>b</I>, <I>a</I>+<I>x</I> > or < <I>b</I>+<I>y</I>), Pappus gives (Prop. 70) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14711 lemma leading to the solution of the <G>neu=sis</G> with regard to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14712 the rhombus (see pp. 190-2, above), and after that the solu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14713 tion by one Heraclitus of the same problem with respect to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14714 a square (Props. 71, 72, pp. 780-4). The problem is, <I>Given a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14715 square ABCD, to draw through B a straight line, meeting CD
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14716 in H and AD produced in E, such that HE is equal to a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14717 length</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14718 <p>The solution depends on a lemma to the effect that, if any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14719 straight line <I>BHE</I> through <I>B</I> meets <I>CD</I> in <I>H</I> and <I>AD</I> pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14720 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14721 duced in <I>E</I>, and if <I>EF</I> be drawn perpendicular to <I>BE</I> meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14722 <I>BC</I> produced in <I>F</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14723 <MATH><I>CF</I><SUP>2</SUP>=<I>BC</I><SUP>2</SUP>+<I>HE</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14724 <pb n=413><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14725 <p>Draw <I>EG</I> perpendicular to <I>BF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14726 <p>Then the triangles <I>BCH, EGF</I> are similar and since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14727 <MATH><I>BG</I>=<I>EG</I></MATH>) equal in all respects; therefore <MATH><I>EF</I>=<I>BH</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14728 <p>Now <MATH><I>BF</I><SUP>2</SUP>=<I>BE</I><SUP>2</SUP>+<I>EF</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14729 or <MATH><I>BC.BF</I>+<I>BF.FC</I>=<I>BH.BE</I>+<I>BE.EH</I>+<I>EF</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14730 <p>But, the angles <I>HCF, HEF</I> being right, <I>H, C, F, E</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14731 concyclic, and <MATH><I>BC.BF</I>=<I>BH.BE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14732 <p>Therefore, by subtraction,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14733 <MATH><I>BF.FC</I>=<I>BE.EH</I>+<I>EF</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14734 =<I>BE.EH</I>+<I>BH</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14735 =<I>BH.HE</I>+<I>EH</I><SUP>2</SUP>+<I>BH</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14736 =<I>EB.BH</I>+<I>EH</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14737 =<I>FB.BC</I>+<I>EH</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14738 <p>Taking away the common part, <I>BC.CF</I>, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14739 <MATH><I>CF</I><SUP>2</SUP>=<I>BC</I><SUP>2</SUP>+<I>EH</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14740 <p>Now suppose that we have to draw <I>BHE</I> through <I>B</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14741 such a way that <MATH><I>HE</I>=<I>k</I></MATH>. Since <I>BC, EH</I> are both given, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14742 have only to determine a length <I>x</I> such that <MATH><I>x</I><SUP>2</SUP>=<I>BC</I><SUP>2</SUP>+<I>k</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14743 produce <I>BC</I> to <I>F</I> so that <MATH><I>CF</I>=<I>x</I></MATH>, draw a semicircle on <I>BF</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14744 diameter, produce <I>AD</I> to meet the semicircle in <I>E</I>, and join
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14745 <I>BE. BE</I> is thus the straight line required.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14746 <p>Prop. 73 (pp. 784-6) proves that, if <I>D</I> be the middle point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14747 of <I>BC</I>, the base of an isosceles triangle <I>ABC</I>, then <I>BC</I> is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14748 shortest of all the straight lines through <I>D</I> terminated by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14749 the straight lines <I>AB, AC</I>, and the nearer to <I>BC</I> is shorter than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14750 the more remote.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14751 <p>There follows a considerable collection of lemmas mostly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14752 showing the equality of certain intercepts made on straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14753 lines through one extremity of the diameter of one of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14754 semicircles having their diameters in a straight line, either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14755 one including or partly including the other, or wholly ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14756 ternal to one another, on the same or opposite sides of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14757 diameter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14758 <pb n=414><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14759 <p>I need only draw two figures by way of illustration.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14760 <p>In the first figure (Prop. 83), <I>ABC, DEF</I> being the semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14761 circles, <I>BEKC</I> is any straight line through <I>C</I> cutting both;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14762 <I>FG</I> is made equal to <I>AD; AB</I> is joined; <I>GH</I> is drawn per-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14763 pendicular to <I>BK</I> produced. It is required to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14764 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14765 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14766 <MATH><I>BE</I>=<I>KH</I></MATH>. (This is obvious when from <I>L</I>, the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14767 semicircle <I>DEF, LM</I> is drawn perpendicular to <I>BK</I>.) If <I>E, K</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14768 coincide in the point <I>M</I>′ of the semicircle so that <I>B</I>′<I>CH</I>′ is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14769 a tangent, then <MATH><I>B</I>′<I>M</I>′=<I>M</I>′<I>H</I>′</MATH> (Props. 83, 84).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14770 <p>In the second figure (Prop. 91) <I>D</I> is the centre of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14771 semicircle <I>ABC</I> and is also the extremity of the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14772 of the semicircle <I>DEF</I>. If <I>BEGF</I> be any straight line through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14773 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14774 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14775 <I>F</I> cutting both semicircles, <MATH><I>BE=EG</I></MATH>. This is clear, since <I>DE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14776 is perpendicular to <I>BG</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14777 <p>The only problem of any difficulty in this section is Prop.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14778 85 (p. 796). Given a semicircle <I>ABC</I> on the diameter <I>AC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14779 and a point <I>D</I> on the diameter, to draw a semicircle passing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14780 through <I>D</I> and having its diameter along <I>DC</I> such that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14781 <I>CEB</I> be drawn touching it at <I>E</I> and meeting the semicircle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14782 <I>ABC</I> in <I>B, BE</I> shall be equal to <I>AD</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14783 <pb n=415><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14784 <p>The problem is reduced to a problem contained in Apollo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14785 nius's <I>Determinate Section</I> thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14786 <p>Suppose the problem solved by the semicircle <I>DEF, BE</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14787 being equal to <I>AD</I>. Join <I>E</I> to the centre <I>G</I> of the semicircle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14788 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14789 <I>DEF</I>. Produce <I>DA</I> to <I>H</I>, making <I>HA</I> equal to <I>AD</I>. Let <I>K</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14790 be the middle point of <I>DC</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14791 <p>Since the triangles <I>ABC, GEC</I> are similar,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14792 <MATH><I>AG</I><SUP>2</SUP>:<I>GC</I><SUP>2</SUP>=<I>BE</I><SUP>2</SUP>:<I>EC</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14793 =<I>AD</I><SUP>2</SUP>:<I>EC</I><SUP>2</SUP>, by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14794 =<I>AD</I><SUP>2</SUP>:<I>GC</I><SUP>2</SUP>-<I>DG</I><SUP>2</SUP> (since <I>DG</I>=<I>GE</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14795 =<I>AG</I><SUP>2</SUP>-<I>AD</I><SUP>2</SUP>:<I>DG</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14796 =<I>HG.DG</I>:<I>DG</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14797 =<I>HG</I>:<I>DG</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14798 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14799 <MATH><I>HG</I>:<I>DG</I>=<I>AD</I><SUP>2</SUP>:<I>GC</I><SUP>2</SUP>-<I>DG</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14800 =<I>AD</I><SUP>2</SUP>:2<I>DC.GK</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14801 <p>Take a straight line <I>L</I> such that <MATH><I>AD</I><SUP>2</SUP>=<I>L</I>.2<I>DC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14802 therefore <MATH><I>HG</I>:<I>DG</I>=<I>L</I>:<I>GK</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14803 or <MATH><I>HG.GK</I>=<I>L.DG</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14804 <p>Therefore, given the two straight lines <I>HD, DK</I> (or the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14805 three points <I>H, D, K</I> on a straight line), we have to find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14806 a point <I>G</I> between <I>D</I> and <I>K</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14807 <MATH><I>HG.GK</I>=<I>L.DG</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14808 which is the second <I>epitagma</I> of the third Problem in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14809 <I>Determinate Section</I> of Apollonius, and therefore may be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14810 taken as solved. (The problem is the equivalent of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14811 <pb n=416><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14812 solution of a certain quadratic equation.) Pappus observes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14813 that the problem is always possible (requires no <G>diorismo/s</G>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14814 and proves that it has only one solution.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14815 <C>(<G>d</G>) <I>Lemmas on the treatise</I> ‘<I>On contacts</I>’ <I>by Apollonius</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14816 <p>These lemmas are all pretty obvious except two, which are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14817 important, one belonging to Book I of the treatise, and the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14818 to Book II. The two lemmas in question have already been set
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14819 out à propos of the treatise of Apollonius (see pp. 182-5, above).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14820 As, however, there are several cases of the first (Props. 105,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14821 107, 108, 109), one case (Prop. 108, pp. 836-8), different from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14822 that before given, may be put down here: <I>Given a circle and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14823 two points D, E within it, to draw straight lines through D, E
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14824 to a point A on the circumference in such a way that, if they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14825 meet the circle again in B, C, BC shall be parallel to DE</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14826 <p>We proceed by analysis. Suppose the problem solved and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14827 <I>DA, EA</I> drawn (‘inflected’) to <I>A</I> in such a way that, if <I>AD</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14828 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14829 <I>AE</I> meet the circle again in <I>B, C,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14830 BC</I> is parallel to <I>DE</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14831 <p>Draw the tangent at <I>B</I> meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14832 <I>ED</I> produced in <I>F</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14833 <p>Then <MATH>&angle;<I>FBD</I>=&angle;<I>ACB</I>=&angle;<I>AED</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14834 therefore <I>A, E, B, F</I> are concyclic,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14835 and consequently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14836 <MATH><I>FD.DE</I>=<I>AD.DB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14837 <p>But the rectangle <I>AD.DB</I> is given, since it depends only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14838 on the position of <I>D</I> in relation to the circle, and the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14839 is given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14840 <p>Therefore the rectangle <I>FD.DE</I> is given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14841 <p>And <I>DE</I> is given; therefore <I>FD</I> is given, and therefore <I>F</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14842 <p>If follows that the tangent <I>FB</I> is given in position, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14843 therefore <I>B</I> is given. Therefore <I>BDA</I> is given and conse-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14844 quently <I>AE</I> also.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14845 <p>To solve the problem, therefore, we merely take <I>F</I> on <I>ED</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14846 produced such that <MATH><I>FD.DE</I>=</MATH> the given rectangle made by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14847 the segments of any chord through <I>D</I>, draw the tangent <I>FB</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14848 join <I>BD</I> and produce it to <I>A</I>, and lastly draw <I>AE</I> through to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14849 <I>C</I>; <I>BC</I> is then parallel to <I>DE</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14850 <pb n=417><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14851 <p>The other problem (Prop. 117, pp. 848-50) is, as we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14852 seen, equivalent to the following: <I>Given a circle and three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14853 points D, E, F in a straight line external to it, to inscribe in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14854 the circle a triangle ABC such that its sides pass severally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14855 through the three points D, E, F</I>. For the solution, see
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14856 pp. 182-4, above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14857 <p>(<G>e</G>) The Lemmas to the <I>Plane Loci</I> of Apollonius (Props.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14858 119-26, pp. 852-64) are mostly propositions in geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14859 algebra worked out by the methods of Eucl., Books II and VI.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14860 We may mention the following:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14861 <p>Prop. 122 is the well-known proposition that, if <I>D</I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14862 middle point of the side <I>BC</I> in a triangle <I>ABC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14863 <MATH><I>BA</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>=2(<I>AD</I><SUP>2</SUP>+<I>DC</I><SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14864 <p>Props. 123 and 124 are two cases of the same proposition,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14865 the enunciation being marked by an expression which is also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14866 found in Euclid's <I>Data</I>. Let <I>AB:BC</I> be a given ratio, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14867 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14868 let the rectangle <I>CA.AD</I> be given; then, if <I>BE</I> is a mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14869 proportional between <I>DB, BC</I>, ‘the square on <I>AE</I> is greater
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14870 by the rectangle <I>CA.AD</I> than in the ratio of <I>AB</I> to <I>BC</I> to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14871 square on <I>EC</I>’, by which is meant that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14872 <MATH><I>AE</I><SUP>2</SUP>=<I>CA.AD</I>+<I>AB</I>/<I>BC</I> <I>EC</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14873 or <MATH>(<I>AE</I><SUP>2</SUP>-<I>CA.AD</I>):<I>EC</I><SUP>2</SUP>=<I>AB</I>:<I>BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14874 <p>The algebraical equivalent may be expressed thus (if <MATH><I>AB</I>=<I>a</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14875 <I>BC</I>=<I>b</I>, <I>AD</I>=<I>c</I>, <I>BE</I>=<I>x</I></MATH>):
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14876 <p>If <MATH><I>x</I>=√(<I>a</I>-<I>c</I>)<I>b</I></MATH>, then <MATH>((<I>a</I>&mnplus;<I>x</I>)<SUP>2</SUP>-(<I>a</I>-<I>b</I>)<I>c</I>)/(<I>x</I>&mnplus;<I>b</I>)<SUP>2</SUP>=<I>a</I>/<I>b</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14877 <p>Prop. 125 is remarkable: If <I>C, D</I> be two points on a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14878 line <I>AB</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14879 <MATH><I>AD</I><SUP>2</SUP>+<I>AC</I>/<I>BC</I>.<I>DB</I><SUP>2</SUP>=<I>AC</I><SUP>2</SUP>+<I>AC.CB</I>+<I>AB</I>/<I>BC</I>.<I>CD</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14880 <pb n=418><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14881 <p>This is equivalent to the general relation between four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14882 points on a straight line discovered by Simson and therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14883 wrongly known as Stewart's theorem:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14884 <MATH><I>AD</I><SUP>2</SUP>.<I>BC</I>+<I>BD</I><SUP>2</SUP>.<I>CA</I>+<I>CD</I><SUP>2</SUP>.<I>AB</I>+<I>BC.CA.AB</I>=0</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14885 <p>(Simson discovered this theorem for the more general case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14886 where <I>D</I> is a point outside the line <I>ABC</I>.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14887 <p>An algebraical equivalent is the identity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14888 <MATH>(<I>d</I>-<I>a</I>)<SUP>2</SUP>(<I>b</I>-<I>c</I>)+(<I>d</I>-<I>b</I>)<SUP>2</SUP>(<I>c</I>-<I>a</I>)+(<I>d</I>-<I>c</I>)<SUP>2</SUP>(<I>a</I>-<I>b</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14889 +(<I>b</I>-<I>c</I>)(<I>c</I>-<I>a</I>)(<I>a</I>-<I>b</I>)=0</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14890 <p>Pappus's proof of the last-mentioned lemma is perhaps
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14891 worth giving.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14892 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14893 <p><I>C, D</I> being two points on the straight line <I>AB</I>, take the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14894 point <I>F</I> on it such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14895 <MATH><I>FD</I>:<I>DB</I>=<I>AC</I>:<I>CB</I></MATH>. (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14896 <p>Then <MATH><I>FB</I>:<I>BD</I>=<I>AB</I>:<I>BC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14897 and <MATH>(<I>AB</I>-<I>FB</I>):(<I>BC</I>-<I>BD</I>)=<I>AB</I>:<I>BC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14898 or <MATH><I>AF</I>:<I>CD</I>=<I>AB</I>:<I>BC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14899 and therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14900 <MATH><I>AF.CD</I>:<I>CD</I><SUP>2</SUP>=<I>AB</I>:<I>BC</I></MATH>. (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14901 <p>From (1) we derive
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14902 <MATH><I>AC</I>/<I>CB</I>.<I>DB</I><SUP>2</SUP>=<I>FD.DB</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14903 and from (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14904 <MATH><I>AB</I>/<I>BC</I>.<I>CD</I><SUP>2</SUP>=<I>AF.CD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14905 <p>We have now to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14906 <MATH><I>AD</I><SUP>2</SUP>+<I>BD.DF</I>=<I>AC</I><SUP>2</SUP>+<I>AC.CB</I>+<I>AF.CD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14907 or <MATH><I>AD</I><SUP>2</SUP>+<I>BD.DF</I>=<I>CA.AB</I>+<I>AF.CD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14908 <pb n=419><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14909 i.e. (if <I>DA.AC</I> be subtracted from each side)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14910 that <MATH><I>AD.DC</I>+<I>FD.DB</I>=<I>AC.DB</I>+<I>AF.CD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14911 i.e. (if <I>AF.CD</I> be subtracted from each side)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14912 that <MATH><I>FD.DC</I>+<I>FD.DB</I>=<I>AC.DB</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14913 or <MATH><I>FD.CB</I>=<I>AC.DB</I></MATH>:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14914 which is true, since, by (1) above, <MATH><I>FD</I>:<I>DB</I>=<I>AC</I>:<I>CB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14915 <C>(<G>z</G>) <I>Lemmas to the ‘Porisms’ of Euclid</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14916 <p>The 38 Lemmas to the <I>Porisms</I> of Euclid form an important
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14917 collection which, of course, has been included in one form or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14918 other in the ‘restorations’ of the original treatise. Chasles<note>Chasles, <I>Les trois livres de Porismes d'Euclide</I>, Paris, 1860, pp. 74 sq.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14919 in particular gives a classification of them, and we cannot
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14920 do better than use it in this place: ‘23 of the Lemmas relate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14921 to rectilineal figures, 7 refer to the harmonic ratio of four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14922 points, and 8 have reference to the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14923 <p>‘Of the 23 relating to rectilineal figures, 6 deal with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14924 quadrilateral cut by a transversal; 6 with the equality of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14925 the anharmonic ratios of two systems of four points arising
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14926 from the intersections of four straight lines issuing from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14927 one point with two other straight lines; 4 may be regarded as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14928 expressing a property of the hexagon inscribed in two straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14929 lines; 2 give the relation between the areas of two triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14930 which have two angles equal or supplementary; 4 others refer
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14931 to certain systems of straight lines; and the last is a case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14932 of the problem of the <I>Cutting-off of an area</I>.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14933 <p>The lemmas relating to the quadrilateral and the transversal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14934 are 1, 2, 4, 5, 6 and 7 (Props. 127, 128, 130, 131, 132, 133).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14935 Prop. 130 is a general proposition about any transversal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14936 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14937 whatever, and is equivalent to one of the equations by which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14938 we express the involution of six points. If <I>A</I>, <I>A</I>′; <I>B</I>, <I>B</I>′;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14939 <I>C, C</I>′ be the points in which the transversal meets the pairs of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14940 <pb n=420><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14941 opposite sides and the two diagonals respectively, Pappus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14942 result is equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14943 <MATH>(<I>AB</I>.<I>B</I>′<I>C</I>)/(<I>A</I>′<I>B</I>′.<I>BC</I>′)=<I>CA</I>/<I>C</I>′<I>A</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14944 Props. 127, 128 are particular cases in which the transversal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14945 is parallel to a side; in Prop. 131 the transversal passes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14946 through the points of concourse of opposite sides, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14947 result is equivalent to the fact that the two diagonals divide
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14948 into proportional parts the straight line joining the points of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14949 concourse of opposite sides; Prop. 132 is the particular case
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14950 of Prop. 131 in which the line joining the points of concourse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14951 of opposite sides is parallel to a diagonal; in Prop. 133 the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14952 transversal passes through one only of the points of concourse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14953 of opposite sides and is parallel to a diagonal, the result being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14954 <MATH><I>CA</I><SUP>2</SUP>=<I>CB.CB</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14955 <p>Props. 129, 136, 137, 140, 142, 145 (Lemmas 3, 10, 11, 14, 16,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14956 19) establish the equality of the anharmonic ratios which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14957 four straight lines issuing from a point determine on two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14958 transversals; but both transversals are supposed to be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14959 from the same point on one of the four straight lines. Let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14960 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14961 <I>AB, AC, AD</I> be cut by transversals <I>HBCD, HEFG</I>. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14962 required to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14963 <MATH>(<I>HE.FG</I>)/(<I>HG.EF</I>)=(<I>HB.CD</I>)/(<I>HD.BC</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14964 Pappus gives (Prop. 129) two methods of proof which are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14965 practically equivalent. The following is the proof ‘by com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14966 pound ratios’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14967 <p>Draw <I>HK</I> parallel to <I>AF</I> meeting <I>DA</I> and <I>AE</I> produced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14968 <pb n=421><head>THE <I>COLLECTION.</I> BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14969 in <I>K, L</I>; and draw <I>LM</I> parallel to <I>AD</I> meeting <I>GH</I> pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14970 duced in <I>M</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14971 <p>Then <MATH>(<I>HE.FG</I>)/(<I>HG.EF</I>)=(<I>HE</I>/<I>EF</I>).(<I>FG</I>/<I>HG</I>)=(<I>LH</I>/<I>AF</I>).(<I>AF</I>/<I>HK</I>)=<I>LH</I>/<I>HK</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14972 <p>In exactly the same way, if <I>DH</I> produced meets <I>LM</I> in <I>M</I>′
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14973 we prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14974 <MATH>(<I>HB.CD</I>)/(<I>HD.BC</I>)=<I>LH</I>/<I>HK</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14975 Therefore <MATH>(<I>HE.FG</I>)/(<I>HG.EF</I>)=(<I>HB.CD</I>)/(<I>HD.BC</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14976 <p>(The proposition is proved for <I>HBCD</I> and any other trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14977 versal not passing through <I>H</I> by applying our proposition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14978 twice, as usual.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14979 <p>Props. 136, 142 are the reciprocal; Prop. 137 is a particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14980 case in which one of the transversals is parallel to one of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14981 straight lines, Prop. 140 a reciprocal of Prop. 137, Prop. 145
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14982 another case of Prop. 129.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14983 <p>The Lemmas 12, 13, 15, 17 (Props. 138, 139, 141, 143) are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14984 equivalent to the property of the hexagon inscribed in two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14985 straight lines, viz. that, if the vertices of a hexagon are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14986 situate, three and three, on two straight lines, the points of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14987 concourse of opposite sides are in a straight line; in Props.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14988 138, 141 the straight lines are parallel, in Props. 139, 143 not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14989 parallel.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14990 <p>Lemmas 20, 21 (Props. 146, 147) prove that, when one angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14991 of one triangle is equal or supplementary to one angle of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14992 another triangle, the areas of the triangles are in the ratios
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14993 of the rectangles contained by the sides containing the equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14994 or supplementary angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14995 <p>The seven Lemmas 22, 23, 24, 25, 26, 27, 34 (Props. 148-53
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14996 and 160) are propositions relating to the segments of a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14997 line on which two intermediate points are marked. Thus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14998 <p>Props. 148, 150.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	14999 <p>If <I>C, D</I> be two points on <I>AB</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15000 <p>(<I>a</I>) if <MATH>2<I>AB.CD</I>=<I>CB</I><SUP>2</SUP>, <I>AD</I><SUP>2</SUP>=<I>AC</I><SUP>2</SUP>+<I>DB</I><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15001 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15002 <p>(<I>b</I>) if <MATH>2<I>AC.BD</I>=<I>CD</I><SUP>2</SUP>, <I>AB</I><SUP>2</SUP>=<I>AD</I><SUP>2</SUP>+<I>CB</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15003 <pb n=422><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15004 <p>Props. 149, 151.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15005 <p>If <MATH><I>AB.BC</I>=<I>BD</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15006 then <MATH>(<I>AD</I>±<I>DC</I>)<I>BD</I>=<I>AD.DC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15007 (<I>AD</I>±<I>DC</I>)<I>BC</I>=<I>DC</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15008 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15009 and <MATH>(<I>AD</I>±<I>DC</I>)<I>BA</I>=<I>AD</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15010 <p>Props. 152, 153.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15011 <p>If <MATH><I>AB</I>:<I>BC</I>=<I>AD</I><SUP>2</SUP>:<I>DC</I><SUP>2</SUP></MATH>, then <MATH><I>AB.BC</I>=<I>BD</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15012 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15013 <p>Prop. 160.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15014 <p>If <MATH><I>AB</I>:<I>BC</I>=<I>AD</I>:<I>DC</I></MATH>, then, if <I>E</I> be the middle point of <I>AC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15015 <MATH><I>BE.ED</I>=<I>EC</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15016 <I>BD.DE</I>=<I>AD.DC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15017 <I>EB.BD</I>=<I>AB.BC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15018 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15019 <p>The Lemmas about the circle include the harmonic proper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15020 ties of the pole and polar, whether the pole is external to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15021 circle (Prop. 154) or internal (Prop. 161). Prop. 155 is a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15022 problem, Given a segment of a circle on <I>AB</I> as base, to inflect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15023 straight lines <I>AC</I>, <I>BC</I> to the segment in a given ratio to one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15024 another.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15025 <p>Prop. 156 is one which Pappus has already used earlier
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15026 in the <I>Collection</I>. It proves that the straight lines drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15027 from the extremities of a chord (<I>DE</I>) to any point (<I>F</I>) of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15028 circumference divide harmonically the diameter (<I>AB</I>) perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15029 dicular to the chord. Or, if <I>ED</I>, <I>FK</I> be parallel chords, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15030 <I>EF</I>, <I>DK</I> meet in <I>G</I>, and <I>EK</I>, <I>DF</I> in <I>H</I>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15031 <MATH><I>AH</I>:<I>HB</I>=<I>AG</I>:<I>GB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15032 <pb n=423><head>THE <I>COLLECTION</I>. BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15033 <p>Since <I>AB</I> bisects <I>DE</I> perpendicularly, <MATH>(arc <I>AE</I>)=(arc <I>AD</I>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15034 and <MATH>&angle;<I>EFA</I>=&angle;<I>AFD</I></MATH>, or <I>AF</I> bisects the angle <I>EFD</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15035 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15036 <p>Since the angle <I>AFB</I> is right, <I>FB</I> bisects &angle;<I>HFG</I>, the supple-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15037 ment of &angle;<I>EFD</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15038 <p>Therefore (Eucl. VI. 3) <MATH><I>GB</I>:<I>BH</I>=<I>GF</I>:<I>FH</I>=<I>GA</I>:<I>AH</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15039 and, alternately and inversely, <MATH><I>AH</I>:<I>HB</I>=<I>AG</I>:<I>GB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15040 <p>Prop. 157 is remarkable in that (without any mention of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15041 a conic) it is practically identical with Apollonius's <I>Conics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15042 III. 45 about the foci of a central conic. Pappus's theorem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15043 is as follows. Let <I>AB</I> be the diameter of a semicircle, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15044 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15045 from <I>A</I>, <I>B</I> let two straight lines <I>AE</I>, <I>BD</I> be drawn at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15046 angles to <I>AB</I>. Let any straight line <I>DE</I> meet the two perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15047 diculars in <I>D</I>, <I>E</I> and the semicircle in <I>F</I>. Further, let <I>FG</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15048 drawn at right angles to <I>DE</I>, meeting <I>AB</I> produced in <I>G</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15049 <p>It is to be proved that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15050 <MATH><I>AG.GB</I>=<I>AE.BD</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15051 <p>Since <I>F</I>, <I>D</I>, <I>G</I>, <I>B</I> are concyclic, <MATH>&angle;<I>BDG</I>=&angle;<I>BFG</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15052 <pb n=424><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15053 <p>And, since <I>AFB</I>, <I>EFG</I> are both right angles, <MATH>&angle;<I>BFG</I>=&angle;<I>AFE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15054 <p>But, since <I>A</I>, <I>E</I>, <I>G</I>, <I>F</I> are concyclic, <MATH>&angle;<I>AFE</I>=&angle;<I>AGE</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15055 <p>Therefore <MATH>&angle;<I>BDG</I>=&angle;<I>AGE</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15056 and the right-angled triangles <I>DBG</I>, <I>GAE</I> are similar.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15057 <p>Therefore <MATH><I>AG</I>:<I>AE</I>=<I>BD</I>:<I>GB</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15058 or <MATH><I>AG.GB</I>=<I>AE.DB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15059 <p>In Apollonius <I>G</I> and the corresponding point <I>G</I>′ on <I>BA</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15060 produced which is obtained by drawing <I>F</I>′<I>G</I>′ perpendicular to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15061 <I>ED</I> (where <I>DE</I> meets the circle again in <I>F</I>′) are the foci
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15062 of a central conic (in this case a hyperbola), and <I>DE</I> is any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15063 tangent to the conic; the rectangle <I>AE.BD</I> is of course equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15064 to the square on half the conjugate axis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15065 <p>(<G>h</G>) The Lemmas to the <I>Conics</I> of Apollonius (pp. 918-1004)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15066 do not call for any extended notice. There are a large number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15067 of propositions in geometrical algebra of the usual kind,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15068 relating to the segments of a straight line marked by a number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15069 of points on it; propositions about lines divided into proportional
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15070 segments and about similar figures; two propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15071 relating to the construction of a hyperbola (Props. 204, 205)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15072 and a proposition (208) proving that two hyperbolas with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15073 same asymptotes do not meet one another. There are also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15074 two propositions (221, 222) equivalent to an obvious trigono-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15075 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15076 metrical formula. Let <I>ABCD</I> be a rectangle, and let any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15077 straight line through <I>A</I> meet <I>DC</I> produced in <I>E</I> and <I>BC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15078 (produced if necessary) in <I>F</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15079 <p>Then <MATH><I>EA.AF</I>=<I>ED.DC</I>+<I>CB.BF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15080 <pb n=425><head>THE <I>COLLECTION</I>. BOOK VII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15081 <p>For <MATH><I>EA</I><SUP>2</SUP>+<I>AF</I><SUP>2</SUP>=<I>ED</I><SUP>2</SUP>+<I>DA</I><SUP>2</SUP>+<I>AB</I><SUP>2</SUP>+<I>BF</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15082 =<I>ED</I><SUP>2</SUP>+<I>BC</I><SUP>2</SUP>+<I>CD</I><SUP>2</SUP>+<I>BF</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15083 <p>Also <MATH><I>EA</I><SUP>2</SUP>+<I>AF</I><SUP>2</SUP>=<I>EF</I><SUP>2</SUP>+2<I>EA.AF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15084 <p>Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15085 <MATH>2<I>EA.AF</I>=<I>EA</I><SUP>2</SUP>+<I>AF</I><SUP>2</SUP>-<I>EF</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15086 =<I>ED</I><SUP>2</SUP>+<I>BC</I><SUP>2</SUP>+<I>CD</I><SUP>2</SUP>+<I>BF</I><SUP>2</SUP>-<I>EF</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15087 =(<I>ED</I><SUP>2</SUP>+<I>CD</I><SUP>2</SUP>)+(<I>BC</I><SUP>2</SUP>+<I>BF</I><SUP>2</SUP>)-<I>EF</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15088 =<I>EC</I><SUP>2</SUP>+2<I>ED.DC</I>+<I>CF</I><SUP>2</SUP>+2<I>CB.BF</I>-<I>EF</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15089 =2<I>ED.DC</I>+2<I>CB.BF</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15090 i.e. <MATH><I>EA.AF</I>=<I>ED.DC</I>+<I>CB.BF</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15091 <p>This is equivalent to <MATH>sec <G>q</G> cosec <G>q</G>=tan <G>q</G>+cot <G>q</G></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15092 <p>The algebraical equivalents of some of the results obtained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15093 by the usual geometrical algebra may be added.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15094 <p>Props. 178, 179, 192-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15095 <MATH>(<I>a</I>+2<I>b</I>)<I>a</I>+(<I>b</I>+<I>x</I>)(<I>b</I>-<I>x</I>)=(<I>a</I>+<I>b</I>+<I>x</I>)(<I>a</I>+<I>b</I>-<I>x</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15096 <p>Prop. 195. <MATH>4<I>a</I><SUP>2</SUP>=2{(<I>a</I>-<I>x</I>)(<I>a</I>+<I>x</I>)+(<I>a</I>-<I>y</I>)(<I>a</I>+<I>y</I>)+<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15097 <p>Prop. 196.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15098 <MATH>(<I>a</I>+<I>b</I>-<I>x</I>)<SUP>2</SUP>+(<I>a</I>+<I>b</I>+<I>x</I>)<SUP>2</SUP>=(<I>x</I>-<I>b</I>)<SUP>2</SUP>+(<I>x</I>+<I>b</I>)<SUP>2</SUP>+2(<I>a</I>+2<I>b</I>)<I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15099 <p>Props. 197, 199, 198.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15100 <MATH><BRACE><note>then <I>x</I>=<I>y</I>.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15101 <p>If (<I>x</I>+<I>y</I>+<I>a</I>)<I>a</I>+<I>x</I><SUP>2</SUP>=(<I>a</I>+<I>x</I>)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15102 or if (<I>x</I>+<I>y</I>+<I>a</I>)<I>a</I>+<I>x</I><SUP>2</SUP>=(<I>a</I>+<I>y</I>)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15103 or if (<I>x</I>+<I>y</I>-<I>a</I>)<I>a</I>+(<I>x</I>-<I>a</I>)<SUP>2</SUP>=<I>y</I><SUP>2</SUP>,</BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15104 <p>Props. 200, 201. If <MATH>(<I>a</I>+<I>b</I>)<I>x</I>=<I>b</I><SUP>2</SUP></MATH>, then <MATH>(2<I>b</I>+<I>a</I>)/<I>a</I>=(<I>b</I>+<I>x</I>)/(<I>b</I>-<I>x</I>)</MATH> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15105 <MATH>(2<I>b</I>+<I>a</I>)<I>a</I>=(<I>a</I>+<I>b</I>)(<I>a</I>+<I>b</I>-<I>x</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15106 <p>Prop. 207. If <MATH>(<I>a</I>+<I>b</I>)<I>b</I>=2<I>a</I><SUP>2</SUP></MATH>, then <MATH><I>a</I>=<I>b</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15107 <p>(<G>q</G>) The two Lemmas to the <I>Surface-Loci</I> of Euclid have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15108 already been mentioned as significant. The first has the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15109 appearance of being a general enunciation, such as Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15110 <pb n=426><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15111 is fond of giving, to cover a class of propositions. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15112 enunciation may be translated as follows: ‘If <I>AB</I> be a straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15113 line, and <I>CD</I> a straight line parallel to a straight line given in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15114 position, and if the ratio <I>AD.DB</I>:<I>DC</I><SUP>2</SUP> be given, the point <I>C</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15115 lies on a conic section. If now <I>AB</I> be no longer given in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15116 position, and the points <I>A</I>, <I>B</I> are no longer given but lie
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15117 (respectively) on straight lines <I>AE</I>, <I>EB</I> given in position, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15118 point <I>C</I> raised above (the plane containing <I>AE</I>, <I>EB</I>) lies on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15119 a surface given in position. And this was proved.’ Tannery
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15120 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15121 was the first to explain this intelligibly;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15122 and his interpretation only requires the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15123 very slight change in the text of sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15124 stituting <G>eu)qei/ais</G> for <G>eu)qei=a</G> in the phrase
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15125 <G>ge/nhtai de\ pro\s qe/sei eu)qei=a tai=s</G> <I>AE</I>, <I>EB</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15126 It is not clear whether, when <I>AB</I> ceases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15127 to be given in <I>position</I>, it is still given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15128 in <I>length</I>. If it is given in <I>length</I> and <I>A</I>, <I>B</I> move on the lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15129 <I>AE</I>, <I>EB</I> respectively, the surface which is the locus of <I>C</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15130 a complicated one such as Euclid would hardly have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15131 in a position to investigate. But two possible cases are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15132 indicated which he may have discussed, (1) that in which <I>AB</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15133 moves always parallel to itself and varies in length accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15134 ingly, (2) that in which the two lines on which <I>A</I>, <I>B</I> move are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15135 parallel instead of meeting at a point. The loci in these two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15136 cases would of course be a cone and a cylinder respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15137 <p>The second Lemma is still more important, since it is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15138 first statement on record of the focus-directrix property of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15139 the three conic sections. The proof, after Pappus, has been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15140 set out above (pp. 119-21).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15141 <C>(<G>i</G>) <I>An unallocated Lemma</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15142 <p>Book VII ends (pp. 1016-18) with a lemma which is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15143 given under any particular treatise belonging to the <I>Treasury
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15144 of Analysis</I>, but is simply called ‘Lemma to the <G>*)analuo/menos</G>’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15145 If <I>ABC</I> be a triangle right-angled at <I>B</I>, and <I>AB</I>, <I>BC</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15146 divided at <I>F</I>, <I>G</I> so that <MATH><I>AF</I>:<I>FB</I>=<I>BG</I>:<I>GC</I>=<I>AB</I>:<I>BC</I></MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15147 if <I>AEG</I>, <I>CEF</I> be joined and <I>BE</I> joined and produced to <I>D</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15148 then shall <I>BD</I> be perpendicular to <I>AC</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15149 <p>The text is unsatisfactory, for there is a long interpolation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15150 containing an attempt at a proof by <I>reductio ad absurdum</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15151 <pb n=427><head>THE <I>COLLECTION</I>. BOOKS VII, VIII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15152 but the genuine proof is indicated, although it breaks off
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15153 before it is quite complete.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15154 <p>Since <MATH><I>AF</I>:<I>FB</I>=<I>BG</I>:<I>GC</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15155 <I>AB</I>:<I>FB</I>=<I>BC</I>:<I>GC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15156 or <MATH><I>AB</I>:<I>BC</I>=<I>FB</I>:<I>GC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15157 But, by hypothesis, <MATH><I>AB</I>:<I>BC</I>=<I>BG</I>:<I>GC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15158 therefore <MATH><I>BF</I>=<I>BG</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15159 <p>From this point the proof apparently proceeded by analysis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15160 ‘Suppose it done’ (<G>gegone/tw</G>), i.e. suppose the proposition true,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15161 and <I>BED</I> perpendicular to <I>AC</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15162 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15163 <p>Then, by similarity of triangles, <MATH><I>AD</I>:<I>DB</I>=<I>AB</I>:<I>BC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15164 therefore <MATH><I>AF</I>:<I>FB</I>=<I>AD</I>:<I>DB</I></MATH>, and consequently the angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15165 <I>ADB</I> is bisected by <I>DF</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15166 <p>Similarly the angle <I>BDC</I> is bisected by <I>DG</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15167 <p>Therefore each of the angles <I>BDF</I>, <I>BDG</I> is half a right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15168 angle, and consequently the angle <I>FDG</I> is a right angle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15169 <p>Therefore <I>B</I>, <I>G</I>, <I>D</I>, <I>F</I> are concyclic; and, since the angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15170 <I>FDB</I>, <I>BDG</I> are equal, <MATH><I>FB</I>=<I>BG</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15171 <p>This is of course the result above proved.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15172 <p>Evidently the interpolator tried to clinch the argument by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15173 proving that the angle <I>BDA</I> could not be anything but a right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15174 angle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15175 <C>Book VIII.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15176 <p>Book VIII of the <I>Collection</I> is mainly on mechanics, although
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15177 it contains, in addition, some propositions of purely geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15178 interest.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15179 <pb n=428><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15180 <C><I>Historical preface</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15181 <p>It begins with an interesting preface on the claim of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15182 theoretical mechanics, as distinct from the merely practical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15183 or industrial, to be regarded as a mathematical subject.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15184 Archimedes, Philon, Heron of Alexandria are referred to as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15185 the principal exponents of the science, while Carpus of Antioch
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15186 is also mentioned as having applied geometry to ‘certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15187 (practical) arts’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15188 <p>The date of Carpus is uncertain, though it is probable that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15189 he came after Geminus; the most likely date seems to be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15190 first or second century A.D. Simplicius gives the authority of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15191 Iamblichus for the statement that Carpus squared the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15192 by means of a certain curve, which he simply called a curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15193 generated by a double motion.<note>Simplicius on Arist. <I>Categ.</I>, p. 192, Kalbfleisch.</note> Proclus calls him ‘Carpus the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15194 writer on mechanics (<G>o( mhxaniko/s</G>)’, and quotes from a work of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15195 his on Astronomy some remarks about the relation between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15196 problems and theorems and the ‘priority in order’ of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15197 former.<note>Proclus on Eucl. I, pp. 241-3.</note> Proclus also mentions him as having held that an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15198 angle belongs to the category of <I>quantity</I> (<G>poso/n</G>), since it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15199 represents a sort of ‘distance’ between the two lines forming
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15200 it, this distance being ‘extended one way’ (<G>e)f) e(\n diestw/s</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15201 though in a different sense from that in which a line represents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15202 extension one way, so that Carpus's view appeared to be ‘the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15203 greatest possible paradox’<note><I>Ib.</I>, pp. 125. 25-126. 6.</note>; Carpus seems in reality to have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15204 been anticipating the modern view of an angle as representing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15205 <I>divergence</I> rather than distance, and to have meant by <G>e)f) e(\n</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15206 <I>in one sense</I> (rotationally), as distinct from one way or in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15207 dimension (linearly).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15208 <p>Pappus tells us that Heron distinguished the logical, i.e.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15209 theoretical, part of mechanics from the practical or manual
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15210 (<G>xeirourgiko/n</G>), the former being made up of geometry, arith-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15211 metic, astronomy and physics, the latter of work in metal,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15212 architecture, carpentering and painting; the man who had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15213 been trained from his youth up in the <I>sciences</I> aforesaid as well
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15214 as practised in the said <I>arts</I> would naturally prove the best
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15215 architect and inventor of mechanical devices, but, as it is diffi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15216 cult or impossible for the same person to do both the necessary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15217 <pb n=429><head>THE <I>COLLECTION</I>. BOOK VIII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15218 mathematics and the practical work, he who has not the former
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15219 must perforce use the resources which practical experience in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15220 his particular art or craft gives him. Other varieties of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15221 mechanical work included by the ancients under the general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15222 term mechanics were (1) the use of the mechanical powers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15223 or devices for moving or lifting great weights by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15224 a small force, (2) the construction of engines of war for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15225 throwing projectiles a long distance, (3) the pumping of water
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15226 from great depths, (4) the devices of ‘wonder-workers’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15227 (<G>qaumasiourgoi/</G>), some depending on pneumatics (like Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15228 in the <I>Pneumatica</I>), some using strings, &c., to produce move-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15229 ments like those of living things (like Heron in ‘Automata and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15230 Balancings’), some employing floating bodies (like Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15231 in ‘Floating Bodies’), others using water to measure time
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15232 (like Heron in his ‘Water-clocks’), and lastly ‘sphere-making’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15233 or the construction of mechanical imitations of the movements
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15234 of the heavenly bodies with the uniform circular motion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15235 water as the motive power. Archimedes, says Pappus, was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15236 held to be the one person who had understood the cause and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15237 the reason of all these various devices, and had applied his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15238 extraordinarily versatile genius and inventiveness to all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15239 purposes of daily life, and yet, although this brought him
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15240 unexampled fame the world over, so that his name was on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15241 every one's lips, he disdained (according to Carpus) to write
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15242 any mechanical work save a tract on sphere-making, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15243 diligently wrote all that he could in a small compass of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15244 most advanced parts of geometry and of subjects connected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15245 with arithmetic. Carpus himself, says Pappus, as well as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15246 others applied geometry to practical arts, and with reason:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15247 ‘for geometry is in no wise injured, nay it is by nature
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15248 capable of giving substance to many arts by being associated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15249 with them, and, so far from being injured, it may be said,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15250 while itself advancing those arts, to be honoured and adorned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15251 by them in return.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15252 <C><I>The object of the Book</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15253 <p>Pappus then describes the object of the Book, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15254 to set out the propositions which the ancients established by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15255 geometrical methods, besides certain useful theorems dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15256 covered by himself, but in a shorter and clearer form and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15257 <pb n=430><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15258 in better logical sequence than his predecessors had attained.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15259 The sort of questions to be dealt with are (1) a comparison
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15260 between the force required to move a given weight along
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15261 a horizontal plane and that required to move the same weight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15262 upwards on an inclined plane, (2) the finding of two mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15263 proportionals between two unequal straight lines, (3) given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15264 a toothed wheel with a certain number of teeth, to find the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15265 diameter of, and to construct, another wheel with a given num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15266 ber of teeth to work on the former. Each of these things, he says,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15267 will be clearly understood in its proper place if the principles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15268 on which the ‘centrobaric doctrine’ is built up are first set out.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15269 It is not necessary, he adds, to define what is meant by ‘heavy’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15270 and ‘light’ or upward and downward motion, since these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15271 matters are discussed by Ptolemy in his <I>Mathematica</I>; but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15272 the notion of the centre of gravity is so fundamental in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15273 whole theory of mechanics that it is essential in the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15274 place to explain what is meant by the ‘centre of gravity’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15275 of any body.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15276 <C><I>On the centre of gravity</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15277 <p>Pappus then defines the centre of gravity as ‘the point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15278 within a body which is such that, if the weight be conceived
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15279 to be suspended from the point, it will remain at rest in any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15280 position in which it is put’.<note>Pappus, viii, p. 1030. 11-13.</note> The method of determining the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15281 point by means of the intersection, first of planes, and then of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15282 straight lines, is next explained (chaps. 1, 2), and Pappus then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15283 proves (Prop. 2) a proposition of some difficulty, namely that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15284 if <I>D</I>, <I>E</I>, <I>F</I> be points on the sides <I>BC</I>, <I>CA</I>, <I>AB</I> of a triangle <I>ABC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15285 such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15286 <MATH><I>BD</I>:<I>DC</I>=<I>CE</I>:<I>EA</I>=<I>AF</I>:<I>FB</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15287 then the centre of gravity of the triangle <I>ABC</I> is also the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15288 centre of gravity of the triangle <I>DEF</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15289 <p>Let <I>H</I>, <I>K</I> be the middle points of <I>BC</I>, <I>CA</I> respectively;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15290 join <I>AH</I>, <I>BK</I>. Join <I>HK</I> meeting <I>DE</I> in <I>L</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15291 <p>Then <I>AH</I>, <I>BK</I> meet in <I>G</I>, the centre of gravity of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15292 triangle <I>ABC</I>, and <MATH><I>AG</I>=2<I>GH</I>, <I>BG</I>=2<I>GK</I></MATH>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15293 <MATH><I>CA</I>:<I>AK</I>=<I>AB</I>:<I>HK</I>=<I>BG</I>:<I>GK</I>=<I>AG</I>:<I>GH</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15294 <pb n=431><head>THE <I>COLLECTION</I>. BOOK VIII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15295 <p>Now, by hypothesis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15296 <MATH><I>CE</I>:<I>EA</I>=<I>BD</I>:<I>DC</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15297 whence <MATH><I>CA</I>:<I>AE</I>=<I>BC</I>:<I>CD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15298 and, if we halve the antecedents,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15299 <MATH><I>AK</I>:<I>AE</I>=<I>HC</I>:<I>CD</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15300 therefore <MATH><I>AK</I>:<I>EK</I>=<I>HC</I>:<I>HD</I></MATH> or <MATH><I>BH</I>:<I>HD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15301 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15302 whence, <I>componendo</I>, <MATH><I>CE</I>:<I>EK</I>=<I>BD</I>:<I>DH</I></MATH>. (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15303 <p>But <MATH><I>AF</I>:<I>FB</I>=<I>BD</I>:<I>DC</I>=(<I>BD</I>:<I>DH</I>).(<I>DH</I>:<I>DC</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15304 =(<I>CE</I>:<I>EK</I>).(<I>DH</I>:<I>DC</I>)</MATH>. (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15305 <p>Now, <I>ELD</I> being a transversal cutting the sides of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15306 triangle <I>KHC</I>, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15307 <MATH><I>HL</I>:<I>KL</I>=(<I>CE</I>:<I>EK</I>).(<I>DH</I>:<I>DC</I>)</MATH>. (3)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15308 [This is ‘Menelaus's theorem’; Pappus does not, however,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15309 quote it, but proves the relation <I>ad hoc</I> in an added lemma by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15310 drawing <I>CM</I> parallel to <I>DE</I> to meet <I>HK</I> produced in <I>M</I>. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15311 proof is easy, for <MATH><I>HL</I>:<I>LK</I>=(<I>HL</I>:<I>LM</I>).(<I>LM</I>:<I>LK</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15312 =(<I>HD</I>:<I>DC</I>).(<I>CE</I>:<I>EK</I>)</MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15313 <p>It follows from (2) and (3) that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15314 <MATH><I>AF</I>:<I>FB</I>=<I>HL</I>:<I>LK</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15315 and, since <I>AB</I> is parallel to <I>HK</I>, and <I>AH</I>, <I>BK</I> are straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15316 lines meeting in <I>G</I>, <I>FGL</I> is a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15317 <p>[This is proved in another easy lemma by <I>reductio ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15318 absurdum</I>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15319 <pb n=432><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15320 <p>We have next to prove that <MATH><I>EL</I>=<I>LD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15321 <p>Now [again by ‘Menelaus's theorem’, proved <I>ad hoc</I> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15322 drawing <I>CN</I> parallel to <I>HK</I> to meet <I>ED</I> produced in <I>N</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15323 <MATH><I>EL</I>:<I>LD</I>=(<I>EK</I>:<I>KC</I>).(<I>CH</I>:<I>HD</I>)</MATH>. (4)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15324 <p>But, by (1) above, <MATH><I>CE</I>:<I>EK</I>=<I>BD</I>:<I>DH</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15325 therefore <MATH><I>CK</I>:<I>KE</I>=<I>BH</I>:<I>HD</I>=<I>CH</I>:<I>HD</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15326 so that <MATH>(<I>EK</I>:<I>KC</I>).(<I>CH</I>:<I>HD</I>)=1</MATH>, and therefore, from (4),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15327 <MATH><I>EL</I>=<I>LD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15328 <p>It remains to prove that <MATH><I>FG</I>=2<I>GL</I></MATH>, which is obvious by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15329 parallels, since <MATH><I>FG</I>:<I>GL</I>=<I>AG</I>:<I>GH</I>=2:1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15330 <p>Two more propositions follow with reference to the centre
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15331 of gravity. The first is, Given a rectangle with <I>AB</I>, <I>BC</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15332 adjacent sides, to draw from <I>C</I> a straight line meeting the side
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15333 opposite <I>BC</I> in a point <I>D</I> such that, if the trapezium <I>ADCB</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15334 hung from the point <I>D</I>, it will rest with <I>AD</I>, <I>BC</I> horizontal.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15335 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15336 In other words, the centre of gravity must be in <I>DL</I> drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15337 perpendicular to <I>BC</I>. Pappus proves by analysis that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15338 <MATH><I>CL</I><SUP>2</SUP>=3<I>BL</I><SUP>2</SUP></MATH>, so that the problem is reduced to that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15339 dividing <I>BC</I> into parts <I>BL</I>, <I>LC</I> such that this relation holds.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15340 The latter problem is solved (Prop. 6) by taking a point,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15341 say <I>X</I>, in <I>CB</I> such that <MATH><I>CX</I>=3<I>XB</I></MATH>, describing a semicircle on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15342 <I>BC</I> as diameter and drawing <I>XY</I> at right angles to <I>BC</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15343 meet the semicircle in <I>Y</I>, so that <MATH><I>XY</I><SUP>2</SUP>=(3/16)<I>BC</I><SUP>2</SUP></MATH>, and then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15344 dividing <I>CB</I> at <I>L</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15345 <MATH><I>CL</I>:<I>LB</I>=<I>CX</I>:<I>XY</I>(=3/4:1/4√3=√3:1)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15346 <p>The second proposition is this (Prop. 7). Given two straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15347 lines <I>AB</I>, <I>AC</I>, and <I>B</I> a fixed point on <I>AB</I>, if <I>CD</I> be drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15348 <pb n=433><head>THE <I>COLLECTION</I>. BOOK VIII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15349 with its extremities on <I>AC</I>, <I>AB</I> and so that <I>AC</I>:<I>BD</I> is a given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15350 ratio, then the centre of gravity of the triangle <I>ADC</I> will lie
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15351 on a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15352 <p>Take <I>E</I>, the middle point of <I>AC</I>, and <I>F</I> a point on <I>DE</I> such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15353 that <MATH><I>DF</I>=2<I>FE</I></MATH>. Also let <I>H</I> be a point on <I>BA</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15354 <MATH><I>BH</I>=2<I>HA</I></MATH>. Draw <I>FG</I> parallel to <I>AC</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15355 Then <MATH><I>AG</I>=(1/3)<I>AD</I></MATH>, and <MATH><I>AH</I>=(1/3)<I>AB</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15356 therefore <MATH><I>HG</I>=(1/3)<I>BD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15357 <p>Also <MATH><I>FG</I>=(2/3)<I>AE</I>=(1/3)<I>AC</I></MATH>. Therefore,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15358 since the ratio <I>AC</I>:<I>BD</I> is given, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15359 ratio <I>GH</I>:<I>GF</I> is given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15360 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15361 <p>And the angle <I>FGH</I> (=<I>A</I>) is given;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15362 therefore the triangle <I>FGH</I> is given in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15363 species, and consequently the angle <I>GHF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15364 is given. And <I>H</I> is a given point.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15365 Therefore <I>HF</I> is a given straight line, and it contains the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15366 centre of gravity of the triangle <I>ADC</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15367 <C><I>The inclined plane</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15368 <p>Prop. 8 is on the construction of a plane at a given inclination
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15369 to another plane parallel to the horizon, and with this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15370 Pappus leaves theory and proceeds to the practical part.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15371 Prop. 9 (p. 1054. 4 sq.) investigates the problem ‘Given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15372 a weight which can be drawn along a plane parallel to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15373 horizon by a given force, and a plane inclined to the horizon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15374 at a given angle, to find the force required to draw the weight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15375 upwards on the inclined plane’. This seems to be the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15376 or only attempt in ancient times to investigate motion on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15377 an inclined plane, and as such it is curious, though of no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15378 value.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15379 <p>Let <I>A</I> be the weight which can be moved by a force <I>C</I> along
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15380 a horizontal plane. Conceive a sphere with weight equal to <I>A</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15381 placed in contact at <I>L</I> with the given inclined plane; the circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15382 <I>OGL</I> represents a section of the sphere by a vertical plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15383 passing through <I>E</I> its centre and <I>LK</I> the line of greatest slope
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15384 drawn through the point <I>L</I>. Draw <I>EGH</I> horizontal and therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15385 parallel to <I>MN</I> in the plane of section, and draw <I>LF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15386 perpendicular to <I>EH</I>. Pappus seems to regard the plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15387 as rough, since he proceeds to make a system in equilibrium
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15388 <pb n=434><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15389 about <I>FL</I> as if <I>L</I> were the fulcrum of a lever. Now the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15390 weight <I>A</I> acts vertically downwards along a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15391 through <I>E</I>. To balance it, Pappus supposes a weight <I>B</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15392 attached with its centre of gravity at <I>G</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15393 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15394 <p>Then <MATH><I>A</I>:<I>B</I>=<I>GF</I>:<I>EF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15395 =(<I>EL</I>-<I>EF</I>):<I>EF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15396 [=(1-sin <G>q</G>):sin <G>q</G></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15397 where <MATH>&angle;<I>KMN</I>=<G>q</G></MATH>];
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15398 and, since &angle;<I>KMN</I> is given, the ratio <I>EF</I>:<I>EL</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15399 and therefore the ratio (<I>EL</I>-<I>EF</I>):<I>EF</I>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15400 given; thus <I>B</I> is found.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15401 <p>Now, says Pappus, if <I>D</I> is the force which will move <I>B</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15402 along a horizontal plane, as <I>C</I> is the force which will move
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15403 <I>A</I> along a horizontal plane, the sum of <I>C</I> and <I>D</I> will be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15404 force required to move the sphere upwards on the inclined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15405 plane. He takes the particular case where <MATH><G>q</G>=60°</MATH>. Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15406 sin <G>q</G> is approximately 104/120 (he evidently uses 1/2.26/15 for 1/2√3),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15407 and <MATH><I>A</I>:<I>B</I>=16:104</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15408 Suppose, for example, that <MATH><I>A</I>=200</MATH> talents; then <I>B</I> is 1300
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15409 talents. Suppose further that <I>C</I> is 40 man-power; then, since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15410 <MATH><I>D</I>:<I>C</I>=<I>B</I>:<I>A</I>, <I>D</I>=260</MATH> man-power; and it will take <I>D</I>+<I>C</I>, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15411 300 man-power, to move the weight up the plane!
