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<metadata>
<dcterms:identifier>ECHO:KN1CRTZ2.xml</dcterms:identifier>
<dcterms:creator>Harriot, Thomas</dcterms:creator>
<dcterms:title xml:lang="en">Mss. 6785</dcterms:title>
<dcterms:date xsi:type="dcterms:W3CDTF">o. J.</dcterms:date>
<dcterms:language xsi:type="dcterms:ISO639-3">eng</dcterms:language>
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<dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder>
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<log>Automatically generated by bare_xml.py on Tue Nov 15 14:20:53 2011</log>
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<text xml:lang="eng" type="free">
<div xml:id="echoid-div1" type="section" level="1" n="1">
<pb file="add_6785_f001" o="1" n="1"/>
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<div xml:id="echoid-div1" type="page_commentary" level="2" n="1">
<p>
<s xml:id="echoid-s1" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s1" xml:space="preserve">
This page referes to Proposition 21 from Book I of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.21
If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter,
the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of
the transverse side of the figure, as the upright side of the figure is to the transverse,
and to each other as the areas contained by the straight lines cut off, as we have said.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head1" xml:space="preserve">
A,6.</head>
<p xml:lang="lat">
<s xml:id="echoid-s3" xml:space="preserve">
Ergo: <lb/>
per: 21, p. <lb/>
1. lib. Ap:
<lb/>[<emph style="it">tr:
Therefore, by Proposition 21 of Book I of Apollonius
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s4" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>m</mi></mstyle></math> cum sit parallela lineæ <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math> est ordinatim applicata <lb/>
ad diametrum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>; et punctum <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math> est in elipsi. <lb/>
Quod demonstrare oportuit.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>m</mi></mstyle></math>, since it is parallel to the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>g</mi></mstyle></math> is an ordinate to the diameter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>d</mi></mstyle></math>;
and the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi></mstyle></math> is on the ellipse. <lb/>
Which was to be demonstrated.
</emph>]<lb/>
</s>
</p>
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<pb file="add_6785_f005v" o="5v" n="10"/>
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<pb file="add_6785_f006v" o="6v" n="12"/>
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<pb file="add_6785_f008v" o="8v" n="16"/>
<pb file="add_6785_f009" o="9" n="17"/>
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<pb file="add_6785_f010" o="10" n="19"/>
<pb file="add_6785_f010v" o="10v" n="20"/>
<pb file="add_6785_f011" o="11" n="21"/>
<pb file="add_6785_f011v" o="11v" n="22"/>
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<pb file="add_6785_f013" o="13" n="25"/>
<div xml:id="echoid-div2" type="page_commentary" level="2" n="2">
<p>
<s xml:id="echoid-s5" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s5" xml:space="preserve">
This page referes to Propositions I.21 and III.52 of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566).
</s>
<lb/>
<quote>
I.21
If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter,
the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of
the transverse side of the figure, as the upright side of the figure is to the transverse,
and to each other as the areas contained by the straight lines cut off, as we have said.
</quote>
<lb/>
<quote>
III.52
If in an ellipse a rectangle equal to the fourth part of the figure is applied from both sides to the major axis
and deficient by a square figure, and from the points resulting from the application straight lines are deflected
to the line of the section, then they will be equal to the axis.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s7" xml:space="preserve">
per 21, p. <lb/>
1. lib. App.
<lb/>[<emph style="it">tr:
by Proposition 21 of Book I of Apollonius
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s8" xml:space="preserve">
Inde
<lb/>[<emph style="it">tr:
Whence
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s9" xml:space="preserve">
figura <lb/>
vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> figura
<lb/>[<emph style="it">tr:
figure <lb/>
or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> of the figure
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s10" xml:space="preserve">
fiat <lb/>[...]<lb/> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>f</mi><mi>i</mi><mi>g</mi><mi>u</mi><mi>r</mi><mi>a</mi></mstyle></math><lb/>
ut sequitur <lb/>
ponatur: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>W</mi></mstyle></math> dari <lb/>
<lb/>[...]<lb/>
centro igitur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, intervallo ad <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> <lb/>
periferia agatur secabit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math> <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math> est centroides per 52. p. 3. Apol.
<lb/>[<emph style="it">tr:
Let <lb/>[...]<lb/> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>o</mi><mi>f</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>f</mi><mi>i</mi><mi>g</mi><mi>u</mi><mi>r</mi><mi>e</mi><mo>,</mo><mi>a</mi><mi>s</mi><mi>f</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>o</mi><mi>w</mi><mi>s</mi><mo>.</mo></mstyle></math><lb/>
Put <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>W</mi></mstyle></math> to be given <lb/>
<lb/>[...]<lb/>
Therefore the centre is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, the interval taken to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> the periphery will therefore cut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>d</mi></mstyle></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>w</mi></mstyle></math> is the centroid by Proposition 52 of Book 3 of Apollonius.
</emph>]<lb/>
</s>
</p>
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<pb file="add_6785_f018" o="18" n="35"/>
<pb file="add_6785_f018v" o="18v" n="36"/>
<pb file="add_6785_f019" o="19" n="37"/>
<pb file="add_6785_f019v" o="19v" n="38"/>
<pb file="add_6785_f020" o="20" n="39"/>
<pb file="add_6785_f020v" o="20v" n="40"/>
<pb file="add_6785_f021" o="21" n="41"/>
<pb file="add_6785_f021v" o="21v" n="42"/>
<pb file="add_6785_f022" o="22" n="43"/>
<pb file="add_6785_f022v" o="22v" n="44"/>
<pb file="add_6785_f023" o="23" n="45"/>
<head xml:id="echoid-head2" xml:space="preserve">
ptol. lib. II
<lb/>[<emph style="it">tr:
Ptolemy, book 2.
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s11" xml:space="preserve">
Three aequall circles.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s12" xml:space="preserve">
Anguli dati
<lb/>[<emph style="it">tr:
Angles given
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s13" xml:space="preserve">
Anguli quærati
<lb/>[<emph style="it">tr:
Angles sought
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s14" xml:space="preserve">
linea quærata
<lb/>[<emph style="it">tr:
line sought
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f023v" o="23v" n="46"/>
<pb file="add_6785_f024" o="24" n="47"/>
<pb file="add_6785_f024v" o="24v" n="48"/>
<pb file="add_6785_f025" o="25" n="49"/>
<div xml:id="echoid-div3" type="page_commentary" level="2" n="3">
<p>
<s xml:id="echoid-s15" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s15" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Variorum responsorum liber VIII</emph>, Chapter 12, Proposition 7.
</s>
<lb/>
<quote xml:lang="lat">
Propositio VII. <lb/>
Si ab unaquaque extremitatum diametri, sumantur in eadem partem circuli duæ circumferentiae æquales
ab altera autem earundem extremitatum, inscribantur lineæ rectæ ad terminus sumptarum æqualium circumferentiarum;
spatium circuli quod interjacet inter diametrum &amp; proximam inscriptam, adjunctaum sectioni circuli,
quam facit altera inscriptarum, æquale est duobus sectoribus qui sub æqualibus sumptis circumferentiis comprehenduntur.
</quote>
<lb/>
<quote>
If from both ends of a diameter there are taken, in the same part of the circle, two equal arcs,
and moreover from one of those same extremities there are drawn straight lines to the ends of the equal arcs,
then the space inside the circle which is bounded by the diameter and the closest inscribed line and the arc
is equal to the two sectors made by the equal arcs.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head3" xml:space="preserve" xml:lang="lat">
Vieta resps. lib. 8. <lb/>
pag. 21. b.
<lb/>[<emph style="it">tr:
Viète, Responsorum liber VII, page 21v
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s17" xml:space="preserve">
Data. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi><mo>=</mo><mi>C</mi><mi>D</mi></mstyle></math>. <lb/>
consequentia: <lb/>
sector in circumferentia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>B</mi><mi>D</mi></mstyle></math> <lb/>
æqualis est: <lb/>
sector in centro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
Etiam: <lb/>
<lb/>[...]<lb/> <lb/>
Quoniam: <lb/>
sector in circumferentia <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi><mi>C</mi></mstyle></math> + segmento <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math>. <lb/>
æqualis est: <lb/>
Duobus sectoribus in centro <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>E</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi><mi>C</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi><mo>=</mo><mi>D</mi><mi>C</mi></mstyle></math>, then: <lb/>
The sector to the circumference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>B</mi><mi>D</mi></mstyle></math>, is equal to the sector to the centre, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
Also: <lb/>
<lb/>[...]<lb/> <lb/>
Because: <lb/>
The sector to the circumference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi><mi>C</mi></mstyle></math> plus the segment <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math> is equal to
the two sectors to the centre, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mi>E</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi><mi>C</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
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<pb file="add_6785_f028v" o="28v" n="56"/>
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<pb file="add_6785_f029v" o="29v" n="58"/>
<pb file="add_6785_f030" o="30" n="59"/>
<div xml:id="echoid-div4" type="page_commentary" level="2" n="4">
<p>
<s xml:id="echoid-s18" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s18" xml:space="preserve">
This page contains varous scraps of writing in teh following order. <lb/>
Hebrew letters for ...<lb/>
Consonants only from the phrase: 'In the beginning God made heaven and earth'.
'Tomas Haryot' written in the letters of Harriot's ALgonquin alphabet (see Add MS 6782, f. 337). <lb/>
'Ld sn sltrm' (unexplained). <lb/>
Consonants only from the phrase: 'Now full well marvel how a thing in it self so weak'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s20" xml:space="preserve">
n th bgnnng gd md hvn nd rth
</s>
</p>
<p>
<s xml:id="echoid-s21" xml:space="preserve">
[tomas haryot]
</s>
</p>
<p>
<s xml:id="echoid-s22" xml:space="preserve">
Ld sn sltrm
</s>
</p>
<p>
<s xml:id="echoid-s23" xml:space="preserve">
Nw f w mrvl hw a thng n t slf s wk
</s>
</p>
<pb file="add_6785_f030v" o="30v" n="60"/>
<pb file="add_6785_f031" o="31" n="61"/>
<pb file="add_6785_f031v" o="31v" n="62"/>
<pb file="add_6785_f032" o="32" n="63"/>
<div xml:id="echoid-div5" type="page_commentary" level="2" n="5">
<p>
<s xml:id="echoid-s24" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s24" xml:space="preserve">
Lists of books held by the Earl of Northumberland and Sir Walter Ralegh, respectively,
probably from the time when both were imprisoned in the Tower of London.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s26" xml:space="preserve">
In my Lords hands <lb/>
Hollanders Viage <lb/>
Carlile [???] into hell. <lb/>
Æliaus de Acribus <lb/>
Rarum diabolica <lb/>
Taleri [???] <lb/>
with lines <lb/>
in the margent .
</s>
</p>
<p>
<s xml:id="echoid-s27" xml:space="preserve">
In Sr Walters hands <lb/>
vita Adriani <lb/>
[???]
</s>
</p>
<pb file="add_6785_f032v" o="32v" n="64"/>
<pb file="add_6785_f033" o="33" n="65"/>
<div xml:id="echoid-div6" type="page_commentary" level="2" n="6">
<p>
<s xml:id="echoid-s28" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s28" xml:space="preserve">
The reference on this page is to Petrus Ramus (Pierre de la Ramée),
<emph style="it">Scholarum mathematicarum libri unus et triginta</emph> (1569).
(The same diagram appears in the 1599 edition, but there on page 314.)
On pages 319–320 (the last two pages of the 1569 edition) Ramus states and proves what is usually known as
Heron's Rule, for the area of a triangle given all three of its sides.
Harriot's diagram is the same as that given by Ramus, but he translates Ramus 19s verbal proof into symbols. <lb/>
See also Add MS 6785, f.345, for the same problem and a similar diagram, there from Clavius,
<emph style="it">Geometria practica</emph> (1604).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head4" xml:space="preserve" xml:lang="lat">
sc<emph style="super">h</emph>olæ mathematicæ Rami. pag. 320.
<lb/>[<emph style="it">tr:
<emph style="it">Scholarum mathematicorum</emph> of Ramus, page 320.
</emph>]<lb/>
</head>
<pb file="add_6785_f033v" o="33v" n="66"/>
<pb file="add_6785_f034" o="34" n="67"/>
<div xml:id="echoid-div7" type="page_commentary" level="2" n="7">
<p>
<s xml:id="echoid-s30" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s30" xml:space="preserve">
The reference on this page is to Simplicius's commentary on Aristotle's Physics, Book I, in
<emph style="it">Simplicii commentarii in octo Aristoteles Physicae ausculationis libros
cum ipso Aristotelis contextu</emph> (1551).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head5" xml:space="preserve" xml:lang="lat">
Eudemi Quadraturas ex Simplicio in lib. prim. phys. Tex. 10. pag. 13. b.
<lb/>[<emph style="it">tr:
Quadrature of Eudemus, from Simplicius, on the first book of Physics, text 10, page 13v.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s32" xml:space="preserve">
Data ex constructio:
<lb/>[<emph style="it">tr:
Given from the construction:
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f034v" o="34v" n="68"/>
<pb file="add_6785_f035" o="35" n="69"/>
<div xml:id="echoid-div8" type="page_commentary" level="2" n="8">
<p>
<s xml:id="echoid-s33" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s33" xml:space="preserve">
The referece at the top of the page is to Viète,
<emph style="it">Adrianus Romanus responsum</emph>, page 32.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head6" xml:space="preserve" xml:lang="lat">
Ad praxin Adrian. pag. 32. de polygones <lb/>
et plura
<lb/>[<emph style="it">tr:
On carrying out Adrianus , page 32, on polygons and more.
</emph>]<lb/>
</head>
<pb file="add_6785_f035v" o="35v" n="70"/>
<pb file="add_6785_f036" o="36" n="71"/>
<pb file="add_6785_f036v" o="36v" n="72"/>
<pb file="add_6785_f037" o="37" n="73"/>
<head xml:id="echoid-head7" xml:space="preserve">
To find the summe of all <emph style="super">the sines</emph> of minutes in <lb/>
a quarter of a circle
</head>
<p xml:lang="lat">
<s xml:id="echoid-s35" xml:space="preserve">
Nostra methodus. Vietæ methodus
<lb/>[<emph style="it">tr:
My method. Viète's method.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s36" xml:space="preserve">
The half the summe of <lb/>
the sines being. 4. the whole sine being one.
</s>
</p>
<p>
<s xml:id="echoid-s37" xml:space="preserve">
The secant and tangent of 89.59 is æquall to the secant of 89.59.30.
see Lansberg's de triangulis pag. 11. and therefore both <lb/>
the values by demonstration are all one. But the numbers disagreeing do argue that the secant &amp; tangent <lb/>
are not truly calculated in Rheticus his tables out of which those are taken.
</s>
</p>
<pb file="add_6785_f037v" o="37v" n="74"/>
<pb file="add_6785_f038" o="38" n="75"/>
<div xml:id="echoid-div9" type="page_commentary" level="2" n="9">
<p>
<s xml:id="echoid-s38" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s38" xml:space="preserve">
Diagrams pertaining to work on polygons in the surrounding pages.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f038v" o="38v" n="76"/>
<pb file="add_6785_f039" o="39" n="77"/>
<pb file="add_6785_f039v" o="39v" n="78"/>
<pb file="add_6785_f040" o="40" n="79"/>
<p xml:lang="lat">
<s xml:id="echoid-s40" xml:space="preserve">
Numerus laterum <lb/>
latera polyg. <lb/>
complementum <lb/>
semissæ complementum <lb/>
complem. semissium <lb/>
semiss. comp. et illorum semiss.
<lb/>[<emph style="it">tr:
Number of sides <lb/>
Polygonal side <lb/>
Complement <lb/>
Halves of the complement <lb/>
Complememnt of the halves <lb/>
Half of the complement and their halves
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s41" xml:space="preserve">
Examinatur
<lb/>[<emph style="it">tr:
Checked
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f040v" o="40v" n="80"/>
<pb file="add_6785_f041" o="41" n="81"/>
<div xml:id="echoid-div10" type="page_commentary" level="2" n="10">
<p>
<s xml:id="echoid-s42" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s42" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Adrianus Romanus responsum</emph> (1595).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s44" xml:space="preserve">
subtensæ omnes quæ commode <lb/>
per æquatione haberi possunt. <lb/>
Nam. 20. <lb/>
Subtensa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> non est quærenda per methodum <lb/>
Adriani Romani, per alias et cæteræ. Inde <lb/>
investiganda per æquationum linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>i</mi></mstyle></math>. unde <lb/>
una subtensa se habetur per æquationem. <lb/>
altera manifesta ex diagrammate.
<lb/>[<emph style="it">tr:
All the subtended angles that can conveniently be had from the equation, namely, 20. <lb/>
The angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> is not sought by the method of Adrianus Romanus, or by others and the rest.
In that place it is investigated by the equation of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>i</mi></mstyle></math>, whence one angle is had from the equation;
the other is clear from the diagram.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f041v" o="41v" n="82"/>
<pb file="add_6785_f042" o="42" n="83"/>
<pb file="add_6785_f042v" o="42v" n="84"/>
<pb file="add_6785_f043" o="43" n="85"/>
<div xml:id="echoid-div11" type="page_commentary" level="2" n="11">
<p>
<s xml:id="echoid-s45" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s45" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Adrianus Romanus responsum</emph> (1595), page 21.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f043v" o="43v" n="86"/>
<pb file="add_6785_f044" o="44" n="87"/>
<pb file="add_6785_f044v" o="44v" n="88"/>
<pb file="add_6785_f045" o="45" n="89"/>
<pb file="add_6785_f045v" o="45v" n="90"/>
<pb file="add_6785_f046" o="46" n="91"/>
<pb file="add_6785_f046v" o="46v" n="92"/>
<pb file="add_6785_f047" o="47" n="93"/>
<pb file="add_6785_f047v" o="47v" n="94"/>
<pb file="add_6785_f048" o="48" n="95"/>
<pb file="add_6785_f048v" o="48v" n="96"/>
<pb file="add_6785_f049" o="49" n="97"/>
<pb file="add_6785_f049v" o="49v" n="98"/>
<pb file="add_6785_f050" o="50" n="99"/>
<div xml:id="echoid-div12" type="page_commentary" level="2" n="12">
<p>
<s xml:id="echoid-s47" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s47" xml:space="preserve">
On this page, Harriot continues his work on Problem IX from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema IX. <lb/>
Datis duobus circulis, &amp; puncto, per datum punctum circulum describere
quem duo dati circuli contingat.
</quote>
<lb/>
<quote>
IX. Given two circles and a point, through the given point describe a circle that touches the two given circles.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head8" xml:space="preserve" xml:lang="lat">
Apoll: Gallus. problema. 9. casus. 2.
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem IX, case 2.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s49" xml:space="preserve">
restitutus
<lb/>[<emph style="it">tr:
restored
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s50" xml:space="preserve">
In isto casu <lb/>
Si punctum datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi></mstyle></math>, sit intra tangentes <lb/>
et intra circulos cuius diamet: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math> <lb/>
Duo circuli describi possunt. <lb/>
Si extra tangentes; unus tantum. <lb/>
punctum non dabitur in spatio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mi>Z</mi><mi>A</mi><mi>X</mi><mi>t</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>p</mi><mi>H</mi><mi>q</mi><mi>m</mi></mstyle></math>. <lb/>
alias ubicunque.
<lb/>[<emph style="it">tr:
In this case, if the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi></mstyle></math> is inside the tangents and inside the circle with diameter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math>,
two circles can be described. If outside the tangents, one such. <lb/>
The point is not given in the space <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>y</mi><mi>Z</mi><mi>A</mi><mi>X</mi><mi>t</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>m</mi><mi>p</mi><mi>H</mi><mi>q</mi><mi>m</mi></mstyle></math>, otherwise anywhere.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f050v" o="50v" n="100"/>
<pb file="add_6785_f051" o="51" n="101"/>
<div xml:id="echoid-div13" type="page_commentary" level="2" n="13">
<p>
<s xml:id="echoid-s51" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s51" xml:space="preserve">
On this page, Harriot continues his work on Problem IX from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema IX. <lb/>
Datis duobus circulis, &amp; puncto, per datum punctum circulum describere
quem duo dati circuli contingat.
</quote>
<lb/>
<quote>
IX. Given two circles and a point, through the given point describe a circle that touches the two given circles.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head9" xml:space="preserve" xml:lang="lat">
Apoll: Gallus. probl. 9. casus. 3.
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem IX, case 3.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s53" xml:space="preserve">
In isto casu <lb/>
Si punctum datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi></mstyle></math>, sit extra <lb/>
circuli circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math>, et intra <lb/>
tangentes ad partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>: <lb/>
intra tangentis ad partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>: <lb/>
Duo circuli possunt tangere duos <lb/>
datos.
<lb/>[<emph style="it">tr:
In this case, if the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi></mstyle></math> is outside the circle around <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math> and inside the tangents on the side of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>,
and inside the tangents on the side of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>, two circles are possible touching those given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s54" xml:space="preserve">
punctum non dabitur. <lb/>
In circulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math> et intra tangentes <lb/>
ad partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math>. <lb/>
Neque in circulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>E</mi></mstyle></math> et intra tangentes <lb/>
extra <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. <lb/>
Alias ubicunque.
<lb/>[<emph style="it">tr:
The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi></mstyle></math> is not given in the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math> and inside the tangents on the side <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>M</mi></mstyle></math>,
nor in the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>E</mi></mstyle></math> and inside the tangents outside <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. <lb/>
Otherwise anywhere.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f051v" o="51v" n="102"/>
<pb file="add_6785_f052" o="52" n="103"/>
<div xml:id="echoid-div14" type="page_commentary" level="2" n="14">
<p>
<s xml:id="echoid-s55" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s55" xml:space="preserve">
An investigation of Lemma I, preceding Problem IX of Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
Harriot notes some cases missed by Viète.
</s>
<lb/>
<quote xml:lang="lat">
Lemma I.<lb/>
Propositis duobus circulis, invenire punctum in jungente ipsorum centra, a quo,
cum ducetur quævis linea recta ipsos circulos secans, similis erunt segmenta.
</quote>
<lb/>
<quote>
Lemma I. Given two circles, find a point in the line joining their centre, from which,
when any straight line is drawn cutting those circles, the segments wil be similar.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head10" xml:space="preserve" xml:lang="lat">
Apoll: Gallus. Lemma. 1. pag. 6
<lb/>[<emph style="it">tr:
Apollonius Gallus, Lemma I, page 6.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s57" xml:space="preserve">
casus a Vieta <lb/>
omissi
<lb/>[<emph style="it">tr:
cases missed by Viète
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s58" xml:space="preserve">
similes arcus <lb/>
ad easdem partes
<lb/>[<emph style="it">tr:
similar arcs on the same side
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s59" xml:space="preserve">
similes arcus <lb/>
ad contrarias partes
<lb/>[<emph style="it">tr:
similar arcs on opposite sides
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s60" xml:space="preserve">
sunt <lb/>
preterea <lb/>
4 casus <lb/>
de circulis secantibus <lb/>
et 4 <lb/>
de tangentibus <lb/>
si placeat. <lb/>
et 1. <lb/>
de parallelis.
<lb/>[<emph style="it">tr:
besides the 4 cases of cutting the circles, there are 4 of tangents, if one wishes; and 1 of parallels.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f052v" o="52v" n="104"/>
<pb file="add_6785_f053" o="53" n="105"/>
<div xml:id="echoid-div15" type="page_commentary" level="2" n="15">
<p>
<s xml:id="echoid-s61" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s61" xml:space="preserve">
On this page, Harriot examines Problem VI from Viète's
<emph style="it">Apollonius Gallus</emph> (1600), noting cases that were missed by Viète.
</s>
<lb/>
<quote xml:lang="lat">
Problema VI. <lb/>
Datis puncto, linea recta, &amp; circulo, per datum punctam describere circulum,
quem data linea recta &amp; datus circulus contingat.
</quote>
<lb/>
<quote>
VI. Given a point, a line, and a circle, through the given point describe a circle
that touches the given line and the given circle.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head11" xml:space="preserve" xml:lang="lat">
(. – o)	Apoll: Gallus. prob. 6.
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem VI.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s63" xml:space="preserve">
alius casus: <lb/>
a Vieta <lb/>
omissus.
<lb/>[<emph style="it">tr:
another case, missed by Viète
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s64" xml:space="preserve">
Data. <lb/>
punctum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/>
linea <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
circulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi><mi>f</mi></mstyle></math> <lb/>
Quæsitum. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>. circulus.
<lb/>[<emph style="it">tr:
Given: <lb/>
the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, <lb/>
the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, <lb/>
the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>e</mi><mi>f</mi></mstyle></math>. <lb/>
Sought: the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s65" xml:space="preserve">
Casus <lb/>
1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>f</mi><mi>d</mi></mstyle></math>. Vieta. <lb/>
2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>. <lb/>
3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mi>f</mi></mstyle></math>. <lb/>
4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mi>f</mi></mstyle></math>. Vieta.
<lb/>[<emph style="it">tr:
Cases: <lb/>
1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>f</mi><mi>d</mi></mstyle></math>, in Viète. <lb/>
2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi><mi>f</mi></mstyle></math>. <lb/>
3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mi>f</mi></mstyle></math>. <lb/>
4. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>c</mi><mi>f</mi></mstyle></math>, in Viète.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s66" xml:space="preserve">
alter casus: <lb/>
a Vieta <lb/>
omissus.
<lb/>[<emph style="it">tr:
another case missed by Viète
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s67" xml:space="preserve">
In tribus prioribus <lb/>
casibus <lb/>
punctum (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) datum, est extra <lb/>
datum circulum, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> punctum <lb/>
est ad easdem partes lineæ datæ. <lb/>
cum (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) puncto seu cum circulo quæsito.
<lb/>[<emph style="it">tr:
In the three previous cases, the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is outside the given circle,
and the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> is on the same side of the given line as the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> or as the given circle.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s68" xml:space="preserve">
In 4<emph style="super">o</emph> casu, punctum (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>) datum <lb/>
est intra datum circulum <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, sunt ad partes <lb/>
contrarias.
<lb/>[<emph style="it">tr:
In the 4th case, the given point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> is inside the given circle, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> are on opposite sides.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f053v" o="53v" n="106"/>
<pb file="add_6785_f054" o="54" n="107"/>
<div xml:id="echoid-div16" type="page_commentary" level="2" n="16">
<p>
<s xml:id="echoid-s69" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s69" xml:space="preserve">
Some working for Lemma II, preceding Problem IX of Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
Harriot notes some cases missed by Viète.
</s>
<lb/>
<quote xml:lang="lat">
Lemma II.<lb/>
Sint duo circuli, unius ABCD, alter EFGH; jungens autem eorum centra KL secet circulum primum in A, D;
secundum vero in E, H; &amp; in ea sumatur M punctum, a quo acta MGFCB recta secet circulm primum in B, C,
secundum in F, G, &amp; sint similia segmenta, &amp; puncta quidem sectionum A, B sint remotior a ipsis C, D,
&amp; puncta F, E ipsis C, H. Ajo id quod fit sub MG, MB aequari id quod fit sub MH, MA.
</quote>
<lb/>
<quote>
Lemma II. Let there be two circles ABCD and EFGH; moreover the line KL joining their centres
cuts the first circle in A and D, and the second in E and H; and in that line there is taken the point M,
from which the straight line MGFCB cuts the first circle in B and C, and the second in F and G,
and the segments are similar, and the points A, B are further away than C, D, and the points F, E than C, H.
I say that the rectangle formed by MG, MB is equal to that formed by MH, MA.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head12" xml:space="preserve" xml:lang="lat">
Apol: Gallus. Lemma 2. pag. 6
<lb/>[<emph style="it">tr:
Apollonius Gallus, Lemma II, page 6.
</emph>]<lb/>
</head>
<pb file="add_6785_f054v" o="54v" n="108"/>
<pb file="add_6785_f055" o="55" n="109"/>
<div xml:id="echoid-div17" type="page_commentary" level="2" n="17">
<p>
<s xml:id="echoid-s71" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s71" xml:space="preserve">
On this page, Harriot examines Problem IX from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema IX. <lb/>
Datis duobus circulis, &amp; puncto, per datum punctum circulum describere
quem duo dati circuli contingat.
</quote>
<lb/>
<quote>
IX. Given two circles and a point, through the given point describe a circle that touches the two given circles.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head13" xml:space="preserve" xml:lang="lat">
.oo) Apoll: Gall. prob. 9. de casibus <lb/>
impossibilibus
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem IX, on impossible cases.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s73" xml:space="preserve">
Duobus datis <lb/>
circulis et <lb/>
puncto.
<lb/>[<emph style="it">tr:
From two given circles and a points
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s74" xml:space="preserve">
Si tertius tangetur <lb/>
a datis, intra: <lb/>
punctum non dabitur in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
neque in spatio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>k</mi><mi>l</mi><mi>h</mi></mstyle></math>. <lb/>
extra: <lb/>
punctum non dabitur <lb/>
in spatio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>g</mi><mi>h</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>c</mi><mi>l</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
If the third is touched by the given one, internally, then the point may not be in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> nor in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>k</mi><mi>l</mi><mi>h</mi></mstyle></math>;
if extrenally, the point may not be in the spaces <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>g</mi><mi>h</mi><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>c</mi><mi>l</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s75" xml:space="preserve">
Extra et intra: <lb/>
punctum non dabitur <lb/>
in spatio <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>g</mi><mi>h</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>c</mi><mi>l</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Outside or inside, the point may not be in the spaces <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>g</mi><mi>h</mi><mi>d</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>c</mi><mi>l</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s76" xml:space="preserve">
Si unus datorum includat <lb/>
alterum circulum: <lb/>
punctum non dabitur <lb/>
extra maiorem circuli <lb/>
vel intra minorem.
<lb/>[<emph style="it">tr:
If one of the given circles includes the other circle,
the point may not be outside the larger circle nor inside the smaller circle.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f055v" o="55v" n="110"/>
<pb file="add_6785_f056" o="56" n="111"/>
<div xml:id="echoid-div18" type="page_commentary" level="2" n="18">
<p>
<s xml:id="echoid-s77" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s77" xml:space="preserve">
On this page, Harriot examines Problem VIII from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema VIII. <lb/>
Datis duobus punctis, &amp; circulo, per data duo puncta circulum describere, qui datum contingat.
</quote>
<lb/>
<quote>
VIII. Given two points and a circle, through the two given points describe a circle that touches the given one.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head14" xml:space="preserve" xml:lang="lat">
Apoll: Gallus. prob. 8.
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem VIII.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s79" xml:space="preserve">
Desirit Duo casus (et alij) <lb/>
in Vieta: videlicet.
<lb/>[<emph style="it">tr:
There are missing two cases in Viète, namely:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s80" xml:space="preserve">
Datis duobus puncti <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/>
Et circulo, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi><mi>f</mi></mstyle></math>: <lb/>
per datum, circulum contingentem <lb/>
describere.
<lb/>[<emph style="it">tr:
Given the two points <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, and the circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi><mi>f</mi></mstyle></math>,
describe a circle through the given points, touching the circle.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s81" xml:space="preserve">
Sit iam factum: <lb/>
Et sit contactus in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math> <lb/>
agantur rectæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> <lb/>
arcus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>f</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi></mstyle></math> sunt similes <lb/>
ita <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>d</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>b</mi></mstyle></math> contingentium. <lb/>
<lb/>[...]<lb/>
<lb/>[<emph style="it">tr:
And let it be done thus: <lb/>
And let it meet in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <lb/>
Connect the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math>. <lb/>
The arcs <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>b</mi></mstyle></math> are simlar, thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>e</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi><mi>d</mi></mstyle></math>, touching <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>o</mi><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s82" xml:space="preserve">
Datur latera triangulorum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>h</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math> est distantia verticum.
<lb/>[<emph style="it">tr:
Given the sides of triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>f</mi><mi>h</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>h</mi></mstyle></math>, The distance to the vertex is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s83" xml:space="preserve">
In alia charta <lb/>
modus investigandum.
<lb/>[<emph style="it">tr:
The method of investigation is in another sheet.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s84" xml:space="preserve">
Si linea a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> <emph style="super">per se vel producta</emph> non secat circulum <lb/>
datum: fierunt duo circuli contingentes.
