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| author | casties |
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| date | Fri, 07 Dec 2012 17:05:22 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/texts/archimedesOldCVSRepository/archimedes/raw/heath_mathe_02_en_1981.raw Fri Dec 07 17:05:22 2012 +0100 @@ -0,0 +1,22549 @@ +<pb> +<table> +<caption align=top><B>CONTENTS OF VOL II</B></caption> +<tr><td>XII. ARISTARCHUS OF SAMOS</td><td align=right>PAGES 1-15</td></tr> +<tr><td>XIII. ARCHIMEDES</td><td align=right>16-109</td></tr> +<tr><td>Traditions</td></tr> +<tr><td>(<G>a</G>) Astronomy</td><td align=right>17-18</td></tr> +<tr><td>(<G>b</G>) Mechanics</td><td align=right>18</td></tr> +<tr><td>Summary of main achievements</td><td align=right>19-20</td></tr> +<tr><td>Character of treatises</td><td align=right>20-22</td></tr> +<tr><td>List of works still extant</td><td align=right>22-23</td></tr> +<tr><td>Traces of lost works</td><td align=right>23-25</td></tr> +<tr><td>The text of Archimedes</td><td align=right>25-27</td></tr> +<tr><td>Contents of <I>The Method</I></td><td align=right>27-34</td></tr> +<tr><td><I>On the Sphere and Cylinder</I>, I, II</td><td align=right>34-50</td></tr> +<tr><td>Cubic equation arising out of II. 4</td><td align=right>43-46</td></tr> +<tr><td>(i) Archimedes's own solution</td><td align=right>45-46</td></tr> +<tr><td>(ii) Dionysodorus's solution</td><td align=right>46</td></tr> +<tr><td>(iii) Diocles's solution of original problem</td><td align=right>47-49</td></tr> +<tr><td><I>Measurement of a Circle</I></td><td align=right>50-56</td></tr> +<tr><td><I>On Conoids and Spheroids</I></td><td align=right>56-64</td></tr> +<tr><td><I>On Spirals</I></td><td align=right>64-75</td></tr> +<tr><td><I>On Plane Equilibriums</I>, I, II</td><td align=right>75-81</td></tr> +<tr><td><I>The Sand-reckoner</I> (<I>Psammites</I> or <I>Arenarius</I>)</td><td align=right>81-85</td></tr> +<tr><td><I>The Quadrature of the Parabola</I></td><td align=right>85-91</td></tr> +<tr><td><I>On Floating Bodies</I>, I, II</td><td align=right>91-97</td></tr> +<tr><td>The problem of the crown</td><td align=right>92-94</td></tr> +<tr><td>Other works</td></tr> +<tr><td>(<G>a</G>) The Cattle-Problem</td><td align=right>97-98</td></tr> +<tr><td>(<G>b</G>) On semi-regular polyhedra</td><td align=right>98-101</td></tr> +<tr><td>(<G>g</G>) The <I>Liber Assumptorum</I></td><td align=right>101-103</td></tr> +<tr><td>(<G>d</G>) Formula for area of triangle</td><td align=right>103</td></tr> +<tr><td>Eratosthenes</td><td align=right>104-109</td></tr> +<tr><td>Measurement of the Earth</td><td align=right>106-108</td></tr> +<tr><td>XIV. CONIC SECTIONS. APOLLONIUS OF PERGA</td><td align=right>110-196</td></tr> +<tr><td>A. HISTORY OF CONICS UP TO APOLLONIUS</td><td align=right>110-126</td></tr> +<tr><td>Discovery of the conic sections by Menaechmus</td><td align=right>110-111</td></tr> +<tr><td>Menaechmus's probable procedure</td><td align=right>111-116</td></tr> +<tr><td>Works by Aristaeus and Euclid</td><td align=right>116-117</td></tr> +<tr><td>‘Solid loci’ and ‘solid problems’</td><td align=right>117-118</td></tr> +<tr><td>Aristaeus's <I>Solid Loci</I></td><td align=right>118-119</td></tr> +<tr><td>Focus-directrix property known to Euclid</td><td align=right>119</td></tr> +<tr><td>Proof from Pappus</td><td align=right>120-121</td></tr> +<tr><td>Propositions included in Euclid's <I>Conics</I></td><td align=right>121-122</td></tr> +<tr><td>Conic sections in Archimedes</td><td align=right>122-126</td></tr> +<pb n=vi><head>CONTENTS</head> +<tr><td>XIV. CONTINUED.