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15412 <p>Prop. 10 gives, from Heron's <I>Barulcus</I>, the machine con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15413 sisting of a pulley, interacting toothed wheels, and a spiral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15414 screw working on the last wheel and turned by a handle;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15415 Pappus merely alters the proportions of the weight to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15416 force, and of the diameter of the wheels. At the end of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15417 the chapter (pp. 1070-2) he repeats his construction for the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15418 finding of two mean proportionals.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15419 <C><I>Construction of a conic through five points</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15420 <p>Chaps. 13-17 are more interesting, for they contain the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15421 solution of the problem of <I>constructing a conic through five
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15422 given points</I>. The problem arises in this way. Suppose we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15423 are given a broken piece of the surface of a cylindrical column
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15424 such that no portion of the circumference of either of its base
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15425 <pb n=435><head>THE <I>COLLECTION</I>. BOOK VIII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15426 is left intact, and let it be required to find the diameter of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15427 a circular section of the cylinder. We take any two points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15428 <I>A</I>, <I>B</I> on the surface of the fragment and by means of these we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15429 find five points on the surface all lying in one plane section,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15430 in general oblique. This is done by taking five different radii
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15431 and drawing pairs of circles with <I>A</I>, <I>B</I> as centres and with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15432 each of the five radii successively. These pairs of circles with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15433 equal radii, intersecting at points on the surface, determine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15434 five points on the plane bisecting <I>AB</I> at right angles. The five
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15435 points are then represented on any plane by triangulation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15436 <p>Suppose the points are <I>A</I>, <I>B</I>, <I>C</I>, <I>D</I>, <I>E</I> and are such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15437 no two of the lines connecting the different pairs are parallel.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15438 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15439 This case can be reduced to the construction of a conic through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15440 the five points <I>A</I>, <I>B</I>, <I>D</I>, <I>E</I>, <I>F</I> where <I>EF</I> is parallel to <I>AB</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15441 This is shown in a subsequent lemma (chap. 16).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15442 <p>For, if <I>EF</I> be drawn through <I>E</I> parallel to <I>AB</I>, and if <I>CD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15443 meet <I>AB</I> in <I>O</I> and <I>EF</I> in <I>O</I>′, we have, by the well-known
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15444 proposition about intersecting chords,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15445 <MATH><I>CO.OD</I>:<I>AO.OB</I>=<I>CO</I>′.<I>O</I>′<I>D</I>:<I>EO</I>′.<I>O</I>′<I>F</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15446 whence <I>O</I>′<I>F</I> is known, and <I>F</I> is determined.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15447 <p>We have then (Prop. 13) to construct a conic through <I>A</I>, <I>B</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15448 <I>D</I>, <I>E</I>, <I>F</I>, where <I>EF</I> is parallel to <I>AB</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15449 <p>Bisect <I>AB</I>, <I>EF</I> at <I>V</I>, <I>W</I>; then <I>VW</I> produced both ways
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15450 is a diameter. Draw <I>DR</I>, the chord through <I>D</I> parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15451 <pb n=436><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15452 to this diameter. Then <I>R</I> is determined by means of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15453 relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15454 <MATH><I>RG.GD</I>:<I>BG.GA</I>=<I>RH.HD</I>:<I>FH.HE</I></MATH> (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15455 in this way.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15456 <p>Join <I>DB</I>, <I>RA</I>, meeting <I>EF</I> in <I>K</I>, <I>L</I> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15457 <p>Then, by similar triangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15458 <MATH><I>RG.GD</I>:<I>BG.GA</I>=(<I>RH</I>:<I>HL</I>).(<I>DH</I>:<I>HK</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15459 =<I>RH.HD</I>:<I>KH.HL</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15460 <p>Therefore, by (1), <MATH><I>FH.HE</I>=<I>KH.HL</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15461 whence <I>HL</I> is determined, and therefore <I>L</I>. The intersection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15462 of <I>AL</I>, <I>DH</I> determines <I>R</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15463 <p>Next, in order to find the extremities <I>P</I>, <I>P</I>′ of the diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15464 through <I>V</I>, <I>W</I>, we draw <I>ED</I>, <I>RF</I> meeting <I>PP</I>′ in <I>M</I>, <I>N</I> respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15465 <p>Then, as before,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15466 <MATH><I>FW.WE</I>:<I>P</I>′<I>W.WP</I>=<I>FH.HE</I>:<I>RH.HD</I></MATH>, by the ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15467 <MATH>=<I>FW.WE</I>:<I>NW.WM</I></MATH>, by similar triangles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15468 <p>Therefore <MATH><I>P</I>′<I>W.WP</I>=<I>NW.WM</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15469 and similarly we can find the value of <I>P</I>′<I>V.VP</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15470 <p>Now, says Pappus, since <I>P</I>′<I>W.WP</I> and <I>P</I>′<I>V.VP</I> are given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15471 areas and the points <I>V</I>, <I>W</I> are given, <I>P</I>, <I>P</I>′ are given. His
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15472 determination of <I>P</I>, <I>P</I>′ amounts (Prop. 14 following) to an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15473 elimination of one of the points and the finding of the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15474 by means of an equation of the second degree.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15475 <p>Take two points <I>Q</I>, <I>Q</I>′ on the diameter such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15476 <MATH><I>P</I>′<I>V.VP</I>=<I>WV.VQ</I></MATH>, (<G>a</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15477 <MATH><I>P</I>′<I>W.WP</I>=<I>VW.WQ</I>′</MATH>; (<G>b</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15478 <I>Q</I>, <I>Q</I>′ are thus known, while <I>P</I>, <I>P</I>′ remain to be found.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15479 <p>By (<G>a</G>) <MATH><I>P</I>′<I>V</I>:<I>VW</I>=<I>QV</I>:<I>VP</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15480 whence <MATH><I>P</I>′<I>W</I>:<I>VW</I>=<I>PQ</I>:<I>PV</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15481 <p>Therefore, by means of (<G>b</G>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15482 <MATH><I>PQ</I>:<I>PV</I>=<I>Q</I>′<I>W</I>:<I>WP</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15483 <pb n=437><head>THE <I>COLLECTION</I>. BOOK VIII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15484 so that <MATH><I>PQ</I>:<I>QV</I>=<I>Q</I>′<I>W</I>:<I>PQ</I>′</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15485 or <MATH><I>PQ.PQ</I>′=<I>QV.Q</I>′<I>W</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15486 <p>Thus <I>P</I> can be found, and similarly <I>P</I>′.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15487 <p>The conjugate diameter is found by virtue of the relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15488 <MATH>(conjugate diam.)<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>=<I>p</I>:<I>PP</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15489 where <I>p</I> is the latus rectum to <I>PP</I>′ determined by the property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15490 of the curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15491 <MATH><I>p</I>:<I>PP</I>′=<I>AV</I><SUP>2</SUP>:<I>PV.VP</I>′</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15492 <C><I>Problem, Given two conjugate diameters of an ellipse,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15493 to find the axes</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15494 <p>Lastly, Pappus shows (Prop. 14, chap. 17) how, when we are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15495 given two conjugate diameters, we can find the axes. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15496 construction is as follows. Let <I>AB</I>, <I>CD</I> be conjugate diameters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15497 (<I>CD</I> being the greater), <I>E</I> the centre.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15498 <p>Produce <I>EA</I> to <I>H</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15499 <MATH><I>EA.AH</I>=<I>DE</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15500 <p>Through <I>A</I> draw <I>FG</I> parallel to <I>CD</I>. Bisect <I>EH</I> in <I>K</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15501 draw <I>KL</I> at right angles to <I>EH</I> meeting <I>FG</I> in <I>L</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15502 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15503 <p>With <I>L</I> as centre, and <I>LE</I> as radius, describe a circle cutting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15504 <I>GF</I> in <I>G</I>, <I>F</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15505 <p>Join <I>EF</I>, <I>EG</I>, and from <I>A</I> draw <I>AM</I>, <I>AN</I> parallel to <I>EF</I>, <I>EG</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15506 respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15507 <pb n=438><head>PAPPUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15508 <p>Take points <I>P</I>, <I>R</I> on <I>EG</I>, <I>EF</I> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15509 <MATH><I>EP</I><SUP>2</SUP>=<I>GE.EM</I></MATH>, and <MATH><I>ER</I><SUP>2</SUP>=<I>FE.EN</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15510 <p>Then <I>EP</I> is half the major axis, and <I>ER</I> half the minor axis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15511 <p>Pappus omits the proof.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15512 <C><I>Problem of seven hexagons in a circle</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15513 <p>Prop. 19 (chap. 23) is a curious problem. To inscribe seven
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15514 equal regular hexagons in a circle in such a way that one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15515 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15516 is about the centre of the circle, while six others stand on its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15517 sides and have the opposite sides in each case placed as chords
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15518 in the circle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15519 <p>Suppose <I>GHKLNM</I> to be the hexagon so described on <I>HK</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15520 a side of the inner hexagon; <I>OKL</I> will then be a straight line.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15521 Produce <I>OL</I> to meet the circle in <I>P</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15522 <p>Then <MATH><I>OK</I>=<I>KL</I>=<I>LN</I></MATH>. Therefore, in the triangle <I>OLN</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15523 <MATH><I>OL</I>=2<I>LN</I></MATH>, while the included angle <I>OLN</I> (=120°) is also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15524 given. Therefore the triangle is given in species; therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15525 the ratio <I>ON</I>:<I>NL</I> is given, and, since <I>ON</I> is given, the side <I>NL</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15526 of each of the hexagons is given.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15527 <p>Pappus gives the auxiliary construction thus. Let <I>AF</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15528 taken equal to the radius <I>OP</I>. Let <MATH><I>AC</I>=(1/3)<I>AF</I></MATH>, and on <I>AC</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15529 base describe a segment of a circle containing an angle of 60°.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15530 Take <I>CE</I> equal to 4/5<I>AC</I>, and draw <I>EB</I> to touch the circle at <I>B</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15531 <pb n=439><head>THE <I>COLLECTION</I>. BOOK VIII</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15532 Then he proves that, if we join <I>AB</I>, <I>AB</I> is equal to the length
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15533 of the side of the hexagon required.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15534 <p>Produce <I>BC</I> to <I>D</I> so that <MATH><I>BD</I>=<I>BA</I></MATH>, and join <I>DA</I>. <I>ABD</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15535 is then equilateral.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15536 <p>Since <I>EB</I> is a tangent to the segment, <MATH><I>AE.EC</I>=<I>EB</I><SUP>2</SUP></MATH> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15537 <MATH><I>AE</I>:<I>EB</I>=<I>EB</I>:<I>EC</I></MATH>, and the triangles <I>EAB</I>, <I>EBC</I> are similar.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15538 <p>Therefore <MATH><I>BA</I><SUP>2</SUP>:<I>BC</I><SUP>2</SUP>=<I>AE</I><SUP>2</SUP>:<I>EB</I><SUP>2</SUP>=<I>AE</I>:<I>EC</I>=9:4</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15539 and <MATH><I>BC</I>=(2/3)<I>BA</I>=(2/3)<I>BD</I></MATH>, so that <MATH><I>BC</I>=2<I>CD</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15540 <p>But <MATH><I>CF</I>=2<I>CA</I></MATH>; therefore <MATH><I>AC</I>:<I>CF</I>=<I>DC</I>:<I>CB</I></MATH>, and <I>AD</I>, <I>BF</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15541 are parallel.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15542 <p>Therefore <MATH><I>BF</I>:<I>AD</I>=<I>BC</I>:<I>CD</I>=2:1</MATH>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15543 <MATH><I>BF</I>=2<I>AD</I>=2<I>AB</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15544 <p>Also <MATH>&angle;<I>FBC</I>=&angle;<I>BDA</I>=60°</MATH>, so that <MATH>&angle;<I>ABF</I>=120°</MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15545 the triangle <I>ABF</I> is therefore equal and similar to the required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15546 triangle <I>NLO</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15547 <C><I>Construction of toothed wheels and indented screws</I></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15548 <p>The rest of the Book is devoted to the construction (1) of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15549 toothed wheels with a given number of teeth equal to those of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15550 a given wheel, (2) of a cylindrical helix, the <I>cochlias</I>, indented
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15551 so as to work on a toothed wheel. The text is evidently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15552 defective, and at the end an interpolator has inserted extracts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15553 about the mechanical powers from Heron's <I>Mechanics</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15554 <pb><C>XX
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15555 ALGEBRA: DIOPHANTUS OF ALEXANDRIA</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15556 <C>Beginnings learnt from Egypt.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15557 <p>IN algebra, as in geometry, the Greeks learnt the beginnings
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15558 from the Egyptians. Familiarity on the part of the Greeks
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15559 with Egyptian methods of calculation is well attested. (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15560 These methods are found in operation in the Heronian writings
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15561 and collections. (2) Psellus in the letter published by Tannery
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15562 in his edition of Diophantus speaks of ‘the method of arith-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15563 metical calculations used by the Egyptians, by which problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15564 in analysis are handled’; he adds details, doubtless taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15565 from Anatolius, of the technical terms used for different kinds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15566 of numbers, including the powers of the unknown quantity.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15567 (3) The scholiast to Plato's <I>Charmides</I> 165 E says that ‘parts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15568 of <G>logistikh/</G>, the science of calculation, are the so-called Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15569 and Egyptian methods in multiplications and divisions, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15570 the additions and subtractions of fractions’. (4) Plato himself
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15571 in the <I>Laws</I> 819 A-C says that free-born boys should, as is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15572 practice in Egypt, learn, side by side with reading, simple
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15573 mathematical calculations adapted to their age, which should
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15574 be put into a form such as to combine amusement with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15575 instruction: problems about the distribution of, say, apples or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15576 garlands, the calculation of mixtures, and other questions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15577 arising in military or civil life.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15578 <C>‘Hau’-calculations.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15579 <p>The Egyptian calculations here in point (apart from their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15580 method of writing and calculating in fractions, which, with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15581 the exception of 2/3, were always decomposed and written
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15582 as the sum of a diminishing series of aliquot parts or sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15583 multiples) are the <I>hau</I>-calculations. <I>Hau</I>, meaning a <I>heap</I>, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15584 the term denoting the unknown quantity, and the calculations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15585 <pb n=441><head>‘HAU’-CALCULATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15586 in terms of it are equivalent to the solutions of simple equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15587 with one unknown quantity. Examples from the Papyrus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15588 Rhind correspond to the following equations:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15589 <MATH>1/7<I>x</I>+<I>x</I>=19,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15590 2/3<I>x</I>+1/2<I>x</I>+1/7<I>x</I>+<I>x</I>=33,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15591 (<I>x</I>+2/3<I>x</I>)-1/3(<I>x</I>+2/3<I>x</I>)=10</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15592 <p>The Egyptians anticipated, though only in an elementary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15593 form, a favourite method of Diophantus, that of the ‘false
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15594 supposition’ or ‘regula falsi’. An arbitrary assumption is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15595 made as to the value of the unknown, and the true value
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15596 is afterwards found by a comparison of the result of sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15597 stituting the wrong value in the original expression with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15598 actual data. Two examples may be given. The first, from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15599 the Papyrus Rhind, is the problem of dividing 100 loaves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15600 among five persons in such a way that the shares are in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15601 arithmetical progression, and one-seventh of the sum of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15602 first three shares is equal to the sum of the other two. If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15603 <MATH><I>a</I>+4<I>d</I>, <I>a</I>+3<I>d</I>, <I>a</I>+2<I>d</I>, <I>a</I>+<I>d</I></MATH>, <I>a</I> be the shares, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15604 <MATH>3<I>a</I>+9<I>d</I>=7(2<I>a</I>+<I>d</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15605 or <MATH><I>d</I>=5 1/2<I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15606 Ahmes says, without any explanation, ‘make the difference,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15607 as it is, 5 1/2’, and then, assuming <MATH><I>a</I>=1</MATH>, writes the series
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15608 23, 17 1/2, 12, 6 1/2, 1. The addition of these gives 60, and 100 is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15609 1 2/3 times 60. Ahmes says simply ‘multiply 1 2/3 times’ and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15610 thus gets the correct values 38 1/3, 29 1/6, 20, 10 2/3 1/6, 1 2/3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15611 <p>The second example (taken from the Berlin Papyrus 6619)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15612 is the solution of the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15613 <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>=100,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15614 <I>x</I>:<I>y</I>=1:(3/4), or <I>y</I>=3/4<I>x</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15615 <I>x</I> is first assumed to be 1, and <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP></MATH> is thus found to be 25/16.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15616 In order to make 100, 25/16 has to be multiplied by 64 or 8<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15617 The true value of <I>x</I> is therefore 8 times 1, or 8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15618 <C>Arithmetical epigrams in the Greek Anthology.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15619 <p>The simple equations solved in the Papyrus Rhind are just
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15620 the kind of equations of which we find many examples in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15621 <pb n=442><head>ALGEBRA: DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15622 arithmetical epigrams contained in the Greek Anthology. Most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15623 of these appear under the name of Metrodorus, a grammarian,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15624 probably of the time of the Emperors Anastasius I (A.D. 491-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15625 518) and Justin I (A.D. 518-27). They were obviously only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15626 collected by Metrodorus, from ancient as well as more recent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15627 sources. Many of the epigrams (46 in number) lead to simple
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15628 equations, and several of them are problems of dividing a num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15629 ber of apples or nuts among a certain number of persons, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15630 is to say, the very type of problem mentioned by Plato. For
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15631 example, a number of apples has to be determined such that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15632 if four persons out of six receive one-third, one-eighth, one-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15633 fourth and one-fifth respectively of the whole number, while
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15634 the fifth person receives 10 apples, there is one apple left over
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15635 for the sixth person, i.e.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15636 <MATH>1/3<I>x</I>+1/8<I>x</I>+1/4<I>x</I>+1/5<I>x</I>+10+1=<I>x</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15637 Just as Plato alludes to bowls (<G>fia/lai</G>) of different metals,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15638 there are problems in which the weights of bowls have to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15639 be found. We are thus enabled to understand the allusions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15640 Proclus and the scholiast on <I>Charmides</I> 165 E to <G>mhli=tai</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15641 and <G>fiali=tai a)riqmoi/</G>, ‘numbers of apples or of bowls’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15642 It is evident from Plato's allusions that the origin of such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15643 simple algebraical problems dates back, at least, to the fifth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15644 century B.C.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15645 <p>The following is a classification of the problems in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15646 <I>Anthology</I>. (1) Twenty-three are simple equations in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15647 unknown and of the type shown above; one of these is an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15648 epigram on the age of Diophantus and certain incidents of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15649 his life (xiv. 126). (2) Twelve are easy simultaneous equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15650 with two unknowns, like Dioph. I. 6; they can of course be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15651 reduced to a simple equation with one unknown by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15652 an easy elimination. One other (xiv. 51) gives simultaneous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15653 equations in three unknowns
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15654 <MATH><I>x</I>=<I>y</I>+1/3<I>z</I>, <I>y</I>=<I>z</I>+1/3<I>x</I>, <I>z</I>=10+1/3<I>y</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15655 and one (xiv. 49) gives four equations in four unknowns,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15656 <MATH><I>x</I>+<I>y</I>=40, <I>x</I>+<I>z</I>=45, <I>x</I>+<I>u</I>=36, <I>x</I>+<I>y</I>+<I>z</I>+<I>u</I>=60</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15657 <p>With these may be compared Dioph. I. 16-21, as well as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15658 general solution of any number of simultaneous linear equa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15659 <pb n=443><head>EPIGRAMS IN THE GREEK ANTHOLOGY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15660 tions of this type with the same number of unknown quantities
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15661 which was given by Thymaridas, an early Pythagorean, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15662 was called the <G>e)pa/nqhma</G>, ‘flower’ or ‘bloom’ of Thymaridas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15663 (see vol. i, pp. 94-6). (3) Six more are problems of the usual
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15664 type about the filling and emptying of vessels by pipes; e.g.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15665 (xiv. 130) one pipe fills the vessel in one day, a second in two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15666 and a third in three; how long will all three running together
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15667 take to fill it? Another about brickmakers (xiv. 136) is of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15668 the same sort.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15669 <C>Indeterminate equations of the first degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15670 <p>The Anthology contains (4) two <I>indeterminate</I> equations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15671 the first degree which can be solved in positive integers in an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15672 infinite number of ways (xiv. 48, 144); the first is a distribu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15673 tion of apples, 3<I>x</I> in number, into parts satisfying the equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15674 <MATH><I>x</I>-3<I>y</I>=<I>y</I></MATH>, where <I>y</I> is not less than 2; the second. leads to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15675 three equations connecting four unknown quantities:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15676 <MATH><I>x</I>+<I>y</I>=<I>x</I><SUB>1</SUB>+<I>y</I><SUB>1</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15677 <I>x</I>=2<I>y</I><SUB>1</SUB>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15678 <I>x</I><SUB>1</SUB>=3<I>y</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15679 the general solution of which is <MATH><I>x</I>=4<I>k</I>, <I>y</I>=<I>k</I>, <I>x</I><SUB>1</SUB>=3<I>k</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15680 <I>y</I><SUB>1</SUB>=2<I>k</I></MATH>. These very equations, which, however, are made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15681 determinate by assuming that <MATH><I>x</I>+<I>y</I>=<I>x</I><SUB>1</SUB>+<I>y</I><SUB>1</SUB>=100</MATH>, are solved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15682 in Dioph. I. 12.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15683 <p>Enough has been said to show that Diophantus was not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15684 the inventor of Algebra. Nor was he the first to solve inde-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15685 terminate problems of the second degree.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15686 <C>Indeterminate equations of second degree before
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15687 Diophantus.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15688 <p>Take first the problem (Dioph. II. 8) of dividing a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15689 number into two squares, or of finding a right-angled triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15690 with sides in rational numbers. We have already seen that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15691 Pythagoras is credited with the discovery of a general formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15692 for finding such triangles, namely,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15693 <MATH><I>n</I><SUP>2</SUP>+{1/2(<I>n</I><SUP>2</SUP>-1)}<SUP>2</SUP>={1/2(<I>n</I><SUP>2</SUP>+1)}<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15694 <pb n=444><head>ALGEBRA: DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15695 where <I>n</I> is any odd number, and Plato with another formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15696 of the same sort, namely <MATH>(2<I>n</I>)<SUP>2</SUP>+(<I>n</I><SUP>2</SUP>-1)<SUP>2</SUP>=(<I>n</I><SUP>2</SUP>+1)<SUP>2</SUP></MATH>. Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15697 (Lemma following X. 28) finds the following more general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15698 formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15699 <MATH><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>p</I><SUP>2</SUP><I>q</I><SUP>2</SUP>={1/2(<I>mnp</I><SUP>2</SUP>+<I>mnq</I><SUP>2</SUP>)}<SUP>2</SUP>-{1/2(<I>mnp</I><SUP>2</SUP>-<I>mnq</I><SUP>2</SUP>)}<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15700 <p>The Pythagoreans too, as we have seen (vol. i, pp. 91-3),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15701 solved another indeterminate problem, discovering, by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15702 of the series of ‘side-’ and ‘diameter-numbers’, any number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15703 of successive integral solutions of the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15704 <MATH>2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>=±1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15705 <p>Diophantus does not particularly mention this equation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15706 but from the Lemma to VI. 15 it is clear that he knew how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15707 to find any number of solutions when one is known. Thus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15708 seeing that <MATH>2<I>x</I><SUP>2</SUP>-1=<I>y</I><SUP>2</SUP></MATH> is satisfied by <MATH><I>x</I>=1, <I>y</I>=1</MATH>, he would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15709 put
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15710 <MATH>2(1+<I>x</I>)<SUP>2</SUP>-1=a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15711 =(<I>px</I>-1)<SUP>2</SUP></MATH>, say;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15712 whence <MATH><I>x</I>=(4+2<I>p</I>)/(<I>p</I><SUP>2</SUP>-2)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15713 Take the value <MATH><I>p</I>=2</MATH>, and we have <MATH><I>x</I>=4</MATH>, and <MATH><I>x</I>+1=5</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15714 in this case <MATH>2.5<SUP>2</SUP>-1=49=7<SUP>2</SUP></MATH>. Putting <I>x</I>+5 in place of <I>x</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15715 we can find a still higher value, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15716 <C>Indeterminate equations in the Heronian collections.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15717 <p>Some further Greek examples of indeterminate analysis are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15718 now available. They come from the Constantinople manuscript
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15719 (probably of the twelfth century) from which Schöne edited
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15720 the <I>Metrica</I> of Heron; they have been published and translated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15721 by Heiberg, with comments by Zeuthen.<note><I>Bibliotheca mathematica</I>, viii<SUB>s</SUB>, 1907-8, pp. 118-34. See now <I>Geom</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15722 24. 1-13 in Heron, vol. iv (ed. Heiberg), pp. 414-26.</note> Two of the problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15723 (thirteen in number) had been published in a less complete
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15724 form in Hultsch's Heron (<I>Geëponicus</I>, 78, 79); the others
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15725 are new.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15726 <p>I. The first problem is to find two rectangles such that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15727 perimeter of the second is three times that of the first, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15728 the area of the first is three times that of the second. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15729 <pb n=445><head>HERONIAN INDETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15730 number 3 is of course only an illustration, and the problem is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15731 equivalent to the solution of the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15732 <MATH><BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15733 (1) <I>u</I>+<I>v</I>=<I>n</I>(<I>x</I>+<I>y</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15734 (2) <I>xy</I>=<I>n.uv</I></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15735 <p>The solution given in the text is equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15736 <MATH><BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15737 <I>x</I>=2<I>n</I><SUP>3</SUP>-1, <I>y</I>=2<I>n</I><SUP>3</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15738 <I>u</I>=<I>n</I>(4<I>n</I><SUP>3</SUP>-2), <I>v</I>=<I>n</I></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15739 <p>Z<*>uthen suggests that the solution may have been obtained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15740 thus. As the problem is indeterminate, it would be natural
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15741 to start with some hypothesis, e.g. to put <I>v</I>=<I>n</I>. It would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15742 follow from equation (1) that <I>u</I> is a multiple of <I>n</I>, say <I>nz</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15743 We have then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15744 <MATH><I>x</I>+<I>y</I>=1+<I>z</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15745 while, by (2), <MATH><I>xy</I>=<I>n</I><SUP>3</SUP><I>z</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15746 whence <MATH><I>xy</I>=<I>n</I><SUP>3</SUP>(<I>x</I>+<I>y</I>)-<I>n</I><SUP>3</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15747 or <MATH>(<I>x</I>-<I>n</I><SUP>3</SUP>)(<I>y</I>-<I>n</I><SUP>3</SUP>)=<I>n</I><SUP>3</SUP>(<I>n</I><SUP>3</SUP>-1)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15748 <p>An obvious solution is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15749 <MATH><I>x</I>-<I>n</I><SUP>3</SUP>=<I>n</I><SUP>3</SUP>-1, <I>y</I>-<I>n</I><SUP>3</SUP>=<I>n</I><SUP>3</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15750 which gives <MATH><I>z</I>=2<I>n</I><SUP>3</SUP>-1+2<I>n</I><SUP>3</SUP>-1=4<I>n</I><SUP>3</SUP>-2</MATH>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15751 <MATH><I>u</I>=<I>nz</I>=<I>n</I>(4<I>n</I><SUP>3</SUP>-2)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15752 <p>II. The second is a similar problem about two rectangles,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15753 equivalent to the solution of the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15754 <MATH><BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15755 (1) <I>x</I>+<I>y</I>=<I>u</I>+<I>v</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15756 (2) <I>xy</I>=<I>n.uv</I></BRACE></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15757 and the solution given in the text is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15758 <MATH><I>x</I>+<I>y</I>=<I>u</I>+<I>v</I>=<I>n</I><SUP>3</SUP>-1, (3)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15759 <MATH><BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15760 <I>u</I>=<I>n</I>-1, <I>v</I>=<I>n</I>(<I>n</I><SUP>2</SUP>-1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15761 <I>x</I>=<I>n</I><SUP>2</SUP>-1, <I>y</I>=<I>n</I><SUP>2</SUP>(<I>n</I>-1)</BRACE></MATH>. (4)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15762 <p>In this case trial may have been made of the assumptions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15763 <MATH><I>v</I>=<I>nx</I>, <I>y</I>=<I>n</I><SUP>2</SUP><I>u</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15764 <pb n=446><head>ALGEBRA: DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15765 when equation (1) would give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15766 <MATH>(<I>n</I>-1)<I>x</I>=(<I>n</I><SUP>2</SUP>-1)<I>u</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15767 a solution of which is <MATH><I>x</I>=<I>n</I><SUP>2</SUP>-1, <I>u</I>=<I>n</I>-1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15768 <p>III. The fifth problem is interesting in one respect. We are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15769 asked to find a right-angled triangle (in rational numbers)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15770 with area of 5 feet. We are told to multiply 5 by some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15771 square <I>containing 6 as a factor</I>, e.g. 36. This makes 180,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15772 and this is the area of the triangle (9, 40, 41). Dividing each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15773 side by 6, we have the triangle required. The author, then,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15774 is aware that the area of a right-angled triangle with sides in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15775 whole numbers is divisible by 6. If we take the Euclidean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15776 formula for a right-angled triangle, making the sides <I>a.mn</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15777 <MATH><I>a</I>.1/2(<I>m</I><SUP>2</SUP>-<I>n</I><SUP>2</SUP>), <I>a</I>.1/2(<I>m</I><SUP>2</SUP>+<I>n</I><SUP>2</SUP>)</MATH>, where <I>a</I> is any number, and <I>m, n</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15778 are numbers which are both odd or both even, the area is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15779 <MATH>1/4<I>mn</I>(<I>m</I>-<I>n</I>) (<I>m</I>+<I>n</I>)<I>a</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15780 and, as a matter of fact, the number <MATH><I>mn</I>(<I>m</I>-<I>n</I>) (<I>m</I>+<I>n</I>)</MATH> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15781 divisible by 24, as was proved later (for another purpose) by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15782 Leonardo of Pisa.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15783 <p>IV. The last four problems (10 to 13) are of great interest.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15784 They are different particular cases of one problem, that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15785 finding a rational right-angled triangle such that the numerical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15786 sum of its area and its perimeter is a given number. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15787 author's solution depends on the following formulae, where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15788 <I>a, b</I> are the perpendiculars, and <I>c</I> the hypotenuse, of a right-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15789 angled triangle, <I>S</I> its area, <I>r</I> the radius of the inscribed circle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15790 and <MATH><I>s</I>=1/2(<I>a</I>+<I>b</I>+<I>c</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15791 <MATH><I>S</I>=<I>rs</I>=1/2<I>ab</I>, <I>r</I>+<I>s</I>=<I>a</I>+<I>b</I>, <I>c</I>=<I>s</I>-<I>r</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15792 (The proof of these formulae by means of the usual figure,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15793 namely that used by Heron to prove the formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15794 <MATH><I>S</I>=√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)}</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15795 is easy.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15796 <p>Solving the first two equations, in order to find <I>a</I> and <I>b</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15797 we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15798 <MATH><BRACE><note>=1/2[<I>r</I>+<I>s</I>&mnplus;√{(<I>r</I>+<I>s</I>)<SUP>2</SUP>-8<I>rs</I>}],</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15799 <I>a</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15800 <I>b</I></BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15801 which formula is actually used by the author for finding <I>a</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15802 <pb n=447><head>HERONIAN INDETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15803 and <I>b</I>. The method employed is to take the sum of the area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15804 and the perimeter <I>S</I>+2<I>s</I>, separated into its two obvious
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15805 factors <MATH><I>s</I>(<I>r</I>+2)</MATH>, to put <MATH><I>s</I>(<I>r</I>+2)=<I>A</I></MATH> (the given number), and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15806 then to separate <I>A</I> into suitable factors to which <I>s</I> and <I>r</I>+2
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15807 may be equated. They must obviously be such that <I>sr</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15808 area, is divisible by 6. To take the first example where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15809 <I>A</I>=280: the possible factors are 2X140, 4X70, 5X56, 7X40,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15810 8X35, 10X28, 14X20. The suitable factors in this case are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15811 <MATH><I>r</I>+2=8, <I>s</I>=35</MATH>, because <I>r</I> is then equal to 6, and <I>rs</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15812 a multiple of 6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15813 <p>The author then says that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15814 <MATH><I>a</I>=1/2[6+35-√{(6+35)<SUP>2</SUP>-8.6.35}]=1/2(41-1)=20,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15815 <I>b</I>=1/2(41+1)=21,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15816 <I>c</I>=35-6=29</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15817 <p>The triangle is therefore (20, 21, 29) in this case. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15818 triangles found in the other three cases, by the same method,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15819 are (9, 40, 41), (8, 15, 17) and (9, 12, 15).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15820 <p>Unfortunately there is no guide to the date of the problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15821 just given. The probability is that the original formulation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15822 of the most important of the problems belongs to the period
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15823 between Euclid and Diophantus. This supposition best agrees
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15824 with the fact that the problems include nothing taken from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15825 the great collection in the <I>Arithmetica</I>. On the other hand,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15826 it is strange that none of the seven problems above mentioned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15827 is found in Diophantus. The five relating to rational right-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15828 angled triangles might well have been included by him; thus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15829 he finds rational right-angled triangles such that the area <I>plus</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15830 or <I>minus</I> one of the perpendiculars is a given number, but not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15831 the rational triangle which has a given area; and he finds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15832 rational right-angled triangles such that the area <I>plus</I> or <I>minus</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15833 the sum of <I>two</I> sides is a given number, but not the rational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15834 triangle such that the sum of the area and the <I>three</I> sides is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15835 a given number. The omitted problems might, it is true, have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15836 come in the lost Books; but, on the other hand, Book VI would
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15837 have been the appropriate place for them.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15838 <p>The crowning example of a difficult indeterminate problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15839 propounded before Diophantus's time is the Cattle-Problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15840 attributed to Archimedes, described above (pp. 97-8).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15841 <pb n=448><head>ALGEBRA: DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15842 <C>Numerical solution of quadratic equations.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15843 <p>The <I>geometrical</I> algebra of the Greeks has been in evidence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15844 all through our history from the Pythagoreans downwards,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15845 and no more need be said of it here except that its arithmetical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15846 application was no new thing in Diophantus. It is probable,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15847 for example, that the solution of the quadratic equation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15848 discovered first by geometry, was applied for the purpose of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15849 finding <I>numerical</I> values for the unknown as early as Euclid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15850 if not earlier still. In Heron the numerical solution of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15851 equations is well established, so that Diophantus was not the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15852 first to treat equations algebraically. What he did was to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15853 take a step forward towards an algebraic <I>notation</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15854 <p>The date of DIOPHANTUS can now be fixed with fair certainty.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15855 He was later than Hypsicles, from whom he quotes a definition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15856 of a polygonal number, and earlier than Theon of Alexandria,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15857 who has a quotation from Diophantus's definitions. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15858 possible limits of date are therefore, say, 150 B.C. to A.D. 350.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15859 But the letter of Psellus already mentioned says that Anatolius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15860 (Bishop of Laodicea about A.D. 280) dedicated to Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15861 a concise treatise on the Egyptian method of reckoning;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15862 hence Diophantus must have been a contemporary, so that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15863 probably flourished A.D. 250 or not much later.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15864 <p>An epigram in the Anthology gives some personal particulars:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15865 his boyhood lasted 1/6th of his life; his beard grew after 1/12th
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15866 more; he married after 1/7th more, and his son was born 5 years
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15867 later; the son lived to half his father's age, and the father
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15868 died 4 years after his son. Thus, if <I>x</I> was his age when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15869 he died,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15870 <MATH>1/6<I>x</I>+1/12<I>x</I>+1/7<I>x</I>+5+1/2<I>x</I>+4=<I>x</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15871 which gives <MATH><I>x</I>=84</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15872 <C>Works of Diophantus.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15873 <p>The works on which the fame of Diophantus rests are:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15874 <p>(1) the <I>Arithmetica</I> (originally in thirteen Books),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15875 <p>(2) a tract <I>On Polygonal Numbers</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15876 <pb n=449><head>WORKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15877 <p>Six Books only of the former and a fragment of the latter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15878 survive.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15879 <p>Allusions in the <I>Arithmetica</I> imply the existence of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15880 <p>(3) A collection of propositions under the title of <I>Porisms</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15881 in three propositions (3, 5, 16) of Book V, Diophantus quotes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15882 as known certain propositions in the Theory of Numbers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15883 prefixing to the statement of them the words ‘We have it in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15884 the <I>Porisms</I> that ...’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15885 <p>A scholium on a passage of Iamblichus, where Iamblichus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15886 cites a dictum of certain Pythagoreans about the unit being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15887 the dividing line (<G>meqo/rion</G>) between number and aliquot parts,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15888 says ‘thus Diophantus in the <I>Moriastica</I> .... for he describes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15889 as “parts” the progression without limit in the direction of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15890 less than the unit’. The <I>Moriastica</I> may be a separate work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15891 by Diophantus giving rules for reckoning with fractions; but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15892 I do not feel sure that the reference may not simply be to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15893 definitions at the beginning of the <I>Arithmetica</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15894 <C>The <I>Arithmetica</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15895 <C><I>The seven lost Books and their place</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15896 <p>None of the manuscripts which we possess contain more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15897 than six Books of the <I>Arithmetica</I>, the only variations being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15898 that some few divide the six Books into seven, while one or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15899 two give the fragment on Polygonal Numbers as VIII. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15900 missing Books were evidently lost at a very early date.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15901 Tannery suggests that Hypatia's commentary extended only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15902 to the first six Books, and that she left untouched the remain-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15903 ing seven, which, partly as a consequence, were first forgotten
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15904 and then lost (cf. the case of Apollonius's <I>Conics</I>, where the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15905 only Books which have survived in Greek, I-IV, are those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15906 on which Eutocius commented). There is no sign that even
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15907 the Arabians ever possessed the missing Books. The <I>Fakhr&imacr;</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15908 an algebraical treatise by Ab&umacr; Bekr Muh. b. al-Hasan al-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15909 Karkh&imacr; (d. about 1029), contains a collection of problems in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15910 determinate and indeterminate analysis which not only show
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15911 that their author had deeply studied Diophantus but in many
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15912 cases are taken direct from the <I>Arithmetica</I>, sometimes with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15913 a change in constants; in the fourth section of the work,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15914 <pb n=450><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15915 between problems corresponding to problems in Dioph. II
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15916 and III, are 25 problems not found in Diophantus, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15917 internal evidence, and especially the admission of irrational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15918 results (which are always avoided by Diophantus), exclude
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15919 the hypothesis that we have here one of the lost Books.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15920 Nor is there any sign that more of the work than we possess
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15921 was known to Ab&umacr;'l Waf&amacr; al-B&umacr;zj&amacr;n&imacr; (A.D. 940-98) who wrote
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15922 a ‘commentary on the algebra of Diophantus’, as well as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15923 a ‘Book of proofs of propositions used by Diophantus in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15924 work’. These facts again point to the conclusion that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15925 lost Books were lost before the tenth century.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15926 <p>The old view of the place originally occupied by the lost
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15927 seven Books is that of Nesselmann, who argued it with great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15928 ability.<note>Nesselmann, <I>Algebra der Griechen</I>, pp. 264-73.</note> According to him (1) much less of Diophantus is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15929 wanting than would naturally be supposed on the basis of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15930 the numerical proportion of 7 lost to 6 extant Books, (2) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15931 missing portion came, not at the end, but in the middle of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15932 the work, and indeed mostly between the first and second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15933 Books. Nesselmann's general argument is that, if we care-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15934 fully read the last four Books, from the third to the sixth,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15935 we shall find that Diophantus moves in a rigidly defined and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15936 limited circle of methods and artifices, and seems in fact to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15937 at the end of his resources. As regards the possible contents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15938 of the lost portion on this hypothesis, Nesselmann can only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15939 point to (1) topics which we should expect to find treated,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15940 either because foreshadowed by the author himself or as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15941 necessary for the elucidation or completion of the whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15942 subject, (2) the <I>Porisms</I>; under head (1) come, (<I>a</I>) deter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15943 minate equations of the second degree, and (<I>b</I>) indeterminate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15944 equations of the first degree. Diophantus does indeed promise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15945 to show how to solve the general quadratic <MATH><I>ax</I><SUP>2</SUP>±<I>bx</I>±<I>c</I>=0</MATH> so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15946 far as it has rational and positive solutions; the suitable place
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15947 for this would have been between Books I and II. But there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15948 is nothing whatever to show that indeterminate equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15949 of the first degree formed part of the writer's plan. Hence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15950 Nesselmann is far from accounting for the contents of seven
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15951 whole Books; and he is forced to the conjecture that the six
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15952 Books may originally have been divided into even more than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15953 seven Books; there is, however, no evidence to support this.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15954 <pb n=451><head>RELATION OF WORKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15955 <C><I>Relation of the ‘Porisms’ to the Arithmetica</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15956 <p>Did the <I>Porisms</I> form part of the <I>Arithmetica</I> in its original
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15957 form? The phrase in which they are alluded to, and which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15958 occurs three times, ‘We have it in the <I>Porisms</I> that ...’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15959 suggests that they were a distinct collection of propositions concerning
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15960 the properties of certain numbers, their divisibility into a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15961 certain number of squares, and so on; and it is possible that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15962 it was from the same collection that Diophantus took the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15963 numerous other propositions which he assumes, explicitly or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15964 implicitly. If the collection was part of the <I>Arithmetica</I>, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15965 would be strange to quote the propositions under a separate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15966 title ‘The Porisms’ when it would have been more natural
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15967 to refer to particular propositions of particular Books, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15968 more natural still to say <G>tou=to ga\r prode/deiktai</G>, or some such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15969 phrase, ‘for this has been proved’, without any reference to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15970 the particular place where the proof occurred. The expression
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15971 ‘We have it in the <I>Porisms</I>’ (in the plural) would be still
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15972 more inappropriate if the <I>Porisms</I> had been, as Tannery
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15973 supposed, not collected together as one or more Books of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15974 <I>Arithmetica</I>, but scattered about in the work as <I>corollaries</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15975 particular propositions. Hence I agree with the view of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15976 Hultsch that the <I>Porisms</I> were not included in the <I>Arith-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15977 metica</I> at all, but formed a separate work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15978 <p>If this is right, we cannot any longer hold to the view of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15979 Nesselmann that the lost Books were in the middle and not at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15980 the end of the treatise; indeed Tannery produces strong
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15981 arguments in favour of the contrary view, that it is the last
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15982 and most difficult Books which are lost. He replies first to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15983 the assumption that Diophantus could not have proceeded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15984 to problems more difficult than those of Book V. ‘If the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15985 fifth or the sixth Book of the <I>Arithmetica</I> had been lost, who,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15986 pray, among us would have believed that such problems had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15987 ever been attempted by the Greeks? It would be the greatest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15988 error, in any case in which a thing cannot clearly be proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15989 to have been unknown to all the ancients, to maintain that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15990 it could not have been known to some Greek mathematician.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15991 If we do not know to what lengths Archimedes brought the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15992 theory of numbers (to say nothing of other things), let us
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15993 admit our ignorance. But, between the famous problem of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15994 <pb n=452><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15995 cattle and the most difficult of Diophantus's problems, is there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15996 not a sufficient gap to require seven Books to fill it? And,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15997 without attributing to the ancients what modern mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15998 maticians have discovered, may not a number of the things
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	15999 attributed to the Indians and Arabs have been drawn from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16000 Greek sources? May not the same be said of a problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16001 solved by Leonardo of Pisa, which is very similar to those of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16002 Diophantus but is not now to be found in the <I>Arithmetica</I>?
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16003 In fact, it may fairly be said that, when Chasles made his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16004 reasonably probable restitution of the <I>Porisms</I> of Euclid, he,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16005 notwithstanding that he had Pappus's lemmas to help him,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16006 undertook a more difficult task than he would have undertaken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16007 if he had attempted to fill up seven Diophantine Books with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16008 numerical problems which the Greeks may reasonably be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16009 supposed to have solved.’<note>Diophantus, ed. Tannery, vol. ii, p. xx.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16010 <p>It is not so easy to agree with Tannery's view of the relation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16011 of the treatise <I>On Polygonal Numbers</I> to the <I>Arithmetica</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16012 According to him, just as Serenus's treatise on the sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16013 of cones and cylinders was added to the mutilated <I>Conics</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16014 Apollonius consisting of four Books only, in order to make up
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16015 a convenient volume, so the tract on Polygonal Numbers was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16016 added to the remains of the <I>Arithmetica</I>, though forming no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16017 part of the larger work.<note><I>Ib</I>., p. xviii.</note> Thus Tannery would seem to deny
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16018 the genuineness of the whole tract on Polygonal Numbers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16019 though in his text he only signalizes the portion beginning
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16020 with the enunciation of the problem ‘Given a number, to find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16021 in how many ways it can be a polygonal number’ as ‘a vain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16022 attempt by a commentator’ to solve this problem. Hultsch,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16023 on the other hand, thinks that we may conclude that Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16024 phantus really solved the problem. The tract begins, like
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16025 Book I of the <I>Arithmetica</I>, with definitions and preliminary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16026 propositions; then comes the difficult problem quoted, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16027 discussion of which breaks off in our text after a few pages,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16028 and to these it would be easy to tack on a great variety of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16029 other problems.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16030 <p>The name of Diophantus was used, as were the names of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16031 Euclid, Archimedes and Heron in their turn, for the pur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16032 pose of palming off the compilations of much later authors.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16033 <pb n=453><head>RELATION OF WORKS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16034 Tannery includes in his edition three fragments under the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16035 heading ‘Diophantus Pseudepigraphus’. The first, which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16036 not ‘from the Arithmetic of Diophantus’ as its heading states,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16037 is worth notice as containing some particulars of one of ‘two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16038 methods of finding the square root of any square number’;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16039 we are told to begin by writing the number ‘according to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16040 the arrangement of the Indian method’, i.e. in the Indian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16041 numerical notation which reached us through the Arabs. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16042 second fragment is the work edited by C. Henry in 1879 as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16043 <I>Opusculum de multiplicatione et divisione sexagesimalibus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16044 Diophanto vel Pappo attribuendum</I>. The third, beginning
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16045 with <G>*diofa/ntou e)pipedometrika/</G> is a Byzantine compilation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16046 from later reproductions of the <G>gewmetrou/mena</G> and <G>stereo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16047 metrou/mena</G> of Heron. Not one of the three fragments has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16048 anything to do with Diophantus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16049 <C><I>Commentators from Hypatia downwards</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16050 <p>The first commentator on Diophantus of whom we hear
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16051 is Hypatia, the daughter of Theon of Alexandria; she
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16052 was murdered by Christian fanatics in A.D. 415. I have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16053 already mentioned the attractive hypothesis of Tannery that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16054 Hypatia's commentary extended only to our six Books, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16055 that this accounts for their survival when the rest were lost.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16056 It is possible that the remarks of Psellus (eleventh century) at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16057 the beginning of his letter about Diophantus, Anatolius and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16058 the Egyptian method of arithmetical reckoning were taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16059 from Hypatia's commentary.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16060 <p>Georgius Pachymeres (1240 to about 1310) wrote in Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16061 a paraphrase of at least a portion of Diophantus. Sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16062 25-44 of this commentary relating to Book I, Def. 1 to Prop.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16063 11, survive. Maximus Planudes (about 1260-1310) also wrote
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16064 a systematic commentary on Books I, II. Arabian commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16065 tators were Ab&umacr;'l Waf&amacr; al-B&umacr;zj&amacr;n&imacr; (940-98), Qust&amacr; b. L&umacr;q&amacr;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16066 al-Ba'labakk&imacr; (d. about 912) and probably Ibn al-Haitham
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16067 (about 965-1039).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16068 <C><I>Translations and editions</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16069 <p>To Regiomontanus belongs the credit of being the first to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16070 call attention to the work of Diophantus as being extant in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16071 <pb n=454><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16072 Greek. In an <I>Oratio</I> delivered at the end of 1463 as an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16073 introduction to a course of lectures on astronomy which he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16074 gave at Padua in 1463-4 he observed: ‘No one has yet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16075 translated from the Greek into Latin the fine thirteen Books
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16076 of Diophantus, in which the very flower of the whole of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16077 arithmetic lies hid, the <I>ars rei et census</I> which to-day they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16078 call by the Arabic name of Algebra.’ Again, in a letter dated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16079 February 5, 1464, to Bianchini, he writes that he has found at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16080 Venice ‘Diofantus, a Greek arithmetician who has not yet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16081 been translated into Latin’. Rafael Bombelli was the first to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16082 find a manuscript in the Vatican and to conceive the idea of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16083 publishing the work; this was towards 1570, and, with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16084 Antonio Maria Pazzi, he translated five Books out of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16085 seven into which the manuscript was divided. The translation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16086 was not published, but Bombelli took all the problems of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16087 first four Books and some of those of the fifth and embodied
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16088 them in his <I>Algebra</I> (1572), interspersing them with his own
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16089 problems.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16090 <p>The next writer on Diophantus was Wilhelm Holzmann,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16091 who called himself Xylander, and who with extraordinary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16092 industry and care produced a very meritorious Latin trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16093 lation with commentary (1575). Xylander was an enthusiast
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16094 for Diophantus, and his preface and notes are often delightful
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16095 reading. Unfortunately the book is now very rare. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16096 standard edition of Diophantus till recent years was that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16097 Bachet, who in 1621 published for the first time the Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16098 text with Latin translation and notes. A second edition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16099 (1670) was carelessly printed and is untrustworthy as regards
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16100 the text; on the other hand it contained the epoch-making
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16101 notes of Fermat; the editor was S. Fermat, his son. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16102 great blot on the work of Bachet is his attitude to Xylander,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16103 to whose translation he owed more than he was willing to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16104 avow. Unfortunately neither Bachet nor Xylander was able
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16105 to use the best manuscripts; that used by Bachet was Parisinus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16106 2379 (of the middle of the sixteenth century), with the help
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16107 of a transcription of part of a Vatican MS. (Vat. gr. 304 of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16108 the sixteenth century), while Xylander's manuscript was the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16109 Wolfenbüttel MS. Guelferbytanus Gudianus 1 (fifteenth cen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16110 tury). The best and most ancient manuscript is that of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16111 Madrid (Matritensis 48 of the thirteenth century) which was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16112 <pb n=455><head>TRANSLATIONS AND EDITIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16113 unfortunately spoiled by corrections made, especially in Books
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16114 I, II, from some manuscript of the ‘Planudean’ class; where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16115 this is the case recourse must be had to Vat. gr. 191 which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16116 was copied from it before it had suffered the general alteration
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16117 referred to: these are the first two of the manuscripts used by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16118 Tannery in his definitive edition of the Greek text (Teubner,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16119 1893, 1895).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16120 <p>Other editors can only be shortly enumerated. In 1585
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16121 Simon Stevin published a French version of the first four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16122 Books, based on Xylander. Albert Girard added the fifth and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16123 sixth Books, the complete edition appearing in 1625. German
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16124 translations were brought out by Otto Schulz in 1822 and by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16125 G. Wertheim in 1890. Poselger translated the fragment on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16126 Polygonal Numbers in 1810. All these translations depended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16127 on the text of Bachet.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16128 <p>A reproduction of Diophantus in modern notation with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16129 introduction and notes by the present writer (second edition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16130 1910) is based on the text of Tannery and may claim to be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16131 most complete and up-to-date edition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16132 <p>My account of the <I>Arithmetica</I> of Diophantus will be most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16133 conveniently arranged under three main headings (1) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16134 notation and definitions, (2) the principal methods employed,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16135 so far as they can be generally stated, (3) the nature of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16136 contents, including the assumed Porisms, with indications of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16137 the devices by which the problems are solved.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16138 <C><B>Notation and definitions</B>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16139 <p>In his work <I>Die Algebra der Griechen</I> Nesselmann distin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16140 guishes three stages in the evolution of algebra. (1) The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16141 first stage he calls ‘Rhetorical Algebra’ or reckoning by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16142 means of complete words. The characteristic of this stage
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16143 is the absolute want of all symbols, the whole of the calcula-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16144 tion being carried on by means of complete words and forming
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16145 in fact continuous prose. This first stage is represented by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16146 such writers as Iamblichus, all Arabian and Persian algebraists,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16147 and the oldest Italian algebraists and their followers, including
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16148 Regiomontanus. (2) The second stage Nesselmann calls the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16149 ‘Syncopated Algebra’, essentially like the first as regards
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16150 <pb n=456><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16151 literary style, but marked by the use of certain abbreviational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16152 symbols for constantly recurring quantities and operations.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16153 To this stage belong Diophantus and, after him, all the later
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16154 Europeans until about the middle of the seventeenth century
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16155 (with the exception of Vieta, who was the first to establish,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16156 under the name of <I>Logistica speciosa</I>, as distinct from <I>Logistica
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16157 numerosa</I>, a regular system of reckoning with letters denoting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16158 magnitudes as well as numbers). (3) To the third stage
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16159 Nesselmann gives the name of ‘Symbolic Algebra’, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16160 uses a complete system of notation by signs having no visible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16161 connexion with the words or things which they represent,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16162 a complete language of symbols, which entirely supplants the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16163 ‘rhetorical’ system, it being possible to work out a solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16164 without using a single word of ordinary language with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16165 exception of a connecting word or two here and there used for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16166 clearness' sake.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16167 <C><I>Sign for the unknown</I> (=<I>x</I>), <I>and its origin</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16168 <p>Diophantus's system of notation then is merely abbrevia-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16169 tional. We will consider first the representation of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16170 unknown quantity (our <I>x</I>). Diophantus defines the unknown
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16171 quantity as ‘<I>containing an indeterminate or undefined multi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16172 tude of units</I>’ (<G>plh=qos mona/dwn a)o/riston</G>), adding that it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16173 called <G>a)riqmo/s</G>, i.e. <I>number</I> simply, and is denoted by a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16174 sign. This sign is then used all through the book. In the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16175 earliest (the Madrid) MS. the sign takes the form <FIG>, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16176 Marcianus 308 it appears as <B>S</B>. In the printed editions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16177 Diophantus before Tannery's it was represented by the final
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16178 sigma with an accent, <G>s</G>′, which is sufficiently like the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16179 of the two forms. Where the symbol takes the place of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16180 inflected forms <G>a)riqmo/n</G>, <G>a)riqmou=</G>, &c., the termination was put
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16181 above and to the right of the sign like an exponent, e.g. <G>s</G>″ for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16182 <G>a)riqmo/n</G> as <G>t</G>″ for <G>to\n</G>, <G>s<SUP>ou=</SUP></G> for <G>a)riqmou=</G>; the symbol was, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16183 addition, doubled in the plural cases, thus <G>ss<SUP>oi/</SUP>, ss<SUP>ou/s</SUP></G>, &c. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16184 coefficient is expressed by putting the required Greek numeral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16185 immediately after it; thus <G>s<SUP>oi\</SUP> ia</G>=<B>11</B> <G>a)riqmoi/</G>, equivalent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16186 to 11<I>x</I>, <G>s</G>′<G>a</G>=<I>x</I>, and so on. Tannery gives reasons for think-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16187 ing that in the archetype the case-endings did not appear, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16188 <pb n=457><head>NOTATION AND DEFINITIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16189 that the sign was not duplicated for the plural, although such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16190 duplication was the practice of the Byzantines. That the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16191 sign was merely an abbreviation for the word <G>a)riqmo/s</G> and no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16192 algebraical symbol is shown by the fact that it occurs in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16193 manuscripts for <G>a)riqmo/s</G> in the ordinary sense as well as for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16194 <G>a)riqmo/s</G> in the technical sense of the unknown quantity. Nor
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16195 is it confined to Diophantus. It appears in more or less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16196 similar forms in the manuscripts of other Greek mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16197 maticians, e.g. in the Bodleian MS. of Euclid (D'Orville 301)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16198 of the ninth century (in the forms <FIG>, or as a curved line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16199 similar to the abbreviation for <G>kai/</G>), in the manuscripts of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16200 the <I>Sand-reckoner</I> of Archimedes (in a form approximat-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16201 ing to <G>s</G>), where again there is confusion caused by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16202 similarity of the signs for <G>a)riqmo/s</G> and <G>kai/</G>, in a manuscript
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16203 of the <I>Geodaesia</I> included in the Heronian collections edited
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16204 by Hultsch (where it appears in various forms resembling
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16205 sometimes <G>z</G>, sometimes <G>r</G>, sometimes <G>o</G>, and once <G>x</G>, with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16206 case-endings superposed) and in a manuscript of Theon of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16207 Smyrna.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16208 <p>What is the origin of the sign? It is certainly not the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16209 final sigma, as is proved by several of the forms which it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16210 takes. I found that in the Bodleian manuscript of Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16211 it is written in the form <FIG>, larger than and quite unlike the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16212 final sigma. This form, combined with the fact that in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16213 place Xylander's manuscript read <G>ar</G> for the full word, suggested
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16214 to me that the sign might be a simple contraction of the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16215 two letters of <G>a)riqmo/s</G>. This seemed to be confirmed by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16216 Gardthausen's mention of a contraction for <G>ar</G>, in the form <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16217 occurring in a papyrus of A.D. 154, since the transition to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16218 form found in the manuscripts of Diophantus might easily
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16219 have been made through an intermediate form <FIG>. The loss of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16220 the downward stroke, or of the loop, would give a close
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16221 approximation to the forms which we know. This hypothesis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16222 as to the origin of the sign has not, so far as I know, been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16223 improved upon. It has the immense advantage that it makes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16224 the sign for <G>a)riqmo/s</G> similar to the signs for the powers of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16225 the unknown, e.g. <G>d<SUP>g</SUP></G> for <G>du/namis</G>, <G>k<SUP>g</SUP></G> for <G>ku/bos</G>, and to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16226 sign <FIG> for the unit, the sole difference being that the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16227 letters coalesce into one instead of being separate.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16228 <pb n=458><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16229 <C><I>Signs for the powers of the unknown and their reciprocals</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16230 <p>The powers of the unknown, corresponding to our <MATH><I>x</I><SUP>2</SUP>, <I>x</I><SUP>3</SUP> ... <I>x</I><SUP>6</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16231 are defined and denoted as follows:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16232 <C><I>x</I><SUP>2</SUP> is <G>du/namis</G> and is denoted by <G>*d<SUP>g</SUP></G></C>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16233 <C><I>x</I><SUP>3</SUP> ” <G>ku/bos</G> ” ” ” <G>*k<SUP>g</SUP></G></C>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16234 <C><I>x</I><SUP>4</SUP> ” <G>dunamodu/namis</G> ” ” <G>d<SUP>g</SUP>d</G></C>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16235 <C><I>x</I><SUP>5</SUP> ” <G>dunamo/kubos</G> ” ” <G>dk<SUP>g</SUP></G></C>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16236 <C><I>x</I><SUP>6</SUP> ” <G>kubo/kubos</G> ” ” ” <G>k<SUP>g</SUP>k</G></C>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16237 Beyond the sixth power Diophantus does not go. It should
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16238 be noted that, while the terms from <G>ku/bos</G> onwards may be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16239 used for the powers of any ordinary known number as well as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16240 for the powers of the unknown, <G>du/namis</G> is restricted to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16241 square of the unknown; wherever a particular square number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16242 is spoken of, the term is <G>tetra/gwnos a)riqmo/s</G>. The term
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16243 <G>dunamodu/namis</G> occurs once in another author, namely in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16244 <I>Metrica</I> of Heron,<note>Heron, <I>Metrica</I>, p. 48. 11, 19, Schöne.</note> where it is used for the fourth power of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16245 the side of a triangle.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16246 <p>Diophantus has also terms and signs for the reciprocals of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16247 the various powers of the unknown, i.e. for 1/<I>x</I>, 1/<I>x</I><SUP>2</SUP> ....
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16248 As an aliquot part was ordinarily denoted by the corresponding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16249 numeral sign with an accent, e.g. <MATH><G>g</G>′=1/3, <G>ia</G>′=1/11</MATH>, Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16250 has a mark appended to the symbols for <I>x</I>, <I>x</I><SUP>2</SUP> ... to denote the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16251 reciprocals; this, which is used for aliquot parts as well, is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16252 printed by Tannery thus, χ. With Diophantus then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16253 <C><G>a)riqmosto/n</G>, denoted by <G>s</G><SUP>χ</SUP>, is equivalent to 1/<I>x</I>,</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16254 <C><G>dunamosto/n</G>, ” <G>*d</G><SUP><G>g</G>χ</SUP> ” ” 1/<I>x</I><SUP>2</SUP>,</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16255 and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16256 <p>The coefficient of the term in <I>x</I>, <I>x</I><SUP>2</SUP> ... or 1/<I>x</I>, 1/<I>x</I><SUP>2</SUP> ... is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16257 expressed by the ordinary numeral immediately following,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16258 e.g. <MATH><G>dk<SUP>g</SUP> ks</G>=26<I>x</I><SUP>5</SUP>, <G>*d</G><SUP><G>g</G>χ</SUP> <G>sn</G>=250/<I>x</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16259 <p>Diophantus does not need any signs for the operations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16260 multiplication and division. Addition is indicated by mere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16261 juxtaposition; thus <G>k<SUP>g</SUP> a d<SUP></SUP> igse</G> corresponds to <MATH><I>x</I><SUP>3</SUP>+13<I>x</I><SUP>2</SUP>+5<I>x</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16262 <pb n=459><head>NOTATION AND DEFINITIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16263 When there are units in addition, the units are indicated by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16264 the abbreviation <FIG>; thus <G>k<SUP>g</SUP> a d<SUP>g</SUP> ig s e</G> <FIG> <G>b</G> corresponds to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16265 <MATH><I>x</I><SUP>3</SUP>+13<I>x</I><SUP>2</SUP>+5<I>x</I>+2</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16266 <C><I>The sign (<FIG>) for minus and its meaning</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16267 <p>For subtraction alone is a sign used. The full term for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16268 <I>wanting</I> is <G>lei=yis</G>, as opposed to <G>u(/parxis</G>, a <I>forthcoming</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16269 which denotes a <I>positive</I> term. The symbol used to indicate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16270 a <I>wanting</I>, corresponding to our sign for <I>minus</I>, is <FIG>, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16271 is described in the text as a ‘<G>y</G> turned downwards and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16272 truncated’ (<G>*y e)llipe\s ka/tw neu=on</G>). The description is evidently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16273 interpolated, and it is now certain that the sign has nothing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16274 to do with <G>y</G>. Nor is it confined to Diophantus, for it appears
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16275 in practically the same form in Heron's <I>Metrica</I>,<note>Heron, <I>Metrica</I>, p. 156. 8, 10.</note> where in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16276 place the reading of the manuscript is <G>mona/dwn od *t i′d′</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16277 74-1/14. In the manuscripts of Diophantus, when the sign
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16278 is resolved by writing the full word instead of it, it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16279 generally resolved into <G>lei/yei</G>, the dative of <G>lei=yis</G>, but in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16280 other places the symbol is used instead of parts of the verb
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16281 <G>lei/pein</G>, namely <G>lipw/n</G> or <G>lei/yas</G> and once even <G>li/pwsi</G>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16282 sometimes <G>lei/yei</G> in the manuscripts is followed by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16283 <I>accusative</I>, which shows that in these cases the sign was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16284 wrongly resolved. It is therefore a question whether Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16285 phantus himself ever used the dative <G>lei/yei</G> for <I>minus</I> at all.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16286 The use is certainly foreign to classical Greek. Ptolemy has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16287 in two places <G>lei=yan</G> and <G>lei/pousan</G> respectively followed,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16288 properly, by the accusative, and in one case he has <G>to\ a)po\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16289 th=s gl leifqe\n u(po\ tou= a)po\ th=s zg</G> (where the meaning is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16290 <G>zg</G><SUP>2</SUP>-<G>gl</G><SUP>2</SUP>). Hence Heron would probably have written a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16291 participle where the <G>*t</G> occurs in the expression quoted above,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16292 say <G>mona/dwn od leiyasw=n tessarakaide/katon</G>. On the whole,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16293 therefore, it is probable that in Diophantus, and wherever else
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16294 it occurred, <FIG> is a compendium for the root of the verb <G>lei/pein</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16295 in fact a <G>l</G> with <G>i</G> placed in the middle (cf. <FIG>, an abbreviation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16296 for <G>ta/lanton</G>). This is the hypothesis which I put forward
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16297 in 1885, and it seems to be confirmed by the fresh evidence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16298 now available as shown above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16299 <pb n=460><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16300 <p>Attached to the definition of <I>minus</I> is the statement that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16301 ‘a <I>wanting</I> (i.e. a <I>minus</I>) multiplied by a <I>wanting</I> makes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16302 a <I>forthcoming</I> (i.e. a <I>plus</I>); and a <I>wanting</I> (a <I>minus</I>) multi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16303 plied by a <I>forthcoming</I> (a <I>plus</I>) makes a <I>wanting</I> (a <I>minus</I>)’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16304 <p>Since Diophantus uses no sign for <I>plus</I>, he has to put all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16305 the positive terms in an expression together and write all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16306 negative terms together after the sign for <I>minus</I>; e.g. for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16307 <MATH><I>x</I><SUP>3</SUP>-5<I>x</I><SUP>2</SUP>+8<I>x</I>-1</MATH> he necessarily writes <G>k<SUP>g</SUP a s h <> *d<SUP>g</SUP> e <FIG> a</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16308 <p>The Diophantine notation for fractions as well as for large
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16309 numbers has been fully explained with many illustrations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16310 in Chapter II above. It is only necessary to add here that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16311 when the numerator and denominator consist of composite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16312 expressions in terms of the unknown and its powers, he puts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16313 the numerator first followed by <G>e)n mori/w\|</G> or <G>mori/ou</G> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16314 denominator.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16315 Thus <MATH><G>d<SUP>g</SUP> x <FIG><>bfk e)n mori/w\| d<SUP>g</SUP>d a <FIG> <> <> *d<SUP>g</SUP> x</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16316 =(60<I>x</I><SUP>2</SUP>+2520)/(<I>x</I><SUP>4</SUP>+900-60<I>x</I><SUP>2</SUP>)</MATH>, [VI. 12]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16317 and <MATH><G>d<SUP>g</SUP> ie <> <FIG> ls e)n mori/w\| d<SUP>g</SUP> d a <FIG> ls <> d<SUP>g</SUP> ib</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16318 =(15<I>x</I><SUP>2</SUP>-36)/(<I>x</I><SUP>4</SUP>+36-12<I>x</I><SUP>2</SUP>)</MATH> [VI. 14].