<lb/>[<emph style="it">tr:
If the lines from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, either in themselves or produced, do not cut the given circle,
then there will arise two touching circles.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s85" xml:space="preserve">
Si unum punctum sit intra alterum extra <lb/>
circulum datum: casus impossibilis.
<lb/>[<emph style="it">tr:
If one point is inside, the other outside, the given circle, the case is impossible.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s86" xml:space="preserve">
Si ab uno datorum punctum <lb/>
sint duæ tangentes <lb/>
circulum: et altera fit: <lb/>
extra <lb/>
intra
<lb/>[<emph style="it">tr:
If from one of the given points there are two tangents to the circle, and the other is constructed <lb/>
outside <lb/>
inside
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f056v" o="56v" n="112"/>
<pb file="add_6785_f057" o="57" n="113"/>
<pb file="add_6785_f057v" o="57v" n="114"/>
<pb file="add_6785_f058" o="58" n="115"/>
<div xml:id="echoid-div19" type="page_commentary" level="2" n="19">
<p>
<s xml:id="echoid-s87" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s87" xml:space="preserve">
The reference to Pappus is to Commandino's edition of Books III to VIII,
<emph style="it">Mathematicae collecitones</emph> (1558).
The relevant proposition on page 44 is Proposition IV.11.
Harriot's diagram is the same as the one given by Commandino except for his use of lower case letters.
</s>
<lb/>
<quote xml:lang="lat">
Theorema XI. Propositio XI. <lb/>
Sit semicirculus ABC, &amp; inflectatur CBA. ducaturque CD ita, vt CB fit æqualis utrisque simul AB CD,
&amp; perpendiculares BE EF ducantur. Dico AF ipsius BE duplæ esse.
</quote>
<lb/>
<quote>
Let there be a semicircle ABC, curved along CBA. The line CD is drawn so that CB is equal to AB and CD together,
and there are drawn the perpendiculars BE and EF. I say that AF is twice BE.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head15" xml:space="preserve" xml:lang="lat">
Utile ad sinus
<lb/>[<emph style="it">tr:
Useful for sines
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s89" xml:space="preserve">
et ad locum <lb/>
de tactibus
<lb/>[<emph style="it">tr:
and for the place of touching
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s90" xml:space="preserve">
pappus pag. 44.
<lb/>[<emph style="it">tr:
Pappus, page 44.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f058v" o="58v" n="116"/>
<pb file="add_6785_f059" o="59" n="117"/>
<div xml:id="echoid-div20" type="page_commentary" level="2" n="20">
<p>
<s xml:id="echoid-s91" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s91" xml:space="preserve">
Some working for Lemma II, preceding Problem IX of Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
Harriot notes some cases missed by Viète.
</s>
<lb/>
<quote xml:lang="lat">
Lemma II.<lb/>
Sint duo circuli, unius ABCD, alter EFGH; jungens autem eorum centra KL secet circulum primum in A, D;
secundum vero in E, H; &amp; in ea sumatur M punctum, a quo acta MGFCB recta secet circulm primum in B, C,
secundum in F, G, &amp; sint similia segmenta, &amp; puncta quidem sectionum A, B sint remotior a ipsis C, D,
&amp; puncta F, E ipsis C, H. Ajo id quod fit sub MG, MB aequari id quod fit sub MH, MA.
</quote>
<lb/>
<quote>
Lemma II. Let there be two circles ABCD and EFGH; moreover the line KL joining their centres
cuts the first circle in A and D, and the second in E and H; and in that line there is taken the point M,
from which the straight line MGFCB cuts the first circle in B and C, and the second in F and G,
and the segments are similar, and the points A, B are further away than C, D, and the points F, E than C, H.
I say that the rectangle formed by MG, MB is equal to that formed by MH, MA.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head16" xml:space="preserve" xml:lang="lat">
Apoll: Gallus. pag. 6. Lemma. 2.
<lb/>[<emph style="it">tr:
Apollonius Gallus, page 6, Lemma II.
</emph>]<lb/>
</head>
<pb file="add_6785_f059v" o="59v" n="118"/>
<pb file="add_6785_f060" o="60" n="119"/>
<div xml:id="echoid-div21" type="page_commentary" level="2" n="21">
<p>
<s xml:id="echoid-s93" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s93" xml:space="preserve">
The reference to Pappus is to Commandino's edition of Books III to VIII,
<emph style="it">Mathematicae collecitones</emph> (1558).
The proposition on page 41 is Proposition IV.7.
</s>
<lb/>
<quote xml:lang="lat">
Theorema VII. Propositio VII. <lb/>
Sit quadrilaterum ABCD, rectum angulus habens ABC, &amp; datam unamquamque rectarum linearum AB BC CD DA.
ostendum est rectam lineam, quæ BD puncta coniungit, datam esse.
</quote>
<lb/>
<quote>
Let there be a quadrilateral ABCD, having a right angle ABC. Given any one of the lines AB, BC, CD, DA,
it is to be shown that the line which joins BD is given.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head17" xml:space="preserve" xml:lang="lat">
Lemmata, ad <lb/>
locum de tactibus
<lb/>[<emph style="it">tr:
Lemmas, on the place of touching
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s95" xml:space="preserve">
Datis tribus lateribus <lb/>
trianguli; invenire <lb/>
diametrum circuli <lb/>
circumscribentis.
<lb/>[<emph style="it">tr:
Given three sides of a triangle, find the diameter of the circumscribing circle.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s96" xml:space="preserve">
poristicon
<lb/>[<emph style="it">tr:
poristic
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s97" xml:space="preserve">
hinc i.p. Appendiculæ <lb/>
Vietæ.
<lb/>[<emph style="it">tr:
Here see the Appendix of Viète.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s98" xml:space="preserve">
Datis lateribus duorum <lb/>
triangulorum super eandem <lb/>
basim: verticum distantiam <lb/>
invenire. <lb/>
videlicet: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math>
<lb/>[<emph style="it">tr:
Given the sides of two triangles on the same base, to find the vertical distance, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s99" xml:space="preserve">
Sunt etiam alia in pappo <lb/>
pag: 41. &amp; sequentibus.
<lb/>[<emph style="it">tr:
There are also more in Pappus, page 41 and what follows.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f060v" o="60v" n="120"/>
<div xml:id="echoid-div22" type="page_commentary" level="2" n="22">
<p>
<s xml:id="echoid-s100" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s100" xml:space="preserve">
On this page, Harriot continues his work on Problem IX from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema IX. <lb/>
Datis duobus circulis, &amp; puncto, per datum punctum circulum describere
quem duo dati circuli contingat.
</quote>
<lb/>
<quote>
IX. Given two circles and a point, through the given point describe a circle that touches the two given circles.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head18" xml:space="preserve" xml:lang="lat">
Apoll: Gallus. prob. 9. casus. 2.
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem IX, case 2.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s102" xml:space="preserve">
Vietæ constructio inepta est. <lb/>
In altera charta restitutur.
<lb/>[<emph style="it">tr:
Viète's construction is inappropriate; it is restored in the other sheet.
</emph>]<lb/>
[<emph style="it">Note:
The other sheet is Add MS 6785, f. 50.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6785_f061" o="61" n="121"/>
<pb file="add_6785_f061v" o="61v" n="122"/>
<div xml:id="echoid-div23" type="page_commentary" level="2" n="23">
<p>
<s xml:id="echoid-s103" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s103" xml:space="preserve">
Note the diagrams representing combinations of points (.), lines (-), and circles (o).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f062" o="62" n="123"/>
<div xml:id="echoid-div24" type="page_commentary" level="2" n="24">
<p>
<s xml:id="echoid-s105" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s105" xml:space="preserve">
A diagram for Lemma I, preceding Problem IX of Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Lemma I.<lb/>
Propositis duobus circulis, invenire punctum in jungente ipsorum centra, a quo,
cum ducetur quævis linea recta ipsos circulos secans, similis erunt segmenta.
</quote>
<lb/>
<quote>
Lemma I. Given two circles, find a point in the line joining their centre, from which,
when any straight line is drawn cutting those circles, the segments wil be similar.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head19" xml:space="preserve" xml:lang="lat">
Apoll: Gall: pag; 5. b. <lb/>
lem. 1.
<lb/>[<emph style="it">tr:
Apollonius Gallus, page 5v, Lemma I.
</emph>]<lb/>
</head>
<pb file="add_6785_f062v" o="62v" n="124"/>
<div xml:id="echoid-div25" type="page_commentary" level="2" n="25">
<p>
<s xml:id="echoid-s107" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s107" xml:space="preserve">
Note the diagrams representing combinations of points (.), lines (-), and circles (o).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f063" o="63" n="125"/>
<pb file="add_6785_f063v" o="63v" n="126"/>
<pb file="add_6785_f064" o="64" n="127"/>
<div xml:id="echoid-div26" type="page_commentary" level="2" n="26">
<p>
<s xml:id="echoid-s109" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s109" xml:space="preserve">
On this page, Harriot examines problems from Appendix I from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Appendicula I. <lb/>
De problematis, quorum geometricam constructionem se nescire ait Regiomontanus.
</quote>
<lb/>
<quote>
On problems whose geometric construction is necessary according to Regiomontanus.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head20" xml:space="preserve" xml:lang="lat">
In appendiculam ad Appollonium gallum.
<lb/>[<emph style="it">tr:
From the appendix to Apollonius Gallus.
</emph>]<lb/>
</head>
<pb file="add_6785_f064v" o="64v" n="128"/>
<div xml:id="echoid-div27" type="page_commentary" level="2" n="27">
<p>
<s xml:id="echoid-s111" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s111" xml:space="preserve">
On this page, Harriot examines Problem I from Appendix I from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema I. <lb/>
Data base trianguli, altitudine, &amp; rectangulo sub cruribus, invenire triangulum.
</quote>
<lb/>
<quote>
I. Given the base of a triangle, its height, and the product of its legs, to find the triangle.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head21" xml:space="preserve" xml:lang="lat">
prob. 1. Apend: Apoll: Gall.
<lb/>[<emph style="it">tr:
Problem I from the appendix to Apollonius Gallus.
</emph>]<lb/>
</head>
<pb file="add_6785_f065" o="65" n="129"/>
<div xml:id="echoid-div28" type="page_commentary" level="2" n="28">
<p>
<s xml:id="echoid-s113" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s113" xml:space="preserve">
An attempt to count the possible ways of arranging three circles. <lb/>
The first row shows the possible ways of arranging two circles
(separated, touching, intersecting, one inside the other touching, one inside the other but not touching). <lb/>
Below that, a third circle is added, outside the other two but touching both, or between them touching both,
or outside one but touching the other, and so on. <lb/>
The attempt soon becomes erratic, but is an interesting example of Harriot's liking for systematic combinations.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s115" xml:space="preserve">
<sc>
The comment 'better' appears to refer to the re-ordering of the first row as 1, 3, 5, 4, 2
</sc>
optime
<lb/>[<emph style="it">tr:
better
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s116" xml:space="preserve">
Habitudines <lb/>
trium circularum
<lb/>[<emph style="it">tr:
Possibilities for three circles
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f065v" o="65v" n="130"/>
<pb file="add_6785_f066" o="66" n="131"/>
<div xml:id="echoid-div29" type="page_commentary" level="2" n="29">
<p>
<s xml:id="echoid-s117" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s117" xml:space="preserve">
On this page, Harriot examines Problem X from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema IX. <lb/>
Datis tribus circulis, desribere quartum circulum quem illi contingat.
</quote>
<lb/>
<quote>
X. Given three circles and a point, describe a fourth circle that touches them.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head22" xml:space="preserve" xml:lang="lat">
Apoll: Gall. prob. 10. casus quidam
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem X, certain cases.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s119" xml:space="preserve">
8. <lb/>
in una <lb/>
diagram
<lb/>[<emph style="it">tr:
8 in one diagram
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f066v" o="66v" n="132"/>
<pb file="add_6785_f067" o="67" n="133"/>
<pb file="add_6785_f067v" o="67v" n="134"/>
<pb file="add_6785_f068" o="68" n="135"/>
<div xml:id="echoid-div30" type="page_commentary" level="2" n="30">
<p>
<s xml:id="echoid-s120" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s120" xml:space="preserve">
On this page, Harriot examines Problem IX from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema IX. <lb/>
Datis duobus circulis, &amp; puncto, per datum punctum circulum describere
quem duo dati circuli contingat.
</quote>
<lb/>
<quote>
IX. Given two circles and a point, through the given point describe a circle that touches the two given circles.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head23" xml:space="preserve" xml:lang="lat">
Appo: Gal. prob: 9. Casus
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem IX, cases.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s122" xml:space="preserve">
Etsi circuli dati <lb/>
ponuntur inæquales <lb/>
tamen similes casus <lb/>
intelligantur de <lb/>
æqualibus circulis.
<lb/>[<emph style="it">tr:
Although the given circles are supposed unequal, nevertheless similar cases can be understood of equal circles.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s123" xml:space="preserve">
Hic casus <lb/>
impossibiles <lb/>
facile intelligantur.
<lb/>[<emph style="it">tr:
These case are easily understood to be impossible.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f068v" o="68v" n="136"/>
<pb file="add_6785_f069" o="69" n="137"/>
<div xml:id="echoid-div31" type="page_commentary" level="2" n="31">
<p>
<s xml:id="echoid-s124" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s124" xml:space="preserve">
On this page, Harriot examines Problem VIII from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema VIII. <lb/>
Datis duobus punctis, &amp; circulo, per dato puncta circulum describere, qui datum contingat.
</quote>
<lb/>
<quote>
VIII. Given two points and a circle, through the given point describe a circle that touches the given one.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head24" xml:space="preserve" xml:lang="lat">
Apol: Gallus. prob. 8. Casus Â <lb/>
(..o)
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem VIII, cases.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s126" xml:space="preserve">
epitagma
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s127" xml:space="preserve">
Imposs.
<lb/>[<emph style="it">tr:
Impossible.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f069v" o="69v" n="138"/>
<pb file="add_6785_f070" o="70" n="139"/>
<div xml:id="echoid-div32" type="page_commentary" level="2" n="32">
<p>
<s xml:id="echoid-s128" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s128" xml:space="preserve">
On this page, Harriot examines Problem VII from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema VII. <lb/>
Datis duobus circulis, &amp; linea recta, describere tertium circulum,
quem duo dati, &amp; dati linea recta contingat.
</quote>
<lb/>
<quote>
VII. Given two circles and a straight line, describe a third circle
that touches the two given and the given straight line.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head25" xml:space="preserve" xml:lang="lat">
Apol: Gal. 7. prob. Casus <lb/>
7. (oo–)
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem VII, cases.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s130" xml:space="preserve">
ut. 2. partium
<lb/>[<emph style="it">tr:
as two parts
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s131" xml:space="preserve">
æquales circuli
<lb/>[<emph style="it">tr:
equal circles
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s132" xml:space="preserve">
Imposs.
<lb/>[<emph style="it">tr:
Impossible.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f070v" o="70v" n="140"/>
<pb file="add_6785_f071" o="71" n="141"/>
<div xml:id="echoid-div33" type="page_commentary" level="2" n="33">
<p>
<s xml:id="echoid-s133" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s133" xml:space="preserve">
Diagrams for Problem VIII from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema VIII. <lb/>
Datis duobus punctis, &amp; circulo, per dato puncta circulum describere, qui datum contingat.
</quote>
<lb/>
<quote>
VIII. Given two points and a circle, through the given point describe a circle that touches the given one.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head26" xml:space="preserve" xml:lang="lat">
Apol: Galli. prob. 8.
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem VIII.
</emph>]<lb/>
</head>
<pb file="add_6785_f071v" o="71v" n="142"/>
<pb file="add_6785_f072" o="72" n="143"/>
<div xml:id="echoid-div34" type="page_commentary" level="2" n="34">
<p>
<s xml:id="echoid-s135" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s135" xml:space="preserve">
On this page, Harriot outlines the first six problems in Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
Each problem requires the construction of a circle passing through given points and/or touching given lines or circles.
The sketches next to the problem numbers, down the left-hand side of the page, offer a brief summary of each problem.
Problem 2, for example, is to describe a circle through two given points touching a given line.
On the right-hand side of the page, Harriot attempts to enumerate the different cases,
according to the relative positions of points and lines and
whether the lines are inclined to each other or parallel.
</s>
<lb/>
<quote xml:lang="lat">
Problema I. <lb/>
Datis tribus punctis per eadem circulum describere:
oportet autem data puncta non existere tria in eadem linea recta. <lb/>
Problema II. <lb/>
Datis duobus punctis, &amp; linea recta, per data puncta circulum describere,
quem data linea recta contingat. <lb/>
Problema III. <lb/>
Datis tribus lineis rectis, describere circulum quem harum unaquæque contingat.
Oportet autem datas lineas rectas non esse parallelas. <lb/>
Problema IV. <lb/>
Datis duabus lineis rectis, &amp; puncto, per datum punctum circulum describere,
quem datae duae lineae rectae contingat. <lb/>
Problema V. <lb/>
Dato circulo, &amp; duabus lineis describere circulum quem datus circulus,
&amp; datae duae lineæ rectæ contingat. <lb/>
Problema VI. <lb/>
Datis puncto, linea recta, &amp; circulo, per datum punctam describere circulum,
quem data linea recta &amp; datus circulus contingat.
</quote>
<lb/>
<quote>
I. Given three points, describe a circle thorugh them;
moreover it must be the case that the three fiven points are not in a straight line. <lb/>
II. Given two points and a line, through the given points describe a circle
that touches the given line. <lb/>
III. Given three lines, describe a circle that touches each of them.
Moroever it must be the case that the three lines are not parallel. <lb/>
IV. Given two lines and point, through the given point describe a circle
that touches the two lines. <lb/>
V. Given a curcle and two lines, describe a cricle that touches the given circle and the two lines. <lb/>
VI. Given a point, a line, and a circle, through the given point describe a circle
that touches the given line and the given circle.
</quote>
<lb/>
<s xml:id="echoid-s136" xml:space="preserve">
It is very difficult to transcribe this page in a meaningful way,
and the reader is strongly advised to examine the layout of the original.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head27" xml:space="preserve" xml:lang="lat">
Appollonij Galli Casus
<lb/>[<emph style="it">tr:
Cases in Apollonius Gallus
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s138" xml:space="preserve">
prob. 1. casus. 2. <lb/>
Data in recta. Imposs. vel infinit. <lb/>
non in recta.
<lb/>[<emph style="it">tr:
Problem I, 2 cases. <lb/>
Given [points] in a line. Impossible or infinitely many. <lb/>
Not in a line.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s139" xml:space="preserve">
2. 3. casus <lb/>
linea per <lb/>
puncta. <lb/>
est parallela. unus quæsitum <lb/>
inclinans. 1. <lb/>
perpendicularis. 1. <lb/>
duo circuli <lb/>
quæsiti.
<lb/>[<emph style="it">tr:
Problem II. <lb/>
3 cases, line through the points; <lb/>
parallel: one sought; <lb/>
inclined: 1. <lb/>
perpendicular: two circles sought.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s140" xml:space="preserve">
duorum punctum in linea <lb/>
2. ad easdem partes
<lb/>[<emph style="it">tr:
Two points in a line. <lb/>
2 cases: on the same side.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s141" xml:space="preserve">
3. imposs. <lb/>
Ad contrarias partes. Imposs. <lb/>
In directum tria. Duo, linea et punctum.
<lb/>[<emph style="it">tr:
3 impossible cases. <lb/>
On opposite sides. Impossible <lb/>
On the line of all three. Two lines and a point.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s142" xml:space="preserve">
casus. 3. 3. <lb/>
1. Nullæ parallelæ <lb/>
2. tertia perpendic. <lb/>
3. secans oblique <lb/>
4. non secans vel parallela. <lb/>
Imposs. <lb/>
5. Duae in directa. <lb/>
6. tres in directæ
<lb/>[<emph style="it">tr:
Problem III. <lb/>
3 + 3 cses <lb/>
1. None parallel <lb/>
2. third perpendicular <lb/>
3. cutting obliquely <lb/>
4. not cutting or parallel; impossible <lb/>
5. Two in a line <lb/>
6. three in a line
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s143" xml:space="preserve">
4. 4. lineæ parallelæ <lb/>
(Duo circ: quæsiti:) <lb/>
puncto in medio. <lb/>
extra media sed intra. <lb/>
Imposs.
<lb/>[<emph style="it">tr:
Problem IV. <lb/>
4 cases: parallel lines (two circles sought) <lb/>
points in the middle <lb/>
outside the middle but inside; impossible
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s144" xml:space="preserve">
punctum in linea. 2. <lb/>
lineæ inclinantes <lb/>
punctum in medio <lb/>
extra media
<lb/>[<emph style="it">tr:
2 cases: points in the line. <lb/>
lines inclined <lb/>
point in the middle <lb/>
outside the middle
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s145" xml:space="preserve">
1. punctum extra lineas. Imposs.
<lb/>[<emph style="it">tr:
1 case: point outside the lines; impossible
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s146" xml:space="preserve">
5. 8. linea parallelæ <lb/>
circulus in medio <lb/>
2. quæs extra <lb/>
2. quæs. intra <lb/>
1. circulus extra medium <lb/>
sed intra <lb/>
2. quæs extra <lb/>
2. quæs. intra <lb/>
lineæ inclinantes <lb/>
similiter. <lb/>
4.
<lb/>[<emph style="it">tr:
Problem V. <lb/>
8 cases: parallel lines <lb/>
circle in the middle <lb/>
2 sought outside, 2 sought inside. <lb/>
1 case, circle outside the middle but between <lb/>
2 sought outside, 2 sought inside <lb/>
4 cases: lines inclined, similarly
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s147" xml:space="preserve">
circulus extra lineas totaliter <lb/>
Imposs.
<lb/>[<emph style="it">tr:
Circle completely outside the lines; impossible.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s148" xml:space="preserve">
4. [???] circulus extra <emph style="super">lineas parallel.</emph>
sed pars periphæriæ <lb/>
intra. similiter. <lb/>
4. <lb/>[<emph style="it">tr:
4 cases: circle outside parallel lines but part of the circumference inside, similarly.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s149" xml:space="preserve">
4. [???] circuli extra lineas inclinationes <lb/>
sed pars periphæriæ intra. Similiter. <lb/>
4.
<lb/>[<emph style="it">tr:
4 cases: circle outside inclined lines but part between the circumferences, similarly.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s150" xml:space="preserve">
6. 6. ad easdem partes lineæ <lb/>
punctum extra circulum et totus circulus extra <lb/>
2. quæs extra. 1. <lb/>
2. quæs. intra. 1. <lb/>
pars <emph style="super">minor</emph> <emph style="super">major</emph> circuli <lb/>
2. quæs extra. 1. <lb/>
2. quæs. extra. 1. <lb/>
punctum intra circuli dati <lb/>
ad easdem partes <emph style="super">lineæ</emph> <lb/>
cum centro circuli. <lb/>
2. quæs. intra. 1. <lb/>
ad contrarias partes <lb/>
2. quæs extra. 1.
<lb/>[<emph style="it">tr:
Problem VI. <lb/>
6 cases: on the same side of the line, point outside circle and total circle outside <lb/>
2 sought outside, 1 case <lb/>
2 sought inside, 1 case <lb/>
lesser or greater part of the circle <lb/>
2 sought outside, 1 case <lb/>
2 sought inside, 1 case <lb/>
point between the given circles, on the same side of the line as the centre of the circle <lb/>
2 sought inside, 1 case <lb/>
on opposite sides <lb/>
2 sought outside, 1 case
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s151" xml:space="preserve">
punctum in linea. 3.
<lb/>[<emph style="it">tr:
3 cases: point in the circumference
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s152" xml:space="preserve">
punctum in periferia. 3.
<lb/>[<emph style="it">tr:
3 cases: point in the circumference
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s153" xml:space="preserve">
1. punctum et circulus <lb/>
totus contrarias partes lineæ. Impossibilis.
<lb/>[<emph style="it">tr:
1 case: point and total circle on opposite sides of the line; impossible.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f072v" o="72v" n="144"/>
<pb file="add_6785_f073" o="73" n="145"/>
<head xml:id="echoid-head28" xml:space="preserve" xml:lang="lat">
1.)	Anguli trichotomia
<lb/>[<emph style="it">tr:
Trisection of angles
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s154" xml:space="preserve">
Deliniatio <emph style="super">completa</emph> pro <lb/>
3<emph style="super">a</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, est in 2<emph style="super">a</emph> charta.
<lb/>[<emph style="it">tr:
A complete delineation for the third <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is in the 2nd sheet.
</emph>]<lb/>
[<emph style="it">Note:
The second sheet is Add MS 6785, f. 74.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s155" xml:space="preserve">
Sint quatuor continue proportionales <lb/>
<lb/>[...]<lb/> <lb/>
erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> triplex.
<lb/>[<emph style="it">tr:
Let there be four continued proportionals. <lb/>
<lb/>[...]<lb/> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> will be threefold.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s156" xml:space="preserve">
Si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>g</mi></mstyle></math> fit latus trianguli <lb/>
erit: <lb/>
1, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo></mstyle></math> lateri nonanguli <lb/>
unius circuli. <lb/>
2, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo></mstyle></math> lateri nonanguli <lb/>
duarum revoluti<lb/>
onum. <lb/>
3, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mo>=</mo></mstyle></math> lateri nonanguli <lb/>
4<emph style="super">a</emph> revolutionum.
<lb/>[<emph style="it">tr:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>g</mi></mstyle></math> is the side of the triangle, then: <lb/>
1. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is the side of a ninth of an angle of one circle.
2. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is the side of a ninth of an angle of two revolutions.
3. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is the side of a ninth of an angle of four revolutions.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f073v" o="73v" n="146"/>
<head xml:id="echoid-head29" xml:space="preserve" xml:lang="lat">
Trichotomia meus
<lb/>[<emph style="it">tr:
My own trisection
</emph>]<lb/>
</head>
<pb file="add_6785_f074" o="74" n="147"/>
<head xml:id="echoid-head30" xml:space="preserve" xml:lang="lat">
2.)	Anguli trichotomia
<lb/>[<emph style="it">tr:
Trisection
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s157" xml:space="preserve">
Deliniatio pro <lb/>
3<emph style="super">a</emph> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> quæ in 1<emph style="super">ae</emph> charta.
<lb/>[<emph style="it">tr:
A delineation for the 3rd <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> as in the 1st sheet.
</emph>]<lb/>
[<emph style="it">Note:
The first sheet is Add MS 6765, f. 73.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6785_f074v" o="74v" n="148"/>
<pb file="add_6785_f075" o="75" n="149"/>
<pb file="add_6785_f075v" o="75v" n="150"/>
<pb file="add_6785_f076" o="76" n="151"/>
<pb file="add_6785_f076v" o="76v" n="152"/>
<pb file="add_6785_f077" o="77" n="153"/>
<pb file="add_6785_f077v" o="77v" n="154"/>
<pb file="add_6785_f078" o="78" n="155"/>
<pb file="add_6785_f078v" o="78v" n="156"/>
<pb file="add_6785_f079" o="79" n="157"/>
<div xml:id="echoid-div35" type="page_commentary" level="2" n="35">
<p>
<s xml:id="echoid-s158" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s158" xml:space="preserve">
Drawing a regular heptagon inside a circle is the subject of Chapter VII of Viète's
<emph style="it">Variorum responsorum liber VII</emph> (1593). Harriot knew this book well
but there is no reference to Viete on this page.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head31" xml:space="preserve" xml:lang="lat">
a.) pro later Heptagoni <lb/>
et Angulorum
<lb/>[<emph style="it">tr:
For the sides of a heptagon, and the angles.
</emph>]<lb/>
</head>
<pb file="add_6785_f079v" o="79v" n="158"/>
<pb file="add_6785_f080" o="80" n="159"/>
<pb file="add_6785_f080v" o="80v" n="160"/>
<pb file="add_6785_f081" o="81" n="161"/>
<pb file="add_6785_f081v" o="81v" n="162"/>
<pb file="add_6785_f082" o="82" n="163"/>
<pb file="add_6785_f082v" o="82v" n="164"/>
<pb file="add_6785_f083" o="83" n="165"/>
<div xml:id="echoid-div36" type="page_commentary" level="2" n="36">
<p>
<s xml:id="echoid-s160" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s160" xml:space="preserve">
A generalization of Pascal 19s triangle,showing the results of multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and by 3
(see also Add MS 6782, f. 165). <lb/>
Note that in the third table, the entry <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn><mn>4</mn><mo>,</mo><mi>a</mi></mrow><mrow><mn>1</mn><mn>2</mn></mrow></mfrac></mstyle></math> on the diagonal, for example,
is to be read as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>3</mn><mo>×</mo><mn>4</mn><mi>a</mi></mrow><mrow><mn>1</mn><mo>×</mo><mn>2</mn></mrow></mfrac></mstyle></math> and similarly for the other entries. <lb/>
In the lower part of the page are general formulae for the rows,
similar to those that Harriot derived elsewhere for the standard version of Pascal's triangle. <lb/>
See also page 2 of the 'Magisteria' (Add MS 6782, f. 109). <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s162" xml:space="preserve">
Æquipollentia ex utraque <lb/>
parte diagonalis perse <lb/>
patet.
<lb/>[<emph style="it">tr:
Equality (or symmetry) on either side of the diagonal is shown by this
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f083v" o="83v" n="166"/>
<div xml:id="echoid-div37" type="page_commentary" level="2" n="37">
<p>
<s xml:id="echoid-s163" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s163" xml:space="preserve">
See page 1 of the 'Magisteria' (Add MS 6782, f. 108). <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f084" o="84" n="167"/>
<pb file="add_6785_f084v" o="84v" n="168"/>
<div xml:id="echoid-div38" type="page_commentary" level="2" n="38">
<p>
<s xml:id="echoid-s165" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s165" xml:space="preserve">
A general form of Pascal 19s triangle,
in which each entry is the sum of the entry above it and the entry to the left of it.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f085" o="85" n="169"/>
<pb file="add_6785_f085v" o="85v" n="170"/>
<pb file="add_6785_f086" o="86" n="171"/>
<pb file="add_6785_f086v" o="86v" n="172"/>
<pb file="add_6785_f087" o="87" n="173"/>
<div xml:id="echoid-div39" type="page_commentary" level="2" n="39">
<p>
<s xml:id="echoid-s167" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s167" xml:space="preserve">
On this page, Harriot works on Propositions 12 and 13 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
Proposition 12 is mentioned explicitly at the top of the page.
The work continues with Proposition 13 below the dividing line.
</s>
<lb/>
<quote xml:lang="lat">
Propositio XII. <lb/>
Data media trium proportionalium et differentia extremarum, invenire extremas.
</quote>
<lb/>
<quote>
Given three proportionals and the difference of the extremes, to find the extremes.
</quote>
<lb/>
<quote xml:lang="lat">
Propositio XII. <lb/>
Data media trium proportionalium &amp; adgregato extremarum, invenire extremas.
</quote>
<lb/>
<quote>
Given three proportionals and the sum of the extremes, to find the extremes.
</quote>
<lb/>
<s xml:id="echoid-s168" xml:space="preserve">
In both of these propositions, Viète showed how the standard construction for three proportionals
can lead to the given equation.
Harriot works the other way round: beginning from an equation,
he gives a construction that represents the same relationship geometrically.
This is what he means by 'effectio æquationis' or 'the construction of an equation'.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head32" xml:space="preserve" xml:lang="lat">
In Effectiones Geometricas. prop. 12 ex 9 et 10
<lb/>[<emph style="it">tr:
From Effectiones Geometricas, Proposition XII, from pages 9 and 10.
</emph>]<lb/>
</head>
<p xml:lang="">
<s xml:id="echoid-s170" xml:space="preserve">
Data media trium proportionalium et differentia extremarum: invenire <lb/>
extremas.
<lb/>[<emph style="it">tr:
Given the mean of three proportionals and thd difference of the extremes, find the extremes.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s171" xml:space="preserve">
Data. <lb/>
Media. <lb/>
Differentia. <lb/>
Quæsita.