</td></tr> +<tr><td>B. APOLLONIUS OF PERGA</td><td align=right>PAGES 126-196</td></tr> +<tr><td>The text of the <I>Conics</I></td><td align=right>126-128</td></tr> +<tr><td>Apollonius's own account of the <I>Conics</I></td><td align=right>128-133</td></tr> +<tr><td>Extent of claim to originality</td><td align=right>132-133</td></tr> +<tr><td>Great generality of treatment</td><td align=right>133</td></tr> +<tr><td>Analysis of the <I>Conics</I></td><td align=right>133-175</td></tr> +<tr><td>Book I</td><td align=right>133-148</td></tr> +<tr><td>Conics obtained in the most general way from +oblique cone</td><td align=right>134-138</td></tr> +<tr><td>New names, ‘parabola’, ‘ellipse’, ‘hyperbola’</td><td align=right>138-139</td></tr> +<tr><td>Fundamental properties equivalent to Cartesian +equations</td><td align=right>139-141</td></tr> +<tr><td>Transition to new diameter and tangent at its +extremity</td><td align=right>141-147</td></tr> +<tr><td>First appearance of principal axes</td><td align=right>147-148</td></tr> +<tr><td>Book II</td><td align=right>148-150</td></tr> +<tr><td>Book III</td><td align=right>150-157</td></tr> +<tr><td>Book IV</td><td align=right>157-158</td></tr> +<tr><td>Book V</td><td align=right>158-167</td></tr> +<tr><td>Normals as maxima and minima</td><td align=right>159-163</td></tr> +<tr><td>Number of normals from a point</td><td align=right>163-164</td></tr> +<tr><td>Propositions leading immediately to determination +of <I>evolute</I> of conic</td><td align=right>164-166</td></tr> +<tr><td>Construction of normals</td><td align=right>166-167</td></tr> +<tr><td>Book VI</td><td align=right>167-168</td></tr> +<tr><td>Book VII</td><td align=right>168-174</td></tr> +<tr><td>Other works by Apollenius</td><td align=right>175-194</td></tr> +<tr><td>(<G>a</G>) <I>On the Cutting off of a Ratio</I> (<G>lo/gou a)potomh/</G>), +two Books</td><td align=right>175-179</td></tr> +<tr><td>(<G>b</G>) <I>On the Cutting-off of an Area</I> (<G>xwri/ou a)potomh/</G>), +two Books</td><td align=right>179-180</td></tr> +<tr><td>(<G>g</G>) <I>On Determinate Section</I> (<G>diwrisme/nh tomh/</G>), two +Books</td><td align=right>180-181</td></tr> +<tr><td>(<G>d</G>) <I>On Contacts</I> or <I>Tangencies</I> (<G>e)pafai/</G>), two Books</td><td align=right>181-185</td></tr> +<tr><td>(<G>e</G>) <I>Plane Loci</I>, two Books</td><td align=right>185-189</td></tr> +<tr><td>(<G>z</G>) N<G>eu/seis</G> (<I>Vergings</I> or <I>Inclinations</I>), two Books</td><td align=right>189-192</td></tr> +<tr><td>(<G>h</G>) <I>Comparison of dodecahedron with icosahedron</I></td><td align=right>192</td></tr> +<tr><td>(<G>q</G>) <I>General Treatise</I></td><td align=right>192-193</td></tr> +<tr><td>(<G>i</G>) <I>On the Cochlias</I></td><td align=right>193</td></tr> +<tr><td>(<G>k</G>) <I>On Unordered Irrationals</I></td><td align=right>193</td></tr> +<tr><td>(<G>l</G>) <I>On the Burning-mirror</I></td><td align=right>194</td></tr> +<tr><td>(<G>m</G>) <G>*)wkuto/kion</G></td><td align=right>194</td></tr> +<tr><td>Astronomy</td><td align=right>195-196</td></tr> +<tr><td>XV. THE SUCCESSORS OF THE GREAT GEOMETERS</td><td align=right>197-234</td></tr> +<tr><td>Nicomedes</td><td align=right>199</td></tr> +<tr><td>Diocles</td><td align=right>200-203</td></tr> +<tr><td>Perseus</td><td align=right>203-206</td></tr> +<tr><td>Isoperimetric figures. Zenodorus</td><td align=right>206-213</td></tr> +<tr><td>Hypsicles</td><td align=right>213-218</td></tr> +<tr><td>Dionysodorus</td><td align=right>218-219</td></tr> +<tr><td>Posidonius</td><td align=right>219-222</td></tr> +<pb n=vii><head>CONTENTS</head> +<tr><td>Geminus</td><td align=right>PAGES 222-234</td></tr> +<tr><td>Attempt to prove the Parallel-Postulate</td><td align=right>227-230</td></tr> +<tr><td>On <I>Meteorologica</I> of Posidonius</td><td align=right>231-232</td></tr> +<tr><td><I>Introduction to the Phaenomena</I> attributed to Geminus</td><td align=right>232-234</td></tr> +<tr><td>XVI. SOME