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16319 <p>For a <I>term</I> in an algebraical expression, i.e. a power of <I>x</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16320 with a certain coefficient, and the term containing a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16321 number of units, Diophantus uses the word <G>ei)=dos</G>, ‘species’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16322 which primarily means the particular power of the variable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16323 without the coefficient. At the end of the definitions he gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16324 directions for simplifying equations until each side contains
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16325 positive terms only, by the addition or subtraction of coeffi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16326 cients, and by getting rid of the negative terms (which is done
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16327 by adding the necessary quantities to both sides); the object,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16328 he says, is to reduce the equation until one term only is left
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16329 on each side; ‘but’, he adds, ‘I will show you later how, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16330 the case also where two terms are left equal to one term,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16331 such a problem is solved’. We find in fact that, when he has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16332 to solve a quadratic equation, he endeavours by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16333 suitable assumptions to reduce it either to a simple equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16334 or a <I>pure</I> quadratic. The solution of the mixed quadratic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16335 <pb n=461><head>NOTATION AND DEFINITIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16336 in three terms is clearly assumed in several places of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16337 <I>Arithmetica</I>, but Diophantus never gives the necessary ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16338 planation of this case as promised in the preface.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16339 <p>Before leaving the notation of Diophantus, we may observe
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16340 that the form of it limits him to the use of one unknown at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16341 a time. The disadvantage is obvious. For example, where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16342 we can begin with any number of unknown quantities and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16343 gradually eliminate all but one, Diophantus has practically to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16344 perform his eliminations beforehand so as to express every
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16345 quantity occurring in the problem in terms of only one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16346 unknown. When he handles problems which are by nature
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16347 indeterminate and would lead in our notation to an inde-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16348 terminate equation containing two or three unknowns, he has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16349 to assume for one or other of these some particular number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16350 arbitrarily chosen, the effect being to make the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16351 determinate. However, in doing so, Diophantus is careful
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16352 to say that we may for such and such a quantity put any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16353 number whatever, say such and such a number; there is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16354 therefore (as a rule) no real loss of generality. The particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16355 devices by which he contrives to express all his unknowns
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16356 in terms of one unknown are extraordinarily various and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16357 clever. He can, of course, use the same variable <G>s</G> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16358 same problem with different significations <I>successively</I>, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16359 when it is necessary in the course of the problem to solve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16360 a subsidiary problem in order to enable him to make the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16361 coefficients of the different terms of expressions in <I>x</I> such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16362 as will answer his purpose and enable the original problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16363 to be solved. There are, however, two cases, II. 28, 29, where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16364 for the proper working-out of the problem two unknowns are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16365 imperatively necessary. We should of course use <I>x</I> and <I>y</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16366 Diophantus calls the first <G>s</G> as usual; the second, for want
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16367 of a term, he agrees to call in the first instance ‘<I>one unit</I>’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16368 i.e. 1. Then later, having completed the part of the solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16369 necessary to find <I>x</I>, he substitutes its value and uses <G>s</G> over
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16370 again for what he had originally called 1. That is, he has to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16371 put his finger on the place to which the 1 has passed, so as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16372 to substitute <G>s</G> for it. This is a <I>tour de force</I> in the particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16373 cases, and would be difficult or impossible in more complicated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16374 problems.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16375 <pb n=462><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16376 <C>The methods of Diophantus.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16377 <p>It should be premised that Diophantus will have in his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16378 solutions no numbers whatever except ‘rational’ numbers;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16379 he admits fractional solutions as well as integral, but he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16380 excludes not only surds and imaginary quantities but also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16381 negative quantities. Of a negative quantity <I>per se</I>, i.e. with-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16382 out some greater positive quantity to subtract it from, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16383 had apparently no conception. Such equations then as lead
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16384 to imaginary or negative roots he regards as useless for his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16385 purpose; the solution is in these cases <G>a)du/natos</G>, impossible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16386 So we find him (V. 2) describing the equation <MATH>4=4<I>x</I>+20</MATH> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16387 <G>a)/topos</G>, absurd, because it would give <MATH><I>x</I>=-4</MATH>. He does, it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16388 true, make occasional use of a quadratic which would give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16389 a root which is positive but a surd, but only for the purpose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16390 of obtaining limits to the root which are integers or numerical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16391 fractions; he never uses or tries to express the actual root of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16392 such an equation. When therefore he arrives in the course
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16393 of solution at an equation which would give an ‘irrational’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16394 result, he retraces his steps, finds out how his equation has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16395 arisen, and how he may, by altering the previous work,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16396 substitute for it another which shall give a rational result.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16397 This gives rise in general to a subsidiary problem the solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16398 of which ensures a rational result for the problem itself.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16399 <p>It is difficult to give a complete account of Diophantus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16400 methods without setting out the whole book, so great is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16401 variety of devices and artifices employed in the different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16402 problems. There are, however, a few general methods which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16403 do admit of differentiation and description, and these we pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16404 ceed to set out under subjects.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16405 <C>I. Diophantus's treatment of equations.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16406 <C>(A) <I>Determinate equations</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16407 <p>Diophantus solved without difficulty determinate equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16408 of the first and second degrees; of a cubic we find only one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16409 example in the <I>Arithmetica</I>, and that is a very special case.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16410 <p>(1) <I>Pure determinate equations</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16411 <p>Diophantus gives a general rule for this case without regard
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16412 to degree. We have to take like from like on both sides of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16413 <pb n=463><head>DETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16414 equation and neutralize negative terms by adding to both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16415 sides, then take like from like again, until we have one term
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16416 left equal to one term. After these operations have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16417 performed, the equation (after dividing out, if both sides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16418 contain a power of <I>x</I>, by the lesser power) reduces to <MATH><I>Ax<SUP>m</SUP></I>=<I>B</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16419 and is considered solved. Diophantus regards this as giving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16420 one root only, excluding any negative value as ‘impossible’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16421 No equation of the kind is admitted which does not give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16422 a ‘rational’ value, integral or fractional. The value <I>x</I>=0 is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16423 ignored in the case where the degree of the equation is reduced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16424 by dividing out by any power of <I>x</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16425 <p>(2) <I>Mixed quadratic equations</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16426 <p>Diophantus never gives the explanation of the method of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16427 solution which he promises in the preface. That he had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16428 a definite method like that used in the Geometry of Heron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16429 is proved by clear verbal explanations in different propositions.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16430 As he requires the equation to be in the form of two positive
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16431 terms being equal to one positive term, the possible forms for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16432 Diophantus are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16433 <MATH>(<I>a</I>) <I>mx</I><SUP>2</SUP>+<I>px</I>=<I>q</I>, (<I>b</I>) <I>mx</I><SUP>2</SUP>=<I>px</I>+<I>q</I>, (<I>c</I>) <I>mx</I><SUP>2</SUP>+<I>q</I>=<I>px</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16434 It does not appear that Diophantus divided by <I>m</I> in order to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16435 make the first term a square; rather he multiplied by <I>m</I> for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16436 this purpose. It is clear that he stated the roots in the above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16437 cases in a form equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16438 <MATH>(<I>a</I>) (-1/2<I>p</I>+√(1/4<I>p</I><SUP>2</SUP>+<I>mq</I>))/<I>m</I>, (<I>b</I>) (1/2<I>p</I>+√(1/4<I>p</I><SUP>2</SUP>+<I>mq</I>))/<I>m</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16439 (<I>c</I>) (1/2<I>p</I>+√(1/4<I>p</I><SUP>2</SUP>-<I>mq</I>))/<I>m</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16440 The explanations which show this are to be found in VI. 6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16441 in IV. 39 and 31, and in V. 10 and VI. 22 respectively. For
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16442 example in V. 10 he has the equation <MATH>17<I>x</I><SUP>2</SUP>+17<72<I>x</I></MATH>, and he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16443 says ‘Multiply half the coefficient of <I>x</I> into itself and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16444 1296; subtract the product of the coefficient of <I>x</I><SUP>2</SUP> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16445 term in units, or 289. The remainder is 1007, the square root
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16446 of which is not greater than 31. Add half the coefficient of <I>x</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16447 and the result is not greater than 67. Divide by the coefficient
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16448 of <I>x</I><SUP>2</SUP>, and <I>x</I> is not greater than (67)/(17).’ In IV. 39 he has the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16449 <pb n=464><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16450 equation <MATH>2<I>x</I><SUP>2</SUP>>6<I>x</I>+18</MATH> and says, ‘To solve this, take the square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16451 of half the coefficient of <I>x</I>, i.e. 9, and the product of the unit-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16452 term and the coefficient of <I>x</I><SUP>2</SUP>, i.e. 36. Adding, we have 45,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16453 the square root of which is not less than 7. Add half the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16454 coefficient of <I>x</I> [and divide by the coefficient of <I>x</I><SUP>2</SUP>]; whence <I>x</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16455 is not less than 5.’ In these cases it will be observed that 31
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16456 and 7 are not accurate limits, but are the nearest integral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16457 limits which will serve his purpose.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16458 <p>Diophantus always uses the positive sign with the radical,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16459 and there has been much discussion as to whether he knew
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16460 that a quadratic equation has <I>two</I> roots. The evidence of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16461 text is inconclusive because his only object, in every case, is to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16462 get one solution; in some cases the other root would be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16463 negative, and would therefore naturally be ignored as ‘absurd’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16464 or ‘impossible’. In yet other cases where the second root is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16465 possible it can be shown to be useless from Diophantus's point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16466 of view. For my part, I find it difficult or impossible to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16467 believe that Diophantus was unaware of the existence of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16468 real roots in such cases. It is so obvious from the geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16469 form of solution based on Eucl. II. 5, 6 and that contained in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16470 Eucl. VI. 27-9; the construction of VI. 28, too, corresponds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16471 in fact to the <I>negative</I> sign before the radical in the case of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16472 particular equation there solved, while a quite obvious and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16473 slight variation of the construction would give the solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16474 corresponding to the <I>positive</I> sign.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16475 <p>The following particular cases of quadratics occurring in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16476 the <I>Arithmetica</I> may be quoted, with the results stated by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16477 Diophantus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16478 <MATH><I>x</I><SUP>2</SUP>=4<I>x</I>-4; therefore <I>x</I>=2. (IV. 22)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16479 325<I>x</I><SUP>2</SUP>=3<I>x</I>+18; <I>x</I>=(78)/(325) or 6/(25). (IV. 31)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16480 84<I>x</I><SUP>2</SUP>+7<I>x</I>=7; <I>x</I>=1/4. (VI. 6)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16481 84<I>x</I><SUP>2</SUP>-7<I>x</I>=7; <I>x</I>=1/3. (VI. 7)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16482 630<I>x</I><SUP>2</SUP>-73<I>x</I>=6; <I>x</I>=6/(35). (VI. 9)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16483 630<I>x</I><SUP>2</SUP>+73<I>x</I>=6; <I>x</I> is rational. (VI. 8)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16484 5<I>x<x</I><SUP>2</SUP>-60<8<I>x</I>; <I>x</I> not<11 and not>12. (V. 30)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16485 17<I>x</I><SUP>2</SUP>+17<72<I>x</I><19<I>x</I><SUP>2</SUP>+19; <I>x</I> not>(67)/(17) and not<(66)/(19). (V. 10)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16486 22<I>x<x</I><SUP>2</SUP>+60<24<I>x</I>; <I>x</I> not<19 but<21. (V. 30)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16487 <pb n=465><head>DETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16488 In the first and third of the last three cases the limits are not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16489 accurate, but are <I>integral</I> limits which are <I>a fortiori</I> safe.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16490 In the second (66)/(19) should have been (67)/(19), and it would have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16491 more correct to say that, if <I>x</I> is not greater than (67)/(17) and not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16492 less than (67)/(19), the given conditions are <I>a fortiori</I> satisfied.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16493 <p>For comparison with Diophantus's solutions of quadratic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16494 equations we may refer to a few of his solutions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16495 <p>(3) <I>Simultaneous equations involving quadratics</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16496 <p>In I. 27, 28, and 30 we have the following pairs of equations.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16497 <MATH><BRACE>(<G>a</G>) <G>x</G>+<G>h</G>=2<I>a</I> <G>xh</G>=<I>B</I></BRACE>, <BRACE>(<G>b</G>) <G>x</G>+<G>h</G>=2<I>a</I> <G>x</G><SUP>2</SUP>+<G>h</G><SUP>2</SUP>=<I>B</I></BRACE>, <BRACE>(<G>g</G>) <G>x</G>-<G>h</G>=2<I>a</I> <G>xh</G>=<I>B</I></BRA
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16498 E></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16499 <p>I use the Greek letters for the numbers required to be found
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16500 as distinct from the one unknown which Diophantus uses, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16501 which I shall call <I>x</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16502 <p>In (<G>a</G>), he says, let <MATH><G>x</G>-<G>h</G>=2<I>x</I> (<G>x</G>><G>h</G>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16503 <p>It follows, by addition and subtraction, that <MATH><G>x</G>=<I>a</I>+<I>x</I>, <G>h</G>=<I>a</I>-<I>x</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16504 therefore <MATH><G>xh</G>=(<I>a</I>+<I>x</I>) (<I>a</I>-<I>x</I>)=<I>a</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>=<I>B</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16505 and <I>x</I> is found from the pure quadratic equation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16506 <p>In (<G>b</G>) similarly he assumes <MATH><G>x</G>-<G>h</G>=2<I>x</I></MATH>, and the resulting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16507 equation is <MATH><G>x</G><SUP>2</SUP>+<G>h</G><SUP>2</SUP>=(<I>a</I>+<I>x</I>)<SUP>2</SUP>+(<I>a</I>-<I>x</I>)<SUP>2</SUP>=2(<I>a</I><SUP>2</SUP>+<I>x</I><SUP>2</SUP>)=<I>B</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16508 <p>In (<G>g</G>) he puts <MATH><G>x</G>+<G>h</G>=2<I>x</I></MATH> and solves as in the case of (<G>a</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16509 <p>(4) <I>Cubic equation</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16510 <p>Only one very particular case occurs. In VI. 17 the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16511 leads to the equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16512 <MATH><I>x</I><SUP>2</SUP>+2<I>x</I>+3=<I>x</I><SUP>3</SUP>+3<I>x</I>-3<I>x</I><SUP>2</SUP>-1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16513 Diophantus says simply ‘whence <I>x</I> is found to be 4’. In fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16514 the equation reduces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16515 <MATH><I>x</I><SUP>3</SUP>+<I>x</I>=4<I>x</I><SUP>2</SUP>+4</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16516 Diophantus no doubt detected, and divided out by, the common
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16517 factor <MATH><I>x</I><SUP>2</SUP>+1</MATH>, leaving <MATH><I>x</I>=4</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16518 <pb n=466><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16519 <C>(B) <I>Indeterminate equations</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16520 <p>Diophantus says nothing of indeterminate equations of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16521 first degree. The reason is perhaps that it is a principle with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16522 him to admit rational <I>fractional</I> as well as integral solutions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16523 whereas the whole point of indeterminate equations of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16524 first degree is to obtain a solution in <I>integral</I> numbers.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16525 Without this limitation (foreign to Diophantus) such equa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16526 tions have no significance.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16527 <C>(<G>a</G>) <I>Indeterminate equations of the second degree</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16528 <p>The form in which these equations occur is invariably this:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16529 one or two (but never more) functions of <B><I>x</I></B> of the form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16530 <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I></MATH> or simpler forms are to be made rational square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16531 numbers by finding a suitable value for <I>x</I>. That is, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16532 to solve, in the most general case, one or two equations of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16533 form <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16534 <p>(1) <I>Single equation</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16535 <p>The solutions take different forms according to the particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16536 values of the coefficients. Special cases arise when one or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16537 more of them vanish or they satisfy certain conditions.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16538 <p>1. When <I>A</I> or <I>C</I> or both vanish, the equation can always
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16539 be solved rationally.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16540 <p>Form <MATH><I>Bx</I>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16541 <p>Form <MATH><I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16542 <p>Diophantus puts for <I>y</I><SUP>2</SUP> any determinate square <I>m</I><SUP>2</SUP>, and <I>x</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16543 immediately found.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16544 <p>Form <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16545 <p>Diophantus puts for <I>y</I> any multiple of <I>x</I>, as <I>m</I>/<I>n</I><I>x</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16546 <p>2. The equation <MATH><I>Ax</I><SUP>2</SUP>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH> can be rationally solved accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16547 ing to Diophantus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16548 <p>(<I>a</I>) when <I>A</I> is positive and a square, say <I>a</I><SUP>2</SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16549 in this case we put <MATH><I>a</I><SUP>2</SUP><I>x</I><SUP>2</SUP>+<I>C</I>=(<I>ax</I>±<I>m</I>)<SUP>2</SUP></MATH>, whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16550 <MATH><I>x</I>=±(<I>C</I>-<I>m</I><SUP>2</SUP>)/(2<I>ma</I>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16551 (<I>m</I> and the sign being so chosen as to give <I>x</I> a positive value);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16552 <pb n=467><head>INDETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16553 <p>(<G>b</G>) when <I>C</I> is positive and a square, say <I>c</I><SUP>2</SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16554 in this case Diophantus puts <MATH><I>Ax</I><SUP>2</SUP>+<I>c</I><SUP>2</SUP>=(<I>mx</I>±<I>c</I>)<SUP>2</SUP></MATH>, and obtains
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16555 <MATH><I>x</I>=±(2<I>mc</I>)/(<I>A</I>-<I>m</I><SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16556 <p>(<G>g</G>) When one solution is known, any number of other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16557 solutions can be found. This is stated in the Lemma to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16558 VI. 15. It would be true not only of the cases <MATH>±<I>Ax</I><SUP>2</SUP>&mnplus;<I>C</I>=<I>y</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16559 but of the general case <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>. Diophantus, how-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16560 ever, only states it of the case <MATH><I>Ax</I><SUP>2</SUP>-<I>C</I>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16561 <p>His method of finding other (greater) values of <I>x</I> satisfy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16562 ing the equation when one (<I>x</I><SUB>0</SUB>) is known is as follows. If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16563 <MATH><I>Ax</I><SUB>0</SUB><SUP>2</SUP>-<I>C</I>=<I>q</I><SUP>2</SUP></MATH>, he substitutes in the original equation (<I>x</I><SUB>0</SUB>+<I>x</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16564 for <I>x</I> and (<I>q</I>-<I>kx</I>) for <I>y</I>, where <I>k</I> is some integer.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16565 <p>Then, since <MATH><I>A</I>(<I>x</I><SUB>0</SUB>+<I>x</I>)<SUP>2</SUP>-<I>C</I>=(<I>q</I>-<I>kx</I>)<SUP>2</SUP></MATH>, while <MATH><I>Ax</I><SUB>0</SUB><SUP>2</SUP>-<I>C</I>=<I>q</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16566 it follows by subtraction that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16567 <MATH>2<I>x</I>(<I>Ax</I><SUB>0</SUB>+<I>kq</I>)=<I>x</I><SUP>2</SUP>(<I>k</I><SUP>2</SUP>-<I>A</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16568 whence <MATH><I>x</I>=2(<I>Ax</I><SUB>0</SUB>+<I>kq</I>)/(<I>k</I><SUP>2</SUP>-<I>A</I>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16569 and the new value of <I>x</I> is <MATH><I>x</I><SUB>0</SUB>+(2(<I>Ax</I><SUB>0</SUB>+<I>kq</I>))/(<I>k</I><SUP>2</SUP>-<I>A</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16570 <p>Form <MATH><I>Ax</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16571 <p>Diophantus says (VI. 14) that a rational solution of this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16572 case is only possible when <I>A</I> is the sum of two squares.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16573 <p>[In fact, if <I>x</I>=<I>p/q</I> satisfies the equation, and <MATH><I>Ax</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>=<I>k</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16574 we have <MATH><I>Ap</I><SUP>2</SUP>=<I>c</I><SUP>2</SUP><I>q</I><SUP>2</SUP>+<I>k</I><SUP>2</SUP><I>q</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16575 or <MATH><I>A</I>=((<I>cq</I>)/<I>p</I>)<SUP>2</SUP>+((<I>kq</I>)/<I>p</I>)<SUP>2</SUP>.]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16576 <p>Form <MATH><I>Ax</I><SUP>2</SUP>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16577 <p>Diophantus proves in the Lemma to VI. 12 that this equa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16578 tion has an infinite number of solutions when <I>A</I>+<I>C</I> is a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16579 i.e. in the particular case where <I>x</I>=1 is a solution. (He does
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16580 not, however, always bear this in mind, for in III. 10 he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16581 regards the equation <MATH>52<I>x</I><SUP>2</SUP>+12=<I>y</I><SUP>2</SUP></MATH> as impossible, though
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16582 <MATH>52+12=64</MATH> is a square, just as, in III. 11, <MATH>266<I>x</I><SUP>2</SUP>-10=<I>y</I><SUP>2</SUP></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16583 is regarded as impossible.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16584 <p>Suppose that <MATH><I>A</I>+<I>C</I>=<I>q</I><SUP>2</SUP></MATH>; the equation is then solved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16585 <pb n=468><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16586 substituting in the original equation 1+<I>x</I> for <I>x</I> and (<I>q</I>-<I>kx</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16587 for <I>y</I>, where <I>k</I> is some integer.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16588 <p>3. Form <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16589 <p>This can be reduced to the form in which the second term is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16590 wanting by replacing <I>x</I> by <I>z</I>- <I>B</I>/(2<I>A</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16591 <p>Diophantus, however, treats this case separately and less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16592 fully. According to him, a rational solution of the equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16593 <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH> is only possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16594 <p>(<G>a</G>) when <I>A</I> is positive and a square, say <I>a</I><SUP>2</SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16595 <p>(<G>b</G>) when <I>C</I> is positive and a square, say <I>c</I><SUP>2</SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16596 <p>(<G>g</G>) when 1/4<I>B</I><SUP>2</SUP>-<I>AC</I> is positive and a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16597 <p>In case (<G>a</G>) <I>y</I> is put equal to (<I>ax</I>-<I>m</I>), and in case (<G>b</G>) <I>y</I> is put
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16598 equal to (<I>mx</I>-<I>c</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16599 <p>Case (<G>g</G>) is not expressly enunciated, but occurs, as it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16600 were, accidentally (IV. 31). The equation to be solved is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16601 <MATH>3<I>x</I>+18-<I>x</I><SUP>2</SUP>=<I>y</I><SUP>2</SUP></MATH>. Diophantus first assumes <MATH>3<I>x</I>+18-<I>x</I><SUP>2</SUP>=4<I>x</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16602 which gives the quadratic <MATH>3<I>x</I>+18=5<I>x</I><SUP>2</SUP></MATH>; but this ‘is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16603 rational’. Therefore the assumption of 4<I>x</I><SUP>2</SUP> for <I>y</I><SUP>2</SUP> will not do,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16604 ‘and we must find a square [to replace 4] such that 18 times
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16605 (this square+1)+(3/2)<SUP>2</SUP> may be a square’. The auxiliary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16606 equation is therefore <MATH>18(<I>m</I><SUP>2</SUP>+1)+9/4=<I>y</I><SUP>2</SUP></MATH>, or <MATH>72<I>m</I><SUP>2</SUP>+81=a</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16607 square, and Diophantus assumes <MATH>72<I>m</I><SUP>2</SUP>+81=(8<I>m</I>+9)<SUP>2</SUP></MATH>, whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16608 <I>m</I>=18. Then, assuming <MATH>3<I>x</I>+18-<I>x</I><SUP>2</SUP>=(18)<SUP>2</SUP><I>x</I><SUP>2</SUP></MATH>, he obtains the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16609 equation <MATH>325<I>x</I><SUP>2</SUP>-3<I>x</I>-18=0</MATH>, whence <MATH><I>x</I>=(78)/(325)</MATH>, that is, 6/25.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16610 <p>(2) <I>Double equation</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16611 <p>The Greek term is <G>diploi+so/ths, diplh= i)so/ths</G> or <G>diplh= i)/swsis</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16612 Two different functions of the unknown have to be made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16613 simultaneously squares. The general case is to solve in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16614 rational numbers the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16615 <MATH><BRACE><I>mx</I><SUP>2</SUP>+<G>a</G><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP><I>nx</I><SUP>2</SUP>+<G>b</G><I>x</I>+<I>b</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16616 The necessary preliminary condition is that each of the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16617 expressions can be made a square. This is always possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16618 when the first term (in <I>x</I><SUP>2</SUP>) is wanting. We take this simplest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16619 case first.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16620 <pb n=469><head>INDETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16621 <p>1. <I>Double equation of the first degree</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16622 <p>The equations are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16623 <MATH><G>a</G><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16624 <G>b</G><I>x</I>+<I>b</I>=<I>w</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16625 <p>Diophantus has one general method taking slightly different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16626 forms according to the nature of the coefficients.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16627 <p>(<I>a</I>) First method of solution.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16628 <p>This depends upon the identity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16629 <MATH>{1/2(<I>p</I>+<I>q</I>)}<SUP>2</SUP>-{1/2(<I>p</I>-<I>q</I>)}<SUP>2</SUP>=<I>pq</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16630 <p>If the difference between the two expressions in <I>x</I> can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16631 separated into two factors <I>p, q</I>, the expressions themselves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16632 are equated to <MATH>{1/2(<I>p</I>+<I>q</I>)}<SUP>2</SUP></MATH> and <MATH>{1/2(<I>p</I>-<I>q</I>)}<SUP>2</SUP></MATH> respectively. As
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16633 Diophantus himself says in II. 11, we ‘equate either the square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16634 of half the difference of the two factors to the lesser of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16635 expressions, or the square of half the sum to the greater’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16636 <p>We will consider the general case and investigate to what
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16637 particular classes of cases the method is applicable from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16638 Diophantus's point of view, remembering that the final quad-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16639 ratic in <I>x</I> must always reduce to a single equation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16640 <p>Subtracting, we have <MATH>(<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>)=<I>u</I><SUP>2</SUP>-<I>w</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16641 <p>Separate <MATH>(<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>)</MATH> into the factors
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16642 <MATH><I>p</I>, {(<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>)}/<I>p</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16643 <p>We write accordingly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16644 <MATH><I>u</I>±<I>w</I>=((<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>))/<I>p</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16645 <I>u</I>&mnplus;<I>w</I>=<I>p</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16646 <p>Thus <MATH><BRACE><I>u</I><SUP>2</SUP>=<G>a</G><I>x</I>+<I>a</I>=1/4((<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>))/<I>p</I>+<I>p</I><SUP>2</SUP></BRACE></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16647 therefore <MATH>{(<G>a</G>-<G>b</G>)<I>x</I>+<I>a</I>-<I>b</I>+<I>p</I><SUP>2</SUP>}<SUP>2</SUP>=4<I>p</I><SUP>2</SUP>(<G>a</G><I>x</I>+<I>a</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16648 <p>This reduces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16649 <MATH>(<G>a</G>-<G>b</G>)<SUP>2</SUP><I>x</I><SUP>2</SUP>+2<I>x</I>{(<G>a</G>-<G>b</G>)(<I>a</I>-<I>b</I>)-<I>p</I><SUP>2</SUP>(<G>a</G>+<G>b</G>)}
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16650 +(<I>a</I>-<I>b</I>)<SUP>2</SUP>-2<I>p</I><SUP>2</SUP>(<I>a</I>+<I>b</I>)+<I>p</I><SUP>4</SUP>=0</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16651 <pb n=470><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16652 <p>In order that this equation may reduce to a simple equation,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16653 either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16654 <p>(1) the coefficient of <I>x</I><SUP>2</SUP> must vanish, or <MATH><G>a</G>-<G>b</G>=0</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16655 or (2) the absolute term must vanish, that is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16656 <MATH><I>p</I><SUP>4</SUP>-2<I>p</I><SUP>2</SUP>(<I>a</I>+<I>b</I>)+(<I>a</I>-<I>b</I>)<SUP>2</SUP>=0</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16657 or <MATH>{<I>p</I><SUP>2</SUP>-(<I>a</I>+<I>b</I>}<SUP>2</SUP>=4<I>ab</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16658 so that <I>ab</I> must be a square number.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16659 <p>As regards condition (1) we observe that it is really sufficient
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16660 if <G>a</G><I>n</I><SUP>2</SUP>=<G>b</G><I>m</I><SUP>2</SUP>, since, if <G>a</G><I>x</I>+<I>a</I> is a square, (<G>a</G><I>x</I>+<I>a</I>)<I>n</I><SUP>2</SUP> is equally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16661 a square, and, if <G>b</G><I>x</I>+<I>b</I> is a square, so is (<G>b</G><I>x</I>+<I>b</I>)<I>m</I><SUP>2</SUP>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16662 vice versa.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16663 <p>That is, (1) we can solve any pair of equations of the form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16664 <MATH><BRACE><G>a</G><I>m</I><SUP>2</SUP><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16665 <G>a</G><I>n</I><SUP>2</SUP><I>x</I>+<I>b</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16666 <p>Multiply by <I>n</I><SUP>2</SUP>, <I>m</I><SUP>2</SUP> respectively, and we have to solve the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16667 equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16668 <MATH><BRACE><G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>x</I>+<I>an</I><SUP>2</SUP>=<I>u</I>′<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16669 <G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>x</I>+<I>bm</I><SUP>2</SUP>=<I>w</I>′<SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16670 <p>Separate the difference, <I>an</I><SUP>2</SUP>-<I>bm</I><SUP>2</SUP>, into two factors <I>p</I>, <I>q</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16671 put <MATH><I>u</I>′±<I>w</I>′=<I>p</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16672 <I>u</I>′&mnplus;<I>w</I>′=<I>q</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16673 therefore <MATH><I>u</I>′<SUP>2</SUP>=1/4(<I>p</I>+<I>q</I>)<SUP>2</SUP>, <I>w</I>′<SUP>2</SUP>=1/4(<I>p</I>-<I>q</I>)<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16674 and <MATH><G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>x</I>+<I>an</I><SUP>2</SUP>=1/4(<I>p</I>+<I>q</I>)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16675 <G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>x</I>+<I>bm</I><SUP>2</SUP>=1/4(<I>p</I>-<I>q</I>)<SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16676 and from either of these equations we get
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16677 <MATH><I>x</I>=(1/4(<I>p</I><SUP>2</SUP>+<I>q</I><SUP>2</SUP>)-1/2(<I>an</I><SUP>2</SUP>+<I>bm</I><SUP>2</SUP>))/(<G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16678 since <MATH><I>pq</I>=<I>an</I><SUP>2</SUP>-<I>bm</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16679 <p>Any factors <I>p, q</I> can be chosen provided that the resulting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16680 value of <I>x</I> is <I>positive</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16681 <pb n=471><head>INDETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16682 <p>Ex. from Diophantus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16683 <MATH><BRACE>65-6<I>x</I>=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16684 65-24<I>x</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>; (IV. 32)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16685 therefore <MATH><BRACE>260-24<I>x</I>=<I>u</I>′<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16686 65-24<I>x</I>=<I>w</I>′<SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16687 <p>The difference=195=15.13, say;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16688 therefore <MATH>1/4(15-13)<SUP>2</SUP>=65-24<I>x</I></MATH>; that is, <MATH>24<I>x</I>=64</MATH>, and <MATH><I>x</I>=8/3</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16689 <p>Taking now the condition (2) that <I>ab</I> is a square, we see
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16690 that the equations can be solved in the cases where either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16691 <I>a</I> and <I>b</I> are both squares, or the ratio of <I>a</I> to <I>b</I> is the ratio of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16692 a square to a square. If the equations are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16693 <MATH><G>a</G><I>x</I>+<I>c</I><SUP>2</SUP>=<I>u</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16694 <G>b</G><I>x</I>+<I>d</I><SUP>2</SUP>=<I>w</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16695 and factors are taken of the difference between the expressions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16696 as they stand, then, since one factor <I>p</I>, as we saw, satisfies the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16697 equation <MATH>{<I>p</I><SUP>2</SUP>-(<I>c</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>}<SUP>2</SUP>=4<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16698 we must have <MATH><I>p</I>=<I>c</I>±<I>d</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16699 <p>Ex. from Diophantus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16700 <MATH><BRACE>10<I>x</I>+9=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16701 5<I>x</I>+4=<I>w</I><SUP>2</SUP></BRACE></MATH>. (III. 15)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16702 The difference is <MATH>5<I>x</I>+5=5(<I>x</I>+1)</MATH>; the solution is given by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16703 <MATH>(1/2<I>x</I>+3)<SUP>2</SUP>=10<I>x</I>+9, and <I>x</I>=28</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16704 <p>Another method is to multiply the equations by squares
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16705 such that, when the expressions are subtracted, the absolute
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16706 term vanishes. The case can be worked out generally, thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16707 <p>Multiply by <I>d</I><SUP>2</SUP> and <I>c</I><SUP>2</SUP> respectively, and we have to solve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16708 <MATH><BRACE><G>a</G><I>d</I><SUP>2</SUP><I>x</I>+<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16709 <G>b</G><I>c</I><SUP>2</SUP><I>x</I>+<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=<I>w</I><SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16710 <p><MATH>Difference =(<G>a</G><I>d</I><SUP>2</SUP>-<G>b</G><I>c</I><SUP>2</SUP>)<I>x</I>=<I>px.q</I></MATH> say.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16711 <p>Then <I>x</I> is found from the equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16712 <MATH><G>a</G><I>d</I><SUP>2</SUP><I>x</I>+<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=1/4(<I>px</I>+<I>q</I>)<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16713 which gives <MATH><I>p</I><SUP>2</SUP><I>x</I><SUP>2</SUP>+2<I>x</I>(<I>pq</I>-2<G>a</G><I>d</I><SUP>2</SUP>)+<I>q</I><SUP>2</SUP>-4<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=0</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16714 <pb n=472><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16715 or, since <MATH><I>pq</I>=<G>a</G><I>d</I><SUP>2</SUP>-<G>b</G><I>c</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16716 <I>p</I><SUP>2</SUP><I>x</I><SUP>2</SUP>-2<I>x</I>(<G>a</G><I>d</I><SUP>2</SUP>+<G>b</G><I>c</I><SUP>2</SUP>)+<I>q</I><SUP>2</SUP>-4<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=0</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16717 <p>In order that this may reduce to a simple equation, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16718 Diophantus requires, the absolute term must vanish, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16719 <MATH><I>q</I>=2<I>cd</I></MATH>. The method therefore only gives one solution, since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16720 <I>q</I> is restricted to the value 2<I>cd</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16721 <p>Ex. from Diophantus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16722 <MATH><BRACE>8<I>x</I>+4=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16723 6<I>x</I>+4=<I>w</I><SUP>2</SUP></BRACE></MATH>. (IV. 39)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16724 Difference 2<I>x</I>; <I>q</I> necessarily taken to be 2√4 or 4; factors
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16725 therefore 1/2<I>x</I>, 4. Therefore <MATH>8<I>x</I>+4=1/4(1/2<I>x</I>+4)<SUP>2</SUP></MATH>, and <MATH><I>x</I>=112</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16726 <p>(<G>b</G>) Second method of solution of a double equation of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16727 first degree.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16728 <p>There is only one case of this in Diophantus, the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16729 being of the form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16730 <MATH><BRACE><I>hx</I>+<I>n</I><SUP>2</SUP>=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16731 (<I>h</I>+<I>f</I>)<I>x</I>+<I>n</I><SUP>2</SUP>=<I>w</I><SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16732 <p>Suppose <MATH><I>hx</I>+<I>n</I><SUP>2</SUP>=(<I>y</I>+<I>n</I>)<SUP>2</SUP></MATH>; therefore <MATH><I>hx</I>=<I>y</I><SUP>2</SUP>+2<I>ny</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16733 and <MATH>(<I>h</I>+<I>f</I>)<I>x</I>+<I>n</I><SUP>2</SUP>=(<I>y</I>+<I>n</I>)<SUP>2</SUP>+<I>f/h</I>(<I>y</I><SUP>2</SUP>+2<I>ny</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16734 <p>It only remains to make the latter expression a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16735 which is done by equating it to (<I>py</I>-<I>n</I>)<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16736 <p>The case in Diophantus is the same as that last mentioned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16737 (IV. 39). Where I have used <I>y</I>, Diophantus as usual contrives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16738 to use his one unknown a second time.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16739 <p>2. <I>Double equations of the second degree</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16740 <p>The general form is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16741 <MATH><BRACE><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16742 <I>A</I>′<I>x</I><SUP>2</SUP>+<I>B</I>′<I>x</I>+<I>C</I>′=<I>w</I><SUP>2</SUP></BRACE></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16743 but only three types appear in Diophantus, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16744 (1) <MATH><BRACE><G>r</G><SUP>2</SUP><I>x</I><SUP>2</SUP>+<G>a</G><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16745 <G>r</G><SUP>2</SUP><I>x</I><SUP>2</SUP>+<G>b</G><I>x</I>+<I>b</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>, where, except in one case, <I>a</I>=<I>b</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16746 <pb n=473><head>INDETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16747 (2) <MATH><BRACE><I>x</I><SUP>2</SUP>+<G>a</G><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16748 <G>b</G><I>x</I><SUP>2</SUP>+<I>a</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16749 (The case where the absolute terms are in the ratio of a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16750 to a square reduces to this.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16751 <p>In all examples of these cases the usual method of solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16752 applies.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16753 (3) <MATH><BRACE><G>a</G><I>x</I><SUP>2</SUP>+<I>ax</I>=<I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16754 <G>b</G><I>x</I><SUP>2</SUP>+<I>bx</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16755 <p>The usual method does not here serve, and a special artifice
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16756 is required.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16757 <p>Diophantus assumes <MATH><I>u</I><SUP>2</SUP>=<I>m</I><SUP>2</SUP><I>x</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16758 <p>Then <MATH><I>x</I>=<I>a</I>/(<I>m</I><SUP>2</SUP>-<G>a</G>)</MATH> and, by substitution in the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16759 equation, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16760 <MATH><G>b</G>(<I>a</I>/(<I>m</I><SUP>2</SUP>-<G>a</G>))<SUP>2</SUP>+(<I>ba</I>)/(<I>m</I><SUP>2</SUP>-<G>a</G>)</MATH>, which must be made a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16761 or <MATH><I>a</I><SUP>2</SUP><G>b</G>+<I>ba</I>(<I>m</I><SUP>2</SUP>-<G>a</G>)</MATH> must be a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16762 <p>We have therefore to solve the equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16763 <MATH><I>abm</I><SUP>2</SUP>+<I>a</I>(<I>a</I><G>b</G>-<G>a</G><I>b</I>)=<I>y</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16764 which can or cannot be solved by Diophantus's methods
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16765 according to the nature of the coefficients. Thus it can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16766 solved if <MATH>(<I>a</I><G>b</G>-<G>a</G><I>b</I>)/<I>a</I></MATH> is a square, or if <I>a/b</I> is a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16767 Examples in VI. 12, 14.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16768 <C>(<I>b</I>) <I>Indeterminate equations of a degree higher than the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16769 second</I>.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16770 <p>(1) <I>Single equations</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16771 <p>There are two classes, namely those in which expressions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16772 in <I>x</I> have to be made squares or cubes respectively. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16773 general form is therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16774 <MATH><I>Ax</I><SUP><I>n</I></SUP>+<I>Bx</I><SUP><I>n</I>-1</SUP>+...+<I>Kx</I>-<I>L</I>=<I>y</I><SUP>2</SUP> or <I>y</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16775 <p>In Diophantus <I>n</I> does not exceed 6, and in the second class
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16776 of cases, where the expression has to be made a cube, <I>n</I> does
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16777 not generally exceed 3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16778 <pb n=474><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16779 <p>The species of the first class found in the <I>Arithmetica</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16780 as follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16781 <p>1. Equation <MATH><I>Ax</I><SUP>3</SUP>+<I>Bx</I><SUP>2</SUP>+<I>Cx</I>+<I>d</I><SUP>2</SUP>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16782 <p>As the absolute term is a square, we can assume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16783 <MATH><I>y</I>=<I>C</I>/(2<I>d</I>)<I>x</I>+<I>d</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16784 or we might assume <MATH><I>y</I>=<I>m</I><SUP>2</SUP><I>x</I><SUP>2</SUP>+<I>nx</I>+<I>d</I></MATH> and determine <I>m</I>, <I>n</I> so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16785 that the coefficients of <I>x</I>, <I>x</I><SUP>2</SUP> in the resulting equation both
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16786 vanish.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16787 <p>Diophantus has only one case, <MATH><I>x</I><SUP>3</SUP>-3<I>x</I><SUP>2</SUP>+3<I>x</I>+1=<I>y</I><SUP>2</SUP></MATH> (VI. 18),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16788 and uses the first method.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16789 <p>2. Equation <MATH><I>Ax</I><SUP>4</SUP>+<I>Bx</I><SUP>3</SUP>+<I>Cx</I><SUP>2</SUP>+<I>Dx</I>+<I>E</I>=<I>y</I><SUP>2</SUP></MATH>, where either <I>A</I> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16790 <I>E</I> is a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16791 <p>If <I>A</I> is a square (=<I>a</I><SUP>2</SUP>), we may assume <MATH><I>y</I>=<I>ax</I><SUP>2</SUP>+<I>B</I>/(2<I>a</I>)<I>x</I>+<I>n</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16792 determining <I>n</I> so that the term in <I>x</I><SUP>2</SUP> in the resulting equa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16793 tion may vanish. If <I>E</I> is a square (=<I>e</I><SUP>2</SUP>), we may assume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16794 <MATH><I>y</I>=<I>mx</I><SUP>2</SUP>+<I>D</I>/(2<I>e</I>)<I>x</I>+<I>e</I></MATH>, determining <I>m</I> so that the term in <I>x</I><SUP>2</SUP> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16795 resulting equation may vanish. We shall then, in either case,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16796 obtain a simple equation in <I>x</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16797 <p>3. Equation <MATH><I>Ax</I><SUP>4</SUP>+<I>Cx</I><SUP>2</SUP>+<I>E</I>=<I>y</I><SUP>2</SUP></MATH>, but in special cases only where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16798 all the coefficients are squares.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16799 <p>4. Equation <MATH><I>Ax</I><SUP>4</SUP>+<I>E</I>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16800 <p>The case occurring in Diophantus is <MATH><I>x</I><SUP>4</SUP>+97=<I>y</I><SUP>2</SUP></MATH> (V. 29).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16801 Diophantus tries one assumption, <MATH><I>y</I>=<I>x</I><SUP>2</SUP>-10</MATH>, and finds that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16802 this gives <MATH><I>x</I><SUP>2</SUP>=3/(20)</MATH>, which leads to no rational result. He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16803 therefore goes back and alters his assumptions so that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16804 is able to replace the refractory equation by <MATH><I>x</I><SUP>4</SUP>+337=<I>y</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16805 and at the same time to find a suitable value for <I>y</I>, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16806 <MATH><I>y</I>=<I>x</I><SUP>2</SUP>-25</MATH>, which produces a rational result, <MATH><I>x</I>=(12)/5</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16807 <p>5. Equation of sixth degree in the special form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16808 <MATH><I>x</I><SUP>6</SUP>-<I>Ax</I><SUP>3</SUP>+<I>Bx</I>+<I>c</I><SUP>2</SUP>=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16809 <p>Putting <MATH><I>y</I>=<I>x</I><SUP>3</SUP>+<I>c</I></MATH>, we have <MATH>-<I>Ax</I><SUP>2</SUP>+<I>B</I>=2<I>cx</I><SUP>2</SUP></MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16810 <MATH><I>x</I><SUP>2</SUP>=<I>B</I>/(<I>A</I>+2<I>c</I>)</MATH>, which gives a rational solution if <I>B</I>/(<I>A</I>+2<I>c</I>) is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16811 <pb n=475><head>INDETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16812 a square. Where this does not hold (in IV. 18) Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16813 harks back and replaces the equation <MATH><I>x</I><SUP>6</SUP>-16<I>x</I><SUP>3</SUP>+<I>x</I>+64=<I>y</I><SUP>2</SUP></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16814 by another, <MATH><I>x</I><SUP>6</SUP>-128<I>x</I><SUP>3</SUP>+<I>x</I>+4096=<I>y</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16815 <p>Of expressions which have to be made <I>cubes</I>, we have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16816 following cases.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16817 <p>1. <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16818 <p>There are only two cases of this. First, in VI. 1, <MATH><I>x</I><SUP>2</SUP>-4<I>x</I>+4</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16819 has to be made a cube, being already a square. Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16820 naturally makes <I>x</I>-2 a cube.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16821 <p>Secondly, a peculiar case occurs in VI. 17, where a cube has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16822 to be found exceeding a square by 2. Diophantus assumes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16823 (<I>x</I>-1)<SUP>3</SUP> for the cube and (<I>x</I>+1)<SUP>2</SUP> for the square. This gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16824 <MATH><I>x</I><SUP>3</SUP>-3<I>x</I><SUP>2</SUP>+3<I>x</I>-1=<I>x</I><SUP>2</SUP>+2<I>x</I>+3</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16825 or <MATH><I>x</I><SUP>3</SUP>+<I>x</I>=4<I>x</I><SUP>2</SUP>+4</MATH>. We divide out by <I>x</I><SUP>2</SUP>+1, and <MATH><I>x</I>=4</MATH>. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16826 seems evident that the assumptions were made with knowledge
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16827 and intention. That is, Diophantus knew of the solution 27
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16828 and 25 and deliberately led up to it. It is unlikely that he was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16829 aware of the fact, observed by Fermat, that 27 and 25 are the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16830 only integral numbers satisfying the condition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16831 <p>2. <MATH><I>Ax</I><SUP>3</SUP>+<I>Bx</I><SUP>2</SUP>+<I>Cx</I>+<I>D</I>=<I>y</I><SUP>3</SUP></MATH>, where either <I>A</I> or <I>D</I> is a cube
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16832 number, or both are cube numbers. Where <I>A</I> is a cube (<I>a</I><SUP>3</SUP>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16833 we have only to assume <MATH><I>y</I>=<I>ax</I>+<I>B</I>/(3<I>a</I><SUP>2</SUP>)</MATH>, and where <I>D</I> is a cube
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16834 (<I>d</I><SUP>3</SUP>), <MATH><I>y</I>=<I>C</I>/(3<I>d</I><SUP>2</SUP>)<I>x</I>+<I>d</I></MATH>. Where <MATH><I>A</I>=<I>a</I><SUP>3</SUP></MATH> and <MATH><I>D</I>=<I>d</I><SUP>3</SUP></MATH>, we can use
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16835 either assumption, or put <MATH><I>y</I>=<I>ax</I>+<I>d</I></MATH>. Apparently Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16836 used the last assumption only in this case, for in IV. 27 he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16837 rejects as impossible the equation <MATH>8<I>x</I><SUP>3</SUP>-<I>x</I><SUP>2</SUP>+8<I>x</I>-1=<I>y</I><SUP>3</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16838 because the assumption <MATH><I>y</I>=2<I>x</I>-1</MATH> gives a negative value
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16839 <MATH><I>x</I>=-2/(11)</MATH>, whereas either of the above assumptions gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16840 a rational value.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16841 <p>(2) <I>Double equations</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16842 <p>Here one expression has to be made a square and another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16843 a cube. The cases are mostly very simple, e.g. (VI. 19)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16844 <MATH><BRACE>4<I>x</I>+2=<I>y</I><SUP>3</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16845 2<I>x</I>+1=<I>z</I><SUP>2</SUP></BRACE></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16846 thus <MATH><I>y</I><SUP>3</SUP>=2<I>z</I><SUP>2</SUP></MATH>, and <MATH><I>z</I>=2</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16847 <pb n=476><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16848 <p>More complicated is the case in VI. 21:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16849 <MATH><BRACE>2<I>x</I><SUP>2</SUP>+2<I>x</I>=<I>y</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16850 <I>x</I><SUP>3</SUP>+2<I>x</I><SUP>2</SUP>+<I>x</I>=<I>z</I><SUP>3</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16851 <p>Diophantus assumes <MATH><I>y</I>=<I>mx</I></MATH>, whence <MATH><I>x</I>=2/(<I>m</I><SUP>2</SUP>-2)</MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16852 <MATH>(2/(<I>m</I><SUP>2</SUP>-2))<SUP>3</SUP>+2(2/(<I>m</I><SUP>2</SUP>-2))<SUP>2</SUP>+2/(<I>m</I><SUP>2</SUP>-2)=<I>z</I><SUP>3</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16853 or <MATH>(2<I>m</I><SUP>4</SUP>)/((<I>m</I><SUP>2</SUP>-2)<SUP>3</SUP>)=<I>z</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16854 <p>We have only to make 2<I>m</I><SUP>4</SUP>, or 2<I>m</I>, a cube.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16855 <C>II. Method of Limits.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16856 <p>As Diophantus often has to find a series of numbers in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16857 order of magnitude, and as he does not admit negative
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16858 solutions, it is often necessary for him to reject a solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16859 found in the usual course because it does not satisfy the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16860 necessary conditions; he is then obliged, in many cases, to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16861 find solutions lying <I>within certain limits</I> in place of those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16862 rejected. For example:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16863 <p>1. It is required to find a value of <I>x</I> such that some power of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16864 it, <I>x</I><SUP><I>n</I></SUP>, shall lie between two given numbers, say <I>a</I> and <I>b</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16865 <p>Diophantus multiplies both <I>a</I> and <I>b</I> by 2<SUP><I>n</I></SUP>, 3<SUP><I>n</I></SUP>, and so on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16866 successively, until some <I>n</I>th power is seen which lies between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16867 the two products. Suppose that <I>c</I><SUP><I>n</I></SUP> lies between <I>ap</I><SUP><I>n</I></SUP> and <I>bp</I><SUP><I>n</I></SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16868 then we can put <MATH><I>x</I>=<I>c/p</I></MATH>, for <MATH>(<I>c/p</I>)<SUP><I>n</I></SUP></MATH> lies between <I>a</I> and <I>b</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16869 <p>Ex. To find a square between 1 1/4 and 2. Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16870 multiplies by a square 64; this gives 80 and 128, between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16871 which lies 100. Therefore ((10)/8)<SUP>2</SUP> or (25)/(16) solves the problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16872 (IV. 31 (2)).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16873 <p>To find a sixth power between 8 and 16. The sixth powers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16874 of 1, 2, 3, 4 are 1, 64, 729, 4096. Multiply 8 and 16 by 64
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16875 and we have 512 and 1024, between which 729 lies; (729)/(64) is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16876 therefore a solution (VI. 21).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16877 <p>2. Sometimes a value of <I>x</I> has to be found which will give
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16878 <pb n=477><head>METHOD OF LIMITS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16879 some function of <I>x</I> a value intermediate between the values
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16880 of two other functions of <I>x</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16881 <p>Ex. 1. In IV. 25 a value of <I>x</I> is required such that 8/(<I>x</I><SUP>2</SUP>+<I>x</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16882 shall lie between <I>x</I> and <I>x</I>+1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16883 <p>One part of the condition gives <MATH>8><I>x</I><SUP>3</SUP>+<I>x</I><SUP>2</SUP></MATH>. Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16884 accordingly assumes <MATH>8=(<I>x</I>+1/3)<SUP>3</SUP>=<I>x</I><SUP>3</SUP>+<I>x</I><SUP>2</SUP>+1/3<I>x</I>+1/(27)</MATH>, which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16885 <MATH>> <I>x</I><SUP>3</SUP>+<I>x</I><SUP>2</SUP></MATH>. Thus <MATH><I>x</I>+1/3=2</MATH> or <MATH><I>x</I>=5/3</MATH> satisfies one part of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16886 the condition. Incidentally it satisfies the other, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16887 <MATH>8/(<I>x</I><SUP>2</SUP>+<I>x</I>)<<I>x</I>+1</MATH>. This is a piece of luck, and Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16888 is satisfied with it, saying nothing more.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16889 <p>Ex. 2. We have seen how Diophantus concludes that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16890 <MATH>1/5 (<I>x</I><SUP>2</SUP>-60)><I>x</I>>1/8(<I>x</I><SUP>2</SUP>-60)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16891 then <I>x</I> is not less than 11 and not greater than 12 (V. 30).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16892 <p>The problem further requires that <I>x</I><SUP>2</SUP>-60 shall be a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16893 Assuming <MATH><I>x</I><SUP>2</SUP>-60=(<I>x</I>-<I>m</I>)<SUP>2</SUP></MATH>, we find <MATH><I>x</I>=(<I>m</I><SUP>2</SUP>+60)/2<I>m</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16894 <p>Since <I>x</I>>11 and<12, says Diophantus, it follows that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16895 <MATH>24<I>m</I>><I>m</I><SUP>2</SUP>+60>22<I>m</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16896 from which he concludes that <I>m</I> lies between 19 and 21.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16897 Putting <MATH><I>m</I>=20</MATH>, he finds <MATH><I>x</I>=11 1/2</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16898 <C>III. Method of approximation to Limits.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16899 <p>Here we have a very distinctive method called by Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16900 <G>pariso/ths</G> or <G>pariso/thtos a)gwgh/</G>. The object is to solve such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16901 problems as that of finding two or three square numbers the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16902 sum of which is a given number, while each of them either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16903 approximates to one and the same number, or is subject to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16904 limits which may be the same or different.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16905 <p>Two examples will best show the method.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16906 <p>Ex. 1. Divide 13 into two squares each of which > 6 (V. 9).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16907 <p>Take half of 13, i.e. 6 1/2, and find what <I>small</I> fraction 1/<I>x</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16908 added to it will give a square;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16909 thus <MATH>6 1/2+1/(<I>x</I><SUP>2</SUP>)</MATH>, or <MATH>26+1/(<I>y</I><SUP>2</SUP>)</MATH>, must be a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16910 <pb n=478><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16911 <p>Diophantus assumes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16912 <MATH>26+1/<I>y</I><SUP>2</SUP>=(5+1/<I>y</I>)<SUP>2</SUP></MATH>, or <MATH>26<I>y</I><SUP>2</SUP>+1=(5<I>y</I>+1)<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16913 whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16914 <MATH><I>y</I>=10</MATH>, and <MATH>1/<I>y</I><SUP>2</SUP>=1/(100)</MATH>, i.e. <MATH>1/<I>x</I><SUP>2</SUP>=1/(400)</MATH>; and <MATH>6 1/2+1/(400)=((51)/(20))<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16915 <p>[The assumption of 5+1/<I>y</I> as the side is not haphazard: 5 is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16916 chosen because it is the most suitable as giving the largest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16917 rational value for <I>y</I>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16918 <p>We have now, says Diophantus, to divide 13 into two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16919 squares each of which is as nearly as possible equal to ((51)/(20))<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16920 <p>Now <MATH>13=3<SUP>2</SUP>+2<SUP>2</SUP></MATH> [it is necessary that the original number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16921 shall be capable of being expressed as the sum of two squares];
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16922 and <MATH>3>(51)/(20) by 9/(20)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16923 while <MATH>2<(51)/(20) by (11)/(20)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16924 <p>But if we took <MATH>3-9/(20), 2+(11)/(20)</MATH> as the sides of two squares,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16925 their sum would be <MATH>2((51)/(20))<SUP>2</SUP>=(5202)/(400)</MATH>, which is > 13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16926 <p>Accordingly we assume <MATH>3-9<I>x</I>, 2+11<I>x</I></MATH> as the sides of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16927 required squares (so that <I>x</I> is not exactly 1/20 but near it).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16928 <p>Thus <MATH>(3-9<I>x</I>)<SUP>2</SUP>+(2+11<I>x</I>)<SUP>2</SUP>=13</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16929 and we find <MATH><I>x</I>=5/(101)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16930 <p>The sides of the required squares are (257)/(101), (258)/(101).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16931 <p>Ex. 2. Divide 10 into three squares each of which > 3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16932 (V. 11).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16933 <p>[The original number, here 10, must of course be expressible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16934 as the sum of three squares.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16935 <p>Take one-third of 10, i.e. 3 1/3, and find what small fraction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16936 1/<I>x</I><SUP>2</SUP> added to it will make a square; i.e. we have to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16937 <MATH>3 1/3+1/(<I>x</I><SUP>2</SUP>)</MATH> a square, i.e. <MATH>30+9/(<I>x</I><SUP>2</SUP>)</MATH> must be a square, or <MATH>30+1/(<I>y</I><SUP>2</SUP>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16938 =</MATH> a square, where <MATH>3/<I>x</I>=1/<I>y</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16939 <p>Diophantus assumes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16940 <MATH>30<I>y</I><SUP>2</SUP>+1=(5<I>y</I>+1)<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16941 the coefficient of <I>y</I>, i.e. 5, being so chosen as to make 1/<I>y</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16942 small as possible;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16943 <pb n=479><head>METHOD OF APPROXIMATION TO LIMITS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16944 therefore <MATH><I>y</I>=2</MATH>, and <MATH>1/<I>x</I><SUP>2</SUP>=1/(36)</MATH>; and <MATH>3 1/3+1/(36)=(121)/(36)</MATH>, a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16945 <p>We have now, says Diophantus, to divide 10 into three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16946 squares with sides as near as may be to (11)/6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16947 <p>Now <MATH>10=9+1=3<SUP>2</SUP>+(3/5)<SUP>2</SUP>+(4/5)<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16948 <p>Bringing 3, 3/5, 4/5 and (11)/6 to a common denominator, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16949 (90)/(30), (18)/(30), (24)/(30) and (55)/(30),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16950 and <MATH>3>(55)/(30) by (35)/(30),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16951 3/5<(55)/(30) by (37)/(30),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16952 4/5<(55)/(30) by (31)/(30)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16953 <p>If now we took 3-(35)/(30), 3/5+(37)/(30), 4/5+(31)/(30) as the sides of squares,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16954 the sum of the squares would be 3((11)/6)<SUP>2</SUP> or (363)/(36), which is > 10.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16955 <p>Accordingly we assume as the sides <MATH>3-35<I>x</I>, 3/5+37<I>x</I>, 4/5+31<I>x</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16956 where <I>x</I> must therefore be not exactly 1/(30) but near it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16957 <p>Solving <MATH>(3-35<I>x</I>)<SUP>2</SUP>+(3/5+37<I>x</I>)<SUP>2</SUP>+(4/5+31<I>x</I>)<SUP>2</SUP>=10</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16958 or <MATH>10-116<I>x</I>+3555<I>x</I><SUP>2</SUP>=10</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16959 we find <MATH><I>x</I>=(116)/(3555)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16960 thus the sides of the required squares are (1321)/(711), (1285)/(711), (1288)/(711);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16961 the squares themselves are (1745041)/(505521), (1651225)/(505521), (1658944)/(505521).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16962 <p>Other instances of the application of the method will be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16963 found in V. 10, 12, 13, 14.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16964 <C>Porisms and propositions in the Theory of Numbers.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16965 <p>I. Three propositions are quoted as occurring in the <I>Porisms</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16966 (‘We have it in the Porisms that ...’); and some other pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16967 positions assumed without proof may very likely have come
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16968 from the same collection. The three propositions from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16969 <I>Porisms</I> are to the following effect.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16970 <p>1. If <I>a</I> is a given number and <I>x</I>, <I>y</I> numbers such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16971 <MATH><I>x</I>+<I>a</I>=<I>m</I><SUP>2</SUP>, <I>y</I>+<I>a</I>=<I>n</I><SUP>2</SUP></MATH>, then, if <MATH><I>xy</I>+<I>a</I></MATH> is also a square, <I>m</I> and <I>n</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16972 differ by unity (V. 3).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16973 <p>[From the first two equations we obtain easily
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16974 <MATH><I>xy</I>+<I>a</I>=<I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP>-<I>a</I>(<I>m</I><SUP>2</SUP>+<I>n</I><SUP>2</SUP>-1)+<I>a</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16975 and this is obviously a square if <MATH><I>m</I><SUP>2</SUP>+<I>n</I><SUP>2</SUP>-1=2<I>mn</I></MATH>, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16976 <MATH><I>m</I>-<I>n</I>=±1</MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16977 <pb n=480><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16978 <p>2. If <I>m</I><SUP>2</SUP>, (<I>m</I> + 1)<SUP>2</SUP> be consecutive squares and a third number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16979 be taken equal to <MATH>2{<I>m</I><SUP>2</SUP> + (<I>m</I> + 1)<SUP>2</SUP>} + 2</MATH>, or <MATH>4(<I>m</I><SUP>2</SUP> + <I>m</I> + 1)</MATH>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16980 three numbers have the property that the product of any two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16981 <I>plus</I> either the sum of those two or the remaining number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16982 gives a square (V. 5).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16983 <p>[In fact, if <I>X, Y, Z</I> denote the numbers respectively,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16984 <MATH><I>XY</I> + <I>X</I> + <I>Y</I> = (<I>m</I><SUP>2</SUP> + <I>m</I> + 1)<SUP>2</SUP>, <I>XY</I> + <I>Z</I> = (<I>m</I><SUP>2</SUP> + <I>m</I> + 2)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16985 <I>YZ</I> + <I>Y</I> + <I>Z</I> = (2<I>m</I><SUP>2</SUP> + 3<I>m</I> + 3)<SUP>2</SUP>, <I>YZ</I> + <I>X</I> = (2<I>m</I><SUP>2</SUP> + 3<I>m</I> + 2)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16986 <I>ZX</I> + <I>Z</I> + <I>X</I> = (2<I>m</I><SUP>2</SUP> + <I>m</I> + 2)<SUP>2</SUP>, <I>ZX</I> + <I>Y</I> = (2<I>m</I><SUP>2</SUP> + <I>m</I> + 1)<SUP>2</SUP></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16987 <p>3. The difference of any two cubes is also the sum of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16988 cubes, i.e. can be transformed into the sum of two cubes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16989 (V. 16).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16990 <p>[Diophantus merely states this without proving it or show-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16991 ing how to make the transformation. The subject of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16992 transformation of sums and differences of cubes was investi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16993 gated by Vieta, Bachet and Fermat.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16994 <p>II. Of the many other propositions assumed or implied by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16995 Diophantus which are not referred to the <I>Porisms</I> we may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16996 distinguish two classes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16997 <p>1. The first class are of two sorts; some are more or less
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16998 of the nature of identical formulae, e.g. the facts that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	16999 expressions <MATH>{1/2(<I>a</I> + <I>b</I>)}<SUP>2</SUP> - <I>ab</I></MATH> and <MATH><I>a</I><SUP>2</SUP>(<I>a</I> + 1)<SUP>2</SUP> + <I>a</I><SUP>2</SUP> + (<I>a</I> + 1)<SUP>2</SUP></MATH> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17000 respectively squares, that <MATH><I>a</I>(<I>a</I><SUP>2</SUP> - <I>a</I>) + <I>a</I> + (<I>a</I><SUP>2</SUP> - <I>a</I>)</MATH> is always a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17001 cube, and that 8 times a triangular number <I>plus</I> 1 gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17002 a square, i.e. <MATH>8.(1/2)<I>x</I>(<I>x</I> + 1) + 1 = (2<I>x</I> + 1)<SUP>2</SUP></MATH>. Others are of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17003 same kind as the first two propositions quoted from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17004 <I>Porisms,</I> e.g.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17005 <p>(1) If <MATH><I>X</I> = <I>a</I><SUP>2</SUP><I>x</I> + 2<I>a, Y</I> = (<I>a</I> + 1)<SUP>2</SUP><I>x</I> + 2(<I>a</I> + 1)</MATH> or, in other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17006 words, if <MATH><I>xX</I> + 1 = (<I>ax</I> + 1)<SUP>2</SUP></MATH> and <MATH><I>xY</I> + 1 = {(<I>a</I> + 1)<I>x</I> + 1}<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17007 then <I>XY</I> + 1 is a square (IV. 20). In fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17008 <MATH><I>XY</I> + 1 = {<I>a</I>(<I>a</I> + 1)<I>x</I> + (2<I>a</I> + 1)}<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17009 <p>(2) If <MATH><I>X</I> ± <I>a</I> = <I>m</I><SUP>2</SUP>, <I>Y</I> ± <I>a</I> = (<I>m</I> + 1)<SUP>2</SUP></MATH>, and <MATH><I>Z</I> = 2(<I>X</I> + <I>Y</I>) - 1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17010 then <MATH><I>YZ</I> ± <I>a, ZX</I> ± <I>a, XY</I> ± <I>a</I></MATH> are all squares (V. 3, 4).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17011 <pb n=481><head>PORISMS AND PROPOSITIONS ASSUMED</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17012 <p>In fact <MATH><I>YZ</I> ± <I>a</I> = {(<I>m</I> + 1) (2<I>m</I> + 1) &mnplus; 2<I>a</I>}<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17013 <I>ZX</I> ± <I>a</I> = {<I>m</I>(2<I>m</I> + 1) &mnplus; 2<I>a</I>}<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17014 <I>XY</I> ± <I>a</I> = {<I>m</I>(<I>m</I> + 1) &mnplus; <I>a</I>}<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17015 <p>(3) If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17016 <MATH><I>X</I> = <I>m</I><SUP>2</SUP> + 2, <I>Y</I> = (<I>m</I> + 1)<SUP>2</SUP> + 2, <I>Z</I> = 2{<I>m</I><SUP>2</SUP> + (<I>m</I> + 1)<SUP>2</SUP> + 1} + 2</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17017 then the six expressions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17018 <MATH><I>YZ</I> - (<I>Y</I> + <I>Z</I>), <I>ZX</I> - (<I>Z</I> + <I>X</I>), <I>XY</I> - (<I>X</I> + <I>Y</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17019 <I>YZ</I> - <I>X, ZX</I> - <I>Y, XY</I> - <I>Z</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17020 are all squares (V. 6).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17021 <p>In fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17022 <MATH><I>YZ</I> - (<I>Y</I> + <I>Z</I>) = (2<I>m</I><SUP>2</SUP> + 3<I>m</I> + 3)<SUP>2</SUP>, <I>YZ</I> - <I>X</I> = (2<I>m</I><SUP>2</SUP> + 3<I>m</I> + 4)<SUP>2</SUP></MATH>, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17023 <p>2. The second class is much more important, consisting of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17024 propositions in the Theory of Numbers which we find first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17025 stated or assumed in the <I>Arithmetica.</I> It was in explana-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17026 tion or extension of these that Fermat's most famous notes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17027 were written. How far Diophantus possessed scientific proofs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17028 of the theorems which he assumes must remain largely a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17029 matter of speculation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17030 <C>(<G>a</G>) <I>Theorems on the composition of numbers as the sum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17031 of two squares.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17032 <p>(1) Any square number can be resolved into two squares in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17033 any number of ways (II. 8).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17034 <p>(2) Any number which is the sum of two squares can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17035 resolved into two other squares in any number of ways (II. 9).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17036 <p>(It is implied throughout that the squares may be fractional
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17037 as well as integral.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17038 <p>(3) If there are two whole numbers each of which is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17039 sum of two squares, the product of the numbers can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17040 resolved into the sum of two squares in two ways.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17041 <p>In fact <MATH>(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>) (<I>c</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP>) = (<I>ac</I> ± <I>bd</I>)<SUP>2</SUP> + (<I>ad</I> &mnplus; <I>bc</I>)<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17042 <p>This proposition is used in III. 19, where the problem is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17043 to find four rational right-angled triangles with the same
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17044 <pb n=482><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17045 hypotenuse. The method is this. Form two right-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17046 triangles from (<I>a, b</I>) and (<I>c, d</I>) respectively, by which Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17047 phantus means, form the right-angled triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17048 <MATH>(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>, <I>a</I><SUP>2</SUP> - <I>b</I><SUP>2</SUP>, 2<I>ab</I>) and (<I>c</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP>, <I>c</I><SUP>2</SUP> - <I>d</I><SUP>2</SUP>, 2<I>cd</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17049 <p>Multiply all the sides in each triangle by the hypotenuse of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17050 the other; we have then two rational right-angled triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17051 with the same hypotenuse <MATH>(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>) (<I>c</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17052 <p>Two others are furnished by the formula above; for we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17053 have only to ‘form two right-angled triangles’ from <MATH>(<I>ac</I> + <I>bd,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17054 ad</I> - <I>bc</I>)</MATH> and from <MATH>(<I>ac</I> - <I>bd, ad</I> + <I>bc</I>)</MATH> respectively. The method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17055 fails if certain relations hold between <I>a, b, c, d.</I> They must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17056 not be such that one number of either pair vanishes, i.e. such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17057 that <MATH><I>ad</I> = <I>bc</I> or <I>ac</I> = <I>bd</I></MATH>, or such that the numbers in either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17058 pair are equal to one another, for then the triangles are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17059 illusory.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17060 <p>In the case taken by Diophantus <MATH><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP> = 2<SUP>2</SUP> + 1<SUP>2</SUP> = 5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17061 <I>c</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP> = 3<SUP>2</SUP> + 2<SUP>2</SUP> = 13</MATH>, and the four right-angled triangles are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17062 (65, 52, 39), (65, 60, 25), (65, 63, 16) and (65, 56, 33).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17063 <p>On this proposition Fermat has a long and interesting note
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17064 as to the number of ways in which a prime number of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17065 form 4<I>n</I> + 1 and its powers can be (<I>a</I>) the hypotenuse of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17066 a rational right-angled triangle, (<I>b</I>) the sum of two squares.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17067 He also extends theorem (3) above: ‘If a prime number which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17068 is the sum of two squares be multiplied by another prime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17069 number which is also the sum of two squares, the product
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17070 will be the sum of two squares in two ways; if the first prime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17071 be multiplied by the square of the second, the product will be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17072 the sum of two squares in three ways; the product of the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17073 and the cube of the second will be the sum of two squares
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17074 in four ways, and so on <I>ad infinitum.</I>’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17075 <p>Although the hypotenuses selected by Diophantus, 5 and 13,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17076 are prime numbers of the form 4<I>n</I> + 1, it is unlikely that he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17077 was aware that prime numbers of the form 4<I>n</I> + 1 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17078 numbers arising from the multiplication of such numbers are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17079 the only classes of numbers which are always the sum of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17080 squares; this was first proved by Euler.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17081 <p>(4) More remarkable is a condition of possibility of solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17082 prefixed to V. 9, ‘To divide 1 into two parts such that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17083 <pb n=483><head>NUMBERS AS THE SUMS OF SQUARES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17084 a given number is added to either part, the result will be a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17085 square.’ The condition is in two parts. There is no doubt as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17086 to the first, ‘The given number must not be odd’ [i.e. no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17087 number of the form 4<I>n</I> + 3 or 4<I>n</I> - 1 can be the sum of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17088 squares]; the text of the second part is corrupt, but the words
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17089 actually found in the text make it quite likely that corrections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17090 made by Hankel and Tannery give the real meaning of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17091 original, ‘nor must the double of the given number <I>plus</I> 1 be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17092 measured by any prime number which is less by 1 than a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17093 multiple of 4’. This is tolerably near the true condition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17094 stated by Fermat, ‘The given number must not be odd, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17095 the double of it increased by 1, when divided by the greatest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17096 square which measures it, must not be divisible by a prime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17097 number of the form 4<I>n</I> - 1.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17098 <C>(<G>b</G>) <I>On numbers which are the sum of three squares.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17099 <p>In V. 11 the number 3<I>a</I> + 1 has to be divisible into three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17100 squares. Diophantus says that <I>a</I> ‘must not be 2 or any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17101 multiple of 8 increased by 2’. That is, ‘<I>a number of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17102 form 24n</I> + <I>7 cannot be the sum of three squares</I>’. As a matter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17103 of fact, the factor 3 in the 24 is irrelevant here, and Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17104 might have said that a number of the form 8<I>n</I> + 7 cannot be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17105 the sum of three squares. The latter condition is true, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17106 does not include <I>all</I> the numbers which cannot be the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17107 three squares. Fermat gives the conditions to which <I>a</I> must be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17108 subject, proving that 3<I>a</I> + 1 cannot be of the form 4<SUP><I>n</I></SUP>(24<I>k</I> + 7)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17109 or 4<SUP><I>n</I></SUP>(8<I>k</I> + 7), where <I>k</I> = 0 or any integer.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17110 <C>(<G>g</G>) <I>Composition of numbers as the sum of four squares.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17111 <p>There are three problems, IV. 29, 30 and V. 14, in which it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17112 is required to divide a number into four squares. Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17113 states no necessary condition in this case, as he does when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17114 it is a question of dividing a number into <I>three</I> or <I>two</I> squares.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17115 Now <I>every number is either a square or the sum of two, three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17116 or four squares</I> (a theorem enunciated by Fermat and proved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17117 by Lagrange who followed up results obtained by Euler), and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17118 this shows that any number can be divided into four squares
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17119 (admitting fractional as well as integral squares), since any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17120 square number can be divided into two other squares, integral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17121 <pb n=484><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17122 or fractional. It is possible, therefore, that Diophantus was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17123 <I>empirically</I> aware of the truth of the theorem of Fermat, but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17124 we cannot be sure of this.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17125 <C>Conspectus of the <I>Arithmetica,</I> with typical solutions.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17126 <p>There seems to be no means of conveying an idea of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17127 extent of the problems solved by Diophantus except by giving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17128 a conspectus of the whole of the six Books. Fortunately this
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17129 can be done by the help of modern notation without occupying
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17130 too many pages.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17131 <p>It will be best to classify the propositions according to their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17132 character rather than to give them in Diophantus's order. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17133 should be premised that <I>x, y, z</I> ... indicating the first, second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17134 and third ... numbers required do not mean that Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17135 indicates any of them by his unknown (<G>s</G>); he gives his un-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17136 known in each case the signification which is most convenient,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17137 his object being to express all his required numbers at once in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17138 terms of the one unknown (where possible), thereby avoiding the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17139 necessity for eliminations. Where I have occasion to specify
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17140 Diophantus's unknown, I shall as a rule call it <G>x</G>, except when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17141 a problem includes a subsidiary problem and it is convenient
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17142 to use different letters for the unknown in the original and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17143 subsidiary problems respectively, in order to mark clearly the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17144 distinction between them. When in the equations expressions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17145 are said to be = <I>u</I><SUP>2</SUP>, <I>v</I><SUP>2</SUP>, <I>w</I><SUP>2</SUP>, <I>t</I><SUP>2</SUP> ... this means simply that they
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17146 are to be made squares. Given numbers will be indicated by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17147 <I>a, b, c ... m, n</I> ... and will take the place of the numbers used
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17148 by Diophantus, which are always specific numbers.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17149 <p>Where the solutions, or particular devices employed, are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17150 specially ingenious or interesting, the methods of solution will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17151 be shortly indicated. The character of the book will be best
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17152 appreciated by means of such illustrations.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17153 <p>[The problems marked with an asterisk are probably
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17154 spurious.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17155 <C>(i) Equations of the first degree with one unknown.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17156 <p>I. 7. <MATH><I>x</I> - <I>a</I> = <I>m</I>(<I>x</I> - <I>b</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17157 <p>I. 8. <MATH><I>x</I> + <I>a</I> = <I>m</I>(<I>x</I> + <I>b</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17158 <pb n=485><head>DETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17159 <p>I. 9. <MATH><I>a</I> - <I>x</I> = <I>m</I>(<I>b</I> - <I>x</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17160 <p>I. 10. <MATH><I>x</I> + <I>b</I> = <I>m</I>(<I>a</I> - <I>x</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17161 <p>I. 11. <MATH><I>x</I> + <I>b</I> = <I>m</I>(<I>x</I> - <I>a</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17162 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17163 <BRACE><note>(<I>a</I> > <I>b</I>)</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17164 I. 39. (<I>a</I> + <I>x</I>)<I>b</I> + (<I>b</I> + <I>x</I>)<I>a</I> = 2(<I>a</I> + <I>b</I>)<I>x</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17165 or (<I>a</I> + <I>b</I>)<I>x</I> + (<I>b</I> + <I>x</I>)<I>a</I> = 2(<I>a</I> + <I>x</I>)<I>b</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17166 or (<I>a</I> + <I>b</I>)<I>x</I> + (<I>a</I> + <I>x</I>)<I>b</I> = 2(<I>b</I> + <I>x</I>)<I>a</I></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17167 <p>Diophantus states this problem in this form, ‘Given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17168 two numbers (<I>a, b</I>), to find a third number (<I>x</I>) such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17169 the numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17170 <MATH>(<I>a</I> + <I>x</I>)<I>b,</I> (<I>b</I> + <I>x</I>)<I>a,</I> (<I>a</I> + <I>b</I>)<I>x</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17171 are in arithmetical progression.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17172 <p>The result is of course different according to the order