<lb/>[<emph style="it">tr:
Given. <lb/>
Mean. <lb/>
Difference. <lb/>
Sought.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s172" xml:space="preserve">
Data Media trium proportionalium et <lb/>
aggregato extremarum: invenire extremas.
<lb/>[<emph style="it">tr:
Given the mean of three proportionals and the sum of the extremes, find the extremes.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s173" xml:space="preserve">
Data. <lb/>
Media. <lb/>
Adgreg. <lb/>
Quæsita.
<lb/>[<emph style="it">tr:
Given. <lb/>
Mean. <lb/>
Sum. <lb/>
Sought.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s174" xml:space="preserve">
Methodus ad exhibenda quæsita <lb/>
in numeris. <lb/>
Dimidium <lb/>
Subtrahe <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>3</mn><mn>6</mn></mrow></msqrt></mstyle></math> id est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>F</mi></mstyle></math> <lb/>
vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>I</mi></mstyle></math>. pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>I</mi></mstyle></math>.
Adde pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>H</mi></mstyle></math> <lb/>
Multiplica <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>H</mi></mstyle></math> <lb/>
per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>I</mi></mstyle></math>. et erit <lb/>
Hoc est. <lb/>
Cuius radix. <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> minus
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>D</mi></mstyle></math> est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>C</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn></mstyle></math>. prima proportionalis. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> plus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn></mstyle></math>. tertia proportionalis.
<lb/>[<emph style="it">tr:
A method of showing the sought quantities in numbers. <lb/>
Halve. <lb/>
Subtract <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>3</mn><mn>6</mn></mrow></msqrt></mstyle></math>, that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>F</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>I</mi></mstyle></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>I</mi></mstyle></math>. <lb/>
Add for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>H</mi></mstyle></math> <lb/>
Multiply <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>H</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>I</mi></mstyle></math> and it will be <lb/>
That is <lb/>
Whose root is <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>6</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>) minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>D</mi></mstyle></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>I</mi><mi>D</mi></mstyle></math>)
is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>C</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>4</mn></mstyle></math>, the first proportional. <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> plus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>5</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn></mstyle></math>, the third proportional.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s175" xml:space="preserve">
Brevius. <lb/>
Et est accurate <lb/>
modus antiquus.
<lb/>[<emph style="it">tr:
More briefly. <lb/>
And it is precisely the ancient method.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s176" xml:space="preserve">
Poste. <lb/>
Etsi modus operandi videtur specie quadam differe antiquo <lb/>
consideranti tamen, et operanti per commpendium; est omnino eadem.
<lb/>[<emph style="it">tr:
Postscript. <lb/>
Although the mode of operation seems in certain respects to differ from the ancient way,
nevertheless examined, and carried out more briefly, it is exactly the same.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f087v" o="87v" n="174"/>
<div xml:id="echoid-div40" type="page_commentary" level="2" n="40">
<p>
<s xml:id="echoid-s177" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s177" xml:space="preserve">
The note from the bottom of Add MS 6785, f. 174 is written again more legibly here on the reverse of the page.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="">
<s xml:id="echoid-s179" xml:space="preserve">
* Etsi modus operandi videtur funde quadam differe antiquo <lb/>
Consideranti tamen, et operanti per commpendium, est omnino eadem.
<lb/>[<emph style="it">tr:
Although the mode of operation seems in certain respects to differ from the ancient way,
nevertheless examined, and carried out more briefly, it is exactly the same.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f088" o="88" n="175"/>
<div xml:id="echoid-div41" type="page_commentary" level="2" n="41">
<p>
<s xml:id="echoid-s180" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s180" xml:space="preserve">
This page gives the three standard cases of quadratic equations
(<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>-</mo><mn>2</mn><mi>c</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math>),
and shows how they relate to divisions of a line into given ratios.
The first case is the subject of Propositions IX and XII from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593). <lb/>
The second case is the subject of Propositions X and XIII. <lb/>
The third case is the subject of Proposition XI. <lb/>
Note on this page two different words used for demonstrating the nature of an equation:
'exegesis' when the equation is written in general notation,
but 'effectio' when it is represented by a geometric construction.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head33" xml:space="preserve" xml:lang="lat">
De effectionibus
<lb/>[<emph style="it">tr:
On constructions
</emph>]<lb/>
</head>
<p xml:lang="">
<s xml:id="echoid-s182" xml:space="preserve">
extrema et <lb/>
media ratio
<lb/>[<emph style="it">tr:
extreme and mean ratio
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s183" xml:space="preserve">
De exegesi per species <lb/>
et per effectiones arith. geomet.
<lb/>[<emph style="it">tr:
On showing [the solution] in general form and by arithmetic or geometric construction.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f088v" o="88v" n="176"/>
<pb file="add_6785_f089" o="89" n="177"/>
<pb file="add_6785_f089v" o="89v" n="178"/>
<pb file="add_6785_f090" o="90" n="179"/>
<pb file="add_6785_f090v" o="90v" n="180"/>
<pb file="add_6785_f091" o="91" n="181"/>
<pb file="add_6785_f091v" o="91v" n="182"/>
<div xml:id="echoid-div42" type="page_commentary" level="2" n="42">
<p>
<s xml:id="echoid-s184" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s184" xml:space="preserve">
This page contains some rough work on Proposition 16 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Data prima trium proportionalium, &amp; ea cujus quadratum æquale est
adgregato quadratorum secundæ &amp; tertiæ, dantur secunda &amp; tertia.
</quote>
<lb/>
<quote>
Given the first of three proportionals and that quantity whose square is equal to
the sum of the squares of the second and third, the second and third are given.
</quote>
<lb/>
<s xml:id="echoid-s185" xml:space="preserve">
Harriot's diagram is a partial copy of Viète's.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head34" xml:space="preserve" xml:lang="lat">
In 16. p. effectionum
<lb/>[<emph style="it">tr:
From page 16 of the Effectionum
</emph>]<lb/>
</head>
<p xml:lang="">
<s xml:id="echoid-s187" xml:space="preserve">
Proportionales. Etiam proportionales.
<lb/>[<emph style="it">tr:
Proportionals. Also proportionals.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s188" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> est differentia inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>B</mi></mstyle></math>. <lb/>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>E</mi></mstyle></math> est media proportinonalis data. <lb/>
Hoc est: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> est prima proportionalis, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>G</mi><mi>E</mi></mstyle></math> ea cuius quadratum est <lb/>
æquale adgregatum quadratorum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>G</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math>. <lb/>
Ita ut ista proportio per interpretationem est vale cum 12<emph style="super">a</emph>.
<lb/>[<emph style="it">tr:
Proportionals. Also proportionals.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f092" o="92" n="183"/>
<pb file="add_6785_f092v" o="92v" n="184"/>
<pb file="add_6785_f093" o="93" n="185"/>
<pb file="add_6785_f093v" o="93v" n="186"/>
<pb file="add_6785_f094" o="94" n="187"/>
<div xml:id="echoid-div43" type="page_commentary" level="2" n="43">
<p>
<s xml:id="echoid-s189" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s189" xml:space="preserve">
The problem on this page is from Propositions 12 and 14 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
Harriot does not mention the <emph style="it">Effectionum</emph> explicitly here
but the notation is essentially Viète's, except reduced to lower case letters.
</s>
<lb/>
<s xml:id="echoid-s190" xml:space="preserve">
Note Harriot's use of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>=</mo></mstyle></math> symbol for what we now write as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>±</mo></mstyle></math>. <lb/>
Note also that once he has arrived at an equation, he regards the problem as solved.
The rest is merely 'mechanicen', or practical calculation.
</s>
<lb/>
<s xml:id="echoid-s191" xml:space="preserve">
Harriot's source for the method of Diophantus must have been the edition of Wilhelm Xylander,
<emph style="it">Diophanti Alexandrini rerum arithmeticarum libri sex</emph> (1575).
Mahomet was by now all that was remembered of the name of Muhammad ibn Musa al-Khwarizmi.
His name appears in this form in Bombelli's <emph style="it">Algebra</emph> (1572), for example.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head35" xml:space="preserve" xml:lang="lat">
Data media trium proportionalium et differentia extremarum: <lb/>
invenire extremas.
<lb/>[<emph style="it">tr:
Given the mean of three proportionals and thd difference of the extremes, find the extremes.
</emph>]<lb/>
</head>
<p xml:lang="">
<s xml:id="echoid-s193" xml:space="preserve">
Sit media data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. et differentia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/>
Et ponatur unus terminus ignotus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
Ergo alter erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mi>b</mi></mstyle></math>. hoc est <lb/>
vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>+</mo><mi>b</mi></mstyle></math>. <lb/>
Ergo: Resolutio. <lb/>
Ergo per Mechanicen. <lb/>
Hoc est Minor. <lb/>
Maior.
<lb/>[<emph style="it">tr:
Let the given mean be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> and the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, and denote one of the unkonwn extrmes by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
Therefore the other will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>±</mo><mi>b</mi></mstyle></math>, that is, either <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>-</mo><mi>b</mi><mn>4</mn><mi>o</mi><mi>r</mi></mstyle></math> a + b <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>.</mo></mstyle></math><lb/>
Hence the solution. <lb/>
Therefore by calculation <lb/>
That is, the lesser extreme <lb/>
the greater
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s194" xml:space="preserve">
Et emitandas applicationes in fine mechanicas, melius est <lb/>
ponere vel notare in principio, dimidium differentiæ <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. tum differentia <lb/>
tota erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>,</mo><mi>c</mi></mstyle></math>. <lb/>
Ergo per resolutionem <lb/>
solutio fit <lb/>
Ergo per Mechanicen
<lb/>[<emph style="it">tr:
To force out the divisions in the final calculation, it is better to put or denote from the beginning
half the difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, then the total difference will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi></mstyle></math>.
Hence from the solution, the equation will be. <lb/>
Therefore by calculation.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s195" xml:space="preserve">
Mechanicum secundum Diophantum <lb/>
et Mahometen. <lb/>
Adde utrique parte æquationis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. <lb/>
<lb/>[...]<lb/> <lb/>
Et per Anithesin.
<lb/>[<emph style="it">tr:
Calculation according to Diohantus and Mahomet. <lb/>
Add <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math> to each side of the equation. <lb/>
<lb/>[...]<lb/> <lb/>
And by antithesis.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f094v" o="94v" n="188"/>
<pb file="add_6785_f095" o="95" n="189"/>
<pb file="add_6785_f095v" o="95v" n="190"/>
<pb file="add_6785_f096" o="96" n="191"/>
<div xml:id="echoid-div44" type="page_commentary" level="2" n="44">
<p>
<s xml:id="echoid-s196" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s196" xml:space="preserve">
This page contains further work on Propostion 12 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
At the end, Harriot makes the same observation as on Add MS 6785, f. 94,
that the method of solving the equation is essentially the same as the 'ancient' method,
that is, the traditional numerical method taught in every algebra text.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head36" xml:space="preserve" xml:lang="lat">
Alia operatio per <emph style="st">[???]</emph> <emph style="super">solam</emph>
proportionem [???] ad illam [???] quod <lb/>
prop. 12. effectionum geometricarum facit:
<lb/>[<emph style="it">tr:
Another method using a single proportion [???] to that [???] done in
Proposition 12 of the <emph style="it">Effectionum geomtericarum</emph>.
</emph>]<lb/>
</head>
<p xml:lang="">
<s xml:id="echoid-s198" xml:space="preserve">
Dico quod: <lb/>
Nam: <lb/>
per const: <lb/>
et per invers: <lb/>
<lb/>[...]<lb/>
<lb/>[<emph style="it">tr:
I say that: <lb/>
For: <lb/>
By constrcution <lb/>
And by inversion
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s199" xml:space="preserve">
A lemma. <lb/>
Ergo. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, vel maior, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/>
mminor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/>
æqualis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>h</mi></mstyle></math> maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>; et <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>h</mi></mstyle></math> maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>a</mi></mstyle></math> et <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>h</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>h</mi></mstyle></math> maior, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>a</mi></mstyle></math>. <lb/>
quod contra hypothesin <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> non maior <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>h</mi></mstyle></math> minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>; et <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>h</mi></mstyle></math> minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>a</mi></mstyle></math>, et <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>h</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>h</mi></mstyle></math>, minor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>a</mi></mstyle></math>. <lb/>
quod contra hypothesin <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> non minor <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
Ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>=</mo><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
A. Lemma. <lb/>
Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is either greater than, less than, or equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
Suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>;
then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>a</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>h</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>h</mi></mstyle></math> is greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo></mstyle></math> 2ca <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo><mi>w</mi><mi>h</mi><mi>i</mi><mi>c</mi><mi>h</mi><mi>i</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>s</mi><mi>t</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>h</mi><mi>y</mi><mi>p</mi><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>s</mi><mi>i</mi><mi>s</mi><mo>.</mo></mstyle></math><lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is not greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
Suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>;
then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>a</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>h</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>h</mi></mstyle></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mo>+</mo></mstyle></math> 2ca <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>,</mo><mi>w</mi><mi>h</mi><mi>i</mi><mi>c</mi><mi>h</mi><mi>i</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>s</mi><mi>t</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>h</mi><mi>y</mi><mi>p</mi><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>s</mi><mi>i</mi><mi>s</mi><mo>.</mo></mstyle></math><lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> is not less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mo>=</mo><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s200" xml:space="preserve">
praxis ista per compendium <lb/>
eadem omnino est cum antiquo
<lb/>[<emph style="it">tr:
The practice of this more briefly is exactly the same as the ancient method.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f096v" o="96v" n="192"/>
<pb file="add_6785_f097" o="97" n="193"/>
<pb file="add_6785_f097v" o="97v" n="194"/>
<pb file="add_6785_f098" o="98" n="195"/>
<pb file="add_6785_f098v" o="98v" n="196"/>
<pb file="add_6785_f099" o="99" n="197"/>
<pb file="add_6785_f099v" o="99v" n="198"/>
<pb file="add_6785_f100" o="100" n="199"/>
<pb file="add_6785_f100v" o="100v" n="200"/>
<pb file="add_6785_f101" o="101" n="201"/>
<p>
<s xml:id="echoid-s201" xml:space="preserve">
To devide a square <lb/>
into two other squares <lb/>
which shall have a ratio geven.
</s>
</p>
<p>
<s xml:id="echoid-s202" xml:space="preserve">
To devide a square into 2 other squares <lb/>
whose sides shall have a ratio geven.
</s>
</p>
<p>
<s xml:id="echoid-s203" xml:space="preserve">
To devide a cube into other cubes <lb/>
which shall have a ratio geven.
</s>
</p>
<pb file="add_6785_f101v" o="101v" n="202"/>
<pb file="add_6785_f102" o="102" n="203"/>
<pb file="add_6785_f102v" o="102v" n="204"/>
<pb file="add_6785_f103" o="103" n="205"/>
<pb file="add_6785_f103v" o="103v" n="206"/>
<pb file="add_6785_f104" o="104" n="207"/>
<pb file="add_6785_f104v" o="104v" n="208"/>
<pb file="add_6785_f105" o="105" n="209"/>
<pb file="add_6785_f105v" o="105v" n="210"/>
<pb file="add_6785_f106" o="106" n="211"/>
<pb file="add_6785_f106v" o="106v" n="212"/>
<pb file="add_6785_f107" o="107" n="213"/>
<head xml:id="echoid-head37" xml:space="preserve" xml:lang="lat">
f.)	Effectiones geometricæ <lb/>
Quadrata in specie
<lb/>[<emph style="it">tr:
Geometrical constructions <lb/>
Squares in general notation
</emph>]<lb/>
</head>
<head xml:id="echoid-head38" xml:space="preserve" xml:lang="lat">
Multiplicatio in radicalibus
<lb/>[<emph style="it">tr:
Multiplication in radicals
</emph>]<lb/>
</head>
<pb file="add_6785_f107v" o="107v" n="214"/>
<pb file="add_6785_f108" o="108" n="215"/>
<div xml:id="echoid-div45" type="page_commentary" level="2" n="45">
<p>
<s xml:id="echoid-s204" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s204" xml:space="preserve">
On this page, Harriot works on the first part of Proposition 14 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Propositio XIV. <lb/>
Quadratum Ã  media proportionali inter hypotenusam trianguli rectanguli &amp; perpendiculum ejusdem,
proportionale est inter quadratum perpendiculi &amp; quadratum idem perpendiculi continuatum basis quadrato.
</quote>
<lb/>
<quote>
The square of the mean proportional between the hypotenuse of a right-angled triangle and its perpendicular,
is the proportional between the square of the perpendicular and the square of the same perpendicular
together with the square of the base.
</quote>
<lb/>
<s xml:id="echoid-s205" xml:space="preserve">
Viète demonstrated this proposition geometrically and showed that it can be represented by the quartic equation
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>A</mi><mn>4</mn></msup></mrow><mo>+</mo><mrow><msup><mi>B</mi><mn>2</mn></msup></mrow><mrow><msup><mi>A</mi><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mi>D</mi><mn>4</mn></msup></mrow></mstyle></math> (in modern notation), where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is the perpendicular, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> the base, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> the mean.
As in the earlier pages in this set, Harriot works the other way round,
beginning from the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> and then deriving the corresponding construction.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head39" xml:space="preserve" xml:lang="lat">
g.)	Effectiones geometricæ
<lb/>[<emph style="it">tr:
Geometrical constructions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s207" xml:space="preserve">
1) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> <lb/>
Et intelligatur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr:
1) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mn>2</mn><mi>c</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math>; and it may be understood that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s208" xml:space="preserve">
Notatio pro effectione geometrica
<lb/>[<emph style="it">tr:
Notation for the geometric construction
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f108v" o="108v" n="216"/>
<pb file="add_6785_f109" o="109" n="217"/>
<div xml:id="echoid-div46" type="page_commentary" level="2" n="46">
<p>
<s xml:id="echoid-s209" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s209" xml:space="preserve">
On this page, Harriot works on the second part of Proposition 14 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Propositio XIV. <lb/>
Idem quadratum Ã  media proportionali inter hypotenusam trianguli rectanguli &amp; perpendiculum ejusdem,
proportionale est inter quadratum hpotenusæ &amp; quadratum idem hypotenusæ multatum basis quadrato.
</quote>
<lb/>
<quote>
The square of the mean proportional between the hypotenuse of a right-angled triangle and its perpendicular,
is the proportional between the square of the hypotenuse and the square of the same hypotenuse
minus the square of the base.
</quote>
<lb/>
<s xml:id="echoid-s210" xml:space="preserve">
Viète demonstrated this proposition geometrically and showed that it can be represented by the quartic equation
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>A</mi><mn>4</mn></msup></mrow><mo>-</mo><mrow><msup><mi>B</mi><mn>2</mn></msup></mrow><mrow><msup><mi>A</mi><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mi>D</mi><mn>4</mn></msup></mrow></mstyle></math> (in modern notation), where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is the hypotenuse, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> the base, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> the mean.
As in the earlier pages in this set, Harriot works the other way round,
beginning from the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> and then deriving the corresponding construction.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head40" xml:space="preserve" xml:lang="lat">
h.)	Effectiones geometricæ
<lb/>[<emph style="it">tr:
Geometrical constructions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s212" xml:space="preserve">
2.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>2</mn><mi>c</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> <lb/>
Et intelligatur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr:
2.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mn>2</mn><mi>c</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math>; and it may be understood that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s213" xml:space="preserve">
Notatio pro effectione geometrica
<lb/>[<emph style="it">tr:
Notation for the geometric construction
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f109v" o="109v" n="218"/>
<pb file="add_6785_f110" o="110" n="219"/>
<div xml:id="echoid-div47" type="page_commentary" level="2" n="47">
<p>
<s xml:id="echoid-s214" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s214" xml:space="preserve">
On this page, Harriot works on Proposition 15 from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Propositio XV. <lb/>
Quadratum Ã  media proportionali inter basin trianguli rectanguli &amp; perpendiculum ejusdem,
proportionale est inter quadratum basi, &amp; quadratum hypotenusae multatum ipso basis quadrato.
Vel etiam inter quadratum perpendiculi, &amp; quadratum hypotenusae multatum ipso perpendiculi quadrato.
</quote>
<lb/>
<quote>
The square of the mean proportional between the base of a right-angled triangle and its perpendicular,
is the proportional between the square of the base and the square of the hypotenuse, reduced by the square of the base.
Or also between the square of the perpendicular and the square of the hypotenuse,
reduced by the square of the perpendicular.
</quote>
<lb/>
<s xml:id="echoid-s215" xml:space="preserve">
Viète demonstrated this proposition geometrically and showed that it can be represented by the quartic equation
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>B</mi><mn>2</mn></msup></mrow><mrow><msup><mi>A</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi>A</mi><mn>4</mn></msup></mrow><mo>=</mo><mrow><msup><mi>D</mi><mn>4</mn></msup></mrow></mstyle></math> (in modern notation), where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math> is the base or perpendicular, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> the hypotenuse,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math> the mean.
As in the earlier pages in this set, Harriot works the other way round,
beginning from the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> and then deriving the corresponding construction.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head41" xml:space="preserve" xml:lang="lat">
i.)	Effectiones geometricæ
<lb/>[<emph style="it">tr:
Geometrical constructions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s217" xml:space="preserve">
3.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>,</mo><mi>c</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> <lb/>
Et intelligatur. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr:
3.) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>b</mi><mi>a</mi><mo>-</mo><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math>; and it may be understood that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mo>=</mo><mi>b</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s218" xml:space="preserve">
Notatio pro effectione geometrica
<lb/>[<emph style="it">tr:
Notation for the geometric construction
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f110v" o="110v" n="220"/>
<pb file="add_6785_f111" o="111" n="221"/>
<div xml:id="echoid-div48" type="page_commentary" level="2" n="48">
<p>
<s xml:id="echoid-s219" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s219" xml:space="preserve">
Further work on the two equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>+</mo><mi>b</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi><mi>a</mi><mi>a</mi><mo>-</mo><mi>b</mi><mi>b</mi><mi>a</mi><mi>a</mi><mo>=</mo><mi>d</mi><mi>d</mi><mi>d</mi><mi>d</mi></mstyle></math>,
which appear in Add MS 6785, f. 108 and f. 109, in connection with Proposition XIV from Viète's
<emph style="it">Effectionum geometricarum canonica recensio</emph> (1593).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head42" xml:space="preserve" xml:lang="lat">
k.)
</head>
<pb file="add_6785_f111v" o="111v" n="222"/>
<pb file="add_6785_f112" o="112" n="223"/>
<pb file="add_6785_f112v" o="112v" n="224"/>
<pb file="add_6785_f113" o="113" n="225"/>
<pb file="add_6785_f113v" o="113v" n="226"/>
<pb file="add_6785_f114" o="114" n="227"/>
<pb file="add_6785_f114v" o="114v" n="228"/>
<pb file="add_6785_f115" o="115" n="229"/>
<pb file="add_6785_f115v" o="115v" n="230"/>
<pb file="add_6785_f116" o="116" n="231"/>
<pb file="add_6785_f116v" o="116v" n="232"/>
<pb file="add_6785_f117" o="117" n="233"/>
<pb file="add_6785_f117v" o="117v" n="234"/>
<pb file="add_6785_f118" o="118" n="235"/>
<pb file="add_6785_f118v" o="118v" n="236"/>
<pb file="add_6785_f119" o="119" n="237"/>
<pb file="add_6785_f119v" o="119v" n="238"/>
<pb file="add_6785_f120" o="120" n="239"/>
<head xml:id="echoid-head43" xml:space="preserve" xml:lang="lat">
Suppostiones
<lb/>[<emph style="it">tr:
Suppositions
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s221" xml:space="preserve">
1.) Gradus <emph style="st">circuli est</emph> periphæriæ est, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>6</mn><mn>0</mn></mrow></mfrac></mstyle></math> totus periphæriæ.
<lb/>[<emph style="it">tr:
A degree of a circumference is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>6</mn><mn>0</mn></mrow></mfrac></mstyle></math> of the total circumference.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s222" xml:space="preserve">
2.) Gradus anguli sphærici est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>6</mn><mn>0</mn></mrow></mfrac></mstyle></math> quatuor rectorum sphæricorum.
<lb/>[<emph style="it">tr:
A degree of a spherical angle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>6</mn><mn>0</mn></mrow></mfrac></mstyle></math> of four spherical right angles.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s223" xml:space="preserve">
3.) Gradus superficiei sphæricæ est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>6</mn><mn>0</mn></mrow></mfrac></mstyle></math> totius superficiei sphæricæ, <lb/>
et est figura biangularis, comprehensa duabus
<emph style="super">semi-</emph> periphæris ex <emph style="st">circuli</emph> maximis <lb/>
cuius uterque angulus est gradus anguli.
<lb/>[<emph style="it">tr:
A degree of a spherical surface is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>6</mn><mn>0</mn></mrow></mfrac></mstyle></math> of the total spherical surface,
and is a biangular figure, contained by two maximum semi-circumferences,
in which either angle is the degree of the angle.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s224" xml:space="preserve">
4.) Biangulum [???] est figura biangularis comprehensa duabus <lb/>
semi periphæris <emph style="super">ex</emph> maximis. Et dicitur dari quando unus angulorum <lb/>
datur. Quoniam ut talis angulus ad 360 ita superficies bianguli <lb/>
ad totam superficiei sphæra.
<lb/>[<emph style="it">tr:
A biangle [???] is a biangular figure contained by two maximum semi-circumferences.
And it is said to be given when one of its angles is given.
Because as such an angle is to 360 degrees, so is the surface of the biangulum to the total surface of the sphere.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s225" xml:space="preserve">
5.) Ex demonstratis Archimedæis, superficies sphæræ est æqualis illa <lb/>
circulo plano cuius semidiameter est sphæræ diameter &amp;c.
<lb/>[<emph style="it">tr:
From the demonstration of Archimedes, that the surface of a sphere is equal to that of a plane circle
whose semidiameter is the diameter of the sphere.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f120v" o="120v" n="240"/>
<pb file="add_6785_f121" o="121" n="241"/>
<pb file="add_6785_f121v" o="121v" n="242"/>
<pb file="add_6785_f122" o="122" n="243"/>
<pb file="add_6785_f122v" o="122v" n="244"/>
<pb file="add_6785_f123" o="123" n="245"/>
<pb file="add_6785_f123v" o="123v" n="246"/>
<p xml:lang="lat">
<s xml:id="echoid-s226" xml:space="preserve">
Chymica <lb/>
quære
</s>
</p>
<pb file="add_6785_f124" o="124" n="247"/>
<pb file="add_6785_f124v" o="124v" n="248"/>
<pb file="add_6785_f125" o="125" n="249"/>
<pb file="add_6785_f125v" o="125v" n="250"/>
<p>
<s xml:id="echoid-s227" xml:space="preserve">
James
</s>
</p>
<pb file="add_6785_f126" o="126" n="251"/>
<pb file="add_6785_f126v" o="126v" n="252"/>
<pb file="add_6785_f127" o="127" n="253"/>
<head xml:id="echoid-head44" xml:space="preserve">
1.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s228" xml:space="preserve">
Memoranda ut quaeratur <lb/>
An Bombellicæ æquationes possunt solvi per numeros triangulates.
<lb/>[<emph style="it">tr:
Note and query: whether Bombelli's equations can be solved by triangulated numbers
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f127v" o="127v" n="254"/>
<pb file="add_6785_f128" o="128" n="255"/>
<pb file="add_6785_f128v" o="128v" n="256"/>
<pb file="add_6785_f129" o="129" n="257"/>
<pb file="add_6785_f129v" o="129v" n="258"/>
<pb file="add_6785_f130" o="130" n="259"/>
<pb file="add_6785_f130v" o="130v" n="260"/>
<pb file="add_6785_f131" o="131" n="261"/>
<pb file="add_6785_f131v" o="131v" n="262"/>
<pb file="add_6785_f132" o="132" n="263"/>
<pb file="add_6785_f132v" o="132v" n="264"/>
<pb file="add_6785_f133" o="133" n="265"/>
<head xml:id="echoid-head45" xml:space="preserve" xml:lang="lat">
Facere quadrato-quadratum, æquale duobus datis.
<lb/>[<emph style="it">tr:
To make a square-square, equal to two given ones.
</emph>]<lb/>
</head>
<pb file="add_6785_f133v" o="133v" n="266"/>
<pb file="add_6785_f134" o="134" n="267"/>
<div xml:id="echoid-div49" type="page_commentary" level="2" n="49">
<p>
<s xml:id="echoid-s229" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s229" xml:space="preserve">
On this page Harriot examines Proposition V from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
See also Add MS 6785, f. 143.
</s>
<lb/>
<quote xml:lang="lat">
Propositio V. <lb/>
Datis duabus lineis rectis, invenire inter easdem duas medias continue, proportionales.
</quote>
<lb/>
<quote>
Given two straight lines, to find two mean proportionals between them.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head46" xml:space="preserve" xml:lang="lat">
5<emph style="super">ia</emph>. pr. <lb/>
De Inveniendis Duabus Medijs continuÃ² proportionalibus inter datas.
<lb/>[<emph style="it">tr:
5th proposition <lb/>
On finding two mean continued proportionals between given quantities.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s231" xml:space="preserve">
Data <lb/>
constructa <lb/>
parallela <lb/>
quæsita
<lb/>[<emph style="it">tr:
Given <lb/>
Constructed <lb/>
Parallels <lb/>
Sought
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s232" xml:space="preserve">
Species continue propotionales
<lb/>[<emph style="it">tr:
The continued proportionals in general form.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s233" xml:space="preserve">
Resolutiones 1.
<lb/>[<emph style="it">tr:
Solutions 1.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s234" xml:space="preserve">
Resolutiones 2.
<lb/>[<emph style="it">tr:
Solutions 2.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f134v" o="134v" n="268"/>
<pb file="add_6785_f135" o="135" n="269"/>
<div xml:id="echoid-div50" type="page_commentary" level="2" n="50">
<p>
<s xml:id="echoid-s235" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s235" xml:space="preserve">
On this page Harriot examines Proposition VI from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
See also Add MS 6785, f. 143.
</s>
<lb/>
<quote xml:lang="lat">
Propositio VI. <lb/>
Dato triangulo rectangulo, invenire aliud triangulum rectangulum majus, &amp; aeque altum;
ut quod fit sub differentia basium ipsorum &amp; differentia hypotenusarum,
aequale fit dato cuicumque recti-lineo.
</quote>
<lb/>
<quote>
Given a right-angled triangle, to find another larger right-angled triangle, with equal height,
so that the product of the difference of the bases and the difference of the hypotenuses
is equal to a given rectangle.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head47" xml:space="preserve" xml:lang="lat">
In prop: 6. Supplementi.
<lb/>[<emph style="it">tr:
From proposition 6 of the Supplement.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s237" xml:space="preserve">
Data <lb/>
prima <lb/>
quarta <lb/>
quatuor <lb/>
parallela <lb/>
Quæsita <lb/>
continue proportionales
<lb/>[<emph style="it">tr:
Given <lb/>
first <lb/>
fourth <lb/>
four <lb/>
parallel <lb/>
Sought <lb/>
continued proportionals
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s238" xml:space="preserve">
Conclusio ex inferiore demonstratione
<lb/>[<emph style="it">tr:
Conclusion from the demonstration below.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s239" xml:space="preserve">
Demonstratio per compositionem. <lb/>
Sint primo constructio quatuor proportionales <lb/>
per 5<emph style="super">tam</emph> prop. <lb/>[...]<lb/> Unde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>β</mi></mstyle></math> <lb/>
est æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>λ</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>Z</mi></mstyle></math> parallelæ. Iam fiat <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>D</mi></mstyle></math> æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>β</mi></mstyle></math>. <lb/>
et ducatur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>μ</mi></mstyle></math> <emph style="super">parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>α</mi></mstyle></math></emph>
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>μ</mi></mstyle></math> sit parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>α</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. Ergo angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>β</mi><mi>μ</mi></mstyle></math> æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>A</mi><mi>α</mi></mstyle></math>.