HANDBOOKS</td><td align=right>235-244</td></tr> +<tr><td>Cleomedes, <I>De motu circulari</I></td><td align=right>235-238</td></tr> +<tr><td>Nicomachus</td><td align=right>238</td></tr> +<tr><td>Theon of Smyrna, <I>Expositio nerum mathematicarum ad +legendum Platonem utilium</I></td><td align=right>238-244</td></tr> +<tr><td>XVII. TRIGONOMETRY: HIPPARCHUS, MENELAUS, PTO- +LEMY</td><td align=right>245-297</td></tr> +<tr><td>Theodosius</td><td align=right>245-246</td></tr> +<tr><td>Works by Theodosius</td><td align=right>246</td></tr> +<tr><td>Contents of the <I>Sphaerica</I></td><td align=right>246-252</td></tr> +<tr><td>No actual trigonometry in Theodosius</td><td align=right>250-252</td></tr> +<tr><td>The beginnings of trigonometry</td><td align=right>252-253</td></tr> +<tr><td>Hipparchus</td><td align=right>253-260</td></tr> +<tr><td>The work of Hipparchus</td><td align=right>254-256</td></tr> +<tr><td>First systematic use of trigonometry</td><td align=right>257-259</td></tr> +<tr><td>Table of chords</td><td align=right>259-260</td></tr> +<tr><td>Menelaus</td><td align=right>260-273</td></tr> +<tr><td>The <I>Sphaerica</I> of Menelaus</td><td align=right>261-273</td></tr> +<tr><td>(<G>a</G>) ‘Menelaus's theorem’ for the sphere</td><td align=right>266-268</td></tr> +<tr><td>(<G>b</G>) Deductions from Menelaus's theorem</td><td align=right>268-269</td></tr> +<tr><td>(<G>g</G>) Anharmonic property of four great circles +through one point</td><td align=right>269-270</td></tr> +<tr><td>(<G>d</G>) Propositions analogous to Eucl. VI. 3</td><td align=right>270</td></tr> +<tr><td>Claudius Ptolemy</td><td align=right>273-297</td></tr> +<tr><td>The M<G>aqhmatikh\ su/ntaxis</G> (Arab. <I>Almagest</I>)</td><td align=right>273-286</td></tr> +<tr><td>Commentaries</td><td align=right>274</td></tr> +<tr><td>Translations and editions</td><td align=right>274-275</td></tr> +<tr><td>Summary of contents</td><td align=right>275-276</td></tr> +<tr><td>Trigonometry in Ptolemy</td><td align=right>276-286</td></tr> +<tr><td>(<G>a</G>) Lemma for finding sin 18° and sin 36°</td><td align=right>277-278</td></tr> +<tr><td>(<G>b</G>) Equivalent of <MATH>sin<SUP>2</SUP><G>q</G>+cos<SUP>2</SUP><G>q</G>=1</MATH></td><td align=right>278</td></tr> +<tr><td>(<G>g</G>) ‘Ptolemy's theorem’, giving the equivalent of +<MATH>sin(<G>q</G>-<G>f</G>)=sin<G>q</G>cos<G>f</G>-cos<G>q</G>sin<G>f</G></MATH></td><td align=right>278-280</td></tr> +<tr><td>(<G>d</G>) Equivalent of <MATH>sin<SUP>2</SUP>1/2<G>q</G>=1/2(1-cos<G>q</G>)</MATH></td><td align=right>280-281</td></tr> +<tr><td>(<G>e</G>) Equivalent of <MATH>cos(<G>q</G>+<G>f</G>)=cos<G>q</G>cos<G>f</G>-sin<G>q</G>sin<G>f</G></MATH></td><td align=right>281</td></tr> +<tr><td>(<G>z</G>) Method of interpolation based on formula +<MATH>sin<G>a</G>/sin<G>b</G><<G>a</G>/<G>b</G>(1/2<G>p</G>><G>a</G>><G>b</G>)</MATH></td><td align=right>281-282</td></tr> +<tr><td>(<G>h</G>) Table of chords</td><td align=right>283</td></tr> +<tr><td>(<G>q</G>) Further use of proportional increase</td><td align=right>283-284</td></tr> +<tr><td>(<G>i</G>) Plane trigonometry in effect used</td><td align=right>284</td></tr> +<tr><td>Spherical trigonometry: formulae in solution of +spherical triangles</td><td align=right>284-286</td></tr> +<tr><td>The <I>Analemma</I></td><td align=right>286-292</td></tr> +<tr><td>The <I>Planisphaerium</I></td><td align=right>292-293</td></tr> +<tr><td>The <I>Optics</I></td><td align=right>293-295</td></tr> +<tr><td>A mechanical work, <G>*peri\ r(opw=n</G></td><td align=right>295</td></tr> +<tr><td>Attempt to prove the Parallel-Postulate</td><td align=right>295-297</td></tr> +<pb n=viii><head>CONTENTS</head> +<tr><td>XVIII. MENSURATION: HERON OF ALEXANDRIA</td><td align=right>PAGES 298-354</td></tr> +<tr><td>Controversies