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17173 of magnitude of the three expressions. If <I>a</I> > <I>b</I> (5 and 3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17174 are the numbers in Diophantus), then <MATH>(<I>a</I> + <I>x</I>)<I>b</I> < (<I>b</I> + <I>x</I>)<I>a</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17175 there are consequently three alternatives, since <MATH>(<I>a</I> + <I>x</I>)<I>b</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17176 must be either the least or the middle, and <MATH>(<I>b</I> + <I>x</I>)<I>a</I></MATH> either
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17177 the middle or the greatest of the three products. We may
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17178 have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17179 <MATH>(<I>a</I> + <I>x</I>)<I>b</I> < (<I>a</I> + <I>b</I>)<I>x</I> < (<I>b</I> + <I>x</I>)<I>a</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17180 or (<I>a</I> + <I>b</I>)<I>x</I> < (<I>a</I> + <I>x</I>)<I>b</I> < (<I>b</I> + <I>x</I>)<I>a</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17181 or (<I>a</I> + <I>x</I>)<I>b</I> < (<I>b</I> + <I>x</I>)<I>a</I> < (<I>a</I> + <I>b</I>)<I>x</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17182 and the corresponding equations are as set out above.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17183 <C>(ii) Determinate systems of equations of the first degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17184 <MATH>I. 1. <I>x</I> + <I>y</I> = <I>a, x</I> - <I>y</I> = <I>b.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17185 <BRACE>I. 2. <I>x</I> + <I>y</I> = <I>a, x</I> = <I>my,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17186 I. 4. <I>x</I> - <I>y</I> = <I>a, x</I> = <I>my.</I></BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17187 <p><MATH>I. 3. <I>x</I> + <I>y</I> = <I>a, x</I> = <I>my</I> + <I>b.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17188 <MATH><BRACE>I. 5. <I>x</I> + <I>y</I> = <I>a,</I> (1/<I>m</I>)<I>x</I> + (1/<I>n</I>)<I>y</I>=<I>b,</I> subject to necessary condition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17189 I. 6. <I>x</I> + <I>y</I> = <I>a,</I> (1/<I>m</I>)<I>x</I> - (1/<I>n</I>)<I>y</I>=<I>b,</I> ” ” ”</BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17190 <pb n=486><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17191 <MATH><BRACE>I. 12. <I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> = <I>y</I><SUB>1</SUB> + <I>y</I><SUB>2</SUB> = <I>a, x</I><SUB>1</SUB> = <I>my</I><SUB>2</SUB>, <I>y</I><SUB>1</SUB> = <I>nx</I><SUB>2</SUB> (<I>x</I><SUB>1</SUB> >
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17192 <I>x</I><SUB>2</SUB>, <I>y</I><SUB>1</SUB> > <I>y</I><SUB>2</SUB>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17193 <BRACE><note>(<I>x</I><SUB>1</SUB> > <I>x</I><SUB>2</SUB>, <I>y</I><SUB>1</SUB> > <I>y</I><SUB>2</SUB>, <I>z</I><SUB>1</SUB> > <I>z</I><SUB>2</SUB>).</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17194 I. 13. <I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> = <I>y</I><SUB>1</SUB> + <I>y</I><SUB>2</SUB> = <I>z</I><SUB>1</SUB> + <I>z</I><SUB>2</SUB> = <I>a</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17195 <I>x</I><SUB>1</SUB> = <I>my</I><SUB>2</SUB>, <I>y</I><SUB>1</SUB> = <I>nz</I><SUB>2</SUB>, <I>z</I><SUB>1</SUB> = <I>px</I><SUB>2</SUB></BRACE></BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17196 <p>I. 15. <MATH><I>x</I> + <I>a</I> = <I>m</I>(<I>y</I> - <I>a</I>), <I>y</I> + <I>b</I> = <I>n</I>(<I>x</I> - <I>b</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17197 <p>[Diophantus puts <MATH><I>y</I> = <G>x</G> + <I>a</I></MATH>, where <G>x</G> is his unknown.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17198 <MATH><BRACE>I. 16. <I>y</I> + <I>z</I> = <I>a, z</I> + <I>x</I> = <I>b, x</I> + <I>y</I> =<I>c.</I> [Dioph. puts <G>x</G> = <I>x</I> + <I>y</I> + <I>z</I>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17199 I. 17. <I>y</I> + <I>z</I> + <I>w</I> = <I>a, z</I> + <I>w</I> + <I>x</I> = <I>b, w</I> + <I>x</I> + <I>y</I> = <I>c, x</I> + <I>y</I> + <I>z</I> = <I>d.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17200 [<I>x</I> + <I>y</I> + <I>z</I> + <I>w</I> = <G>x</G>.]</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17201 <BRACE>I. 18. <I>y</I> + <I>z</I> - <I>x</I> = <I>a, z</I> + <I>x</I> - <I>y</I> = <I>b, x</I> + <I>y</I> - <I>z</I> = <I>c.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17202 [Dioph. puts 2<G>x</G> = <I>x</I> + <I>y</I> + <I>z</I>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17203 I. 19. <I>y</I> + <I>z</I> + <I>w</I> - <I>x</I> = <I>a, z</I> + <I>w</I> + <I>x</I> - <I>y</I> = <I>b, w</I> + <I>x</I> + <I>y</I> - <I>z</I> = <I>c,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17204 <I>x</I> + <I>y</I> + <I>z</I> - <I>w</I> = <I>d.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17205 [2<G>x</G> = <I>x</I> + <I>y</I> + <I>z</I> + <I>w.</I>]</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17206 <p>I. 20. <I>x</I> + <I>y</I> + <I>z</I> = <I>a, x</I> + <I>y</I> = <I>mz, y</I> + <I>z</I> = <I>nx.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17207 <p>I. 21. <I>x</I> = <I>y</I> + (1/<I>m</I>)<I>z, y</I> = <I>z</I> + (1/<I>n</I>)<I>x, z</I> = <I>a</I> + (1/<I>p</I>)<I>y</I> (where <I>x > y > z</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17208 with necessary condition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17209 <p>II. 18*. <I>x</I> - ((1/<I>m</I>)<I>x</I> + <I>a</I>) + ((1/<I>p</I>)<I>z</I> + <I>c</I>) = <I>y</I> - ((1/<I>n</I>)<I>y</I> + <I>b</I>) + ((1/<I>m</I>)<I>x</I> + <I>a</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17210 = <I>z</I> - ((1/<I>p</I>)<I>z</I> + <I>c</I>) + ((1/<I>n</I>)<I>y</I> + <I>b</I>), <I>x</I> + <I>y</I> + <I>z</I> = <I>a.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17211 <p>[Solution wanting.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17212 <C>(iii) Determinate systems of equations reducible to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17213 first degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17214 <p><MATH>I. 26. <I>ax</I> = <G>a</G><SUP>2</SUP>, <I>bx</I> = <G>a</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17215 <p>I. 29. <I>x</I> + <I>y</I> = <I>a, x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>b.</I> [Dioph. puts 2<G>x</G> = <I>x</I> - <I>y.</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17216 <BRACE>I. 31. <I>x</I> = <I>my, x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> + <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17217 I. 32. <I>x</I> = <I>my, x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> - <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17218 I. 33. <I>x</I> = <I>my, x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> + <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17219 I. 34. <I>x</I> = <I>my, x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> - <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17220 I. 34. Cor. 1. <I>x</I> = <I>my, xy</I> = <I>n</I>(<I>x</I> + <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17221 Cor. 2. <I>x</I> = <I>my, xy</I> = <I>n</I>(<I>x</I> - <I>y</I>)</BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17222 <pb n=487><head>DETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17223 <MATH><BRACE>I. 35. <I>x</I> = <I>my, y</I><SUP>2</SUP> = <I>nx.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17224 I. 36. <I>x</I> = <I>my, y</I><SUP>2</SUP> = <I>ny</I></BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17225 <p>I. 37. <I>x</I> = <I>my, y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> + <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17226 <p>I. 38. <I>x</I> = <I>my, y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> - <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17227 <p>I. 38. Cor. <I>x</I> = <I>my, x</I><SUP>2</SUP> = <I>ny.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17228 ” <I>x</I> = <I>my, x</I><SUP>2</SUP> = <I>nx.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17229 ” <I>x</I> = <I>my, x</I><SUP>2</SUP> = <I>n</I>(<I>x</I> + <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17230 ” <I>x</I> = <I>my, x</I><SUP>2</SUP> = <I>n</I>(<I>x</I> - <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17231 <p>II. 6*. <I>x</I> - <I>y</I> = <I>a, x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>x</I> - <I>y</I> + <I>b.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17232 <p>IV. 36. <I>yz</I> = <I>m</I>(<I>y</I> + <I>z</I>), <I>zx</I> = <I>n</I>(<I>z</I> + <I>x</I>), <I>xy</I> = <I>p</I>(<I>x</I> + <I>y</I>).</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17233 <p>[Solved by means of Lemma: see under (vi) Inde-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17234 terminate equations of the first degree.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17235 <C>(iv) Determinate systems reducible to equations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17236 second degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17237 <MATH><BRACE>I. 27. <I>x</I> + <I>y</I> = <I>a, xy</I> = <I>b.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17238 [Dioph. states the necessary condition, namely that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17239 (1/4)<I>a</I><SUP>2</SUP> - <I>b</I> must be a square, with the words <G>e)/sti de\ tou=to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17240 plasmatiko/n</G>, which no doubt means ‘this is of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17241 nature of a formula (easily obtained)’. He puts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17242 <I>x</I> - <I>y</I> = 2<G>x</G>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17243 I. 30. <I>x</I> - <I>y</I> = <I>a, xy</I> = <I>b</I></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17244 <p>[Necessary condition (with the same words) <MATH>4<I>b</I> + <I>a</I><SUP>2</SUP> =</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17245 a square. <MATH><I>x</I> + <I>y</I> is put = 2<G>x</G></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17246 <p>I. 28. <MATH><I>x</I> + <I>y</I> = <I>a, x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>b</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17247 <p>[Necessary condition <MATH>2<I>b</I> - <I>a</I><SUP>2</SUP> = a square. <I>x</I> - <I>y</I> = 2<G>x</G></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17248 <MATH><BRACE>IV. 1. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> = <I>a, x</I> + <I>y</I> = <I>b.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17249 [Dioph. puts <I>x</I> - <I>y</I> = 2<G>x</G>, whence <I>x</I>=(1/2)<I>b</I> + <G>x</G>, <I>y</I> = (1/2)<I>b</I> - <G>x</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17250 The numbers <I>a, b</I> are so chosen that (<I>a</I> - (1/4)<I>b</I><SUP>3</SUP>)/3<I>b</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17251 a square.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17252 IV. 2. <I>a</I><SUP>3</SUP> - <I>y</I><SUP>3</SUP> = <I>a, x</I> - <I>y</I> = <I>b.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17253 [<I>x</I> + <I>y</I> = 2<G>x</G>.]</BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17254 <pb n=488><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17255 <p>IV. 15. <MATH>(<I>y</I> + <I>z</I>)<I>x</I> = <I>a,</I> (<I>z</I> + <I>x</I>)<I>y</I> = <I>b,</I> (<I>x</I> + <I>y</I>)<I>z</I> = <I>c</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17256 <p>[Dioph. takes the third number <I>z</I> as his unknown;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17257 thus <MATH><I>x</I> + <I>y</I> = <I>c/z.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17258 <p>Assume <I>x</I> = <I>p/z, y</I> = <I>q/z.</I> Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17259 <I>pq/z</I><SUP>2</SUP> + <I>p</I> = <I>a,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17260 <I>pq/z</I><SUP>2</SUP> + <I>q</I> = <I>b.</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17261 <p>These equations are inconsistent unless <MATH><I>p</I> - <I>q</I> = <I>a</I> - <I>b</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17262 We have therefore to determine <I>p, q</I> by dividing <I>c</I> into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17263 two parts such that their difference <MATH>= <I>a</I> - <I>b</I></MATH> (cf. I. 1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17264 <p>A very interesting use of the ‘false hypothesis’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17265 (Diophantus first takes two <I>arbitrary</I> numbers for <I>p, q</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17266 such that <MATH><I>p</I> + <I>q</I> = <I>c</I></MATH>, and finds that the values taken have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17267 to be corrected).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17268 <p>The final equation being <MATH><I>pq/z</I><SUP>2</SUP> + <I>p</I> = <I>a</I></MATH>, where <I>p, q</I> are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17269 determined in the way described, <MATH><I>z</I><SUP>2</SUP> = <I>pq</I>/(<I>a</I> - <I>p</I>)</MATH> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17270 <MATH><I>pq</I>/(<I>b</I> - <I>q</I>)</MATH>, and the numbers <I>a, b, c</I> have to be such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17271 either of these expressions gives a square.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17272 <p>IV. 34. <MATH><I>yz</I> + (<I>y</I> + <I>z</I>) = <I>a</I><SUP>2</SUP> - 1, <I>zx</I> + (<I>z</I> + <I>x</I>) = <I>b</I><SUP>2</SUP> - 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17273 <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>c</I><SUP>2</SUP> - 1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17274 <p>[Dioph. states as the necessary condition for a rational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17275 solution that each of the three constants to which the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17276 three expressions are to be equal must be some square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17277 diminished by 1. The true condition is seen in our
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17278 notation by transforming the equations <MATH><I>yz</I> + (<I>y</I> + <I>z</I>) = <G>a</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17279 <I>zx</I> + (<I>z</I> + <I>x</I>) = <G>b</G>, <I>xy</I> + (<I>x</I> + <I>y</I>) = <G>g</G></MATH> into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17280 <MATH>(<I>y</I> + 1) (<I>z</I> + 1) = <G>a</G> + 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17281 (<I>z</I> + 1) (<I>x</I> + 1) = <G>b</G> + 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17282 (<I>x</I> + 1) (<I>y</I> + 1) = <G>g</G> + 1,</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17283 <pb n=489><head>DETERMINATE EQUATIONS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17284 whence <MATH><I>x</I> + 1 = √{((<G>b</G> + 1) (<G>g</G> + 1))/(<G>a</G> + 1)}</MATH> &c.;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17285 and it is only necessary that <MATH>(<G>a</G> + 1) (<G>b</G> + 1) (<G>g</G> + 1)</MATH> should
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17286 be a square, not that <I>each</I> of the expressions <MATH><G>a</G> + 1, <G>b</G> + 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17287 <G>g</G> + 1</MATH> should be a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17288 <p>Dioph. finds in a Lemma (see under (vi) below) a solu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17289 tion <G>e)n a)ori/stw\|</G> (indeterminately) of <MATH><I>xy</I> + (<I>x</I> + <I>y</I>) = <I>k</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17290 which practically means finding <I>y</I> in terms of <I>x.</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17291 <p>IV. 35. <MATH><I>yz</I> - (<I>y</I> + <I>z</I>) = <I>a</I><SUP>2</SUP> - 1, <I>zx</I> - (<I>z</I> + <I>x</I>) = <I>b</I><SUP>2</SUP> - 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17292 <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>c</I><SUP>2</SUP> - 1</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17293 <p>[The remarks on the last proposition apply <I>mutatis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17294 mutandis.</I> The lemma in this case is the indeterminate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17295 solution of <MATH><I>xy</I> - (<I>x</I> + <I>y</I>) = <I>k</I></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17296 <p>IV. 37. <MATH><I>yz</I> = <I>a</I>(<I>x</I> + <I>y</I> + <I>z</I>), <I>zx</I> = <I>b</I>(<I>x</I> + <I>y</I> + <I>z</I>), <I>xy</I> = <I>c</I>(<I>x</I> + <I>y</I> + <I>z</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17297 <p>[Another interesting case of ‘false hypothesis’. Dioph.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17298 first gives <MATH><I>x</I> + <I>y</I> + <I>z</I></MATH> an <I>arbitrary</I> value, then finds that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17299 the result is not rational, and proceeds to solve the new
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17300 problem of finding a value of <MATH><I>x</I> + <I>y</I> + <I>z</I></MATH> to take the place of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17301 the first value.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17302 <p>If <MATH><I>w</I> = <I>x</I> + <I>y</I> + <I>z</I></MATH>, we have <MATH><I>x</I> = <I>cw/y, z</I> = <I>aw/y</I></MATH>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17303 <MATH><I>zx</I> = <I>acw</I><SUP>2</SUP>/<I>y</I><SUP>2</SUP> = <I>bw</I></MATH> by hypothesis; therefore <MATH><I>y</I><SUP>2</SUP> = (<I>ac/b</I>)<I>w</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17304 <p>For a rational solution this last expression must be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17305 a square. Suppose, therefore, that <MATH><I>w</I> = (<I>ac/b</I>)<G>x</G><SUP>2</SUP></MATH>, and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17306 <MATH><I>x</I> + <I>y</I> + <I>z</I> = (<I>ac/b</I>)<G>x</G><SUP>2</SUP>, <I>y</I> = (<I>ac/b</I>)<G>x</G>, <I>z</I> = <I>a</I><G>x</G>, <I>x</I> = <I>c</I><G>x</G></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17307 <p>Eliminating <I>x, y, z,</I> we obtain <MATH><G>x</G> = (<I>bc</I> + <I>ca</I> + <I>ab</I>)/<I>ac</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17308 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17309 <MATH><I>x</I> = (<I>bc</I> + <I>ca</I> + <I>ab</I>)/<I>a, y</I> = (<I>bc</I> + <I>ca</I> + <I>ab</I>)/<I>b,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17310 <I>z</I> = (<I>bc</I> + <I>ca</I> + <I>ab</I>)/<I>c</I></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17311 Lemma to V. 8. <MATH><I>yz</I> = <I>a</I><SUP>2</SUP>, <I>zx</I> = <I>b</I><SUP>2</SUP>, <I>xy</I> = <I>c</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17312 <pb n=490><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17313 <C>(v) Systems of equations apparently indeterminate but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17314 really reduced, by arbitrary assumptions, to deter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17315 minate equations of the first degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17316 <p><MATH>I. 14. <I>xy</I> = <I>m</I>(<I>x</I> + <I>y</I>). [Value of <I>y</I> arbitrarily assumed.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17317 <BRACE><note>[<I>x</I> assumed = 2<I>y.</I>]</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17318 II. 3*. <I>xy</I> = <I>m</I>(<I>x</I> + <I>y</I>), and <I>xy</I> = <I>m</I>(<I>x</I> - <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17319 II. 1*. (cf. I. 31). <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> + <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17320 II. 2*. (cf. I. 34). <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> - <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17321 II. 4*. (cf. I. 32). <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> - <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17322 II. 5*. (cf. I. 33). <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> + <I>y</I>).</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17323 <p>II. 7*. <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> - <I>y</I>) + <I>a.</I> [Dioph. assumes <I>x</I> - <I>y</I> = 2.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17324 <BRACE>I. 22. <I>x</I> - (1/<I>m</I>)<I>x</I> + (1/<I>p</I>)<I>z</I> = <I>y</I> - (1/<I>n</I>)<I>y</I> + (1/<I>m</I>)<I>x</I> = <I>z</I> - (1/<I>p</I>)<I>z</I> + (1/<I>n</I>)<I>y.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17325 [Value of <I>y</I> assumed.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17326 I. 23. <I>x</I> - (1/<I>m</I>)<I>x</I> + (1/<I>q</I>)<I>w</I> = <I>y</I> - (1/<I>n</I>)<I>y</I> + (1/<I>m</I>)<I>x</I> = <I>z</I> - (1/<I>p</I>)<I>z</I> + (1/<I>n</I>)<I>y</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17327 = <I>w</I> - (1/<I>q</I>)<I>w</I> + (1/<I>p</I>)<I>z.</I> [Value of <I>y</I> assumed.]</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17328 <BRACE>I. 24. <I>x</I> + (1/<I>m</I>)(<I>y</I> + <I>z</I>) = <I>y</I> + (1/<I>n</I>)(<I>z</I> + <I>x</I>) = <I>z</I> + (1/<I>p</I>)(<I>x</I> + <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17329 [Value of <I>y</I> + <I>z</I> assumed.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17330 I. 25. <I>x</I> + (1/<I>m</I>)(<I>y</I> + <I>z</I> + <I>w</I>) = <I>y</I> + (1/<I>n</I>)(<I>z</I> + <I>w</I> + <I>x</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17331 = <I>z</I> + (1/<I>p</I>)(<I>w</I> + <I>x</I> + <I>y</I>) = <I>w</I> + (1/<I>q</I>)(<I>x</I> + <I>y</I> + <I>z</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17332 [Value of <I>y</I> + <I>z</I> + <I>w</I> assumed.]</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17333 <p>II. 17*. (cf. I. 22). <I>x</I> - ((1/<I>m</I>)<I>x</I> + <I>a</I>) + ((1/<I>p</I>)<I>z</I> + <I>c</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17334 = <I>y</I> - ((1/<I>n</I>)<I>y</I> + <I>b</I>) + ((1/<I>m</I>)<I>x</I> + <I>a</I>) = <I>z</I> - ((1/<I>p</I>)<I>z</I> + <I>c</I>) + ((1/<I>n</I>)<I>y</I> + <I>b</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17335 <p>[Ratio of <I>x</I> to <I>y</I> assumed.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17336 <pb n=491><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17337 <p>IV. 33. <MATH><I>x</I> + (1/<I>z</I>)<I>y</I> = <I>m</I>(<I>y</I> - (1/<I>z</I>)<I>y</I>), <I>y</I> + (1/<I>z</I>)<I>x</I> = <I>n</I>(<I>x</I> - (1/<I>z</I>)<I>x</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17338 <p>[Dioph. assumes (1/<I>z</I>)<I>y</I> = 1.]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17339 <C>(vi) Indeterminate equations of the first degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17340 <MATH><BRACE><note>[Solutions <G>e)n a)ori/stw\|</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17341 <I>y</I> practically found
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17342 in terms of <I>x.</I>]</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17343 Lemma to IV. 34. <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17344 ” ” IV. 35. <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17345 ” ” IV. 36. <I>xy</I> = <I>m</I>(<I>x</I> + <I>y</I>).</BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17346 <C>(vii) Indeterminate analysis of the second degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17347 <MATH><BRACE>II. 8. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>a</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17348 [<I>y</I><SUP>2</SUP> = <I>a</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP> must be a square = (<I>mx</I> - <I>a</I>)<SUP>2</SUP>, say.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17349 II. 9. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>. [Put <I>x</I> = <G>x</G> + <I>a, y</I> = <I>m</I><G>x</G> - <I>b.</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17350 II. 10. <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17351 [Put <I>x</I> = <I>y</I> + <I>m,</I> choosing <I>m</I> such that <I>m</I><SUP>2</SUP> < <I>a.</I>]</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17352 <BRACE>II. 11. <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>b</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17353 II. 12. <I>a</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>b</I> - <I>x</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17354 II. 13. <I>x</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>x</I> - <I>b</I> = <I>v</I><SUP>2</SUP></BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17355 <p>[Dioph. solves II. 11 and 13, (1) by means of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17356 ‘double equation’ (see p. 469 above), (2) without a double
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17357 equation by putting <MATH><I>x</I> = <G>x</G><SUP>2</SUP> ± <I>a</I></MATH> and equating <MATH>(<G>x</G><SUP>2</SUP> ± <I>a</I>) ± <I>b</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17358 to <MATH>(<G>x</G> - <I>m</I>)<SUP>2</SUP></MATH>. In II. 12 he puts <MATH><I>x</I> = <I>a</I> - <G>x</G><SUP>2</SUP></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17359 <p>II. 14 = III. 21. <MATH><I>x</I> + <I>y</I> = <I>a, x</I> + <I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>z</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17360 <p>[Diophantus takes <I>z</I> as the unknown, and puts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17361 <MATH><I>u</I><SUP>2</SUP> = (<I>z</I> + <I>m</I>)<SUP>2</SUP>, <I>v</I><SUP>2</SUP> = (<I>z</I> + <I>n</I>)<SUP>2</SUP></MATH>. Therefore <MATH><I>x</I> = 2<I>mz</I> + <I>m</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17362 <MATH><I>y</I> = 2<I>nz</I> + <I>n</I><SUP>2</SUP></MATH>, and <I>z</I> is found, by substitution in the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17363 equation, to be <MATH>(<I>a</I> - (<I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP>))/(2(<I>m</I> + <I>n</I>))</MATH>. In order that the solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17364 may be rational, <I>m, n</I> must satisfy a certain condition.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17365 Dioph. takes them such that <MATH><I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> < <I>a</I></MATH>, but it is suffi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17366 cient, if <I>m</I> > <I>n,</I> that <I>a</I> + <I>mn</I> should be > <I>n</I><SUP>2</SUP>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17367 <p>II. 15 = III. 20. <MATH><I>x</I> + <I>y</I> = <I>a, z</I><SUP>2</SUP> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> - <I>y</I> = <I>v</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17368 <p>[The solution is similar, and a similar remark applies
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17369 to Diophantus's implied condition.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17370 <pb n=492><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17371 <p><MATH>II. 16. <I>x</I> = <I>my, a</I><SUP>2</SUP> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>a</I><SUP>2</SUP> + <I>y</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17372 <p>II. 19. <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>y</I><SUP>2</SUP> - <I>z</I><SUP>2</SUP>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17373 <BRACE>II. 20. <I>x</I><SUP>2</SUP> + <I>y</I> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + <I>x</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17374 [Assume <I>y</I> = 2<I>mx</I> + <I>m</I><SUP>2</SUP>, and one condition is satisfied.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17375 II. 21. <I>x</I><SUP>2</SUP> - <I>y</I> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - <I>x</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17376 [Assume <I>x</I> = <G>x</G> + <I>m, y</I> = 2<I>m</I><G>x</G> + <I>m</I><SUP>2</SUP>, and one condition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17377 is satisfied.]</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17378 <BRACE>II. 22. <I>x</I><SUP>2</SUP> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + (<I>x</I> + <I>y</I>) = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17379 [Put <I>x</I> + <I>y</I> = 2<I>mx</I> + <I>m</I><SUP>2</SUP>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17380 II. 23. <I>x</I><SUP>2</SUP> - (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - (<I>x</I> + <I>y</I>) = <I>v</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17381 <BRACE>II. 24. (<I>x</I> + <I>y</I>)<SUP>2</SUP> + <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I>)<SUP>2</SUP> + <I>y</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17382 [Assume <I>x</I> = (<I>m</I><SUP>2</SUP> - 1)<G>x</G><SUP>2</SUP>, <I>y</I> = (<I>n</I><SUP>2</SUP> - 1)<G>x</G><SUP>2</SUP>, <I>x</I> + <I>y</I> = <G>x</G>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17383 II. 25. (<I>x</I> + <I>y</I>)<SUP>2</SUP> - <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I>)<SUP>2</SUP> - <I>y</I> = <I>v</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17384 <BRACE>II. 26. <I>xy</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>xy</I> + <I>y</I> = <I>v</I><SUP>2</SUP>, <I>u</I> + <I>v</I> = <I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17385 [Put <I>y</I> = <I>m</I><SUP>2</SUP><I>x</I> - 1.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17386 II. 27. <I>xy</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>xy</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, <I>u</I> + <I>v</I> = <I>a.</I></BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17387 <BRACE>II. 28. <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17388 II. 29. <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17389 <p>II. 30. <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>v</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17390 <p>[Since <MATH><I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> ± 2<I>mn</I></MATH> is a square, assume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17391 <MATH><I>xy</I> = (<I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP>)<G>x</G><SUP>2</SUP> and <I>x</I> + <I>y</I> = 2<I>mn</I><G>x</G><SUP>2</SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17392 put <I>x</I> = <I>p<G>x</G>, y</I> = <I>q</I><G>x</G>, where <I>pq</I> = <I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP>; then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17393 (<I>p</I> + <I>q</I>)<G>x</G> = 2<I>mn</I><G>x</G><SUP>2</SUP></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17394 <p>II. 31. <MATH><I>xy</I> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17395 <p>[Suppose <MATH><I>w</I><SUP>2</SUP> = 2.2<I>m.m</I></MATH>, which is a square, and use
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17396 formula <MATH>(2<I>m</I>)<SUP>2</SUP> + <I>m</I><SUP>2</SUP> ± 2.2<I>m.m</I> = a</MATH> square.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17397 <MATH><BRACE>II. 32. <I>y</I><SUP>2</SUP> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> + <I>y</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17398 [<I>y</I> = <G>x</G>, <I>z</I> = (2<I>a</I><G>x</G> + <I>a</I><SUP>2</SUP>), <I>x</I> = 2<I>b</I>(2<I>a</I><G>x</G> + <I>a</I><SUP>2</SUP>) + <I>b</I><SUP>2</SUP>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17399 II. 33. <I>y</I><SUP>2</SUP> - <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> - <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> - <I>y</I> = <I>w</I><SUP>2</SUP>.</BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17400 <pb n=493><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17401 <MATH><BRACE>II. 34. <I>x</I><SUP>2</SUP> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17402 <I>z</I><SUP>2</SUP> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>w</I><SUP>2</SUP>.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17403 [Since {(1/2)(<I>m</I> - <I>n</I>)}<SUP>2</SUP> + <I>mn</I> is a square, take any number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17404 separable into two factors (<I>m, n</I>) in three ways. This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17405 gives three values, say, <I>p, q, r</I> for (1/2)(<I>m</I> - <I>n</I>). Put
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17406 <I>x</I> = <I>p<G>x</G>, y</I> = <I>q<G>x</G>, z</I> = <I>r</I><G>x</G>, and <I>x</I> + <I>y</I> + <I>z</I> = <I>mn</I><G>x</G><SUP>2</SUP>; therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17407 (<I>p</I> + <I>q</I> + <I>r</I>)<G>x</G> = <I>mn</I><G>x</G><SUP>2</SUP>, and <G>x</G> is found.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17408 II. 35. <MATH><I>x</I><SUP>2</SUP> - (<I>x</I> + <I>y</I> + <I>z</I>) = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - (<I>x</I> + <I>y</I> + <I>z</I>) = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17409 <I>z</I><SUP>2</SUP> - (<I>x</I> + <I>y</I> + <I>z</I>) = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17410 [Use the formula {1/2(<I>m</I> + <I>n</I>)}<SUP>2</SUP> - <I>mn</I> = a square and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17411 proceed similarly.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17412 III. 1*. (<I>x</I> + <I>y</I> + <I>z</I>) - <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>) - <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17413 (<I>x</I> + <I>y</I> + <I>z</I>) - <I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17414 <BRACE>III. 2*. (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> + <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> + <I>y</I> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17415 (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> + <I>z</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17416 III. 3*. (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> - <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> - <I>y</I> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17417 (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> - <I>z</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17418 III. 4*. <I>x</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>y</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17419 <I>z</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> = <I>w</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17420 <p>III. 5. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>y</I> + <I>z</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> - <I>y</I> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17421 <I>x</I> + <I>y</I> - <I>z</I> = <I>w</I><SUP>2</SUP>.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17422 <p>[The first solution of this problem assumes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17423 <MATH><I>t</I><SUP>2</SUP> = <I>x</I> + <I>y</I> + <I>z</I> = (<G>x</G> + 1)<SUP>2</SUP>, <I>w</I><SUP>2</SUP> = 1, <I>u</I><SUP>2</SUP> = <G>x</G><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17424 whence <I>x, y, z</I> are found in terms of <G>x</G>, and <MATH><I>z</I> + <I>x</I> - <I>y</I></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17425 is then made a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17426 <p>The alternative solution, however, is much more ele-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17427 gant, and can be generalized thus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17428 <p>We have to find <I>x, y, z</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17429 <MATH><BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17430 - <I>x</I> + <I>y</I> + <I>z</I> = a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17431 <I>x</I> - <I>y</I> + <I>z</I> = a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17432 <I>x</I> + <I>y</I> - <I>z</I> = a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17433 <I>x</I> + <I>y</I> + <I>z</I> = a square</BRACE></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17434 <p>Equate the first three expressions to <I>a</I><SUP>2</SUP>, <I>b</I><SUP>2</SUP>, <I>c</I><SUP>2</SUP>, being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17435 squares such that their sum is also a square = <I>k</I><SUP>2</SUP>, say.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17436 <pb n=494><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17437 <p>Then, since the sum of the first three expressions is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17438 itself equal to <MATH><I>x</I> + <I>y</I> + <I>z</I></MATH>, we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17439 <MATH><I>x</I> = (1/2)(<I>b</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>), <I>y</I> = (1/2)(<I>c</I><SUP>2</SUP> + <I>a</I><SUP>2</SUP>), <I>z</I> = (1/2)(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>).]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17440 <p>III. 6. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17441 <p>III. 7. <I>x</I> - <I>y</I> = <I>y</I> - <I>z, y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17442 <BRACE>III. 8. <I>x</I> + <I>y</I> + <I>z</I> + <I>a</I> = <I>t</I><SUP>2</SUP>, <I>y</I> + <I>z</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> + <I>a</I> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17443 <I>x</I> + <I>y</I> + <I>a</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17444 III. 9. <I>x</I> + <I>y</I> + <I>z</I> - <I>a</I> = <I>t</I><SUP>2</SUP>, <I>y</I> + <I>z</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> - <I>a</I> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17445 <I>x</I> + <I>y</I> - <I>a</I> = <I>w</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17446 <p>III. 10. <I>yz</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>a</I> = <I>w</I><SUP>2</SUP>.</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17447 <p>[Suppose <MATH><I>yz</I> + <I>a</I> = <I>m</I><SUP>2</SUP></MATH>, and let <MATH><I>y</I> = (<I>m</I><SUP>2</SUP> - <I>a</I>)<G>x</G>, <I>z</I> = 1/<G>x</G></MATH>:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17448 also let <MATH><I>zx</I> + <I>a</I> = <I>n</I><SUP>2</SUP></MATH>; therefore <MATH><I>x</I> = (<I>n</I><SUP>2</SUP> - <I>a</I>)<G>x</G></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17449 We have therefore to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17450 <MATH>(<I>m</I><SUP>2</SUP> - <I>a</I>) (<I>n</I><SUP>2</SUP> - <I>a</I>) <G>x</G><SUP>2</SUP> + <I>a</I></MATH> a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17451 <p>Diophantus takes <MATH><I>m</I><SUP>2</SUP> = 25, <I>a</I> = 12, <I>n</I><SUP>2</SUP> = 16</MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17452 arrives at <MATH>52<G>x</G><SUP>2</SUP> + 12</MATH>, which is to be made a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17453 Although <MATH>52.1<SUP>2</SUP> + 12</MATH> is a square, and it follows that any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17454 number of other solutions giving a square are possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17455 by substituting <MATH>1 + <G>h</G></MATH> for <G>x</G> in the expression, and so on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17456 Diophantus says that the equation could easily be solved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17457 if 52 was a square, and proceeds to solve the problem of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17458 finding two squares such that each increased by 12 will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17459 give a square, in which case their product also will be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17460 a square. In other words, we have to find <I>m</I><SUP>2</SUP> and <I>n</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17461 such that <MATH><I>m</I><SUP>2</SUP> - <I>a, n</I><SUP>2</SUP> - <I>a</I></MATH> are both squares, which, as he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17462 says, is easy. We have to find two pairs of squares
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17463 differing by <I>a.</I> If
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17464 <MATH><I>a</I> = <I>pq</I> = <I>p′q′,</I> {(1/2)(<I>p</I> - <I>q</I>)}<SUP>2</SUP> + <I>a</I> = {(1/2)(<I>p</I> + <I>q</I>)}<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17465 and {(1/2)(<I>p′</I> - <I>q′</I>)}<SUP>2</SUP> + <I>a</I> = {(1/2)(<I>p′</I> + <I>q′</I>)}<SUP>2</SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17466 let, then, <I>m</I><SUP>2</SUP> = {(1/2)(<I>p</I> + <I>q</I>)}<SUP>2</SUP>, <I>n</I><SUP>2</SUP> = {(1/2)(<I>p′</I> + <I>q′</I>)}<SUP>2</SUP>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17467 <p>III. 11. <I>yz</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> - <I>a</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>a</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17468 <p>[The solution is like that of III. 10 <I>mutatis mutandis.</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17469 <BRACE>III. 12. <I>yz</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> + <I>y</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>z</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17470 III. 13. <I>yz</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>z</I> = <I>w</I><SUP>2</SUP>.</BRACE></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17471 <pb n=495><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17472 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17473 <p>III. 14. <I>yz</I> + <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>zx</I> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17474 <p>III. 15. <I>yz</I> + (<I>y</I> + <I>z</I>) = <I>u</I><SUP>2</SUP>, <I>zx</I> + (<I>z</I> + <I>x</I>) = <I>v</I><SUP>2</SUP>, <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17475 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17476 <p>[<I>Lemma.</I> If <I>a,</I> <I>a</I> + 1 be two consecutive numbers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17477 <MATH><I>a</I><SUP>2</SUP>(<I>a</I> + 1)<SUP>2</SUP> + <I>a</I><SUP>2</SUP> + (<I>a</I> + 1)<SUP>2</SUP></MATH> is a square. Let
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17478 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17479 <I>y</I> = <I>m</I><SUP>2</SUP>, <I>z</I> = (<I>m</I> + 1)<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17480 <BRACE>Therefore (<I>m</I><SUP>2</SUP> + 2<I>m</I> + 2)<I>x</I> + (<I>m</I> + 1)<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17481 and (<I>m</I><SUP>2</SUP> + 1)<I>x</I> + <I>m</I><SUP>2</SUP></BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17482 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17483 have to be made squares. This is solved as a double-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17484 equation; in Diophantus's problem <I>m</I> = 2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17485 <p><I>Second solution.</I> Let <I>x</I> be the first number, <I>m</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17486 second; then (<I>m</I> + 1) <I>x</I> + <I>m</I> is a square = <I>n</I><SUP>2</SUP>, say; there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17487 fore <MATH><I>x</I> = (<I>n</I><SUP>2</SUP> - <I>m</I>)/(<I>m</I> + 1)</MATH>, while <I>y</I> = <I>m.</I> We have then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17488 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17489 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17490 (<I>m</I> + 1)<I>z</I> + <I>m</I> = a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17491 and ((<I>n</I><SUP>2</SUP> + 1)/(<I>m</I> + 1))<I>z</I> + (<I>n</I><SUP>2</SUP> - <I>m</I>)/(<I>m</I> + 1) = a square</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17492 </MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17493 <p>Diophantus has <I>m</I> = 3, <I>n</I> = 5, so that the expressions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17494 to be made squares are with him
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17495 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17496 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17497 4<I>z</I> + 3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17498 (6 1/2)<I>z</I> + 5 1/2</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17499 </MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17500 This is not possible because, of the corresponding coeffi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17501 cients, neither pair are in the ratio of squares. In order to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17502 substitute, for 6 1/2, 4, coefficients which are in the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17503 of a square to a square he then finds two numbers, say,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17504 <I>p, q</I> to replace 5 1/2, 3 such that <MATH><I>pq</I> + <I>p</I> + <I>q</I></MATH> = a square, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17505 <MATH>(<I>p</I> + 1)/(<I>q</I> + 1)</MATH> = a square. He assumes <G>x</G> and 4<G>x</G> + 3,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17506 which satisfies the second condition, and then solves for <G>x</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17507 which must satisfy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17508 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17509 4<G>x</G><SUP>2</SUP> + 8<G>x</G> + 3 = a square = (2<G>x</G> - 3)<SUP>2</SUP>, say,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17510 which gives <G>x</G> = 3/10, 4<G>x</G> + 3 = 4 1/5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17511 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17512 <p>He then solves, for <I>z,</I> the third number, the double-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17513 equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17514 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17515 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17516 (5 1/5)<I>z</I> + 4 1/5 = square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17517 (13/10)<I>z</I> + 3/10 = square</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17518 </MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17519 <pb n=496><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17520 after multiplying by 25 and 100 respectively, making
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17521 expressions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17522 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17523 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17524 130<I>x</I> + 105
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17525 130<I>x</I> + 30</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17526 </MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17527 <p>In the above equations we should only have to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17528 <I>n</I><SUP>2</SUP> + 1 a square, and then multiply the first by <I>n</I><SUP>2</SUP> + 1 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17529 the second by (<I>m</I> + 1)<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17530 <p>Diophantus, with his notation, was hardly in a position
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17531 to solve, as we should, by writing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17532 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17533 (<I>y</I> + 1) (<I>z</I> + 1) = <I>a</I><SUP>2</SUP> + 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17534 (<I>z</I> + 1) (<I>x</I> + 1) = <I>b</I><SUP>2</SUP> + 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17535 (<I>x</I> + 1) (<I>y</I> + 1) = <I>c</I><SUP>2</SUP> + 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17536 which gives <I>x</I> + 1 = √{(<I>b</I><SUP>2</SUP> + 1) (<I>c</I><SUP>2</SUP> + 1)/(<I>a</I><SUP>2</SUP> + 1)}, &c.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17537 <p>III. 16. <I>yz</I> - (<I>y</I> + <I>z</I>) = <I>u</I><SUP>2</SUP>, <I>zx</I> - (<I>z</I> + <I>x</I>) = <I>v</I><SUP>2</SUP>, <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17538 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17539 <p>[The method is the same <I>mutatis mutandis</I> as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17540 second of the above solutions.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17541 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17542 <BRACE>III. 17. <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>xy</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>y</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17543 III. 18. <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>xy</I> - <I>x</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>y</I> = <I>w</I><SUP>2</SUP></BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17544 <BRACE><note>III. 19. (<I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB>)<SUP>2</SUP> ± <I>x</I><SUB>1</SUB> =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17545 <I>t</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17546 <I>t</I>′<SUP>2</SUP></BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17547 <BRACE><note>(<I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB>)<SUP>2</SUP> ± <I>x</I><SUB>2</SUB> =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17548 <I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17549 <I>u</I>′<SUP>2</SUP></BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17550 <BRACE><note>(<I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB>)<SUP>2</SUP> ± <I>x</I><SUB>3</SUB> =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17551 <I>v</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17552 <I>v</I>′<SUP>2</SUP></BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17553 <BRACE><note>(<I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB>)<SUP>2</SUP> ± <I>x</I><SUB>4</SUB> =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17554 <I>w</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17555 <I>w</I>′<SUP>2</SUP></BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17556 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17557 <p>[Diophantus finds, in the way we have seen (p. 482),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17558 four different rational right-angled triangles with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17559 same hypotenuse, namely (65, 52, 39), (65, 60, 25), (65,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17560 56, 33), (65, 63, 16), or, what is the same thing, a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17561 which is divisible into two squares in four different ways;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17562 this will solve the problem, since, if <I>h, p, b</I> be the three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17563 sides of a right-angled triangle, <I>h</I><SUP>2</SUP> ± 2 <I>pb</I> are both squares.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17564 <pb n=497><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17565 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17566 <p>Put therefore <I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB> = 65<G>x</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17567 and <I>x</I><SUB>1</SUB> = 2.39.52<G>x</G><SUP>2</SUP>, <I>x</I><SUB>2</SUB> = 2.25.60<G>x</G><SUP>2</SUP>, <I>x</I><SUB>3</SUB> = 2.33.56<G>x</G><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17568 <I>x</I><SUB>4</SUB> = 2.16.63<G>x</G><SUP>2</SUP>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17569 this gives 12768<G>x</G><SUP>2</SUP> = 65<G>x</G>, and <G>x</G> = 65/12768.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17570 <BRACE>IV. 4. <I>x</I><SUP>2</SUP> + <I>y</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>u.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17571 IV. 5. <I>x</I><SUP>2</SUP> + <I>y</I> = <I>u,</I> <I>x</I> + <I>y</I> = <I>u</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17572 <p>IV. 13. <I>x</I> + 1 = <I>t</I><SUP>2</SUP>, <I>y</I> + 1 = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>y</I> + 1 = <I>v</I><SUP>2</SUP>, <I>y</I> - <I>x</I> + 1 = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17573 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17574 <p>[Put <MATH><I>x</I> = (<I>m</I><G>x</G> + 1)<SUP>2</SUP> - 1 = <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>m</I><G>x</G></MATH>; the second and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17575 third conditions require us to find two squares with <I>x</I> as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17576 difference. The difference <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>m</I><G>x</G> is separated into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17577 the factors <I>m</I><SUP>2</SUP><G>x</G> + 2<I>m</I>, <G>x</G>; the square of half the differ-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17578 ence = {(1/2)(<I>m</I><SUP>2</SUP> - 1)<G>x</G> + <I>m</I>}<SUP>2</SUP>. Put this equal to <I>y</I> + 1, so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17579 that <MATH><I>y</I> = (1/4)(<I>m</I><SUP>2</SUP> - 1)<SUP>2</SUP><G>x</G><SUP>2</SUP> + <I>m</I>(<I>m</I><SUP>2</SUP> - 1)<G>x</G> + <I>m</I><SUP>2</SUP> - 1</MATH>, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17580 first three conditions are satisfied. The fourth gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17581 <MATH>(1/4)(<I>m</I><SUP>4</SUP> - 6<I>m</I><SUP>2</SUP> + 1)<G>x</G><SUP>2</SUP> + (<I>m</I><SUP>3</SUP> - 3<I>m</I>)<G>x</G> + <I>m</I><SUP>2</SUP></MATH> = a square, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17582 we can equate to (<I>n</I><G>x</G> - <I>m</I>)<SUP>2</SUP>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17583 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17584 <p>IV. 14. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> = (<I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP>) + (<I>y</I><SUP>2</SUP> - <I>z</I><SUP>2</SUP>) + (<I>x</I><SUP>2</SUP> - <I>z</I><SUP>2</SUP>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17585 (<I>x</I>><I>y</I>><I>z.</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17586 <p>IV. 16. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> + <I>y</I> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + <I>z</I> = <I>v</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>x</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17587 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17588 <p>[Put 4<I>m</I><G>x</G> for <I>y,</I> and by means of the factors 2<I>m</I><G>x</G>, 2
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17589 we can satisfy the second condition by making <I>x</I> equal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17590 to half the difference, or <I>m</I><G>x</G> - 1. The third condition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17591 is satisfied by subtracting (4<I>m</I><G>x</G>)<SUP>2</SUP> from some square, say
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17592 (4<I>m</I><G>x</G> + 1)<SUP>2</SUP>; therefore <MATH><I>z</I> = 8<I>m</I><G>x</G> + 1</MATH>. By the first con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17593 dition 13<I>m</I><G>x</G> must be a square. Let it be 169<G>h</G><SUP>2</SUP>; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17594 numbers are therefore 13<G>h</G><SUP>2</SUP> - 1, 52<G>h</G><SUP>2</SUP>, 104<G>h</G><SUP>2</SUP> + 1, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17595 the last condition gives <MATH>10816<G>h</G><SUP>4</SUP> + 221<G>h</G><SUP>2</SUP></MATH> = a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17596 i.e. <MATH>10816<G>h</G><SUP>2</SUP> + 221 = a square = (104<G>h</G> + 1)<SUP>2</SUP></MATH>, say. This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17597 gives the value of <G>h</G>, and solves the problem.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17598 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17599 <p>IV. 17. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> - <I>y</I> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - <I>z</I> = <I>v</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> - <I>x</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17600 <p>IV. 19. <I>yz</I> + 1 = <I>u</I><SUP>2</SUP>, <I>zx</I> + 1 = <I>v</I><SUP>2</SUP>, <I>xy</I> + 1 = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17601 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17602 <p>[We are asked to solve this indeterminately (<G>e)n tw=\|
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17603 a)ori/stw\|</G>). Put for <I>yz</I> some square minus 1, say <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17604 + 2<I>m</I><G>x</G>; one condition is now satisfied. Put <I>z</I> = <G>x</G>, so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17605 that <MATH><I>y</I> = <I>m</I><SUP>2</SUP><G>x</G> + 2<I>m</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17606 <pb n=498><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17607 <p>Similarly we satisfy the second condition by assuming
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17608 <MATH><I>zx</I> = <I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>n</I><G>x</G>; therefore <I>x</I> = <I>n</I><SUP>2</SUP><G>x</G> + 2<I>n</I></MATH>. To satisfy the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17609 third condition, we must have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17610 <MATH>(<I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>mn</I>.&horbar;(<I>m</I> + <I>n</I>)<G>x</G> + 4<I>mn</I>) + 1</MATH> a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17611 We must therefore have 4<I>mn</I> + 1 a square and also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17612 <MATH><I>mn</I>(<I>m</I> + <I>n</I>) = <I>mn</I>√(4<I>mn</I> + 1)</MATH>. The first condition is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17613 satisfied by <MATH><I>n</I> = <I>m</I> + 1</MATH>, which incidentally satisfies the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17614 second condition also. We put therefore <MATH><I>yz</I> = (<I>m</I><G>x</G> + 1)<SUP>2</SUP> - 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17615 and <I>zx</I> = {(<I>m</I> + 1)<G>x</G> + 1}<SUP>2</SUP> - 1</MATH>, and assume that <I>z</I> = <G>x</G>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17616 <MATH><I>y</I> = <I>m</I><SUP>2</SUP><G>x</G> + 2<I>m</I>, <I>x</I> = (<I>m</I> + 1)<SUP>2</SUP><G>x</G> + 2(<I>m</I> + 1)</MATH>, and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17617 shown that the third condition is also satisfied. Thus we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17618 have a solution in terms of the undetermined unknown <G>x</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17619 The above is only slightly generalized from Diophantus.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17620 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17621 <p>IV. 20. <I>x</I><SUB>2</SUB><I>x</I><SUB>3</SUB> + 1 = <I>r</I><SUP>2</SUP>, <I>x</I><SUB>3</SUB><I>x</I><SUB>1</SUB> + 1 = <I>s</I><SUP>2</SUP>, <I>x</I><SUB>1</SUB><I>x</I><SUB>2</SUB> + 1 = <I>t</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17622 <I>x</I><SUB>1</SUB><I>x</I><SUB>4</SUB> + 1 = <I>u</I><SUP>2</SUP>, <I>x</I><SUB>2</SUB><I>x</I><SUB>4</SUB> + 1 = <I>v</I><SUP>2</SUP>, <I>x</I><SUB>3</SUB><I>x</I><SUB>4</SUB> + 1 = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17623 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17624 <p>[This proposition depends on the last, <I>x</I><SUB>1</SUB>, <I>x</I><SUB>2</SUB>, <I>x</I><SUB>3</SUB> being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17625 determined as in that proposition. If <I>x</I><SUB>3</SUB> corresponds to <I>z</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17626 in that proposition, we satisfy the condition <MATH><I>x</I><SUB>3</SUB><I>x</I><SUB>4</SUB> + 1 = <I>w</I><SUP>2</SUP></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17627 by putting <MATH><I>x</I><SUB>3</SUB><I>x</I><SUB>4</SUB> = {(<I>m</I> + 2)<G>x</G> + 1}<SUP>2</SUP> - 1</MATH>, and so find <I>x</I><SUB>4</SUB> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17628 terms of <G>x</G>, after which we have only two conditions more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17629 to satisfy. The condition <I>x</I><SUB>1</SUB><I>x</I><SUB>4</SUB> + 1 = square is auto-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17630 matically satisfied, since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17631 <MATH>{(<I>m</I> + 1)<SUP>2</SUP><G>x</G> + 2(<I>m</I> + 1)} {(<I>m</I> + 2)<SUP>2</SUP><G>x</G> + 2(<I>m</I> + 2)} + 1</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17632 is a square, and it only remains to satisfy <I>x</I><SUB>2</SUB><I>x</I><SUB>4</SUB> + 1 = square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17633 That is,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17634 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17635 (<I>m</I><SUP>2</SUP><G>x</G> + 2<I>m</I>) {(<I>m</I> + 2)<SUP>2</SUP><G>x</G> + 2(<I>m</I> + 2)} + 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17636 = <I>m</I><SUP>2</SUP>(<I>m</I> + 2)<SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>m</I>(<I>m</I> + 2) (2<I>m</I> + 2)<G>x</G> + 4<I>m</I>(<I>m</I> + 2) + 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17637 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17638 has to be made a square, which is easy, since the coefficient
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17639 of <G>x</G><SUP>2</SUP> is a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17640 <p>With Diophantus <I>m</I> = 1, so that <MATH><I>x</I><SUB>1</SUB> = 4<G>x</G> + 4, <I>x</I><SUB>2</SUB> = <G>x</G> + 2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17641 <I>x</I><SUB>3</SUB> = <G>x</G>, <I>x</I><SUB>4</SUB> = 9<G>x</G> + 6, and 9<G>x</G><SUP>2</SUP> + 24<G>x</G> + 13</MATH> has to be made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17642 a square. He equates this to (3<G>x</G> - 4)<SUP>2</SUP>, giving <G>x</G> = 1/16.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17643 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17644 <p>IV. 21. <I>xz</I> = <I>y</I><SUP>2</SUP>, <I>x</I> - <I>y</I> = <I>u</I><SUP>2</SUP>, <I>x</I> - <I>z</I> = <I>v</I><SUP>2</SUP>, <I>y</I> - <I>z</I> = <I>w</I><SUP>2</SUP>. (<I>x</I>><I>y</I>><I>z</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17645 <BRACE>IV. 22. <I>xyz</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>xyz</I> + <I>y</I> = <I>v</I><SUP>2</SUP>, <I>xyz</I> + <I>z</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17646 IV. 23. <I>xyz</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>xyz</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, <I>xyz</I> - <I>z</I> = <I>w</I><SUP>2</SUP></BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17647 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17648 <pb n=499><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17649 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17650 <BRACE>IV. 29. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>w</I><SUP>2</SUP> + <I>x</I> + <I>y</I> + <I>z</I> + <I>w</I> = <I>a</I>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17651 [Since <I>x</I><SUP>2</SUP> + <I>x</I> + 1/4 is a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17652 (<I>x</I><SUP>2</SUP> + <I>x</I>) + (<I>y</I><SUP>2</SUP> + <I>y</I>) + (<I>z</I><SUP>2</SUP> + <I>z</I>) + (<I>w</I><SUP>2</SUP> + <I>w</I>) + 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17653 is the sum of four squares, and we only have to separate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17654 <I>a</I> + 1 into four squares.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17655 IV. 30. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>w</I><SUP>2</SUP> - (<I>x</I> + <I>y</I> + <I>z</I> + <I>w</I>) = <I>a.</I></BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17656 <p>IV. 31. <I>x</I> + <I>y</I> = 1, (<I>x</I> + <I>a</I>) (<I>y</I> + <I>b</I>) = <I>u</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17657 <p>IV. 32. <I>x</I> + <I>y</I> + <I>z</I> = <I>a,</I> <I>xy</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>xy</I> - <I>z</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17658 <p>IV. 39. <I>x</I> - <I>y</I> = <I>m</I>(<I>y</I> - <I>z</I>), <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17659 <p>IV. 40. <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>y</I> - <I>z</I>), <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17660 <BRACE>V. 1. <I>xz</I> = <I>y</I><SUP>2</SUP>, <I>x</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> - <I>a</I> = <I>v</I><SUP>2</SUP>, <I>z</I> - <I>a</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17661 V. 2. <I>xz</I> = <I>y</I><SUP>2</SUP>, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>z</I> + <I>a</I> = <I>w</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17662 <BRACE>V. 3. <I>x</I> + <I>a</I> = <I>r</I><SUP>2</SUP>, <I>y</I> + <I>a</I> = <I>s</I><SUP>2</SUP>, <I>z</I> + <I>a</I> = <I>t</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17663 <I>yz</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>a</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17664 V. 4. <I>x</I> - <I>a</I> = <I>r</I><SUP>2</SUP>, <I>y</I> - <I>a</I> = <I>s</I><SUP>2</SUP>, <I>z</I> - <I>a</I> = <I>t</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17665 <I>yz</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> - <I>a</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>a</I> = <I>w</I><SUP>2</SUP></BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17666 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17667 <p>[Solved by means of the <I>Porisms</I> that, if <I>a</I> be the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17668 given number, the numbers <I>m</I><SUP>2</SUP> - <I>a</I>, (<I>m</I> + 1)<SUP>2</SUP> - <I>a</I> satisfy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17669 the conditions of V. 3, and the numbers <I>m</I><SUP>2</SUP> + <I>a</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17670 (<I>m</I> + 1)<SUP>2</SUP> + <I>a</I> the conditions of V. 4 (see p. 479 above). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17671 third number is taken to be <MATH>2{<I>m</I><SUP>2</SUP>&mnplus;<I>a</I> + (<I>m</I> + 1)<SUP>2</SUP>&mnplus;<I>a</I>} - 1</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17672 and the three numbers automatically satisfy two more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17673 conditions (see p. 480 above). It only remains to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17674 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17675 2{<I>m</I><SUP>2</SUP>&mnplus;<I>a</I> + (<I>m</I> + 1)<SUP>2</SUP>&mnplus;<I>a</I>} - 1±<I>a</I> a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17676 or 4<I>m</I><SUP>2</SUP> + 4<I>m</I>&mnplus;3<I>a</I> + 1 = a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17677 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17678 which is easily solved.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17679 <p>With Diophantus <G>x</G> + 3 takes the place of <I>m</I> in V. 3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17680 and <G>x</G> takes its place in V. 4, while <I>a</I> is 5 in V. 3 and 6
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17681 in V. 4.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17682 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17683 <p>V. 5. <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> = <I>r</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>s</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> =
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17684 <I>t</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17685 <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17686 <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17687 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17688 <p>[Solved by means of the Porism numbered 2 on p. 480.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17689 <pb n=500><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17690 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17691 <p>V. 6. <I>x</I> - 2 = <I>r</I><SUP>2</SUP>, <I>y</I> - 2 = <I>s</I><SUP>2</SUP>, <I>z</I> - 2 = <I>t</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17692 <I>yz</I> - <I>y</I> - <I>z</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> - <I>z</I> - <I>x</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>x</I> - <I>y</I> = <I>w</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17693 <I>yz</I> - <I>x</I> = <I>u</I>′<SUP>2</SUP>, <I>zx</I> - <I>y</I> = <I>v</I>′<SUP>2</SUP>, <I>xy</I> - <I>z</I> = <I>w</I>′<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17694 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17695 <p>[Solved by means of the proposition numbered (3) on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17696 p. 481.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17697 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17698 <p>Lemma 1 to V. 7. <I>xy</I> + <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17699 <BRACE><note>V. 7. <I>x</I><SUP>2</SUP> ± (<I>x</I> + <I>y</I> + <I>z</I>) =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17700 <I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17701 <I>u</I>′<SUP>2</SUP>,</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17702 <BRACE><I>y</I><SUP>2</SUP>±(<I>x</I> + <I>y</I> + <I>z</I>) =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17703 <I>v</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17704 <I>v</I>′<SUP>2</SUP>,</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17705 <BRACE><note><I>z</I><SUP>2</SUP>±(<I>x</I> + <I>y</I> + <I>z</I>) =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17706 <I>w</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17707 <I>w</I>′<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17708 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17709 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17710 <p>[Solved by means of the subsidiary problem (Lemma 2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17711 of finding three rational right-angled triangles with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17712 equal area. If <I>m, n</I> satisfy the condition in Lemma 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17713 i.e. if <MATH><I>mn</I> + <I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> = <I>p</I><SUP>2</SUP></MATH>, the triangles are ‘formed’ from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17714 the pairs of numbers (<I>p, m</I>), (<I>p, n</I>), (<I>p, m</I> + <I>n</I>) respec-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17715 tively. Diophantus assumes this, but it is easy to prove.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17716 In his case <I>m</I> = 3, <I>n</I> = 5, so that <I>p</I> = 7. Now, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17717 a right-angled triangle, (hypotenuse)<SUP>2</SUP>±four times area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17718 is a square. We equate, therefore, <I>x</I> + <I>y</I> + <I>z</I> to four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17719 times the common area multiplied by <G>x</G><SUP>2</SUP>, and the several
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17720 numbers <I>x, y, z</I> to the three hypotenuses multiplied by <G>x</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17721 and equate the two values. In Diophantus's case the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17722 triangles are (40, 42, 58), (24, 70, 74) and (15, 112, 113),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17723 and 245<G>x</G> = 3360<G>x</G><SUP>2</SUP>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17724 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17725 <BRACE><note>V. 8. <I>yz</I>±(<I>x</I> + <I>y</I> + <I>z</I>) =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17726 <I>u</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17727 <I>u</I>′<SUP>2</SUP>,</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17728 <BRACE><note><I>zx</I> ± (<I>x</I> + <I>y</I> + <I>z</I>) =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17729 <I>v</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17730 <I>v</I>′<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17731 <BRACE><note><I>xy</I> ± (<I>x</I> + <I>y</I> + <I>z</I>) =</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17732 <I>w</I><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17733 <I>w</I>′<SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17734 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17735 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17736 <p>[Solved by means of the same three rational right-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17737 angled triangles found in the Lemma to V. 7, together
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17738 with the Lemma that we can solve the equations <I>yz</I> = <I>a</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17739 <I>zx</I> = <I>b</I><SUP>2</SUP>, <I>xy</I> = <I>c</I><SUP>2</SUP>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17740 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17741 <BRACE>V. 9. (Cf. II. 11). <I>x</I> + <I>y</I> = 1, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>a</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17742 V. 11. <I>x</I> + <I>y</I> + <I>z</I> = 1, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>z</I> + <I>a</I> = <I>w</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17743 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17744 <p>[These are the problems of <G>pariso/thtos a)gwgh/</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17745 <pb n=501><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17746 described above (pp. 477-9). The problem is ‘to divide
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17747 unity into two (or three) parts such that, if one and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17748 same given number be added to each part, the results are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17749 all squares’.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17750 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17751 <BRACE>V. 10. <I>x</I> + <I>y</I> = 1, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>b</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17752 V. 12. <I>x</I> + <I>y</I> + <I>z</I> = 1, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>b</I> = <I>v</I><SUP>2</SUP>, <I>z</I> + <I>c</I> = <I>w</I><SUP>2</SUP></BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17753 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17754 <p>[These problems are like the preceding except that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17755 <I>different</I> given numbers are added. The second of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17756 two problems is not worked out, but the first is worth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17757 reproducing. We must take the particular figures used
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17758 by Diophantus, namely <I>a</I> = 2, <I>b</I> = 6. We have then to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17759 divide 9 into two squares such that one of them lies
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17760 between 2 and 3. Take two squares lying between 2
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17761 and 3, say 289/144, 361/144. We have then to find a square <G>x</G><SUP>2</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17762 lying between them; if we can do this, we can make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17763 9 - <G>x</G><SUP>2</SUP> a square, and so solve the problem.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17764 <p>Put <MATH>9 - <G>x</G><SUP>2</SUP> = (3 - <I>m</I><G>x</G>)<SUP>2</SUP></MATH>, say, so that <MATH><G>x</G> = 6<I>m</I>/(<I>m</I><SUP>2</SUP> + 1)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17765 and <I>m</I> has to be determined so that <G>x</G> lies between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17766 17/12 and 19/12.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17767 <p>Therefore <MATH>17/12 < (6<I>m</I>)/(<I>m</I><SUP>2</SUP> + 1) < 19/12</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17768 <p>Diophantus, as we have seen, finds <I>a fortiori</I> integral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17769 limits for <I>m</I> by solving these inequalities, making <I>m</I> not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17770 greater than 67/17 and not less than 66/19 (see pp. 463-5 above).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17771 He then takes <I>m</I> = 3 1/2 and puts <MATH>9 - <G>x</G><SUP>2</SUP> = (3 - (3 1/2)<G>x</G>)<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17772 which gives <G>x</G> = 84/53.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17773 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17774 <BRACE>V. 13. <I>x</I> + <I>y</I> + <I>z</I> = <I>a</I>, <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17775 V. 14. <I>x</I> + <I>y</I> + <I>z</I> + <I>w</I> = <I>a</I>, <I>x</I> + <I>y</I> + <I>z</I> = <I>s</I><SUP>2</SUP>, <I>y</I> + <I>z</I> + <I>w</I> = <I>t</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17776 <I>z</I> + <I>w</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>w</I> + <I>x</I> + <I>y</I> = <I>v</I><SUP>2</SUP></BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17777 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17778 <p>[The method is the same.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17779 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17780 <BRACE>V. 21. <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17781 <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17782 V. 22. <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17783 <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> - <I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17784 V. 23. <I>x</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP> <I>z</I><SUP>2</SUP> -
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17785 <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>.</BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17786 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17787 <p>[Solved by means of right-angled triangles in rational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17788 numbers.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17789 <pb n=502><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17790 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17791 <BRACE>V. 24. <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + 1 = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + 1 = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + 1 = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17792 V. 25. <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> - 1 = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> - 1 = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> - 1 = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17793 V. 26. 1 - <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, 1 - <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>, 1 - <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP></BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17794 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17795 <p>[These reduce to the preceding set of three problems.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17796 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17797 <BRACE>V. 27. <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>a</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17798 V. 28. <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> - <I>a</I> = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> - <I>a</I> =
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17799 <I>w</I><SUP>2</SUP></BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17800 <p>V. 30. <I>mx</I> + <I>ny</I> = <I>u</I><SUP>2</SUP>, <I>u</I><SUP>2</SUP> + <I>a</I> = (<I>x</I> + <I>y</I>)<SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17801 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17802 <p>[This problem is enunciated thus. ‘A man buys a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17803 certain number of measures of wine, some at 8 drachmas,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17804 some at 5 drachmas each. He pays for them a <I>square</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17805 number of drachmas; and if 60 is added to this number,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17806 the result is a square, the side of which is equal to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17807 whole number of measures. Find the number bought at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17808 each price.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17809 <p>Let <G>x</G> = the whole number of measures; therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17810 <G>x</G><SUP>2</SUP> - 60 was the number of drachmas paid, and <G>x</G><SUP>2</SUP> - 60
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17811 = a square, say (<G>x</G> - <I>m</I>)<SUP>2</SUP>; hence <MATH><G>x</G> = (<I>m</I><SUP>2</SUP> + 60)/2<I>m</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17812 <p>Now 1/5 of the price of the five-drachma measures + 1/3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17813 of that of the eight-drachma measures = <G>x</G>; therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17814 <G>x</G><SUP>2</SUP> - 60, the total price, has to be divided into two parts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17815 such that 1/5 of one + 1/8 of the other = <G>x</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17816 <p>We cannot have a real solution of this unless
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17817 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17818 <G>x</G> > (1/8)(<G>x</G><SUP>2</SUP> - 60) and < (1/5)(<G>x</G><SUP>2</SUP> - 60);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17819 therefore 5<G>x</G> < <G>x</G><SUP>2</SUP> - 60 < 8<G>x</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17820 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17821 <p>Diophantus concludes, as we have seen (p. 464 above),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17822 that <G>x</G> is not less than 11 and not greater than 12.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17823 <p>Therefore, from above, since <MATH><G>x</G> = (<I>m</I><SUP>2</SUP> + 60)/2<I>m</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17824 22<I>m</I> < <I>m</I><SUP>2</SUP> + 60 < 24<I>m</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17825 and Diophantus concludes that <I>m</I> is not less than 19 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17826 not greater than 21. He therefore puts <I>m</I> = 20.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17827 <p>Therefore <MATH><G>x</G> = (<I>m</I><SUP>2</SUP> + 60)/2<I>m</I> = 11 1/2, <G>x</G><SUP>2</SUP> = 132 1/4, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17828 <G>x</G><SUP>2</SUP> - 60 = 72 1/4</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17829 <p>We have now to divide 72 1/4 into two parts such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17830 1/5 of one part + 1/8 of the other = 11 1/2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17831 <pb n=503><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17832 <p>Let the first part = 5<I>z</I>; therefore 1/3 (second part)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17833 = 11 1/2 - <I>z,</I> or second part = 92 - 8<I>z.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17834 <p>Therefore <MATH>5<I>z</I> + 92 - 8<I>z</I> = 72 1/4, and <I>z</I> = 79/12</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17835 therefore the number of five-drachma measures is 79/12 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17836 the number of eight-drachma measures 59/12.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17837 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17838 <BRACE><note>(see p. 467
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17839 above.)</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17840 Lemma 2 to VI. 12. <I>ax</I><SUP>2</SUP> + <I>b</I> = <I>u</I><SUP>2</SUP> (where <I>a</I> + <I>b</I> = <I>c</I><SUP>2</SUP>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17841 Lemma to VI. 15. <I>ax</I><SUP>2</SUP> - <I>b</I> = <I>u</I><SUP>2</SUP> (where <I>ad</I><SUP>2</SUP> - <I>b</I> = <I>c</I><SUP>2</SUP>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17842 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17843 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17844 [III. 15]. <I>xy</I> + <I>x</I> + <I>y</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + 1 = (<I>v</I><SUP>2</SUP>/<I>w</I><SUP>2</SUP>)(<I>y</I> + 1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17845 [III. 16]. <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>x</I> - 1 = (<I>v</I><SUP>2</SUP>/<I>w</I><SUP>2</SUP>)(<I>y</I> - 1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17846 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17847 <p>[IV. 32]. <I>x</I> + 1 = (<I>u</I><SUP>2</SUP>/<I>v</I><SUP>2</SUP>)(<I>x</I> - 1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17848 <p>[V. 21]. <I>x</I><SUP>2</SUP> + 1 = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + 1 = <I>v</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + 1 = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17849 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17850 <C>(viii) Indeterminate analysis of the third degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17851 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17852 <p>IV. 3. <I>x</I><SUP>2</SUP><I>y</I> = <I>u, xy</I> = <I>u</I><SUP>3</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17853 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17854 IV. 6. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>2</SUP> = <I>u</I><SUP>3</SUP>, <I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17855 IV. 7. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>3</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17856 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17857 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17858 <BRACE><note>(really reducible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17859 to the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17860 degree.)</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17861 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17862 IV. 8. <I>x</I> + <I>y</I><SUP>3</SUP> = <I>u</I><SUP>3</SUP>, <I>x</I> + <I>y</I> = <I>u.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17863 IV. 9. <I>x</I> + <I>y</I><SUP>3</SUP> = <I>u, x</I> + <I>y</I> = <I>u</I><SUP>3</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17864 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17865 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17866 IV. 10. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> = <I>x</I> + <I>y.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17867 <BRACE><note>the same problem.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17868 IV. 11. <I>x</I><SUP>3</SUP> - <I>y</I><SUP>3</SUP> = <I>x</I> - <I>y.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17869 IV. 12. <I>x</I><SUP>3</SUP> + <I>y</I> = <I>y</I><SUP>3</SUP> + <I>x.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17870 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17871 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17872 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17873 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17874 <p>[We may give as examples the solutions of IV. 7,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17875 IV. 8, IV. 11.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17876 <p>IV. 7. Since <I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = a cube, suppose <MATH><I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>x</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17877 To make <I>x</I><SUP>3</SUP> + <I>y</I><SUP>2</SUP> a square, put <MATH><I>x</I><SUP>3</SUP> = <I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> = 2 <I>ab</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17878 which also satisfies <MATH><I>x</I><SUP>3</SUP> - <I>y</I><SUP>2</SUP> = <I>z</I><SUP>2</SUP></MATH>. We have then to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17879 2<I>ab</I> a square. Let <I>a</I> = <G>x</G>, <I>b</I> = 2<G>x</G>; therefore <MATH><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP> = 5<G>x</G><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17880 2<I>ab</I> = 4<G>x</G><SUP>2</SUP>, <I>y</I> = 2<G>x</G>, <I>z</I> = <G>x</G>, and we have only to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17881 5<G>x</G><SUP>2</SUP> a cube. <G>x</G> = 5, and <I>x</I><SUP>3</SUP> = 125, <I>y</I><SUP>2</SUP> = 100, <I>z</I><SUP>2</SUP> = 25.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17882 <pb n=504><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17883 <p>IV. 8. Suppose <I>x</I> = <G>x</G>, <I>y</I><SUP>3</SUP> = <I>m</I><SUP>3</SUP><G>x</G><SUP>3</SUP>; therefore <MATH><I>u</I> = (<I>m</I> + 1)<G>x</G></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17884 must be the side of the cube <I>m</I><SUP>3</SUP><G>x</G><SUP>3</SUP> + <G>x</G>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17885 <MATH><I>m</I><SUP>3</SUP><G>x</G><SUP>2</SUP> + 1 = (<I>m</I><SUP>3</SUP> + 3<I>m</I><SUP>2</SUP> + 3<I>m</I> + 1)<G>x</G><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17886 To solve this, we must have 3<I>m</I><SUP>2</SUP> + 3<I>m</I> + 1 (the difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17887 between consecutive cubes) a square. Put
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17888 <MATH>3<I>m</I><SUP>2</SUP> + 3<I>m</I> + 1 = (1 - <I>nm</I>)<SUP>2</SUP>, and <I>m</I> = (3 + 2<I>n</I>)/(<I>n</I><SUP>2</SUP> - 3)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17889 <p>IV. 11. Assume <MATH><I>x</I> = (<I>m</I> + 1)<G>x</G>, <I>y</I> = <I>m</I><G>x</G></MATH>, and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17890 to make (3<I>m</I><SUP>3</SUP> + 3<I>m</I><SUP>2</SUP> + 1)<G>x</G><SUP>2</SUP> equal to 1, i.e. we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17891 only to make 3<I>m</I><SUP>2</SUP> + 3<I>m</I> + 1 a square.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17892 <p>IV. 18. <MATH><I>x</I><SUP>3</SUP> + <I>y</I> = <I>u</I><SUP>3</SUP>, <I>y</I><SUP>2</SUP> + <I>x</I> = <I>v</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17893 <p>IV. 24. <MATH><I>x</I> + <I>y</I> = <I>a, xy</I> = <I>u</I><SUP>3</SUP> - <I>u</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17894 <p>[<MATH><I>y</I> = <I>a</I> - <I>x</I></MATH>; therefore <I>ax</I> - <I>x</I><SUP>2</SUP> has to be made a cube
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17895 <I>minus</I> its side, say (<I>mx</I> - 1)<SUP>3</SUP> - (<I>mx</I> - 1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17896 <p>Therefore <MATH><I>ax</I> - <I>x</I><SUP>2</SUP> = <I>m</I><SUP>3</SUP><I>x</I><SUP>3</SUP> - 3<I>m</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + 2<I>mx</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17897 To reduce this to a simple equation, we have only to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17898 put <I>m</I> = (1/2)<I>a.</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17899 <p>IV. 25. <MATH><I>x</I> + <I>y</I> + <I>z</I> = <I>a, xyz</I> = {(<I>x</I> - <I>y</I>) + (<I>x</I> - <I>z</I>) + (<I>y</I> - <I>z</I>)}<SUP>3</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17900 (<I>x</I> > <I>y</I> > <I>z</I>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17901 <p>[The cube = 8(<I>x</I> - <I>z</I>)<SUP>3</SUP>. Let <MATH><I>x</I> = (<I>m</I> + 1)<G>x</G>, <I>z</I> = <I>m</I><G>x</G></MATH>, so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17902 that <MATH><I>y</I> = 8<G>x</G> / (<I>m</I><SUP>2</SUP> + <I>m</I>)</MATH>, and we have only to contrive that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17903 <MATH>8 / (<I>m</I><SUP>2</SUP> + <I>m</I>)</MATH> lies between <I>m</I> and <I>m</I> + 1. Dioph. takes the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17904 first limit <MATH>8 > <I>m</I><SUP>3</SUP> + <I>m</I><SUP>2</SUP></MATH>, and puts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17905 <MATH>8 = (<I>m</I> + 1/3)<SUP>3</SUP> or <I>m</I><SUP>3</SUP> + <I>m</I><SUP>2</SUP> + 1/3<I>m</I> + 1/27</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17906 whence <I>m</I> = 5/3; therefore <I>x</I> = (8/3)<G>x</G>, <I>y</I> = (9/5)<G>x</G>, <I>z</I> = 5/3<G>x</G>. Or,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17907 multiplying by 15, we have <I>x</I> = 40<G>x</G>, <I>y</I> = 27<G>x</G>, <I>z</I> = 25<G>x</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17908 The first equation then gives <G>x</G>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17909 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17910 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17911 IV. 26. <I>xy</I> + <I>x</I> = <I>u</I><SUP>3</SUP>, <I>xy</I> + <I>y</I> = <I>v</I><SUP>3</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17912 IV. 27. <I>xy</I> - <I>x</I> = <I>u</I><SUP>3</SUP>, <I>xy</I> - <I>y</I> = <I>v</I><SUP>3</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17913 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17914 <p>IV. 28. <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>3</SUP>, <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>v</I><SUP>3</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17915 [<I>x</I> + <I>y</I> = 1/2(<I>u</I><SUP>3</SUP> - <I>v</I><SUP>3</SUP>), <I>xy</I> = 1/2(<I>u</I><SUP>3</SUP> + <I>v</I><SUP>3</SUP>); therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17916 (<I>x</I> - <I>y</I>)<SUP>2</SUP> = 1/4(<I>u</I><SUP>3</SUP> - <I>v</I><SUP>3</SUP>)<SUP>2</SUP> - 2(<I>u</I><SUP>3</SUP> + <I>v</I><SUP>3</SUP>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17917 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17918 which latter expression has to be made a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17919 <pb n=505><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17920 <p>Diophantus assumes <MATH><I>u</I> = <G>x</G> + 1, <I>v</I> = <G>x</G> - 1</MATH>, whence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17921 <MATH>1/4(6<G>x</G><SUP>2</SUP> + 2)<SUP>2</SUP> - 2(2<G>x</G><SUP>3</SUP> + 6<G>x</G>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17922 must be a square, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17923 <MATH>9<G>x</G><SUP>4</SUP> - 4<G>x</G><SUP>3</SUP> + 6<G>x</G><SUP>2</SUP> - 12<G>x</G> + 1 = a square = (3<G>x</G><SUP>2</SUP> - 6<G>x</G> + 1)<SUP>2</SUP></MATH>, say;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17924 therefore 32<G>x</G><SUP>3</SUP> = 36<G>x</G><SUP>2</SUP>, and <G>x</G> = 9/8. Thus <I>u, v</I> are found,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17925 and then <I>x, y.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17926 <p>The second (alternative) solution uses the formula that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17927 <MATH><G>x</G>(<G>x</G><SUP>2</SUP> - <G>x</G>) + (<G>x</G><SUP>2</SUP> - <G>x</G>) + <G>x</G> =</MATH> a cube. Put <MATH><I>x</I> = <G>x</G>, <I>y</I> = <G>x</G><SUP>2</SUP> - <G>x</G></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17928 and one condition is satisfied. We then only have to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17929 make <MATH><G>x</G>(<G>x</G><SUP>2</SUP> - <G>x</G>) - <G>x</G> - (<G>x</G><SUP>2</SUP> - <G>x</G>) or <G>x</G><SUP>3</SUP> - 2<G>x</G><SUP>2</SUP></MATH> a cube (less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17930 <G>x</G><SUP>3</SUP>), i. e. <MATH><G>x</G><SUP>3</SUP> - 2<G>x</G><SUP>2</SUP> = ((1/2)<G>x</G>)<SUP>3</SUP></MATH>, say.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17931 <p>IV. 38. <MATH>(<I>x</I> + <I>y</I> + <I>z</I>)<I>x</I> = (1/2)<I>u</I>(<I>u</I> + 1), (<I>x</I> + <I>y</I> + <I>z</I>)<I>y</I> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17932 (<I>x</I> + <I>y</I> + <I>z</I>)<I>z</I> = <I>w</I><SUP>3</SUP>, [<I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>]</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17933 <p>[Suppose <MATH><I>x</I> + <I>y</I> + <I>z</I> = <G>x</G><SUP>2</SUP></MATH>; then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17934 <MATH><I>x</I> = (<I>u</I>(<I>u</I> + 1))/2<G>x</G><SUP>2</SUP>, <I>y</I> = <I>v</I><SUP>2</SUP>/<G>x</G><SUP>2</SUP>, <I>z</I> = <I>w</I><SUP>3</SUP>/<G>x</G><SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17935 therefore <MATH><G>x</G><SUP>4</SUP> = (1/2)<I>u</I>(<I>u</I> + 1) + <I>v</I><SUP>2</SUP> + <I>w</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17936 Diophantus puts 8 for <I>w</I><SUP>3</SUP>, but we may take any cube, as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17937 <I>m</I><SUP>3</SUP>; and he assumes <MATH><I>v</I><SUP>2</SUP> = (<G>x</G><SUP>2</SUP> - 1)<SUP>2</SUP></MATH>, for which we might
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17938 substitute (<G>x</G><SUP>2</SUP> - <I>n</I><SUP>2</SUP>)<SUP>2</SUP>. We then have the triangular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17939 number <MATH>(1/2)<I>u</I>(<I>u</I> + 1) = 2<I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP> - <I>n</I><SUP>4</SUP> - <I>m</I><SUP>3</SUP></MATH>. Since 8 times a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17940 triangular number <I>plus</I> 1 gives a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17941 <MATH>16<I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP> - 8<I>n</I><SUP>4</SUP> - 8<I>m</I><SUP>3</SUP> + 1 = a square = (4<I>n</I><G>x</G> - <I>k</I>)<SUP>2</SUP></MATH>, say,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17942 and the problem is solved.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17943 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17944 V. 15. <MATH>(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>x</I> = <I>u</I><SUP>3</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>y</I> = <I>v</I><SUP>3</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17945 (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>z</I> = <I>w</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17946 [Let <MATH><I>x</I> + <I>y</I> + <I>z</I> = <G>x</G>, <I>u</I><SUP>3</SUP> = <I>m</I><SUP>3</SUP><G>x</G><SUP>3</SUP>, <I>v</I><SUP>3</SUP> = <I>n</I><SUP>3</SUP><G>x</G><SUP>3</SUP>, <I>w</I><SUP>3</SUP> =
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17947 <I>p</I><SUP>3</SUP><G>x</G><SUP>3</SUP></MATH>; therefore <MATH><G>x</G> = {(<I>m</I><SUP>3</SUP> - 1) + (<I>n</I><SUP>3</SUP> - 1) + (<I>p</I><SUP>3</SUP> - 1)}<G>x</G><SUP>3</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17948 and we have to find three cubes <I>m</I><SUP>3</SUP>, <I>n</I><SUP>3</SUP>, <I>p</I><SUP>3</SUP> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17949 <MATH><I>m</I><SUP>3</SUP> + <I>n</I><SUP>3</SUP> + <I>p</I><SUP>3</SUP> - 3 =</MATH> a square. Diophantus assumes as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17950 the sides of the cubes (<I>k</I> + 1), (2 - <I>k</I>), 2; this gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17951 <pb n=506><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17952 <MATH>9<I>k</I><SUP>2</SUP> - 9<I>k</I> + 14 = a square = (3<I>k</I> - <I>l</I>)<SUP>2</SUP></MATH>, say; and <I>k</I> is found.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17953 Retracing our steps, we find <G>x</G> and therefore <I>x, y, z.</I>]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17954 V. 16. <MATH>(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>x</I> = <I>u</I><SUP>3</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>y</I> = <I>v</I><SUP>3</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17955 (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>z</I> = <I>w</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17956 V. 17. <MATH><I>x</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>u</I><SUP>3</SUP>, <I>y</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>v</I><SUP>3</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17957 <I>z</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>w</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17958 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17959 <p>V. 18. <MATH><I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>y</I> = <I>v</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17960 (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>z</I> = <I>w</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17961 <p>[Put <MATH><I>x</I> + <I>y</I> + <I>z</I> = <G>x</G><SUP>2</SUP>, <I>x</I> = (<I>p</I><SUP>2</SUP> - 1)<G>x</G><SUP>6</SUP>, <I>y</I> = (<I>q</I><SUP>2</SUP> - 1)<G>x</G><SUP>6</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17962 <I>z</I> = (<I>r</I><SUP>2</SUP> - 1)<G>x</G><SUP>6</SUP></MATH>, whence <MATH><G>x</G><SUP>2</SUP> = (<I>p</I><SUP>2</SUP> - 1 + <I>q</I><SUP>2</SUP> - 1 + <I>r</I><SUP>2</SUP> - 1)<G>x</G><SUP>6</SUP></MATH>, so
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17963 that <I>p</I><SUP>2</SUP> - 1 + <I>q</I><SUP>2</SUP> - 1 + <I>r</I><SUP>2</SUP> - 1 must be made a fourth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17964 power. Diophantus assumes <MATH><I>p</I><SUP>2</SUP> = (<I>m</I><SUP>2</SUP> - 1)<SUP>2</SUP>, <I>q</I><SUP>2</SUP> = (<I>m</I> + 1)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17965 <I>r</I><SUP>2</SUP> = (<I>m</I> - 1)<SUP>2</SUP></MATH>, since <MATH><I>m</I><SUP>4</SUP> - 2<I>m</I><SUP>2</SUP> + <I>m</I><SUP>2</SUP> + 2<I>m</I> + <I>m</I><SUP>2</SUP> - 2<I>m</I> = <I>m</I><SUP>4</SUP></MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17966 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17967 <p>V. 19. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>x</I> = <I>u</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17968 (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>y</I> = <I>v</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>z</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17969 <p>V. 19a. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>x</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>u</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17970 <I>y</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>v</I><SUP>2</SUP>, <I>z</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17971 <p>V. 19. b, c. <I>x</I> + <I>y</I> + <I>z</I> = <I>a,</I> (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> ± <I>x</I> = <I>u</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17972 (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> ± <I>y</I> = <I>v</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> ± <I>z</I> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17973 <p>V. 20. <I>x</I> + <I>y</I> + <I>z</I> = 1/<I>m</I>, <I>x</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>u</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17974 <I>y</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>v</I><SUP>2</SUP>, <I>z</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>w</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17975 <p>[IV. 8]. <I>x</I> - <I>y</I> = 1, <I>x</I><SUP>3</SUP> - <I>y</I><SUP>3</SUP> = <I>u</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17976 <p>[IV. 9, 10]. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> = (<I>u</I><SUP>2</SUP>/<I>v</I><SUP>2</SUP>)(<I>x</I> + <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17977 <p>[IV. 11]. <I>x</I><SUP>3</SUP> - <I>y</I><SUP>3</SUP> = (<I>u</I><SUP>2</SUP>/<I>v</I><SUP>2</SUP>)(<I>x</I> - <I>y</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17978 <p>[V. 15]. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> + <I>z</I><SUP>3</SUP> - 3 = <I>u</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17979 <p>[V. 16]. 3 - (<I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> + <I>z</I><SUP>3</SUP>) = <I>u</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17980 <p>[V. 17]. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> + <I>z</I><SUP>3</SUP> + 3 = <I>u</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17981 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17982 <pb n=507><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17983 <C>(ix) Indeterminate analysis of the fourth degree.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17984 <p>V. 29. <MATH><I>x</I><SUP>4</SUP> + <I>y</I><SUP>4</SUP> + <I>z</I><SUP>4</SUP> = <I>u</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17985 <p>[‘Why’, says Fermat, ‘did not Diophantus seek <I>two</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17986 fourth powers such that their sum is a square. This