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi><mi>A</mi></mstyle></math>, angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>μ</mi><mi>D</mi><mi>β</mi></mstyle></math>, et tertius angulo tertio. <lb/>
Ergo triangula <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>α</mi><mi>β</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>μ</mi><mi>β</mi></mstyle></math> simila et æqualia. Et producta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>μ</mi></mstyle></math> transibit per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>,
alias <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>α</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>C</mi></mstyle></math> non sunt æquales. Sit producta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>γ</mi></mstyle></math> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. <lb/>
Et ducatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> <emph style="super">parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>γ</mi></mstyle></math></emph>. Sit inde <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>δ</mi></mstyle></math> parallela <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>Z</mi></mstyle></math>.
Ergo anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>δ</mi><mi>E</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi><mi>E</mi></mstyle></math>, æqualis, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>ɛ</mi><mi>γ</mi></mstyle></math>.
et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>ɛ</mi></mstyle></math> æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>H</mi></mstyle></math>. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>E</mi></mstyle></math>. <lb/>
et æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>γ</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>A</mi></mstyle></math>. et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>λ</mi></mstyle></math> æqualis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>ɛ</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>θ</mi></mstyle></math>.
Et quia <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>β</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>D</mi></mstyle></math> æqualis inter parallelas, æqualis etiam <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>λ</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>λ</mi><mi>C</mi></mstyle></math>.
Conclusio igitur <lb/>
facile colligitur et manifesta. vel triplex ut supra.
<lb/>[<emph style="it">tr:
Demonstration by construction. <lb/>
Let there be first constructed four proportionals by the 5th proposition. <lb/>
Whence <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>β</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>λ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>Z</mi></mstyle></math> are parallel. <lb/>
Now construct <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>D</mi></mstyle></math> equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>β</mi></mstyle></math>, and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>μ</mi></mstyle></math> parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>α</mi></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>μ</mi></mstyle></math> is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>α</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>.
Therefore the angule <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>β</mi><mi>μ</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>A</mi><mi>α</mi></mstyle></math>,
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi><mi>A</mi></mstyle></math> to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>μ</mi><mi>D</mi><mi>β</mi></mstyle></math>, and the third angle to the third.
Therefore the triangles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>α</mi><mi>β</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>μ</mi><mi>β</mi></mstyle></math> are similar and qual. <lb/>
And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>μ</mi></mstyle></math> produced will pass through <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi></mstyle></math>,
otherwis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>α</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>C</mi></mstyle></math> are not equal. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>γ</mi></mstyle></math> be produced towars <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. <lb/>
And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>E</mi></mstyle></math> is constrcuted parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>γ</mi></mstyle></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>δ</mi></mstyle></math> be parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>Z</mi></mstyle></math>.
Therefore angles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>δ</mi><mi>E</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi><mi>E</mi></mstyle></math> are equal, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>ɛ</mi><mi>γ</mi></mstyle></math>;
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>ɛ</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>H</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>δ</mi><mi>E</mi></mstyle></math>;
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>γ</mi></mstyle></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>A</mi></mstyle></math>; and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>λ</mi></mstyle></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>ɛ</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>γ</mi><mi>θ</mi></mstyle></math>.
And because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>β</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>D</mi></mstyle></math> are equal between parallels, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>λ</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>λ</mi><mi>C</mi></mstyle></math> are also equal. <lb/>
Therefore the conclusion is easily gathered and shown, or three times, as above.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f135v" o="135v" n="270"/>
<pb file="add_6785_f136" o="136" n="271"/>
<div xml:id="echoid-div51" type="page_commentary" level="2" n="51">
<p>
<s xml:id="echoid-s240" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s240" xml:space="preserve">
On this page Harriot continues to work on Proposition VI from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
See also Add MS 6785, f. 135
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head48" xml:space="preserve" xml:lang="lat">
Prop. 6. Supplementi, in partem posteriorum
<lb/>[<emph style="it">tr:
Proposition 6 of the Supplement, on the last part.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s242" xml:space="preserve">
Hoc est: <lb/>
Quatuor continue proportionalium, quorum <lb/>
prima minima: <lb/>
Quadratum quartæ, minus quadrao primæ <lb/>
Æquatur: <lb/>
Quadrato, compositæ ex quarta <lb/>
et duplæ secundæ, minus quadrato <lb/>
compositæ ex prima et duplo tertiæ. <lb/>
Hæc demonstrantur per <lb/>
compositione altera charta.
<lb/>[<emph style="it">tr:
That is: for four continued proportionals, of which the first is the least,
the square of the fourth minus the square of the first is equal to
the square composed of the fourth and twice the second,
minus the square composed of the first and twice the third. <lb/>
This is demonstrated by construction on the previous sheet.
</emph>]<lb/>
[<emph style="it">Note:
The previous sheet is Add MS 6785, f. 135.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6785_f136v" o="136v" n="272"/>
<pb file="add_6785_f137" o="137" n="273"/>
<pb file="add_6785_f137v" o="137v" n="274"/>
<pb file="add_6785_f138" o="138" n="275"/>
<head xml:id="echoid-head49" xml:space="preserve">
Mesographa <lb/>
Heronis. <lb/>
Philonis Bizantij. <lb/>
Appolonij.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s243" xml:space="preserve">
Cum Annotatione nostra de <lb/>
faciliori praxi.
<lb/>[<emph style="it">tr:
With my annotations for easier practice.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s244" xml:space="preserve">
where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is <lb/>
double to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> <lb/>
as here:
</s>
</p>
<p>
<s xml:id="echoid-s245" xml:space="preserve">
Note: <lb/>
whether these <lb/>
lines be æquall:
</s>
</p>
<p>
<s xml:id="echoid-s246" xml:space="preserve">
then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>h</mi></mstyle></math> would be parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>c</mi></mstyle></math> <lb/>
&amp; the problem performed nearly <lb/>
and also other wayes &amp;c.
</s>
</p>
<p>
<s xml:id="echoid-s247" xml:space="preserve">
Although Eutocius prefereth philo Bizantius his pratice of finding two <lb/>
mean proportionalls Before that of Herons.
</s>
<s xml:id="echoid-s248" xml:space="preserve">
Because the number being devided <lb/>
into small æquall parts, it may <emph style="super">now</emph> easily be seene when <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi><mi>g</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>c</mi></mstyle></math> be æquall, <lb/>
then by often applying of the compasses to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>f</mi></mstyle></math> &amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>e</mi><mi>g</mi></mstyle></math> æquall.
</s>
<s xml:id="echoid-s249" xml:space="preserve">
Yet in <lb/>
my improvement the pratice would be better &amp; more easy thus.
</s>
<s xml:id="echoid-s250" xml:space="preserve">
Let the figures <lb/>
nombring the æquall parts beginne at <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>. &amp; let their numeration runne towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>; <lb/>
&amp; the like from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math> towards <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>.
</s>
<s xml:id="echoid-s251" xml:space="preserve">
Then will the shape of a rectangle or gnomon <lb/>
keepe <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>e</mi></mstyle></math> always at rectangles with the ruler <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>g</mi></mstyle></math>. <emph style="st">and then</emph> and moving <lb/>
the ruler with the gnomon keeping the poynt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> also in the line till you find <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi><mi>g</mi></mstyle></math> æquall; then is that performed <emph style="st">whi</emph> also which they now have <lb/>
&amp; <emph style="st">[???]</emph> <emph style="super">thus</emph>
easier in practice because that æquallity is sooner found <lb/>
<emph style="st">because</emph> the figures go <emph style="super">in</emph> both wayes a like,
which in philoes practice cannot <lb/>
be observed but with <emph style="st">with</emph> as much difficulty almost, if not so much as <lb/>
that of <emph style="st">her</emph> Herons.
</s>
</p>
<pb file="add_6785_f138v" o="138v" n="276"/>
<pb file="add_6785_f139" o="139" n="277"/>
<div xml:id="echoid-div52" type="page_commentary" level="2" n="52">
<p>
<s xml:id="echoid-s252" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s252" xml:space="preserve">
The text referred to here is Bernard Salignac,
<emph style="it">Mesolabii expositio </emph> (1574).
The diagram relates to Proposition 4, on pages 20–22.
Harriot 19s lettering matches Salignac 19s diagrams on pages 21 and 22,
but is more complete and shows construction lines.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head50" xml:space="preserve" xml:lang="lat">
De Mesolabio B. Salignaci
<lb/>[<emph style="it">tr:
On the mesolabium of B. Salignac
</emph>]<lb/>
</head>
<pb file="add_6785_f139v" o="139v" n="278"/>
<pb file="add_6785_f140" o="140" n="279"/>
<pb file="add_6785_f140v" o="140v" n="280"/>
<pb file="add_6785_f141" o="141" n="281"/>
<pb file="add_6785_f141v" o="141v" n="282"/>
<pb file="add_6785_f142" o="142" n="283"/>
<pb file="add_6785_f142v" o="142v" n="284"/>
<pb file="add_6785_f143" o="143" n="285"/>
<div xml:id="echoid-div53" type="page_commentary" level="2" n="53">
<p>
<s xml:id="echoid-s254" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s254" xml:space="preserve">
On this page Harriot examines Proposition V from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
See also Add MS 6785, 134.
</s>
<lb/>
<quote xml:lang="lat">
Propositio V. <lb/>
Datis duabus lineis rectis, invenire inter easdem duas medias continue, proportionales.
</quote>
<lb/>
<quote>
Given two straight lines, to find two mean proportionals between them.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head51" xml:space="preserve" xml:lang="lat">
In 5<emph style="super">am</emph> Supplementi. De medias proportionales inter Datas.
<lb/>[<emph style="it">tr:
From the 5th proposition of the Supplement. On two mean proportionals between given quantities.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s256" xml:space="preserve">
per constructione <lb/>
<lb/>[...]<lb/> <lb/>
lineæ extreiores <lb/>
lineæ interiores <lb/>
Ergo per 4<emph style="super">am</emph> prop <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>K</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>B</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>I</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>. continuæ proportionales.
<lb/>[<emph style="it">tr:
by construction<lb/>
<lb/>[...]<lb/> <lb/>
external lines <lb/>
internal lines <lb/>
Therefore by the 4th proposition, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>K</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi><mi>I</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> are continued proportionals.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f143v" o="143v" n="286"/>
<pb file="add_6785_f144" o="144" n="287"/>
<div xml:id="echoid-div54" type="page_commentary" level="2" n="54">
<p>
<s xml:id="echoid-s257" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s257" xml:space="preserve">
This is the first of a set of twenty-one pages exploring propositions from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
In the <emph style="it">Effectionum geometricarum</emph> (1593),
whic preceded it, Viète gave Euclidean constructions to demonstrate relationships between proportional lines,
and showed that they corresponded to quadratic, or sometimes quartic, equations.
This, however, gave him only a limited range of constructions, or equations,
insufficient for the requirements of the analytic art by which he meant to leave no problem unsolved
(<foreign xml:lang="lat">nulla non problema solvere</foreign>,
<emph style="it">In artem analyticen isagoge</emph>, (1591), final sentence).
The <emph style="it">Supplementum geometricarum</emph> was intended to remedy this shortcoming
(<foreign xml:lang="lat">defecta Geometriæ</foreign>)
by offering constructions that went beyone the limitations of ruler and compass.
Thus the first statement of the book is:
</s>
<lb/>
<quote xml:lang="lat">
A quovis puncto ad duas quavis lineas rectam ducere,
interceptam ab iis præfinito possibili quocumque intersegmento.
</quote>
<lb/>
<quote>
To draw a straight from any point to any two straight lines,
the intercept between them being any possible predefined distance.
</quote>
<lb/>
<s xml:id="echoid-s258" xml:space="preserve">
Such constructions are sometimes knownn as <foreign xml:lang="lat">neusis</foreign> constructions. <lb/>
On this page, Harriot examines Propositions 3 and 4.
</s>
<lb/>
<quote xml:lang="lat">
Propositio III. <lb/>
Si duae lineae rectae Ã  puncto extra circulum eductae ipsum secent,
pars autem exterior primae fit proportionalis inter partem exteriorem secundae &amp; partem interiorem ejusdem:
erit quoque pars exterior secundae proportionalis inter partem exteriorem primae &amp; partem interiorem ejusdem.
</quote>
<lb/>
<quote>
If two straight lines drawn from a point outside a circle cut it in such a way that
the external part of the first is a proportional between the external and internal parts of the second,
the external part of the second will be a proportional between the external and internal parts of the first.
</quote>
<lb/>
<quote xml:lang="lat">
Propositio IV. <lb/>
Si duae lineae rectae Ã  puncto extra circulum eductae ipsum secent
quod autem fit sub partibus exterioribus eductarum, aequale fit ei quod fit sub intertioribus:
exteriores partes permutatim sumptae, erunt continue proportionales inter partes interiors.
</quote>
<lb/>
<quote>
If two straight lines drawn from a point outside a circle cut it,
and moreover the product of the external parts is equal to that of the internal parts,
the external parts taken in turn will be continued proportionals between the internal parts.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head52" xml:space="preserve" xml:lang="lat">
In prop: 3am Supplementi.
<lb/>[<emph style="it">tr:
From the 3rd proposition of the Supplement
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s260" xml:space="preserve">
sunt partes ablatæ a <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math> et in eadem ratione <lb/>
partes reliquæ sunt <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi></mstyle></math> quæ per 19,5 sunt <emph style="super">etiam</emph> in eadem ratione. <lb/>
Ergo ut supra. <lb/>
Ergo si <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math> sit media proportionalis inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>D</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> erit inter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>E</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
the parts are taken from <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math> in the same ratio. <lb/>
the remaining parts are <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>F</mi></mstyle></math>, which by
[Euclid's <emph style="it">Elements</emph>] V.19 are also in the same ratio. <lb/>
Therefore as above. <lb/>
Therefore if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math> is the mean proportional between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>D</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>C</mi></mstyle></math> will be between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>E</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head53" xml:space="preserve" xml:lang="lat">
From the fourth.
<lb/>[<emph style="it">tr:
Proposition 3 of the Supplement
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s261" xml:space="preserve">
continue proportionales ut supra <lb/>
<lb/>[...]<lb/> <lb/>
Et per synæresin <lb/>
Ex ratione constructionis <lb/>
<lb/>[...]<lb/> <lb/>
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>D</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math>. continuæ proportionales.
<lb/>[<emph style="it">tr:
continued proportionals as above <lb/>
<lb/>[...]<lb/> <lb/>
And by synæresis <lb/>
By reason of the construction <lb/>
<lb/>[...]<lb/> <lb/>
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>C</mi><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>E</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mi>F</mi></mstyle></math> are continued proportionals.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f144v" o="144v" n="288"/>
<pb file="add_6785_f145" o="145" n="289"/>
<pb file="add_6785_f145v" o="145v" n="290"/>
<pb file="add_6785_f146" o="146" n="291"/>
<pb file="add_6785_f146v" o="146v" n="292"/>
<pb file="add_6785_f147" o="147" n="293"/>
<pb file="add_6785_f147v" o="147v" n="294"/>
<pb file="add_6785_f148" o="148" n="295"/>
<pb file="add_6785_f148v" o="148v" n="296"/>
<pb file="add_6785_f149" o="149" n="297"/>
<pb file="add_6785_f149v" o="149v" n="298"/>
<pb file="add_6785_f150" o="150" n="299"/>
<pb file="add_6785_f150v" o="150v" n="300"/>
<pb file="add_6785_f151" o="151" n="301"/>
<pb file="add_6785_f151v" o="151v" n="302"/>
<pb file="add_6785_f152" o="152" n="303"/>
<pb file="add_6785_f152v" o="152v" n="304"/>
<pb file="add_6785_f153" o="153" n="305"/>
<div xml:id="echoid-div55" type="page_commentary" level="2" n="55">
<p>
<s xml:id="echoid-s262" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s262" xml:space="preserve">
This page contains symbolic versions of Euclid Book II, Propositions 12 to 14.
These propositions in full are as follows: <lb/>
II.12.In obtuse-angle triangles the square on the side opposite the obtuse angle
is greater than the sum of the squares on the sides containing the obtuse angle
by twice the rectangle contained by one of the sides about the obtuse angle,
namely that on which the perpendicular falls, and the straight line cut off outside
by the perpendicular towards the obtuse angle. <lb/>
II.13. In acute-angled triangles the square on the side opposite the acute angle
is less than the sum of the squares on the sides containing the acute angle
by twice the rectangle contained by one of the sides about the acute angle,
namely that on which the perpendicular falls, and the straight line cut off within
by the perpendicular towards the acute angle. <lb/>
II.14. To construct a square equal to a given rectilinear figure.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head54" xml:space="preserve" xml:lang="lat">
d)	propositiones 2<emph style="super">i</emph> Euclidis
<lb/>[<emph style="it">tr:
Propositions from the second book of Euclid
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s264" xml:space="preserve">
Invenire quod:
<lb/>[<emph style="it">tr:
To show that:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s265" xml:space="preserve">
pro.12
<lb/>[<emph style="it">tr:
Proposition 12
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s266" xml:space="preserve">
Invenire quod:
<lb/>[<emph style="it">tr:
To show that:
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s267" xml:space="preserve">
p.13
<lb/>[<emph style="it">tr:
Proposition 13
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s268" xml:space="preserve">
p.14. et ultima
<lb/>[<emph style="it">tr:
Proposition 14, the last
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s269" xml:space="preserve">
Facere quadratum, æquale rectangulo
<lb/>[<emph style="it">tr:
Make a square equal to the rectangle.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s270" xml:space="preserve">
(cuiuslibet parallelogrammi rectanguli <lb/>
unæqualium laterum: <lb/>
si maius latus ponatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math> <lb/>
minus latus erit, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math>)
<lb/>[<emph style="it">tr:
For any rectangular parallelogram of unequal sides,
if the longer side is supposed <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>+</mo><mi>c</mi></mstyle></math>, the shorter will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s271" xml:space="preserve">
Finis 2<emph style="super">i</emph> libri
<lb/>[<emph style="it">tr:
The end of the second book.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f153v" o="153v" n="306"/>
<pb file="add_6785_f154" o="154" n="307"/>
<div xml:id="echoid-div56" type="page_commentary" level="2" n="56">
<p>
<s xml:id="echoid-s272" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s272" xml:space="preserve">
This page contains a symbolic version of Euclid Book II, Proposition 11.
Harriot observes that this is the same as Book VI, Proposition 30.
These propositions in full are as follows: <lb/>
II.11. To cut a given straight line so that the rectangle contained by the whole
and one of the segments equals the square on the remaining segment. <lb/>
VI.30. To cut a given finite straight line in extreme and mean ratio.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head55" xml:space="preserve" xml:lang="lat">
c)	propositiones 2<emph style="super">i</emph> Euclidis
<lb/>[<emph style="it">tr:
Propositions from the second book of Euclid
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s274" xml:space="preserve">
11.p) est etiam : e.6 p. 30
<lb/>[<emph style="it">tr:
Proposition 11 is also Proposition VI.30
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s275" xml:space="preserve">
Data <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math> <lb/>
Facere <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi><mo>.</mo><mi>A</mi><mi>H</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>.</mo><mi>H</mi><mi>B</mi></mstyle></math>
(hoc est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>,</mo><mi>A</mi><mi>H</mi><mo>:</mo><mi>A</mi><mi>H</mi><mo>,</mo><mi>H</mi><mi>B</mi></mstyle></math> ut e,6. p. 30)
<lb/>[<emph style="it">tr:
Given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, make <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi><mo>.</mo><mi>A</mi><mi>H</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>.</mo><mi>H</mi><mi>B</mi></mstyle></math>. (that is, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>:</mo><mi>A</mi><mi>H</mi><mo>=</mo><mi>A</mi><mi>H</mi><mo>:</mo><mi>H</mi><mi>B</mi></mstyle></math> as in Proposition VI.30.)
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s276" xml:space="preserve">
zetetica
<lb/>[<emph style="it">tr:
zetetic
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f154v" o="154v" n="308"/>
<pb file="add_6785_f155" o="155" n="309"/>
<div xml:id="echoid-div57" type="page_commentary" level="2" n="57">
<p>
<s xml:id="echoid-s277" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s277" xml:space="preserve">
This page contains symbolic versions of Euclid Book II, Propositions 9 and 10.
These propositions in full are as follows: <lb/>
II.9. If a straight line is cut into equal and unequal segments, then the sum of the squares
on the unequal segments of the whole is double the sum of the square on the half
and the square on the straight line between the points of section. <lb/>
II.10. If a straight line is bisected, and a straight line is added to it in a straight line,
then the square on the hole with the added straight line and the square on the added straight line
both together are double the sum of the square on the half and the square described on the straight line
made up of the half and the added straight line as on one straight line.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head56" xml:space="preserve" xml:lang="lat">
b)	propositiones 2<emph style="super">i</emph> Euclidis
<lb/>[<emph style="it">tr:
Propositions from the second book of Euclid
</emph>]<lb/>
</head>
<pb file="add_6785_f155v" o="155v" n="310"/>
<pb file="add_6785_f156" o="156" n="311"/>
<div xml:id="echoid-div58" type="page_commentary" level="2" n="58">
<p>
<s xml:id="echoid-s279" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s279" xml:space="preserve">
This page contains symbolic versions of Euclid Book II, Propositions 1 to 8.
These propositions in full are as follows: <lb/>
II.1. If there are two straight lines, and one of them is cut into any number of segments whatever,
then the rectangle contained by the two straight lines equals the sum of the rectangles contained by
the uncut straight line and each of the segments. <lb/>
II.2. If a straight line is cut at random, then the sum of the rectangles contained by the whole
and each of the segments equals the square on the whole. <lb/>
II.3. If a straight line is cut a random, then the rectangle contained by the whole and one of the segments
equals the sum of the rectangle contained by the segments and the square on the aforesaid segment. <lb/>
II.4. If a straight line is cut at random, the square on the whole equals the square on the segments
plus twice the rectangle contained by the segments. <lb/>
II.5. If a straight line is cut into equal and unequal segments, then the rectangle contained by
the unequal segments of the whole together with the square on the straight line between the points of section
equals the square on the half. <lb/>
II.6. If a straight line is bisected and a straight line is added to it in a straight line,
then the rectangle contained by the whole width with the added straight line
and the added straight line together with the square on the half
equals the square on the straight line made up of the half and the added straight line. <lb/>
II.7. If a straight line is cut at random, then the sum of the square on the whole
and that on one of the segments equals twice the rectangle contained by the hole
and the said segment plus the square on the remaining segment. <lb/>
II.8. If a straight line is cut at random, then four times the rectangle contained by the whole
and one of the segments plus the square on the remaining segment
equals the square described on the whole and the aforesaid segment as on one straight line.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head57" xml:space="preserve" xml:lang="lat">
a)	propositiones 2<emph style="super">i</emph> Euclidis
<lb/>[<emph style="it">tr:
Propositions from the second book of Euclid
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s281" xml:space="preserve">
[<emph style="it">Note:
For the full version of Harriot's verse beginning 'If more by more' verse, see Add MS 6784, f. 321v.
 </emph>]<lb/>
If more <lb/>
by more <lb/>
&amp;c.
</s>
</p>
<pb file="add_6785_f156v" o="156v" n="312"/>
<pb file="add_6785_f157" o="157" n="313"/>
<div xml:id="echoid-div59" type="page_commentary" level="2" n="59">
<p>
<s xml:id="echoid-s282" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s282" xml:space="preserve">
This page contains symbolic versions of Euclid Book II, Propositions 1 to 4.
For full versions of these propositions see the commentary to Add MS 6785, f. 156.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head58" xml:space="preserve" xml:lang="lat">
Euclid. lib. 2
<lb/>[<emph style="it">tr:
Euclid Book II
</emph>]<lb/>
</head>
<pb file="add_6785_f157v" o="157v" n="314"/>
<pb file="add_6785_f158" o="158" n="315"/>
<div xml:id="echoid-div60" type="page_commentary" level="2" n="60">
<p>
<s xml:id="echoid-s284" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s284" xml:space="preserve">
This page contains symbolic versions of Euclid Book II, Propositions 5 to 7.
For full versions of these propositions see the commentary to Add MS 6785, f. 156.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head59" xml:space="preserve" xml:lang="lat">
Euclid. lib. 2
<lb/>[<emph style="it">tr:
Euclid Book II
</emph>]<lb/>
</head>
<pb file="add_6785_f158v" o="158v" n="316"/>
<div xml:id="echoid-div61" type="page_commentary" level="2" n="61">
<p>
<s xml:id="echoid-s286" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s286" xml:space="preserve">
This page contains symbolic versions of Euclid Book II, Propositions 8 to 10.
For full versions of these propositions see the commentary to Add MS 6785, f. 156 and f. 155.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head60" xml:space="preserve" xml:lang="lat">
lib. 2. El.
<lb/>[<emph style="it">tr:
Euclid Book II
</emph>]<lb/>
</head>
<pb file="add_6785_f159" o="159" n="317"/>
<div xml:id="echoid-div62" type="page_commentary" level="2" n="62">
<p>
<s xml:id="echoid-s288" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s288" xml:space="preserve">
This page contains symbolic versions of Euclid Book II, Propositions 12 and 13.
For full versions of these propositions see the commentary to Add MS 6785, f. 153.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head61" xml:space="preserve" xml:lang="lat">
Euclid. lib. 2
<lb/>[<emph style="it">tr:
Euclid Book II
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s290" xml:space="preserve">
prop. 11. problema analytica
<lb/>[<emph style="it">tr:
Proposition 11 is an analytic problem
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s291" xml:space="preserve">
prop. 12. analytice
<lb/>[<emph style="it">tr:
proposition 12 analytically
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s292" xml:space="preserve">
13. analytice
<lb/>[<emph style="it">tr:
proposition 13 analytically
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f159v" o="159v" n="318"/>
<pb file="add_6785_f160" o="160" n="319"/>
<pb file="add_6785_f160v" o="160v" n="320"/>
<pb file="add_6785_f161" o="161" n="321"/>
<pb file="add_6785_f161v" o="161v" n="322"/>
<pb file="add_6785_f162" o="162" n="323"/>
<div xml:id="echoid-div63" type="page_commentary" level="2" n="63">
<p>
<s xml:id="echoid-s293" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s293" xml:space="preserve">
Pascal's triangle, dot patterns for linear and triangular numbers,
and a sketchy attempt to write the numbers of the triangle in general form.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s295" xml:space="preserve">
[???] <lb/>
unum <lb/>
totum <lb/>
idem
<lb/>[<emph style="it">tr:
[???] <lb/>
one <lb/>
all <lb/>
the same
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s296" xml:space="preserve">
a half <lb/>
a third <lb/>
a quarter <lb/>
1 fifth part
</s>
</p>
<pb file="add_6785_f162v" o="162v" n="324"/>
<pb file="add_6785_f163" o="163" n="325"/>
<pb file="add_6785_f163v" o="163v" n="326"/>
<pb file="add_6785_f164" o="164" n="327"/>
<pb file="add_6785_f164v" o="164v" n="328"/>
<pb file="add_6785_f165" o="165" n="329"/>
<pb file="add_6785_f165v" o="165v" n="330"/>
<pb file="add_6785_f166" o="166" n="331"/>
<pb file="add_6785_f166v" o="166v" n="332"/>
<pb file="add_6785_f167" o="167" n="333"/>
<pb file="add_6785_f167v" o="167v" n="334"/>
<pb file="add_6785_f168" o="168" n="335"/>
<pb file="add_6785_f168v" o="168v" n="336"/>
<pb file="add_6785_f169" o="169" n="337"/>
<pb file="add_6785_f169v" o="169v" n="338"/>
<pb file="add_6785_f170" o="170" n="339"/>
<pb file="add_6785_f170v" o="170v" n="340"/>
<pb file="add_6785_f171" o="171" n="341"/>
<pb file="add_6785_f171v" o="171v" n="342"/>
<pb file="add_6785_f172" o="172" n="343"/>
<pb file="add_6785_f172v" o="172v" n="344"/>
<pb file="add_6785_f173" o="173" n="345"/>
<pb file="add_6785_f173v" o="173v" n="346"/>
<pb file="add_6785_f174" o="174" n="347"/>
<pb file="add_6785_f174v" o="174v" n="348"/>
<pb file="add_6785_f175" o="175" n="349"/>
<pb file="add_6785_f175v" o="175v" n="350"/>
<pb file="add_6785_f176" o="176" n="351"/>
<pb file="add_6785_f176v" o="176v" n="352"/>
<pb file="add_6785_f177" o="177" n="353"/>
<pb file="add_6785_f177v" o="177v" n="354"/>
<pb file="add_6785_f178" o="178" n="355"/>
<pb file="add_6785_f178v" o="178v" n="356"/>
<div xml:id="echoid-div64" type="page_commentary" level="2" n="64">
<p>
<s xml:id="echoid-s297" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s297" xml:space="preserve">
Sums of some infinite progressions. <lb/>
The first example is
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>1</mn><mn>6</mn></mrow></mfrac><mo>+</mo><mo>…</mo><mo>=</mo><mn>4</mn></mstyle></math>.
From the similar examples shown on Add MS 6789, f. 44, we may assume that Harriot summed this as follows: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>+</mo><mo>…</mo><mo>=</mo><mn>2</mn></mstyle></math>, <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>+</mo><mo>…</mo><mo>=</mo><mn>1</mn></mstyle></math>, <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>+</mo><mo>…</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math>, <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>…</mo></mstyle></math>.
The sums of these series form a geometric progression
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mo>…</mo><mo>=</mo><mn>4</mn></mstyle></math>. <lb/>
The second example is similar to the first. <lb/>
The third example is
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mn>9</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mn>1</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mn>0</mn></mrow><mrow><mn>8</mn><mn>1</mn></mrow></mfrac><mo>+</mo><mo>…</mo><mo>=</mo><mn>3</mn><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>.
This can be rewritten as the sum of two separate series: <lb>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>9</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>8</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mn>6</mn></mrow><mrow><mn>8</mn><mn>1</mn></mrow></mfrac><mo>+</mo><mo>…</mo></mstyle></math>.
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>9</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><mn>7</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>8</mn><mn>1</mn></mrow></mfrac><mo>+</mo><mo>…</mo></mstyle></math>.
The first is a geometric progression whose sum is 3. <lb/>
The second can be summed as in the first example, to give <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mstyle></math>. </lb>
Thus the total sum is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>. <lb/>
The fourth example is
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mn>3</mn></mrow><mrow><mn>3</mn><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><mn>5</mn></mrow><mrow><mn>2</mn><mn>1</mn><mn>6</mn></mrow></mfrac><mo>+</mo><mo>…</mo></mstyle></math>.
This can be rewritten as the sum of two separate series: <lb>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mn>3</mn><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mn>2</mn><mn>1</mn><mn>6</mn></mrow></mfrac><mo>+</mo><mo>…</mo></mstyle></math>.
and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mn>3</mn><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mn>7</mn></mrow><mrow><mn>2</mn><mn>1</mn><mn>6</mn></mrow></mfrac><mo>+</mo><mo>…</mo></mstyle></math>.
The first is a geometric progression whose sum is 3. <lb/>
The second is a geomteric porgression whose sum is 2. </lb>
Thus the total sum is 5. <lb/>
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f179" o="179" n="357"/>
<div xml:id="echoid-div65" type="page_commentary" level="2" n="65">
<p>
<s xml:id="echoid-s299" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s299" xml:space="preserve">
The text referred to here is Thomas Finck,
<emph style="it">Geometriae rotundi libri XIIII</emph> (1583), pages 1 to 4.
There Finck discusses two propositions of Ptolemy. The second is:
</s>
<lb/>
<s xml:id="echoid-s300" xml:space="preserve">
<emph style="it" xml:lang="lat">
15. Si quatuor rectarum duae faciant angulum, reliquae ab harum terminis in se reflexae priores secent:
ratio unius ad segmentum suum, vel segmentorum inter se fit e ratione ita conterminarum,
ut prima facientium conterminetur antecedentis factae principio,
secunda huius consequentis fini contermina terminetur in finem consequentis factae.
Sint enim quatuor rectae ae. ai. eu. io. quarum duae priores faciant angulum ad a. reliquae
ab harum terminis reflexae secent se in y. atque duae priores in u &amp; o. <lb/>
Dico primo rationem ia ad au esse factam e ratione io ad oy. &amp; ye ad eu.
Qui primus Ptolemaei casus est. <lb/>
Secundo rationem iu ad ua esse factam e ratione iy ad yo. &amp; oe ad ea.
Qui secundus est Ptolemaei casus.
</emph>
</s>
<lb/>
<s xml:id="echoid-s301" xml:space="preserve">
Finck proves these and notes four further variations given by Theon;
these are listed by Harriot as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><mi>e</mi><mi>u</mi></mrow><mrow><mi>u</mi><mi>y</mi></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><mi>a</mi><mi>i</mi></mrow><mrow><mi>i</mi><mi>u</mi></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><mi>u</mi><mi>e</mi></mrow><mrow><mi>e</mi><mi>y</mi></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mrow><mi>e</mi><mi>y</mi></mrow><mrow><mi>y</mi><mi>u</mi></mrow></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head62" xml:space="preserve" xml:lang="lat">
In propositionem ptolomaicum initio Finkij
<lb/>[<emph style="it">tr:
On a proposition of Ptolemy, from the beginning of Finck.
</emph>]<lb/>
</head>
<pb file="add_6785_f179v" o="179v" n="358"/>
<pb file="add_6785_f180" o="180" n="359"/>
<div xml:id="echoid-div66" type="page_commentary" level="2" n="66">
<p>
<s xml:id="echoid-s303" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s303" xml:space="preserve">
The reference on this sheet is to page 35 of Commandino's
<emph style="it">Liber de centro gravitatis solidorum </emph> (1565).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head63" xml:space="preserve" xml:lang="lat">
De frustra pyramedis <lb/>
Vel Coni. Ut Comandinus <lb/>
pag: 35.
<lb/>[<emph style="it">tr:
On the frustrum of a pyramid or cone, as Commandinus, page 35.
</emph>]<lb/>
</head>
<pb file="add_6785_f180v" o="180v" n="360"/>
<pb file="add_6785_f181" o="181" n="361"/>
<pb file="add_6785_f181v" o="181v" n="362"/>
<pb file="add_6785_f182" o="182" n="363"/>
<pb file="add_6785_f182v" o="182v" n="364"/>
<pb file="add_6785_f183" o="183" n="365"/>
<pb file="add_6785_f183v" o="183v" n="366"/>
<pb file="add_6785_f184" o="184" n="367"/>
<div xml:id="echoid-div67" type="page_commentary" level="2" n="67">
<p>
<s xml:id="echoid-s305" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s305" xml:space="preserve">
On this page Harriot investigates Propositions 20 and 21 from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Proposition XX. <lb/>
Constituere triangulum aæquicrurum, ut differntia inter basin &amp; alterum e cruribus fit ad basin,
sicut quadratum cruris ad quadratum compositae ex crure &amp; base.
</quote>
<lb/>
<quote>
To construct an isosceles triangle so that the difference between the base and either of the legs
is to the base as the square of a leg is to the square of the sum of a leg and the base.
</quote>
<lb/>
<quote xml:lang="lat">
Proposition XXI. <lb/>
Si fuerit triangulum aequicrurum, fit autem differentia inter basin &amp; alterum e cruribus ad basin,
sicut quadratum cruris ad quadratum compositæ ex crure &amp; base:
quae a termino basis ducetur ad crus linea recta ipsi cruri æquale, secabit bisariam angulum ad basin.
</quote>
<lb/>
<quote>
If there is an isosceles triangle, and moreover the [ratio of the] difference between the base
and either of the legs, to the base, is equal to the square of the leg to the square of the sum of a leg
and the base, then a line drawn from the [end of the] base to the leg, equal [in length] to that leg,
will bisect the angle at the base.
</quote>
<lb/>
<s xml:id="echoid-s306" xml:space="preserve">
There is a reference to Propostion 19 of the <emph style="it">Supplementum</emph> (see Add MS 6785, f. 186).
There are also several references to propostions from Euclid's <emph style="it">Elements</emph>.
</s>
<lb/>
<quote>
III.20 The angle at the centre of a circle is double the angle at the circumference,
when they have the same part of the circumference for a base.
</quote>
<lb/>
<quote>
III.36 If from a point without a circle two straight lines be drawn to it,
one of which is a tangent to the circle, and the other cuts it;
the rectangle under the whole cutting line and the external segment is equal to the square of the tangent.
</quote>
<lb/>
<quote>
VI.12 To find a fourth proportional to three given lines.
</quote>
<lb/>
<quote>
VI.14 Equal parallograms which have one angle each eaual have the sides about the equal angles reciprocally proportional.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head64" xml:space="preserve">
prop. 20. Supplementi.
<lb/>[<emph style="it">tr:
Proposition 20 from the Supplementum
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s308" xml:space="preserve">
Constituere triangulum aæquicrurum; ut differntia inter basin et alterum <lb/>
e cruribus fit ad basin, sicut quadratum cruris ad quadratum compositæ <lb/>
ex crure et Base.
<lb/>[<emph style="it">tr:
To construct an isosceles triangle so that the difference between the base and either of the legs
is to the base, as the square of a leg is to the square of the sum of a leg and the base.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s309" xml:space="preserve">
per 19,p fiat <lb/>
Et ponatur in circumferentia, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>E</mi><mo>=</mo><mi>A</mi><mi>B</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. <lb/>
Et ducantur recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math> <lb/>
Triangulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi><mi>D</mi></mstyle></math> est quod quærebatur.
<lb/>[<emph style="it">tr:
By Proposition 19, <lb/>
And in the circumference, put <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>E</mi><mo>=</mo><mi>A</mi><mi>B</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>C</mi></mstyle></math>. and constructing the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi></mstyle></math>,
the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>E</mi><mi>D</mi></mstyle></math> is as required.
</emph>]<lb/>
</s>
</p>
<head xml:id="echoid-head65" xml:space="preserve">
prop. 21.
<lb/>[<emph style="it">tr:
Proposition 21
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s310" xml:space="preserve">
Si fuerit triangulum aequicrurum: fit autem differentia inter basin et alterum <lb/>
e cruribus ad basin; sicut quadratum cruris ad quadratum compositæ ex crure et base. <lb/>
Quae a termino basis ducetur ad crus linea recta ipsi cruri æquale: secabit <lb/>
bisariam angulum ad basin.
<lb/>[<emph style="it">tr:
If there is an isosceles triangle, and moreover the [ratio of the] difference between the base
and either of the legs, to the base, is equal to the square of the leg to the square of the sum of a leg
and the base, then a line drawn from the [end of the] base to the leg, equal [in length] to that leg,
will bisect the angle at the base.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s311" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>F</mi></mstyle></math> secat bisariam angulum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
Nam ex hypothesi. <lb/>
<lb/>[...]<lb/> <lb/>
Et per 36,3 el <lb/>
Ergo. per 14, 6 el <lb/>
<lb/>[...]<lb/> <lb/>
Consequenter: <lb/>
Et subducendo <lb/>
Ergo: per 2,6: el: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>C</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>A</mi></mstyle></math> sunt parallelæ <lb/>
Et: Angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>C</mi><mi>D</mi></mstyle></math> æqualis angulo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
Sed. per 20,3: Angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>D</mi></mstyle></math> est duplus anguli <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>C</mi><mi>D</mi></mstyle></math> Hoc est: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>A</mi><mi>D</mi></mstyle></math> <lb/>
Ergo angulus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>D</mi></mstyle></math> sectis est bisariam a recta <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>F</mi></mstyle></math>. <lb/>
Quod erat Demonstrandum.
<lb/>[<emph style="it">tr:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>F</mi></mstyle></math> bisects the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
For from the hypothesis <lb/>
<lb/>[...]<lb/> <lb/>
And by Elements III.36 <lb/>
Therefore by Elements VI.14 <lb/>
<lb/>[...]<lb/> <lb/>
Consequently: <lb/>
And subtracting <lb/>
Therefore, by Elements VI.2, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>C</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>A</mi></mstyle></math> are parallel. <lb/>
And angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>C</mi><mi>D</mi></mstyle></math> is equal to angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
But by Elements III.20, angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>D</mi></mstyle></math> is twice angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>C</mi><mi>D</mi></mstyle></math>, that is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>F</mi><mi>A</mi><mi>D</mi></mstyle></math>. <lb/>
Therefore angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>A</mi><mi>D</mi></mstyle></math> is cut in two by the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>F</mi></mstyle></math>. <lb/>
Which was to be shown.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f184v" o="184v" n="368"/>
<pb file="add_6785_f185" o="185" n="369"/>
<div xml:id="echoid-div68" type="page_commentary" level="2" n="68">
<p>
<s xml:id="echoid-s312" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s312" xml:space="preserve">
On this page Harriot gives a statement and diagram for Proposition 24 from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Proposition XXIV. <lb/>
In dato circulo heptagonum æquilaterum &amp; æquiangulum describere.
</quote>
<lb/>
<quote>
To describe a regular heptagon in a given circle.
</quote>
<lb/>
<s xml:id="echoid-s313" xml:space="preserve">
There are two references to equations found in connection with Proposition 19 of the <emph style="it">Supplementum</emph>;
these are to be found on Add MS 6785, f. 187.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head66" xml:space="preserve">
Explicatio aeqationum quae habentur post 24 propositionem Supplementi.
<lb/>[<emph style="it">tr:
An explanation of the equation to be found after Proposition 19 in the Supplementum
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s315" xml:space="preserve">
Sit triangulum æquicrurum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>N</mi><mi>I</mi></mstyle></math> <lb/>
cuius angulus ad verticem <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> <lb/>
sesquialter est utriusque angulorum <lb/>
ad basim. oportet invenire <lb/>
basis quantitatem <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>A</mi></mstyle></math> in numeris.
<lb/>[<emph style="it">tr:
Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>N</mi><mi>I</mi></mstyle></math> be an isosceles triangle with vertical angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>,
which is one and a half times either angle at the base. <lb/>
There must be found <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>A</mi></mstyle></math>, the length of the base, in numbers.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s316" xml:space="preserve">
In 19<emph style="super">a</emph> propositione, secundum illatum ita est: <lb/>
sit ergo pro basi <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>A</mi></mstyle></math>, nota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>. et pro cruro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>N</mi></mstyle></math> cui æquatur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>. <lb/>
nota <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. et forma æquationis ita erit. <lb/>
Aliter per reductionem. <lb/>
In eadem 19<emph style="it">a</emph> propositione demonstratur ista Analogia: <lb/>
Notatur igitur loco <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>I</mi><mi>D</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>+</mo><mn>3</mn><mi>I</mi><mi>A</mi></mstyle></math>, litera <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math>. Et pro <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> et contra. <lb/>
et analogia ita erit: <lb/>
Ergo resoluta analogia æquatio ita erit:
<lb/>[<emph style="it">tr:
In Proposition 19, the second result is: <lb/>
therfore let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi><mi>A</mi></mstyle></math> be the base, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>N</mi></mstyle></math> the side, which is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math>. <lb/>
And the form of the equation will be: <lb/>
Otherwise, by reduction: <lb/>
In the same Propostion 19, there is demonstrated this ratio: <lb/>
Therefore there may be put the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi></mstyle></math> in place of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>3</mn><mi>I</mi><mi>D</mi></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi><mo>+</mo><mn>3</mn><mi>I</mi><mi>A</mi></mstyle></math>. And <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Z</mi></mstyle></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>B</mi></mstyle></math>, and conversely. <lb/>
And the ratio will be: <lb/>
Therfore, having resolved the ratio, the equation will be:
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f185v" o="185v" n="370"/>
<pb file="add_6785_f186" o="186" n="371"/>
<div xml:id="echoid-div69" type="page_commentary" level="2" n="69">
<p>
<s xml:id="echoid-s317" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s317" xml:space="preserve">
On this page Harriot gives a statement and diagram for Proposition 19 from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593). See also Add MS 6785, f. 187.
</s>
<lb/>
<quote xml:lang="lat">
Proposition XIX. <lb/>
Diametrum circuli ita continuare, ut fit continuatio ad semidiametrum adjunctum continuationi,
sicut quadratum semidiametri ad quadratum continuatae diametri.
</quote>
<lb/>
<quote>
To extend the diameter of a circle so that the extension is to the semidiameter
together with the extension as the square of the semidiameter to the square of the extended diameter.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head67" xml:space="preserve">
prop. 19. Supplementi.
<lb/>[<emph style="it">tr:
Proposition 19 from the Supplementum
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s319" xml:space="preserve">
Diametrum circuli ita continuare, ut fit continuatio ad semidiametrum adjunctum continuationi,
sicut quadratum semidiametri ad quadratum continuatae diametri.
<lb/>[<emph style="it">tr:
To extend the diameter of a circle so that the extension is to the semidiameter
together with the extension as the square of the semidiameter to the square of the extended diameter.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f186v" o="186v" n="372"/>
<pb file="add_6785_f187" o="187" n="373"/>
<div xml:id="echoid-div70" type="page_commentary" level="2" n="70">
<p>
<s xml:id="echoid-s320" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s320" xml:space="preserve">
On this page Harriot investigates Proposition 19 from Viète's
<emph style="it">Supplementum geometriæ</emph> (1593).
</s>
<lb/>
<quote xml:lang="lat">
Proposition XIX. <lb/>
Diametrum circuli ita continuare, ut fit continuatio ad semidiametrum adjunctum continuationi,
sicut quadratum semidiametri ad quadratum continuatae diametri.
</quote>
<lb/>
<quote>
To extend the diameter of a circle so that the extension is to the semidiameter
together with the extension as the square of the semidiameter to the square of the extended diameter.
</quote>
<lb/>
<s xml:id="echoid-s321" xml:space="preserve">
There is a reference to Proposition 10 from the <emph style="it">Supplementum</emph> (see Add MS 6784, f. 354),
and there are also two references to propositions from Euclid's <emph style="it">Elements</emph>.
</s>
<lb/>
<quote>
II.4 If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the parts,
together with twice the rectangle contained by the parts.
</quote>
<lb/>
<quote>
XIII.12
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head68" xml:space="preserve">
prop. 19. Supplementi.
<lb/>[<emph style="it">tr:
Proposition 19 from the Supplementum
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s323" xml:space="preserve">
per 16,p <lb/>
<lb/>[...]<lb/> <lb/>
Ducantur per <lb/>
Dividantur per 3: tum
<lb/>[<emph style="it">tr:
If two triangles are each isosceles, equal to one another in their legs,
By Proposition 16. <lb/>
<lb/>[...]<lb/> <lb/>
Multiplying by <lb/>
Dividing by 3, then:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s324" xml:space="preserve">
per <lb/>
12,13 <lb/>
el
<lb/>[<emph style="it">tr:
by Proposition XIII.12 of the Elements
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s325" xml:space="preserve">
Ista æquatio [???] fit ex æquatione supra <lb/>
Scilicet <lb/>
Ducantibus per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>L</mi></mstyle></math> <lb/>
Itaque per istam, et primam <lb/>
et 10<emph style="super">am</emph> æquationem supra: <lb/>
Primum illatum
<lb/>[<emph style="it">tr:
This equation arises from the equation above, namely: <lb/>
Having multiplied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>L</mi></mstyle></math> <lb/>
Therfore by this, and the first, and the equation of the 10th above: <lb/>
The first result.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s326" xml:space="preserve">
Ducantur per 27. Ergo: <lb/>
Fiat reductio ad <lb/>
analogiam et erunt: <lb/>
per 4,2, el <lb/>
<lb/>[...]<lb/> <lb/>
Ducantur per 9. et erunt: <lb/>
per superiorem analogiam et <lb/>
æquationis erunt: <lb/>
resolutio Anaolgia: erunt: <lb/>
Secundum illatum
<lb/>[<emph style="it">tr:
Multiplying by 27, therefore: <lb/>
Carry out the reduction to the ratio, and then: <lb/>
by Proposition II.4 of the Elements <lb/>
<lb/>[...]<lb/> <lb/>
Multiplying by 9, and then: <lb/>
by the ratio above and the equation: <lb/>
The second result.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s327" xml:space="preserve">
Fiat reductio ad analogiam: et erunt: <lb/>
per 4,2, el <lb/>
Ergo tandem propositum
<lb/>[<emph style="it">tr:
Carry out the reduction of the ratio and then: <lb/>
by Proposition II.4 <lb/>
Therefore finally the proposition.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f187v" o="187v" n="374"/>
<pb file="add_6785_f188" o="188" n="375"/>
<pb file="add_6785_f188v" o="188v" n="376"/>
<pb file="add_6785_f189" o="189" n="377"/>
<pb file="add_6785_f189v" o="189v" n="378"/>
<pb file="add_6785_f190" o="190" n="379"/>
<head xml:id="echoid-head69" xml:space="preserve" xml:lang="lat">
De infinitis
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<head xml:id="echoid-head70" xml:space="preserve" xml:lang="lat">
Ex Linea Quadrataria producta. <lb/>
Consequentiones quædam miranda.
<lb/>[<emph style="it">tr:
From the production of a quadrate line, <lb/>
certain marvellous consequences.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s328" xml:space="preserve">
* Eadem evenient si motus <lb/>
BD sit in maior vel <lb/>
minori ratione.
<lb/>[<emph style="it">tr:
The same hapapens if the motion of BD is in a greater or smaller ratio.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s329" xml:space="preserve">
* Si ponatur BD movendi <lb/>
ad situm CE in spacio unius <lb/>
horæ: ac etiam eodem tempore <lb/>
AB producta movendi ad <lb/>
situm AK per circulationem.
<lb/>[<emph style="it">tr:
If it is supposed that BD moves towards the position of CE in the space of one hour,
and also in the same time that AB produced moves towards the position of AK by circulation.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s330" xml:space="preserve">
Haec consequentur:
<lb/>[<emph style="it">tr:
These things follow:
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s331" xml:space="preserve">
1. Ex communi sectione duarum <lb/>
linearum dictarum sit curva <lb/>
linea infinita BFGH &amp;c. <lb/>
acta designata in termino <lb/>
illius horæ.
<lb/>[<emph style="it">tr:
1. From the point of intersection of the two said lines comes the infnite curved line BFGH etc.
and the path is traced out in one hour.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s332" xml:space="preserve">
2. Illa curva cum linea CE <lb/>
non concurrebat ante terminum <lb/>
horæ: et in ipse termino <lb/>
concurrit: et si dicti motus <lb/>
continuentur, ultra terminum <lb/>
horæ non fit ulterior productio <lb/>
nec sectio.
<lb/>[<emph style="it">tr:
2. The curve and the line CE do not meet before the end of one hour; and at the end they do meet;
and if the said motions are continued, beyond the end of the hour
there will be no further lengthening or cutting.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s333" xml:space="preserve">
3. Eodem instanti scilicet <lb/>
termino horæ, AB mota et <lb/>
in termino motus: tum secat <lb/>
lineam CE: cum habet situ <lb/>
AK scilicet parallelum ad <lb/>
CE.
<lb/>[<emph style="it">tr:
3. In that same instant at the end of the hour, AB is moving and at the end of its motion;
then it cuts the line CE; while of course the position of AK is parallel to CE.
</emph>]<lb/>
</s>
<s xml:id="echoid-s334" xml:space="preserve">
Ita ut re hac racionatione sequitur duas <lb/>
lineas parallelas in infinite distantia secare.
<lb/>[<emph style="it">tr:
Thus by this reasoning it follows that two parallel lines cut at an infinite distance.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s335" xml:space="preserve">
Et hoc mirandum quod <emph style="st">[???]</emph>
<emph style="super">dum</emph> generatur illa curva terminus in acta <lb/>
productionis magis ac magis distantia linea AK, et in termino horæ <lb/>
habet suam maximam distantia ac etiam concursum cum AK et CE.
<lb/>[<emph style="it">tr:
And this is marvellous, that while the end of the curve is generated,
in the act of production it becomes more and more distant from the line AK,
and at the end of the hour has its maximum distance and yet meets with AK and CE.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f190v" o="190v" n="380"/>
<head xml:id="echoid-head71" xml:space="preserve" xml:lang="lat">
De infinitis.
<lb/>[<emph style="it">tr:
On infinity
</emph>]<lb/>
</head>
<head xml:id="echoid-head72" xml:space="preserve" xml:lang="lat">
Quod <emph style="super">quædam</emph> superficies infinitae longitudinis erit æqualis <lb/>
cuisdam finitæ.
<lb/>[<emph style="it">tr:
How a certain surface infinite in length may be equal in length to one finite.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s336" xml:space="preserve">
in infinitis
<lb/>[<emph style="it">tr:
on infinity
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f191" o="191" n="381"/>
<div xml:id="echoid-div71" type="page_commentary" level="2" n="71">
<p>
<s xml:id="echoid-s337" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s337" xml:space="preserve">
The demonstration on this page relies on <emph style="it">Elements</emph>, Book V, Proposition 12: <lb/>
If any number of magnitudes be proportional,
as one of the antecedents is to one of the consequents,
so will all the antecedents be to all the consequents.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head73" xml:space="preserve" xml:lang="lat">
De Continue proportionalibus. Et Infinitis.
<lb/>[<emph style="it">tr:
On continued propotions. An infinity.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s339" xml:space="preserve">
Si fuerint quantitates continue proportionales; Erit ut terminus <lb/>
rationis maior, ad terminum rationis minorem: ita differentia <lb/>
compositæ ex omnibus et minimæ, ad differentiae compositæ ex <lb/>
omnibus et maximæ.
<lb/>[<emph style="it">tr:
If there are quantities in continued proportion:
as the greater term of the ratio is to the lesser term of the ratio,
so will be the difference between the sum and the least
to the difference between the sum and the greatest.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s340" xml:space="preserve">
Sint continue proportionales. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let there be continued proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>g</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s341" xml:space="preserve">
Differentiæ compositæ ex omnibus <lb/>
et minimæ.
<lb/>[<emph style="it">tr:
The difference between the sum and the least.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s342" xml:space="preserve">
Hoc est, omnes antecedentes.
<lb/>[<emph style="it">tr:
That is, all before it.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s343" xml:space="preserve">
Differentæ compositæ ex omnibus <lb/>
et maximæ.
<lb/>[<emph style="it">tr:
The difference between the sum and the greatest.
</emph>]<lb/>
</s>
<s xml:id="echoid-s344" xml:space="preserve">
Hoc est, omnis <lb/>
consequentes, quæ ita sub ante-<lb/>
cedentibus disponantur.
<lb/>[<emph style="it">tr:
That is, all the consequents, which are thus placed under the antecedents.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s345" xml:space="preserve">
Inde manifesta Theorematis demonstratio; quia ut <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> ita (per <lb/>
synthetica) omnes antecedentes ad omnes consequentes. (per 12. pr. li. 5)
<lb/>[<emph style="it">tr:
Thus the demonstration of the theorem is clear; because as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> is to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> so,
by construction, are all the antecedents to all the consequents (by Book 5, Proposition 12).
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s346" xml:space="preserve">
Datis <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, dabitur compositæ ex omnibus <lb/>
quæ significetur per <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Given <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, ..., <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>k</mi></mstyle></math>, there is given the sum of all of them, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s347" xml:space="preserve">
Sit tota composita ducenda per primam <lb/>
quantitatem affirmatam et secundum negatum: <lb/>
facta erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>k</mi></mstyle></math>. cæteræ partes <lb/>
intermediæ factæ eliduntur quoniam æquales <lb/>
affirmatæ et negatæ scilicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>c</mi><mi>b</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let the total be multiplied by the first quantity taken positively and the second taken negatively:
it will make <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>k</mi></mstyle></math>. The other intermediate terms will be destroyed
because of equal positive and negative quantities like <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mi>c</mi><mi>b</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>+</mo><mi>b</mi><mi>c</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s348" xml:space="preserve">
Ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>k</mi></mrow><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfrac><mo>=</mo><mi>a</mi></mstyle></math>. compositæ ex omnibus. patet igitur demonstratio.
<lb/>[<emph style="it">tr:
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>b</mi><mo>-</mo><mi>c</mi><mi>k</mi></mrow><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfrac><mo>=</mo><mi>a</mi></mstyle></math> is the sum of all of them. The demonstration is therefore clear.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s349" xml:space="preserve">
alia Notatio quantitati.
<lb/>[<emph style="it">tr:
another notation for quantities.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s350" xml:space="preserve">
Hinc in infinitis progressionibus cum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math> hoc est ultima quantitatis in infinitum <lb/>
abeat, tres isti termini sint proport[ionales] videlicet: <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> co[mpositæ om]nibus.
<lb/>[<emph style="it">tr:
Hence, in an infinite porgression, since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>u</mi></mstyle></math>, that is, the last quantity, disappears to infinity,
these three terms are clearly proportional:
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> the sum of all.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f191v" o="191v" n="382"/>
<pb file="add_6785_f192" o="192" n="383"/>
<pb file="add_6785_f192v" o="192v" n="384"/>
<pb file="add_6785_f193" o="193" n="385"/>
<pb file="add_6785_f193v" o="193v" n="386"/>
<pb file="add_6785_f194" o="194" n="387"/>
<pb file="add_6785_f194v" o="194v" n="388"/>
<pb file="add_6785_f195" o="195" n="389"/>
<div xml:id="echoid-div72" type="page_commentary" level="2" n="72">
<p>
<s xml:id="echoid-s351" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s351" xml:space="preserve">
On this page, Harriot works a problem from Cardano's
<emph style="it">De regula aliza liber</emph>, Chapter, pages 82–83.
Cardano states the problem as follows: <lb/>
<emph style="it" xml:lang="lat">
De difficillimo problemate quod facillimum uidetur. CAP. XLI. <lb/>
Nihil est admirabilius quam cum sub facili quaestione latet difficillimus scrupulus,
huiusmodi est hic: quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> cum latere <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> est 10, &amp; quadratum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> cum latere <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> est 8,
quaeritur quantum fit unum horum seu latus seu quadratum: Quia ergo <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> est 10,
&amp; <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> 1 quad. erit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> 10 m: 1 quad. igitur bc 100 m: 20 quad. p: 1 pos. p: 1 quad. quad.
&amp; hoc est aequale 8, quare 1 quad. quad. p: 92, aequatur 20 quad. m: 1 pos. adde 19 quad.
utrinque fient 1 quad. quad. p: 19, quadrat. p: 92, aequalia 39. quad, m: 1 pos. detrahe <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>,
erunt 1 quad. quad. p: 19 quad. p: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>0</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> aequalia 39 quad. m: 1 pos. m: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>,
inde adde 2 pos. p: 1 quad. utrinque ut in Arte magna, &amp; videbis difficillimam quaestionem.
</emph>
</s>
<lb/>
<s xml:id="echoid-s352" xml:space="preserve">
Chapter 41. On a very difficult question that seems easy. <lb/>
Nothing is more remarkable than when under an easy question there lies a very hard stone,
of which kind is this: the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> with the side <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is 10,
and the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> with the side <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> is 8; it is required to find one of those sides or squares.
Therefore because <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi><mi>c</mi></mstyle></math> is 10, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>b</mi></mstyle></math> is one square, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> will be 10 minus a square.
Therefore the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> is 100 minus 20 squares plus 1 square-square.
Therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi><mi>d</mi></mstyle></math> will be 100 minus 20 squares plus 1 unknown plus 1 square-square,
and this is equal to 8, whence 1 square-square plus 92 equals 20 squares minus 1 unknown.
Add 19 squares to each side, making 1 square-square plus 19 squares plus 92 equal to 39 squares minus 1 unknown.
Subtract <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, to give 1 square-square plus 19 squares plus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>9</mn><mn>0</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> equal to 39 squares
minus 1 unknown minus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>, whence add 2 unknowns plus one square to each side as in the Ars magna,
and you will see this very difficult question.
</s>
<lb/>
<s xml:id="echoid-s353" xml:space="preserve">
Harriot lets the side of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math> and then works through these instructions precisely.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head74" xml:space="preserve" xml:lang="lat">
Cardan. de Aliza. pa. 83
<lb/>[<emph style="it">tr:
Cardano, <emph style="it">De regula aliza liber</emph>, page 83
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s355" xml:space="preserve">
Data
<lb/>[<emph style="it">tr:
Given
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s356" xml:space="preserve">
Quæritur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi></mstyle></math> vel <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>g</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Sought, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mi>f</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi><mi>g</mi></mstyle></math>.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s357" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr:
Another way
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s358" xml:space="preserve">
Et per <lb/>
Antithesin
<lb/>[<emph style="it">tr:
And by Antithesis
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f195v" o="195v" n="390"/>
<pb file="add_6785_f196" o="196" n="391"/>
<pb file="add_6785_f196v" o="196v" n="392"/>
<pb file="add_6785_f197" o="197" n="393"/>
<div xml:id="echoid-div73" type="page_commentary" level="2" n="73">
<p>
<s xml:id="echoid-s359" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s359" xml:space="preserve">
This question is a slight variation on a question from Girolamo Cardano,
<emph style="it">De arithmetica liber X</emph> (= <emph style="it">Artis magnae</emph>Artis magnae) (1570),
Chapter 39, page 143, Question I:
</s>
<lb/>
<s xml:id="echoid-s360" xml:space="preserve">
<emph style="it" xml:lang="lat">
Quaestio I. <lb/>
Exemplum. Inuenias tres numeros in continua proportione,
quorum quadratum primi fit aequale secundo &amp; tertio,
&amp; quadratum tertij fit aequale quadratis primi &amp; secundi.
</emph>
</s>
<lb/>
<s xml:id="echoid-s361" xml:space="preserve">
Question 1. <lb/>
Example. Find three numbers in continual proportion,
such that the square of the first is equal to the second and third,
and the square of the third is equal to the square of the first and second.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head75" xml:space="preserve" xml:lang="lat">
Cardan. Arith, lib. 10. cap. 39. pag. 143.
<lb/>[<emph style="it">tr:
Cardano, <emph style="it">De arithmetica liber X</emph>, Chapter 39, page 143.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s363" xml:space="preserve">
Invenire tres numeros <lb/>
continue proportionales <lb/>
quorum tertius sit aequalis <lb/>
primo et secundo, et <lb/>
quadratum primi sit aequale <lb/>
aggregato secundi et tertij.
<lb/>[<emph style="it">tr:
Find three numbers in continual proportion of which the third is equal to the first and second,
and the square of the first is equal to the sum of the second and third.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s364" xml:space="preserve">
Argumentatio prima
<lb/>[<emph style="it">tr:
First argument
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s365" xml:space="preserve">
Ergo <emph style="st">prima</emph> dispositio terminorum proportionalium ex prima conditione illata <lb/>
ita se habet.
</s>
<lb/>[<emph style="it">tr:
Therefore the arrangement of proportional terms according to the first condition is had thus.
</emph>]<lb/>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s366" xml:space="preserve">
Argumentatio secunda, seu de secunda conditione
<lb/>[<emph style="it">tr:
Second argument, or from the second condition
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s367" xml:space="preserve">
Unde tres proportionales quæsiti.
<lb/>[<emph style="it">tr:
Whence the three proportionals sought
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s368" xml:space="preserve">
Examen fit in altera charta.