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17987 problem is, in fact, impossible, as by my method I am
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17988 able to prove with all rigour.’ No doubt Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17989 knew this truth empirically. Let <MATH><I>x</I><SUP>2</SUP> = <G>x</G><SUP>2</SUP>, <I>y</I><SUP>2</SUP> = <I>p</I><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17990 <I>z</I><SUP>2</SUP> = <I>q</I><SUP>2</SUP></MATH>. Therefore <MATH><G>x</G><SUP>4</SUP> + <I>p</I><SUP>4</SUP> + <I>q</I><SUP>4</SUP> = a square = (<G>x</G><SUP>2</SUP> - <I>r</I>)<SUP>2</SUP></MATH>, say;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17991 therefore <MATH><G>x</G><SUP>2</SUP> = (<I>r</I><SUP>2</SUP> - <I>p</I><SUP>4</SUP> - <I>q</I><SUP>4</SUP>)/2<I>r</I></MATH>, and we have to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17992 this expression a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17993 <p>Diophantus puts <MATH><I>r</I> = <I>p</I><SUP>2</SUP> + 4, <I>q</I><SUP>2</SUP> = 4</MATH>, so that the expres-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17994 sion reduces to <MATH>8<I>p</I><SUP>2</SUP>/(2<I>p</I><SUP>2</SUP> + 8) or 4<I>p</I><SUP>2</SUP>/(<I>p</I><SUP>2</SUP> + 4)</MATH>. To make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17995 this a square, let <MATH><I>p</I><SUP>2</SUP> + 4 = (<I>p</I> + 1)<SUP>2</SUP></MATH>, say; therefore <I>p</I> = 1 1/2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17996 and <I>p</I><SUP>2</SUP> = 2 1/4, <I>q</I><SUP>2</SUP> = 4, <I>r</I> = 6 1/4; or (multiplying by 4)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17997 <I>p</I><SUP>2</SUP> = 9, <I>q</I><SUP>2</SUP> = 16, <I>r</I> = 25, which solves the problem.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17998 <p>[V. 18]. <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> - 3 = <I>u</I><SUP>4</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	17999 <p>(See above under V. 18.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18000 <C>(x) Problems of constructing right-angled triangles with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18001 sides in rational numbers and satisfying various
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18002 other conditions.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18003 <p>[I shall in all cases call the hypotenuse <I>z,</I> and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18004 other two sides <I>x, y,</I> so that the condition <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>z</I><SUP>2</SUP></MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18005 applies in all cases, in addition to the other conditions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18006 specified.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18007 <p>[Lemma to V. 7]. <MATH><I>xy</I> = <I>x</I><SUB>1</SUB><I>y</I><SUB>1</SUB> = <I>x</I><SUB>2</SUB><I>y</I><SUB>2</SUB></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18008 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18009 VI. 1. <MATH><I>z</I> - <I>x</I> = <I>u</I><SUP>3</SUP>, <I>z</I> - <I>y</I> = <I>v</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18010 [Form a right-angled triangle from <G>x</G>, <I>m,</I> so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18011 <MATH><I>z</I> = <G>x</G><SUP>2</SUP> + <I>m</I><SUP>2</SUP>, <I>x</I> = 2<I>m</I><G>x</G>, <I>y</I> = <G>x</G><SUP>2</SUP> - <I>m</I><SUP>2</SUP></MATH>; thus <MATH><I>z</I> - <I>y</I> = 2<I>m</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18012 and, as this must be a cube, we put <I>m</I> = 2; therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18013 <MATH><I>z</I> - <I>x</I> = <G>x</G><SUP>2</SUP> - 4<G>x</G> + 4</MATH> must be a cube, or <G>x</G> - 2 = a cube,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18014 say <I>n</I><SUP>3</SUP>, and <MATH><G>x</G> = <I>n</I><SUP>3</SUP> + 2</MATH>.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18015 VI. 2. <MATH><I>z</I> + <I>x</I> = <I>u</I><SUP>3</SUP>, <I>z</I> + <I>y</I> = <I>v</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18016 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18017 <pb n=508><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18018 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18019 VI. 3. <MATH>(1/2)<I>xy</I> + <I>a</I> = <I>u</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18020 [Suppose the required triangle to be <I>h</I><G>x</G>, <I>p</I><G>x</G>, <I>b</I><G>x</G>; there-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18021 fore (1/2)<I>pb</I><G>x</G><SUP>2</SUP> + <I>a</I> = a square = <I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP>, say, and the ratio of <I>a</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18022 to <I>n</I><SUP>2</SUP> - (1/2)<I>pb</I> must be the ratio of a square to a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18023 To find <I>n, p, b</I> so as to satisfy this condition, form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18024 a right-angled triangle from <I>m,</I> 1/<I>m</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18025 i. e. <MATH>(<I>m</I><SUP>2</SUP> + 1/<I>m</I><SUP>2</SUP>, 2, <I>m</I><SUP>2</SUP> - 1/<I>m</I><SUP>2</SUP>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18026 therefore <MATH>(1/2)<I>pb</I> = <I>m</I><SUP>2</SUP> - 1/<I>m</I><SUP>2</SUP></MATH>. Assume <MATH><I>n</I><SUP>2</SUP> = (<I>m</I> + 2<I>a/m</I>)<SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18027 therefore <MATH><I>n</I><SUP>2</SUP> - (1/2)<I>pb</I> = 4<I>a</I> + (4<I>a</I><SUP>2</SUP> + 1)/<I>m</I><SUP>2</SUP></MATH>; and <MATH>(4<I>a</I> + (4<I>a</I><SUP>2</SUP> + 1)/<I>m</I><SUP>2</SUP>) / <I>a</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18028 or <MATH>4<I>a</I><SUP>2</SUP> + (<I>a</I>(4<I>a</I><SUP>2</SUP> + 1))/<I>m</I><SUP>2</SUP></MATH>, has to be made a square. Put
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18029 <MATH>4<I>a</I><SUP>2</SUP><I>m</I><SUP>2</SUP> + <I>a</I>(4<I>a</I><SUP>2</SUP> + 1) = (2<I>am</I> + <I>k</I>)<SUP>2</SUP></MATH>, and we have a solution.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18030 Diophantus has <I>a</I> = 5, leading to <MATH>100<I>m</I><SUP>2</SUP> + 505 = a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18031 = (10<I>m</I> + 5)<SUP>2</SUP></MATH>, say, which gives <I>m</I> = 24/5 and <I>n</I> = 413/60.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18032 <I>h, p, b</I> are thus determined in such a way that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18033 <MATH>(1/2)<I>pb</I><G>x</G><SUP>2</SUP> + <I>a</I> = <I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP></MATH> gives a rational solution.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18034 VI. 4. <MATH>(1/2)<I>xy</I> - <I>a</I> = <I>u</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18035 VI. 5. <MATH><I>a</I> - (1/2)<I>xy</I> = <I>u</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18036 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18037 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18038 VI. 6. <MATH>(1/2)<I>xy</I> + <I>x</I> = <I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18039 [Assume the triangle to be <I>h</I><G>x</G>, <I>p</I><G>x</G>, <I>b</I><G>x</G>, so that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18040 <MATH>(1/2)<I>pb</I><G>x</G><SUP>2</SUP> + <I>p</I><G>x</G> = <I>a</I></MATH>, and for a rational solution of this equa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18041 tion we must have ((1/2)<I>p</I>)<SUP>2</SUP> + <I>a</I>((1/2)<I>pb</I>) a square. Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18042 assumes <I>p</I> = 1, <I>b</I> = <I>m,</I> whence (1/2)<I>am</I> + 1/4 or 2<I>am</I> + 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18043 = a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18044 But, since the triangle is rational, <I>m</I><SUP>2</SUP> + 1 = a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18045 That is, we have a double equation. Difference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18046 <MATH>= <I>m</I><SUP>2</SUP> - 2<I>am</I> = <I>m</I>(<I>m</I> - 2<I>a</I>)</MATH>. Put
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18047 <MATH>2<I>am</I> + 1 = {1/2(<I>m</I> - &hordar;(<I>m</I> - 2<I>a</I>)}<SUP>2</SUP> = <I>a</I><SUP>2</SUP>, and <I>m</I> = (<I>a</I><SUP>2</SUP> - 1)/2<I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18048 The sides of the auxiliary triangle are thus determined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18049 in such a way that the original equation in <G>x</G> is solved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18050 rationally.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18051 VI. 7. <MATH>(1/2)<I>xy</I> - <I>x</I> = <I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18052 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18053 <pb n=509><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18054 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18055 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18056 VI. 8. (1/2)<I>xy</I> + (<I>x</I> + <I>y</I>) = <I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18057 VI. 9. (1/2)<I>xy</I> - (<I>x</I> + <I>y</I>) = <I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18058 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18059 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18060 <p>[With the same assumptions we have in these cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18061 to make {1/2 (<I>p</I> + <I>b</I>)}<SUP>2</SUP> + <I>a</I> ((1/2)<I>pb</I>) a square. Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18062 assumes as before 1, <I>m</I> for the values of <I>p, b,</I> and obtains
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18063 the double equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18064 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18065 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18066 1/4 (<I>m</I> + 1)<SUP>2</SUP> + (1/2)<I>am</I> = square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18067 <I>m</I><SUP>2</SUP> + 1 = square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18068 </BRACE>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18069 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18070 <I>m</I><SUP>2</SUP> + (2<I>a</I> + 2)<I>m</I> + 1 = square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18071 or <I>m</I><SUP>2</SUP> + 1 = square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18072 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18073 </MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18074 solving in the usual way.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18075 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18076 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18077 VI. 10. (1/2)<I>xy</I> + <I>x</I> + <I>z</I> = <I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18078 VI. 11. (1/2)<I>xy</I> - (<I>x</I> + <I>z</I>) = <I>a.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18079 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18080 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18081 <p>[In these cases the auxiliary right-angled triangle has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18082 to be found such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18083 <MATH>{1/2 (<I>h</I> + <I>p</I>)}<SUP>2</SUP> + <I>a</I>((1/2)<I>pb</I>) =</MATH> a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18084 <p>Diophantus assumes it formed from 1, <I>m</I> + 1; thus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18085 <MATH>1/4 (<I>h</I> + <I>p</I>)<SUP>2</SUP> = 1/4 {<I>m</I><SUP>2</SUP> + 2<I>m</I> + 2 + <I>m</I><SUP>2</SUP> + 2<I>m</I>}<SUP>2</SUP> = (<I>m</I><SUP>2</SUP> + 2<I>m</I> + 1)<SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18086 and <I>a</I>((1/2)<I>pb</I>) = <I>a</I> (<I>m</I> + 1) (<I>m</I><SUP>2</SUP> + 2<I>m</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18087 Therefore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18088 <MATH><I>m</I><SUP>4</SUP> + (<I>a</I> + 4)<I>m</I><SUP>3</SUP> + (3<I>a</I> + 6)<I>m</I><SUP>2</SUP> + (2<I>a</I> + 4)<I>m</I> + 1</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18089 = a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18090 <MATH>= {1 + (<I>a</I> + 2)<I>m</I> - <I>m</I><SUP>2</SUP>}<SUP>2</SUP></MATH>, say;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18091 and <I>m</I> is found.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18092 <p>Lemma 1 to VI. 12. <MATH><I>x</I> = <I>u</I><SUP>2</SUP>, <I>x</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, (1/2)<I>xy</I> + <I>y</I> = <I>w</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18093 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18094 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18095 VI. 12. (1/2)<I>xy</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> + <I>y</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18096 VI. 13. (1/2)<I>xy</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> - <I>y</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18097 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18098 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18099 <p>[These problems and the two following are interesting,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18100 but their solutions run to some length; therefore only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18101 one case can here be given. We will take VI. 12 with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18102 its Lemma 1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18103 <pb n=510><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18104 <I>Lemma</I> 1. If a rational right-angled triangle be formed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18105 from <I>m, n,</I> the perpendicular sides are 2<I>mn, m</I><SUP>2</SUP> - <I>n</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18106 We will suppose the greater of the two to be 2<I>mn.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18107 The first two relations are satisfied by making <I>m</I> = 2<I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18108 Form, therefore, a triangle from <G>x</G>, 2<G>x</G>. The third con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18109 dition then gives <MATH>6<G>x</G><SUP>4</SUP> + 3<G>x</G><SUP>2</SUP> = a square or 6<G>x</G><SUP>2</SUP> + 3 =</MATH> a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18110 square. One solution is <G>x</G> = 1 (and there are an infinite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18111 number of others to be found by means of it). If <G>x</G> = 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18112 the triangle is formed from 1, 2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18113 <p>VI. 12. Suppose the triangle to be (<I>h</I><G>x</G>, <I>b</I><G>x</G>, <I>p</I><G>x</G>). Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18114 <MATH>((1/2)<I>pb</I>)<G>x</G><SUP>2</SUP> + <I>p</I><G>x</G> = a square = (<I>k</I><G>x</G>)<SUP>2</SUP></MATH>, say, and <MATH><G>x</G> = <I>p</I>/(<I>k</I><SUP>2</SUP> - (1/2)<I>pb</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18115 This value must be such as to make <MATH>((1/2)<I>pb</I>)<G>x</G><SUP>2</SUP> + <I>b</I><G>x</G></MATH> a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18116 also. By substitution of the value of <G>x</G> we get
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18117 <MATH>{<I>bpk</I><SUP>2</SUP> + (1/2)<I>p</I><SUP>2</SUP><I>b</I>(<I>p</I> - <I>b</I>)}/(<I>k</I><SUP>2</SUP> - (1/2)<I>pb</I>)<SUP>2</SUP></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18118 so that <I>bpk</I><SUP>2</SUP> + (1/2)<I>p</I><SUP>2</SUP><I>b</I>(<I>p</I> - <I>b</I>) must be a square; or, if <I>p,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18119 the greater perpendicular, is made a square number,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18120 <I>bk</I><SUP>2</SUP> + (1/2)<I>pb</I>(<I>p</I> - <I>b</I>) has to be made a square. This by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18121 Lemma 2 (see p. 467 above) can be made a square if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18122 <I>b</I> + (1/2)<I>pb</I>(<I>p</I> - <I>b</I>) is a square. <I>How to solve these problems,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18123 says Diophantus, <I>is shown in the Lemmas.</I> It is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18124 clear how they were applied, but, in fact, his solution
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18125 is such as to make <I>p, p</I> - <I>b,</I> and <I>b</I> + (1/2)<I>pb</I> all squares,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18126 namely <I>b</I> = 3, <I>p</I> = 4, <I>h</I> = 5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18127 <p>Accordingly, putting for the original triangle 3<G>x</G>, 4<G>x</G>, 5<G>x</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18128 we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18129 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18130 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18131 6<G>x</G><SUP>2</SUP> + 4<G>x</G> = a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18132 6<G>x</G><SUP>2</SUP> + 3<G>x</G> = a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18133 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18134 </MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18135 <p>Assuming <MATH>6<G>x</G><SUP>2</SUP> + 4<G>x</G> = <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP></MATH>, we have <MATH><G>x</G> = 4 / (<I>m</I><SUP>2</SUP> - 6)</MATH>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18136 the second condition gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18137 <MATH>96/(<I>m</I><SUP>4</SUP> - 12<I>m</I><SUP>2</SUP> + 36) + 12/(<I>m</I><SUP>2</SUP> - 6) =</MATH> a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18138 or <MATH>12<I>m</I><SUP>2</SUP> + 24 =</MATH> a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18139 <p>This can be solved, since <I>m</I> = 1 satisfies it (Lemma 2).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18140 <p>A solution is <I>m</I><SUP>2</SUP> = 25, whence <G>x</G> = 4/19.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18141 <MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18142 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18143 VI. 14. (1/2)<I>xy</I> - <I>z</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> - <I>x</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18144 VI. 15. (1/2)<I>xy</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> + <I>x</I> = <I>v</I><SUP>2</SUP>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18145 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18146 </MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18147 <pb n=511><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18148 <p>[The auxiliary right-angled triangle in this case must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18149 be such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18150 <MATH><I>m</I><SUP>2</SUP><I>hp</I> - (1/2)<I>pb.p</I>(<I>h</I> - <I>p</I>)</MATH> is a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18151 <p>If, says Diophantus (VI. 14), we form a triangle from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18152 the numbers <I>X</I><SUB>1</SUB>, <I>X</I><SUB>2</SUB> and suppose that <I>p</I> = 2<I>X</I><SUB>1</SUB><I>X</I><SUB>2</SUB>, and if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18153 we then divide out by (<I>X</I><SUB>1</SUB> - <I>X</I><SUB>2</SUB>)<SUP>2</SUP>, which is equal to <I>h</I> - <I>p,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18154 we must find a square <MATH><I>k</I><SUP>2</SUP>[= <I>m</I><SUP>2</SUP> √(<I>X</I><SUB>1</SUB> - <I>X</I><SUB>2</SUB>)<SUP>2</SUP>]</MATH> such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18155 <MATH><I>k</I><SUP>2</SUP><I>hp</I> - (1/2)<I>pb.p</I></MATH> is a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18156 <p>The problem, says Diophantus, can be solved if <I>X</I><SUB>1</SUB>, <I>X</I><SUB>2</SUB>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18157 are ‘similar plane numbers’ (numbers such as <I>ab, m</I><SUP>2</SUP>/<I>n</I><SUP>2</SUP> <I>ab</I>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18158 This is stated without proof, but it can easily be verified
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18159 that, if <I>k</I><SUP>2</SUP> = <I>X</I><SUB>1</SUB><I>X</I><SUB>2</SUB>, the expression is a square. Dioph.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18160 takes 4, 1 as the numbers, so that <I>k</I><SUP>2</SUP> = 4. The equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18161 for <I>m</I> becomes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18162 <MATH>8 . 17<I>m</I><SUP>2</SUP> - 4 . 15 . 8 . 9 =</MATH> a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18163 or <MATH>136<I>m</I><SUP>2</SUP> - 4320 =</MATH> a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18164 The solution <I>m</I><SUP>2</SUP> = 36 (derived from the fact that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18165 <MATH><I>k</I><SUP>2</SUP> = <I>m</I><SUP>2</SUP>/(<I>X</I><SUB>1</SUB> - <I>X</I><SUB>2</SUB>)<SUP>2</SUP>, or 4 = <I>m</I><SUP>2</SUP>/3<SUP>2</SUP>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18166 satisfies the condition that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18167 <MATH><I>m</I><SUP>2</SUP><I>hp</I> - (1/2)<I>pb.p</I>(<I>h</I> - <I>p</I>)</MATH> is a square.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18168 <p>VI. 16. <MATH><G>x</G> + <G>h</G> = <I>x,</I> <G>x</G>/<G>h</G> = <I>y</I>/<I>z</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18169 <p>[To find a rational right-angled triangle such that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18170 number representing the (portion intercepted within
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18171 the triangle of the) bisector of an acute angle is rational.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18172 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18173 Let the bisector be 5<G>x</G>, the segment <I>BD</I> of the base 3<G>x</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18174 so that the perpendicular is 4<G>x</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18175 Let <I>CB</I> = 3<I>n.</I> Then <I>AC</I> : <I>AB</I> = <I>CD</I> : <I>DB,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18176 <pb n=512><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18177 so that <I>AC</I> = 4(<I>n</I> - <G>x</G>). Therefore (Eucl. I. 47)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18178 <MATH>16(<I>n</I><SUP>2</SUP> - 2<I>n</I><G>x</G> + <G>x</G><SUP>2</SUP>) = 16<G>x</G><SUP>2</SUP> + 9<I>n</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18179 so that <MATH><G>x</G> = 7<I>n</I><SUP>2</SUP>/32<I>n</I> = (7/32)<I>n</I></MATH>. [Dioph. has <I>n</I> = 1.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18180 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18181 VI. 17. <MATH>(1/2)<I>xy</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>y</I> + <I>z</I> = <I>v</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18182 [Let <G>x</G> be the area (1/2)<I>xy,</I> and let <I>z</I> = <I>k</I><SUP>2</SUP> - <G>x</G>. Since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18183 <I>xy</I> = 2<G>x</G>, suppose <I>x</I> = 2, <I>y</I> = <G>x</G>. Therefore 2 + <I>k</I><SUP>2</SUP> must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18184 be a cube. As we have seen (p. 475), Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18185 takes (<I>m</I> - 1)<SUP>3</SUP> for the cube and (<I>m</I> + 1)<SUP>2</SUP> for <I>k</I><SUP>2</SUP>, giving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18186 <MATH><I>m</I><SUP>3</SUP> - 3<I>m</I><SUP>2</SUP> + 3<I>m</I> - 1 = <I>m</I><SUP>2</SUP> + 2<I>m</I> + 3</MATH>, whence <I>m</I> = 4. There-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18187 fore <I>k</I> = 5, and we assume <MATH>(1/2)<I>xy</I> = <G>x</G>, <I>z</I> = 25 - <G>x</G></MATH>, with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18188 <I>x</I> = 2, <I>y</I> = <G>x</G> as before. Then we have to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18189 <MATH>(25 - <G>x</G>)<SUP>2</SUP> = 4 + <G>x</G><SUP>2</SUP></MATH>, and <G>x</G> = 621/50.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18190 VI. 18. <MATH>(1/2)<I>xy</I> + <I>z</I> = <I>u</I><SUP>3</SUP>, <I>x</I> + <I>y</I> + <I>z</I> = <I>v</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18191 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18192 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18193 VI. 19. <MATH>(1/2)<I>xy</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>y</I> + <I>z</I> = <I>v</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18194 [Here a right-angled triangle is formed from one odd
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18195 number, say 2<G>x</G> + 1, according to the Pythagorean for-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18196 mula <MATH><I>m</I><SUP>2</SUP> + {(1/2)(<I>m</I><SUP>2</SUP> - 1)}<SUP>2</SUP> = {(1/2)(<I>m</I><SUP>2</SUP> + 1)}<SUP>2</SUP></MATH>, where <I>m</I> is an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18197 odd number. The sides are therefore 2<G>x</G> + 1, 2<G>x</G><SUP>2</SUP> + 2<G>x</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18198 2<G>x</G><SUP>2</SUP> + 2<G>x</G> + 1. Since the perimeter = a cube,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18199 <MATH>4<G>x</G><SUP>2</SUP> + 6<G>x</G> + 2 = (4<G>x</G> + 2) (<G>x</G> + 1) =</MATH> a cube.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18200 Or, if we divide the sides by <G>x</G> + 1, 4<G>x</G> + 2 has to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18201 made a cube.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18202 Again <MATH>(1/2)<I>xy</I> + <I>x</I> = ((2<G>x</G><SUP>3</SUP> + 3<G>x</G><SUP>2</SUP> + <G>x</G>)/((<G>x</G> + 1)<SUP>2</SUP>)) + ((2<G>x</G> + 1)/(<G>x</G> + 1)) =</MATH> a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18203 which reduces to <MATH>2<G>x</G> + 1 =</MATH> a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18204 But 4<G>x</G> + 2 is a cube. We therefore put 8 for the cube,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18205 and <G>x</G> = 1 1/2.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18206 VI. 20. <MATH>(1/2)<I>xy</I> + <I>x</I> = <I>u</I><SUP>3</SUP>, <I>x</I> + <I>y</I> + <I>z</I> = <I>v</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18207 </BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18208 <p>VI. 21. <MATH><I>x</I> + <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>v</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18209 <p>[Form a right-angled triangle from <G>x</G>, 1, i. e. (2<G>x</G>, <G>x</G><SUP>2</SUP> - 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18210 <G>x</G><SUP>2</SUP> + 1). Then 2<G>x</G><SUP>2</SUP> + 2<G>x</G> must be a square, and <G>x</G><SUP>3</SUP> + 2<G>x</G><SUP>2</SUP> + <G>x</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18211 <pb n=513><head>INDETERMINATE ANALYSIS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18212 a cube. Put <MATH>2<G>x</G><SUP>2</SUP> + 2<G>x</G> = <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP></MATH>, so that <G>x</G> = 2/(<I>m</I><SUP>2</SUP> - 2),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18213 and we have to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18214 <MATH>8/(<I>m</I><SUP>2</SUP> - 2)<SUP>3</SUP> + 8/(<I>m</I><SUP>2</SUP> - 2)<SUP>2</SUP> + 2/(<I>m</I><SUP>2</SUP> - 2), or (2<I>m</I><SUP>4</SUP>)/(<I>m</I><SUP>2</SUP> - 2)<SUP>3</SUP></MATH>, a cube.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18215 Make 2<I>m</I> a cube = <I>n</I><SUP>3</SUP>, so that 2<I>m</I><SUP>4</SUP> = <I>m</I><SUP>3</SUP><I>n</I><SUP>3</SUP>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18216 <I>m</I> = (1/2)<I>n</I><SUP>3</SUP>; therefore <MATH><G>x</G> = 8/(<I>n</I><SUP>6</SUP> - 8)</MATH>, and <G>x</G> must be made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18217 greater than 1, in order that <G>x</G><SUP>2</SUP> - 1 may be positive.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18218 <p>Therefore 8 < <I>n</I><SUP>6</SUP> < 16;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18219 this is satisfied by <I>n</I><SUP>6</SUP> = 729/64 or <I>n</I><SUP>3</SUP> = 27/8, and <I>m</I> = 27/16.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18220 <p>VI. 22. <MATH><I>x</I> + <I>y</I> + <I>z</I> = <I>u</I><SUP>3</SUP>, (1/2)<I>xy</I> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>v</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18221 <p>[(1) First seek a rational right-angled triangle such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18222 that its perimeter and its area are given numbers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18223 say <I>p, m.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18224 <p>Let the perpendiculars be 1/<G>x</G>, 2<I>m</I><G>x</G>; therefore the hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18225 tenuse = <I>p</I> - (1/<G>x</G>) - 2<I>m</I><G>x</G>, and (Eucl. I. 47)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18226 <MATH>(1/<G>x</G><SUP>2</SUP>) + 4<I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + (<I>p</I><SUP>2</SUP> + 4<I>m</I>) - (2<I>p</I>/<G>x</G>) - 4<I>mp</I><G>x</G> = (1/<G>x</G><SUP>2</SUP>) + 4<I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18227 or <I>p</I><SUP>2</SUP> + 4<I>m</I> = 4<I>mp</I><G>x</G> + (2<I>p</I>/<G>x</G>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18228 that is, (<I>p</I><SUP>2</SUP> + 4<I>m</I>)<G>x</G> = 4<I>mp</I><G>x</G><SUP>2</SUP> + 2<I>p</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18229 <p>(2) In order that this may have a rational solution,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18230 <MATH>{(1/2)(<I>p</I><SUP>2</SUP> + 4<I>m</I>)}<SUP>2</SUP> - 8<I>p</I><SUP>2</SUP><I>m</I></MATH> must be a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18231 i.e. <MATH>4<I>m</I><SUP>2</SUP> - 6<I>p</I><SUP>2</SUP><I>m</I> + (1/4)<I>p</I><SUP>4</SUP> =</MATH> a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18232 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18233 or <MATH><I>m</I><SUP>2</SUP> - (3/2)<I>p</I><SUP>2</SUP><I>m</I> + (1/16)<I>p</I><SUP>4</SUP> =</MATH> a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18234 Also, by the second condition, <MATH><I>m</I> + <I>p</I> =</MATH> a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18235 </BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18236 <p>To solve this, we must take for <I>p</I> some number which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18237 is both a square and a cube (in order that it may be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18238 possible, by multiplying the second equation by some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18239 square, to make the constant term equal to the constant
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18240 <pb n=514><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18241 term in the first). Diophantus takes <I>p</I> = 64, making
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18242 the equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18243 <BRACE>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18244 <MATH><I>m</I><SUP>2</SUP> - 6144<I>m</I> + 1048576 =</MATH> a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18245 <MATH><I>m</I> + 64 =</MATH> a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18246 </BRACE>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18247 Multiplying the second by 16384, and subtracting the two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18248 expressions, we have as the difference <I>m</I><SUP>2</SUP> - 22528<I>m.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18249 <p>Diophantus observes that, if we take <I>m, m</I> - 22528 as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18250 the factors, we obtain <I>m</I> = 7680, an impossible value for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18251 the area of a right-angled triangle of perimeter <I>p</I> = 64.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18252 <p>We therefore take as factors 11<I>m,</I> (1/11)<I>m</I> - 2048, and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18253 equating the square of half the difference (= (60/11)<I>m</I> + 1024)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18254 to 16384<I>m</I> + 1048576, we have <I>m</I> = 39424/225.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18255 <p>(3) Returning to the original problem, we have to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18256 substitute this value for <I>m</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18257 <MATH>(64 - (1/<G>x</G>) - 2<I>m</I><G>x</G>)<SUP>2</SUP> = (1/<G>x</G><SUP>2</SUP>) + 4<I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18258 and we obtain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18259 <MATH>78848<G>x</G><SUP>2</SUP> - 8432<G>x</G> + 225 = 0</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18260 the solution of which is rational, namely <G>x</G> = 25/448 (or 9/176).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18261 Diophantus naturally takes the first value, though the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18262 second gives the same triangle.]
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18263 <p>VI. 23. <MATH><I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP> + <I>u, z</I><SUP>2</SUP>/<I>x</I> = <I>v</I><SUP>3</SUP> + <I>v</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18264 <p>VI. 24. <MATH><I>z</I> = <I>u</I><SUP>3</SUP> + <I>u, x</I> = <I>v</I><SUP>3</SUP> - <I>v, y</I> = <I>w</I><SUP>3</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18265 <p>[VI. 6, 7]. <MATH>((1/2)<I>x</I>)<SUP>2</SUP> + (1/2)<I>mxy</I> = <I>u</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18266 <p>[VI. 8, 9]. <MATH>{(1/2)(<I>x</I> + <I>y</I>)}<SUP>2</SUP> + (1/2)<I>mxy</I> = <I>u</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18267 <p>[VI. 10, 11]. <MATH>{(1/2)(<I>z</I> + <I>x</I>)}<SUP>2</SUP> + (1/2)<I>mxy</I> = <I>u</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18268 <p>[VI. 12.] <MATH><I>y</I> + (<I>x</I> - <I>y</I>) . (1/2)<I>xy</I> = <I>u</I><SUP>2</SUP>, <I>x</I> = <I>v</I><SUP>2</SUP>. (<I>x</I> > <I>y.</I>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18269 <p>[VI. 14, 15]. <MATH><I>u</I><SUP>2</SUP><I>zx</I> - (1/2)<I>xy</I> . <I>x</I>(<I>z</I> - <I>x</I>) = <I>v</I><SUP>2</SUP>. (<I>u</I><SUP>2</SUP> < or > (1/2)<I>xy.</I>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18270 <C>The treatise on Polygonal Numbers.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18271 <p>The subject of Polygonal Numbers on which Diophantus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18272 also wrote is, as we have seen, an old one, going back to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18273 <pb n=515><head>THE TREATISE ON POLYGONAL NUMBERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18274 Pythagoreans, while Philippus of Opus and Speusippus carried
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18275 on the tradition. Hypsicles (about 170 B.C.) is twice men-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18276 tioned by Diophantus as the author of a ‘definition’ of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18277 a polygonal number which, although it does not in terms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18278 mention any polygonal number beyond the pentagonal,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18279 amounts to saying that the <I>n</I>th <I>a</I>-gon (1 counting as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18280 first) is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18281 <MATH>(1/2)<I>n</I> {2 + (<I>n</I> - 1) (<I>a</I> - 2)}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18282 Theon of Smyrna, Nicomachus and Iamblichus all devote
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18283 some space to polygonal numbers. Nicomachus in particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18284 gives various rules for transforming triangles into squares,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18285 squares into pentagons, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18286 <p>1. If we put two consecutive triangles together, we get a square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18287 In fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18288 <MATH>(1/2)(<I>n</I> - 1)<I>n</I> + (1/2)<I>n</I>(<I>n</I> + 1) = <I>n</I><SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18289 <p>2. A pentagon is obtained from a square by adding to it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18290 a triangle the side of which is 1 less than that of the square;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18291 similarly a hexagon from a pentagon by adding a triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18292 the side of which is 1 less than that of the pentagon, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18293 <p>In fact
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18294 <MATH>(1/2)<I>n</I> {2 + (<I>n</I> - 1) (<I>a</I> - 2)} + (1/2)(<I>n</I> - 1)<I>n</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18295 = (1/2)<I>n</I>[2 + (<I>n</I> - 1) {(<I>a</I> + 1) - 2}]</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18296 <p>3. Nicomachus sets out the first triangles, squares, pentagons,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18297 hexagons and heptagons in a diagram thus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18298 <table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18299 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18300 <td>Triangles</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18301 <td>1</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18302 <td>3</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18303 <td> 6</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18304 <td>10</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18305 <td>15</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18306 <td>21</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18307 <td> 28</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18308 <td> 36</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18309 <td> 45</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18310 <td> 55,</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18311 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18312 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18313 <td>Squares</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18314 <td>1</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18315 <td>4</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18316 <td> 9</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18317 <td>16</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18318 <td>25</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18319 <td>36</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18320 <td> 49</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18321 <td> 64</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18322 <td> 81</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18323 <td>100,</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18324 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18325 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18326 <td>Pentagons</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18327 <td>1</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18328 <td>5</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18329 <td>12</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18330 <td>22</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18331 <td>35</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18332 <td>51</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18333 <td> 70</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18334 <td> 92</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18335 <td>117</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18336 <td>145,</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18337 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18338 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18339 <td>Hexagons</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18340 <td>1</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18341 <td>6</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18342 <td>15</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18343 <td>28</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18344 <td>45</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18345 <td>66</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18346 <td> 91</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18347 <td>120</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18348 <td>153</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18349 <td>190,</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18350 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18351 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18352 <td>Heptagons</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18353 <td>1</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18354 <td>7</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18355 <td>18</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18356 <td>34</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18357 <td>55</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18358 <td>81</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18359 <td>112</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18360 <td>148</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18361 <td>189</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18362 <td>235,</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18363 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18364 </table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18365 and observes that:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18366 <p>Each polygon is equal to the polygon immediately above it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18367 in the diagram <I>plus</I> the triangle with 1 less in its side, i. e. the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18368 triangle in the preceding column.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18369 <pb n=516><head>DIOPHANTUS OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18370 <p>4. The vertical columns are in arithmetical progression, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18371 common difference being the triangle in the preceding column.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18372 <p>Plutarch, a contemporary of Nicomachus, mentions another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18373 method of transforming triangles into squares. <I>Every tri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18374 angular number taken eight times and then increased by 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18375 gives a square.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18376 <p>In fact, <MATH>8.(1/2)<I>n</I>(<I>n</I> + 1) + 1 = (2<I>n</I> + 1)<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18377 <p>Only a fragment of Diophantus's treatise <I>On Polygonal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18378 Numbers</I> survives. Its character is entirely different from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18379 that of the <I>Arithmetica.</I> The method of proof is strictly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18380 geometrical, and has the disadvantage, therefore, of being long
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18381 and involved. He begins with some preliminary propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18382 of which two may be mentioned. Prop. 3 proves that, if <I>a</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18383 the first and <I>l</I> the last term in an arithmetical progression
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18384 of <I>n</I> terms, and if <I>s</I> is the sum of the terms, <MATH>2<I>s</I> = <I>n</I>(<I>l</I> + <I>a</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18385 Prop. 4 proves that, if 1, 1 + <I>b,</I> 1 + 2<I>b,</I> ... 1 + (<I>n</I> - 1)<I>b</I> be an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18386 A. P., and <I>s</I> the sum of the terms,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18387 <MATH>2<I>s</I> = <I>n</I> {2 + (<I>n</I> - 1)<I>b</I>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18388 <p>The main result obtained in the fragment as we have it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18389 is a generalization of the formula <MATH>8.(1/2)<I>n</I>(<I>n</I> + 1) + 1 = (2<I>n</I> + 1)<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18390 Prop. 5 proves the fact stated in Hypsicles's definition and also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18391 (the generalization referred to) that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18392 <MATH>8<I>P</I>(<I>a</I> - 2) + (<I>a</I> - 4)<SUP>2</SUP> =</MATH> a square,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18393 where <I>P</I> is any polygonal number with <I>a</I> angles.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18394 <p>It is also proved that, if <I>P</I> be the <I>n</I>th <I>a</I>-gonal number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18395 (1 being the first),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18396 <MATH>8<I>P</I>(<I>a</I> - 2) + (<I>a</I> - 4)<SUP>2</SUP> = {2 + (2<I>n</I> - 1) (<I>a</I> - 2)}<SUP>2</SUP></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18397 <p>Diophantus deduces rules as follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18398 <p>1. <I>To find the number from its side.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18399 <MATH><I>P</I> = ({2 + (2<I>n</I> - 1) (<I>a</I> - 2)}<SUP>2</SUP> - (<I>a</I> - 4)<SUP>2</SUP>)/(8(<I>a</I> - 2))</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18400 <p>2. <I>To find the side from the number.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18401 <MATH><I>n</I> = (1/2)((√{8<I>P</I>(<I>a</I> - 2) + (<I>a</I> - 4)<SUP>2</SUP>} - 2)/(<I>a</I> - 2) + 1)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18402 <pb n=517><head>THE TREATISE ON POLYGONAL NUMBERS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18403 <p>The last proposition, which breaks off in the middle, is:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18404 <p><I>Given a number, to find in how many ways it can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18405 polygonal.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18406 <p>The proposition begins in a way which suggests that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18407 Diophantus first proved geometrically that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18408 <MATH>8<I>P</I>(<I>a</I> - 2) + (<I>a</I> - 4)<SUP>2</SUP> = {2 + (2<I>n</I> - 1) (<I>a</I> - 2)}<SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18409 then <MATH>2<I>P</I> = <I>n</I>{2 + (<I>n</I> - 1) (<I>a</I> - 2)}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18410 Wertheim. (in his edition of Diophantus) has suggested a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18411 restoration of the complete proof of this proposition, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18412 I have shown (in my edition) how the proof can be made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18413 shorter. Wertheim adds an investigation of the main pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18414 blem, but no doubt opinions will continue to differ as to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18415 whether Diophantus actually solved it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18416 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18417 <C>XXI
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18418 COMMENTATORS AND BYZANTINES</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18419 <p>WE have come to the last stage of Greek mathematics; it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18420 only remains to include in a last chapter references to com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18421 mentators of more or less note who contributed nothing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18422 original but have preserved, among observations and explana-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18423 tions obvious or trivial from a mathematical point of view,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18424 valuable extracts from works which have perished, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18425 historical allusions which, in the absence of original docu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18426 ments, are precious in proportion to their rarity. Nor must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18427 it be forgotten that in several cases we probably owe to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18428 commentators the fact that the masterpieces of the great
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18429 mathematicians have survived, wholly or partly, in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18430 original Greek or at all. This may have been the case even
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18431 with the works of Archimedes on which Eutocius wrote com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18432 mentaries.. It was no doubt these commentaries which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18433 aroused in the school of Isidorus of Miletus (the colleague
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18434 of Anthemius as architect of Saint Sophia at Constantinople)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18435 a new interest in the works of Archimedes and caused them
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18436 to be sought out in the various libraries or wherever they had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18437 lain hid. This revived interest apparently had the effect of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18438 evoking new versions of the famous works commented upon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18439 in a form more convenient for the student, with the Doric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18440 dialect of the original eliminated; this translation of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18441 Doric into the more familiar dialect was systematically
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18442 carried out in those books only which Eutocius commented
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18443 on, and it is these versions which alone survive. Again,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18444 Eutocius's commentary on Apollonius's <I>Conics</I> is extant for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18445 the first four Books, and it is probably owing to their having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18446 been commented on by Eutocius, as well as to their being
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18447 more elementary than the rest, that these four Books alone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18448 <pb n=519><head>SERENUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18449 survive in Greek. Tannery, as we have seen, conjectured
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18450 that, in like manner, the first six of the thirteen Books of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18451 Diophantus's <I>Arithmetica</I> survive because Hypatia wrote
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18452 commentaries on these Books only and did not reach the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18453 others.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18454 <p>The first writer who calls for notice in this chapter is one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18455 who was rather more than a commentator in so far as he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18456 wrote a couple of treatises to supplement the <I>Conics</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18457 Apollonius, I mean SERENUS. Serenus came from Antinoeia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18458 or Antinoupolis, a city in Egypt founded by Hadrian (A.D.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18459 117-38). His date is uncertain, but he most probably be-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18460 longed to the fourth century A.D., and came between Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18461 and Theon of Alexandria. He tells us himself that he wrote
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18462 a commentary on the <I>Conics</I> of Apollonius.<note>Serenus, <I>Opuscula,</I> ed. Heiberg, p. 52. 25-6.</note> This has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18463 perished and, apart from a certain proposition ‘of Serenus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18464 the philosopher, from the Lemmas’ preserved in certain manu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18465 scripts of Theon of Smyrna (to the effect that, if a number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18466 rectilineal angles be subtended at a point on a diameter of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18467 circle which is not the centre, by equal arcs of that circle, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18468 angle nearer to the centre is always less than the angle more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18469 remote), we have only the two small treatises by him entitled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18470 <I>On the Section of a Cylinder</I> and <I>On the Section of a Cone.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18471 These works came to be connected, from the seventh century
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18472 onwards, with the <I>Conics</I> of Apollonius, on account of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18473 affinity of the subjects, and this no doubt accounts for their
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18474 survival. They were translated into Latin by Commandinus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18475 in 1566; the first Greek text was brought out by Halley along
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18476 with his Apollonius (Oxford 1710), and we now have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18477 definitive text edited by Heiberg (Teubner 1896).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18478 <C>(<G>a</G>) <I>On the Section of a Cylinder.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18479 <p>The occasion and the object of the tract <I>On the Section of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18480 a Cylinder</I> are stated in the preface. Serenus observes that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18481 many persons who were students of geometry were under the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18482 erroneous impression that the oblique section of a cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18483 was different from the oblique section of a cone known as an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18484 ellipse, whereas it is of course the same curve. Hence he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18485 thinks it necessary to establish, by a regular geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18486 <pb n=520><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18487 proof, that the said oblique sections cutting all the generators
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18488 are equally ellipses whether they are sections of a cylinder or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18489 of a cone. He begins with ‘a more general definition’ of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18490 cylinder to include any oblique circular cylinder. ‘If in two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18491 equal and parallel circles which remain fixed the diameters,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18492 while remaining parallel to one another throughout, are moved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18493 round in the planes of the circles about the centres, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18494 remain fixed, and if they carry round with them the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18495 joining their extremities on the same side until they bring it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18496 back again to the same place, let the surface described by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18497 straight line so carried round be called a <I>cylindrical surface.</I>’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18498 The <I>cylinder</I> is the figure contained by the parallel circles and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18499 the cylindrical surface intercepted by them; the parallel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18500 circles are the <I>bases,</I> the <I>axis</I> is the straight line drawn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18501 through their centres; the generating straight line in any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18502 position is a <I>side.</I> Thirty-three propositions follow. Of these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18503 Prop. 6 proves the existence in an oblique cylinder of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18504 parallel circular sections subcontrary to the series of which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18505 the bases are two, Prop. 9 that the section by any plane not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18506 parallel to that of the bases or of one of the subcontrary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18507 sections but cutting all the generators is not a circle; the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18508 next propositions lead up to the main results, namely those in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18509 Props. 14 and 16, where the said section is proved to have the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18510 property of the ellipse which we write in the form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18511 <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I>=<I>CD</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18512 and in Prop. 17, where the property is put in the Apollonian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18513 form involving the <I>latus rectum,</I> <MATH><I>QV</I><SUP>2</SUP>=<I>PV.VR</I></MATH> (see figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18514 on p. 137 above), which is expressed by saying that the square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18515 on the semi-ordinate is equal to the rectangle applied to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18516 <I>latus rectum PL,</I> having the abscissa <I>PV</I> as breadth and falling
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18517 short by a rectangle similar to the rectangle contained by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18518 diameter <I>PP′</I> and the <I>latus rectum PL</I> (which is determined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18519 by the condition <MATH><I>PL.PP′</I>=<I>DD′</I><SUP>2</SUP></MATH> and is drawn at right angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18520 to <I>PV</I>). Prop. 18 proves the corresponding property with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18521 reference to the conjugate diameter <I>DD′</I> and the correspond-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18522 ing <I>latus rectum,</I> and Prop. 19 gives the main property in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18523 form <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P′V</I>=<I>Q′V′</I><SUP>2</SUP>:<I>PV′.P′V′</I></MATH>. Then comes the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18524 proposition that ‘it is possible to exhibit a cone and a cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18525 which are alike cut in one and the same ellipse’ (Prop. 20).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18526 <pb n=521><head>SERENUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18527 Serenus then solves such problems as these: Given a cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18528 (or cylinder) and an ellipse on it, to find the cylinder (cone)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18529 which is cut in the same ellipse as the cone (cylinder)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18530 (Props. 21, 22); given a cone (cylinder), to find a cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18531 (cone) and to cut both by one and the same plane so that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18532 sections thus made shall be similar ellipses (Props. 23, 24).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18533 Props. 27, 28 deal with similar elliptic sections of a scalene
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18534 cylinder and cone; there are two pairs of infinite sets of these
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18535 similar to any one given section, the first pair being those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18536 which are parallel and subcontrary respectively to the given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18537 section, the other pair subcontrary to one another but not to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18538 either of the other sets and having the conjugate diameter
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18539 occupying the corresponding place to the transverse in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18540 other sets, and vice versa.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18541 <p>In the propositions (29-33) from this point to the end of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18542 the book Serenus deals with what is really an optical pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18543 blem. It is introduced by a remark about a certain geometer,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18544 Peithon by name, who wrote a tract on the subject of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18545 parallels. Peithon, not being satisfied with Euclid's treat-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18546 ment of parallels, thought to define parallels by means of an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18547 illustration, observing that parallels are such lines as are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18548 shown on a wall or a roof by the shadow of a pillar with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18549 a light behind it. This definition, it appears, was generally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18550 ridiculed; and Serenus seeks to rehabilitate Peithon, who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18551 was his friend, by showing that his statement is after all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18552 mathematically sound. He therefore proves, with regard to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18553 the cylinder, that, if any number of rays from a point outside
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18554 the cylinder are drawn touching it on both sides, all the rays
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18555 pass through the sides of a parallelogram (a section of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18556 cylinder parallel to the axis)—Prop. 29—and if they are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18557 produced farther to meet any other plane parallel to that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18558 of the parallelogram the points in which they meet the plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18559 will lie on two parallel lines (Prop. 30); he adds that the lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18560 will not <I>seem</I> parallel (<I>vide</I> Euclid's <I>Optics,</I> Prop. 6). The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18561 problem about the rays touching the surface of a cylinder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18562 suggests the similar one about any number of rays from an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18563 external point touching the surface of a <I>cone</I>; these meet the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18564 surface in points on a triangular section of the cone (Prop. 32)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18565 and, if produced to meet a plane parallel to that of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18566 triangle, meet that plane in points forming a similar triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18567 <pb n=522><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18568 (Prop. 33). Prop. 31 preceding these propositions is a par-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18569 ticular case of the constancy of the anharmonic ratio of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18570 pencil of four rays. If two sides <I>AB, AC</I> of a triangle meet
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18571 a transversal through <I>D,</I> an external point, in <I>E, F</I> and another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18572 ray <I>AG</I> between <I>AB</I> and <I>AC</I> cuts <I>DEF</I> in a point <I>G</I> such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18573 that <MATH><I>ED</I>:<I>DF</I>=<I>EG</I>:<I>GF</I></MATH>, then any other transversal through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18574 <I>D</I> meeting <I>AB, AG, AC</I> in <I>K, L, M</I> is also divided harmoni-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18575 cally, i.e. <MATH><I>KD</I>:<I>DM</I>=<I>KL</I>:<I>LM</I></MATH>. To prove the succeeding pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18576 positions, 32 and 33, Serenus uses this proposition and a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18577 reciprocal of it combined with the harmonic property of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18578 pole and polar with reference to an ellipse.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18579 <C>(<G>b</G>) <I>On the Section of a Cone.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18580 <p>The treatise <I>On the Section of a Cone</I> is even less important,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18581 although Serenus claims originality for it. It deals mainly
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18582 with the areas of triangular sections of right or scalene cones
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18583 made by planes passing through the vertex and either through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18584 the axis or not through the axis, showing when the area of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18585 a certain triangle of a particular class is a maximum, under
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18586 what conditions two triangles of a class may be equal in area,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18587 and so on, and solving in some easy cases the problem of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18588 finding triangular sections of given area. This sort of investi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18589 gation occupies Props. 1-57 of the work, these propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18590 including various lemmas required for the proofs of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18591 substantive theorems. Props. 58-69 constitute a separate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18592 section of the book dealing with the volumes of right cones
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18593 in relation to their heights, their bases and the areas of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18594 triangular sections through the axis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18595 <p>The essence of the first portion of the book up to Prop. 57
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18596 is best shown by means of modern notation. We will call <I>h</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18597 the height of a right cone, <I>r</I> the radius of the base; in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18598 case of an oblique cone, let <I>p</I> be the perpendicular from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18599 vertex to the plane of the base, <I>d</I> the distance of the foot of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18600 this perpendicular from the centre of the base, <I>r</I> the radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18601 of the base.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18602 <p>Consider first the right cone, and let 2<I>x</I> be the base of any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18603 triangular section through the vertex, while of course 2<I>r</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18604 the base of the triangular section through the axis. Then, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18605 <I>A</I> be the area of the triangular section with base 2<I>x,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18606 <MATH><I>A</I>=<I>x</I>√(<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18607 <pb n=523><head>SERENUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18608 <p>Observing that the sum of <I>x</I><SUP>2</SUP> and <MATH><I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP></MATH> is constant, we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18609 see that <I>A</I><SUP>2</SUP>, and therefore <I>A,</I> is a maximum when
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18610 <MATH><I>x</I><SUP>2</SUP>=<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP></MATH>, or <MATH><I>x</I><SUP>2</SUP>=1/2(<I>r</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18611 and, since <I>x</I> is not greater than <I>r,</I> it follows that, for a real
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18612 value of <I>x</I> (other than <I>r</I>), <I>h</I> is less than <I>r,</I> or the cone is obtuse-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18613 angled. When <I>h</I> is not less than <I>r,</I> the maximum triangle is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18614 the triangle through the axis and vice versa (Props. 5, 8);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18615 when <MATH><I>h</I>=<I>r</I></MATH>, the maximum triangle is also right-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18616 (Prop. 13).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18617 <p>If the triangle with base 2<I>c</I> is equal to the triangle through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18618 the axis, <MATH><I>h</I><SUP>2</SUP><I>r</I><SUP>2</SUP>=<I>c</I><SUP>2</SUP>(<I>r</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP>)</MATH>, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18619 <MATH>(<I>r</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>)(<I>c</I><SUP>2</SUP>-<I>h</I><SUP>2</SUP>)=0</MATH>, and,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18620 since <MATH><I>c</I> < <I>r</I>, <I>h</I>=<I>c</I></MATH>, so that <MATH><I>h</I> < <I>r</I></MATH> (Prop. 10). If <I>x</I> lies between <I>r</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18621 and <I>c</I> in this case, <MATH>(<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>)(<I>x</I><SUP>2</SUP>-<I>h</I><SUP>2</SUP>) > 0</MATH> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18622 <MATH><I>x</I><SUP>2</SUP>(<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP>) > <I>h</I><SUP>2</SUP><I>r</I><SUP>2</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18623 and the triangle with base 2<I>x</I> is greater than either of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18624 equal triangles with bases 2<I>r,</I> 2<I>c,</I> or 2<I>h</I> (Prop. 11).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18625 <p>In the case of the scalene cone Serenus compares individual
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18626 triangular sections belonging to one of three classes with other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18627 sections of the same class as regards their area. The classes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18628 are:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18629 <p>(1) axial triangles, including all sections through the axis;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18630 <p>(2) isosceles sections, i.e. the sections the bases of which are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18631 perpendicular to the projection of the axis of the cone on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18632 plane of the base;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18633 <p>(3) a set of triangular sections the bases of which are (<I>a</I>) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18634 diameter of the circular base which passes through the foot of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18635 the perpendicular from the vertex to the plane of the base, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18636 (<I>b</I>) the chords of the circular base parallel to that diameter.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18637 <p>After two preliminary propositions (15, 16) and some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18638 lemmas, Serenus compares the areas of the first class of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18639 triangles through the axis. If, as we said, <I>p</I> is the perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18640 dicular from the vertex to the plane of the base, <I>d</I> the distance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18641 of the foot of this perpendicular from the centre of the base,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18642 and <G>q</G> the angle which the base of any axial triangle with area
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18643 <I>A</I> makes with the base of the axial triangle passing through
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18644 <I>p</I> the perpendicular,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18645 <MATH><I>A</I>=<I>r</I>√(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>sin<SUP>2</SUP><G>q</G>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18646 <p>This area is a minimum when <G>q</G>=0, and increases with <G>q</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18647 <pb n=524><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18648 until <MATH><G>Q</G>=1/2<G>p</G></MATH> when it is a maximum, the triangle being then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18649 isosceles (Prop. 24).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18650 <p>In Prop. 29 Serenus takes up the third class of sections with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18651 bases parallel to <I>d.</I> If the base of such a section is 2<I>x,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18652 <MATH><I>A</I>=<I>x</I>√(<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>p</I><SUP>2</SUP>)</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18653 and, as in the case of the right cone, we must have for a real
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18654 maximum value
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18655 <MATH><I>x</I><SUP>2</SUP>=1/2(<I>r</I><SUP>2</SUP>+<I>p</I><SUP>2</SUP>)</MATH>, while <MATH><I>x</I><<I>r</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18656 so that, for a real value of <I>x</I> other than <I>r, p</I> must be less than
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18657 <I>r,</I> and, if <I>p</I> is not less than <I>r,</I> the maximum triangle is that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18658 which is perpendicular to the base of the cone and has 2<I>r</I> for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18659 its base (Prop. 29). If <MATH><I>p</I> < <I>r</I></MATH>, the triangle in question is not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18660 the maximum of the set of triangles (Prop. 30).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18661 <p>Coming now to the isosceles sections (2), we may suppose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18662 2<G>q</G> to be the angle subtended at the centre of the base by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18663 base of the section in the direction away from the projection
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18664 of the vertex. Then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18665 <MATH><I>A</I>=<I>r</I> sin<G>q</G>√{<I>p</I><SUP>2</SUP>+(<I>d</I>+<I>r</I>cos<G>q</G>)<SUP>2</SUP>}</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18666 <p>If <I>A</I><SUB>0</SUB> be the area of the isosceles triangle through the axis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18667 we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18668 <MATH><I>A</I><SUB>0</SUB><SUP>2</SUP>-<I>A</I><SUP>2</SUP>=<I>r</I><SUP>2</SUP>(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>)-<I>r</I><SUP>2</SUP>sin<SUP>2</SUP><G>q</G>(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>+<I>r</I><SUP>2</SUP>cos<SUP>2</SUP><G>q</G>+
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18669 2<I>dr</I>cos<G>q</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18670 =<I>r</I><SUP>2</SUP>(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>)cos<SUP>2</SUP><G>q</G>-<I>r</I><SUP>4</SUP>sin<SUP>2</SUP><G>q</G>cos<SUP>2</SUP><G>q</G>-2<I>dr</I><SUP>3</SUP>cos<G>q</G>sin<SUP>2</SUP><G>q</G></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18671 <p>If <MATH><I>A</I>=<I>A</I><SUB>0</SUB></MATH>, we must have for triangles on the side of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18672 centre of the base of the cone towards the vertex of the cone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18673 (since cos <G>q</G> is negative for such triangles)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18674 <MATH><I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP><<I>r</I><SUP>2</SUP>sin<SUP>2</SUP><G>q</G></MATH>, and <I>a fortiori</I> <MATH><I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP><<I>r</I><SUP>2</SUP></MATH> (Prop. 35).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18675 <p>If <MATH><I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>≥<I>r</I><SUP>2</SUP></MATH>, <I>A</I><SUB>0</SUB> is always greater than <I>A,</I> so that <I>A</I><SUB>0</SUB> is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18676 maximum isosceles triangle of the set (Props. 31, 32).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18677 <p>If <I>A</I> is the area of any one of the isosceles triangles with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18678 bases on the side of the centre of the base of the cone away
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18679 from the projection of the vertex, cos <G>q</G> is positive and <I>A</I><SUB>0</SUB> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18680 proved to be neither the minimum nor the maximum triangle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18681 of this set of triangles (Props. 36, 40-4).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18682 <p>In Prop. 45 Serenus returns to the set of triangular sections
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18683 through the axis, proving that the feet of the perpendiculars
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18684 from the vertex of the cone on their bases all lie on a circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18685 the diameter of which is the straight line joining the centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18686 <pb n=525><head>SERENUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18687 the base of the cone to the projection of the vertex on its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18688 plane; the areas of the axial triangles are therefore propor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18689 tional to the generators of the cone with the said circle as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18690 base and the same vertex as the original cone. Prop. 50 is to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18691 the effect that, if the axis of the cone is equal to the radius of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18692 the base, the least axial triangle is a mean proportional
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18693 between the greatest axial triangle and the isosceles triangular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18694 section perpendicular to the base; that is, with the above nota-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18695 tion, if <MATH><I>r</I>=√(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>)</MATH>, then <MATH><I>r</I>√(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>):<I>rp</I>=<I>rp</I>:<I>p</I>√(<I>r</I><SUP>2</SUP>-<I>d</I><SUP>2</SUP>)</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18696 which is indeed obvious.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18697 <p>Prop. 57 is interesting because of the lemmas leading to it.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18698 It proves that the greater axial triangle in a scalene cone has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18699 the greater perimeter, and conversely. This is proved by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18700 means of the lemma (Prop. 54), applied to the variable sides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18701 of axial triangles, that if <MATH><I>a</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP>+<I>c</I><SUP>2</SUP></MATH> and <MATH><I>a</I> > <I>b</I>≥<I>c</I> > <I>d</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18702 then <MATH><I>a</I>+<I>d</I> < <I>b</I>+<I>c</I></MATH> (<I>a, d</I> are the sides other than the base of one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18703 axial triangle, and <I>b, c</I> those of the other axial triangle com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18704 pared with it; and if <I>ABC, ADE</I> be two axial triangles and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18705 <I>O</I> the centre of the base, <MATH><I>BA</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>=<I>DA</I><SUP>2</SUP>+<I>AE</I><SUP>2</SUP></MATH> because each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18706 of these sums is equal to <MATH>2<I>AO</I><SUP>2</SUP>+2<I>BO</I><SUP>2</SUP></MATH>, Prop. 17). This proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18707 tion again depends on the lemma (Props. 52, 53) that, if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18708 straight lines be ‘inflected’ from the ends of the base of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18709 a segment of a circle to the curve (i.e. if we join the ends
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18710 of the base to any point on the curve) the line (i.e. the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18711 the chords) is greatest when the point taken is the middle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18712 point of the arc, and diminishes as the point is taken farther
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18713 and farther from that point.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18714 <p>Let <I>B</I> be the middle point of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18715 arc of the segment <I>ABC, D, E</I> any
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18716 other points on the curve towards
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18717 <I>C</I>; I say that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18718 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18719 <MATH><I>AB</I>+<I>BC</I> > <I>AD</I>+<I>DC</I> > <I>AE</I>+<I>EC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18720 <p>With <I>B</I> as centre and <I>BA</I> as radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18721 describe a circle, and produce <I>AB,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18722 AD, AE</I> to meet this circle in <I>F, G,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18723 H.</I> Join <I>FC, GC, HC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18724 <p>Since <MATH><I>AB</I>=<I>BC</I>=<I>BF</I></MATH>, we have <MATH><I>AF</I>=<I>AB</I>+<I>BC</I></MATH>. Also the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18725 angles <I>BFC, BCF</I> are equal, and each of them is half of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18726 the angle <I>ABC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18727 <pb n=526><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18728 <p>Again <MATH>&angle;<I>AGC</I>=&angle;<I>AFC</I>=1/2&angle;<I>ABC</I>=1/2&angle;<I>ADC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18729 therefore the angles <I>DGC, DCG</I> are equal and <MATH><I>DG</I>=<I>DC</I></MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18730 therefore <MATH><I>AG</I>=<I>AD</I>+<I>DC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18731 <p>Similarly <MATH><I>EH</I>=<I>EC</I></MATH> and <MATH><I>AH</I>=<I>AE</I>+<I>EC</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18732 <p>But, by Eucl. III. 7 or 15, <MATH><I>AF</I> > <I>AG</I> > <I>AH</I></MATH>, and so on;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18733 therefore <MATH><I>AB</I>+<I>BC</I> > <I>AD</I>+<I>DC</I> > <I>AE</I>+<I>EC</I></MATH>, and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18734 <p>In the particular case where the segment <I>ABC</I> is a semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18735 circle <MATH><I>AB</I><SUP>2</SUP>+<I>BC</I><SUP>2</SUP>=<I>AC</I><SUP>2</SUP>=<I>AD</I><SUP>2</SUP>+<I>DC</I><SUP>2</SUP></MATH>, &c., and the result of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18736 Prop. 57 follows.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18737 <p>Props. 58-69 are propositions of this sort: In equal right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18738 cones the triangular sections through the axis are reciprocally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18739 proportional to their bases and conversely (Props. 58, 59);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18740 right cones of equal height have to one another the ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18741 duplicate of that of their axial triangles (Prop. 62); right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18742 cones which are reciprocally proportional to their bases have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18743 axial triangles which are to one another reciprocally in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18744 triplicate ratio of their bases and conversely (Props. 66, 67);
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18745 and so on.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18746 <p>THEON OF ALEXANDRIA lived towards the end of the fourth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18747 century A.D. Suidas places him in the reign of Theodosius I
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18748 (379-95); he tells us himself that he observed a solar eclipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18749 at Alexandria in the year 365, and his notes on the chrono-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18750 logical tables of Ptolemy extend down to 372.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18751 <C>Commentary on the <I>Syntaxis.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18752 <p>We have already seen him as the author of a commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18753 on Ptolemy's <I>Syntaxis</I> in eleven Books. This commentary is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18754 not calculated to give us a very high opinion of Theon's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18755 mathematical calibre, but it is valuable for several historical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18756 notices that it gives, and we are indebted to it for a useful
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18757 account of the Greek method of operating with sexagesimal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18758 fractions, which is illustrated by examples of multiplication,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18759 division, and the extraction of the square root of a non-square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18760 number by way of approximation. These illustrations of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18761 numerical calculation have already been given above (vol. i,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18762 <pb n=527><head>THEON OF ALEXANDRIA</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18763 pp. 58-63). Of the historical notices we may mention the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18764 following. (1) Theon mentions the treatise of Menelaus <I>On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18765 Chords in a Circle,</I> i.e. Menelaus's Table of Chords, which came
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18766 between the similar Tables of Hipparchus and Ptolemy. (2) A
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18767 quotation from Diophantus furnishes incidentally a lower limit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18768 for the date of the <I>Arithmetica.</I> (3) It is in the commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18769 on Ptolemy that Theon tells us that the second part of Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18770 VI. 33 relating to <I>sectors</I> in equal circles was inserted by him-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18771 self in his edition of the <I>Elements,</I> a notice which is of capital
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18772 importance in that it enables the Theonine manuscripts of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18773 Euclid to be distinguished from the ante-Theonine, and is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18774 therefore the key to the question how far the genuine text
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18775 of Euclid was altered in Theon's edition. (4) As we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18776 seen (pp. 207 sq.), Theon, à propos of an allusion of Ptolemy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18777 to the theory of isoperimetric figures, has preserved for us
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18778 several propositions from the treatise by Zenodorus on that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18779 subject.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18780 <C>Theon's edition of Euclid's <I>Elements.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18781 <p>We are able to judge of the character of Theon's edition of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18782 Euclid by a comparison between the Theonine manuscripts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18783 and the famous Vatican MS. 190, which contains an earlier
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18784 edition than Theon's, together with certain fragments of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18785 ancient papyri. It appears that, while Theon took some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18786 trouble to follow older manuscripts, it was not so much his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18787 object to get the most authoritative text as to make what he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18788 considered improvements of one sort or other. (1) He made
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18789 alterations where he found, or thought he found, mistakes in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18790 the original; while he tried to remove some real blots, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18791 altered other passages too hastily when a little more considera-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18792 tion would have shown that Euclid's words are right or could
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18793 be excused, and offer no difficulty to an intelligent reader.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18794 (2) He made emendations intended to improve the form or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18795 diction of Euclid; in general they were prompted by a desire
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18796 to eliminate anything which was out of the common in expres-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18797 sion or in form, in order to reduce the language to one and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18798 same standard or norm. (3) He bestowed, however, most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18799 attention upon additions designed to supplement or explain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18800 the original; (<I>a</I>) he interpolated whole propositions where he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18801 thought them necessary or useful, e.g. the addition to VI. 33
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18802 <pb n=528><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18803 already referred to, a second case to VI. 27, a porism or corollary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18804 to II. 4, a second porism to III. 16, the proposition VII. 22,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18805 a lemma after X. 12, besides alternative proofs here and there;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18806 (<I>b</I>) he added words for the purpose of making smoother and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18807 clearer, or more precise, things which Euclid had expressed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18808 with unusual brevity, harshness, or carelessness; (<I>c</I>) he sup-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18809 plied intermediate steps where Euclid's argument seemed too
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18810 difficult to follow. In short, while making only inconsider-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18811 able additions to the content of the <I>Elements,</I> he endeavoured
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18812 to remove difficulties that might be felt by learners in study-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18813 ing the book, as a modern editor might do in editing a classical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18814 text-book for use in schools; and there is no doubt that his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18815 edition was approved by his pupils at Alexandria for whom it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18816 was written, as well as by later Greeks, who used it almost
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18817 exclusively, with the result that the more ancient text is only
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18818 preserved complete in one manuscript.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18819 <C>Edition of the <I>Optics</I> of Euclid.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18820 <p>In addition to the <I>Elements,</I> Theon edited the <I>Optics</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18821 Euclid; Theon's recension as well as the genuine work is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18822 included by Heiberg in his edition. It is possible that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18823 <I>Catoptrica</I> included by Heiberg in the same volume is also by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18824 Theon.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18825 <p>Next to Theon should be mentioned his daughter HYPATIA,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18826 who is mentioned by Theon himself as having assisted in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18827 revision of the commentary on Ptolemy. This learned lady
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18828 is said to have been mistress of the whole of pagan science,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18829 especially of philosophy and medicine, and by her eloquence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18830 and authority to have attained such influence that Christianity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18831 considered itself threatened, and she was put to death by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18832 a fanatical mob in March 415. According to Suidas she wrote
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18833 commentaries on Diophantus, on the Astronomical Canon (of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18834 Ptolemy) and on the Conics of Apollonius. These works
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18835 have not survived, but it has been conjectured (by Tannery)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18836 that the remarks of Psellus (eleventh century) at the begin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18837 ning of his letter about Diophantus, Anatolius, and the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18838 Egyptian method of arithmetical reckoning were taken bodily
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18839 from some manuscript of Diophantus containing an ancient
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18840 and systematic commentary which may very well have been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18841 that of Hypatia. Possibly her commentary may have extended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18842 <pb n=529><head>HYPATIA. PORPHYRY</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18843 only to the first six Books, in which case the fact that Hypatia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18844 wrote a commentary on them may account for the survival of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18845 these Books while the rest of the thirteen were first forgotten
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18846 and then lost.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18847 <p>It will be convenient to take next the series of Neo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18848 Platonist commentators. It does not appear that Ammonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18849 Saccas (about A.D. 175-250), the founder of Neo-Platonism, or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18850 his pupil Plotinus (A.D. 204-69), who first expounded the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18851 doctrines in systematic form, had any special connexion with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18852 mathematics, but PORPHYRY (about 232-304), the disciple of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18853 Plotinus and the reviser and editor of his works, appears to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18854 have written a commentary on the <I>Elements.</I> This we gather
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18855 from Proclus, who quotes from Porphyry comments on Eucl.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18856 I. 14 and 26 and alternative proofs of I. 18, 20. It is possible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18857 that Porphyry's work may have been used later by Pappus in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18858 writing his own commentary, and Proclus may have got his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18859 references from Pappus, but the form of these references sug-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18860 gests that he had direct access to the original commentary of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18861 Porphyry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18862 <p>IAMBLICHUS (died about A.D. 330) was the author of a com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18863 mentary on the <I>Introductio arithmetica</I> of Nicomachus, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18864 of other works which have already been mentioned. He was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18865 a pupil of Porphyry as well as of Anatolius, also a disciple of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18866 Porphyry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18867 <p>But the most important of the Neo-Platonists to the his-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18868 torian of mathematics is PROCLUS (A.D. 410-85). Proclus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18869 received his early training at Alexandria, where Olympio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18870 dorus was his instructor in the works of Aristotle, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18871 mathematics was taught him by one Heron (of course a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18872 different Heron from the ‘<I>mechanicus</I> Hero’ of the <I>Metrica,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18873 &c.). He afterwards went to Athens, where he learnt the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18874 Neo-Platonic philosophy from Plutarch, the grandson of Nes-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18875 torius, and from his pupil Syrianus, and became one of its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18876 most prominent exponents. He speaks everywhere with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18877 highest respect of his masters, and was in turn regarded with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18878 extravagant veneration by his contemporaries, as we learn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18879 from Marinus, his pupil and biographer. On the death of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18880 Syrianus he was put at the head of the Neo-Platonic school.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18881 He was a man of untiring industry, as is shown by the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18882 <pb n=530><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18883 number of books which he wrote, including a large number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18884 commentaries, mostly on the dialogues of Plato (e.g. the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18885 <I>Timaeus,</I> the <I>Republic,</I> the <I>Parmenides,</I> the <I>Cratylus</I>). He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18886 was an acute dialectician and pre-eminent among his contem-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18887 poraries in the range of his learning; he was a competent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18888 mathematician; he was even a poet. At the same time he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18889 was a believer in all sorts of myths and mysteries, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18890 a devout worshipper of divinities both Greek and Oriental.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18891 He was much more a philosopher than a mathematician. In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18892 his commentary on the <I>Timaeus,</I> when referring to the ques-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18893 tion whether the sun occupies a middle place among the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18894 planets, he speaks as no real mathematician could have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18895 spoken, rejecting the view of Hipparchus and Ptolemy because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18896 <G>o( qeourgo/s</G> (<I>sc.</I> the Chaldean, says Zeller) thinks otherwise,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18897 ‘whom it is not lawful to disbelieve’. Martin observes too,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18898 rather neatly, that ‘for Proclus the Elements of Euclid had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18899 the good fortune not to be contradicted either by the Chaldean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18900 Oracles or by the speculations of Pythagoreans old and new’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18901 <C>Commentary on Euclid, Book I.</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18902 <p>For us the most important work of Proclus is his commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18903 tary on Euclid, Book I, because it is one of the main sources