<lb/>[<emph style="it">tr:
To be tested in another sheet
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s369" xml:space="preserve">
Aggregatum I + II + III <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>2</mn><mn>5</mn></mrow></msqrt><mo>+</mo><mn>1</mn><mn>1</mn></mstyle></math> vel duplum III
<lb/>[<emph style="it">tr:
The sum of the first, second, and third is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mrow><mn>1</mn><mn>2</mn><mn>5</mn></mrow></msqrt><mo>+</mo><mn>1</mn><mn>1</mn></mstyle></math>, or twice the third.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f197v" o="197v" n="394"/>
<pb file="add_6785_f198" o="198" n="395"/>
<pb file="add_6785_f198v" o="198v" n="396"/>
<pb file="add_6785_f199" o="199" n="397"/>
<pb file="add_6785_f199v" o="199v" n="398"/>
<pb file="add_6785_f200" o="200" n="399"/>
<pb file="add_6785_f200v" o="200v" n="400"/>
<pb file="add_6785_f201" o="201" n="401"/>
<div xml:id="echoid-div74" type="page_commentary" level="2" n="74">
<p>
<s xml:id="echoid-s370" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s370" xml:space="preserve">
Further calculations of Pythagorean triples from the parameters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>,
continued from Add MS 6785, f. 405 and f. 403.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head76" xml:space="preserve">
6)
</head>
<pb file="add_6785_f201v" o="201v" n="402"/>
<pb file="add_6785_f202" o="202" n="403"/>
<div xml:id="echoid-div75" type="page_commentary" level="2" n="75">
<p>
<s xml:id="echoid-s372" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s372" xml:space="preserve">
Further calculations of Pythagorean triples from the parameters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>,
continued from Add MS 6785, f. 405.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head77" xml:space="preserve">
5)
</head>
<pb file="add_6785_f202v" o="202v" n="404"/>
<pb file="add_6785_f203" o="203" n="405"/>
<div xml:id="echoid-div76" type="page_commentary" level="2" n="76">
<p>
<s xml:id="echoid-s374" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s374" xml:space="preserve">
This folio shows the calculation of Pythagorean triples from the parameters <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math>. <lb/>
Those marked with asterisks do not appear in the lists given by Stifel,
which were reproduced by Harriot on Add MS 6782, f. 84.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head78" xml:space="preserve">
4)
</head>
<pb file="add_6785_f203v" o="203v" n="406"/>
<pb file="add_6785_f204" o="204" n="407"/>
<div xml:id="echoid-div77" type="page_commentary" level="2" n="77">
<p>
<s xml:id="echoid-s376" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s376" xml:space="preserve">
The reference at the top of the page is to Zetetic IV.1 from Viète's
<emph style="it">Zeteticorum libri quinque</emph>.
This is also Proposition II.8 from the <emph style="it">Arithmetica of Diophantus</emph>.
In Zeteticum IV.1, Viète wrote as follows:
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum I <lb/>
Invenire numero duo quadrata, aequalia dato quadrato. <lb/>
<lb/>[...]<lb/> <lb/>
Eoque recidit Analysis Diophantæa, secundum quam oporteat B quadratum, in duo quadrata dispescere.
Latus primi quadrati esto A, secundi B – S in A/R. Primi lateris in quadratum, est A quadratum.
Secundi, B quad. – S in A in B2/R + S quad. in A quad./R quad. Quae duo quadrata ideo aequalia sunt B quadrato. <lb/>
Aequalitas igitur ordinetur. S in R in B2/S quad. + R quadr. aequabitur A lateri primi singularis quadrati.
Et latus secundi fit R quad. in B, – S quad. in B/S quad. + R quad.
Nempe triangulum rectangulum numero effingitur a lateribus duobus S &amp; R,
&amp; fit hypotenusa similis S quad + R quad. basis similis S quadrato – R quadrato. Perendiculum simile S in R2.
Itaque ad dispectionem B quadrati fit, ut S quadr. + R quadr. ad B hypotenusam similis trianguli,
ita R quadr. – S quadr. ad basim, latus unus singularis quadrati,
&amp; ita S in R2 ad perpendiculum, latus alterius.
</quote>
<lb/>
<quote>
To find in numbers, two squares equal to a given square. <lb/>
<lb/>[...]<lb/> <lb/>
The same is taught in the analysis of Diophantus, according to which it is required to divide the square of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>
into two other squares. Let the side of the first be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>, of the second <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>.
The square of the first side is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mstyle></math>, of the second <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>-</mo><mfrac><mrow><mn>2</mn><mi>s</mi><mi>a</mi><mi>b</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>+</mo><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mstyle></math>.
Which two squares are therefore equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mstyle></math>. <lb/>
The equalisation is carried out. [We obtain] <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>s</mi><mi>r</mi><mi>b</mi></mrow><mrow><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mfrac></mstyle></math>.
And the second side is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow><mi>b</mi><mo>-</mo><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow><mi>b</mi></mrow><mrow><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mrow></mfrac></mstyle></math>.
That is, a right-angled triangle in numbers may be constructed from the two terms <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>,
and the hypotenuse will be proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mstyle></math>, the base to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mstyle></math>
[actually <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow></mstyle></math>], the perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>r</mi><mi>s</mi></mstyle></math>.
Thus, the division of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mstyle></math> [into two other squares]
gives a triangle with hypotenuse proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mstyle></math>,
base proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mstyle></math>, [axtually <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi>s</mi><mn>2</mn></msup></mrow></mstyle></math>],
the side of one square, and perpendicular proportional to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>s</mi><mi>r</mi></mstyle></math>, the side of the other.
</quote>
<lb/>
<s xml:id="echoid-s377" xml:space="preserve">
For the first time in this run of pages, Harriot refers directly to Diophantus,
not only in the heading but also in the course of his working.
It seems likely that he had now turned directly to Problem II.8 of the <emph style="it">Arithemtica</emph>
in the edition by Wilhelm Xylander,
<emph style="it">Diophanti Alexandrini rerum arithmeticarum libri sex</emph> (1575).
There he would have found that Diophantus gave only a single numerical example,
with none of the generality that Viète had introduced.
</s>
<lb/>
<s xml:id="echoid-s378" xml:space="preserve">
Supposing the initial square was 16,
Diophantus took the side of the first square to be some number which, following Xylander, we may call <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math>,
with side <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>Q</mi></mstyle></math>, and the side of the second square to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>N</mi><mo>-</mo><mn>4</mn></mstyle></math>.
In Viète's more general notation, 16 was replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>N</mi></mstyle></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>.
The side of the second square, in Diophantus's method, was thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>A</mi><mo>-</mo><mi>B</mi></mstyle></math>.
Harriot was therefore correct in his assertion that Viète, who wrote the second side as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mo>-</mo><mfrac><mrow><mi>S</mi><mi>A</mi></mrow><mrow><mi>R</mi></mrow></mfrac></mstyle></math>,
had actually proceeded differently from Diophantus.
</s>
<lb/>
<s xml:id="echoid-s379" xml:space="preserve">
Diophantus was working purely in numbers, but for Viète,
who was relating the problem to lengths of sides of a triangle,
it was clearly more natural to take the side of the second square to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mo>-</mo><mfrac><mrow><mi>S</mi><mi>A</mi></mrow><mrow><mi>R</mi></mrow></mfrac></mstyle></math>.
Viete did not specify the relative sizes of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math>,
but it must be the case that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>S</mi><mi>A</mi></mrow><mrow><mi>R</mi></mrow></mfrac><mo>&lt;</mo><mi>B</mi></mstyle></math> if the triangle is to be constructed.
In the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi><mo>&gt;</mo><mi>R</mi></mstyle></math>,
Harriot argues that one should instead use the method of Diophantus (who gave the example where <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi><mo>:</mo><mi>R</mi><mo>=</mo><mn>2</mn><mo>:</mo><mn>1</mn></mstyle></math>),
in which case one requires <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>S</mi><mi>A</mi></mrow><mrow><mi>R</mi></mrow></mfrac><mo>&gt;</mo><mi>B</mi></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head79" xml:space="preserve" xml:lang="lat">
3.) Diophantus. lib. 2. 8. Zet. 4. 1.
<lb/>[<emph style="it">tr:
Diophantus, Book II, Proposition 8
Zetetica, Book IV, Zetetic 1
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s381" xml:space="preserve">
Dividere <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> in duo quadrata numeri. <lb/>
<lb/>[<emph style="it">tr:
Divide <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> into two square numbers.
</emph>]<lb/>
</s>
<s xml:id="echoid-s382" xml:space="preserve">
Sit 1. <reg norm="quadratus" type="abbr">quad</reg>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math> <lb/>
<lb/>[<emph style="it">tr:
Let the first square be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi><mi>a</mi></mstyle></math>.
</emph>]<lb/>
</s>
<s xml:id="echoid-s383" xml:space="preserve">
Erit: 2. <reg norm="quadratus" type="abbr">quad</reg>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math> <lb/>
<lb/>[<emph style="it">tr:
The second square will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi><mo>-</mo><mi>a</mi><mi>a</mi></mstyle></math>. <lb/>
</emph>]<lb/>
</s>
<s xml:id="echoid-s384" xml:space="preserve">
Qæritur latus <lb/>
huius 2<emph style="super">i</emph> et fit: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>
vel secundi <reg norm="Diophantus" type="abbr">Diophant</reg>: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>-</mo><mi>b</mi></mstyle></math>
<lb/>[<emph style="it">tr:
The side of this second square is sought; and it will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>,
or according to Diophantus <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>-</mo><mi>b</mi></mstyle></math>
</emph>]<lb/>
</s></p>
<p xml:lang="lat">
<s xml:id="echoid-s385" xml:space="preserve">
latus secundi fuerit. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math> <lb/>
<lb/>[<emph style="it">tr:
the side of the second will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>
</emph>]<lb/>
</s>
<s xml:id="echoid-s386" xml:space="preserve">
(sive ex hypothesi quod <lb/>
est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>&gt;</mo><mi>s</mi></mstyle></math>) <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math> formeretur pro lateri 2<emph style="super">i</emph> <lb/>
<lb/>[<emph style="it">tr:
(or from the hypothesis that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>&gt;</mo><mi>s</mi></mstyle></math>), the side of the second square will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>
</emph>]<lb/>
</s>
<s xml:id="echoid-s387" xml:space="preserve">
Et in illa <lb/>
forma <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>s</mi><mi>s</mi><mo>+</mo><mi>b</mi><mi>r</mi><mi>r</mi></mrow><mrow><mi>s</mi><mi>s</mi><mo>+</mo><mi>r</mi><mi>r</mi></mrow></mfrac><mo>=</mo><mi>b</mi></mstyle></math>. latus dati.
<lb/>[<emph style="it">tr:
And in that form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>s</mi><mi>s</mi><mo>+</mo><mi>b</mi><mi>r</mi><mi>r</mi></mrow><mrow><mi>s</mi><mi>s</mi><mo>+</mo><mi>r</mi><mi>r</mi></mrow></mfrac><mo>=</mo><mi>b</mi></mstyle></math> is the side of the given square.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s388" xml:space="preserve">
Sed si ponatur quod <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> sit <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>&gt;</mo><mi>r</mi></mstyle></math>; <emph style="st">[???]</emph> ut posuit Vieta <lb/>
latus 2<emph style="super">i</emph> [???] est <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>-</mo><mi>b</mi></mstyle></math>. ut Diophantus.
<lb/>[<emph style="it">tr:
But if we put <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi><mo>&gt;</mo><mi>r</mi></mstyle></math>, as Viète supposed, the side of the second square is <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mo>-</mo><mi>b</mi></mstyle></math>, as in Diophantus.
</emph>]<lb/>
</s>
<s xml:id="echoid-s389" xml:space="preserve">
Vieta igitur emendandi.
<lb/>[<emph style="it">tr:
Viete is therefore to be corrected.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f204v" o="204v" n="408"/>
<pb file="add_6785_f205" o="205" n="409"/>
<div xml:id="echoid-div78" type="page_commentary" level="2" n="78">
<p>
<s xml:id="echoid-s390" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s390" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Zetetic 1.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum I <lb/>
Invenire numero duo quadrata, aequalia dato quadrato.
</quote>
<lb/>
<quote>
To find in numbers, two squares equal to a given square.
</quote>
<lb/>
<s xml:id="echoid-s391" xml:space="preserve">
Harriot's pagination indicates that this page follows Add MS 6785, f. 207,
but now <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> have been replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>s</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi></mstyle></math>. <lb/>
In the second half of the page, Harriot addresses the problem posed in Zeteticum IV.1:
to divide a square into two other squares.
This is also Problem II.8 in the <emph style="it">Arithmetica</emph> of Diophantus.
Viète referred to the working by Diophantus, but Harriot refers only to Viète. <lb/>
Following Viète, Harriot denoted the side of the given square by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>,
the side of the first unknown square by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>,
and the side of the other by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mfrac><mrow><mi>s</mi><mi>a</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mstyle></math>.
The side of the second square is thus found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mi>b</mi><mi>r</mi><mi>r</mi><mo>-</mo><mi>b</mi><mi>s</mi><mi>s</mi></mrow><mrow><mi>r</mi><mi>r</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow></mfrac></mstyle></math>.
In the 1591 edition of Viète's <emph style="it">Zetetica</emph>,
there is a switch between <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>R</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>S</mi></mstyle></math> at this point, so that the second side is given as
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mrow><msup><mi>S</mi><mn>2</mn></msup></mrow><mi>B</mi><mo>-</mo><mrow><msup><mi>R</mi><mn>2</mn></msup></mrow><mi>B</mi></mrow><mrow><mrow><msup><mi>S</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>R</mi><mn>2</mn></msup></mrow></mrow></mfrac></mstyle></math>. This is the error Harriot refers to.
It was corrected in the 1646 edition of Viète's collected works,
the <emph style="it">Opera mathematica</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head80" xml:space="preserve" xml:lang="lat">
2.) Zet. lib. 4. 1.
<lb/>[<emph style="it">tr:
Zetetica, Book IV, Zeteticum 1
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s393" xml:space="preserve">
Dividere <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> in duo
<lb/>[<emph style="it">tr:
To divide <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>b</mi></mstyle></math> into two [squares].
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s394" xml:space="preserve">
latus 2<emph style="super">i</emph>
<lb/>[<emph style="it">tr:
the side of the second <emph style="it">square</emph>
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s395" xml:space="preserve">
Contra Vieta, igitur emendandus <lb/>
vide chartam (3)
<lb/>[<emph style="it">tr:
Contrary to Viete, therefore to be amended <lb/>
see sheet (3).
</emph>]<lb/>
[<emph style="it">Note:
Sheet 3 is Add MS 6782, f. 204.
 </emph>]<lb/>
</s>
</p>
<pb file="add_6785_f205v" o="205v" n="410"/>
<pb file="add_6785_f206" o="206" n="411"/>
<div xml:id="echoid-div79" type="page_commentary" level="2" n="79">
<p>
<s xml:id="echoid-s396" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s396" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book III, Zetetic 9.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum IX <lb/>
Invenitur triangulum rectangulum numero. <lb/>
Enim vero, <lb/>
Adsumptis duobus lateribus rationalibus, hypotenusa fit similis adgregata quadratorum, basis differentia corumdem,
perpendiculum duplo sub lateribus rectangulo. <lb/>
Sint duo latero B &amp; D. Sunt igitur proportionalia tria latera B, D, D quadratum/B. Omnia in B.
Sunt tria proprtionalia Bq. Bin D. Dq. A quibus proportionalibus fit per antecdicta,
hypotenusa trianguli similis Bq + Dq. basis Bq=Dq. perpendiculum B in D2. Et alioqui jam ordinatum est.
Quadratum ab adgregato quadratorum, aequare quadratum a differentia quadratorum,
adjunctum quadrato dupli rectanguli sub lateribus. <lb/>
Sit B2. D3. Hypotenusa fiet similis 13, basis 5, perpendiculum 12.
</quote>
<lb/>
<quote>
To find a right-angled triangle in numbers. <lb/>
Taking two rational sides, the hypotenuse is similar to the sum of the squares, the base to their difference,
the perpendicular to twice the product. <lb/>
Let the two sides be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>. There are therefore three proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mfrac><mrow><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow></mrow><mrow><mi>B</mi></mrow></mfrac></mstyle></math>.
[Multiply] all by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. There are three proportionals <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>d</mi><mn>2</mn></msup></mrow></mstyle></math>.
From which proportionals it comes about, from what has been said before
[see Zeteticum III.8],
that the hypotenuse of the triangle is similar to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>B</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow></mstyle></math>, the base to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>B</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi>D</mi><mn>2</mn></msup></mrow></mstyle></math>,
the perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>B</mi><mi>D</mi></mstyle></math>. And now the rest is in order. The square of the sum of squares is equal to
the square of the difference of squares added to the square of twice the product. <lb/>
Suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi><mo>=</mo><mn>2</mn></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mo>=</mo><mn>3</mn></mstyle></math>. The hypotenuse is similar to 13, the base to 5, the perpendicular to 12.
</quote>
<lb/>
<s xml:id="echoid-s397" xml:space="preserve">
Harriot followed the same instructions, replacing Viète's <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi></mstyle></math>, by <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>,
reserving the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> for the base of the triangle.
He denotes the quantities <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mo>+</mo><mi>d</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>c</mi><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math> by
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>h</mi></mstyle></math> (hypotenuse), <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>p</mi></mstyle></math> (perpendicular), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math> (base), respectively,
and demonstrates that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>h</mi><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mi>p</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow></mstyle></math>, as required. <lb/>
Note his use of what looks like an = sign in the first appearance of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mo>-</mo><mi>d</mi><mi>d</mi></mstyle></math>.
This indicates that the positive difference is to be taken if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi><mo>&gt;</mo><mi>c</mi></mstyle></math>.
In modern notation, Harriot's <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi><mo>=</mo><mi>d</mi><mi>d</mi></mstyle></math> would be written <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo lspace="0em" rspace="0em" maxsize="1">|</mo><mrow><msup><mi>c</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><msup><mi>d</mi><mn>2</mn></msup></mrow><mo lspace="0em" rspace="0em" maxsize="1">|</mo></mstyle></math>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head81" xml:space="preserve" xml:lang="lat">
1.) Zet. lib. 3. 9.
<lb/>[<emph style="it">tr:
Zetetica, Book III, Zeteticum 9
</emph>]<lb/>
</head>
<pb file="add_6785_f206v" o="206v" n="412"/>
<pb file="add_6785_f207" o="207" n="413"/>
<div xml:id="echoid-div80" type="page_commentary" level="2" n="80">
<p>
<s xml:id="echoid-s399" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s399" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book III, Zetetic 10.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum X <lb/>
Dato adgregato quadratorum Ã  singulis tribus proportionalibus,
atque ea in serie extremarum una, invenitur altera extrema.
</quote>
<lb/>
<quote>
Given the sum the sum of squares of each of three proportionals, and one of the extremes of the sequence,
the other extreme may be found.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head82" xml:space="preserve" xml:lang="lat">
Zet. lib. 3. 10.
<lb/>[<emph style="it">tr:
Zetetica, Book 3, Zeteticum 10.
</emph>]<lb/>
</head>
<pb file="add_6785_f207v" o="207v" n="414"/>
<pb file="add_6785_f208" o="208" n="415"/>
<div xml:id="echoid-div81" type="page_commentary" level="2" n="81">
<p>
<s xml:id="echoid-s401" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s401" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Zetetic 6.
This is also Proposition II.10 from the <emph style="it">Arithmetica</emph> of Diophantus,
but Harriot refers only to Viète's version of it.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum VI <lb/>
Invenire numero duo quadrata, distantia dato intervallo.
</quote>
<lb/>
<quote>
To find in numbers two squares having a given difference between them.
</quote>
<lb/>
<s xml:id="echoid-s402" xml:space="preserve">
Viète used the letter <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> for the given difference.
Harriot followed Viète's working but in his own lower-case notation.
The 1591 edition of the <emph style="it">Zetetica</emph> at one point mistakenly gives 'maior' instead of 'minor'.
This is the error that Harriot points out.
It was corrected in the 1646 edition of Viète's <emph style="it">Opera mathematica</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head83" xml:space="preserve" xml:lang="lat">
Zet. lib. 4. 6.
<lb/>[<emph style="it">tr:
Zetetica, Book IV, Zeteticum 6.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s404" xml:space="preserve">
Invenire duo quadrata <lb/>
distantia data intervallo
<lb/>[<emph style="it">tr:
To find two squares a given distance apart.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s405" xml:space="preserve">
et ut differentia vel <lb/>
summa laterum sit æqualis, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math> <lb/>
dato numero.
<lb/>[<emph style="it">tr:
and so that the difference or sum of the sides is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, a given number.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s406" xml:space="preserve">
Sit datum intervallum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>s</mi></mstyle></math> <lb/>
et intelligatur pro quadrato basis <lb/>
Et sit differentia inter hypotenusum <lb/>
et perpendiculum quælibet quantitas <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>.
<lb/>[<emph style="it">tr:
Let the given interval be <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mi>s</mi></mstyle></math>, and it is understood to be the square of the base;
And let the difference between the hypotenuse and the perpendicular be any quantity.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s407" xml:space="preserve">
Menda in Vieta
<lb/>[<emph style="it">tr:
Wrong in Viète.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s408" xml:space="preserve">
Ergo latera trianguli
<lb/>[<emph style="it">tr:
Therefore the sides of the triangle:
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f208v" o="208v" n="416"/>
<pb file="add_6785_f209" o="209" n="417"/>
<div xml:id="echoid-div82" type="page_commentary" level="2" n="82">
<p>
<s xml:id="echoid-s409" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s409" xml:space="preserve">
The reference at the top of this page is to Viète's
<emph style="it">Zeteticorum libri quinque</emph>, Book IV, Zetetic 6.
This is also Proposition II.10 from the <emph style="it">Arithmetica</emph> of Diophantus.
The page appears to be a continuation of Add MS 6785, f. 208.
</s>
<lb/>
<quote xml:lang="lat">
Zeteticum VI <lb/>
Invenire numero duo quadrata, distantia dato intervallo.
</quote>
<lb/>
<quote>
To find in numbers two squares having a given difference between them.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head84" xml:space="preserve" xml:lang="lat">
Zet. lib. 4. 6.
<lb/>[<emph style="it">tr:
Zetetica, Book IV, Zeteticum 6.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s411" xml:space="preserve">
ut in altera <lb/>
charta supra
<lb/>[<emph style="it">tr:
as in the other sheet above
</emph>]<lb/>
[<emph style="it">Note:
The other sheet is Add MS 6785, f. 208.
 </emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s412" xml:space="preserve">
Menda in Vieta
<lb/>[<emph style="it">tr:
Wrong in Viète.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f209v" o="209v" n="418"/>
<pb file="add_6785_f210" o="210" n="419"/>
<p>
<s xml:id="echoid-s413" xml:space="preserve">
To devide a number into 2 such partes <lb/>
that the difference of there squares be <lb/>
æquall to an <emph style="super">other</emph> nomber geven.
</s>
</p>
<p>
<s xml:id="echoid-s414" xml:space="preserve">
The nomber to be devided let be. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>. <lb/>
The second nober geven. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>c</mi></mstyle></math>. <lb/>
The first part. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>. <lb/>
The second. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mo>-</mo><mi>a</mi></mstyle></math>.
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s415" xml:space="preserve">
Oportet
<lb/>[<emph style="it">tr:
It must be that
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s416" xml:space="preserve">
the first.
</s>
<lb/>
<s xml:id="echoid-s417" xml:space="preserve">
the second;
</s>
<lb/>
<s xml:id="echoid-s418" xml:space="preserve">
the roote of the 2 <lb/>
nomber geven.
</s>
</p>
<p>
<s xml:id="echoid-s419" xml:space="preserve">
<foreign xml:lang="lat">Aliter</foreign> the greater parte. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>a</mi></mstyle></math>.
</s>
<lb/>
<s xml:id="echoid-s420" xml:space="preserve">
The greater parte.
</s>
<lb/>
<s xml:id="echoid-s421" xml:space="preserve">
The lesser parte.
</s>
<lb/>
<s xml:id="echoid-s422" xml:space="preserve">
The roote of the <lb/>
nomber geven.
</s>
</p>
<p>
<s xml:id="echoid-s423" xml:space="preserve">
<foreign xml:lang="lat">Examinatur</foreign>. true.
</s>
</p>
<pb file="add_6785_f210v" o="210v" n="420"/>
<div xml:id="echoid-div83" type="page_commentary" level="2" n="83">
<p>
<s xml:id="echoid-s424" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s424" xml:space="preserve">
The first row contains numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow></mstyle></math>. The second row contains numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><mo>-</mo><mn>1</mn></mstyle></math>.
Those that are prime, for example, 3, 7, and 31, give rise to perfect numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo maxsize="1">(</mo><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><mo>-</mo><mn>1</mn><mo maxsize="1">)</mo></mstyle></math>,
as stated in Euclid IX.36. See also Add MS 6786, f. 230v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f211" o="211" n="421"/>
<pb file="add_6785_f211v" o="211v" n="422"/>
<pb file="add_6785_f212" o="212" n="423"/>
<pb file="add_6785_f212v" o="212v" n="424"/>
<div xml:id="echoid-div84" type="page_commentary" level="2" n="84">
<p>
<s xml:id="echoid-s426" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s426" xml:space="preserve">
Calculation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>+</mo><mi>d</mi><mo>-</mo><mi>f</mi><mo maxsize="1">)</mo><mo maxsize="1">(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>-</mo><mi>d</mi><mo>+</mo><mi>f</mi><mo maxsize="1">)</mo><mo>…</mo></mstyle></math>. <lb/>
See also
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f213" o="213" n="425"/>
<pb file="add_6785_f213v" o="213v" n="426"/>
<pb file="add_6785_f214" o="214" n="427"/>
<pb file="add_6785_f214v" o="214v" n="428"/>
<pb file="add_6785_f215" o="215" n="429"/>
<pb file="add_6785_f215v" o="215v" n="430"/>
<pb file="add_6785_f216" o="216" n="431"/>
<div xml:id="echoid-div85" type="page_commentary" level="2" n="85">
<p>
<s xml:id="echoid-s428" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s428" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Variorum responsorum liber VIII</emph>, Chapter XVIII, Proposition 2, Corollary.
</s>
<lb/>
<quote xml:lang="lat">
Corollarium. <lb/>
Itaque quadratum circulo inscriptum erit ad circulum,
sicut latus illius quadrati ad potestatem diametri altissimam adplicatam
ad id quod fit continue sub apotomis laterum octogoni, hexdecagoni,
polygoni triginta duorum laterum, sexagintao quatuor, centum viginti octo, ducentorum quinquaginta sex,
&amp; reliquorum omnium in ea ratione angulorum laterumve subdupla.
</quote>
<lb/>
<quote>
Thus a square inscribed in a circle will be to the circle as the side of the square to
the greatest power of the diameter applied to that which is successively under the apotome of the sides of
octagons, hexdecagons, polygons with thirty-two sides, sixty-four,
one hundred and twenty eight, two hundred and fifty six,
and so on, all in the ratio of halved angles and sides.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head85" xml:space="preserve" xml:lang="lat">
Responsorum. pag. 30. <lb/>
in Corollarium.
<lb/>[<emph style="it">tr:
Responsorum, page 30, on the Corollary
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s430" xml:space="preserve">
per propositione antecedente
<lb/>[<emph style="it">tr:
by the preceding proposition
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s431" xml:space="preserve">
As <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> to <lb/>[...]<lb/> so let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> be to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>γ</mi></mstyle></math>. <lb/>
So will the square <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mo>\</mo><mi>v</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>p</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>l</mi><mi>n</mi></mstyle></math> be to the oblong <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>β</mi><mi>θ</mi></mstyle></math>. <lb/>
And as ... to ... <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>O</mi></mstyle></math>. <lb/>
Therefore if <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>α</mi><mi>β</mi></mstyle></math> be æquall to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> <lb/>
these æquations will also follow: <lb/>
And therfore the oblong made of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>B</mi></mstyle></math> and <lb/>[...]<lb/> <lb/>
is æquall to the circle.
</s>
</p>
<p>
<s xml:id="echoid-s432" xml:space="preserve">
The oblong or square therefore æquall to the circle is <lb/>
Devide it by the semidiameter <lb/>
And the Quotient wilbe the semiperimeter thus:
</s>
</p>
<pb file="add_6785_f216v" o="216v" n="432"/>
<pb file="add_6785_f217" o="217" n="433"/>
<div xml:id="echoid-div86" type="page_commentary" level="2" n="86">
<p>
<s xml:id="echoid-s433" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s433" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Variorum responsorum liber VIII</emph>, Chapter XVIII, Proposition 2.
</s>
<lb/>
<quote xml:lang="lat">
Propositio II. <lb/>
Si eidem circulo inscribantur polygona ordinata in infnitu,
&amp; numerus laterum primi fit ad numerum laterum secundi subduplus,
ad numerum vero laterum tertii subquadruplus, quarti suboctuplus, quinti subdexdecuplus,
&amp; ea deinceps continua ratione subdupla. ...
</quote>
<lb/>
<quote>
If in the same circle there are inscribed polygons ordered indefinitely,
and the number of sides of the first is half the number of sides of the second,
and a quarter the number of sides of the third, and an eighth of the fourth,
and a sixteenth of the fifth, and so on continually halving. ...
</quote>
<lb/>
<s xml:id="echoid-s434" xml:space="preserve">
There are also references to Propositions VI.14 and XI.34 from Euclid's <emph style="it">Elements.</emph>
</s>
<lb/>
<quote>
VI.14 Equal parallograms which have an angle in each equal, have the sides about the equal angles reciprocally proportional.
</quote>
<lb/>
<quote>
XI.34
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head86" xml:space="preserve" xml:lang="lat">
Responsorum. pag. 30
prop. 2.
<lb/>[<emph style="it">tr:
Responsorum, page 30, Proposition 2.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s436" xml:space="preserve">
Apotome lateris <lb/>
Diameter circuli <lb/>
polygon ipsa
<lb/>[<emph style="it">tr:
Apotome of the side <lb/>
Diameter of the circle <lb/>
the polygon itself
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s437" xml:space="preserve">
Est igitur per 14.p.6.l. vel 34.p.11.li.
<lb/>[<emph style="it">tr:
Therefore by Elements VI.14 or XI.34
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s438" xml:space="preserve">
Est igitur per 14.p.6.l.
<lb/>[<emph style="it">tr:
Therefore by Elements VI.14
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s439" xml:space="preserve">
Est sic de cæteris.
<lb/>[<emph style="it">tr:
And so on for the rest.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f217v" o="217v" n="434"/>
<pb file="add_6785_f218" o="218" n="435"/>
<div xml:id="echoid-div87" type="page_commentary" level="2" n="87">
<p>
<s xml:id="echoid-s440" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s440" xml:space="preserve">
The reference on this page is to Viète's
<emph style="it">Adrianus Romanus resposum</emph> (1595), pages 110, 111.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head87" xml:space="preserve" xml:lang="lat">
Secundum Adrianum Romanum. pag. 110. et 111.
<lb/>[<emph style="it">tr:
According to Adrianus Romanus, pages 110 and 111.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s442" xml:space="preserve">
Latus trigintanguli <lb/>
Latus sexagintanguli <lb/>
Ex. pag. 28. Adriani <lb/>
Latus trigintanguli <lb/>
subtensa 132 <lb/>
subtensa 24 <lb/>
seu latus quindecagoni.
<lb/>[<emph style="it">tr:
Side of a thirtieth of an angle <lb/>
Side of a sixtieth of an angle <lb/>
From page 28 of Adrianus <lb/>
Side of a thirtieth of an angle <lb/>
Subtended side of 132 degrees <lb/>
Subtended side of 24 degrees, a quindecagon.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f218v" o="218v" n="436"/>
<pb file="add_6785_f219" o="219" n="437"/>
<pb file="add_6785_f219v" o="219v" n="438"/>
<pb file="add_6785_f220" o="220" n="439"/>
<pb file="add_6785_f220v" o="220v" n="440"/>
<pb file="add_6785_f221" o="221" n="441"/>
<pb file="add_6785_f221v" o="221v" n="442"/>
<pb file="add_6785_f222" o="222" n="443"/>
<pb file="add_6785_f222v" o="222v" n="444"/>
<pb file="add_6785_f223" o="223" n="445"/>
<pb file="add_6785_f223v" o="223v" n="446"/>
<pb file="add_6785_f224" o="224" n="447"/>
<pb file="add_6785_f224v" o="224v" n="448"/>
<pb file="add_6785_f225" o="225" n="449"/>
<pb file="add_6785_f225v" o="225v" n="450"/>
<pb file="add_6785_f226" o="226" n="451"/>
<pb file="add_6785_f226v" o="226v" n="452"/>
<pb file="add_6785_f227" o="227" n="453"/>
<pb file="add_6785_f227v" o="227v" n="454"/>
<pb file="add_6785_f228" o="228" n="455"/>
<pb file="add_6785_f228v" o="228v" n="456"/>
<pb file="add_6785_f229" o="229" n="457"/>
<pb file="add_6785_f229v" o="229v" n="458"/>
<pb file="add_6785_f230" o="230" n="459"/>
<pb file="add_6785_f230v" o="230v" n="460"/>
<pb file="add_6785_f231" o="231" n="461"/>
<pb file="add_6785_f231v" o="231v" n="462"/>
<pb file="add_6785_f232" o="232" n="463"/>
<pb file="add_6785_f232v" o="232v" n="464"/>
<div xml:id="echoid-div88" type="page_commentary" level="2" n="88">
<p>
<s xml:id="echoid-s443" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s443" xml:space="preserve">
This sheet refers to Stevin's <emph style="it">L 19arithmétique ... aussi l 19algebre</emph> (1585), page 214.