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18904 of our information as to the history of elementary geometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18905 Its great value arises mainly from the fact that Proclus had
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18906 access to a number of historical and critical works which are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18907 now lost except for fragments preserved by Proclus and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18908 others.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18909 <C>(<G>a</G>) <I>Sources of the Commentary.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18910 <p>The historical work the loss of which is most deeply to be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18911 deplored is the <I>History of Geometry</I> by Eudemus. There
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18912 appears to be no reason to doubt that the work of Eudemus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18913 was accessible to Proclus at first hand. For the later writers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18914 Simplicius and Eutocius refer to it in terms such as leave no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18915 doubt that <I>they</I> had it before them. Simplicius, quoting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18916 Eudemus as the best authority on Hippocrates's quadratures
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18917 of lunes, says he will set out what Eudemus says ‘word for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18918 word’, adding only a little explanation in the shape of refer-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18919 ences to Euclid's <I>Elements</I> ‘owing to the <I>memorandum-like
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18920 style of Eudemus,</I> who sets out his explanations in the abbre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18921 <pb n=531><head>PROCLUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18922 viated form usual with ancient writers. Now in the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18923 book of the history of geometry he writes as follows’.<note>Simplicius on Arist. <I>Phys.</I>, p. 60. 28, Diels.</note> In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18924 like manner Eutocius speaks of the paralogisms handed down
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18925 in connexion with the attempts of Hippocrates and Antiphon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18926 to square the circle, ‘with which I imagine that all persons
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18927 are accurately acquainted who have <I>examined</I> (<G>e)peskemme/nous</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18928 the geometrical history of Eudemus and know the <I>Ceria
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18929 Aristotelica</I>’.<note>Archimedes, ed. Heib., vol. iii, p. 228. 17-19.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18930 <p>The references by Proclus to Eudemus by name are not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18931 indeed numerous; they are five in number; but on the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18932 hand he gives at least as many other historical data which can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18933 with great probability be attributed to Eudemus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18934 <p>Proclus was even more indebted to Geminus, from whom
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18935 he borrows long extracts, often mentioning him by name—there
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18936 are some eighteen such references—but often omitting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18937 to do so. We are able to form a tolerably certain judge-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18938 ment as to the origin of the latter class of passages on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18939 strength of the similarity of the subjects treated and the views
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18940 expressed to those found in the acknowledged extracts. As
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18941 we have seen, the work of Geminus mainly cited seems to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18942 have borne the title <I>The Doctrine</I> or <I>Theory of the Mathematics,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18943 which was a very comprehensive work dealing, in a portion of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18944 it, with the ‘classification of mathematics’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18945 <p>We have already discussed the question of the authorship
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18946 of the famous historical summary given by Proclus. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18947 divided, as every one knows, into two distinct parts between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18948 which comes the remark, ‘Those who compiled histories
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18949 bring the development of this science up to this point. Not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18950 much younger than these is Euclid, who’, &c. The ultimate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18951 source at any rate of the early part of the summary must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18952 presumably have been the great work of Eudemus above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18953 mentioned.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18954 <p>It is evident that Proclus had before him the original works
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18955 of Plato, Aristotle, Archimedes and Plotinus, the <G>*summikta/</G> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18956 Porphyry and the works of his master Syrianus, as well as a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18957 group of works representing the Pythagorean tradition on its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18958 mystic, as distinct from its mathematical, side, from Philo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18959 laus downwards, and comprisin&gacute; the more or less apocryphal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18960 <pb n=532><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18961 <G>i(ero\s lo/gos</G> of Pythagoras, the <I>Oracles</I> (<G>lo/gia</G>) and Orphic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18962 verses.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18963 <p>The following will be a convenient summary of the other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18964 works used by Proclus, and will at the same time give an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18965 indication of the historical value of his commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18966 Euclid, Book I:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18967 <p>Eudemus: <I>History of Geometry.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18968 <p>Geminus: <I>The Theory of the Mathematical Sciences.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18969 <p>Heron: <I>Commentary on the Elements of Euclid.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18970 <p>Porphyry: ” ” ”
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18971 <p>Pappus: ” ” ”
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18972 <p>Apollonius of Perga: A work relating to elementary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18973 geometry.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18974 <p>Ptolemy: <I>On the parallel-postulate.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18975 <p>Posidonius: A book controverting Zeno of Sidon.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18976 <p>Carpus: <I>Astronomy.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18977 <p>Syrianus: A discussion on the <I>angle.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18978 <C>(<G>b</G>) <I>Character of the Commentary.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18979 <p>We know that in the Neo-Platonic school the pupils learnt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18980 mathematics; and it is clear that Proclus taught this subject,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18981 and that this was the origin of his commentary. Many
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18982 passages show him as a master speaking to scholars; in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18983 place he speaks of ‘my hearers’.<note>Proclus on Eucl. I, p. 210. 19.</note> Further, the pupils whom
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18984 he was addressing were <I>beginners</I> in mathematics; thus in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18985 passage he says that he omits ‘for the present’ to speak of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18986 discoveries of those who employed the curves of Nicomedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18987 and Hippias for trisecting an angle, and of those who used the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18988 Archimedean spiral for dividing an angle in a given ratio,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18989 because these things would be ‘too difficult for beginners’.<note><I>Ib.,</I> p. 272. 12.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18990 But there are signs that the commentary was revised and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18991 re-edited for a larger public; he speaks for instance in one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18992 place of ‘those who will come across his work’.<note><I>Ib.,</I> p. 84. 9.</note> There are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18993 also passages, e.g. passages about the cylindrical helix, con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18994 choids and cissoids, which would not have been understood by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18995 the beginners to whom he lectured.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18996 <pb n=533><head>PROCLUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18997 <p>The commentary opens with two Prologues. The first is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18998 on mathematics in general and its relation to, and use in,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	18999 philosophy, from which Proclus passes to the classification of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19000 mathematics. Prologue II deals with geometry generally and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19001 its subject-matter according to Plato, Aristotle and others.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19002 After this section comes the famous summary (pp. 64-8)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19003 ending with a eulogium of Euclid, with particular reference
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19004 to the admirable discretion shown in the selection of the pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19005 positions which should constitute the <I>Elements</I> of geometry,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19006 the ordering of the whole subject-matter, the exactness and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19007 the conclusiveness of the demonstrations, and the power with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19008 which every question is handled. Generalities follow, such as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19009 the discussion of the nature of <I>elements</I>, the distinction between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19010 theorems and problems according to different authorities, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19011 finally a division of Book I into three main sections, (1) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19012 construction and properties of triangles and their parts and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19013 the comparison between triangles in respect of their angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19014 and sides, (2) the properties of parallels and parallelograms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19015 and their construction from certain data, and (3) the bringing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19016 of triangles and parallelograms into relation as regards area.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19017 <p>Coming to the Book itself, Proclus deals historically and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19018 critically with all the definitions, postulates and axioms in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19019 order. The notes on the postulates and axioms are preceded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19020 by a general discussion of the principles of geometry, hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19021 theses, postulates and axioms, and their relation to one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19022 another; here as usual Proclus quotes the opinions of all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19023 important authorities. Again, when he comes to Prop. 1, he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19024 discusses once more the difference between theorems and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19025 problems, then sets out and explains the formal divisions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19026 a proposition, the <I>enunciation</I> (<G>pro/tasis</G>), the <I>setting-out</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19027 (<G>e)/kqesis</G>), the <I>definition</I> or <I>specification</I> (<G>diorismo/s</G>), the <I>con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19028 struction</I> (<G>kataskeuh/</G>), the <I>proof</I> (<G>a)po/deixis</G>), the <I>conclusion</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19029 (<G>sumpe/rasma</G>), and finally a number of other technical terms,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19030 e.g. things said to be <I>given</I>, in the various senses of this term,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19031 the <I>lemma</I>, the <I>case</I>, the <I>porism</I> in its two senses, the <I>objection</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19032 (<G>e)/nstasis</G>), the <I>reduction</I> of a problem, <I>reductio ad absurdum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19033 analysis</I> and <I>synthesis.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19034 <p>In his comments on the separate propositions Proclus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19035 generally proceeds in this way: first he gives explanations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19036 regarding Euclid's proofs, secondly he gives a few different
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19037 <pb n=534><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19038 cases, mainly for the sake of practice, and thirdly he addresses
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19039 himself to refuting objections which cavillers had taken or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19040 might take to particular propositions or arguments. He does
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19041 not seem to have had any notion of correcting or improving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19042 Euclid; only in one place does he propose anything of his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19043 own to get over a difficulty which he finds in Euclid; this is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19044 where he tries to prove the parallel-postulate, after giving
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19045 Ptolemy's attempt to prove it and pointing out objections to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19046 Ptolemy's proof.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19047 <p>The book is evidently almost entirely a compilation, though
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19048 a compilation ‘in the better sense of the term’. The <I>onus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19049 probandi</I> is on any one who shall assert that anything in it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19050 Proclus's own; very few things can with certainty be said to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19051 be so. Instances are (1) remarks on certain things which he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19052 quotes from Pappus, since Pappus was the last of the com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19053 mentators whose works he seems to have used, (2) a defence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19054 of Geminus against Carpus, who criticized Geminus's view of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19055 the difference between theorems and problems, and perhaps
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19056 (3) criticisms of certain attempts by Apollonius to improve on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19057 Euclid's proofs and constructions; but the only substantial
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19058 example is (4) the attempted proof of the parallel-postulate,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19059 based on an ‘axiom’ to the effect that, ‘if from one point two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19060 straight lines forming an angle be produced <I>ad infinitum</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19061 distance between them when so produced <I>ad infinitum</I> exceeds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19062 any finite magnitude (i. e. length)’, an assumption which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19063 purports to be the equivalent of a statement in Aristotle.<note><I>De caelo</I>, i. 5, 271 b 28-30.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19064 Philoponus says that Proclus as well as Ptolemy wrote a whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19065 book on the parallel-postulate.<note>Philoponus on <I>Anal. Post.</I> i. 10, p. 214 a 9-12, Brandis.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19066 <p>It is still not quite certain whether Proclus continued his
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19067 commentaries beyond Book I. He certainly intended to do so,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19068 for, speaking of the trisection of an angle by means of certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19069 curves, he says, ‘we may perhaps more appropriately examine
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19070 these things on the third Book, where the writer of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19071 Elements bisects a given circumference’, and again, after
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19072 saying that of all parallelograms which have the same peri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19073 meter the square is the greatest ‘and the rhomboid least of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19074 all’, he adds, ‘But this we will prove in another place, for it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19075 is more appropriate to the discussion of the hypotheses of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19076 <pb n=535><head>PROCLUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19077 second Book’. But at the time when the commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19078 Book I was written he was evidently uncertain whether he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19079 would be able to continue it, for at the end he says, ‘For my
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19080 part, if I should be able to discuss the other Books in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19081 same way, I should give thanks to the gods; but, if other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19082 cares should draw me away, I beg those who are attracted by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19083 this subject to complete the exposition of the other Books as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19084 well, following the same method and addressing themselves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19085 throughout to the deeper and more sharply defined questions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19086 involved’.<note>Proclus on Eucl. I, p. 432. 9-15.</note> Wachsmuth, finding a Vatican manuscript contain-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19087 ing a collection of scholia on Books I, II, V, VI, X, headed <G>*ei)s ta\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19088 eu)klei/dou stoixei=a prolambano/mena e)k tw=n pro/klou spora/dhn
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19089 kai\ kat' e)pitomh/n</G>, and seeing that the scholia on Book I were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19090 extracts from the extant commentary of Proclus, concluded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19091 that those on the other Books were also from Proclus; but
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19092 the <G>pro</G>- in <G>prolambano/mena</G> rather suggests that only the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19093 scholia to Book I are from Proclus. Heiberg found and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19094 published in 1903 a scholium to X. 9, in which Proclus is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19095 expressly quoted as the authority, but he does not regard
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19096 this circumstance as conclusive. On the other hand, Heiberg
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19097 has noted two facts which go against the view that Proclus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19098 wrote on the later Books: (1) the scholiast's copy of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19099 Proclus was not much better than our manuscripts; in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19100 particular, it had the same lacunae in the notes to I. 36,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19101 37, and I. 41-3; this makes it improbable that the scholiast
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19102 had further commentaries of Proclus which have vanished
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19103 for us; (2) there is no trace in the scholia of the notes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19104 which Proclus promised in the passages already referred to.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19105 All, therefore, that we can say is that, while the Wachsmuth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19106 scholia <I>may</I> be extracts from Proclus, it is on the whole
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19107 improbable.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19108 <C><I>Hypotyposis of Astronomical Hypotheses.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19109 <p>Another extant work of Proclus which should be referred
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19110 to is his <I>Hypotyposis of Astronomical Hypotheses</I>, a sort of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19111 readable and easy introduction to the astronomical system
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19112 of Hipparchus and Ptolemy. It has been well edited by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19113 Manitius (Teubner, 1909). Three things may be noted as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19114 <pb n=536><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19115 regards this work. It contains<note>Proclus, <I>Hypotyposis</I>, c. 4, pp. 120-22.</note> a description of the method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19116 of measuring the sun's apparent diameter by means of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19117 Heron's water-clock, which, by comparison with the corre-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19118 sponding description in Theon's commentary to the <I>Syntaxis</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19119 of Ptolemy, is seen to have a common source with it. That
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19120 source is Pappus, and, inasmuch as Proclus has a figure (repro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19121 duced by Manitius in his text from one set of manuscripts)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19122 corresponding to the description, while the text of Theon has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19123 no figure, it is clear that Proclus drew directly on Pappus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19124 who doubtless gave, in his account of the procedure, a figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19125 taken from Heron's own work on water-clocks. A simple
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19126 proof of the equivalence of the epicycle and eccentric hypo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19127 theses is quoted by Proclus from one Hilarius of Antioch.<note><I>Ib.</I>, c. 3, pp. 76, 17 sq.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19128 An interesting passage is that in chap. 4 (p. 130, 18) where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19129 Sosigenes the Peripatetic is said to have recorded in his work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19130 ‘on reacting spheres’ that an <I>annular</I> eclipse of the sun is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19131 sometimes observed at times of perigee; this is, so far as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19132 I know, the only allusion in ancient times to annular eclipses,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19133 and Proclus himself questions the correctness of Sosigenes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19134 statement.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19135 <C>Commentary on the <I>Republic.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19136 <p>The commentary of Proclus on the <I>Republic</I> contains some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19137 passages of great interest to the historian of mathematics.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19138 The most important is that<note><I>Procli Diadochi in Platonis Rempublicam Commentarii</I>, ed. Kroll,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19139 vol. ii, p. 27.</note> in which Proclus indicates that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19140 Props. 9, 10 of Euclid, Book II, are Pythagorean proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19141 tions invented for the purpose of proving geometrically the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19142 fundamental property of the series of ‘side-’ and ‘diameter-’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19143 numbers, giving successive approximations to the value of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19144 √2 (see vol. i, p. 93). The explanation<note><I>Ib.</I>, vol. ii, pp. 36-42.</note> of the passage in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19145 Plato about the Geometrical Number is defective and dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19146 appointing, but it contains an interesting reference to one
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19147 Paterius, of date presumably intermediate between Nestorius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19148 and Proclus. Paterius is said to have made a calculation, in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19149 units and submultiples, of the lengths of different segments of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19150 <pb n=537><head>PROCLUS. MARINUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19151 straight lines in a figure formed by taking a triangle with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19152 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19153 sides 3, 4, 5 as <I>ABC</I>, then drawing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19154 <I>BD</I> from the right angle <I>B</I> perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19155 dicular to <I>AC</I>, and lastly drawing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19156 perpendiculars <I>DE, DF</I> to <I>AB, BC.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19157 A diagram in the text with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19158 lengths of the segments shown along-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19159 side them in the usual numerical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19160 notation shows that Paterius obtained from the data <I>AB</I>=3,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19161 <I>BC</I>=4, <I>CA</I>=5 the following:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19162 <MATH><I>DC</I>=<G>ge</G>′=3 1/5</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19163 <MATH><I>BD</I>=<G>bg</G>′<G>ie</G>′=2 1/3 1/15 [=2 2/5]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19164 <MATH><I>AD</I>=<G>asd</G>′<G>k</G>′=1 1/2 1/4 1/20 [=1 4/5]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19165 <MATH><I>FC</I>=<G>bsk</G>′<G>r</G>′=2 1/2 1/20 1/100 [=2 14/25]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19166 <MATH><I>FB</I>=<G>ag</G>′<G>ie</G>′<G>ke</G>′=1 1/3 1/15 1/25 [=1 11/25]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19167 <MATH><I>BE</I>=<G>asg</G>′<G>ie</G>′<G>n</G>′=1 1/2 1/3 1/15 1/50 [=1 23/25]</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19168 <MATH><I>EA</I>=<G>aie</G>′<G>oe</G>′=1 1/15 1/75 [=1 2/25]</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19169 <p>This is an example of the Egyptian method of stating frac-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19170 tions preceding by some three or four centuries the exposition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19171 of the same method in the papyrus of Akhm&imacr;m.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19172 <p>MARINUS of Neapolis, the pupil and biographer of Proclus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19173 wrote a commentary or rather introduction to the <I>Data</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19174 Euclid.<note>See Heiberg and Menge's Euclid, vol. vi, pp. 234-56.</note> It is mainly taken up with a discussion of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19175 question <G>ti/ to\ dedome/non</G>, what is meant by <I>given</I>? There
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19176 were apparently many different definitions of the term <I>given</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19177 by earlier and later authorities. Of those who tried to define
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19178 it in the simplest way by means of a single <I>differentia</I>, three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19179 are mentioned by name. Apollonius in his work on <G>neu/seis</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19180 and his ‘general treatise’ (presumably that on elementary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19181 geometry) described the <I>given</I> as <I>assigned</I> or <I>fixed</I> (<G>tetag-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19182 me/non</G>), Diodorus called it <I>known</I> (<G>gnw/rimon</G>); others regarded
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19183 it as <I>rational</I> (<G>r(hto/n</G>) and Ptolemy is classed with these, rather
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19184 oddly, because ‘he called those things given the measure of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19185 which is given either exactly or approximately’. Others
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19186 <pb n=538><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19187 combined two of these ideas and called it <I>assigned</I> or <I>fixed</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19188 and <I>procurable</I> or capable of being found (<G>po/rimon</G>); others
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19189 ‘fixed and known’, and a third class ‘known and procurable’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19190 These various views are then discussed at length.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19191 <p>DOMNINUS of Larissa, a pupil of Syrianus at the same time
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19192 as Proclus, wrote a <I>Manual of Introductory Arithmetic</I> <G>e)gxei-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19193 ri/dion a)riqmhtikh=s ei)sagwgh=s</G>, which was edited by Boissonade<note><I>Anecdota Graeca</I>, vol. iv, pp. 413-29.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19194 and is the subject of two articles by Tannery,<note><I>Mémoires scientifiques</I>, vol. ii, nos. 35, 40.</note> who also left
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19195 a translation of it, with prolegomena, which has since been
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19196 published.<note><I>Revue des études grecques</I>, 1906, pp. 359-82; <I>Mémoires scientifiques</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19197 vol. iii, pp. 256-81.</note> It is a sketch of the elements of the theory of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19198 numbers, very concise and well arranged, and is interesting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19199 because it indicates a serious attempt at a reaction against the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19200 <I>Introductio arithmetica</I> of Nicomachus and a return to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19201 doctrine of Euclid. Besides Euclid, Nicomachus and Theon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19202 of Smyrna, Domninus seems to have used another source,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19203 now lost, which was also drawn upon by Iamblichus. At the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19204 end of this work Domninus foreshadows a more complete
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19205 treatise on the theory of numbers under the title <I>Elements of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19206 Arithmetic</I> (<G>a)riqmhtikh\ stoixei/wsis</G>), but whether this was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19207 ever written or not we do not know. Another tract
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19208 attributed to Domninus <G>pw=s e)/sti lo/gon e)k lo/gou a)felei=n</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19209 (how a ratio can be taken out of a ratio) has been published
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19210 with a translation by Ruelle<note><I>Revue de Philologie</I>, 1883, p. 83 sq.</note>; if it is not by Domninus, it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19211 probably belongs to the same period.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19212 <p>A most honourable place in our history must be reserved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19213 for SIMPLICIUS, who has been rightly called ‘the excellent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19214 Simplicius, the Aristotle-commentator, to whom the world can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19215 never be grateful enough for the preservation of the frag-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19216 ments of Parmenides, Empedocles, Anaxagoras, Melissus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19217 Theophrastus and others’ (v. Wilamowitz-Möllendorff). He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19218 lived in the first half of the sixth century and was a pupil,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19219 first of Ammonius of Alexandria, and then of Damascius,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19220 the last head of the Platonic school at Athens. When in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19221 year 529 the Emperor Justinian, in his zeal to eradicate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19222 paganism, issued an edict forbidding the teaching of philo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19223 <pb n=539><head>DOMNINUS. SIMPLICIUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19224 sophy at Athens, the last members of the school, including
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19225 Damascius and Simplicius, migrated to Persia, but returned
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19226 about 533 to Athens, where Simplicius continued to teach for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19227 some time though the school remained closed.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19228 <C><I>Extracts from Eudemus.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19229 <p>To Simplicius we owe two long extracts of capital impor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19230 tance for the history of mathematics and astronomy. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19231 first is his account, based upon and to a large extent quoted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19232 textually from Eudemus's <I>History of Geometry</I>, of the attempt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19233 by Antiphon to square the circle and of the quadratures of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19234 lunes by Hippocrates of Chios. It is contained in Simplicius's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19235 commentary on Aristotle's <I>Physics</I>,<note>Simpl. <I>in Phys.</I>, pp. 54-69, ed. Diels.</note> and has been the subject
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19236 of a considerable literature extending from 1870, the date
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19237 when Bretschneider first called attention to it, to the latest
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19238 critical edition with translation and notes by Rudio (Teubner,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19239 1907). It has already been discussed (vol. i, pp. 183-99).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19240 <p>The second, and not less important, of the two passages is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19241 that containing the elaborate and detailed account of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19242 system of concentric spheres, as first invented by Eudoxus for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19243 explaining the apparent motion of the sun, moon, and planets,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19244 and of the modifications made by Callippus and Aristotle. It
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19245 is contained in the commentary on Aristotle's <I>De caelo</I><note>Simpl. on Arist. <I>De caelo</I>, p. 488. 18-24 and pp. 493-506, ed. Heiberg.</note>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19246 Simplicius quotes largely from Sosigenes the Peripatetic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19247 (second century A.D.), observing that he in his turn drew
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19248 from Eudemus, who dealt with the subject in the second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19249 book of his <I>History of Astronomy.</I> It is this passage of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19250 Simplicius which, along with a passage in Aristotle's <I>Meta-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19251 physics</I>,<note><I>Metaph.</I> <*>. 8, 1073 b 17-1074 a 14.</note> enabled Schiaparelli to reconstruct Eudoxus's system
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19252 (see vol. i, pp. 329-34). Nor must it be forgotten that it is in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19253 Simplicius's commentary on the <I>Physics</I><note>Simpl. <I>in Phys.</I>, pp. 291-2, ed. Diels.</note> that the extract
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19254 from Geminus's summary of the <I>Meteorologica</I> of Posidonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19255 occurs which was used by Schiaparelli to support his view
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19256 that it was Heraclides of Pontus, not Aristarchus of Samos,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19257 who first propounded the heliocentric hypothesis.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19258 <p>Simplicius also wrote a commentary on Euclid's <I>Elements</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19259 Book I, from which an-Nair&imacr;z&imacr;, the Arabian commentator,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19260 <pb n=540><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19261 made valuable extracts, including the account of the attempt of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19262 ‘Aganis’ to prove the parallel-postulate (see pp. 228-30 above).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19263 <p>Contemporary with Simplicius, or somewhat earlier, was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19264 EUTOCIUS, the commentator on Archimedes and Apollonius.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19265 As he dedicated the commentary on Book I <I>On the Sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19266 and Cylinder</I> to Ammonius (a pupil of Proclus and teacher
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19267 of Simplicius), who can hardly have been alive after A.D. 510,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19268 Eutocius was probably born about A.D. 480. His date used
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19269 to be put some fifty years later because, at the end of the com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19270 mentaries on Book II <I>On the Sphere and Cylinder</I> and on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19271 the <I>Measurement of a Circle</I>, there is a note to the effect that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19272 ‘the edition was revised by Isidorus of Miletus, the mechanical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19273 engineer, <I>our teacher</I>’. But, in view of the relation to Ammo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19274 nius, it is impossible that Eutocius can have been a pupil of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19275 Isidorus, who was younger than Anthemius of Tralles, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19276 architect of Saint Sophia at Constantinople in 532, whose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19277 work was continued by Isidorus after Anthemius's death
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19278 about A.D. 534. Moreover, it was to Anthemius that Eutocius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19279 dedicated, separately, the commentaries on the first four
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19280 Books of Apollonius's <I>Conics</I>, addressing Anthemius as ‘my
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19281 dear friend’. Hence we conclude that Eutocius was an elder
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19282 contemporary of Anthemius, and that the reference to Isidorus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19283 is by an editor of Eutocius's commentaries who was a pupil of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19284 Isidorus. For a like reason, the reference in the commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19285 on Book II <I>On the Sphere and Cylinder</I><note>Archimedes, ed. Heiberg, vol. iii, p. 84. 8-11.</note> to a <G>diabh/ths</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19286 invented by Isidorus ‘our teacher’ for drawing a parabola
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19287 must be considered to be an interpolation by the same editor.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19288 <p>Eutocius's commentaries on Archimedes apparently ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19289 tended only to the three works, <I>On the Sphere and Cylinder,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19290 Measurement of a Circle</I> and <I>Plane Equilibriums</I>, and those
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19291 on the <I>Conics</I> of Apollonius to the first four Books only.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19292 We are indebted to these commentaries for many valuable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19293 historical notes. Those deserving special mention here are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19294 (1) the account of the solutions of the problem of the duplica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19295 tion of the cube, or the finding of two mean proportionals,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19296 by ‘Plato’, Heron, Philon, Apollonius, Diocles, Pappus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19297 Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes, (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19298 the fragment discovered by Eutocius himself containing the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19299 <pb n=541><head>EUTOCIUS. ANTHEMIUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19300 missing solution, promised by Archimedes in <I>On the Sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19301 and Cylinder</I>, II. 4, of the auxiliary problem amounting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19302 to the solution by means of conics of the cubic equation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19303 <MATH>(<I>a</I>-<I>x</I>)<I>x</I><SUP>2</SUP>=<I>bc</I><SUP>2</SUP></MATH>, (3) the solutions (<I>a</I>) by Diocles of the original
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19304 problem of II. 4 without bringing in the cubic, (<I>b</I>) by Diony-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19305 sodorus of the auxiliary cubic equation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19306 <p>ANTHEMIUS of Tralles, the architect, mentioned above, was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19307 himself an able mathematician, as is seen from a fragment of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19308 a work of his, <I>On Burning-mirrors.</I> This is a document of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19309 considerable importance for the history of conic sections.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19310 Originally edited by L. Dupuy in 1777, it was reprinted in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19311 Westermann's <G>*paradoxogra/foi</G> (<I>Scriptores rerum mirabilium
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19312 Graeci</I>), 1839, pp. 149-58. The first and third portions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19313 the fragment are those which interest us.<note>See <I>Bibliotheca mathematica</I>, vii<SUB>3</SUB>, 1907, pp. 225-33.</note> The first gives
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19314 a solution of the problem, To contrive that a ray of the sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19315 (admitted through a small hole or window) shall fall in a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19316 given spot, without moving away at any hour and season.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19317 This is contrived by constructing an elliptical mirror one focus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19318 of which is at the point where the ray of the sun is admitted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19319 while the other is at the point to which the ray is required
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19320 to be reflected at all times. Let <I>B</I> be the hole, <I>A</I> the point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19321 to which reflection must always take place, <I>BA</I> being in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19322 meridian and parallel to the horizon. Let <I>BC</I> be at right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19323 angles to <I>BA</I>, so that <I>CB</I> is an equinoctial ray; and let <I>BD</I> be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19324 the ray at the summer solstice, <I>BE</I> a winter ray.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19325 <p>Take <I>F</I> at a convenient distance on <I>BE</I> and measure <I>FQ</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19326 equal to <I>FA.</I> Draw <I>HFG</I> through <I>F</I> bisecting the angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19327 <I>AFQ</I>, and let <I>BG</I> be the straight line bisecting the angle <I>EBC</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19328 between the winter and the equinoctial rays. Then clearly,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19329 since <I>FG</I> bisects the angle <I>QFA</I>, if we have a plane mirror in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19330 the position <I>HFG</I>, the ray <I>BFE</I> entering at <I>B</I> will be reflected
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19331 to <I>A.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19332 <p>To get the equinoctial ray similarly reflected to <I>A</I>, join <I>GA</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19333 and with <I>G</I> as centre and <I>GA</I> as radius draw a circle meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19334 <I>BC</I> in <I>K.</I> Bisect the angle <I>KGA</I> by the straight line <I>GLM</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19335 meeting <I>BK</I> in <I>L</I> and terminated at <I>M</I>, a point on the bisector
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19336 of the angle <I>CBD.</I> Then <I>LM</I> bisects the angle <I>KLA</I> also, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19337 <I>KL</I>=<I>LA</I>, and <I>KM</I>=<I>MA</I>. If then <I>GLM</I> is a plane mirror,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19338 the ray <I>BL</I> will be reflected to <I>A.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19339 <pb n=542><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19340 <p>By taking the point <I>N</I> on <I>BD</I> such that <I>MN</I>=<I>MA</I>, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19341 bisecting the angle <I>NMA</I> by the straight line <I>MOP</I> meeting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19342 <I>BD</I> in <I>O</I>, we find that, if <I>MOP</I> is a plane mirror, the ray <I>BO</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19343 is reflected to <I>A.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19344 <p>Similarly, by continually bisecting angles and making more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19345 mirrors, we can get any number of other points of impact. Mak-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19346 ing the mirrors so short as to form a continuous curve, we get
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19347 the curve containing all points such that the sum of the distances
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19348 of each of them from <I>A</I> and <I>B</I> is constant and equal to <I>BQ, BK</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19349 or <I>BN.</I> ‘If then’, says Anthemius, ‘we stretch a string passed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19350 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19351 round the points <I>A, B</I>, and through the first point taken on the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19352 rays which are to be reflected, the said curve will be described,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19353 which is part of the so-called “ellipse”, with reference to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19354 which (i.e. by the revolution of which round <I>BA</I>) the surface
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19355 of impact of the said mirror has to be constructed.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19356 <p>We have here apparently the first mention of the construc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19357 tion of an ellipse by means of a string stretched tight round
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19358 the foci. Anthemius's construction depends upon two pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19359 positions proved by Apollonius (1) that the sum of the focal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19360 distances of any point on the ellipse is constant, (2) that the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19361 focal distances of any point make equal angles with the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19362 tangent at that point, and also (3) upon a proposition not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19363 found in Apollonius, namely that the straight line joining
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19364 <pb n=543><head>ANTHEMIUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19365 the focus to the intersection of two tangents bisects the angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19366 between the straight lines joining the focus to the two points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19367 of contact respectively.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19368 <p>In the third portion of the fragment Anthemius proves that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19369 parallel rays can be reflected to one single point from a para-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19370 bolic mirror of which the point is the focus. The <I>directrix</I> is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19371 used in the construction, which follows, <I>mutatis mutandis</I>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19372 same course as the above construction in the case of the ellipse.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19373 <p>As to the supposition of Heiberg that Anthemius may also
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19374 be the author of the <I>Fragmentum mathematicum Bobiense</I>, see
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19375 above (p. 203).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19376 <C><I>The Papyrus of Akhm&imacr;m.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19377 <p>Next in chronological order must apparently be placed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19378 the Papyrus of Akhm&imacr;m, a manual of calculation written
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19379 in Greek, which was found in the metropolis of Akhm&imacr;m,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19380 the ancient Panopolis, and is now in the Musée du
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19381 Gizeh. It was edited by J. Baillet<note><I>Mémoires publiés par les membres de la Mission archéologique française
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19382 au Caire</I>, vol. ix, part 1, pp. 1-89.</note> in 1892. Accord-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19383 ing to the editor, it was written between the sixth and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19384 ninth centuries by a Christian. It is interesting because
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19385 it preserves the Egyptian method of reckoning, with proper
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19386 fractions written as the sum of primary fractions or sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19387 multiples, a method which survived alongside the Greek and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19388 was employed, and even exclusively taught, in the East. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19389 advantage of this papyrus, as compared with Ahmes's, is that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19390 we can gather the formulae used for the decomposition of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19391 ordinary proper fractions into sums of submultiples. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19392 formulae for decomposing a proper fraction into the sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19393 two submultiples may be shown thus:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19394 (1) <MATH><I>a</I>/(<I>bc</I>)=1/(<I>c</I>.(<I>b</I>+<I>c</I>)/<I>a</I>)+1/(<I>b</I>.(<I>b</I>+<I>c</I>)/<I>a</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19395 Examples <MATH>2/11=1/6 1/66, 3/110=1/70 1/77, 18/323=1/34 1/38</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19396 (2) <MATH><I>a</I>/(<I>bc</I>)=1/(<I>c</I>.(<I>b</I>+<I>mc</I>)/<I>a</I>)+1/(<I>b</I>.((<I>b</I>+<I>mc</I>)/<I>a</I>).(1/<I>m</I>)</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19397 <pb n=544><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19398 Ex. <MATH>7/176=1/(11((16+3.11)/7))+1/(16((16+3.11)/7)1/3)=1/77+3/112</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19399 and again <MATH>3/112=1/(7((16+2.7)/3))+1/(16((16+2.7)/3)1/2)=1/70 1/80</MATH>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19400 (3) <MATH><I>a</I>/(<I>cdf</I>)=1/(<I>c</I>.((<I>cd</I>+<I>df</I>)/<I>a</I>)+1/(<I>f</I>.((<I>cd</I>+<I>df</I>)/<I>a</I>))</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19401 Example.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19402 <MATH>28/1320=28/(10.12.11)=1/(10.((120+132)/28))+1/(11.((120+132)/28))=1/90 1/99</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19403 <p>The object is, of course, to choose the factors of the denomi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19404 nator, and the multiplier <I>m</I> in (2), in such a way as to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19405 the two denominators on the right-hand side integral.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19406 <p>When the fraction has to be decomposed into a sum of three
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19407 or more submultiples, we take out an obvious submultiple
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19408 first, then if necessary a second, until one of the formulae
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19409 will separate what remains into two submultiples. Or we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19410 take out a part which is not a submultiple but which can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19411 divided into two submultiples by one of the formulae.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19412 <p>For example, to decompose 31/616. The factors of 616 are 8.77
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19413 or 7.88. Take out 1/88, and <MATH>31/616=1/88 24/616=1/88 3/77=1/88 1/77 2/77</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19414 and <MATH>2/77=1/63 1/99</MATH> by formula (1), so that <MATH>31/616=1/63 1/77 1/88 1/99</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19415 <p>Take 239/6460. The factors of 6460 are 85.76 or 95.68. Take
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19416 out 1/85, and <MATH>239/6460=1/85 163/6460</MATH>. Again take out 1/95, and we have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19417 1/85 1/95 95/6460 or 1/85 1/95 1/68. The actual problem here is to find
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19418 (1/323)rd of 11 1/2 1/3 1/10 1/60, which latter expression reduces to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19419 (1/20).239.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19420 <p>The sort of problems solved in the book are (1) the division
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19421 of a number into parts in the proportion of certain given
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19422 numbers, (2) the solution of simple equations such as this:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19423 From a certain treasure we take away (1/13)th, then from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19424 remainder (1/17)th of that remainder, and we find 150 units left;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19425 what was the treasure? <MATH>[{<I>x</I>-(1/<I>a</I>)<I>x</I>-1/<I>b</I>(<I>x</I>-(1/<I>a</I>)<I>x</I>)-...}=<I>R</I>]</MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19426 <pb n=545><head>THE PAPYRUS OF AKHM&Imacr;M. PSELLUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19427 (3) subtractions such as: From 2/3 subtract 1/10 1/11 1/20 1/22 1/30 1/33
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19428 1/40 1/44 1/50 1/55 1/60 1/66 1/70 1/77 1/88 1/90 1/99 1/100 1/110. Answer, 1/10 1/50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19429 <p>The book ends with long tables of results obtained (1) by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19430 multiplying successive numbers, tens, hundreds and thousands
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19431 up to 10,000 by 2/3, 1/3, 1/4, 1/5, 1/6, &c., up to 1/10, (2) by multiplying
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19432 all the successive numbers 1, 2, 3 ... <I>n</I> by 1/<I>n</I>, where <I>n</I> is succes-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19433 sively 11, 12, ... and 20; the results are all arranged as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19434 sums of integers and submultiples.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19435 <p>The <I>Geodaesia</I> of a Byzantine author formerly called, with-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19436 out any authority, ‘Heron the Younger’ was translated into
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19437 Latin by Barocius in 1572, and the Greek text was published
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19438 with a French translation by Vincent.<note><I>Notices et extraits</I>, xix, pt. 2, Paris, 1858.</note> The place of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19439 author's observations was the hippodrome at Constantinople,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19440 and the date apparently about 938. The treatise was modelled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19441 on Heron of Alexandria, especially the <I>Dioptra</I>, while some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19442 measurements of areas and volumes are taken from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19443 <I>Metrica.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19444 <p>MICHAEL PSELLUS lived in the latter part of the eleventh
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19445 century, since his latest work bears the date 1092. Though
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19446 he was called ‘first of philosophers’, it cannot be said that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19447 what survives of his mathematics suits this title. Xylander
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19448 edited in 1556 the Greek text, with a Latin translation, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19449 a book purporting to be by Psellus on the four mathematical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19450 sciences, arithmetic, music, geometry and astronomy, but it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19451 evident that it cannot be entirely Psellus's own work, since
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19452 the astronomical portion is dated 1008. The arithmetic con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19453 tains no more than the names and classification of numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19454 and ratios. The geometry has the extraordinary remark that,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19455 while opinions differed as to how to find the area of a circle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19456 the method which found most favour was to take the area as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19457 the geometric mean between the inscribed and circumscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19458 squares; this gives <G>p</G>=√8=2.8284271! The only thing of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19459 Psellus which has any value for us is the letter published by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19460 Tannery in his edition of Diophantus.<note>Diophantus, vol. ii, pp. 37-42.</note> In this letter Psellus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19461 says that both Diophantus and Anatolius (Bishop of Laodicea
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19462 about A.D. 280) wrote on the Egyptian method of reckoning,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19463 <pb n=546><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19464 and that Anatolius's account, which was different and more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19465 succinct, was dedicated to Diophantus (this enables us to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19466 determine Diophantus's date approximately). He also notes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19467 the difference between the Diophantine and Egyptian names
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19468 for the successive powers of <G>a)riqmo/s</G>: the next power after
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19469 the fourth (<G>dunamodu/namis</G>=<I>x</I><SUP>4</SUP>), i.e. <I>x</I><SUP>5</SUP>, the Egyptians called
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19470 ‘the first undescribed’ (<G>a)/logos prw=tos</G>) or the ‘fifth number’;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19471 the sixth, <I>x</I><SUP>6</SUP>, they apparently (like Diophantus) called the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19472 cube-cube; but with them the seventh, <I>x</I><SUP>7</SUP>, was the ‘second
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19473 undescribed’ or the ‘seventh number’, the eighth (<I>x</I><SUP>8</SUP>) was the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19474 ‘quadruple square’ (<G>tetraplh= du/namis</G>), the ninth (<I>x</I><SUP>9</SUP>) the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19475 ‘extended cube’ (<G>ku/bos e)xelikto/s</G>). Tannery conjectures that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19476 all these remarks were taken direct from an old commentary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19477 on Diophantus now lost, probably Hypatia's.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19478 <p>GEORGIUS PACHYMERES (1242-1310) was the author of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19479 work on the Quadrivium (<G>*su/ntagma tw=n tessa/rwn maqhma/twn</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19480 or <G>*tetra/biblon</G>). The arithmetical portion contains, besides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19481 excerpts from Nicomachus and Euclid, a paraphrase of Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19482 phantus, Book I, which Tannery published in his edition of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19483 Diophantus<note>Diophantus, vol. ii, pp. 78-122.</note>; the musical section with part of the preface was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19484 published by Vincent,<note><I>Notices et extraits</I>, xvii, 1858, pp. 362-533.</note> and some fragments from Book IV by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19485 Martin in his edition of the <I>Astronomy</I> of Theon of Smyrna.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19486 <p>MAXIMUS PLANUDES, a monk from Nicomedia, was the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19487 envoy of the Emperor Andronicus II at Venice in the year
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19488 1297, and lived probably from about 1260 to 1310. He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19489 wrote scholia on the first two Books of Diophantus, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19490 are extant and are included in Tannery's edition of Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19491 phantus.<note>Diophantus, vol. ii, pp. 125-255.</note> They contain nothing of particular interest except
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19492 a number of conspectuses of the working-out of problems of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19493 Diophantus written in Diophantus's own notation but with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19494 steps in separate lines, and with abbreviations on the left of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19495 words indicating the operations (e.g. <G>e)/kq</G>.=<G>e)/kqesis, tetr</G>.=
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19496 <G>tetragwnismo/s, su/nq</G>.=<G>su/nqesis</G>, &c.); the result is to make
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19497 the work almost as easy to follow as it is in our notation.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19498 <p>Another work of Planudes is called <G>yhfofori/a kat' )indou/s</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19499 or <I>Arithmetic after the Indian method</I>, and was edited as <I>Das</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19500 <pb n=547><head>PSELLUS. PACHYMERES. PLANUDES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19501 <I>Rechenbuch des Maximus Planudes</I> in Greek by Gerhardt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19502 (Halle, 1865) and in a German translation by H. Waeschke
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19503 (Halle, 1878). There was, however, an earlier book under the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19504 similar title <G>)arxh\ th=s mega/lhs kai\ )indikh=s yhfifori/as</G> (<I>sic</I>),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19505 written in 1252, which is extant in the Paris MS. Suppl. Gr.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19506 387; and Planudes seems to have raided this work. He
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19507 begins with an account of the symbols which, he says, were
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19508 <p>‘invented by certain distinguished astronomers for the most
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19509 convenient and accurate expression of numbers. There are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19510 nine of these symbols (our 1, 2, 3, 4, 5, 6, 7, 8, 9), to which is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19511 added another called <I>Tzifra</I> (cypher), written 0 and denoting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19512 zero. The nine signs as well as this one are Indian.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19513 <p>But this is, of course, not the first occurrence of the Indian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19514 numerals; they were known, except the zero, to Gerbert
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19515 (Pope Sylvester II) in the tenth century, and were used by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19516 Leonardo of Pisa in his <I>Liber abaci</I> (written in 1202 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19517 rewritten in 1228). Planudes used the Persian form of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19518 numerals, differing in this from the writer of the treatise of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19519 1252 referred to, who used the form then current in Italy.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19520 It scarcely belongs to Greek mathematics to give an account
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19521 of Planudes's methods of subtraction, multiplication, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19522 <C><I>Extraction of the square root.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19523 <p>As regards the extraction of the square root, he claims to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19524 have invented a method different from the Indian method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19525 and from that of Theon. It does not appear, however, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19526 there was anything new about it. Let us try to see in what
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19527 the supposed new method consisted.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19528 <p>Planudes describes fully the method of extracting the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19529 square root of a number with several digits, a method which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19530 is essentially the same as ours. This appears to be what he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19531 refers to later on as ‘the Indian method’. Then he tells us
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19532 how to find a first approximation to the root when the number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19533 is not a complete square.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19534 <p>‘Take the square root of the next lower actual square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19535 number, and double it: then, from the number the square root
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19536 of which is required, subtract the next lower square number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19537 so found, and to the remainder (as numerator) give as de-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19538 nominator the double of the square root already found.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19539 <pb n=548><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19540 <p>The example given is √(18). Since <MATH>4<SUP>2</SUP>=16</MATH> is the next
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19541 lower square, the approximate square root is 4+2/2.4 or 4 1/4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19542 The formula used is, therefore, <MATH>√(<I>a</I><SUP>2</SUP>+<I>b</I>)=<I>a</I>+<I>b</I>/(2<I>a</I>)</MATH> approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19543 mately. (An example in larger numbers is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19544 <MATH>√(1690196789)=41112+245/82224</MATH> approximately.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19545 Planudes multiplies 4 1/4 by itself and obtains 18 1/16, which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19546 shows that the value 4 1/4 is not accurate. He adds that he will
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19547 explain later a method which is more exact and nearer the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19548 truth, a method ‘which I claim as a discovery made by me
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19549 with the help of God’. Then, coming to the method which he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19550 claims to have discovered, Planudes applies it to √6. The
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19551 object is to develop this in units and sexagesimal fractions.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19552 Planudes begins by multiplying the 6 by 3600, making 21600
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19553 second-sixtieths, and finds the square root of 21600 to lie
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19554 between 146 and 147. Writing the 146′ as 2 26′, he proceeds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19555 to find the rest of the approximate square root (2 26′ 58″ 9‴)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19556 by the same procedure as that used by Theon in extracting
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19557 the square root of 4500 and 2 28′ respectively. The differ-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19558 ence is that in neither of the latter cases does Theon multiply
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19559 by 3600 so as to reduce the units to second-sixtieths, but he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19560 begins by taking the approximate square root of 2, viz. 1, just
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19561 as he does that of 4500 (viz. 67). It is, then, the multiplication
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19562 by 3600, or the reduction to second-sixtieths to start with, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19563 constitutes the difference from Theon's method, and this must
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19564 therefore be what Planudes takes credit for as a new dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19565 covery. In such a case as √(2 28′) or √3, Theon's method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19566 has the inconvenience that the number of <I>minutes</I> in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19567 second term (34′ in the one case and 43′ in the other) cannot
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19568 be found without some trouble, a difficulty which is avoided
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19569 by Planudes's expedient. Therefore the method of Planudes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19570 had its advantage in such a case. But the discovery was not
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19571 new. For it will be remembered that Ptolemy (and doubtless
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19572 Hipparchus before him) expressed the chord in a circle sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19573 tending an angle of 120° at the centre (in terms of 120th parts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19574 of the diameter) as 103<SUP><I>p</I></SUP> 55′ 23″, which indicates that the first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19575 step in calculating √3 was to multiply it by 3600, making
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19576 10800, the nearest square below which is 103<SUP>2</SUP> (=10609). In
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19577 <pb n=549><head>PLANUDES. MOSCHOPOULOS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19578 the scholia to Eucl., Book X, the same method is applied.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19579 Examples have been given above (vol. i, p. 63). The supposed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19580 new method was therefore not only already known to the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19581 scholiast, but goes back, in all probability, to Hipparchus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19582 <C><I>Two problems.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19583 <p>Two problems given at the end of the Manual of Planudes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19584 are worth mention. The first is stated thus: ‘A certain man
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19585 finding himself at the point of death had his desk or safe
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19586 brought to him and divided his money among his sons with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19587 the following words, “I wish to divide my money equally
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19588 between my sons: the first shall have one piece and (1/7)th of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19589 rest, the second 2 and (1/7)th of the remainder, the third 3 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19590 (1/7)th of the remainder.” At this point the father died without
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19591 getting to the end either of his money or the enumeration of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19592 his sons. I wish to know how many sons he had and how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19593 much money.’ The solution is given as (<I>n</I>-1)<SUP>2</SUP> for the number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19594 of coins to be divided and (<I>n</I>-1) for the number of his sons;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19595 or rather this is how it might be stated, for Planudes takes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19596 <I>n</I>=7 arbitrarily. Comparing the shares of the first two we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19597 must clearly have
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19598 <MATH>1+1/<I>n</I>(<I>x</I>-1)=2+1/<I>n</I>{<I>x</I>-(1+(<I>x</I>-1)/<I>n</I>+2)}</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19599 which gives <MATH><I>x</I>=(<I>n</I>-1)<SUP>2</SUP></MATH>; therefore each of (<I>n</I>-1) sons received
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19600 (<I>n</I>-1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19601 <p>The other problem is one which we have already met with,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19602 that of finding two rectangles of equal perimeter such that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19603 the area of one of them is a given multiple of the area of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19604 the other. If <I>n</I> is the given multiple, the rectangles are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19605 (<I>n</I><SUP>2</SUP>-1, <I>n</I><SUP>3</SUP>-<I>n</I><SUP>2</SUP>) and (<I>n</I>-1, <I>n</I><SUP>3</SUP>-<I>n</I>) respectively. Planudes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19606 states the solution correctly, but how he obtained it is not clear.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19607 <p>We find also in the Manual of Planudes the ‘proof by nine’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19608 (i.e. by casting out nines), with a statement that it was dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19609 covered by the Indians and transmitted to us through the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19610 Arabs.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19611 <p>MANUEL MOSCHOPOULOS, a pupil and friend of Maximus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19612 Planudes, lived apparently under the Emperor Andronicus II
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19613 (1282-1328) and perhaps under his predecessor Michael VIII
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19614 (1261-82) also. A man of wide learning, he wrote (at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19615 <pb n=550><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19616 instance of Nicolas Rhabdas, presently to be mentioned) a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19617 treatise on <I>magic squares</I>; he showed, that is, how the num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19618 bers 1, 2, 3 ... <I>n</I><SUP>2</SUP> could be placed in the <I>n</I><SUP>2</SUP> compartments of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19619 a square, divided like a chess-board into <I>n</I><SUP>2</SUP> small squares, in such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19620 a way that the sum of the numbers in each horizontal and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19621 each vertical row of compartments, as well as in the rows
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19622 forming the diagonals, is always the same, namely 1/2<I>n</I>(<I>n</I><SUP>2</SUP>+1).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19623 Moschopoulos gives rules of procedure for the cases in which
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19624 <MATH><I>n</I>=2<I>m</I>+1</MATH> and <MATH><I>n</I>=4<I>m</I></MATH> respectively, and these only, in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19625 treatise as we have it; he promises to give the case where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19626 <MATH><I>n</I>=4<I>m</I>+2</MATH> also, but does not seem to have done so, as the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19627 two manuscripts used by Tannery have after the first two cases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19628 the words <G>te/los tou= au)tou=</G>. The treatise was translated by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19629 De la Hire,<note><I>Mém. de l'Acad. Royale des Sciences</I>, 1705.</note> edited by S. Günther,<note><I>Vermischte Untersuchungen zur Gesch. d. Math.</I>, Leipzig, 1876.</note> and finally edited in an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19630 improved text with translation by Tannery.<note>‘Le traité de Manuel Moschopoulos sur les carrés magiques’ in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19631 <I>Annuaire de l'Association pour l'encouragement des études grecques</I>, xx,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19632 1886, pp. 88-118.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19633 <p>The work of Moschopoulos was dedicated to Nicolas Arta-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19634 vasdus, called RHABDAS, a person of some importance in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19635 history of Greek arithmetic. He edited, with some additions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19636 of his own, the Manual of Planudes; this edition exists in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19637 the Paris MS. 2428. But he is also the author of two letters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19638 which have been edited by Tannery in the Greek text with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19639 French translation.<note>‘Notices sur les deux lettres arithmétiques de Nicolas Rhabdas’ in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19640 <I>Notices et extraits des manuscrits de la Bibliothèque Nationale</I>, xxxii, pt. 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19641 1886, pp. 121-252.</note> The date of Rhabdas is roughly fixed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19642 by means of a calculation of the date of Easter ‘in the current
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19643 year’ contained in one of the letters, which shows that its
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19644 date was 1341. It is remarkable that each of the two letters
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19645 has a preface which (except for the words <G>th\n dh/lwsin tw=n e)n
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19646 toi=s a)riqmoi=s zhthma/twn</G> and the name or title of the person
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19647 to whom it is addressed) copies word for word the first thir-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19648 teen lines of the preface to Diophantus's <I>Arithmetica</I>, a piece
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19649 of plagiarism which, if it does not say much for the literary
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19650 resource of Rhabdas, may indicate that he had studied Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19651 phantus. The first of the two letters has the heading ‘A con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19652 cise and most clear exposition of the science of calculation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19653 written at Byzantium of Constantine, by Nicolas Artavasdus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19654 <pb n=551><head>MOSCHOPOULOS. RHABDAS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19655 of Smyrna, arithmetician and geometer, <G>tou= *(pabda=</G>, at the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19656 instance of the most revered Master of Requests, Georgius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19657 Chatzyces, and most easy for those who desire to study it.’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19658 A long passage, called <G>e)/kfrasis tou= daktulikou= me/trou</G>, deals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19659 with a method of finger-notation, in which the fingers of each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19660 hand held in different positions are made to represent num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19661 bers.<note>A similar description occurs in the works of the Venerable Bede
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19662 (‘De computo vel loquela digitorum’, forming chapter i of <I>De temporum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19663 ratione</I>), where expressions are also quoted from St. Jerome (d. 420 A.D.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19664 as showing that he too was acquainted with the system (<I>The Miscellaneous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19665 Works of the Venerable Bede</I>, ed. J. A. Giles, vol. vi, 1843, pp. 141-3).</note> The fingers of the left hand serve to represent all the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19666 units and tens, those of the right all the hundreds and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19667 thousands up to 9000; ‘for numbers above these it is neces-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19668 sary to use writing, the hands not sufficing to represent such
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19669 numbers.’ The numbers begin with the little fingers of each
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19670 hand; if we call the thumb and the fingers after it the 1st,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19671 2nd, 3rd, 4th, and 5th fingers in the German style, the succes-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19672 sive signs may be thus described, premising that, where fingers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19673 are not either bent or ‘half-closed’ (<G>klino/menoi</G>) or ‘closed’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19674 (<G>sustello/menoi</G>), they are supposed to be held out straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19675 (<G>e)kteino/menoi</G>).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19676 <p>(<I>a</I>) <I>On the left hand</I>:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19677 <p>for 1, half-close the 5th finger only;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19678 <p>” 2, ” ” 4th and 5th fingers only;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19679 <p>” 3, ” ” 3rd, 4th and 5th fingers only;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19680 <p>” 4, ” ” 3rd and 4th fingers only;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19681 <p>” 5, ” ” 3rd finger only;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19682 <p>” 6, ” ” 4th ” ”
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19683 <p>” 7, close the 5th finger only;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19684 <p>” 8, ” ” 4th and 5th fingers only;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19685 <p>” 9, ” ” 3rd, 4th and 5th fingers only.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19686 <p>(<I>b</I>) The same operations on the <I>right hand</I> give the <I>thou-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19687 sands</I>, from 1000 to 9000.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19688 <p>(<I>c</I>) <I>On the left hand</I>:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19689 <p>for 10, apply the tip of the forefinger to the first joint of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19690 the thumb so that the resulting figure resembles <G>s</G>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19691 <pb n=552><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19692 <p>for 20, stretch out the forefinger straight and vertical,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19693 keep fingers 3, 4, 5 together but separate from it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19694 and inclined slightly to the palm; in this position
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19695 touch the forefinger with the thumb;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19696 <p>” 30, join the tips of the forefinger and thumb;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19697 <p>” 40, place the thumb on the knuckle of the forefinger
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19698 behind, making a figure like the letter <G>*g</G>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19699 <p>” 50, make a like figure with the thumb on the knuckle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19700 of the forefinger <I>inside</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19701 <p>” 60, place the thumb inside the forefinger as for 50 and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19702 bring the forefinger down over the thumb, touch-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19703 ing the ball of it;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19704 <p>” 70, rest the forefinger round the tip of the thumb,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19705 making a curve like a spiral;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19706 <p>” 80, fingers 3, 4, 5 being held together and inclined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19707 at an angle to the palm, put the thumb across the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19708 palm to touch the third phalanx of the middle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19709 finger (3) and in this position bend the forefinger
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19710 above the first joint of the thumb;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19711 <p>” 90, close the forefinger only as completely as possible.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19712 <p>(<I>d</I>) The same operations on the <I>right hand</I> give the <I>hun-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19713 dreds</I>, from 100 to 900.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19714 <p>The first letter also contains tables for addition and sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19715 traction and for multiplication and division; as these are said
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19716 to be the ‘invention of Palamedes’, we must suppose that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19717 such tables were in use from a remote antiquity. Lastly, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19718 first letter contains a statement which, though applied to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19719 particular numbers, expresses a theorem to the effect that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19720 <MATH>(<I>a</I><SUB>0</SUB>+10<I>a</I><SUB>1</SUB>+...+10<I><SUP>m</SUP>a<SUB>m</SUB></I>)(<I>b</I><SUB>0</SUB>+10<I>b</I><SUB>1</SUB>+...+10<I><SUP>n</SUP>b<SUB>n</SUB></I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19721 is not>10<SUP>(<I>m</I>+<I>n</I>+2)</SUP></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19722 where <I>a</I><SUB>0</SUB>, <I>a</I><SUB>1</SUB> ... <I>b</I><SUB>0</SUB>, <I>b</I><SUB>1</SUB> ... are any numbers from 0 to 9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19723 <p>In the second letter of Rhabdas we find simple algebraical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19724 problems of the same sort as those of the <I>Anthologia Graeca</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19725 and the Papyrus of Akhm&imacr;m. Thus there are five problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19726 leading to equations of the type
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19727 <MATH><I>x</I>/<I>m</I>+<I>x</I>/<I>n</I>+...=<I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19728 <pb n=553><head>RHABDAS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19729 <p>Rhabdas solves the equation <MATH><I>x</I>/<I>m</I>+<I>x</I>/<I>n</I>=<I>a</I></MATH>, practically as we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19730 should, by multiplying up to get rid of fractions, whence he
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19731 obtains <MATH><I>x</I>=<I>mna</I>/(<I>m</I>+<I>n</I>)</MATH>. Again he solves the simultaneous
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19732 equations <MATH><I>x</I>+<I>y</I>=<I>a</I>, <I>mx</I>=<I>ny</I></MATH>; also the pair of equations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19733 <MATH><I>x</I>+<I>y</I>/<I>m</I>=<I>y</I>+<I>x</I>/<I>n</I>=<I>a</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19734 Of course, <I>m, n, a</I> ... have particular numerical values in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19735 all cases.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19736 <C><I>Rhabdas's Rule for approximating to the square root of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19737 a non-square number.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19738 <p>We find in Rhabdas the equivalent of the Heronian formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19739 for the approximation to the square root of a non-square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19740 number <MATH><I>A</I>=<I>a</I><SUP>2</SUP>+<I>b</I></MATH>, namely
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19741 <MATH><G>a</G>=<I>a</I>+<I>b</I>/(2<I>a</I>)</MATH>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19742 he further observes that, if <G>a</G> be an approximation by excess,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19743 then <MATH><G>a</G><SUB>1</SUB>=<I>A</I>/<G>a</G></MATH> is an approximation by defect, and 1/2(<G>a</G>+<G>a</G><SUB>1</SUB>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19744 is an approximation nearer than either. This last form is of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19745 course exactly Heron's formula <MATH><G>a</G>=1/2(<I>a</I>+<I>A</I>/<I>a</I>)</MATH>. The formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19746 was also known to Barlaam (presently to be mentioned), who
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19747 also indicates that the procedure can be continued indefinitely.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19748 <p>It should here be added that there is interesting evidence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19749 of the Greek methods of approximating to square roots in two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19750 documents published by Heiberg in 1899.<note>‘Byzantinische Analekten’ in <I>Abh. zur Gesch. d. Math.</I> ix. Heft, 1899,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19751 pp. 163 sqq.</note> The first of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19752 these documents (from a manuscript of the fifteenth century
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19753 at Vienna) gives the approximate square root of certain non-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19754 square numbers from 2 to 147 in integers and proper fractions.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19755 The numerals are the Greek alphabetic numerals, but they are
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19756 given place-value like our numerals: thus <G>ah</G>=18, <G>adz</G>=147,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19757 <MATH>(<G>ag</G)>/(<G>bh</G)>=13/28</MATH>, and so on: 0 is indicated by <*> or, sometimes, by.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19758 All these square roots, such as √(21)=4 21/36, √(35)=5 11/12,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19759 √(112)=10 49/84, and so on, can be obtained (either exactly or,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19760 in a few cases, by neglecting or adding a small fraction in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19761 <pb n=554><head>COMMENTATORS AND BYZANTINES</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19762 numerator of the fractional part of the root) in one or other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19763 of the following ways:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19764 <p>(1) by taking the nearest square to the given number <I>A</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19765 say <I>a</I><SUP>2</SUP>, and using the Heronian formulae
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19766 <MATH><G>a</G><SUB>1</SUB>=1/2(<I>a</I>+<I>A</I>/<I>a</I>), <G>a</G><SUB>2</SUB>=1/2(<G>a</G><SUB>1</SUB>+<I>A</I>/<G>a</G><SUB>1</SUB>)</MATH>, &c.;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19767 <p>(2) by using one or other of the following approximations,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19768 where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19769 <MATH><I>a</I><SUP>2</SUP><<I>A</I><(<I>a</I>+1)<SUP>2</SUP>, and <I>A</I>=<I>a</I><SUP>2</SUP>+<I>b</I>=(<I>a</I>+1)<SUP>2</SUP>-<I>c</I></MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19770 namely, <MATH><I>a</I>+<I>b</I>/(2<I>a</I>), <I>a</I>+<I>b</I>/((2<I>a</I>)+<I>b</I>/(2<I>a</I>))</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19771 <MATH>(<I>a</I>+1)-<I>c</I>/2(<I>a</I>+1), (<I>a</I>+1)-<I>c</I>/(2(<I>a</I>+1)-<I>c</I>/2(<I>a</I>+1))</MATH>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19772 or a combination of two of these with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19773 <p>(3) the formula that, if <MATH><I>a</I>/<I>b</I><<I>c</I>/<I>d</I></MATH>, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19774 <MATH><I>a</I>/<I>b</I><(<I>ma</I>+<I>nc</I>)/(<I>mb</I>+<I>nd</I>)<<I>c</I>/<I>d</I></MATH>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19775 <p>It is clear that it is impossible to deny to the Greeks the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19776 knowledge of these simple formulae.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19777 <p>Three more names and we have done.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19778 <p>IOANNES PEDIASIMUS, also called Galenus, was Keeper of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19779 Seal to the Patriarch of Constantinople in the reign of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19780 Andronicus III (1328-41). Besides literary works of his,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19781 some notes on difficult points in arithmetic and a treatise on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19782 the duplication of the cube by him are said to exist in manu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19783 scripts. His <I>Geometry</I>, which was edited by Friedlein in 1866,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19784 follows very closely the mensuration of Heron.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19785 <p>BARLAAM, a monk of Calabria, was abbot at Constantinople
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19786 and later Bishop of Geraci in the neighbourhood of Naples;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19787 he died in 1348. He wrote, in Greek, arithmetical demon-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19788 strations of propositions in Euclid, Book II,<note>Edited with Latin translation by Dasypodius in 1564, and included
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19789 in Heiberg and Menge's Euclid, vol. v, <I>ad fin.</I></note> and a <I>Logistic</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19790 six Books, a laborious manual of calculation in whole numbers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19791 <pb n=555><head>PEDIASIMUS. BARLAAM. ARGYRUS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19792 ordinary fractions and sexagesimal fractions (printed at
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19793 Strassburg in 1592 and at Paris in 1600). Barlaam, as we
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19794 have seen, knew the Heronian formulae for finding successive
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19795 approximations to square roots, and was aware that they could
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19796 be indefinitely continued.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19797 <p>ISAAC ARGYRUS, a monk, who lived before 1368, was one of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19798 a number of Byzantine translators of Persian astronomical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19799 works. In mathematics he wrote a <I>Geodaesia</I> and scholia to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19800 the first six Books of Euclid's <I>Elements.</I> The former is con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19801 tained in the Paris MS. 2428 and is called ‘a method of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19802 geodesy or the measurement of surfaces, exact and shortened’;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19803 the introductory letter addressed to one Colybos is followed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19804 by a compilation of extracts from the <I>Geometrica</I> and <I>Stereo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19805 metrica</I> of Heron. He is apparently the author of some
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19806 further additions to Rhabdas's revision of the Manual of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19807 Planudes contained in the same manuscript. A short tract
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19808 of his ‘On the discovery of the square roots of non-rational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19809 square numbers’ is mentioned as contained in two other manu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19810 scripts at Venice and Rome respectively (Codd. Marcianus Gr.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19811 333 and Vaticanus Gr. 1058), where it is followed by a table
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19812 of the square roots of all numbers from 1 to 102 in sexa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19813 gesimal fractions (e.g. √2=1 24′ 51″ 48‴, √3=1 43′ 56″ 0‴).<note>Heiberg, ‘Byzantinische Analekten’, in <I>Abh. zur Gesch. d. Math.</I> ix,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19814 pp. 169-70.</note>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19815 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19816 <C><B>APPENDIX</B></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19817 <C><I>On Archimedes's proof of the subtangent-property of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19818 a spiral.</I></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19819 <p>THE section of the treatise <I>On Spirals</I> from Prop. 3 to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19820 Prop. 20 is an elaborate series of propositions leading up
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19821 to the proof of the fundamental property of the subtangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19822 corresponding to the tangent at any point on any turn of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19823 spiral. Libri, doubtless with this series of propositions in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19824 mind, remarks (<I>Histoire des sciences mathématiques en Italie,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19825 i, p. 31) that ‘Après vingt siècles de travaux et de décou-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19826 vertes, les intelligences les plus puissantes viennent encore
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19827 échouer contre la synthèse difficile du <I>Traité des Spirales</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19828 d'Archimède.’ There is no foundation for this statement,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19829 which seems to be a too hasty generalization from a dictum,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19830 apparently of Fontenelle, in the <I>Histoire de l'Académie des
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19831 Sciences pour l'année 1704</I> (p. 42 of the edition of 1722),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19832 who says of the proofs of Archimedes in the work <I>On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19833 Spirals</I>: ‘Elles sont si longues, et si difficiles à embrasser,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19834 que, comme on l'a pû voir dans la Préface de l'Analyse des
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19835 Infiniment petits, M. Bouillaud a avoué qu'il ne les avoit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19836 jamais bien entendues, et que Viète les a injustement soup-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19837 çonnées de paralogisme, parce qu'il n'avoit pû non plus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19838 parvenir à les bien entendre. Mais toutes les preuves qu'on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19839 peut donner de leur difficulté et de leur obscurité tournent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19840 à la gloire d'Archimède; car quelle vigueur d'esprit, quelle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19841 quantité de vûes différentes, quelle opiniâtreté de travail n'a-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19842 t-il pas fallu pour lier et pour disposer un raisonnement que
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19843 quelques-uns de nos plus grands géomètres ne peuvent suivre,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19844 tout lié et tout disposé qu'il est?’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19845 <p>P. Tannery has observed<note><I>Bulletin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19846 des sciences mathématiques,</I> 1895, Part i, pp. 265-71.</note> that, as a matter of fact, no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19847 mathematicians of real authority who have applied or ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19848 tended Archimedes's methods (such men as Huygens, Pascal,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19849 Roberval and Fermat, who alone could have expressed an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19850 opinion worth having), have ever complained of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19851 <pb n=557><head>APPENDIX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19852 ‘obscurity’ of Archimedes; while, as regards Vieta, he has
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19853 shown that the statement quoted is based on an entire mis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19854 apprehension, and that, so far from suspecting a fallacy in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19855 Archimedes's proofs, Vieta made a special study of the treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19856 <I>On Spirals</I> and had the greatest admiration for that work.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19857 <p>But, as in many cases in Greek geometry where the analy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19858 sis is omitted or even (as Wallis was tempted to suppose) of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19859 set purpose hidden, the reading of the completed synthetical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19860 proof leaves a certain impression of mystery; for there is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19861 nothing in it to show <I>why</I> Archimedes should have taken
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19862 precisely this line of argument, or how he evolved it. It is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19863 a fact that, as Pappus said, the subtangent-property can be
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19864 established by purely ‘plane’ methods, without recourse to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19865 a ‘solid’ <G>neu=sis</G> (whether actually solved or merely assumed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19866 capable of being solved). If, then, Archimedes chose the more
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19867 difficult method which we actually find him employing, it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19868 scarcely possible to assign any reason except his definite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19869 predilection for the form of proof by <I>reductio ad absurdum</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19870 based ultimately on his famous ‘Lemma’ or Axiom.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19871 <p>It seems worth while to re-examine the whole question of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19872 the discovery and proof of the property, and to see how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19873 Archimedes's argument compares with an easier ‘plane’ proof
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19874 suggested by the figures of some of the very propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19875 proved by Archimedes in the treatise.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19876 <p>In the first place, we may be sure that the property was
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19877 not discovered by the steps leading to the proof as it stands.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19878 I cannot but think that Archimedes divined the result by an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19879 argument corresponding to our use of the differential calculus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19880 for determining tangents. He must have considered the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19881 instantaneous direction of the motion of the point <I>P</I> describ-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19882 ing the spiral, using for this purpose the parallelogram of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19883 velocities. The motion of <I>P</I> is compounded of two motions,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19884 one along <I>OP</I> and the other at right angles to it. Comparing
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19885 the distances traversed in an instant of time in the two direc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19886 tions, we see that, corresponding to a small increase in the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19887 radius vector <I>r,</I> we have a small distance traversed perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19888 dicularly to it, a tiny arc of a circle of radius <I>r</I> subtended by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19889 the angle representing the simultaneous small increase of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19890 angle <G>q</G> (<I>AOP</I>). Now <I>r</I> has a constant ratio to <G>q</G> which we call
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19891 <I>a</I> (when <G>q</G> is the circular measure of the angle <G>q</G>). Consequently
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19892 <pb n=558><head>APPENDIX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19893 the small increases of <I>r</I> and <G>q</G> are in that same ratio. There-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19894 fore what we call the tangent of the angle <I>OPT</I> is <I>r/a,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19895 i.e. <I>OT/r</I> = <I>r/a</I>; and <I>OT</I> = <I>r</I><SUP>2</SUP>/<I>a,</I> or <I>r</I><G>q</G>, that is, the arc of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19896 circle of radius <I>r</I> subtended by the angle <G>q</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19897 <p>To <I>prove</I> this result Archimedes would doubtless begin by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19898 an <I>analysis</I> of the following sort. Having drawn <I>OT</I> perpen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19899 dicular to <I>OP</I> and of length equal to the arc <I>ASP,</I> he had to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19900 prove that the straight line joining <I>P</I> to <I>T</I> is the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19901 at <I>P.</I> He would evidently take the line of trying to show
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19902 that, if <I>any</I> radius vector to the spiral is drawn, as <I>OQ′,</I> on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19903 either side of <I>OP, Q′</I> is always on the side of <I>TP</I> towards <I>O,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19904 or, if <I>OQ′</I> meets <I>TP</I> in <I>F, OQ′</I> is always less than <I>OF.</I> Suppose
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19905 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19906 that in the above figure <I>OR′</I> is any radius vector between <I>OP</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19907 and <I>OS</I> on the ‘backward’ side of <I>OP,</I> and that <I>OR′</I> meets the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19908 circle with radius <I>OP</I> in <I>R,</I> the tangent to it at <I>P</I> in <I>G,</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19909 spiral in <I>R′,</I> and <I>TP</I> in <I>F′.</I> We have to prove that <I>R, R′</I> lie
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19910 on opposite sides of <I>F′,</I> i.e. that <I>RR′</I> > <I>RF′</I>; and again, sup-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19911 posing that <I>any</I> radius vector <I>OQ′</I> on the ‘forward’ side of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19912 <I>OP</I> meets the circle with radius <I>OP</I> in <I>Q,</I> the spiral in <I>Q′</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19913 <I>TP</I> produced in <I>F,</I> we have to prove that <I>QQ′</I> < <I>QF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19914 <p>Archimedes then had to prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19915 (1) <I>F′R : RO</I> < <I>RR′ : RO,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19916 (2) <I>FQ : QO</I> > <I>QQ′ : QO.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19917 <p>Now (1) is equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19918 <I>F′R : RO</I> < (arc <I>RP</I>) : (arc <I>ASP</I>), since <I>RO</I> = <I>PO.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19919 <pb n=559><head>APPENDIX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19920 <p>But (arc <I>ASP</I>) = <I>OT,</I> by hypothesis;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19921 therefore it was necessary to prove, <I>alternando,</I> that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19922 (3) <I>F′R</I> : (arc <I>RP</I>) < <I>RO : OT,</I> or <I>PO : OT,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19923 i.e. < <I>PM : MO,</I> where <I>OM</I> is perpendicular to <I>SP.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19924 <p>Similarly, in order to satisfy (2), it was necessary to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19925 prove that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19926 (4) <I>FQ</I> : (arc <I>PQ</I>) > <I>PM : MO.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19927 <p>Now, as a matter of fact, (3) is <I>a fortiori</I> satisfied if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19928 <I>F′R : (chord RP)</I> < <I>PM : MO</I>;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19929 but in the case of (4) we cannot substitute the <I>chord PQ</I> for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19930 the arc <I>PQ,</I> and we have to substitute <I>PG′,</I> where <I>G′</I> is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19931 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19932 <CAP>FIG. 1.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19933 point in which the tangent at <I>P</I> to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19934 the circle meets <I>OQ</I> produced; for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19935 of course <I>PG′</I> > (arc <I>PQ</I>), so that (4)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19936 is <I>a fortiori</I> satisfied if
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19937 <I>FQ : PG′</I> > <I>PM : MO.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19938 <p>It is remarkable that Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19939 uses for his proof of the two cases Prop.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19940 8 and Prop. 7 respectively, and makes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19941 no use of Props 6 and 9, whereas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19942 the above argument points precisely to the use of the figures
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19943 of the two latter propositions only.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19944 <p>For in the figure of Prop: 6 (Fig. 1), if <I>OFP</I> is any radius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19945 cutting <I>AB</I> in <I>F,</I> and if <I>PB</I> produced cuts <I>OT,</I> the parallel to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19946 <I>AB</I> through <I>O,</I> in <I>H,</I> it is obvious, by parallels, that
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19947 <I>PF</I> : (chord <I>PB</I>) = <I>OP : PH.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19948 <p>Also <I>PH</I> becomes greater the farther <I>P</I> moves from <I>B</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19949 towards <I>A,</I> so that the ratio <I>PF : PB</I> diminishes continually,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19950 while it is always less than <I>OB : BT</I> (where <I>BT</I> is the tangent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19951 at <I>B</I> and meets <I>OH</I> in <I>T</I>), i.e. always less than <I>BM : MO.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19952 <p>Hence the relation (3) is always satisfied for any point <I>R′</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19953 the spiral on the ‘backward’ side of <I>P.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19954 <p>But (3) is equivalent to (1), from which it follows that <I>F′R</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19955 is always less than <I>RR′,</I> so that <I>R′</I> always lies on the side
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19956 of <I>TP</I> towards <I>O.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19957 <pb n=560><head>APPENDIX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19958 <p>Next, for the point <I>Q′</I> on the ‘forward’ side of the spiral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19959 from <I>P,</I> suppose that in the figure of Prop. 9 or Prop. 7 (Fig. 2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19960 any radius <I>OP</I> of the circle meets <I>AB produced</I> in <I>F,</I> and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19961 <FIG>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19962 <CAP>FIG. 2.</CAP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19963 the tangent at <I>B</I> in <I>G</I>; and draw <I>BPH, BGT</I> meeting <I>OT,</I> the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19964 parallel through <I>O</I> to <I>AB,</I> in <I>H, T.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19965 <table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19966 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19967 <td>Then</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19968 <td><I>PF : BG</I></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19969 <td>> <I>FG : BG,</I> since <I>PF</I> > <I>FG,</I></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19970 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19971 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19972 <td></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19973 <td></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19974 <td>> <I>OG : GT,</I> by parallels,</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19975 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19976 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19977 <td></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19978 <td></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19979 <td>> <I>OB : BT, a fortiori,</I></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19980 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19981 <tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19982 <td></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19983 <td></td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19984 <td>> <I>BM : MO</I>;</td>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19985 </tr>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19986 </table>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19987 and obviously, as <I>P</I> moves away from <I>B</I> towards <I>OT,</I> i.e. as <I>G</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19988 moves away from <I>B</I> along <I>BT,</I> the ratio <I>OG : GT</I> increases
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19989 continually, while, as shown, <I>PF : BG</I> is always > <I>BM : MO,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19990 and, <I>a fortiori,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19991 <I>PF</I> : (arc <I>PB</I>) > <I>BM : MO.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19992 <p>That is, (4) is always satisfied for any point <I>Q′</I> of the spiral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19993 ‘forward’ of <I>P,</I> so that (2) is also satisfied, and <I>QQ′</I> is always
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19994 less than <I>QF.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19995 <p>It will be observed that no <G>neu=sis</G>, and nothing beyond
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19996 ‘plane’ methods, is required in the above proof, and Pappus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19997 criticism of Archimedes's proof is therefore justified.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19998 <p>Let us now consider for a moment what Archimedes actually
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	19999 does. In Prop. 8, which he uses to prove our proposition in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20000 the ‘backward’ case (<I>R′, R, F′</I>), he shows that, if <I>PO : OV</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20001 is any ratio whatever less than <I>PO : OT</I> or <I>PM : MO,</I> we can
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20002 find points <I>F′, G</I> corresponding to any ratio <I>PO : OV′</I> where
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20003 <I>OT</I> < <I>OV′</I> < <I>OV,</I> i.e. we can find a point <I>F′</I> corresponding to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20004 a ratio still nearer to <I>PO : OT</I> than <I>PO : OV</I> is. This proves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20005 that the ratio <I>RF′ : PG,</I> while it is always less than <I>PM : MO,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20006 <pb n=561><head>APPENDIX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20007 approaches that ratio without limit as <I>R</I> approach<*>s <I>P.</I> But
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20008 the proof does not enable us to say that <I>RF′ : (chord PR),</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20009 which is > <I>RF′ : PG,</I> is also always less than <I>PM : MO.</I> At
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20010 first sight, therefore, it would seem that the proof must fail.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20011 Not so, however; Archimedes is nevertheless able to prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20012 that, if <I>PV</I> and not <I>PT</I> is the tangent at <I>P</I> to the spiral, an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20013 absurdity follows. For his proof establishes that, if <I>PV</I> is the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20014 tangent and <I>OF′</I> is drawn as in the proposition, then
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20015 <I>F′O : RO</I> < <I>OR′ : OP,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20016 or <I>F′O</I> < <I>OR′,</I> ‘which is impossible’. Why this is impossible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20017 does not appear in Props. 18 and 20, but it follows from the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20018 argument in Prop. 13, which proves that a tangent to the spiral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20019 cannot meet the curve again, and in fact that the spiral is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20020 everywhere concave towards the origin.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20021 <p>Similar remarks apply to the proof by Archimedes of the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20022 impossibility of the other alternative supposition (that the tan-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20023 gent at <I>P</I> meets <I>OT</I> at a point <I>U</I> nearer to <I>O</I> than <I>T</I> is).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20024 <p>Archimedes's proof is therefore in both parts perfectly valid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20025 in spite of any appearances to the contrary. The only draw-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20026 back that can be urged seems to be that, if we assume the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20027 tangent to cut <I>OT</I> at a point <I>very near</I> to <I>T</I> on either side,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20028 Archimedes's construction brings us perilously near to infini-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20029 tesimals, and the proof may appear to hang, as it were, on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20030 a thread, albeit a thread strong enough to carry it. But it is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20031 remarkable that he should have elaborated such a difficult
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20032 proof by means of Props. 7, 8 (including the ‘solid’ <G>neu=sis</G> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20033 Prop. 8), when the figures of Props. 6 and 7 (or 9) themselves
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20034 suggest the direct proof above given, which is independent of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20035 any <G>neu=sis</G>.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20036 <p>P. Tannery,<note>Tannery,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20037 <I>Mémoires scientifiques,</I> i, 1912, pp. 300-16.</note> in a paper on Pappus's criticism of the proof as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20038 unnecessarily involving ‘solid’ methods, has given another
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20039 proof of the subtangent-property based on ‘plane’ methods
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20040 only; but I prefer the method which I have given above
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20041 because it corresponds more closely to the preliminary proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20042 tions actually given by Archimedes.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20043 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20044 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20045 <C><B>INDEX OF GREEK WORDS</B></C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20046 <C>[The pages are those of the first volume except where otherwise stated.]</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20047 <p><G>a)/bac, a)ba/kion</G> 47.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20048 <p><G>a)gewme/trhtos, -on</G>: <G>a)gewme/trhtos mh-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20049 dei\s ei)si/tw</G> (Plato) iii. 355.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20050 <p><G>a)dieci/thtos, -on</G>, that cannot be gone
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20051 through, i.e. infinite 343.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20052 <p><G>a)du/natos, -on</G> ii. 462: <G>a)pagwgh\ ei)s
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20053 a)du/naton</G>, &c. 372.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20054 <p><G>ai)/thma</G>, postulate 373.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20055 <p><G>a)kousmatikoi/</G> 11.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20056 <p><G>a)/logos, -on</G>, irrational 84, 90: <G>peri\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20057 a)lo/gwn grammw=n kai\ nastw=n</G> (Demo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20058 critus) 156-7, 181: <G>a)/logoi w(/sper
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20059 grammai/</G> (Plato) 157.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20060 <p><G>a)na/lhmma</G> ii. 287.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20061 <p><G>a)na/logon</G>, proportional: used as ad-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20062 jective 85.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20063 <p><G>a)naluo/menos</G> (<G>to/pos</G>), <I>Treasury of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20064 Analysis</I> 421-2, ii. 399, 400, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20065 426.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20066 <p><G>a)na/palin</G>, inversely 385: <G>a)na/palin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20067 lu/sis</G> ii. 400.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20068 <p><G>a)nastre/yanti</G> (<G>a)nastre/fw</G>), <I>conver-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20069 tendo</I> 386.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20070 <p><G>a)nastrofh/</G>, conversion <I>ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20071 <p><G>a)nastrofiko\s</G> (<G>to/pos</G>), a class of locus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20072 ii. 185.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20073 <p><G>*)avaforiko/s</G> by Hypsicles 419, ii. 213.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20074 <p><G>a)neli/ttein</G> ii. 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20075 <p><G>a)/cwn</G>, axis 341.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20076 <p><G>a)o/ristos, -on</G>, undefined: <G>plh=qos mo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20077 na/dwn a)o/riston</G> (= unknown, <I>x</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20078 94, ii. 456: <G>e)n a)ori/stw\|</G> ii. 489, 491.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20079 <p><G>a)pagwgh/</G>, <I>reduction</I> 372: <G>a)p. ei)s
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20080 a)du/naton</G>, <I>reductio ad absurdum</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20081 372.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20082 <p><G>a)po/deicis</G>, <I>proof</I> 370, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20083 <p><G>a)pokatastatiko/s, -h/, -o/n</G>, recurring 108.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20084 <p><G>a)po/stasis</G>, distance or <I>dimension</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20085 305 <I>n.,</I> or <I>interval</I> 306 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20086 <p><G>a)/rbhlos</G>, ‘shoemaker's knife’ ii. 23,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20087 ii. 101-2, ii. 371-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20088 <p><G>a)riqmhtikh/</G>, theory of numbers, opp.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20089 to <G>logistikh/</G> 13-16.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20090 <p><G>a)riqmhtiko/s, -h/, -o/n</G>: <G>a)riqmhtikh\ ei)sa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20091 gwgh/</G> of Nicomachus 97.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20092 <p><G>a)riqmo/s</G>, number: definitions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20093 ‘number’ 69-70: in Diophantus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20094 used for unknown quantity (<I>x</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20095 94, ii. 456.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20096 <p><G>a)riqmosto/n</G>: reciprocal of <G>a)riqmo/s</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20097 (=<I>x</I>) in Diophantus ii. 458.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20098 <p><G>a(rpedona/ptai</G>, ‘rope-stretchers’ 121-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20099 2, 178.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20100 <p><G>a)/rrhtos, -on</G>, irrational 157.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20101 <p><G>a)rtia/kis a)/rtios</G>, <I>even-times-even</I> 71,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20102 with Neo-Pythagoreans =2<SUP><I>u</I></SUP>, 72.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20103 <p><G>a)rtia/kis peritto/s</G>, <I>even-times-odd</I> 72.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20104 <p><G>a)rtiope/rittos</G>, <I>even-odd,</I> restricted
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20105 by Neo-Pythagoreans to form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20106 2(2<I>m</I> + 1), 72.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20107 <p><G>a)/rtios, -a, on</G>, <I>even</I> 70.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20108 <p><G>*)arxai/</G>, a lost work of Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20109 ii. 81.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20110 <p><G>*)astroqesi/ai</G> of Eratosthenes ii. 109.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20111 <p><G>a)strola/bon o)/rganon</G> of Hipparchus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20112 ii. 256.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20113 <p><G>a)su/mmetros, -on</G>, incommensurable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20114 157.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20115 <p><G>a)su/mptwtos, -on</G>, non-secant ii. 227.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20116 <p><G>a)su/nqetos, -on</G>, incomposite 72.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20117 <p><G>a)/tomos, -on</G>, indivisible 181: Aristo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20118 telian <G>peri\ a)to/mwn grammw=n</G> 157,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20119 346-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20120 <p><G>a)/topos, -on</G>, absurd ii. 462.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20121 <p><G>au)ca/nein</G>: <G>tri\s au)chqei/s</G> (Plato) 306-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20122 <p><G>au)/ch, tri/th</G>, 297: <G>ku/bwn au)/ch</G>, 297.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20123 <p><G>au)/chsis</G> 305-6 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20124 <p><G>au)tomatopoihtikh/</G> ii. 308.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20125 <p><G>a(yi/s</G>, segment of circle less than a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20126 semicircle ii. 314.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20127 <p><G>*baroulko/s</G> of Heron ii. 309, ii. 346-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20128 <p><G>*belopoii+ka/</G> of Heron 18, ii. 298, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20129 302, ii. 308-9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20130 <p><G>bia/zein</G>: <G>bebiasme/nos</G>, forced or un-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20131 natural ii. 362.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20132 <pb n=564><head>INDEX OF GREEK WORDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20133 <p><G>bwmi/skos</G>, ‘little altar’, properly a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20134 wedge-shaped solid ii. 319, ii. 333:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20135 measurement of(Heron), ii. 332-3:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20136 (=<G>sfhni/skos</G>) of a certain kind
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20137 of solid number 107, ii. 240, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20138 315.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20139 <p><G>gewdaisi/a</G>=mensuration 16.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20140 <p><G>*gewmetrou/mena</G> of Heron ii. 318, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20141 453.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20142 <p><G>glwxi/s</G> (arrow-head), Pythagorean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20143 name for angle 166.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20144 <p><G>gnwmonikh/</G> 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20145 <p><G>gnw/mwn</G>, <I>gnomon</I>, q.v.: <G>kata\ gnw/mona</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20146 =perpendicular 78, 175.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20147 <p><G>gnw/rimos, -on</G>, known: <G>gnw/rimon</G>, an
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20148 alternative term for <G>dedome/non</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20149 <I>given</I> ii. 537.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20150 <p><G>gnwri/mws</G>, ‘in the recognized manner’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20151 ii. 79.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20152 <p><G>gra/mma</G>, ‘figure’ or proposition, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20153 theorem of Eucl. I. 47, 144.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20154 <p><G>grammh/</G>: <G>dia\</G> or <G>e)k tw=n grammw=n</G> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20155 theoretical proof ii. 257, 258.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20156 <p><G>grammiko/s, -h/, -o/n</G>, linear: used of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20157 prime numbers 73: <G>grammikai\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20158 e)pista/seis</G>, ‘Considerations on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20159 Curves’, by Demetrius ii. 359:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20160 <G>grammikw=s</G>, graphically 93.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20161 <p><G>gra/fein</G>, to draw or write on 159,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20162 173: also to <I>prove</I> 203 <I>n.</I>, 339.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20163 <p><G>dedome/nos, -h, -on</G>, <I>given</I>: senses of,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20164 ii. 537-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20165 <p><G>deiknu/nai</G>, to prove 328.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20166 <p><G>dei=n</G>: <G>dei= dh/</G> 371.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20167 <p><G>deu/teros</G>, <I>secondary</I>: of composite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20168 numbers 72: <G>deute/ra muria/s</G> (=
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20169 10,000<SUP>2</SUP>) 40.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20170 <p><G>diabh/ths</G>, compasses 308, ii. 540.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20171 <p><G>diairei=n</G>: <G>dielo/nti</G>, <I>separando</I> or <I>divi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20172 dendo</I> (in transformation of ratios)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20173 386.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20174 <p><G>diai/resis</G>: <G>lo/gou</G>, <I>separation</I> of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20175 ratio 386: <G>peri\ diaire/sewn bibli/on</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20176 <I>On divisions</I> (<I>of figures</I>), by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20177 Euclid 425.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20178 <p><G>dia/stasis</G>, dimension: <G>peri\ diasta/-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20179 sews</G>, a work of Ptolemy ii. 295.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20180 <p><G>dia/sthma</G>, interval 215: distance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20181 239.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20182 <p><G>di/aulos</G>, ‘race-course’: representa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20183 tions of square and oblong num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20184 bers as sums of terms 114.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20185 <p><G>dido/nai</G>: <G>dedome/non</G>, <I>given</I>, senses ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20186 537-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20187 <p><G>diecodiko\s</G> (<G>to/pos</G>), a species of locus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20188 ii. 185.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20189 <p><G>dii+sta/nai</G>: <G>e)f) e(\n diestw/s</G>, extended
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20190 one way ii. 428.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20191 <p><G>diko/louros, -on</G>, twice-truncated 107.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20192 <p><G>di/optra</G>, dioptra, q.v.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20193 <p><G>dioptrikh/</G> 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20194 <p><G>diori/zein</G>: <G>diwrisme/nh tomh/</G>, <I>Deter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20195 minate Section</I>, by Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20196 ii. 180.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20197 <p><G>diorismo/s</G>, definition, delimitation:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20198 two senses (1) a constituent part
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20199 of a theorem or problem 370,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20200 (2) a statement of conditions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20201 possibility of a problem 303, 319-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20202 20, 371, 377, 395, 396, 428, ii. 45-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20203 6, ii. 129-32, ii. 168, ii. 230.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20204 <p><G>diploi+so/ths</G>, double-equation (Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20205 phantus) ii. 468.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20206 <p><G>diplou=s, -h=, -ou=n</G>: <G>diplh= muria/s</G> =
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20207 10,000<SUP>2</SUP> (Apollonius) 40: <G>diplh=
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20208 i)so/ths, diplh= i)/swsis</G>, double-equa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20209 tion (Diophantus) ii. 468.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20210 <p><G>doki/s</G>, <I>beam</I>, a class of solid number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20211 107, ii. 240.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20212 <p><G>doko/s</G> = <G>doki/s</G> ii. 315.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20213 <p><G>draxmh/</G>, sign for, 31, 49, 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20214 <p><G>du/namis</G>: incommensurable side of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20215 square containing a non-square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20216 number of units of area 203-4:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20217 square or square root 209 <I>n.</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20218 297: square of unknown quantity
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20219 (= <I>x</I><SUP>2</SUP>) (Diophantus) ii. 457-8:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20220 <G>duna/mei</G>, ‘in square’ 187, 308:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20221 <G>tetraplh= du/namis</G> = eighth power
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20222 (Egypt) ii. 546; <I>power</I> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20223 mechanics 445.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20224 <p><G>dunamodu/namis</G>, square - square =
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20225 fourth power (Heron) ii. 458:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20226 fourth power of unknown (Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20227 phantus) ii. 458, ii. 546.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20228 <p><G>dunamo/kubos</G>, square-cube, = fifth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20229 power of unknown (Diophantus)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20230 ii. 458.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20231 <p><G>dunamosto/n, dunamodunamosto/n</G>, &c.,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20232 reciprocals of powers of unknown
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20233 (Diophantus) ii. 458.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20234 <p><G>du/nasqai</G>, to be equivalent ‘in square’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20235 to, i. e. to be the side of a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20236 equal to (a given area): <G>duname/nh</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20237 305-6 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20238 <p><G>dunasteuome/nh</G>, opp. to <G>duname/nh</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20239 305-6 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20240 <p><G>ei)=dos</G>, ‘figure’ of a conic ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20241 139: ‘species’ = particular power
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20242 <pb n=565><head>INDEX OF GREEK WORDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20243 of unknown, or term, in an equa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20244 tion (Diophantus) ii. 460.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20245 <p><G>ei(=s, mi/a, e(/n</G>, one: <G>e(/na plei/w</G>, ‘several
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20246 ones’ (definition of ‘number’)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20247 70.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20248 <p><G>ei)shgei=sqai</G>, to introduce or explain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20249 213.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20250 <p><G>e)/kqesis</G>, <I>setting-out</I> 370, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20251 <p><G>*)ekpeta/smata</G> of Democritus 178,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20252 181.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20253 <p><G>e(kth/moros</G> (<G>ku/klos</G>) ii. 288.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20254 <p><G>e)/lleiyis</G>, <I>falling-short</I> (in application
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20255 of areas), name given to <I>ellipse</I> by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20256 Apollonius 150, ii. 138.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20257 <p><G>e)lliph/s, -e/s</G>, <I>defective</I> (of numbers),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20258 contrasted with <I>perfect</I> 74, 101:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20259 <G>*y e)llipe\s ka/tw neu=on</G> ii. 459.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20260 <p><G>e)nalla/c</G>, alternately (in proportions)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20261 385.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20262 <p><G>e)/nnoia</G>, notion: <G>koinai\ e)/nnoiai</G>, com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20263 mon notions = axioms 336.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20264 <p><G>e)/nstasis</G>, <I>objection</I> 372, ii. 311, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20265 <p><G>e)/ntasis</G>, bulging out 6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20266 <p><G>e)celigmo/s</G> ii. 234.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20267 <p><G>e)ch/ghsis</G>, elucidation ii. 223, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20268 231-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20269 <p><G>e(chkosto/n</G>, or <G>prw=ton e(c.</G>, a 60th (=
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20270 a <I>minute</I>), <G>deu/teron e(c.</G>, a <I>second</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20271 &c. 45.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20272 <p><G>e)pa/nqhma</G>, (‘bloom’) of Thymaridas:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20273 a system of linear equations solved
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20274 94.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20275 <p><G>e)pafh/</G>, contact: <G>*)epafai/</G>, <I>Contacts</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20276 or <I>Tangencies</I>, by Apollonius ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20277 181.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20278 <p><G>e)pi/</G>, on: <G>to\ shmei=on e)f) w(=\|</G> (or <G>ou(=</G>) <I>K,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20279 archaic for ‘the point <I>K</I>’ 199:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20280 <G>h( e)f) h(=\|</G> <I>AB</I>, ‘the straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20281 <I>AB</I>’ <I>ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20282 <p><G>e)pimerh/s</G>, <I>superpartiens</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20283 = ratio 1 +(&horbar;(<I>m</I> + <I>n</I>)), 102.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20284 <p><G>e)pimo/rios</G>, <I>superparticularis</I> = ratio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20285 of form (<I>n</I> + 1)/<I>n</I>, 90, 101: <G>e)pi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20286 mo/rion dia/sthma</G> 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20287 <p><G>e)pipedometrika/</G> ii. 453.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20288 <p><G>e)pishmasi/ai</G>, weather indications
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20289 177 <I>n.</I>, ii. 234.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20290 <p><G>e)pi/tritos</G> = ratio 4/3, 101: <G>e)pi/tritos
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20291 puqmh/n</G> (Plato) 306-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20292 <p><G>e)/sxatos</G>: <G>ta\ e)/sxata</G>, extremities
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20293 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20294 <p><G>e(teromh/khs, -es</G>, <I>oblong</I>; of numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20295 of form <I>m</I>(<I>m</I> + 1), 82, 108.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20296 <p><G>eu)qugrammiko\s</G> (<G>a)riqmo/s</G>) = <I>prime</I> 72.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20297 <p><G>e)fektiko/s</G>, a class of locus ii. 185,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20298 ii. 193.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20299 <p><G>e)fo/dion</G>, <I>Method</I> ii. 246.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20300 <p><G>zugo/n</G>, lever or balance: <G>peri\ zugw=n</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20301 a work of Archimedes ii. 23-4,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20302 ii. 351.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20303 <p><G>h(mio/lios, -a, -on</G>, ratio of 3/2, 101.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20304 <p><G>h(miwbe/lion</G>, (1/2).obol, sign for, 31, 49, 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20305 <p><G>qaumatopoii+kh/</G> 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20306 <p><G>qeologou/mena a)riqmhtikh=s</G> 97.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20307 <p><G>qe/sis</G>, position: <G>para\ qe/sei\|</G> (<I>sc.</I> <G>dedo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20308 me/nhn</G>), parallel to a straight line
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20309 given in position ii. 193: <G>pro\s
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20310 qe/sei eu)qei/ais</G>, on straight lines
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20311 given in position ii. 426.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20312 <p><G>qureo/s</G>, <I>shield</I>, old name for ellipse
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20313 439, ii. 111, ii. 125.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20314 <p><G>i)/llesqai</G>: <G>i)llome/nhn</G> used by Plato of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20315 the earth 314-15.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20316 <p><G>i)sa/kis i)/sos</G>, equal an equal number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20317 of times, or equal multiplied by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20318 equal 204.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20319 <p><G>i)so/metros, -on</G>, of equal contour:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20320 <G>peri\ i)some/trwn sxhma/twn</G>, by Zeno-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20321 dorus ii. 207, ii. 390.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20322 <p><G>i)so/pleuros, -on</G>, equilateral: of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20323 square number (Plato) 204.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20324 <p><G>i)sorropi/a</G>, equilibrium : <G>peri\ i)sorro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20325 piw=n</G>, work by Archimedes ii. 24,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20326 ii. 351.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20327 <p><G>i)/sos</G>, equal: <G>di) i)/sou</G>, <I>ex aequali</I> (in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20328 proportions) 386 : <G>di) i)/sou e)n te-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20329 taragme/nh\| a)nalogi/a\|</G> 386.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20330 <p><G>i)so/ths</G> or <G>i)/swsis</G>, equation ii. 468.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20331 <p><G>i(stori/a</G>, inquiry, Pythagoras's name
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20332 for geometry 166.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20333 <p><G>i)sxu/s</G>, power (in mechanics) 445.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20334 <p><G>kampth/r</G>, turning-point in race-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20335 course 114.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20336 <p><G>kampu/los, -h, -on</G>, curved 249, 341.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20337 <p><G>kanonikh/</G>, <I>Canonic</I>, q.v.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20338 <p><G>kanw/n</G>, ruler 239: <I>Table</I> (astron.),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20339 <G>*proxei/rwn kano/nwn dia/tasis kai\
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20340 yhfofori/a</G>, work by Ptolemy ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20341 293: <I>canon</I> (in music), v. <G>*katatomh/.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20342 katagra/fein</G>: <I>to inscribe in</I> or <I>on</I> (c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20343 gen.) 131.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20344 <p><G>*kata/logoi</G>, work by Eratosthenes ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20345 108.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20346 <p><G>kataskeua/zein</G> 193 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20347 <p><G>kataskeuh/</G>, <I>construction</I> (constituent
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20348 part of proposition) 370, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20349 <pb n=566><head>INDEX OF GREEK WORDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20350 <p><G>*katasterismoi/</G>, work by Eratosthe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20351 nes ii. 108.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20352 <p><G>*katatomh\ kano/nos</G>, <I>Sectio canonis,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20353 attributed to Euclid 17, 444.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20354 <p><G>katono/macis tw=n a)riqmw=n</G>, naming of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20355 numbers (Archimedes) ii. 23.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20356 <p><G>katoptrikh/</G>, theory of mirrors 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20357 <p><G>kentrobarika/</G>, problems on centre of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20358 gravity ii. 24, ii. 350.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20359 <p><G>ke/ntron</G>, centre: <G>h( e)k tou= ke/ntrou</G> =
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20360 radius 381.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20361 <p><G>keratoeidh\s</G> (<G>gwni/a</G>) 178, 382.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20362 <p><G>*khri/a</G> of Sporus 234.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20363 <p><G>kla/ein</G>, inflect: <G>kekla/sqai</G> 337.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20364 <p><G>kogxoeidh\s grammh/</G>, <I>conchoid</I> 238.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20365 <p><G>koilogw/nion</G> ii. 211.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20366 <p><G>ko/louros, -on</G>, truncated ii. 333: (of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20367 pyramidal number) 107.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20368 <p><G>ko/skinon</G>, <I>sieve</I> (of Eratosthenes) 16,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20369 100, ii. 105.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20370 <p><G>koxloeidh\s grammh/</G>, <I>cochloid</I> 238.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20371 <p><G>kubo/kubos</G>, cube-cube, = sixth power
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20372 of unknown (Diophantus) ii. 458.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20373 <p><G>kubokubosto/n</G>, reciprocal of <G>kubo/-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20374 kubos</G> ii. 458.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20375 <p><G>ku/bos</G>, cube: <G>ku/bwn au)/ch</G> (Plato)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20376 297: cube of unknown (Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20377 phantus) ii. 458: <G>ku/bos e)celikto/s</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20378 = ninth power of unknown
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20379 (Egyptian) ii. 546.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20380 <p><G>kuklikh\ qewri/a</G>, <I>De motu circulari,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20381 by Cleomedes ii. 235.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20382 <p><G>kukliko/s, -h/, -o/n</G>, <I>circular,</I> used of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20383 square numbers ending in 5 or 6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20384 108.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20385 <p><G>lei/pein</G>: forms used to express <I>minus,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20386 and sign for (Diophantus), ii. 459.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20387 <p><G>lei=yis</G>, <I>wanting</I> (Diophantus): <G>lei/yei</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20388 = <I>minus</I> ii. 459.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20389 <p><G>le/cis</G>: <G>kata\ le/cin</G>, word for word
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20390 183.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20391 <p><G>lepto/n</G>, a <I>fraction</I> (Heron) 43: = a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20392 <I>minute</I> (Ptolemy) 45.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20393 <p><G>lh=mma</G>, <I>lemma</I> 373.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20394 <p><G>logismo/s</G>, calculation 13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20395 <p><G>logistikh/</G>, art of calculation, opp.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20396 to <G>a)riqmhtikh/</G> 13-16, 53.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20397 <p><G>lo/gos</G>, ratio: <G>lo/gou a)potomh/</G>, <I>sectio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20398 rationis,</I> by Apollonius ii. 175.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20399 <p><G>maqh/mata</G>, subjects of instruction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20400 10-11: term first appropriated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20401 to mathematics by Pythagoreans
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20402 11: <G>peri\ tw=n maqhma/twn</G>, a work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20403 by Protagoras 179.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20404 <p><G>maqhmatiko/s, -h/, -o/n</G>: <G>maqhmatikoi/</G> in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20405 Pythagorean school, opp. to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20406 <G>a)kousmatikoi/</G> 11: <G>*maqhmatikh\ su/n-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20407 tacis</G> of Ptolemy ii. 273-4: <G>maqh-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20408 matika/, ta\</G> (Plato) 288.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20409 <p><G>meqo/rion</G>, boundary ii. 449.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20410 <p><G>mei/ouron proeskarifeume/non</G> (Heron),
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20411 curtailed and pared in front (cf.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20412 scarify), of a long, narrow, tri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20413 angular prism (Heib.) ii. 319.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20414 <p><G>me/ros</G>: <G>me/rh</G>, <I>parts</I> (= proper frac-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20415 tion) dist. from <G>me/ros</G> (aliquot
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20416 part) 42 (cf. p. 294).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20417 <p><G>mesola/bon</G>, <I>mean-finder</I> (of Erato-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20418 sthenes) ii. 104. ii. 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20419 <p><G>mete/wros, -on</G>: <G>peri\ metew/rwn</G>, work
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20420 by Posidonius ii. 219, ii. 231-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20421 <p><G>metewroskopikh/</G> 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20422 <p><G>*metrh/seis</G>, <I>Mensurae</I> (Heronian) ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20423 319.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20424 <p><G>mh=kos</G>, length: used by Plato of side
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20425 of square containing a square
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20426 number of units of area 204.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20427 <p><G>mhli/ths</G> (<G>a)riqmo/s</G>), term for problems
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20428 about numbers of apples (e.g.) 14,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20429 ii. 442.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20430 <p><G>*mikro\s a)stronomou/menos</G> (<G>to/pos</G>), <I>Little
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20431 Astronomy</I> ii. 273.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20432 <p><G>mna=</G>, mina (= 1000 drachmae): <G>*m</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20433 stands for, 31.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20434 <p><G>moi=ra</G>, fraction: 1/360th of circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20435 ference or a <I>degree</I> 45, 61: <G>moi=ra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20436 topikh/, xronikh/</G> (in Hypsicles) ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20437 214.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20438 <p><G>mona/s</G>, monad or unit 43: definitions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20439 of, 69: <G>mona/dwn su/sthma</G> = number,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20440 69: <G>deuterwdoume/nh mona/s</G> = 10,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20441 <G>triwdoume/rh m.</G> = 100, &c. (Iambl.)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20442 114: <G>mona\s qe/sin e)/xousa</G> = point
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20443 69, 283.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20444 <p><G>mo/rion</G>, part or fraction: <G>mori/ou</G> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20445 <G>e)n mori/w\|</G> = divided by (Diophan-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20446 tus) 44.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20447 <p><G>muria/s</G> (with or without <G>prw/th</G> or
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20448 <G>a(plh=</G>) myriad (10,000), <G>m. deute/ra</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20449 or <G>diplh=</G> 10,000<SUP>2</SUP>, &c. 40.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20450 <p><G>nasto/n</G> (solid?) 156, 178.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20451 <p><G>neu/ein</G>, to verge (towards) 196, 239,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20452 337, ii. 65.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20453 <p><G>neu=sis</G>, <I>inclinatio</I> or ‘verging’, a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20454 type of problem 235-41, 260, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20455 199, ii. 385: <G>neu/seis</G> in Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20456 medes ii 65-8: two books of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20457 <G>neu/seis</G> by Apollonius ii. 189-92
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20458 ii. 401, ii. 412-13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20459 <pb n=567><head>INDEX OF GREEK WORDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20460 <p><G>nu/ssa</G>, goal or end of race-course
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20461 114.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20462 <p><G>o)bole/s</G>, obol: sign for, 31, 49, 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20463 <p><G>*)olumpioni=kai</G>, work by Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20464 ii. 109.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20465 <p><G>o)/nuc</G>, a wedge-shaped figure ii. 319,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20466 ii. 333.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20467 <p><G>o)rganopoii+kh/</G> 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20468 <p><G>o)/rqios, -a, -on</G>, right or perpendi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20469 cular: <G>o)rqi/a pleura/</G>, <I>latus rectum</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20470 ii. 139: <G>o)rqi/a dia/metros</G>, ‘erect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20471 diameter’, in double hyperbola,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20472 ii. 134.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20473 <p><G>o(ri/zein</G>: <G>w(risme/nos</G>, defined, i. e. de-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20474 terminate 94, 340.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20475 <p><G>o(ri/zwn</G> (<G>ku/klos</G>), dividing circle:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20476 <I>horizon</I> (Eucl.) 351. 352.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20477 <p><G>o(/ros</G>, (1) definition 373: (2) limit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20478 or boundary 293: (3) <I>term</I> (in a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20479 proportion) 306 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20480 <p><G>ou)demi/a</G> or <G>ou)de/n</G>, sign for (O), 39, 45.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20481 <p><G>pa= bw= kai\ kinw= ta\n ga=n</G>, saying of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20482 Archimedes ii. 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20483 <p><G>par' h(\n du/nantai</G> (<G>ai( katago/menai tetag-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20484 me/nws</G>), expression for <I>parameter</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20485 of ordinates ii. 139.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20486 <p><G>parabolh/</G>, application: <G>p. tw=n xwri/wn</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20487 application of areas 150: <G>ta\ e)k
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20488 th=s parabolh=s gino/mena shmei=a</G>, the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20489 foci of a central conic, ii. 156:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20490 <I>parabola</I> (the conic) 150, ii. 138.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20491 <p><G>*paradocogra/foi</G> ii. 541.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20492 <p><G>para/docos grammh/</G>, paradoxical curve
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20493 (of Menelaus) ii. 260-1, ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20494 <p><G>para/phgma</G> 177, ii. 234.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20495 <p><G>paraspa=n</G>, to pull awry: <G>parespa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20496 sme/nos</G> ii. 398.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20497 <p><G>pariso/ths</G>, nearness to equality, ap-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20498 proximation: <G>pariso/thtos a)gwgh/</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20499 (Diophantus) ii. 477, ii. 500.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20500 <p><G>pe/lekus</G>, axe-shaped figure ii. 315.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20501 <p><G>pempa/zein</G>, to ‘five’ (= count) 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20502 <p><G>pe/ntaqlos</G> 176, ii. 104.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20503 <p><G>perai/nousa poso/ths</G> = unit, 69.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20504 <p><G>pe/ras</G>, limit or extremity 293:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20505 limiting surface 166: <G>pe/ras sug-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20506 klei=on</G>, definition of figure ii. 221.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20507 <p><G>perissa/rtios</G>, <I>odd-even</I>: with Neo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20508 Pythagoreans is of form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20509 2<SUP><I>n</I>+1</SUP> (2<I>m</I>+1), 72.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20510 <p><G>perisso/s, -h/, -o/n</G>, <I>odd,</I> q.v.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20511 <p><G>pettei/a</G> 19.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20512 <p><G>phli/kos, -h, -on</G>, how great (of mag-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20513 nitude) 12.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20514 <p><G>phliko/ths</G>, size 384.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20515 <p><G>pla/gios, -a, -on</G>, transverse: <G>plagi/a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20516 dia/metros</G> or <G>pleura/</G> ii. 139.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20517 <p><G>plasmatiko/s, -o/n</G>, (easily) formable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20518 ii. 487.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20519 <p><G>*platwniko/s</G>, a work by Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20520 ii. 104.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20521 <p><G>plh=qos</G>, multitude: <G>plh=qos e(/n</G> = unit,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20522 69: <G>plh=qos w(risme/non</G> = number,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20523 70: <G>plh=qos mora/dwn a)o/riston</G>, def.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20524 of unknown ‘quantity’ 94, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20525 456.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20526 <p><G>plinqi/s</G>, a <I>brick,</I> a solid number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20527 a certain form 107, ii. 240, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20528 315.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20529 <p><G>pollaplasiepimerh/s</G>, <I>multiplex super-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20530 partiens,</I> = ratio of form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20531 <I>p</I> + <I>m</I>/(<I>m</I>+<I>n</I>), 103.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20532 <p><G>pollaplasiepimo/rios</G>, <I>multiplex su-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20533 perparticularis,</I> = ratio of form
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20534 <I>m</I> + 1/<I>n,</I> 103.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20535 <p><G>pollapla/sios, -a, -on</G>, multiple 101.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20536 <p><G>polu/spastos</G>, a compound pulley ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20537 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20538 <p><G>po/rimos, -on</G> (<G>pori/zein</G>), procurable:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20539 one sense of <G>dedomenos</G> ii. 538.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20540 <p><G>po/risma</G>, porism: (1) = corollary,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20541 (2) a certain type of proposition
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20542 372-3, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20543 <p><G>poso/n</G>, quantity, of number, 12.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20544 <p><G>poso/ths</G>, quantity 69, 70: number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20545 defined as <G>poso/thtos xu/ma e)k mona/-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20546 dwn sugkei/menon</G> 70.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20547 <p><G>promh/khs</G>, <I>prolate</I> (= oblong) 203:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20548 but distinguished from <G>e(teromh/khs</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20549 83, 108.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20550 <p><G>prosagw/gion</G> 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20551 <p><G>pro/tasis</G> = <I>enunciation</I> 370, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20552 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20553 <p><G>prw=tos</G>, <I>prime</I> 72.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20554 <p><G>ptw=sis</G>, <I>case</I> 372.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20555 <p><G>puqmh/n</G>, base; = digit 55-7, 115-17:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20556 <G>e)pi/tritos piqmh/n</G> 306-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20557 <p><G>purami/s</G>, pyramid 126.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20558 <p><G>pu/reion, pu/rion</G>, burning mirror:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20559 <G>peri\ purei/wn</G>, work by Diocles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20560 264, ii. 200; <G>peri\ tou= puri/ou</G>, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20561 Apollonius ii. 194.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20562 <p><G>r(hto/s, -h/, -o/n</G>, rational: used in sense
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20563 of ‘given’ ii. 537.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20564 <p><G>r(oph/</G>: <G>peri\ r(opw=n</G>, a mechanical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20565 work by Ptolemy ii. 295.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20566 <pb n=568><head>INDEX OF GREEK WORDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20567 <p><G>sa/linon</G> of Archimedes ii. 23, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20568 103.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20569 <p><G>sh/kwma</G> 49.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20570 <p><G>ska/fh</G>, a form of sun-dial ii. 1, ii. 4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20571 <p><G>skhnografikh/</G>, scene-painting 18, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20572 224.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20573 <p><G>*sofi/a</G>, nickname of Democritus 176.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20574 <p><G>spei=ra</G>, <I>spire</I> or <I>tore</I> ii. 117: varie-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20575 ties of (<G>diexh/s, sunexh/s, e)mpepleg-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20576 me/nh</G> or <G>e)palla/ttouta</G>), ii. 204.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20577 <p><G>sta/qmh</G>, plumb-line 78, 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20578 <p><G>stath/r</G>, sign for, 31.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20579 <p><G>stereometri/a</G>, solid geometry 12-13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20580 <p><G>stereometrou/mena</G> ii. 453.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20581 <p><G>sthli/s</G>, column, a class of solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20582 number. 107.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20583 <p><G>stigmh/</G>, point 69: <G>stigmh\ a)/qetos</G> =
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20584 unit, 69.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20585 <p><G>stoixeiw/ths, -o(</G>, the writer of Ele-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20586 ments (<G>stoixei=on</G>), used of Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20587 357.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20588 <p><G>stroggu/los, -on</G>, round or circular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20589 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20590 <p><G>sumpe/rasma</G>, <I>conclusion</I> (of proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20591 tion) 370, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20592 <p><G>su/nqesis</G> (<G>lo/gou</G>), composition (of a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20593 ratio) 385.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20594 <p><G>su/ntacis</G>, collection: <G>*mega/lh su/n-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20595 tacis</G> of Ptolemy 348, called
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20596 <G>*maqhmatikh\ su/ntacis</G> ii. 273.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20597 <p><G>suntiqe/nai</G>: <G>sunqe/nti</G> = <I>componendo</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20598 (in proportion) 385.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20599 <p><G>su/stasis</G>, construction 151, 158.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20600 <p><G>sfairiko/s, -h/, -o/n</G>, spherical: used of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20601 cube numbers ending in 5 or 6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20602 107-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20603 <p><G>sfhki/skos</G>, <I>stake,</I> a form of solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20604 number, 107.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20605 <p><G>sfhni/skos</G>, <I>wedge,</I> a solid of a certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20606 form, measurement of, ii. 332-3:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20607 a solid number, 107, ii. 315, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20608 319.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20609 <p><G>sxe/sis</G>, relation 384.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20610 <p><G>sxhmatopoiei=n</G>, to form a figure ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20611 226.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20612 <p><G>*ta/lanton</G>, sign for (T), 31, 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20613 <p><G>tara/ssein</G>: (<G>di' i)/sou</G>) <G>e)n tetaragme/nh\|
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20614 a)nalogi/a\|</G>, <I>in disturbed proportion</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20615 386.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20616 <p><G>ta/ssein</G>: <G>tetagme/non</G>, <I>assigned = da-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20617 tum</I> ii. 192, ii. 537: <G>ai( katago/menai
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20618 tetagme/nws</G> (<G>eu)qei=ai</G>), (straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20619 lines) drawn <I>ordinate-wise</I> = or-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20620 dinates ii. 139: <G>tetagme/nws kat-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20621 h=xqai</G> ii. 134.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20622 <p><G>ta/xos</G>, speed: <G>peri\ taxw=n</G>, work by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20623 Eudoxus 329.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20624 <p><G>te/leios, -a, -on</G>, perfect: <G>te/leios a)riq-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20625 mo/s</G> 74, 101.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20626 <p><G>tetarthmo/rion</G>, 1/4 of obol, sign for, 31,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20627 49, 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20628 <p><G>tetragwni/zein</G>, to square: <G>h( tetragw-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20629 ni/zousa</G> (<G>grammh/</G>), the <I>quadratrix</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20630 225, ii. 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20631 <p><G>tetragwnismo/s</G>, <I>squaring</I> 173.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20632 <p><G>tetraktu/s</G> 75, 99 <I>n.,</I> 313, ii. 241.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20633 <p><G>tetraplh= du/namis</G> = 8th power of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20634 unknown (Egyptian term) ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20635 546.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20636 <p><G>tmh=ma</G>, segment: used of lunes as
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20637 well as segments of circles 184:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20638 segments or sectors 187-9: <G>tmh/-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20639 mata</G> = 1/360th parts of circum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20640 ference and 1/120th parts of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20641 diameter of circle (Ptolemy) 45.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20642 <p><G>tomeu/s</G>, shoemaker's knife, term for
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20643 <I>sector</I> of circle 381.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20644 <p><G>tomh/</G>, section: <G>ta\ peri\ th\n tomh/n</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20645 (Proclus) 324-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20646 <p><G>to/pos</G>, locus: classifications of loci
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20647 218-19, ii. 185: <G>to/poi pro\s gram-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20648 mai=s, to/poi pro\s e)pifanei/ais</G> (<G>-a\|</G>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20649 218-19, 439: <G>to/poi pro\s meso/thtas</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20650 ii. 105: <G>to/pos a)naluo/menos</G>, <I>Trea-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20651 sury of Analysis,</I> q. v.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20652 <p><G>to/rnos</G>, circle-drawer 78, 308.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20653 <p><G>tri/gwnos a)riqmo/s</G>, triangular number,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20654 15-16.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20655 <p><G>triko/louros</G>, thrice-truncated 107.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20656 <p><G>tri/pleuron</G>, <I>three-side,</I> Menelaus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20657 term for spherical triangle ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20658 262.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20659 <p><G>triw/bolon</G>, sign for, 49.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20660 <p><G>u(/dria w(roskopei=a</G>, water-clocks ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20661 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20662 <p><G>u(/parcis</G>, forthcoming: <I>positive</I> term,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20663 dist. from negative (<G>lei=yis</G>) ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20664 459.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20665 <p><G>u(pepimerh/s</G>, <I>subsuperpartiens,</I> reci-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20666 procal of <G>e)pimerh/s</G> 102.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20667 <p><G>u(pepimo/rios</G>, <I>subsuperparticularis,</I> re-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20668 ciprocal of <G>e)pimo/rios</G> 101.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20669 <p><G>u(perbolh/</G>, <I>exceeding</I> (in application
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20670 of areas): name given to <I>hyper-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20671 bola</I> 150, ii. 138.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20672 <p><G>u(perte/leios, u(pertelh/s</G>, <I>over-perfect</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20673 (number) 74, 100.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20674 <p><G>(*upoqe/seis tw=n planwme/nwn</G>, work by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20675 Ptolemy ii. 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20676 <p><G>u(popollapla/sios, u(popollaplasiepi-</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20677 <pb n=569><head>INDEX OF GREEK WORDS</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20678 <G>merh/s, u(popollaplasiepimo/rios</G>, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20679 101-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20680 <p><G>u(potei/nein</G>, subtend 193 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20681 <p><G>u(/splhc</G>, starting-point (of race-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20682 course) 114.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20683 <p><G>*fa/seis a)planw=n a)ste/rwn</G>, work by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20684 Ptolemy, ii. 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20685 <p><G>fiali/ths</G> (<G>a)riqmo/s</G>), (number) of bowls
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20686 (in simple algebraical problems)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20687 14, ii. 442.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20688 <p><G>*filokali/a</G>, by Geminus ii. 223.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20689 <p><G>xalkou=s</G> ((1/8)th of obol), sign for, 31:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20690 48, 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20691 <p><G>xei/r</G>, <I>manus,</I> in sense of number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20692 men 27.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20693 <p><G>xeiroba/llistra</G> ii. 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20694 <p><G>xroia/</G>, colour or skin: Pythagorean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20695 name for surface 166, 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20696 <p><G>*xronografi/ai</G>, work by Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20697 ii. 109.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20698 <p><G>xrw=ma</G>, colour (in relation to sur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20699 face) 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20700 <p><G>xwri/on</G>, area 300 <I>n.</I>: <G>xwri/ou a)potomh/</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20701 <I>sectio spatii,</I> by Apollonius ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20702 179.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20703 <p><G>yhfofori/a kat) )indou/s</G> ii. 546.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20704 <p><G>*)wkuto/kion</G> of Apollonius 234, ii. 194,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20705 ii. 253.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20706 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20707 <C><B>ENGLISH INDEX</B>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20708 [The pages are those of the first volume except where otherwise stated.]</C>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20709 <p>Abacus 46-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20710 <p>‘Abdelmelik al-Sh&imacr;r&amacr;z&imacr; ii. 128.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20711 <p>Abraham Echellensis ii. 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20712 <p>Ab&umacr; Bekr Mu&hdot;. b. al-&Hdot;asan al-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20713 Karkh&imacr;, <I>see</I> al-Karkh&imacr;.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20714 <p>Ab&umacr; 'l Fat&hdot; al-I⋅fah&amacr;n&imacr; ii. 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20715 <p>Ab&umacr; 'l Waf&amacr; al-B&umacr;zj&amacr;n&imacr; ii. 328, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20716 450, ii. 453.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20717 <p>Ab&umacr; Nasr Man⋅&umacr;r ii. 262.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20718 <p><I>Achilles</I> of Zeno 275-6, 278-80.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20719 <p>Adam, James, 305-7, 313.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20720 <p>Addition in Greek notation 52.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20721 <p>Adrastus ii. 241, 243, 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20722 <p>Aëtius 158-9, 163, ii. 2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20723 <p>‘Aganis’: attempt to prove paral-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20724 lel-postulate 358, ii. 228-30.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20725 <p>Agatharchus 174.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20726 <p>Ahmes (Papyrus Rhind) 125, 130,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20727 ii. 441.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20728 <p>Akhm&imacr;m, Papyrus of, ii. 543-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20729 <p>Albertus Pius ii. 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20730 <p>Al-Ch&amacr;zin&imacr; ii. 260-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20731 <p>Alexander the ‘Aetolian’ ii. 242.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20732 <p>Alexander Aphrodisiensis 184, 185,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20733 186, 222, 223, ii. 223, ii. 231.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20734 <p>Alexeieff, ii. 324-5 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20735 <p><I>Al-Fakhri,</I> by al-Karkh&imacr; 109, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20736 449-50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20737 <p>Algebra: beginnings in Egypt ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20738 440: <I>hau</I>-calculations ii. 440-1:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20739 Pythagorean, 91-7: <I>epanthema</I> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20740 Thymaridas 94-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20741 <p>Algebra, geometrical, 150-4: ap-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20742 plication of areas (q.v.) 150-3:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20743 scope of geometrical algebra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20744 153-4: method of proportion <I>ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20745 <p>Al-&Hdot;ajj&amacr;j, translator of Euclid,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20746 362: of Ptolemy ii. 274.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20747 <p>Alhazen, problem of, ii. 294.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20748 <p><I>Al-K&amacr;f&imacr;</I> of al-Karkh&imacr; 111.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20749 <p>Al-Karkh&imacr;: on sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20750 1<SUP>3</SUP>+2<SUP>3</SUP> +...+ <I>n</I><SUP>3</SUP>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20751 109-10, 111, ii. 51, ii. 449.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20752 <p>Allman, G. J. 134, 183.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20753 <p><I>Almagest</I> ii. 274.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20754 <p>Alphabet, Greek: derived from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20755 Phoenician, 31-2: Milesian, 33-4:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20756 <I>quasi</I>-numerical use of alphabet,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20757 35-6 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20758 <p>Alphabetic numerals 31-40, 42-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20759 <p>Amasis 4, 129.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20760 <p>Amenemhat I 122, III 122.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20761 <p>Ameristus 140, 141, 171.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20762 <p>Amyclas (better Amyntas) 320-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20763 <p>Amyntas 320-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20764 <p><I>Analemma</I> of Ptolemy ii. 286-92:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20765 of Diodorus ii. 287.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20766 <p>Analysis: already used by Pytha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20767 goreans 168: supposed invention
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20768 by Plato 291-2: absent from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20769 Euclid's <I>Elements</I> 371-2: defined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20770 by Pappus ii. 400.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20771 <p>Anatolius 11, 14, 97, ii. 448, ii. 545-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20772 <p>Anaxagoras: explanation of eclipses
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20773 7, 162, 172: moon borrows light
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20774 from sun 138, 172, ii. 244: cen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20775 trifugal force and centripetal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20776 tendency 172-3: geometry 170:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20777 tried to square circle 173, 220:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20778 on perspective 174: in <I>Erastae</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20779 22, 174.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20780 <p>Anaximander 67, 177: introduced
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20781 <I>gnomon</I> 78, 139, 140: astronomy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20782 139, ii. 244: distances of sun and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20783 moon 139: first map of inhabited
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20784 earth <I>ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20785 <p>Anaximenes ii. 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20786 <p>Anchor-ring, <I>see</I> Tore.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20787 <p>Anderson, Alex., ii. 190.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20788 <p>Angelo Poliziano ii. 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20789 <p>Angle ‘<I>of</I> a segment’ and ‘<I>of</I> a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20790 semicircle’ 179: ‘angle of con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20791 tact’ 178-9, ii. 202.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20792 <p>Anharmonic property, of arcs of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20793 great circles ii. 269-70: of straight
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20794 lines ii. 270, ii. 420-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20795 <pb n=571><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20796 <p>Anthemius of Tralles 243, ii. 194,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20797 ii. 200-3, ii. 518, ii. 540, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20798 541-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20799 <p>Antiphon 184, 219, 221-2, 224,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20800 271.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20801 <p><I>&Amacr;pastamba-&Sacute;ulba-S&umacr;tra</I> 145-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20802 <p>Apelt, E. F. 330.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20803 <p>Apelt, O. 181 <I>n.,</I> 182.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20804 <p><I>Apices</I> 47.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20805 <p>Apollodorus, author of <I>Chronica,</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20806 176.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20807 <p>Apollodorus <G>o( logistiko/s</G>: distich of,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20808 131, 133, 134, 144, 145.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20809 <p>Apollonius of Perga ii. 1, ii. 126.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20810 <p>Arithmetic: <G>w)kuto/kion</G> 234, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20811 194, ii. 253 (approximation to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20812 <G>p</G>, <I>ib.</I>), ‘tetrads’ 40, continued
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20813 multiplications 54-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20814 <p>Astronomy ii. 195-6: A. and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20815 Tycho Brahe 317, ii. 196: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20816 epicycles and eccentrics ii. 195-6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20817 ii. 243: trigonometry ii. 253.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20818 <p><I>Conics</I> ii. 126-75: text ii. 126-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20819 8, Arabic translations ii. 127,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20820 profaces ii. 128-32, characteris-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20821 tics ii. 132-3: conics obtained
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20822 from oblique cone ii. 134-8,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20823 prime property equivalent to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20824 Cartesian equation (oblique axes)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20825 ii. 139, new names, <I>parabola,</I> &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20826 150, 167, ii. 138, transformation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20827 of coordinates ii. 141-7, tangents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20828 ii. 140-1, asymptotes ii. 148-9,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20829 rectangles under segments of in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20830 tersecting chords ii. 152-3, har-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20831 monic properties ii. 154-5, focal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20832 properties (central conics) ii. 156-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20833 7, normals as maxima and mini-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20834 ma ii. 159-67, construction of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20835 normals ii. 166-7, number of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20836 normals through point ii. 163-4,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20837 propositions giving evolute ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20838 164-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20839 <p><I>On contacts</I> ii. 181-5 (lemmas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20840 to, ii. 416-17), three-circle pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20841 blem ii. 182-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20842 <p><I>Sectio rationis</I> ii. 175-9 (lemmas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20843 to, ii. 404-5).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20844 <p><I>Sectio spatii</I> ii. 179-80, ii. 337,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20845 ii. 339.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20846 <p><I>Determinate section</I> ii. 180-1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20847 (lemmas to, ii. 405-12).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20848 <p>Comparison of dodecahedron
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20849 and icosahedron 419-20, ii. 192.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20850 <p>Duplication of cube 262-3, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20851 194.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20852 <p>‘General treatise’ ii. 192-3, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20853 253: on Book I of Euclid 358.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20854 <p><G>neu/seis</G> ii. 68, ii. 189-92 (lemmas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20855 to, ii. 412-16), rhombus-problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20856 ii. 190-2, square - problem ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20857 412-13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20858 <p><I>Plane Loci</I> ii. 185-9 (lemmas to,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20859 ii. 417-19).