On that page, Stevin wrote:
'Et multipliant √bino. 1 + √2, par √bino. 3 + √5, le produict sera √quadrino. √3 + √6 + √5 + √10.'
Harriot has re-written the same calculation in his own notation: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><msqrt><mo maxsize="1">[</mo></msqrt><mn>2</mn><mo maxsize="1">]</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></mstyle></math>, and so on.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p>
<s xml:id="echoid-s445" xml:space="preserve">
Stevin. <lb/>
pa. 214.
</s>
</p>
<pb file="add_6785_f233" o="233" n="465"/>
<pb file="add_6785_f233v" o="233v" n="466"/>
<pb file="add_6785_f234" o="234" n="467"/>
<pb file="add_6785_f234v" o="234v" n="468"/>
<pb file="add_6785_f235" o="235" n="469"/>
<pb file="add_6785_f235v" o="235v" n="470"/>
<pb file="add_6785_f236" o="236" n="471"/>
<pb file="add_6785_f236v" o="236v" n="472"/>
<pb file="add_6785_f237" o="237" n="473"/>
<head xml:id="echoid-head88" xml:space="preserve" xml:lang="lat">
Aliter: De quadrilatero <lb/>
sive ptolomaico
<lb/>[<emph style="it">tr:
Another way: on quadrilaterals, or by Ptolemy
</emph>]<lb/>
</head>
<pb file="add_6785_f237v" o="237v" n="474"/>
<pb file="add_6785_f238" o="238" n="475"/>
<pb file="add_6785_f238v" o="238v" n="476"/>
<pb file="add_6785_f239" o="239" n="477"/>
<div xml:id="echoid-div89" type="page_commentary" level="2" n="89">
<p>
<s xml:id="echoid-s446" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s446" xml:space="preserve">
See Add MS 6786, f. 240 for the continuation.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head89" xml:space="preserve">
Simon Stevin. Novemb. 25, 1612
</head>
<p xml:lang="lat">
<s xml:id="echoid-s448" xml:space="preserve">
Datur quadrilaterum circulo inscribendum cuius <emph style="super">area</emph> est 1344 <lb/>
ambitus 162, &amp; productum ex duabus diagonalibus seu <lb/>
dimetientibus 2912.
</s>
<s xml:id="echoid-s449" xml:space="preserve">
Exprimantur latera &amp; dimetientes <lb/>
situsque ipsius quadrilateri, ac circuli etiam diametrus.
</s>
</p>
<pb file="add_6785_f239v" o="239v" n="478"/>
<pb file="add_6785_f240" o="240" n="479"/>
<p xml:lang="lat">
<s xml:id="echoid-s450" xml:space="preserve">
Diameter circuli. 65.
</s>
</p>
<pb file="add_6785_f240v" o="240v" n="480"/>
<pb file="add_6785_f241" o="241" n="481"/>
<pb file="add_6785_f241v" o="241v" n="482"/>
<pb file="add_6785_f242" o="242" n="483"/>
<pb file="add_6785_f242v" o="242v" n="484"/>
<pb file="add_6785_f243" o="243" n="485"/>
<pb file="add_6785_f243v" o="243v" n="486"/>
<pb file="add_6785_f244" o="244" n="487"/>
<pb file="add_6785_f244v" o="244v" n="488"/>
<pb file="add_6785_f245" o="245" n="489"/>
<pb file="add_6785_f245v" o="245v" n="490"/>
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<pb file="add_6785_f246v" o="246v" n="492"/>
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<pb file="add_6785_f248" o="248" n="495"/>
<pb file="add_6785_f248v" o="248v" n="496"/>
<p>
<s xml:id="echoid-s451" xml:space="preserve">
Joannes Thinnagel [???] <lb/>
Cæsariæ Mtrs [???] <lb/>
Supplirium <lb/>
Johannis (Erici) Frink
</s>
</p>
<pb file="add_6785_f249" o="249" n="497"/>
<pb file="add_6785_f249v" o="249v" n="498"/>
<pb file="add_6785_f250" o="250" n="499"/>
<head xml:id="echoid-head90" xml:space="preserve" xml:lang="lat">
w.1.) Invenire 4<emph style="sup">or</emph> numerus. b. c. d. f. <lb/>
ita ut tres sequentes comparationes sint veræ.
<lb/>[<emph style="it">tr:
To find four numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>d</mi></mstyle></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>f</mi></mstyle></math>, such that the three following comparisons are true.
</emph>]<lb/>
</head>
<pb file="add_6785_f250v" o="250v" n="500"/>
<pb file="add_6785_f251" o="251" n="501"/>
<head xml:id="echoid-head91" xml:space="preserve">
w.2.)
</head>
<pb file="add_6785_f251v" o="251v" n="502"/>
<head xml:id="echoid-head92" xml:space="preserve">
w.4.)
</head>
<pb file="add_6785_f252" o="252" n="503"/>
<pb file="add_6785_f252v" o="252v" n="504"/>
<pb file="add_6785_f253" o="253" n="505"/>
<pb file="add_6785_f253v" o="253v" n="506"/>
<pb file="add_6785_f254" o="254" n="507"/>
<pb file="add_6785_f254v" o="254v" n="508"/>
<pb file="add_6785_f255" o="255" n="509"/>
<pb file="add_6785_f255v" o="255v" n="510"/>
<pb file="add_6785_f256" o="256" n="511"/>
<pb file="add_6785_f256v" o="256v" n="512"/>
<pb file="add_6785_f257" o="257" n="513"/>
<head xml:id="echoid-head93" xml:space="preserve">
w.3.
</head>
<pb file="add_6785_f257v" o="257v" n="514"/>
<pb file="add_6785_f258" o="258" n="515"/>
<pb file="add_6785_f258v" o="258v" n="516"/>
<pb file="add_6785_f259" o="259" n="517"/>
<pb file="add_6785_f259v" o="259v" n="518"/>
<pb file="add_6785_f260" o="260" n="519"/>
<pb file="add_6785_f260v" o="260v" n="520"/>
<pb file="add_6785_f261" o="261" n="521"/>
<pb file="add_6785_f261v" o="261v" n="522"/>
<pb file="add_6785_f262" o="262" n="523"/>
<pb file="add_6785_f262v" o="262v" n="524"/>
<pb file="add_6785_f263" o="263" n="525"/>
<pb file="add_6785_f263v" o="263v" n="526"/>
<pb file="add_6785_f264" o="264" n="527"/>
<pb file="add_6785_f264v" o="264v" n="528"/>
<pb file="add_6785_f265" o="265" n="529"/>
<pb file="add_6785_f265v" o="265v" n="530"/>
<pb file="add_6785_f266" o="266" n="531"/>
<pb file="add_6785_f266v" o="266v" n="532"/>
<pb file="add_6785_f267" o="267" n="533"/>
<pb file="add_6785_f267v" o="267v" n="534"/>
<pb file="add_6785_f268" o="268" n="535"/>
<pb file="add_6785_f268v" o="268v" n="536"/>
<pb file="add_6785_f269" o="269" n="537"/>
<pb file="add_6785_f269v" o="269v" n="538"/>
<pb file="add_6785_f270" o="270" n="539"/>
<pb file="add_6785_f270v" o="270v" n="540"/>
<pb file="add_6785_f271" o="271" n="541"/>
<pb file="add_6785_f271v" o="271v" n="542"/>
<pb file="add_6785_f272" o="272" n="543"/>
<pb file="add_6785_f272v" o="272v" n="544"/>
<pb file="add_6785_f273" o="273" n="545"/>
<pb file="add_6785_f273v" o="273v" n="546"/>
<pb file="add_6785_f274" o="274" n="547"/>
<pb file="add_6785_f274v" o="274v" n="548"/>
<pb file="add_6785_f275" o="275" n="549"/>
<pb file="add_6785_f275v" o="275v" n="550"/>
<pb file="add_6785_f276" o="276" n="551"/>
<head xml:id="echoid-head94" xml:space="preserve" xml:lang="lat">
1) De quadrilatero, <lb/>
et cæteris multilateris
<lb/>[<emph style="it">tr:
On quadrilaterals and other multilaterals
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s452" xml:space="preserve">
sides
</s>
<lb/>
<s xml:id="echoid-s453" xml:space="preserve">
diagonals
</s>
<lb/>
<s xml:id="echoid-s454" xml:space="preserve" xml:lang="lat">
trianguli
<lb/>[<emph style="it">tr:
triangular numbers
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s455" xml:space="preserve">
or 2 in 3; 2 in 4; 2 in 5; 2 in 6 ; 2 in 7 combined
</s>
<lb/>
<s xml:id="echoid-s456" xml:space="preserve">
or the diagonalls thus
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s457" xml:space="preserve">
solidus angulus ex <lb/>
quotlibet datis non <lb/>
potest constitui sine <lb/>
hoc problemata.
<lb/>[<emph style="it">tr:
It is not possible to construct a solid angle from any given number [of plane angles] without this problem.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s458" xml:space="preserve">
[<emph style="it">Note:
The relevant passage in Commandino's <emph style="it">Euclidis elementorum XV</emph> (1572)
is to be found on page 201v (not 102), as part of a Scholium following Book XI, Proposition 23.
Commandino's statement is: <lb/>
<foreign xml:lang="lat">
Ex planis quotlibet datis angulis, quorum uno reliqui sint maiores quomodocumque sumpti,
solidum angulum constituere, oportet autem datos angulos quatuor rectis esse minores.
</foreign>
From any number of given plane angles, given one of which, the rest are greater, however taken,
to make a solid angle it is necessary that the given angles are less than four right angles.
 </emph>]<lb/>
vide. Euclid. lib. 11 <lb/>
pag. 102. comand.
<lb/>[<emph style="it">tr:
See Euclid Book XI, page 202 of Commandino.
</emph>]<lb/>
</s>
</p>
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<pb file="add_6785_f277" o="277" n="553"/>
<pb file="add_6785_f277v" o="277v" n="554"/>
<pb file="add_6785_f278" o="278" n="555"/>
<pb file="add_6785_f278v" o="278v" n="556"/>
<pb file="add_6785_f279" o="279" n="557"/>
<pb file="add_6785_f279v" o="279v" n="558"/>
<pb file="add_6785_f280" o="280" n="559"/>
<pb file="add_6785_f280v" o="280v" n="560"/>
<pb file="add_6785_f281" o="281" n="561"/>
<pb file="add_6785_f281v" o="281v" n="562"/>
<pb file="add_6785_f282" o="282" n="563"/>
<pb file="add_6785_f282v" o="282v" n="564"/>
<pb file="add_6785_f283" o="283" n="565"/>
<pb file="add_6785_f283v" o="283v" n="566"/>
<pb file="add_6785_f284" o="284" n="567"/>
<pb file="add_6785_f284v" o="284v" n="568"/>
<pb file="add_6785_f285" o="285" n="569"/>
<pb file="add_6785_f285v" o="285v" n="570"/>
<pb file="add_6785_f286" o="286" n="571"/>
<pb file="add_6785_f286v" o="286v" n="572"/>
<pb file="add_6785_f287" o="287" n="573"/>
<pb file="add_6785_f287v" o="287v" n="574"/>
<pb file="add_6785_f288" o="288" n="575"/>
<pb file="add_6785_f288v" o="288v" n="576"/>
<pb file="add_6785_f289" o="289" n="577"/>
<head xml:id="echoid-head95" xml:space="preserve" xml:lang="lat">
b.2.)	2<emph style="super">o</emph>. De quadrilatero.	pro diagonijs. <lb/>
per ptolomaicum
<lb/>[<emph style="it">tr:
On quadrilaterals; for finding the diagonals, as Ptolemy.
</emph>]<lb/>
</head>
<pb file="add_6785_f289v" o="289v" n="578"/>
<pb file="add_6785_f290" o="290" n="579"/>
<head xml:id="echoid-head96" xml:space="preserve" xml:lang="lat">
3<emph style="super">i</emph> Aliter. De quadrilatero.	 Sive Ptolomaico
<lb/>[<emph style="it">tr:
Another way. On quadrilaterals. Or by Ptolemy.
</emph>]<lb/>
</head>
<pb file="add_6785_f290v" o="290v" n="580"/>
<pb file="add_6785_f291" o="291" n="581"/>
<pb file="add_6785_f291v" o="291v" n="582"/>
<pb file="add_6785_f292" o="292" n="583"/>
<pb file="add_6785_f292v" o="292v" n="584"/>
<pb file="add_6785_f293" o="293" n="585"/>
<pb file="add_6785_f293v" o="293v" n="586"/>
<pb file="add_6785_f294" o="294" n="587"/>
<div xml:id="echoid-div90" type="page_commentary" level="2" n="90">
<p>
<s xml:id="echoid-s459" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s459" xml:space="preserve">
Lists of Pythagorean triples, including some where one side length is a mixed fraction.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head97" xml:space="preserve" xml:lang="lat">
7) Latera triangulorum rectangulorum <lb/>
rational
<lb/>[<emph style="it">tr:
Rational sides of right-angled triangles
</emph>]<lb/>
</head>
<pb file="add_6785_f294v" o="294v" n="588"/>
<pb file="add_6785_f295" o="295" n="589"/>
<div xml:id="echoid-div91" type="page_commentary" level="2" n="91">
<p>
<s xml:id="echoid-s461" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s461" xml:space="preserve">
On this page, Harriot examines Problem IX from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Problema IX. <lb/>
Datis duobus circulis, &amp; puncto, per datum punctum circulum describere
quem duo dati circuli contingat.
</quote>
<lb/>
<quote>
IX. Given two circles and a point, through the given point describe a circle that touches the two given circles.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head98" xml:space="preserve" xml:lang="lat">
Appoll. Gall. problema. 9. <lb/>
Casus. 1.
<lb/>[<emph style="it">tr:
Apollonius Gallus, Problem IX, case 1.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s463" xml:space="preserve">
In isto casu: <lb/>
Si punctum datum <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi></mstyle></math> sit extra <lb/>
circulum circa <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math>, et intra <lb/>
tangentes ad partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>: <lb/>
vel extra eundem circulum et <lb/>
intra tangentes ad partes <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>. <lb/>
Duo circuli possunt <lb/>
tangere duos datos.
<lb/>[<emph style="it">tr:
In this case, if the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>I</mi></mstyle></math> is outside the circle around <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>H</mi></mstyle></math>, and inside the tangents on the side of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi></mstyle></math>,
or outside the same circle and inside the tangents on the sides of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>H</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>M</mi></mstyle></math>,
then two circles can touch the two given.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s464" xml:space="preserve">
Punctum non dabiter in circulis <lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math> et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>H</mi></mstyle></math>, neque in spatio intra <lb/>
illos circulos vidilicet <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>E</mi></mstyle></math> <lb/>
et intra tangentes. <lb/>
Alias utercunque: et unus tantum <lb/>
circulus tangens describitur nisi in <lb/>
locis supra limitatis.
<lb/>[<emph style="it">tr:
The point will not be given in the circles <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>A</mi><mi>D</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>E</mi><mi>H</mi></mstyle></math>, nor in the space inside those circles, namely <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>D</mi><mi>E</mi></mstyle></math>,
and inside the tangents. Any other way, and
one such tangent circle will be descibed unless in the places delineated above.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f295v" o="295v" n="590"/>
<pb file="add_6785_f296" o="296" n="591"/>
<pb file="add_6785_f296v" o="296v" n="592"/>
<pb file="add_6785_f297" o="297" n="593"/>
<pb file="add_6785_f297v" o="297v" n="594"/>
<pb file="add_6785_f298" o="298" n="595"/>
<pb file="add_6785_f298v" o="298v" n="596"/>
<pb file="add_6785_f299" o="299" n="597"/>
<pb file="add_6785_f299v" o="299v" n="598"/>
<pb file="add_6785_f300" o="300" n="599"/>
<div xml:id="echoid-div92" type="page_commentary" level="2" n="92">
<p>
<s xml:id="echoid-s465" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s465" xml:space="preserve">
The text reffered to here is Bernard Slaignac,
<emph style="it">Tractatus arithmetici partium et alligationis</emph> (1575).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head99" xml:space="preserve" xml:lang="lat">
Ad Demonstrandi secundum praemissum cap. 2. in tractatum Alligationis <lb/>
Bernardi Salignaci.
<lb/>[<emph style="it">tr:
A demonstration of the second premise of Chapter 2 in Bernard Salignac's treatise on alligation
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s467" xml:space="preserve">
præmissum 2<emph style="super">m</emph>
<lb/>[<emph style="it">tr:
second premise
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s468" xml:space="preserve">
exemplum sequens
<lb/>[<emph style="it">tr:
the example following
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s469" xml:space="preserve">
Exempli, proprietates <lb/>
2<emph style="super">a</emph>, illustratio <lb/>[<emph style="it">tr:
Examples, illustrating the properties of the second.
</emph>]<lb/>
<lb/>[<emph style="it">tr:
second premise
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f300v" o="300v" n="600"/>
<pb file="add_6785_f301" o="301" n="601"/>
<pb file="add_6785_f301v" o="301v" n="602"/>
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<pb file="add_6785_f302v" o="302v" n="604"/>
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<pb file="add_6785_f303v" o="303v" n="606"/>
<pb file="add_6785_f304" o="304" n="607"/>
<pb file="add_6785_f304v" o="304v" n="608"/>
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<pb file="add_6785_f315v" o="315v" n="630"/>
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<pb file="add_6785_f317" o="317" n="633"/>
<pb file="add_6785_f317v" o="317v" n="634"/>
<pb file="add_6785_f318" o="318" n="635"/>
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<pb file="add_6785_f319" o="319" n="637"/>
<pb file="add_6785_f319v" o="319v" n="638"/>
<pb file="add_6785_f320" o="320" n="639"/>
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<pb file="add_6785_f322v" o="322v" n="644"/>
<pb file="add_6785_f323" o="323" n="645"/>
<pb file="add_6785_f323v" o="323v" n="646"/>
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<pb file="add_6785_f324v" o="324v" n="648"/>
<pb file="add_6785_f325" o="325" n="649"/>
<pb file="add_6785_f325v" o="325v" n="650"/>
<pb file="add_6785_f326" o="326" n="651"/>
<pb file="add_6785_f326v" o="326v" n="652"/>
<pb file="add_6785_f327" o="327" n="653"/>
<pb file="add_6785_f327v" o="327v" n="654"/>
<pb file="add_6785_f328" o="328" n="655"/>
<pb file="add_6785_f328v" o="328v" n="656"/>
<pb file="add_6785_f329" o="329" n="657"/>
<pb file="add_6785_f329v" o="329v" n="658"/>
<pb file="add_6785_f330" o="330" n="659"/>
<pb file="add_6785_f330v" o="330v" n="660"/>
<pb file="add_6785_f331" o="331" n="661"/>
<pb file="add_6785_f331v" o="331v" n="662"/>
<pb file="add_6785_f332" o="332" n="663"/>
<pb file="add_6785_f332v" o="332v" n="664"/>
<pb file="add_6785_f333" o="333" n="665"/>
<pb file="add_6785_f333v" o="333v" n="666"/>
<div xml:id="echoid-div93" type="page_commentary" level="2" n="93">
<p>
<s xml:id="echoid-s470" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s470" xml:space="preserve">
This page and the next (Add MD 6785, f. 334 )contain a summary of earlier propositions used
in each proposition in Book I of Euclid's <emph style="it">Elements</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f334" o="334" n="667"/>
<div xml:id="echoid-div94" type="page_commentary" level="2" n="94">
<p>
<s xml:id="echoid-s472" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s472" xml:space="preserve">
The continuation of Add MS 6785, f. 333v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f334v" o="334v" n="668"/>
<pb file="add_6785_f335" o="335" n="669"/>
<div xml:id="echoid-div95" type="page_commentary" level="2" n="95">
<p>
<s xml:id="echoid-s474" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s474" xml:space="preserve">
This page contains an analysis of earlier propositions used in each proposition
in Book IV of Euclid's <emph style="it">Elements</emph>.
(For similar analyses of Books I and III, see Add MS 6785, f. 337, f. 338, f. 336.) <lb/>
The first wide column shows all the propositions, postulates, definitions, and axioms
used in each proposition in Book IV, in the order in which they occur, including repetitions. <lb/>
The second wide column shows the earlier propositions used in each proposition in Book IV,
in numerical order without repetitions. <lb/>
Proposition 1, for example, relies on Proposition 3 from Book I, and Proposition 15 from Book III.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s476" xml:space="preserve">
In lib. 4<emph style="super">o</emph>
<lb/>[<emph style="it">tr:
In the fourth book
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s477" xml:space="preserve">
Definitiones 7
<lb/>[<emph style="it">tr:
Definitions 7
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s478" xml:space="preserve">
propositiones 16
<lb/>[<emph style="it">tr:
propositions 16
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s479" xml:space="preserve">
pr.
<lb/>[<emph style="it">tr:
problems
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s480" xml:space="preserve">
Th.
<lb/>[<emph style="it">tr:
theorems
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s481" xml:space="preserve">
pro-<lb/>
pos.
<lb/>[<emph style="it">tr:
propositions
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f335v" o="335v" n="670"/>
<pb file="add_6785_f336" o="336" n="671"/>
<div xml:id="echoid-div96" type="page_commentary" level="2" n="96">
<p>
<s xml:id="echoid-s482" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s482" xml:space="preserve">
This page contains an analysis of earlier propositions used in each proposition
in Book III of Euclid's <emph style="it">Elements</emph>.
(For a similar analysis of Book I see Add MS 6785, f. 337, f, 338.) <lb/>
The first wide column shows all the propositions, postulates, definitions, and axioms
used in each proposition in Book III, in the order in which they occur, including repetitions. <lb/>
The second wide column shows the earlier propositions used in each proposition in Book III,
in numerical order without repetitions. <lb/>
Proposition 2, for example, relies on Propositions 5, 16, 19 from Book I, and Proposition 1 from Book III.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s484" xml:space="preserve">
In lib. 3<emph style="super">o</emph>
<lb/>[<emph style="it">tr:
In the third book
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s485" xml:space="preserve">
Definitiones 10
<lb/>[<emph style="it">tr:
Definitions 10
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s486" xml:space="preserve">
propositiones 37
<lb/>[<emph style="it">tr:
propositions 37
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s487" xml:space="preserve">
pr.
<lb/>[<emph style="it">tr:
problems
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s488" xml:space="preserve">
Th.
<lb/>[<emph style="it">tr:
theorems
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s489" xml:space="preserve">
pro-<lb/>
pos.
<lb/>[<emph style="it">tr:
propositions
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f336v" o="336v" n="672"/>
<pb file="add_6785_f337" o="337" n="673"/>
<div xml:id="echoid-div97" type="page_commentary" level="2" n="97">
<p>
<s xml:id="echoid-s490" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s490" xml:space="preserve">
This page contains an analysis of the contents and structure of
Book I of Euclid's <emph style="it">Elements</emph>.
The list at the top of the page gives the number of definitions, postulates, axioms, and propositions
given by various authors.
The table in the lower half of the page lists the individual propositions,
each classified as a Problem or a Theorem,
with notes of the postulates, definitions, and axioms used in each. <lb/>
The first wide column lists all the postulates, definitions, and axioms used in each proposition,
in the order in which they occur, including repetitions. <lb/>
The second wide column lists the postulates, definitions, and axioms in numerical order without repetitions. <lb/>
The third wide column lists previous propositions used. <lb/>
The final narrow column shows the total number of previous propositions used. <lb/>
Proposition 2, for example uses Definition 15; Postulates 1, 2, 3; Axioms 1, 3; and Proposition 1. <lb/>
<lb/>
The editions of Euclid referred to on this page are: <lb/>
Federico Commandino, <emph style="it">Euclidis Elementorum libri XV</emph> (1572) <lb/>
Christophor Clavius, <emph style="it">Euclidis Elementorum libri XV</emph> (1574, 1589, 1591, 1603, 1607),
to which was added a supposed sixteenth book, <emph style="it">De solidorum regularium comparatione</emph>
by Francis Flussas Candalla (François de Foix comte de Candale).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head100" xml:space="preserve" xml:lang="lat">
In Libro primo Elementorum Euclidis
<lb/>[<emph style="it">tr:
In Book I of Euclid's <emph style="it">Elements</emph>
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s492" xml:space="preserve">
Definitiones
<lb/>[<emph style="it">tr:
Definitions
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s493" xml:space="preserve">
Graeco codice et apud Comandinum. 35 <lb/>
Secundum Flussatum et Clavium. 36
<lb/>[<emph style="it">tr:
In a Greek codex and in Commandino 35; and in Flussas and Clavius 36.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s494" xml:space="preserve">
Postulata.
<lb/>[<emph style="it">tr:
Postulates
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s495" xml:space="preserve">
In graecis quibusdum codicibus et <lb/>
et [???] Gemini. 3
<lb/>[<emph style="it">tr:
In certain Greek codices and in [???] Gemini. 3
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s496" xml:space="preserve">
In alijs graecis codicibus et apud <lb/>
Comandinum. 5
<lb/>[<emph style="it">tr:
In other Greek codices and in Commandino. 5
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s497" xml:space="preserve">
Apud Clavium. 4
<lb/>[<emph style="it">tr:
In Clavius. 4
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s498" xml:space="preserve">
Axiomata
<lb/>[<emph style="it">tr:
Axioms
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s499" xml:space="preserve">
In Graecis quibusdum. 12
<lb/>[<emph style="it">tr:
In certain Greek authors 12
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s500" xml:space="preserve">
Apud Comandinum. 10
<lb/>[<emph style="it">tr:
In Commandino 10
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s501" xml:space="preserve">
Apud Clavium. 20
<lb/>[<emph style="it">tr:
In Clavius 20
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s502" xml:space="preserve">
Problemata. 14
<lb/>[<emph style="it">tr:
Problems 14
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s503" xml:space="preserve">
Theoremata. 34
<lb/>[<emph style="it">tr:
Theorems 34
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s504" xml:space="preserve">
propositiones. 48
<lb/>[<emph style="it">tr:
Propositions 48
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s505" xml:space="preserve">
probl.
<lb/>[<emph style="it">tr:
problems
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s506" xml:space="preserve">
Theor.
<lb/>[<emph style="it">tr:
theorems
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s507" xml:space="preserve">
prop.
<lb/>[<emph style="it">tr:
propositions
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f337v" o="337v" n="674"/>
<pb file="add_6785_f338" o="338" n="675"/>
<div xml:id="echoid-div98" type="page_commentary" level="2" n="98">
<p>
<s xml:id="echoid-s508" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s508" xml:space="preserve">
This page continues the table begun on Add MS 6785, f. 337, to the end of Book I of Euclid's
<emph style="it">Elements</emph>.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head101" xml:space="preserve" xml:lang="lat">
Lib. I.
<lb/>[<emph style="it">tr:
Book I
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s510" xml:space="preserve">
pro-<lb/>
blema
<lb/>[<emph style="it">tr:
problems
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s511" xml:space="preserve">
The.
<lb/>[<emph style="it">tr:
theorems
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s512" xml:space="preserve">
p.
<lb/>[<emph style="it">tr:
propositions
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f338v" o="338v" n="676"/>
<pb file="add_6785_f339" o="339" n="677"/>
<pb file="add_6785_f339v" o="339v" n="678"/>
<pb file="add_6785_f340" o="340" n="679"/>
<head xml:id="echoid-head102" xml:space="preserve" xml:lang="lat">
vide proclum. lib. 4<emph style="it">o</emph>. <lb/>
pag. 222.
<lb/>[<emph style="it">tr:
See Proclus, Book 4, page 222.
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s513" xml:space="preserve">
Trochus
<lb/>[<emph style="it">tr:
Toy wheel
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s514" xml:space="preserve">
Et sic infinitum
<lb/>[<emph style="it">tr:
And thus infinitely
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f340v" o="340v" n="680"/>
<pb file="add_6785_f341" o="341" n="681"/>
<pb file="add_6785_f341v" o="341v" n="682"/>
<pb file="add_6785_f342" o="342" n="683"/>
<head xml:id="echoid-head103" xml:space="preserve">
1) 1.Δ.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s515" xml:space="preserve">
hoc est quadrato superficiei <lb/>
trianguli ABC
<lb/>[<emph style="it">tr:
that is, the square of the surface of triangle ABC
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f342v" o="342v" n="684"/>
<pb file="add_6785_f343" o="343" n="685"/>
<head xml:id="echoid-head104" xml:space="preserve">
2.) 2.Δ.
</head>
<p xml:lang="lat">
<s xml:id="echoid-s516" xml:space="preserve">
[<emph style="it">Note:
Sheet (1Δ) is the previous one, Add MS 6785, f. 342.
 </emph>]<lb/>
eadem species ut in (1Δ)
<lb/>[<emph style="it">tr:
the same form as in 1Δ
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s517" xml:space="preserve">
ergo. perpendiculum AD. et superficies Δ<emph style="super">i</emph>, ABC
habet eadem species ut (1Δ)
<lb/>[<emph style="it">tr:
therefore the perpendicular AD and the surface of the triangle ABC have the same form as in 1Δ
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s518" xml:space="preserve">
Quæritur perpendiculum AD. aliter
<lb/>[<emph style="it">tr:
The perpendicular AD is sought another way
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s519" xml:space="preserve">
idem ut supra. (1Δ.)
<lb/>[<emph style="it">tr:
the same as above (1Δ)
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f343v" o="343v" n="686"/>
<pb file="add_6785_f344" o="344" n="687"/>
<head xml:id="echoid-head105" xml:space="preserve">
2.2<emph style="super">o</emph>
</head>
<pb file="add_6785_f344v" o="344v" n="688"/>
<pb file="add_6785_f345" o="345" n="689"/>
<div xml:id="echoid-div99" type="page_commentary" level="2" n="99">
<p>
<s xml:id="echoid-s520" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s520" xml:space="preserve">
The diagram on this page is copied from Clavius, <emph style="it">Geometria practica</emph> (1604), page 176.
On pages 175 to 178 Clavius states and proves what is usually known as Heron's Rule,
for the area of a triangle given its sides.
Harriot translates Clavius 19s verbal proof into symbols, and adds variants of his own. <lb/>
In the second edition of the <emph style="it">Geometria practica</emph>, of 1606,
the same text appears on pages 158 to 161, with the diagram on page 159. <lb/>
See Also Add MS 6785, f. 33, for the same problem and a similar diagram, there from Ramus,
<emph style="it">Scholarum mathematicarum libri unus et triginta</emph> (1569).
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head106" xml:space="preserve" xml:lang="lat">
2.3<emph style="super">o</emph> Cla. pa. 176. Geom. pract.
<lb/>[<emph style="it">tr:
Clavius, page 176, <emph style="it">Geometria practica</emph>
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s522" xml:space="preserve">
Clavius
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s523" xml:space="preserve">
Ego aliter, et brevissimia
<lb/>[<emph style="it">tr:
Another way of my own, much shorter
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s524" xml:space="preserve">
Ego aliter
<lb/>[<emph style="it">tr:
Another way of mine
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f345v" o="345v" n="690"/>
<pb file="add_6785_f346" o="346" n="691"/>
<head xml:id="echoid-head107" xml:space="preserve">
2.4<emph style="super">o</emph>
</head>
<pb file="add_6785_f346v" o="346v" n="692"/>
<pb file="add_6785_f347" o="347" n="693"/>
<head xml:id="echoid-head108" xml:space="preserve">
3.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s525" xml:space="preserve">
superficies Δ<emph style="super">i</emph>
<lb/>[<emph style="it">tr:
surface of the triangle
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s526" xml:space="preserve">
dimidium laterum
<lb/>[<emph style="it">tr:
half of the side
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s527" xml:space="preserve">
recte ita
<lb/>[<emph style="it">tr:
correctly thus
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f347v" o="347v" n="694"/>
<pb file="add_6785_f348" o="348" n="695"/>
<head xml:id="echoid-head109" xml:space="preserve">
4.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s528" xml:space="preserve">
non
<lb/>[<emph style="it">tr:
not
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s529" xml:space="preserve">
AD perpend
<lb/>[<emph style="it">tr:
AD perpendicula
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s530" xml:space="preserve">
corallarium ad sinus vel chordus.
<lb/>[<emph style="it">tr:
pertaining to the sine or chord.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s531" xml:space="preserve">
Dato diameter et duabus subtensis <lb/>
datur tertia.
<lb/>[<emph style="it">tr:
Given the diamter and two chords, one is given the third.
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s532" xml:space="preserve">
dupla superficies Δ
<lb/>[<emph style="it">tr:
twice the area of the triangle
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f348v" o="348v" n="696"/>
<pb file="add_6785_f349" o="349" n="697"/>
<head xml:id="echoid-head110" xml:space="preserve">
5.)
</head>
<p xml:lang="lat">
<s xml:id="echoid-s533" xml:space="preserve">
Data:
<lb/>[<emph style="it">tr:
Given
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s534" xml:space="preserve">
Triangula
<lb/>[<emph style="it">tr:
Triangle
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s535" xml:space="preserve">
Quæritur
<lb/>[<emph style="it">tr:
Sought
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s536" xml:space="preserve">
Dantur
<lb/>[<emph style="it">tr:
Given
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s537" xml:space="preserve">
ergo et, CG differentia
<lb/>[<emph style="it">tr:
therfore also CG, the difference
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s538" xml:space="preserve">
Dantur
<lb/>[<emph style="it">tr:
Given
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s539" xml:space="preserve">
Ergo
<lb/>[<emph style="it">tr:
Therefore
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s540" xml:space="preserve">
Notum igitur
<lb/>[<emph style="it">tr:
Note therefore
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f349v" o="349v" n="698"/>
<pb file="add_6785_f350" o="350" n="699"/>
<div xml:id="echoid-div100" type="page_commentary" level="2" n="100">
<p>
<s xml:id="echoid-s541" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s541" xml:space="preserve">
Franciscus Flussas Candalla (François de Foix, comte de Candale) added a sixteenth book
to Clavius's <emph style="it">Euclidis Elementorum</emph> (1574, 1589, 1591, 1603, 1607).
In it he claimed to compare the propositions of Book XV in the same way that
Book XIV compared the propositions of Book XIII.
The reference to Proposition 37 is puzzling, however, since Book XVI contains only 31 propositions. <lb/>
See also Add MS 6783, f. 318v.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head111" xml:space="preserve" xml:lang="lat">
1.) De triangulis poristica
</head>
<p xml:lang="">
<s xml:id="echoid-s543" xml:space="preserve">
Vide Flussata <lb/>
lib.16.p.37. <lb/>
cor.2
<lb/>[<emph style="it">tr:
See Flussas, Book 16, Proposition 37, Corollary 2.
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s544" xml:space="preserve">
poristicum 1<emph style="super">a</emph> arithmetica proportio
<lb/>[<emph style="it">tr:
first proof; arithmetic proportion
</emph>]<lb/>
</s>
</p>
<p xml:lang="">
<s xml:id="echoid-s545" xml:space="preserve">
poristicum alterum
<lb/>[<emph style="it">tr:
another proof
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f350v" o="350v" n="700"/>
<pb file="add_6785_f351" o="351" n="701"/>
<head xml:id="echoid-head112" xml:space="preserve">
6.)
</head>
<p>
<s xml:id="echoid-s546" xml:space="preserve">
The 6 sides of a <lb/>
pyramis being geven <lb/>
to find the solsidity.
</s>
</p>
<p>
<s xml:id="echoid-s547" xml:space="preserve">
The sides <lb/>
b, c, d, f, g, h.
</s>
</p>
<p>
<s xml:id="echoid-s548" xml:space="preserve">
I would have the <lb/>
solidity geven without <lb/>
the table of sines. <lb/>
or <lb/>
in specie of <lb/>
the sides only <lb/>
if it may be.
</s>
<lb/>
<s xml:id="echoid-s549" xml:space="preserve">
as the superficies <lb/>
of a triangle
</s>
</p>
<p>
<s xml:id="echoid-s550" xml:space="preserve">
As the superficies of a trinagle is <emph style="st">had</emph> argued by a circle inscribed, two sides <lb/>
produced, &amp; like triangles: In like manner remember to try to argue <lb/>
the solidity of a pyramis, by a sphære inscribed, three planes produced &amp; <lb/>
like pyramides.
</s>
<s xml:id="echoid-s551" xml:space="preserve">
Now followeth the way by <emph style="super">the</emph> perpendicular only.
</s>
</p>
<p>
<s xml:id="echoid-s552" xml:space="preserve">
If the vertex of a <emph style="super">triangular</emph> pyramis to the base be understood <lb/>
a perpendicular falling, &amp; from the end or poynt in the base <lb/>
be drawne perpendiculars to the sides of the triangles of <lb/>
the base:
</s>
</p>
<p>
<s xml:id="echoid-s553" xml:space="preserve">
pappus <lb/>
lib.6. <lb/>
pr. 43.
</s>
</p>
<p>
<s xml:id="echoid-s554" xml:space="preserve">
Then if from the vertex be drawne lines to the sayd poyntes <lb/>
in the triangular base; <emph style="st">where</emph> those lines shalbe also perpendicular.
</s>
</p>
<p>
<s xml:id="echoid-s555" xml:space="preserve">
Therefore:
</s>
<lb/>
<s xml:id="echoid-s556" xml:space="preserve">
Let two perpendiculars be drawne aθ and αx <lb/>
&amp; suppose the perpendicular from the vertex to the playne of <lb/>
the base be αε.
</s>
</p>
<p>
<s xml:id="echoid-s557" xml:space="preserve">
Then Drawe the lines θε, εx, θx.
</s>
</p>
<p>
<s xml:id="echoid-s558" xml:space="preserve">
θδ &amp; δθ with the anlge θδx are knowne, <lb/>
therefore θx with his angle adjacent are also knwone.
</s>
<lb/>
<s xml:id="echoid-s559" xml:space="preserve">
Therefore in the triangle θxε, besides the side θx the two angles <lb/>
adiacent are also knowne. therefore also the sides θε and εx.
</s>
</p>
<p>
<s xml:id="echoid-s560" xml:space="preserve">
Then in the triangle αxε having αεx a right angle; &amp; the <lb/>
two sides εx &amp; xα being knowne, αε cannot be unknowne.
</s>
<lb/>
<s xml:id="echoid-s561" xml:space="preserve">
And therefore the solidity of the pyramis wilbe also knowne.
</s>
</p>
<pb file="add_6785_f351v" o="351v" n="702"/>
<pb file="add_6785_f352" o="352" n="703"/>
<head xml:id="echoid-head113" xml:space="preserve">
6.2<emph style="super">o</emph>)
</head>
<pb file="add_6785_f352v" o="352v" n="704"/>
<pb file="add_6785_f353" o="353" n="705"/>
<head xml:id="echoid-head114" xml:space="preserve">
7.)
</head>
<pb file="add_6785_f353v" o="353v" n="706"/>
<pb file="add_6785_f354" o="354" n="707"/>
<pb file="add_6785_f354v" o="354v" n="708"/>
<pb file="add_6785_f355" o="355" n="709"/>
<div xml:id="echoid-div101" type="page_commentary" level="2" n="101">
<p>
<s xml:id="echoid-s562" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s562" xml:space="preserve">
On this page, Harriot examines Problem V from Appendix II from Viète's
<emph style="it">Apollonius Gallus</emph> (1600).
</s>
<lb/>
<quote xml:lang="lat">
Appendicula II. <lb/>
De problemata quorum factionem geometricam non tradunt astronomi, itaque infeliciter resolvunt. <lb/>
Problema V. <lb/>
Dato triangulo, invenire punctum, a quo ad apices dati trianguli actæ tres lineæ rectæ imperatam teneant rationem. <lb/>
</quote>
<lb/>
<quote>
Appendix II. <lb/>
On problems whose geometric construction the astronomers do not teach,
thereby resolving them imperfectly. <lb/>
Problem V. <lb/>
Given a triangle, to find a point from which there may be drawn three straight lines
to the vertices of the given triangle, keeping a fixed ratio.
</quote>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head115" xml:space="preserve" xml:lang="lat">
Quinta et ultima propositio appendiculæ 2<emph style="super"/> <emph style="it"/>
Appollonij Galli
<lb/>[<emph style="it">tr:
Fifth and last proposition from Appendiula II of Apollonius Gallus
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s564" xml:space="preserve">
Dato triangulo: invenire punctum, a quo ad apices dati <lb/>
triangli actæ tres lineæ rectæ imperatam teneant <lb/>
rationem. Si sit possibile.
<lb/>[<emph style="it">tr:
Given a triangle, to find a point from which to the given vertices of the triangle
there are constructed three straight lines in a determined ratio. If it is possible.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f355v" o="355v" n="710"/>
<pb file="add_6785_f356" o="356" n="711"/>
<pb file="add_6785_f356v" o="356v" n="712"/>
<pb file="add_6785_f357" o="357" n="713"/>
<pb file="add_6785_f357v" o="357v" n="714"/>
<pb file="add_6785_f358" o="358" n="715"/>
<pb file="add_6785_f358v" o="358v" n="716"/>
<pb file="add_6785_f359" o="359" n="717"/>
<pb file="add_6785_f359v" o="359v" n="718"/>
<pb file="add_6785_f360" o="360" n="719"/>
<pb file="add_6785_f360v" o="360v" n="720"/>
<pb file="add_6785_f361" o="361" n="721"/>
<div xml:id="echoid-div102" type="page_commentary" level="2" n="102">
<p>
<s xml:id="echoid-s565" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s565" xml:space="preserve">
The reference at the top of the page is to a lemma given by Federico Commandino to Proposition 43
of Book X of Euclid's <emph style="it">Elements</emph>, in his
<emph style="it">Euclidis Elementorum XV</emph> (1572). The original proposition is: <lb/>
X.43 A first bimedial straightline is divided at one point only. <lb/>
There are also references to Book II, Propositions 5 and 9 (marked as 5,2. and 9,2.): <lb/>
II.5. If a straight line is cut into equal and unequal segments,
then the rectangle contained by the unequal segments of the whole
together with the square on the straight line between the points of section equals the square on the half. <lb/>
II.9. If a straight line is cut into equal and unequal segments,
then the sum of the squares on the unequal segments of the whole is double the sum of the square on the half
and the square on the straight line between the points of section.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head116" xml:space="preserve" xml:lang="lat">
Comandinus <lb/>
Lemma ad 43 p. 10. lib. el.
<lb/>[<emph style="it">tr:
Commandino, Lemma to Proposition 43, Book X of the Elements
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s567" xml:space="preserve">
est minus
<lb/>[<emph style="it">tr:
is less than
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s568" xml:space="preserve">
maius
<lb/>[<emph style="it">tr:
greater
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s569" xml:space="preserve">
conclusio
<lb/>[<emph style="it">tr:
conclusion
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s570" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr:
Another way
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s571" xml:space="preserve">
maius
<lb/>[<emph style="it">tr:
greater
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s572" xml:space="preserve">
prima conclusio
<lb/>[<emph style="it">tr:
first conclusion
</emph>]<lb/>
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s573" xml:space="preserve">
sed <lb/>
maius est <lb/>[...]<lb/> per prima demonstratio
<lb/>[<emph style="it">tr:
but <lb/>[...]<lb/> is greater than <lb/>[...]<lb/> by the first demonstration
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s574" xml:space="preserve">
ergo <lb/>
minus est <lb/>[...]<lb/> secunda conclusio
<lb/>[<emph style="it">tr:
therefore <lb/>[...]<lb/> is less than <lb/>[...]<lb/> second conclusion
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f361v" o="361v" n="722"/>
<pb file="add_6785_f362" o="362" n="723"/>
<pb file="add_6785_f362v" o="362v" n="724"/>
<pb file="add_6785_f363" o="363" n="725"/>
<div xml:id="echoid-div103" type="page_commentary" level="2" n="103">
<p>
<s xml:id="echoid-s575" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s575" xml:space="preserve">
The table analyses the first 56 propositions of Book I of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566),
showing the 'parents' of each proposition.
(For a similar analysis of Book I of Euclid's <emph style="it">Elements</emph> see Add MS 6785, f. 337.)
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head117" xml:space="preserve" xml:lang="lat">
Appoll. lib. 1.
<lb/>[<emph style="it">tr:
Apollonius, Book I
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s577" xml:space="preserve">
prop. parentes.
<lb/>[<emph style="it">tr:
Proposition. Parents.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f363v" o="363v" n="726"/>
<pb file="add_6785_f364" o="364" n="727"/>
<div xml:id="echoid-div104" type="page_commentary" level="2" n="104">
<p>
<s xml:id="echoid-s578" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s578" xml:space="preserve">
The table analyses the first 56 propositions of Book I of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566),
showing the 'offspring' of each proposition.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head118" xml:space="preserve" xml:lang="lat">
Appoll. lib. 1.
<lb/>[<emph style="it">tr:
Apollonius, Book I
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s580" xml:space="preserve">
prop. soboles.
<lb/>[<emph style="it">tr:
Proposition. Offspring.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f364v" o="364v" n="728"/>
<pb file="add_6785_f365" o="365" n="729"/>
<div xml:id="echoid-div105" type="page_commentary" level="2" n="105">
<p>
<s xml:id="echoid-s581" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s581" xml:space="preserve">
The table analyses the first 11 (of 53) propositions of Book II of Apollonius, as edited by Commandino in
<emph style="it">Conicorum libri quattuor</emph> (1566),
showing their 'parent' propositions in Book I and Book II.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head119" xml:space="preserve" xml:lang="lat">
Appoll. lib. 2.
<lb/>[<emph style="it">tr:
Apollonius, Book II
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s583" xml:space="preserve">
parentes. <lb/>
lib. 1
<lb/>[<emph style="it">tr:
parents, Book I
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s584" xml:space="preserve">
prop.
<lb/>[<emph style="it">tr:
proposition
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s585" xml:space="preserve">
parentes <lb/>
lib. 2.
<lb/>[<emph style="it">tr:
parents, Book II
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f365v" o="365v" n="730"/>
<pb file="add_6785_f366" o="366" n="731"/>
<pb file="add_6785_f366v" o="366v" n="732"/>
<pb file="add_6785_f367" o="367" n="733"/>
<pb file="add_6785_f367v" o="367v" n="734"/>
<pb file="add_6785_f368" o="368" n="735"/>
<pb file="add_6785_f368v" o="368v" n="736"/>
<pb file="add_6785_f369" o="369" n="737"/>
<pb file="add_6785_f369v" o="369v" n="738"/>
<pb file="add_6785_f370" o="370" n="739"/>
<pb file="add_6785_f370v" o="370v" n="740"/>
<pb file="add_6785_f371" o="371" n="741"/>
<pb file="add_6785_f371v" o="371v" n="742"/>
<pb file="add_6785_f372" o="372" n="743"/>
<pb file="add_6785_f372v" o="372v" n="744"/>
<pb file="add_6785_f373" o="373" n="745"/>
<pb file="add_6785_f373v" o="373v" n="746"/>
<pb file="add_6785_f374" o="374" n="747"/>
<pb file="add_6785_f374v" o="374v" n="748"/>
<pb file="add_6785_f375" o="375" n="749"/>
<pb file="add_6785_f375v" o="375v" n="750"/>
<pb file="add_6785_f376" o="376" n="751"/>
<pb file="add_6785_f376v" o="376v" n="752"/>
<pb file="add_6785_f377" o="377" n="753"/>
<pb file="add_6785_f377v" o="377v" n="754"/>
<pb file="add_6785_f378" o="378" n="755"/>
<pb file="add_6785_f378v" o="378v" n="756"/>
<pb file="add_6785_f379" o="379" n="757"/>
<pb file="add_6785_f379v" o="379v" n="758"/>
<pb file="add_6785_f380" o="380" n="759"/>
<pb file="add_6785_f380v" o="380v" n="760"/>
<pb file="add_6785_f381" o="381" n="761"/>
<pb file="add_6785_f381v" o="381v" n="762"/>
<pb file="add_6785_f382" o="382" n="763"/>
<pb file="add_6785_f382v" o="382v" n="764"/>
<pb file="add_6785_f383" o="383" n="765"/>
<div xml:id="echoid-div106" type="page_commentary" level="2" n="106">
<p>
<s xml:id="echoid-s586" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s586" xml:space="preserve">
On the left of the page, Harriot tests several quadratic equations for two roots,
one positive and one negative in each case. <lb/>
It is not clear what is going on at A and B since <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mn>6</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mo>-</mo><mn>4</mn></mstyle></math>
are not correct solutions to the given equations. <lb/>
'W.W.' is presumably Walter Warner.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s588" xml:space="preserve">
Ad Impossibilia <emph style="super">W.W.</emph> responsa.
<lb/>[<emph style="it">tr:
On impossibility, response to W.W.
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f383v" o="383v" n="766"/>
<pb file="add_6785_f384" o="384" n="767"/>
<pb file="add_6785_f384v" o="384v" n="768"/>
<p>
<s xml:id="echoid-s589" xml:space="preserve">
4. If more be more &amp; lesse be lesse <lb/>
3. Lesse by lesse brings lesse of lesse <lb/>
2. Lesse by more brings lesse of more <lb/>
1. More by more must needes bring more. <lb/>
Novemb. 23. 1558.
</s>
</p>
<pb file="add_6785_f385" o="385" n="769"/>
<p>
<s xml:id="echoid-s590" xml:space="preserve">
Double solutions
</s>
<lb/>
<s xml:id="echoid-s591" xml:space="preserve">
Single.
</s>
</p>
<pb file="add_6785_f385v" o="385v" n="770"/>
<div xml:id="echoid-div107" type="page_commentary" level="2" n="107">
<p>
<s xml:id="echoid-s592" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s592" xml:space="preserve">
This page shows a table of vales for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>z</mi></mstyle></math> (in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>2</mn><mi>x</mi><mo>-</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle></math>),
for <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>r</mi><mo>=</mo></mstyle></math> 1, 2, 3, 4, 5, 6, 7. The required value, 5, does not appear in the column of possible values.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<pb file="add_6785_f386" o="386" n="771"/>
<pb file="add_6785_f386v" o="386v" n="772"/>
<pb file="add_6785_f387" o="387" n="773"/>
<pb file="add_6785_f387v" o="387v" n="774"/>
<pb file="add_6785_f388" o="388" n="775"/>
<pb file="add_6785_f388v" o="388v" n="776"/>
<pb file="add_6785_f389" o="389" n="777"/>
<head xml:id="echoid-head120" xml:space="preserve" xml:lang="lat">
Ad Quintem lib. Euclidis
<lb/>[<emph style="it">tr:
On the fifith book of Euclid
</emph>]<lb/>
</head>
<p xml:lang="lat">
<s xml:id="echoid-s594" xml:space="preserve">
Quicquid intelligatur; et dicitur, vel dici potest, esse: appellatur [???] <lb/>
modus vel forma essendi qui vel qua <lb/>
Malus quo, vel forma qua utiquid dicitur esse; appellatur entitas <lb/>
vel essentia.
</s>
</p>
<pb file="add_6785_f389v" o="389v" n="778"/>
<pb file="add_6785_f390" o="390" n="779"/>
<pb file="add_6785_f390v" o="390v" n="780"/>
<pb file="add_6785_f391" o="391" n="781"/>
<pb file="add_6785_f391v" o="391v" n="782"/>
<pb file="add_6785_f392" o="392" n="783"/>
<pb file="add_6785_f392v" o="392v" n="784"/>
<div xml:id="echoid-div108" type="page_commentary" level="2" n="108">
<p>
<s xml:id="echoid-s595" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s595" xml:space="preserve">
The text at the top of the page uses Stevin's notation, 5(2), for example, for what we would now write as <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>5</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle></math>. <lb/>
At the bottom of the page are two references to Stevin's <emph style="it">L 19arithmétique … aussi l 19algebre</emph>,
pages 289 and 293.
On page 289 Stevin deals with equations of the form: square = number – roots.
On page 293 he deals with the form: square = roots – number.
Stevin's example is 1(2) = 6(1) – 5 (in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>5</mn></mstyle></math>), which has two real roots, 1 and 5.
Harriot's example <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>z</mi><mo>=</mo><mn>2</mn><mi>r</mi><mo>-</mo><mn>5</mn></mstyle></math> (in modern notation <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>5</mn></mstyle></math>) has no real roots.
The annotation 'W.W.' is presumably a reference to Harriot's friend Walter Warner.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<p xml:lang="lat">
<s xml:id="echoid-s597" xml:space="preserve">
to find a number which being multiplied by 3. &amp; the product mulltiplied into it self <lb/>
may be equal to the first number multiplied by it self, <emph style="st">after</emph> and the product by 5.
</s>
<lb/>
<s xml:id="echoid-s598" xml:space="preserve">
Suppose the number 1(1) to be multiplied by 3 to be 3(1) which multiplied into it self makes 9(2)
</s>
<lb/>
<s xml:id="echoid-s599" xml:space="preserve">
after, multiplie the first supposed number being 1(1) into it self which is 1(2) and the same <lb/>
1(2) multiplie by 5 the product shalbe 5(2) which must be equal to 9(2) which <lb/>
equation is impossible
</s>
</p>
<p xml:lang="lat">
<s xml:id="echoid-s600" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>z</mi><mo>=</mo><mn>7</mn><mo>-</mo><mn>8</mn><mi>r</mi></mstyle></math>. 289. Stevin.
</s>
<lb/>
<s xml:id="echoid-s601" xml:space="preserve">
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>1</mn><mi>z</mi><mo>=</mo><mn>2</mn><mi>r</mi><mo>-</mo><mn>5</mn></mstyle></math> imposib. W.W. 293
</s>
</p>
<pb file="add_6785_f393" o="393" n="785"/>
<pb file="add_6785_f393v" o="393v" n="786"/>
<pb file="add_6785_f394" o="394" n="787"/>
<pb file="add_6785_f394v" o="394v" n="788"/>
<pb file="add_6785_f395" o="395" n="789"/>
<pb file="add_6785_f395v" o="395v" n="790"/>
<pb file="add_6785_f396" o="396" n="791"/>
<pb file="add_6785_f396v" o="396v" n="792"/>
<pb file="add_6785_f397" o="397" n="793"/>
<p>
<s xml:id="echoid-s602" xml:space="preserve">
[<emph style="it">Note:
'W.W.' is presumably a reference to Walter Warner.
The same equation appears again on the other side of this page, Add MS 6785, f. 397v.
 </emph>]<lb/>
<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mn>1</mn><mn>6</mn></mstyle></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mn>6</mn><mn>4</mn></mstyle></math> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mn>8</mn><mn>0</mn><mi>r</mi><mo>-</mo><mn>1</mn><mi>z</mi><mi>z</mi></mstyle></math> imposs. W.W.
</s>
</p>
<pb file="add_6785_f397v" o="397v" n="794"/>
<pb file="add_6785_f398" o="398" n="795"/>
<pb file="add_6785_f398v" o="398v" n="796"/>
<pb file="add_6785_f399" o="399" n="797"/>
<pb file="add_6785_f399v" o="399v" n="798"/>
<pb file="add_6785_f400" o="400" n="799"/>
<pb file="add_6785_f400v" o="400v" n="800"/>
<pb file="add_6785_f401" o="401" n="801"/>
<pb file="add_6785_f401v" o="401v" n="802"/>
<pb file="add_6785_f402" o="402" n="803"/>
<pb file="add_6785_f402v" o="402v" n="804"/>
<pb file="add_6785_f403" o="403" n="805"/>
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<head xml:id="echoid-head121" xml:space="preserve" xml:lang="lat">
De infnitis
</head>
<p xml:lang="lat">
<s xml:id="echoid-s603" xml:space="preserve">
An sit maximum <emph style="st">finitis</emph> et minimum finitum. <lb/>
An sit minimum et maximum infinitum. <lb/>
An ex finito generetur infinitum. <lb/>
An ex finitis componatur infinitum. <lb/>
An resolvatur finitum in indidivisibilia. <lb/>
<emph style="st">vel</emph> An componatur finitum ex indivisibilibus. <lb/>
An a finito ad infinitum sint transitis per maximum finitum. <lb/>
An æquale et inæquale possit [???] omniciarii de infinitis. <lb/>
An æquale et inæquale attribuatur indivisibilibus.
</s>
</p>
<p>
<s xml:id="echoid-s604" xml:space="preserve">
Much ado about nothing. <lb/>
Great warres &amp; no blows. <lb/>
Who is the foole now.
</s>
</p>
<pb file="add_6785_f436v" o="436v" n="872"/>
<pb file="add_6785_f437" o="437" n="873"/>
<head xml:id="echoid-head122" xml:space="preserve" xml:lang="lat">
1. Achilles
</head>
<pb file="add_6785_f437v" o="437v" n="874"/>
<pb file="add_6785_f438" o="438" n="875"/>
<pb file="add_6785_f438v" o="438v" n="876"/>
<pb file="add_6785_f439" o="439" n="877"/>
<div xml:id="echoid-div109" type="page_commentary" level="2" n="109">
<p>
<s xml:id="echoid-s605" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s605" xml:space="preserve">
This page is based on Euclid, Proposition XIII.18: <lb/>
XIII.18 To set out the sides of the five figures and compare them with one another.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head123" xml:space="preserve" xml:lang="lat">
Euclid.lib.13.pr.18 De lateribus corporum regularium
<lb/>[<emph style="it">tr:
Euclid Book XIII, Proposition 18: On the sides of regular solids
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s607" xml:space="preserve">
secetur <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> extrema et <lb/>
media ratione
<lb/>[<emph style="it">tr:
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>f</mi></mstyle></math> be cut in extreme and mean ratio
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s608" xml:space="preserve">
Collectio
<lb/>[<emph style="it">tr:
Collection
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s609" xml:space="preserve">
Dimetiens spæræ
<lb/>[<emph style="it">tr:
Meaure of a sphere
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s610" xml:space="preserve">
Latus pyramidis
<lb/>[<emph style="it">tr:
Side of a pyramid
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s611" xml:space="preserve">
Latus octaedri
<lb/>[<emph style="it">tr:
Side of an octahedron
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s612" xml:space="preserve">
Latus cubi
<lb/>[<emph style="it">tr:
Side of a cube
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s613" xml:space="preserve">
Latus Icosaedri
<lb/>[<emph style="it">tr:
Side of an icosahedron
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s614" xml:space="preserve">
Latus Dodecaedri
<lb/>[<emph style="it">tr:
Side of a dodecahedron
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f439v" o="439v" n="878"/>
<pb file="add_6785_f440" o="440" n="879"/>
<pb file="add_6785_f440v" o="440v" n="880"/>
<pb file="add_6785_f441" o="441" n="881"/>
<div xml:id="echoid-div110" type="page_commentary" level="2" n="110">
<p>
<s xml:id="echoid-s615" xml:space="preserve">[<emph style="it">Note:
<p>
<s xml:id="echoid-s615" xml:space="preserve">
This page and the next explore sides of polygons inscribed in circles.
Harriot refers to two propoistions from Euclid Book XIII: <lb/>
XIII.12 If an equilateral triangle is inscribed in a circle,
then the square on the side of the triangle is triple the square on the radius of the circle. <lb/>
XIII.9 If the side of the hexagon and that of the decagon inscribed in the same circle be added together,
the whole straight line has been cut in extreme and mean ratio,
and its greater segment is the side of the hexagon. <lb/>
There is also a reference to Euclid X. 13: <lb/>
X.13 If two magnitudes be commensurable, and one of them be incommensurable with some magnitude,
the remaning one will also be incommensuarble with the same.
</s>
</p>
</emph>]
<lb/><lb/></s></p></div>
<head xml:id="echoid-head124" xml:space="preserve" xml:lang="lat">
De lateribus polygonum in circulo
<lb/>[<emph style="it">tr:
On the sides of polygons in circles
</emph>]<lb/>
</head>
<p>
<s xml:id="echoid-s617" xml:space="preserve">
Euclid.lib.13.pr.12
Euclid Book XIII, Propostion 12.
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s618" xml:space="preserve">
latus trianguli
<lb/>[<emph style="it">tr:
The side of a triangle
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s619" xml:space="preserve">
lib.13.pr.9
<lb/>[<emph style="it">tr:
Book XIII, Proposition 9
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s620" xml:space="preserve">
Apot. 5<emph style="super">a</emph>. Decagonalibus
<lb/>[<emph style="it">tr:
A fifth apotome, for decagons
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s621" xml:space="preserve">
cuius quadratum <lb/>[...]<lb/> Apot 1<emph style="super">a</emph>
<lb/>[<emph style="it">tr:
whose square is a first apotome
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s622" xml:space="preserve">
ergo etiam <lb/>[...]<lb/> decagoni latus
<lb/>[<emph style="it">tr:
therefore also <lb/>[...]<lb/> the side of a decagon
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s623" xml:space="preserve">
lateri quadrati
<lb/>[<emph style="it">tr:
sides of squares
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s624" xml:space="preserve">
sint <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math>, et <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math>, lateri pentagoni <lb/>
ergo: <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> latus pentagoni
<lb/>[<emph style="it">tr:
let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>c</mi></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>c</mi><mi>d</mi></mstyle></math> be sides of a pentagon, therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle><mi>b</mi><mi>d</mi></mstyle></math> is the side of a pentagon
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s625" xml:space="preserve">
ut supra latus decagoni
<lb/>[<emph style="it">tr:
as above for the side of a decagon
</emph>]<lb/>
</s>
<lb/>
<lb/>[...]<lb/>
<lb/>
<s xml:id="echoid-s626" xml:space="preserve">
Minor. Latus pentagoni.
<lb/>[<emph style="it">tr:
Lesser. Side of a pentagon.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s627" xml:space="preserve">
per 10.p.13.eucl.
<lb/>[<emph style="it">tr:
by Proposition X.13 of Euclid
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s628" xml:space="preserve">
lateris pentagoni
<lb/>[<emph style="it">tr:
side of a pentagon
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s629" xml:space="preserve">
Lateri pentagoni. <lb/>
Minor
<lb/>[<emph style="it">tr:
Side of a pentagon; lesser.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s630" xml:space="preserve">
verte paginam pro <lb/>
ambitiosa radice.
<lb/>[<emph style="it">tr:
turn the page for the complicated root
</emph>]<lb/>
</s>
</p>
<pb file="add_6785_f441v" o="441v" n="882"/>
<p>
<s xml:id="echoid-s631" xml:space="preserve">
pro ambitiosa latere pentagoni.
<lb/>[<emph style="it">tr:
for the complicated pentagonal root
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s632" xml:space="preserve">
lateris pentagoni
<lb/>[<emph style="it">tr:
side of a pentagon
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s633" xml:space="preserve">
cuius radix
<lb/>[<emph style="it">tr:
whose root
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s634" xml:space="preserve">
Latus pentagoni. <lb/>
Minor.
<lb/>[<emph style="it">tr:
Side of a pentagon; lesser.
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s635" xml:space="preserve">
Aliter
<lb/>[<emph style="it">tr:
Another way.
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s636" xml:space="preserve">
cuius radix
<lb/>[<emph style="it">tr:
whose root is
</emph>]<lb/>
</s>
<lb/>
<s xml:id="echoid-s637" xml:space="preserve">
eadem quæ supra
<lb/>[<emph style="it">tr:
the same as above
</emph>]<lb/>
</s>
</p>
<p>
<s xml:id="echoid-s638" xml:space="preserve">
Radix igitur vera, <lb/>
sed ambitiosa nimis.
<lb/>[<emph style="it">tr:
Therefore the true root, but exceedingly complicated.
</emph>]<lb/>
</s>
</p>
</div>
</text>
</echo>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Thu, 02 May 2013 11:29:00 +0200
parents	22d6a63640c6
children