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20860 <p><I>On cochlias</I> 232, ii. 193, ‘sister
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20861 of cochloid’ 225, 231-2, <I>On irra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20862 tionals</I> ii. 193, <I>On the burning-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20863 mirror</I> ii. 194, ii. 200-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20864 <p>Application of areas 150-3: method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20865 attributed to Pythagoras 150,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20866 equivalent to solution of general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20867 quadratic 150-2, 394-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20868 <p>Approximations to √2 (by means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20869 of ‘side-’ and ‘diameter-’ num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20870 bers) 91-3, (Indian) 146: to √3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20871 (Ptolemy) 45, 62-3, (Archimedes)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20872 ii. 51-2: to <G>p</G> 232-5, ii. 194, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20873 253: to surds (Heron) ii. 323-6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20874 cf. ii. 547-9, ii. 553-4: to cube
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20875 root (Heron) ii. 341-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20876 <p>Apuleius of Madaura 97, 99.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20877 <p>Archibald, R. C. 425 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20878 <p>Archimedes 3, 52, 54, 180, 199, 202,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20879 203 <I>n.,</I> 213, 217, 224-5, 229, 234,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20880 272, ii. 1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20881 <p>Traditions ii. 16-17, engines ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20882 17, mechanics ii. 18, general
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20883 estimate ii. 19-20.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20884 <p>Works: character of, ii. 20-2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20885 works extant ii. 22-3, lost ii. 23-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20886 5, 103; text ii. 25-7, MSS. ii. 26,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20887 editions ii. 27: <I>The Method</I> ii. 20,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20888 21, 22, 27-34, ii. 246, ii. 317-18:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20889 <I>On the Sphere and Cylinder</I> ii. 34-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20890 50: <I>Measurement of a circle</I> ii. 50-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20891 6, ii. 253: <I>On Conoids and Sphe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20892 roids</I> ii. 56-64: <I>On Spirals</I> 230-1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20893 ii. 64-75 (cf. ii. 377-9), ii. 556-61:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20894 <I>Sand-reckoner</I> ii. 81-5: <I>Quadra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20895 ture of Parabola</I> ii. 85-91: me-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20896 chanical works, titles ii. 23-4,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20897 <I>Plane equilibriums</I> ii. 75-81: <I>On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20898 Floating Bodies</I> ii. 91-7, problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20899 of crown ii. 92-4: <I>Liber assump-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20900 torum</I> ii. 101-3: Cattle-problem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20901 14, 15, ii. 23, ii. 97-8, ii. 447:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20902 <I>Catoptrica</I> 444, ii. 24.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20903 <p>Arithmetic: octads 40-1, frac-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20904 tions 42, value of <G>p</G> 232-3, 234,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20905 ii. 50-6: approximations to √3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20906 ii. 51-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20907 <p>Astronomy ii. 17-18, sphere-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20908 <pb n=572><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20909 making ii. 18, on Aristarchus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20910 hypothesis ii. 3-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20911 <p>Conics, propositions in, 438-9,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20912 ii. 122-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20913 <p>Cubic equation solved by conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20914 ii. 45-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20915 <p>On Democritus 180, 327,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20916 equality of angles of incidence
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20917 and reflection ii. 353-4, integral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20918 calculus anticipated ii. 41-2, 61,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20919 62-3, 74, 89-90: Lemma or Axiom
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20920 of A. 326-8, ii. 35: <G>neu/seis</G> in, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20921 65-8 (Pappus on, ii. 68): on semi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20922 regular solids ii. 98-101: triangle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20923 area in terms of sides ii. 103:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20924 trisection of any angle 240-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20925 <p>Archytas 2, 170, 212-16, ii. 1: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20926 <G>maqh/mata</G> 11, on <I>logistic</I> 14, on 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20927 as odd-even 71: on means 85, 86:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20928 no mean proportional between <I>n</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20929 and <I>n</I> + 1, 90, 215: on music 214:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20930 mechanics 213: solution of pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20931 blem of two mean proportionals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20932 214, 219, 245, 246-9, 334, ii. 261.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20933 <p>Argyrus, Isaac, 224 <I>n.,</I> ii. 555.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20934 <p>Aristaeus: comparison of five regu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20935 lar solids 420: <I>Solid Loci</I> (conics)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20936 438, ii. 116, 118-19.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20937 <p>Aristaeus of Croton 86.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20938 <p>Aristarchus of Samos 43, 139, ii. 1-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20939 15, ii. 251: date ii. 2: <G>ska/fh</G> of,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20940 ii. 1: anticipated Copernicus ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20941 2-3: other hypotheses ii. 3, 4:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20942 treatise <I>On sizes and distances of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20943 Sun and Moon</I> ii. 1, 3, 4-15, tri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20944 gonometrical purpose ii. 5: num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20945 bers in, 39: fractions in, 43.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20946 <p>Aristonophus, vase of, 162.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20947 <p>Aristophanes 48, 161, 220.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20948 <p>Aristotelian treatise on indivisible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20949 lines 157, 346-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20950 <p>Aristotherus 348.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20951 <p>Aristotle 5, 120, 121: on origin of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20952 science 8: on mathematical sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20953 jects 16-17: on first principles, de-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20954 finitions, postulates, axioms 336-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20955 <p>Arithmetic: reckoning by tens
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20956 26-7, why 1 is odd-even 71: 2
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20957 even and prime 73: on Pytha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20958 goreans and numbers 67-9: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20959 the gnomon 77-8, 83.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20960 <p>Astronomy: Pythagorean sys-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20961 tem 164-5, on hypothesis of con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20962 centric spheres 329, 335, ii. 244,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20963 on Plato's view about the earth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20964 314-15.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20965 <p>On the continuous and infinite
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20966 342-3: proof of incommensura-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20967 bility of diagonal 91: on principle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20968 of exhaustion 340: on Zeno's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20969 paradoxes 272, 275-7, 278-9, 282:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20970 on Hippocrates 22: encomium on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20971 Democritus 176.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20972 <p>Geometry: illustrations from,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20973 335, 336, 338-40, on parallels
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20974 339, proofs differing from Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20975 338-9, propositions not in Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20976 340, on quadratures 184-5, 221,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20977 223, 224 <I>n.,</I> 271, on quadrature
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20978 by lunes (Hippocrates) 184-5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20979 198-9: on Plato and regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20980 solids 159: curves and solids in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20981 A. 341.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20982 <p>Mechanics 344-6, 445-6: paral-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20983 lelogram of velocities 346: ‘Aris-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20984 totle's wheel’ ii. 347-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20985 <p>Aristoxenus 24 <I>n.,</I> 66.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20986 <p>Arithmetic (1) = theory of numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20987 (opp. to <G>logistikh/</G>) 13-16: early
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20988 ‘Elements of Arithmetic’ 90, 216:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20989 systematic treatises, Nicomachus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20990 <I>Introd. Ar.</I> 97-112, Theon of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20991 Smyrna 112-3, Iam blichus, Comm.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20992 on Nicomachus 113-15, Domninus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20993 ii. 538. (2) Practical arithmetic:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20994 originated with Phoenicians 120-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20995 1, in primary education 19-20.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20996 <p>Arithmetic mean, defined 85.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20997 <p><I>Arithmetica</I> of Diophantus 15-16,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20998 ii. 449-514.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	20999 <p>Arithmetical operations: <I>see</I> Addi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21000 tion, Subtraction, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21001 <p><I>Arrow</I> of Zeno 276, 280-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21002 <p>&Amacr;ryabha&tdot;&tdot;a 234.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21003 <p>Asclepius of Tralles 99.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21004 <p>Astronomy in elementary education
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21005 19: as secondary subject 20-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21006 <p>Athelhard of Bath, first translator
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21007 of Euclid 362-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21008 <p>Athenaeus 144, 145.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21009 <p>Athenaeus of Cyzicus 320-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21010 <p>‘Attic’ (or ‘Herodianic’) numerals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21011 30-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21012 <p>August, E. F. 299, 302, 361.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21013 <p>Autolycus of Pitane 348: works
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21014 <I>On the moving Sphere</I> 348-52, <I>On
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21015 Risings and Settings</I> 352-3: rela-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21016 tion to Euclid 351-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21017 <p>Auverus, C. ii. 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21018 <p>Axioms: Aristotle on, 336: =<I>Com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21019 mon Notions</I> in Euclid 376: Axiom
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21020 of Archimedes 326-8, ii. 35.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21021 <pb n=573><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21022 <p>Babylonians: civilization of, 8, 9:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21023 system of numerals 28-9: sexa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21024 gesimal fractions 29: ‘perfect
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21025 proportion’ 86.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21026 <p>Bachet, editor of Diophantus ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21027 454-5, ii. 480.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21028 <p>Bacon, Roger: on Euclid 367-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21029 <p>Baillet, J. ii. 543.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21030 <p>Baldi, B. ii. 308.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21031 <p>Barlaam ii. 324 <I>n.,</I> ii. 554-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21032 <p>Barocius ii. 545.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21033 <p>Barrow, I., edition of Euclid, 369-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21034 70: on Book V 384.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21035 <p>Bathycles 142.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21036 <p>B&amacr;udh&amacr;yana &Sacute;. S. 146.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21037 <p>Baynard, D. ii. 128.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21038 <p>Benecke, A. 298, 302-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21039 <p>Benedetti, G. B. 344, 446.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21040 <p>Bertrand, J. ii. 324 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21041 <p>Bessarion ii. 27.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21042 <p>Besthorn, R. O. 362, ii. 310.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21043 <p>Billingsley, Sir H. 369.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21044 <p>Björnbo, A. A. 197 <I>n.,</I> 363, ii. 262.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21045 <p>Blass, C. 298.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21046 <p>Blass, F. 182.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21047 <p>Boeckh, A. 50, 78, 315.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21048 <p>Boëtius 37, 47, 90: translation of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21049 Euclid 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21050 <p>Boissonade ii. 538.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21051 <p>Bombelli, Rafael, ii. 454.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21052 <p>Borchardt, L. 125, 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21053 <p>Borelli, G. A. ii. 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21054 <p>Bouillaud (Bullialdus) ii. 238, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21055 556.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21056 <p>Braunmühl, A. von, ii. 268-9 <I>n.,</I> ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21057 288, ii. 291.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21058 <p>Breton (de Champ), P. 436, ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21059 <p>Bretschneider, C. A. 149, 183, 324-5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21060 ii. 539.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21061 <p>Brochard, V. 276-7, 279 <I>n.,</I> 282.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21062 <p>Brougham, Lord, 436.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21063 <p>Brugsch, H. K. 124.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21064 <p>Bryson 219, 223-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21065 <p>Burnet, J. 203 <I>n.,</I> 285, 314-15.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21066 <p>Butcher, S. H. 299, 300.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21067 <p>Buzengeiger ii. 324 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21068 <p>Cajori, F. 283 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21069 <p>Calculation, practical: the abacus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21070 46-8, addition and subtraction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21071 52, multiplication (i) Egyptian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21072 52-3 (Russian ? 53 <I>n.</I>), (ii) Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21073 53-8, division 58-60, extraction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21074 of square root 60-3, of cube root
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21075 63-4, ii. 341-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21076 <p>Callimachus 141-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21077 <p>Callippus: Great Year 177: system of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21078 concentric spheres 329, 335, ii. 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21079 <p>Cambyses 5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21080 <p>Camerarius, Joachim, ii. 274.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21081 <p>Camerer, J. G. ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21082 <p>Campanus, translator of Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21083 363-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21084 <p><I>Canonic</I> = theory of musical inter-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21085 vals 17.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21086 <p>Cantor, G. 279.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21087 <p>Cantor, M. 37-8, 123, 127, 131, 135,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21088 182, ii. 203, ii. 207.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21089 <p>Carpus of Antioch 225, 232, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21090 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21091 <p><I>Case</I> (<G>ptw=sis</G>) 372, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21092 <p>Cassini ii. 206.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21093 <p>Casting out nines 115-17, ii. 549.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21094 <p><I>Catoptric,</I> theory of mirrors 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21095 <p><I>Catoptrica</I>: treatises by Euclid (?)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21096 442, by Theon (?) 444, by Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21097 medes 444, and Heron 444, ii. 294,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21098 ii. 310, ii. 352-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21099 <p>Cattle-problem of Archimedes 14,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21100 15, ii. 23, ii. 97-8, ii. 447.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21101 <p>Cavalieri, B. 180, ii. 20.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21102 <p>Censorinus 177.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21103 <p>Centre of gravity: definitions ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21104 302, ii. 350-1, ii. 430.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21105 <p><I>Ceria Aristotelica</I> ii. 531.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21106 <p>Chalcidius ii. 242, 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21107 <p>Chaldaeans: measurement of angles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21108 by <I>ells</I> ii. 215-16: order of planets
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21109 ii. 242.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21110 <p>Charmandrus ii. 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21111 <p>Chasles, M. ii. 19, 20: on Porisms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21112 435-7, ii. 419.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21113 <p>Chords, Tables of, 45, ii. 257, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21114 259-60.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21115 <p>Chrysippus 179: definition of unit 69.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21116 <p>Cicero 144, 359, ii. 17, 19.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21117 <p>Circle: division into degrees ii. 214-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21118 15: squaring of, 173, 220-35,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21119 Antiphon 221-2, Bryson 223-4,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21120 by Archimedes's spiral 225, 230-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21121 1, Nicomedes, Dinostratus, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21122 quadratrix 225-9, Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21123 225, Carpus 225; approximations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21124 to <G>p</G> 124, 232-5, ii. 194, ii. 253,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21125 ii. 545.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21126 <p>Cissoid of Diocles 264-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21127 <p>Clausen, Th. 200.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21128 <p>Cleanthes ii. 2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21129 <p>Cleomedes: ‘paradoxical’ eclipse 6:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21130 <I>De motu circulari</I> ii. 235-8, 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21131 <p>Cleonides 444.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21132 <p><I>Cochlias</I> 232, ii. 193.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21133 <pb n=574><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21134 <p><I>Cochloids</I> 238-40: ‘sister of coch-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21135 loid’ 225, 231-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21136 <p>Coins and weights, notation for, 31.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21137 <p>Columella ii. 303.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21138 <p>Commandinus, F., translator of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21139 Euclid, 365, 425, Apollonius ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21140 127, <I>Analemma</I> of Ptolemy ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21141 287, <I>Planispherium</I> ii. 292, Heron's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21142 <I>Pneumatica</I> ii. 308, Pappus ii. 360,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21143 Serenus ii. 519.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21144 <p>Conchoid of Nicomedes 238-40.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21145 <p><I>Conclusion</I> 370, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21146 <p>Cone: Democritus on, 179-80, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21147 110: volume of, 176, 180, 217,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21148 327, 413, ii. 21, ii. 332: volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21149 of frustum ii. 334: division of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21150 frustum in given ratio ii. 340-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21151 <p>Conic sections: discovered by Me-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21152 naechmus 252-3, ii. 110-16: Eu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21153 clid's <I>Conics</I> and Aristaeus's <I>Solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21154 Loci</I> 438, ii. 116-19: propositions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21155 included in Euclid's <I>Conics</I> ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21156 121-2 (focus-directrix property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21157 243-4, ii. 119-21), conics in Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21158 medes ii. 122-6: names due to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21159 Apollonius 150, ii. 138: Apollo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21160 nius's <I>Conics</I> ii. 126-75: conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21161 in <I>Fragmentum Bobiense</I> ii. 200-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21162 203: in Anthemius ii. 541-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21163 <p>Conon of Samos ii. 16, ii. 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21164 <p><I>Construction</I> 370, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21165 <p><I>Conversion</I> of ratio (<I>convertendo</I>) 386.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21166 <p>Cook-Wilson, J. 300 <I>n.,</I> ii. 370.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21167 <p>Counter-earth 164.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21168 <p>Croesus 4, 129.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21169 <p>Ctesibius 213: relation to Philon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21170 and Heron ii. 298-302.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21171 <p>Cube: called ‘geometrical har-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21172 mony’ (Philolaus) 85-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21173 <p>Cube, duplication of: history of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21174 problem 244-6: reduction by Hip-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21175 pocrates to problem of two mean
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21176 proportionals 2, 183, 200, 245:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21177 solutions, by Archytas 246-9, Eu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21178 doxus 249-51, Menaechmus 251-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21179 5, ‘Plato’ 255-8, Eratosthenes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21180 258-60, Nicomedes 260-2, Apol-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21181 lonius, Philon, Heron 262-4, Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21182 cles 264-6, Sporus and Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21183 266-8: approximation by plane
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21184 method 268-70.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21185 <p>Cube root, extraction of, 63-4:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21186 Heron's case ii. 341-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21187 <p>Cubic equations, solved by conics,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21188 237-8, ii. 45-6, ii. 46; particular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21189 case in Diophantus ii. 465, ii. 512.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21190 <p>Curtze, M. 75 <I>n.,</I> ii. 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21191 <p>Cyrus 129.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21192 <p><I>Dactylus,</I> 1/24th of ell, ii. 216.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21193 <p>Damastes of Sigeum 177.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21194 <p>Damianus ii. 294.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21195 <p>Darius-vase 48-9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21196 <p>D'Armagnac, G. ii. 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21197 <p>Dasypodius ii. 554 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21198 <p>De la Hire ii. 550.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21199 <p><I>De levi et ponderoso</I> 445-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21200 <p>Decagon inscribed in circle, side of,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21201 416: area of, ii. 328.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21202 <p>Dee, John, 369, 425.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21203 <p>Definitions: Pythagorean 166: in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21204 Plato 289, 292-4: Aristotle on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21205 337: in Euclid 373: <I>Definitions</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21206 of Heron, ii. 314-16.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21207 <p>Demetrius of Alexandria ii. 260, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21208 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21209 <p>Democritus of Abdera 12, 119, 121,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21210 182: date 176, travels 177: Aris-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21211 totle's encomium 176: list of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21212 works (1) astronomical 177, (2)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21213 mathematical 178: on irrational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21214 lines and solids 156-7, 181: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21215 angle of contact 178-9: on cir-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21216 cular sections of cone 179-80, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21217 110: first discovered volume of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21218 cone and pyramid 176, 180, 217,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21219 ii. 21: atoms mathematically di-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21220 visible <I>ad inf.</I> 181: <G>*)ekpeta/smata</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21221 178, 181: on perspective 174: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21222 Great Year 177.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21223 <p>Dercyllides ii. 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21224 <p>Descartes 75 <I>n.,</I> 279.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21225 <p>Dicaearchus ii. 242.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21226 <p><I>Dichotomy</I> of Zeno 275, 278 80.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21227 <p>Diels, H., 142 <I>n.,</I> 176, 178, 184, 188.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21228 <p>Digamma: from Phoenician Vau
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21229 32: signs for, <I>ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21230 <p><I>Digit</I> 27.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21231 <p>Dinostratus 225, 229, 320-1, ii. 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21232 <p>Diocles: inventor of cissoid 264-6:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21233 solution of Archimedes <I>On Sph.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21234 and Cyl.</I> II. 4, ii. 47-8: on burn-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21235 ing-mirrors ii. 200-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21236 <p>Diodorus (math.): on parallel-pos-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21237 tulate 358: <I>Analemma</I> of, ii. 287,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21238 ii. 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21239 <p>Diodorus Siculus 121, 141, 142, 176.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21240 <p>Diogenes Laertius 144, 145, 177, 291.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21241 <p>Dionysius, Plato's master, 22.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21242 <p>Dionysius, a friend of Heron, ii. 306.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21243 <p>Dionysodorus ii. 46, ii. 218-19, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21244 334-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21245 <p>Diophantus of Alexandria: date ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21246 <pb n=575><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21247 448: works and editions ii. 448-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21248 55: <I>Arithmetica</I> 15-16: fractions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21249 in, 42-4: notation and definitions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21250 ii. 455-61: signs for unknown (<I>x</I>)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21251 and powers ii. 456-9, for <I>minus</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21252 ii. 459: methods ii. 462-79: de-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21253 terminate equations ii. 462-5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21254 484-90: indeterminate analysis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21255 ii. 466-76, 491-514: ‘Porisms’ ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21256 449, 450, 451, ii. 479-80: propo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21257 sitions in theory of numbers ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21258 481-4: conspectus of <I>Arithmetica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21259 ii. 484-514: <I>On Polygonal Num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21260 bers</I> 16, 84, ii. 514-17: ‘<I>Moriastica</I>’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21261 ii. 449.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21262 <p>Dioptra 18, ii. 256: Heron's <I>Dioptra</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21263 ii. 345-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21264 <p>Division: Egyptian method 53,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21265 Greek 58-60: example with sexa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21266 gesimal fractions (Theon of Alex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21267 andria) 59-60.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21268 <p><I>Divisions (of Figures), On,</I> by Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21269 425-30: similar problems in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21270 Heron ii. 336-40.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21271 <p>Dodecagon, area of, ii. 328.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21272 <p>Dodecahedron: discovery attributed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21273 to Pythagoras or Pythagoreans
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21274 65, 141, 158-60, 162: early occur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21275 rence 160: inscribed in sphere
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21276 (Euclid) 418-19, (Pappus) ii. 369:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21277 Apollonius on, 419-20: volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21278 of, ii. 335.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21279 <p>Domninus ii. 538.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21280 <p>Dositheus ii. 34.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21281 <p>Duhem, P. 446.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21282 <p>Dupuis, J. ii. 239.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21283 <p>Earth: measurements of, ii. 82,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21284 (Eratosthenes) ii. 106-7, (Posido-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21285 nius) ii. 220.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21286 <p>Ecliptic: obliquity discovered by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21287 Oenopides 174, ii. 244: estimate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21288 of inclination (Eratosthenes, Pto-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21289 lemy) ii. 107-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21290 <p>Ecphantus 317, ii. 2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21291 <p>Edfu, Temple of Horus 124.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21292 <p>Egypt: priests 4-5, 8-9: relations
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21293 with Greece 8; origin of geometry
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21294 in, 120-2: orientation of temples
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21295 122.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21296 <p>Egyptian mathematics: numeral
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21297 system 27-8, fractions 28, multi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21298 plication, &c. 14-15, 52-3: geo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21299 metry (mensuration) 122-8: tri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21300 angle (3, 4, 5) right-angled 122,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21301 147: value of <G>p</G> 124, 125: measure-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21302 ment of pyramids 126-8: maps
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21303 (regional) 139: algebra in Papyrus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21304 Rhind, &c. ii. 440-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21305 <p>Eisenlohr, A. 123, 126, 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21306 <p>Eisenmann, H. J. ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21307 <p>Elements: as known to Pytha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21308 goreans 166-8: progress in, down
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21309 to Plato 170-1, 175-6, 201-2, 209-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21310 13, 216-17: writers of Elements,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21311 Hippocrates of Chios 170-1, 201-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21312 2, Leon, Theudius 320-1: other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21313 contributors to, Leodamas, Ar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21314 chytas 170, 212-13, Theaetetus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21315 209-12, 354, Hermotimus of Colo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21316 phon 320, Eudoxus 320, 323-9,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21317 354: <I>Elements</I> of Euclid 357-419:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21318 the so-called ‘Books XIV, XV’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21319 419-21.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21320 <p><I>Ell,</I> as measure of angles ii. 215-16.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21321 <p>Empedocles: on Pythagoras 65.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21322 <p>Eneström, G. ii. 341-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21323 <p>Enneagon: Heron's measurement
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21324 of side ii. 259, of area ii. 328-9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21325 <p><I>Epanthema</I> of Thymaridas (system
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21326 of simple equations) 94: other
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21327 types reduced to, 94-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21328 <p>Equations: simple, in Papyrus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21329 Rhind, &c. ii. 441: in <I>epanthema</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21330 of Thymaridas and in lamblichus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21331 94-6: in Greek anthology ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21332 441-3: indeterminate, <I>see</I> Inde-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21333 terminate Analysis: <I>see also</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21334 Quadratic, Cubic.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21335 <p>Eratosthenes ii. 1, 16: date, &c.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21336 ii. 104: <I>sieve</I> (<G>ko/skinon</G>) for finding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21337 primes 16, 100, ii. 105: on dupli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21338 cation of cube 244-6, 251, 258-60:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21339 the <I>Platonicus</I> ii. 104-5: <I>On Means</I>
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21341 earth</I> ii. 106-7, ii. 242, ii. 346:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21342 astronomy ii. 107-9: chronology
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21343 and <I>Geographica</I> ii. 109: on <I>Octaë-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21344 teris ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21345 <p>Erycinus ii. 359, 365-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21346 <p>Euclid 2-3, 93, 131: date, &c. 354-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21347 6: stories of, 25, 354, 357: rela-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21348 tion to predecessors 354, 357:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21349 Pappus on, 356-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21350 <p>Arithmetic: classification and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21351 definitions of numbers 72-3, 397,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21352 ‘perfect’ numbers 74, 402: for-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21353 mula for right-angled triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21354 in rational numbers 81-2, 405.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21355 <p><I>Conics</I> 438-9, ii. 121-2, focus-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21356 directrix property ii. 119-21: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21357 ellipse 439, ii. 111, ii. 125.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21358 <pb n=576><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21359 <p><I>Data</I> 421-5, <I>Divisions (of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21360 figures)</I> 425-30, ii. 336, 339.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21361 <p><I>Elements</I>: text 360-1, Theon's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21362 edition 358, 360, ii. 527-8, trans-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21363 lation by Boëtius 359, Arabic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21364 translations 362, ancient com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21365 mentaries 358-9, <I>editio princeps</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21366 of Greek text 360, Greek texts of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21367 Gregory, Peyrard, August, Hei-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21368 berg 360-1: Latin translations,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21369 Athelhard 362-3, Gherard 363,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21370 Campanus 363-4, Commandinus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21371 365: first printed editions, Rat-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21372 dolt 364-5, Zamberti 365: first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21373 introduction into England 363:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21374 first English editions, Billingsley,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21375 &c. 369-70: Euclid in Middle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21376 Ages 365-9, at Universities 368-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21377 9: analysis of, 373-419: arrange-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21378 ment of postulates and axioms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21379 361: I. 47, how originally proved
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21381 375, ii. 227-30, ii. 295-7, ii. 534:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21382 so-called ‘Books XIV, XV’ 419-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21383 21.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21384 <p>Mechanics 445-6: Music 444-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21385 5, <I>Sectio canonis</I> 17, 90, 215,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21386 444-5: <I>Optics</I> 17-18, 441-4: <I>Phae-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21387 nomena</I> 349, 351-2, 440-1, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21388 249: <I>Porisms</I> 431-8, lemmas to,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21389 ii. 419-24: <I>Pseudaria</I> 430-1: <I>Sur-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21390 face-Loci</I> 243-4, 439-40, lemmas
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21392 <p>Eudemus 201, 209, 222: <I>History of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21393 Geometry</I> 118, 119, 120, 130, 131,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21394 135, 150, 171: on Hippocrates's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21395 lunes 173, 182, 183-98: <I>History
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21396 of Astronomy</I> 174, 329, ii. 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21397 <p>Eudoxus 24, 118, 119, 121, 320,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21398 322-4: new theory of proportion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21399 (that of Eucl. V. ii) 2, 153, 216,
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21401 haustion 2, 176, 202, 206, 217,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21402 222, 326, 327-9: problem of two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21403 mean proportionals 245, 246, 249-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21404 51: discovered three new means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21405 86: ‘general theorems’ 323-4:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21406 <I>On speeds,</I> theory of concentric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21407 spheres 329-34, ii. 244: <I>Phaeno-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21408 mena</I> and <I>Mirror</I> 322.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21409 <p>Eugenius Siculus, Admiral, ii. 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21410 <p>Euler, L. 75 <I>n.,</I> ii. 482, ii. 483.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21411 <p>Euphorbus (= Pythagoras) 142.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21412 <p>Eurytus 69.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21413 <p>Eutocius 52, 57-8, ii. 25, ii. 45, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21414 126, ii. 518, ii. 540-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21415 <p>Exhaustion, method of, 2, 176, 202,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21416 217, 222, 326, 327-9: develop-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21417 ment of, by Archimedes 224, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21418 35-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21419 <p>False hypothesis: Egyptian use ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21420 441: in Diophantus ii. 488, 489.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21421 <p>Fermat, P. 75 <I>n.,</I> ii. 20, ii. 185, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21422 454, ii. 480, ii. 481-4: on Porisms
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21423 435.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21424 <p>Fontenelle ii. 556.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21425 <p>Fractions: Egyptian (submultiples
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21426 except 2/3) 27-8, 41: Greek sys-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21427 tems 42-4: Greek notation <I>ib.</I>:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21428 sexagesimal fractions, Babylo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21429 nian 29, in Greek 44-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21430 <p>‘Friendly’ numbers 75.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21431 <p>Galilei 344, 446.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21432 <p><I>Geëponicus, Liber,</I> 124, ii. 309, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21433 318, ii. 344.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21434 <p>Geminus 119, ii. 222-34: on arith-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21435 metic and logistic 14: on divi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21436 sions of optics, &c. 17-18: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21437 original steps in proof of Eucl. I
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21438 32, 135-6: on parallels 358:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21439 attempt to prove parallel-postu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21440 late ii. 227-30: on original way
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21441 of producing the three conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21442 ii. 111: encyclopaedic work on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21443 mathematics ii. 223-31: on Posi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21444 donius's <I>Meteorologica</I> ii. 231-2:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21445 <I>Introduction to Phaenomena</I> ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21446 232-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21447 <p>Geodesy (<G>gewdaisi/a</G>) = mensuration
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21448 (as distinct from geometry) 16-17.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21449 <p>Geometric mean, defined (Archytas)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21450 85: one mean between two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21451 squares (or similar numbers), two
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21452 between cubes (or similar solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21453 numbers) 89-90, 112, 201, 297,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21454 400: no rational mean between
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21455 consecutive numbers 90, 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21456 <p>‘Geometrical harmony’ (Philolaus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21457 name for cube) 85-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21458 <p>Geometry: origin in Egypt 120-2:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21459 geometry in secondary education
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21460 20-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21461 <p>Georgius Pachymeres ii. 453, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21462 546.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21463 <p>Gerbert (Pope Sylvester II) 365-7:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21464 geometry of, 366: ii. 547.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21465 <p>Gerhardt, C. J. ii. 360, ii. 547.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21466 <p>Gherard of Cremona, translator of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21467 Euclid and an-Nair&imacr;z&imacr; 363, 367,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21468 ii. 309: of Menelaus ii. 252, ii. 262.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21469 <pb n=577><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21470 <p>Ghetaldi, Marino, ii. 190.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21471 <p>Girard, Albert, 435, ii. 455.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21472 <p><I>Gnomon</I>: history of term 78-9:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21473 gnomons of square numbers 77-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21474 8, of oblong numbers 82-3, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21475 polygonal numbers 79: in appli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21476 cation of areas 151-2: use by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21477 al-Karkh&imacr; 109-10: in Euclid 379:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21478 sun-dial with vertical needle 139.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21479 <p>Gomperz, Th. 176.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21480 <p>Govi, G. ii. 293 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21481 <p>Gow, J. 38.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21482 <p>Great Year, of Oenopides 174-5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21483 of Callippus and Democritus 177.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21484 <p>Gregory, D. 360-1, 440, 441, ii. 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21485 <p>Griffith, F. Ll. 125.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21486 <p>Günther, S. ii. 325 <I>n.,</I> ii. 550.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21487 <p>Guldin's theorem, anticipated by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21488 Pappus ii. 403.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21489 <p>Halicarnassus inscriptions 32 - 3,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21490 34.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21491 <p>Halley, E., editions of Apollonius's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21492 <I>Conics</I> ii. 127-8, and <I>Sectio ratio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21493 nis</I> ii. 175, 179, of Menelaus ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21494 252, ii. 262, of extracts from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21495 Pappus ii. 360, of Serenus ii. 519.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21496 <p>Halma, editor of Ptolemy ii. 274,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21497 275.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21498 <p>Hammer-Jensen, I. ii. 300 <I>n.,</I> ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21499 304 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21500 <p>Hankel, H. 145, 149, 288, 369, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21501 483.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21502 <p>Hardy, G. H. 280.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21503 <p>Harmonic mean (originally ‘sub-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21504 contrary’) 85.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21505 <p><I>Harpedonaptae,</I> ‘rope-stretchers’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21506 121-2, 178.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21507 <p>H&amacr;r&umacr;n ar-Rash&imacr;d 362.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21508 <p><I>Hau</I> - calculations (Egyptian) ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21509 440-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21510 <p>Hecataeus of Miletus 65, 177.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21511 <p>Heiben, J. L. 233 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21512 <p>Heiberg, J. L. 184, 187 <I>n.,</I> 188,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21513 192 <I>n.,</I> 196-7 <I>n.,</I> 315, 361, ii. 203,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21514 ii. 309, 310, 316, 318, 319, ii. 519,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21515 ii. 535, 543, 553, 555 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21516 <p><I>Helceph</I> 111.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21517 <p>Hendecagon in a circle (Heron) ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21518 259, ii. 329.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21519 <p>Henry, C. ii. 453.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21520 <p>Heptagon in a circle, ii. 103:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21521 Heron's measurement of, ii. 328.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21522 <p>Heraclides of Pontus 24, ii. 231-2:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21523 discovered rotation of earth about
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21524 axis 316-17, ii. 2-3, and that Venus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21525 and Mercury revolve about sun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21526 312, 317, ii. 2, ii. 244.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21527 <p>Heraclitus of Ephesus 65.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21528 <p>Heraclitus, mathematician ii. 192,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21529 ii. 359, ii. 412.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21530 <p>Hermannus Secundus ii. 292.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21531 <p>Hermesianax 142 <I>n.,</I> 163.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21532 <p>Hermodorus ii. 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21533 <p>Hermotimus of Colophon 320-1:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21534 Elements and Loci <I>ib.,</I> 354.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21535 <p>‘Herodianic’ (or ‘Attic’) numerals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21536 30-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21537 <p>Herodotus 4, 5, 48, 65, 121, 139.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21538 <p>Heron of Alexandria 121, ii. 198,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21539 ii. 259: controversies on date ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21540 298-307: relation to Ctesibius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21541 and Philon ii. 298-302, to Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21542 ii. 299-300, to Posidonius and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21543 Vitruvius ii. 302-3, to <I>agrimen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21544 sores</I> ii. 303, to Ptolemy ii. 303-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21545 <p>Arithmetic: fractions 42-4, mul-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21546 tiplications 58, approximation to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21547 surds ii. 51, ii. 323-6, approxima-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21548 tion to cube root 64, ii. 341-2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21549 quadratic equations ii. 344, in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21550 determinate problems ii. 344,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21551 444-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21552 <p>Character of works ii. 307-8:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21553 list of treatises ii. 308-10.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21554 <p>Geometry ii. 310-14, <I>Definitions</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21555 ii. 314-16: comm. on Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21556 <I>Elements</I> 358, ii. 310-14: proof of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21557 formula for area of triangle in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21558 terms of sides ii. 321-3: duplica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21559 tion of cube 262-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21560 <p><I>Metrica</I> ii. 320-43: (1) mensu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21561 ration ii. 316-35: triangles ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21562 320-3, quadrilaterals ii. 326,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21563 regular polygons ii. 326-9, circle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21564 and segments ii. 329-31: volumes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21565 ii. 331-5, <G>bwmi/skos</G> ii. 332-3, frus-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21566 tum of cone, sphere and segment
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21567 ii. 334, <I>tore</I> ii. 334-5, five regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21568 solids ii. 335. (2) divisions of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21569 figures ii. 336-43, of frustum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21570 cone ii. 342-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21571 <p><I>Mechanics</I> ii. 346-52: on Ar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21572 chimedes's mechanical works ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21573 23-4, on centre of gravity ii. 350-1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21574 352.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21575 <p><I>Belopoeïca</I> 18, ii. 308-9, <I>Catop-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21576 trica</I> 18, ii. 294, ii. 310, ii. 352-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21577 <p><I>Dioptra</I> ii. 345-6, <I>Pneumatica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21578 and <I>Automata</I> 18, ii. 308, 310.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21579 <p><I>On Water-clocks</I> ii. 429, ii. 536.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21580 <p>Heron, teacher of Proclus ii. 529.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21581 <pb n=578><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21582 <p>‘Heron the Younger’ ii. 545.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21583 <p>Heronas 99.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21584 <p>Hicetas 317.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21585 <p>Hierius 268, ii. 359.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21586 <p>Hieronymus 129.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21587 <p>Hil&amacr;l b. Ab&imacr; Hil&amacr;l al-&Hdot;im⋅&imacr; ii. 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21588 <p>Hiller, E. ii. 239.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21589 <p>Hilprecht, H. V. 29.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21590 <p>Hipparchus ii. 3, 18, 198, 216, 218:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21591 date, &c. ii. 253: work ii. 254-6:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21592 on epicycles and eccentrics ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21593 243, ii. 255: discovery of preces-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21594 sion ii.254: on mean lunar month
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21595 ii. 254-5: catalogue of stars ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21596 255: geography ii. 256: trigono-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21597 metry ii. 257-60, ii. 270.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21598 <p>Hippasus 65, 85, 86, 214: construc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21599 tion of ‘twelve pentagons in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21600 sphere’ 160.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21601 <p>Hippias of Elis: taught mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21602 matics 23: varied accomplish-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21603 ments <I>ib.</I>, lectures in Sparta 24:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21604 inventor of <I>quadratrix</I> 2, 171, 182,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21605 219, 225-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21606 <p>Hippocrates of Chios 2, 182, 211:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21607 taught for money 22: first writer
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21608 of Elements 119, 170, 171: ele-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21609 ments as known to, 201-2:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21610 assumes <G>neu=sis</G> equivalent to solu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21611 tion of quadratic equation 88,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21612 195-6: on quadratures of lunes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21613 170, 171, 173, 182, 183-99, 220,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21614 221: proved theorem of Eucl.XII
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22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21616 of cube to problem of finding
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21617 two mean proportionals 2, 183,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21618 200, 245.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21619 <p>Hippolytus: on <G>puqme/nes</G> (bases) and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21620 ‘rule of nine’ and ‘seven’ 115-16.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21621 <p><I>Hippopede</I> of Eudoxus 333-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21622 <p>Homer 5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21623 <p>‘Horizon’: use in technical sense by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21624 Euclid 352.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21625 <p>Horsley, Samuel, ii. 190, ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21626 <p>Hultsch, F. 204, 230, 349, 350, ii. 51,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21627 ii. 308, ii. 318, 319, ii. 361.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21628 <p>Hunrath, K. ii. 51.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21629 <p>Hunt, A. S. 142.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21630 <p>Hypatia ii. 449, ii. 519, ii. 528-9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21631 <p>Hypotenuse, theorem of square on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21632 142, 144-9: Proclus on discovery
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21633 of, 145: supposed Indian origin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21634 145-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21635 <p>Hypsicles: author of so-called Book
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21636 XIV of Eucl. 419-20, ii. 192: de-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21637 finition of ‘polygonal number’ 84,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21638 ii. 213, ii. 515: <G>*)anaforiko/s</G> ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21639 213-18, first Greek division of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21640 zodiac circle into 360 parts ii. 214.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21641 <p>Iamblichus 4, 69, 72, 73, 74, 75, 86,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21642 107, ii. 515, 529: on <G>e)pa/nqhmt</G> of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21643 Thymaridas, &c. 94-6: works
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21644 113-14: comm. on Nicomachus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21645 113-15: squares and oblong num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21646 bers as ‘race-courses’ 114: pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21647 perty of sum of numbers 3<I>n</I>-2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21648 3<I>n</I>-1, 3<I>n</I> 114-15.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21649 <p>Ibn al-Haitham, on burning-mirrors
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21650 ii. 201: ii. 453.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21651 <p>Icosahedron 159: discovery attri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21652 buted to Theaetetus 162: volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21653 of, ii. 335.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21654 <p>Incommensurable, discovery of, 65,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21655 90-1, 154: proof of incommensu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21656 rability of diagonal of square 91.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21657 <p>Indeterminate analysis: first cases,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21658 right-angled triangles in rational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21659 numbers 80, 81, ‘side-’ and ‘dia-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21660 meter-’ numbers 91-3, ii. 536:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21661 rectangles with area and peri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21662 meter numerically equal 96-7:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21663 indeterminate equations, first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21664 degree ii. 443, second degree ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21665 443-4 (<I>see also</I> Diophantus), in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21666 Heronian collections ii. 344, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21667 444-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21668 <p>India: rational right-angled tri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21669 angles in, 145-6: approximation
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21670 to √2, 146.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21671 <p>Indian Table of Sines ii. 253.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21672 <p>Irrational: discovered by Pythago-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21673 reans 65, 90-1, 154, and with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21674 reference to √2, 155, 168: Demo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21675 crituson, 156-7,181: Theodorus on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21676 203-9: extensions by Theaetetus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21677 209-12, Euclid 402-11, Apollonius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21678 ii. 193.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21679 <p>Isaac Argyrus 224 <I>n.</I>, ii. 555.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21680 <p>Is&hdot;&amacr;q b. Hunain, translator of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21681 Euclid 362, of Menelaus ii. 261,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21682 and Ptolemy ii. 274.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21683 <p>Isidorus Hispalensis 365.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21684 <p>Isidorus of Miletus 421, ii. 25, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21685 518, ii. 540.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21686 <p>Isocrates: on mathematics in edu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21687 cation 21.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21688 <p>Isoperimetric figures ii. 206-13, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21689 390-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21690 <p>Jacob b. Machir ii. 252, ii. 262.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21691 <p>Jacobus Cremonensis ii. 26-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21692 <pb n=579><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21693 <p>Jan, C. 444.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21694 <p>Joachim Camerarius ii. 274.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21695 <p>Joachim, H. H. 348 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21696 <p>Johannes de Sacrobosco 368.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21697 <p>Jordanus Nemorarius ii. 328.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21698 <p>Jourdain, P. E. B. 283 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21699 <p>Kahu&ndot; Papyri 125, 126.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21700 <p>Kant 173.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21701 <p>Keil, B. 34-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21702 <p>Kepler ii. 20, ii. 99.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21703 <p>Köchly, H. A. T. ii. 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21704 <p>Koppa (<G>*\|o</G>for90)=Phoenician Qoph
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21705 32.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21706 <p>Kubitschek, W. 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21707 <p>Lagrange ii. 483.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21708 <p>Laird, A. G. 306 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21709 <p>Laplace 173.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21710 <p>Larfeld, W. 31 <I>n.</I>, 33-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21711 <p>Lawson 436.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21712 <p>Leibniz 279, ii. 20.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21713 <p><I>Lemma</I> 373, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21714 <p>Leodamas of Thasos 120, 170, 212,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21715 291, 319.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21716 <p>Leon 319.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21717 <p>Leon (of Constantinople) ii. 25.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21718 <p>Leonardo of Pisa 367, 426, ii. 547.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21719 <p>Lepsius, C. R. 124.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21720 <p>Leucippus 181.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21721 <p>Libri, G. ii. 556.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21722 <p>‘Linear’ (of numbers) 73.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21723 <p>‘Linear’ loci and problems 218-19.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21724 <p>Lines, classification of, ii. 226.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21725 <p>Livy ii. 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21726 <p>Loci: classification of,218-19, plane,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21727 solid, linear 218: loci on surfaces
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21728 219: ‘solid loci’ ii. 116-19.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21729 <p>Loftus, W. K. 28.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21730 <p>Logistic (opp. to ‘arithmetic’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21731 science of calculation 13-16, 23,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21732 53.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21733 <p><I>Logistica speciosa</I> and <I>numerosa</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21734 (Vieta) ii. 456.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21735 <p>Loria, G. iv-v, 350 <I>n.</I>, ii. 293 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21736 <p>Luca Paciuolo 367, ii. 324 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21737 <p>Lucas, E. 75 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21738 <p>Lucian 75 <I>n.</I>, 77, 99, 161, ii. 18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21739 <p>Lucretius 177.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21740 <p>Magic squares ii. 550.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21741 <p>Magnus, <I>Logistica</I> 234-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21742 <p>Mamercus or Mamertius 140, 141,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21743 171.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21744 <p>al-Ma'm&umacr;n, Caliph 362.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21745 <p>al-Man⋅&umacr;r, Caliph 362.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21746 <p><I>Manus</I>, for number 27.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21747 <p>Marinus 444, ii. 192, ii. 537-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21748 <p>Martianus Capella 359, 365.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21749 <p>Martin, T. H. ii. 238, ii. 546.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21750 <p>Maslama b. A&hdot;mad al-Majr&imacr;&tdot;&imacr; ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21751 292.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21752 <p>Massalia 8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21753 <p>Mastaba tombs 128.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21754 <p>Mathematics: meaning 10-11, clas-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21755 sification of subjects 11-18:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21756 branches of applied mathematics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21757 17-18: mathematics in Greek
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21758 education 18-25.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21759 <p>Maurolycus ii. 262.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21760 <p>Means: arithmetic, geometric, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21761 subcontrary (harmonic) known
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21762 in Pythagoras's time 85: defined
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21763 by Archytas <I>ib.</I>: fourth, fifth, and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21764 sixth discovered, perhaps by Eu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21765 doxus 86, four more by Myonides
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21766 and Euphranor 86: ten means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21767 in Nicomachus and Pappus 87-9,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21768 Pappus's propositions 88-9: no
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21769 rational geom. mean between suc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21770 cessive numbers (Archytas) 90,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21771 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21772 <p>Mechanics, divisions of, 18: writers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21773 on, Archytas 213, Aristotle 344-6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21774 445-6, Archimedes ii. 18, ii. 23-4,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21775 ii. 75-81, Ptolemy ii. 295, Heron ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21776 346-52, Pappus ii. 427-34.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21777 <p>Megethion ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21778 <p>Memus, Johannes Baptista, ii. 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21779 <p>Menaechmus 2, 25, 251-2, 320-1:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21780 discoverer of conic sections 251-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21781 3, ii. 110-16: solved problem of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21782 two mean proportionals 245, 246,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21783 251-5: on ‘problems’ 318.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21784 <p>Menelaus of Alexandria ii. 198, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21785 252-3: date, &c. ii. 260-1: Table of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21786 Chords ii. 257: <I>Sphaerica</I> ii. 261-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21787 73: Menelaus's theorem ii. 266-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21788 8, 270: anharmonic property ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21789 269: <G>para/doxos</G> curve ii. 260-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21790 <p><I>Mensa Pythagorea</I> 47.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21791 <p>Mensuration: in primary education
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21792 19: in Egypt 122-8: in Heron ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21793 316-35.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21794 <p>Meton 220.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21795 <p>Metrodorus ii. 442.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21796 <p><I>Minus</I>, sign for, in Diophantus ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21797 459-60.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21798 <p>Mochus 4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21799 <p>Moschopoulos, Manuel, ii. 549-50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21800 <p>Mu&hdot;ammad Bagdadinus 425.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21801 <p>Multiplication: Egyptian method
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21802 <pb n=580><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21803 52-3, Greek 53-4, ‘Russian’ 53 <I>n.</I>:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21804 examples from Eutocius, Heron,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21805 Theon 57-8: Apollonius's con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21806 tinued multiplications 54-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21807 <p>Multiplication Table 53.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21808 <p><I>Murran</I>, an angular measure ii. 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21809 <p>Musical intervals and numerical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21810 ratios 69, 75-6, 85, 165.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21811 <p>Myriads, ‘first’, ‘second’, &c., nota-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21812 tion for, 39-40.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21813 <p>Nagl, A. 50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21814 <p>an-Nair&imacr;z&imacr;: comm. on Euclid 363,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21815 ii. 224, ii. 228-30, ii. 309-10.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21816 <p>Na⋅&imacr;radd&imacr;n a&tdot;-&Tdot;&umacr;s&imacr;: version of Eu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21817 clid 362, of Apollonius's <I>Conics</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21818 ii. 127: of Ptolemy ii. 275.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21819 <p>Naucratis inscriptions 33.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21820 <p>Nemesius 441.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21821 <p>Neoclides 319.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21822 <p><I>Ner</I> (Babylonian) (=600) 28, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21823 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21824 <p>Nesselmann, G. H. F. ii. 450-1, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21825 455-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21826 <p>Newton 370, ii. 20, ii. 182.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21827 <p>Nicolas Rhabdas 40, ii. 324 <I>n.</I>, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21828 550-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21829 <p>Nicomachus of Gerasa 12, 69, 70,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21830 72, 73, 74, 76, 83, 85, 86, ii. 238,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21831 ii. 515: works of, 97: <I>Introductio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21832 arithmetica</I>: character of treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21833 98-9, contents 99-112, classifica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21834 tion of numbers 99-100: on ‘per-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21835 fect’ numbers 74, 100-1: on ten
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21836 means 87: on a ‘Platonic’ theo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21837 rem 297: sum of series of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21838 natural cubes 109-10.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21839 <p>Nicomedes 225-6, ii. 199: cochloids
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21840 or conchoids 238-40: duplica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21841 tion of cube 260-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21842 <p>Niloxenus 129.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21843 <p>Nine, rule of, 115-16: casting out
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21844 nines ii. 549.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21845 <p>Nipsus, M. Junius, 132.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21846 <p>Nix, L. ii. 128, 131, ii. 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21847 <p>Noël, G. 282.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21848 <p>Number: defined, by Thales 69, by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21849 Moderatus, Eudoxus, Nicoma-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21850 chus, Aristotle 70: classification
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21851 of numbers 70-4: ‘perfect’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21852 ‘over-perfect’ and ‘defective’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21853 numbers 74-5, ‘friendly’ 75,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21854 figured 76-9: ‘oblong’, ‘prolate’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21855 82-3, 108, 114, similar plane and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21856 solid numbers 81-2, 90, solid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21857 numbers classified 106-8: ‘the
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21858 number in the heaven’ (Pytha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21859 gorean) 68, ‘number’ of an object
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21860 69.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21861 <p>Numerals: systems of, decimal, qui-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21862 nary, vigesimal 26: origin of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21863 decimal system 26-7: Egyptian
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21864 27-8; Babylonian systems (1)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21865 &ddot;ecimal 28, (2) sexagesimal 28-9:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21866 Greek (1) ‘Attic’ or ‘Herodianic’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21867 30-1: (2) alphabetic system,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21868 original in Greece 31-7, how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21869 evolved 31-2, date of introduc-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21870 tion 33-5, mode of writing 36-7,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21871 comparison of two systems 37-9:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21872 notation for large numbers, Apol-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21873 lonius's tetrads 40, Archimedes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21874 octads 40-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21875 <p>Nymphodorus 213.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21876 <p>‘Oblong’ numbers 82-3, 108, 114:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21877 gnomons of, 82-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21878 <p>Ocreatus, 111.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21879 <p>Octads, of Archimedes 40-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21880 <p>Octagon, regular, area of, ii. 328.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21881 <p>Octahedron 159, 160, 162: volume
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21882 of, ii. 335.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21883 <p>‘Odd’ number defined 70-1: 1
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21884 called ‘odd-even’ 71: ‘odd-even’,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21885 ‘odd-times-odd’, &c., numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21886 71-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21887 <p>Oenopides of Chios 22, 121: dis-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21888 covered obliquity of ecliptic 138,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21889 174, ii. 244: Great Year of, 174-5:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21890 called perpendicular <I>gnomon-wise</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21891 78, 175: two propositions in ele-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21892 mentary geometry 175.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21893 <p>Olympiodorus 444.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21894 <p>One, the principle of number 69.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21895 <p>Oppermann ii. 324 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21896 <p>Optics: divisions of, 17-18: of Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21897 441-4: of Ptolemy ii. 293-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21898 <p>Oval of Cassini ii. 206.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21899 <p>Oxyrhynchus Papyri 142.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21900 <p>Pamphile, 131, 133, 134.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21901 <p>Pandrosion ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21902 <p>Pappus (<I>see also</I> Table of Contents,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21903 under Chap. XIX) ii. 17-18, ii. 175,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21904 180, 181, 182, 183, 185, 186, 187,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21905 188, 189, 190, ii. 207, 211, 212,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21906 213, ii. 262, ii. 337, ii. 355-439:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21907 on Apollonius's tetrads 40, on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21908 Apollonius's continued multi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21909 plications 54-7: on ten means
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21910 87-9: on mechanical works of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21911 Archimedes ii. 23-4: on conics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21912 <pb n=581><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21913 of Euclid and Apollonius 438,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21914 proof of focus-directrix property
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21915 ii. 120-1: commentary on Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21916 358, ii. 356-7, on Book X 154-5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21917 209, 211, ii. 193: commentary on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21918 Euclid's <I>Data</I> 421-2, ii. 357, on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21919 Diodorus's <I>Analemma</I> ii. 287,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21920 scholia on <I>Syntaxis</I> ii. 274: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21921 classification of problems and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21922 loci (plane, solid, linear) 218-19,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21923 ii.117-18, criticism on Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21924 and Apollonius 288, ii. 68, ii. 167:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21925 on surface-loci 439-40, ii. 425-6:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21926 on Euclid's <I>Porisms</I> 431-3, 436-7,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21927 ii. 270, ii. 419-24: on ‘Treasury
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21928 of Analysis’ 421, 422, 439, ii. 399-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21929 427: on <I>cochloids</I> 238-9: on <I>quad-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21930 ratrix</I> 229-30, ii. 379-80, con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21931 structions for, ii. 380-2: on dupli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21932 cation of cube 266-8, 268-70: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21933 trisection of any angle 241-3,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21934 ii. 385-6, <G>neu=sis</G> with regard to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21935 parallelogram 236-7: on isoperi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21936 metry (cf. Zenodorus) ii. 207, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21937 211-12, ii. 390-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21938 <p>‘Paradoxes’ of Erycinus ii. 365-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21939 <p>Parallelogram of velocities 346, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21940 348-9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21941 <p><I>Parapegma</I> of Democritus 177.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21942 <p>Parmenides 138.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21943 <p>Paterius ii. 536-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21944 <p>Patricius ii. 318, 319.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21945 <p>Pebbles, for calculation 46, 48.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21946 <p>Pentagon, regular: construction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21947 Pythagorean 160-2; area of, ii. 327.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21948 <p>Pentagram, Pythagorean 161-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21949 <p>‘Perfect’ numbers 74-5: list of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21950 first ten <I>ib.</I>: contrasted with
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21951 ‘over-perfect’ and ‘defective’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21952 <I>ib.</I>: 10 with Pythagoreans 75.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21953 <p>‘Perfect’ proportion 86.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21954 <p>Pericles 172.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21955 <p>Pericles, a mathematician ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21956 <p>Perseus 226: spiric sections ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21957 203-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21958 <p>‘Phaenomena’ = observational as-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21959 tronomy 17: 322, 349.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21960 <p>Philippus of Opus 354: works by,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21961 321: on polygonal numbers 84,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21962 ii. 515: astronomy 321.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21963 <p>Philolaus 67, 72, 76, 78, 86, 158,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21964 ii. 1: on odd, even, and even-odd
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21965 numbers 70-1: Pythagorean non-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21966 geocentric astronomy attributed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21967 to, 163-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21968 <p>Philon of Byzantium 213: duplica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21969 tion of cube 262-3: Philon, Ctesi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21970 bius and Heron ii. 298-302.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21971 <p>Philon of Gadara 234.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21972 <p>Philon of Tyana ii. 260.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21973 <p>Philoponus, Joannes, 99, 223, 224 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21974 <p>Phocaeans 7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21975 <p>Phocus of Samos 138.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21976 <p>Phoenician alphabet, how treated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21977 by Greeks 31-2: arithmetic ori-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21978 ginated with Phoenicians 120-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21979 <p>‘Piremus’ or ‘peremus’ in pyramid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21980 126, 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21981 <p>‘Plane’ loci 218.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21982 <p>‘Plane’ problems 218-19.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21983 <p><I>Planisphaerium</I> of Ptolemy ii. 292-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21984 <p>Planudes, Maximus, 117, ii. 453, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21985 546-9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21986 <p>Plato 19, 22, 24, 121, 142 <I>n.</I>, 170, 176:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21987 <G>*qeo\s a)ei\gewmetrei=</G> 10: <G>mhdei\sa)gewme/-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21988 trhtos ei)si/tw</G> iii, 24, 355: on educa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21989 tion in mathematics 19-20, 284:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21990 on mathematical ‘arts’, measure-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21991 ment and weighing 308, instru-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21992 ments for, 308-9, principle of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21993 lever 309: on optics 309, 441:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21994 on music 310: Plato's astronomy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21995 310-15: on arithmetic and logistic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21996 13-14: classification of numbers,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21997 odd, even, &c. 71-2, 292: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21998 number 5040, 294: the Geometri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	21999 cal Number, 305-8: on arithme-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22000 tical problems 15, ii. 442: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22001 geometry 286-8, constructions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22002 alien to true geometry <I>ib.</I>: on-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22003 tology of mathematics 288-9:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22004 hypotheses of mathematics 289-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22005 90: two intellectual methods
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22006 290-2: supposed discovery of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22007 mathematical analysis, 120, 212-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22008 13, 291-2: definitions of various
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22009 species of numbers 292, figure
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22010 292-3, line and straight line 293,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22011 circle and sphere 293-4: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22012 points and indivisible lines 293:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22013 formula for rational right-angled
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22014 triangles 81, 304: ‘rational’ and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22015 ‘irrational diameter of 5’ 93,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22016 306-7: Plato and the irrational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22017 156, 203-5, 304: on solid geo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22018 metry 12-13, 303: on regular and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22019 semi-regular solids 294-7: Plato
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22020 and duplication of cube 245-6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22021 255, 287-8, 303: on geometric
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22022 means between two squares and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22023 two cubes respectively 89, 112,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22024 <pb n=582><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22025 201, 297, 400: on ‘perfect’ pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22026 portion 86: a proposition in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22027 proportion 294: two geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22028 passages in <I>Meno</I> 297-303: pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22029 positions ‘on the <I>section</I>’ 304,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22030 324-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22031 <p>‘Platonic’ figures (the regular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22032 solids) 158, 162, 294-5, 296-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22033 <p>Playfair, John, 436.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22034 <p>Pliny 129, ii. 207.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22035 <p>Plutarch 84, 96, 128, 129, 130, 133,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22036 144, 145, 167, 179, ii. 2, 3, ii. 516:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22037 on Archimedes ii. 17-18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22038 <p>Point: defined as a ‘unit having
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22039 position’ 69, 166: Plato on points
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22040 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22041 <p>Polybius 48, ii. 17 <I>n.</I>, ii. 207.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22042 <p>Polygon: propositions about sum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22043 of exterior or interior angles 144:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22044 measurement of regular polygons
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22045 ii. 326-9.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22046 <p>Polygonal numbers 15, 76, 79, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22047 213, ii. 514-17.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22048 <p>Polyhedra, <I>see</I> Solids.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22049 <p><I>Porism</I> (1) = corollary 372: (2) a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22050 certain type of proposition 373,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22051 431-8: <I>Porisms</I> of Euclid, <I>see</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22052 Euclid: of Diophantus, <I>see</I> Dio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22053 phantus.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22054 <p>Porphyry 145: commentary on Eu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22055 clid's <I>Elements</I> 358, ii. 529.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22056 <p>Poselger, F. T. ii. 455.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22057 <p>Posidonius ii. 219-22: definitions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22058 ii. 221, 226; on parallels 358, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22059 228: <I>versus</I> Zeno of Sidon ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22060 221-2: <I>Meteorologica</I> ii. 219:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22061 measurement of earth ii. 220: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22062 size of sun ii. 108, ii. 220-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22063 <p>Postulates: Aristotle on, 336: in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22064 Euclid 336, 374-5: in Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22065 336, ii. 75.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22066 <p>Powers, R. E. 75 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22067 <p>Prestet, Jean, 75 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22068 <p>Prime numbers and numbers prime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22069 to one another 72-3: defined 73:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22070 2 prime with Euclid and Aristotle,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22071 not Theon of Smyrna and Neo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22072 Pythagoreans <I>ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22073 <p>Problems: classification 218-19:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22074 plane and solid ii. 117-18: pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22075 blems and theorems 318, 431, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22076 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22077 <p>Proclus 12, 99, 175, 183, 213, 224 <I>n.</I>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22078 ii. 529-37: <I>Comm. on Eucl. I.</I> ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22079 530-5: sources ii. 530-2: ‘sum-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22080 mary’ 118-21, 170, object of, 170-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22081 1: on discoveries of Pythagoras
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22082 84-5, 90, 119, 141, 154: on Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22083 I. 47, 145, 147: attempt to prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22084 parallel-postulate 358, ii. 534: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22085 loci 219: on porisms 433-4: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22086 Euclid's music 444: comm. on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22087 <I>Republic</I> 92-3, ii. 536-7: <I>Hypoty-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22088 posis of astronomical hypotheses</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22089 ii. 535-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22090 <p>Prodicus, on secondary education
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22091 20-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22092 <p><I>Prolate</I>, of numbers 108, 204.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22093 <p><I>Proof</I> 370, ii. 533.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22094 <p>Proportion: theory discovered by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22095 Pythagoras 84-5, but his theory
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22096 numerical and applicable to com-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22097 mensurables only 153, 155, 167:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22098 def. of numerical proportion 190:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22099 the ‘perfect’ proportion 86:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22100 Euclid's universally applicable
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22101 theory due to Eudoxus 153, 155,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22102 216, 325-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22103 <p>Proposition, geometrical: formal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22104 divisions of, 370-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22105 <p>Protagoras 202: on mathematics
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22106 23, 179.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22107 <p>Prou, V. ii. 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22108 <p><I>Psammites</I> or <I>Sand-reckoner</I> of Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22109 medes 40, ii. 3, ii. 81-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22110 <p>Psellus, Michael, 223-4 <I>n.</I>, ii. 453,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22111 ii. 545-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22112 <p><I>Pseudaria</I> of Euclid 430-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22113 <p>Pseudo-Boëtius 47.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22114 <p>Pseudo-Eratosthenes: letter on du-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22115 plication of cube 244-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22116 <p>Ptolemies: coins of, with alphabetic
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22117 numerals 34-5: Ptolemy I, story
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22118 of, 354.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22119 <p>Ptolemy, Claudius, 181, ii. 198, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22120 216, ii. 218, ii. 273-97: sexa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22121 gesimal fractions 44-5, approxi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22122 mation to <G>p</G> 233: attempt to prove
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22123 parallel-postulate 358, ii. 295-7:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22124 <I>Syntaxis</I> ii. 273-86, commentaries
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22125 and editions ii. 274-5, contents
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22126 of, ii. 275-6, trigonometry in, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22127 276-86, 290-1, Table of Chords
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22128 ii. 259, ii. 283-4, on obliquity of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22129 ecliptic ii. 107-8: <I>Analemma</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22130 ii. 286-92: <I>Planispherium</I> ii. 292-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22131 3, <I>Optics</I> ii. 293-4, other works ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22132 293: <G>peri\ r(opw=n</G> ii. 295: <G>peri\ dia-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22133 sta/sews</G> <I>ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22134 <p>Pyramids: origin of name 126:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22135 measurements of, in Rhind Papy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22136 rus 126-8: pyramids of Daksh&umacr;r,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22137 <pb n=583><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22138 Gizeh, and M&emacr;d&umacr;m 128: measure-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22139 ment of height by Thales 129-30;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22140 volume of pyramid 176, 180, 217,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22141 ii. 21, &c., volume of frustum ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22142 334.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22143 <p>Pythagoras 65-6, 121, 131, 133, 138:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22144 travels 4-5, story of bribed pupil
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22145 24-5: motto 25, 141: Heraclitus,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22146 Empedocles and Herodotus on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22147 65: Proclus on discoveries of, 84-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22148 5, 90, 119, 141, 154: made mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22149 matics a part of liberal education
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22150 141, called geometry ‘inquiry’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22151 166, used definitions 166: arith-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22152 metic (theory of numbers) 66-80,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22153 figured numbers 76-9: gnomons
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22154 77, 79: ‘friendly’ numbers 75:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22155 formula for right-angled tri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22156 angles in rational numbers 79-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22157 80: founded theory of proportion
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22158 84-5, introduced ‘perfect’ pro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22159 portion 86: discovered depen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22160 dence of musical intervals on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22161 numerical ratios 69, 75-6, 85,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22162 165: astronomy 162-3, earth
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22163 spherical <I>ib.</I>, independent move-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22164 ment of planets 67, 163: Theorem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22165 of Pythagoras 142, 144-9, how
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22166 discovered? 147-9, general proof,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22167 how developed <I>ib.</I>, Pappus's ex-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22168 tension ii. 369-71.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22169 <p>Pythagoreans 2, 11, 220: <I>quadri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22170 vium</I> 11: a Pythagorean first
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22171 taught for money 22: first to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22172 advance mathematics 66: ‘all
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22173 things are numbers’ 67-9: ‘num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22174 ber’ of an object 69, ‘number in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22175 the heaven’ 68: figured numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22176 69: definition of unit 69: 1 is
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22177 odd-even 71: classification of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22178 numbers 72-4: ‘friendly’ num-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22179 bers 75: 10 the ‘perfect’ number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22180 75: oblong numbers 82-3, 108,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22181 114: side-and diameter-numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22182 giving approximations to √2, 91-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22183 3: first cases of indeterminate
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22184 analysis 80, 91, 96-7: sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22185 angles of triangle = 2 <I>R</I>, 135,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22186 143: geometrical theorems attri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22187 buted to, 143-54: invented appli-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22188 cation of areas and geometrical
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22189 algebra 150-4: discovered the in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22190 commensurable 65, 90-1, 154,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22191 with reference to √2 155, 168:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22192 theory of proportion only ap-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22193 plicable to commensurables 153,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22194 155, 167, 216: construction of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22195 regular pentagon 160-2: astro-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22196 nomical system (non-geocentric)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22197 163-5: definitions 166: on order
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22198 of planets ii. 242.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22199 <p><I>Qay en &hdot;eru</I>, height (of pyramid)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22200 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22201 <p>Quadratic equation: solved by Py-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22202 thagorean application of areas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22203 150-2, 167, 394-6, 422-3<I>:</I> nu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22204 merical solutions ii. 344, ii. 448,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22205 ii. 463-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22206 <p><I>Quadratrix</I> 2, 23, 171, 182, 218, 219,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22207 225-30, ii. 379-82.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22208 <p><I>Quadrivium</I> of Pythagoreans 11.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22209 <p>Quinary system of numerals 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22210 <p>Quintilian ii. 207.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22211 <p>Qus&tdot;&amacr; b. L&umacr;q&amacr;, translator of Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22212 362, ii. 453.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22213 <p>Rangabé, A. R. 49-50.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22214 <p>Ratdolt, Erhard, first edition of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22215 Euclid 364-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22216 <p><I>Reductio ad absurdum</I> 372: already
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22217 used by Pythagoreans 168.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22218 <p><I>Reduction</I> (of a problem) 372.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22219 <p>Reflection: equality of angles of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22220 incidence and reflection 442, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22221 294, ii. 353-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22222 <p>Refraction 6-7, 444: first attempt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22223 at a law (Ptolemy) ii. 294.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22224 <p>Regiomontanus 369, ii. 27, ii. 453-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22225 <p><I>Regula Nicomachi</I> 111.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22226 <p>Rhabdas, Nicolas, 40, ii. 324 <I>n.</I>, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22227 550-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22228 <p>Rhind Papyrus: mensuration in,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22229 122-8: algebra in, ii. 440-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22230 <p>Right-angled triangle: inscribed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22231 by Thales in circle 131: theorem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22232 of Eucl. I. 47, attributed to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22233 Pythagoras 142, 144-5, supposed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22234 Indian origin of, 145-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22235 <p>Right-angled triangles in rational
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22236 numbers: Pythagoras's formula
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22237 80, Plato's 81, Euclid's 81-2,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22238 405: triangle (3, 4, 5) known to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22239 Egyptians 122: Indian examples
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22240 146: Diophantus's problems on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22241 ii. 507-14.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22242 <p>Robertson, Abram, ii. 27.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22243 <p>Rodet, L. 234.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22244 <p>Rodolphus Pius ii. 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22245 <p>Roomen, A. van, ii. 182.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22246 <p>Rudio, F. 173, 184, 187-91, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22247 539.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22248 <pb n=584><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22249 <p>Rudolph of Bruges ii. 292.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22250 <p>Ruelle, Ch. Em. ii. 538.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22251 <p>Rüstow, F. W. ii. 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22252 <p>Ruler-and - compasses restriction
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22253 175-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22254 <p>Sachs, Eva, 209 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22255 <p>Salaminian table 48, 50-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22256 <p><I>Salinon</I> ii. 23, ii. 103.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22257 <p>Sampi (<*> = 900) derived from
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22258 Ssade q.v.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22259 <p><I>Sar</I> (Babylonian for 60<SUP>2</SUP>) 28, ii. 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22260 <p><I>Satapatha Br&amacr;hma&ndot;a</I>, 146.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22261 <p>Savile, Sir H., on Euclid 360, 369.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22262 <p><I>Scalene</I>: of triangles 142: of certain
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22263 solid numbers 107: of an odd
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22264 number (Plato) 292: of an oblique
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22265 cone ii. 134.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22266 <p>Schiaparelli, G. 317, 330, ii. 539.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22267 <p>Schmidt, W. ii. 308, 309, 310.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22268 <p>Schöne, H. ii. 308.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22269 <p>Schöne, R. ii. 308, 317.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22270 <p>Scholiast to <I>Charmides</I> 14, 53.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22271 <p>Schooten, F. van, 75 <I>n.</I>, ii. 185.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22272 <p>Schulz, O. ii. 455.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22273 <p>Scopinas ii. 1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22274 <p>Secondary numbers 72.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22275 <p><I>Sectio canonis</I> 17, 215, 444.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22276 <p>Seelhoff, P. 75 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22277 <p>Seleucus ii. 3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22278 <p>Semicircle: angle in, is right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22279 (Thales) 131, 133-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22280 <p>Senkereh, Tables 28, 29.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22281 <p><I>Senti</I>, base (of pyramid) 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22282 <p><I>Se-qet</I>, ‘that which makes the nature’
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22283 (of pyramid) = cotangent of angle
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22284 of slope 127-8, 130, 131.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22285 <p>Serenus ii. 519-26: <I>On section of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22286 cylinder</I> ii. 519-22, <I>On section of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22287 cone</I> ii. 522-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22288 <p>Sesostris (Ramses II) 121.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22289 <p>Sexagesimal system of numerals
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22290 and fractions 28-9: sexagesimal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22291 fractions in Greek 44-5, 59, 61-3,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22292 233, ii. 277-83.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22293 <p>Sextius 220.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22294 <p>Sicily 8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22295 <p>‘Side-’ and ‘diameter-numbers’ 91-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22296 3, 112, 153, 308, 380, ii. 536.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22297 <p>Simon, M. 200.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22298 <p>Simplicius: extract from Eudemus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22299 on Hippocrates's quadrature of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22300 lunes 171, 182-99: on Antiphon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22301 221-2: on Eudoxus's theory of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22302 concentric spheres 329: commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22303 tary on Euclid 358, ii. 539-40: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22304 mechanical works of Archimedes
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22305 ii. 24: ii. 538-40.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22306 <p>Simson, R., edition of Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22307 <I>Elements</I> 365, 369, and of Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22308 <I>Data</I> 421: on Euclid's <I>Porisms</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22309 435-6: restoration of <I>Plane Loci</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22310 of Apollonius ii. 185, ii. 360.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22311 <p>Simus of Posidonia 86.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22312 <p>Sines, Tables of, ii. 253, ii. 259-60.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22313 <p><I>Sinus rectus, sinus versus</I> 367.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22314 <p>Sluse, R. F. de, 96.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22315 <p>Smith, D. E. 49, 133 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22316 <p>‘Solid’ loci and problems 218, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22317 117-18: <I>Solid Loci</I> of Aristaeus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22318 438, ii. 118-19.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22319 <p>‘Solid’ numbers, classified 106-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22320 <p>Solids, Five regular: discovery at-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22321 tributed to Pythagoras or Pytha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22322 goreans 84, 141, 158-60, 168,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22323 alternatively (as regards octahe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22324 dron and icosahedron) to Theae-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22325 tetus 162: all five investigated
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22326 by Theaetetus 159, 162, 212, 217:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22327 Plato on, 158-60: Euclid's con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22328 structions for, 415-19: Pappus's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22329 constructions ii. 368-9: content
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22330 of, ii. 335, ii. 395-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22331 <p>Solon 4, 48.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22332 <p>Sophists: taught mathematics 23.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22333 <p>Sosigenes 316, 329.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22334 <p><I>Soss = sussu</I> = 60 (Babylonian) 28,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22335 ii. 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22336 <p>Speusippus 72, 73, 75, ii. 515: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22337 Pythagorean numbers 76, 318:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22338 on the five regular solids 318: on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22339 <I>theorems ib.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22340 <p><I>Sphaeric</I> 11-12: treatises on, by Au-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22341 tolycus and Euclid 348-52, 440-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22342 1: earlier text-book presupposed
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22343 in Autolycus 349-50: <I>Sphaerica</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22344 of Theodosius ii. 245, 246-52, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22345 Menelaus ii. 252-3, 260, 261-73.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22346 <p>Sphere-making 18: Archimedes on,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22347 ii. 17-18.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22348 <p>Spiric sections ii. 203-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22349 <p>Sporus 226: criticisms on <I>quadra-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22350 trix</I> 229-30: <G>khri/a</G> 234: duplica-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22351 tion of cube 266-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22352 <p>Square root, extraction of, 60-3:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22353 ex. in sexagesimal fractions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22354 (Theon) 61-2, (scholiast to Eu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22355 clid) 63: method of approxima-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22356 ting to surds ii. 51-2, ii. 323-6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22357 ii. 547-9, ii. 553-4.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22358 <p>Square numbers 69: formation by
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22359 adding successive gnomons (odd
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22360 <pb n=585><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22361 numbers) 77: any square is sum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22362 of two triangular numbers 83-4:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22363 8 times a triangular number
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22364 +1 = square, 84, ii. 516.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22365 <p><I>Ssade</I>, Phoenician sibilant (signs
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22366 <> <> <> <>) became <*> (900) 32.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22367 <p>‘Stadium,’ 1/60th of 30°, ii. 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22368 <p><I>Stadium</I> of Zeno 276-7, 281-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22369 <p>Star-pentagon, or <I>pentagram</I>, of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22370 Pythagoreans 161-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22371 <p>Stereographic projection (Ptolemy)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22372 ii. 292.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22373 <p>Stevin, S. ii. 455.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22374 <p>‘Stigma,’ name for numeral <G>s</G>,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22375 originally <*> (digamma) 32.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22376 <p>Strabo 121, ii. 107, ii. 220.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22377 <p>Strato ii. 1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22378 <p>Subcontrary (= harmonic) mean,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22379 defined 85.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22380 <p>Subtraction in Greek notation 52.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22381 <p>Surds: Theodorus on, 22-3, 155-6,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22382 203-9, 304: Theaetetus's general-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22383 ization 203-4, 205, 209, 304: see
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22384 also ‘Approximations’.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22385 <p><I>Surface-Loci</I> 219, ii. 380-5: Euclid's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22386 439-40, ii. 119, ii. 425-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22387 <p><I>S&umacr;rya-Siddh&amacr;nta</I> ii. 253.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22388 <p><I>Sussu = soss</I> (Babylonian for 60) 28,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22389 ii. 215.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22390 <p>Synesius of Cyrene ii. 293.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22391 <p>Synthesis 371-2: defined by Pappus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22392 ii. 400.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22393 <p>Syracuse 8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22394 <p>Table of Chords 45, ii. 259-60, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22395 283.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22396 <p><I>T&amacr;ittir&imacr;ya Samhit&amacr;</I> 146.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22397 <p>Tannery, P. 15, 44, 87, 89, 119, 132,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22398 180, 182, 184, 188, 196 <I>n.</I>, 232,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22399 279, 326, 440, ii. 51, ii. 105, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22400 204-5, ii. 215, ii. 218, ii. 253,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22401 ii. 317, ii. 453, ii. 483, ii. 519,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22402 ii. 538, ii. 545, 546, ii. 550, ii. 556,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22403 ii. 561.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22404 <p>Teles on secondary education 21.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22405 <p>Teos inscription 32, 34.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22406 <p>Tetrads of Apollonius 40.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22407 <p>Tetrahedron: construction 416, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22408 368: volume of, ii. 335.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22409 <p>Th&amacr;bit b. Qurra: translator of Eu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22410 clid 362, 363: of Archimedes's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22411 <I>Liber assumptorum</I> ii. 22: of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22412 Apollonius's <I>Conics</I> V-VII ii. 127:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22413 of Menelaus's <I>Elements of Geo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22414 metry</I> ii. 260: of Ptolemy ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22415 274-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22416 <p>Thales 2, 4, 67: one of Seven Wise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22417 Men 128, 142: introduced geo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22418 metry into Greece 128: geometri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22419 cal theorems attributed to, 130-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22420 7: measurement of height of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22421 pyramid 129-30, and of distance
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22422 of ship from shore 131-3: defini-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22423 tion of number 69: astronomy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22424 137-9, ii. 244: predicted solar
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22425 eclipse 137-8.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22426 <p>Theaetetus 2, 119, 170: on surds
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22427 22-3, 155, 203-4, 205, 209, 304:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22428 investigated regular solids 159,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22429 162, 212, 217: on irrationals 209-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22430 12, 216-17.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22431 <p>Themistius 221, 223, 224.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22432 <p>Theodorus of Cyrene: taught mathe-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22433 matics 22-3: on surds 22-3, 155-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22434 6, 203-9, 304.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22435 <p>Theodosius ii. 245-6: <I>Sphaerica</I> 349-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22436 50, ii. 246-52: other works ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22437 246: no trigonometry in, ii. 250.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22438 <p><I>Theologumena arithmetices</I> 96, 97,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22439 318.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22440 <p>Theon of Alexandria: examples of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22441 multiplication and division 58,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22442 59-60: extraction of square root
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22443 61-3: edition of Euclid's <I>Elements</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22444 360-1, ii. 527-8: of <I>Optics</I> 441,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22445 ii. 528: <I>Catoptrica ib.</I>: commen-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22446 tary on <I>Syntaxis</I> 58, 60, ii. 274,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22447 ii. 526-7.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22448 <p>Theon of Smyrna 12, 72, 73, 74, 75,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22449 76, 79, 83, 87, ii. 515: treatise
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22450 of, ii. 238-44: on ‘side-’ and
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22451 ‘diameter-numbers’ 91-3, 112:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22452 forms of numbers which cannot
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22453 be squares 112-13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22454 <p>Theophrastus 158, 163: on Plato's
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22455 view of the earth 315.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22456 <p>Theudius 320-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22457 <p>Theuth, Egyptian god, reputed in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22458 ventor of mathematics 121.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22459 <p>Thévenot, M. ii. 308.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22460 <p>Thrasyllus 97, 176, 177, ii. 241, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22461 243.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22462 <p>Thucydides ii. 207.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22463 <p>Thymaridas: definition of unit 69:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22464 ‘rectilinear’ = prime numbers
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22465 72: <G>e)pa/nqhma</G>, a system of simple
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22466 equations solved 94.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22467 <p>Timaeus of Locri 86.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22468 <p>Tittel ii. 300, 301, 304.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22469 <p><I>Tore</I> (or anchor-ring): use by Ar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22470 chytas 219, 247-9: sections of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22471 (Perseus), ii. 203-6: volume of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22472 <pb n=586><head>ENGLISH INDEX</head>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22473 (Dionysodorus and Heron), ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22474 218-19, ii. 334-5.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22475 <p>Torelli, J. ii. 27.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22476 <p>Transversal: Menelaus's theorem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22477 for spherical and plane triangles
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22478 ii. 266-70: lemmas relating to
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22479 quadrilateral and transversal
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22480 (Pappus) ii. 419-20.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22481 <p>‘Treasury of Analysis’ 421, 422,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22482 439, ii. 399-427.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22483 <p>Triangle: theorem about sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22484 angles Pythagorean 135, 143,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22485 Geminus and Aristotle on, 135-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22486 <p>Triangle, spherical: called <G>tri/pleu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22487 ron</G> (Menelaus) ii. 262: proposi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22488 tions analogous to Euclid's on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22489 plane triangles ii. 262-5: sum of
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22490 angles greater than two right
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22491 angles ii. 264.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22492 <p>Triangular numbers 15, 69: forma-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22493 tion 76-7: 8 times triangular
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22494 number +1 = a square 84, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22495 516.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22496 <p>Trigonometry ii 5, ii. 198, ii. 257-9,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22497 ii. 265-73, ii. 276-86, ii. 290-1.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22498 <p>Trisection of any angle: solutions
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22499 235-44: Pappus on, ii. 385-6.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22500 <p>Tschirnhausen, E. W. v., 200.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22501 <p>Tycho Brahe 317, ii. 2, ii. 196.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22502 <p>Tzifra (=0) ii. 547.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22503 <p><I>Ukha-thebt</I> (side of base in pyramid)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22504 126, 127.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22505 <p>Unit: definitions (Pythagoreans,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22506 Euclid, Thymaridas, Chrysippus)
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22507 69.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22508 <p>Usener, H. 184, 188.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22509 <p>Valla, G.: translator of extracts
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22510 from Euclid 365, and from Archi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22511 medes ii. 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22512 <p>Venatorius, Thomas Gechauff: <I>ed.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22513 princeps</I> of Archimedes ii. 27.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22514 <p>Venturi, G. ii. 308.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22515 <p>Vieta 200, 223, ii. 182, ii. 456, ii. 480,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22516 ii. 557.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22517 <p>Vigesimal system (of numerals) 26.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22518 <p>Vincent, A. J. H. 50, 436, ii. 308,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22519 ii. 545, ii. 546.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22520 <p>Vitruvius 18, 147, 174, 213, ii. 1,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22521 ii. 245: Vitruvius and Heron,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22522 ii. 302-3.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22523 <p>Viviani, V. ii. 261.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22524 <p>Vogt, H., 156 <I>n.</I>, 203 <I>n.</I>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22525 <p>Wescher, C. ii. 309.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22526 <p>Wilamowitz - Moellendorff, U. v.,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22527 158 <I>n.</I>, 245, ii. 128.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22528 <p>Xenocrates 24, 319: works on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22529 Numbers 319: upheld ‘indivisible
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22530 lines’ 181.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22531 <p>Xenophon, on arithmetic in educa-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22532 tion 19.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22533 <p>Xylander (W. Holzmann) ii. 454-5,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22534 ii. 545.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22535 <p>Ya&hdot;y&amacr; b. Kh&amacr;lid b. Barmak ii. 274.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22536 <p>Zamberti, B., translator of Euclid
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22537 365, 441.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22538 <p>Zeno of Elea 271-3: arguments on
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22539 motion 273-83.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22540 <p>Zeno of Sidon on Eucl. I. 1, 359, ii.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22541 221-2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22542 <p>Zenodorus ii. 207-13.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22543 <p>Zero in Babylonian notation 29:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22544 <*> in Ptolemy 39, 45.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22545 <p>Zeuthen, H. G. 190, 206-9, 210-11,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22546 398, 437, ii. 52, ii. 105, ii. 203,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22547 ii. 290-1, ii. 405, ii. 444.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22548 <p>Zodiac circle: obliquity discovered
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/ casties parents: diff changeset	22549 by Oenopides 138, 174, ii. 244.

Mercurial > hg > mpdl-xml-content

annotate texts/archimedesOldCVSRepository/archimedes/raw/heath_mathe_02_en_1981.raw @ 11:35edd67cabf9