as to Heron's date</td><td align=right>298-306</td></tr> +<tr><td>Character of works</td><td align=right>307-308</td></tr> +<tr><td>List of treatises</td><td align=right>308-310</td></tr> +<tr><td>Geometry</td></tr> +<tr><td>(<G>a</G>) Commentary on Euclid's <I>Elements</I></td><td align=right>310-314</td></tr> +<tr><td>(<G>b</G>) The <I>Definitions</I></td><td align=right>314-316</td></tr> +<tr><td>Mensuration</td><td align=right>316-344</td></tr> +<tr><td>The <I>Metrica, Geometrica, Stereometrica, Geodaesia, +Mensurae</I></td><td align=right>316-320</td></tr> +<tr><td>Contents of the <I>Metrica</I></td><td align=right>320-344</td></tr> +<tr><td>Book I. Measurement of areas</td><td align=right>320-331</td></tr> +<tr><td>(<G>a</G>) Area of scalene triangle</td><td align=right>320-321</td></tr> +<tr><td>Proof of formula <MATH>▵=√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>(<I>s</I>-<I>c</I>)}</MATH></td><td align=right>321-323</td></tr> +<tr><td>(<G>b</G>) Method of approximating to the square root +of a non-square number</td><td align=right>323-326</td></tr> +<tr><td>(<G>g</G>) Quadrilaterals</td><td align=right>326</td></tr> +<tr><td>(<G>d</G>) Regular polygons with 3, 4, 5, 6, 7, 8, 9, 10, +11, or 12 sides</td><td align=right>326-329</td></tr> +<tr><td>(<G>e</G>) The circle</td><td align=right>329</td></tr> +<tr><td>(<G>z</G>) Segment of a circle</td><td align=right>330-331</td></tr> +<tr><td>(<G>h</G>) Ellipse, parabolic segment, surface of cylinder, +right cone, sphere and segment of sphere</td><td align=right>331</td></tr> +<tr><td>Book II. Measurement of volumes</td><td align=right>331-335</td></tr> +<tr><td>(<G>a</G>) Cone, cylinder, parallelepiped(prism), pyramid +and frustum</td><td align=right>332</td></tr> +<tr><td>(<G>b</G>) Wedge-shaped solid (<G>bwmi/skos</G> or <G>sfhni/skos</G>)</td><td align=right>332-334</td></tr> +<tr><td>(<G>g</G>) Frustum of cone, sphere, and segment thereof</td><td align=right>334</td></tr> +<tr><td>(<G>d</G>) Anchor-ring or tore</td><td align=right>334-335</td></tr> +<tr><td>(<G>e</G>) The two special solids of Archimedes's ‘Method’</td><td align=right>335</td></tr> +<tr><td>(<G>z</G>) The five regular solids</td><td align=right>335</td></tr> +<tr><td>Book III. Divisions of figures</td><td align=right>336-343</td></tr> +<tr><td>Approximation to the cube root of a non-cube +number</td><td align=right>341-342</td></tr> +<tr><td>Quadratic equations solved in Heron</td><td align=right>344</td></tr> +<tr><td>Indeterminate problems in the <I>Geometrica</I></td><td align=right>344</td></tr> +<tr><td>The <I>Dioptra</I></td><td align=right>345-346</td></tr> +<tr><td>The <I>Mechanics</I></td><td align=right>346-352</td></tr> +<tr><td>Aristotle's Wheel</td><td align=right>347-348</td></tr> +<tr><td>The parallelogram of velocities</td><td align=right>348-349</td></tr> +<tr><td>Motion on an inclined plane</td><td align=right>349-350</td></tr> +<tr><td>On the centre of gravity</td><td align=right>350-351</td></tr> +<tr><td>The five mechanical powers</td><td align=right>351</td></tr> +<tr><td>Mechanics in daily life: queries and answers</td><td align=right>351-352</td></tr> +<tr><td>Problems on the centre of gravity, &c</td><td align=right>352</td></tr> +<tr><td>The <I>Catoptrica</I></td><td align=right>352-354</td></tr> +<tr><td>Heron's proof of equality of angles of incidence and +reflection</td><td align=right>353-354</td></tr> +<tr><td>XIX. PAPPUS OF ALEXANDRIA</td><td align=right>355-439</td></tr> +<tr><td>Date of Pappus</td><td align=right>356</td></tr> +<tr><td>Works (commentaries) other than the <I>Collection</I></td><td align=right>356-357</td></tr> +<pb n=ix><head>CONTENTS</head> +<tr><td>The <I>Synagoge</I> or <I>Collection</I></td><td align=right>PAGES 357-439</td></tr> +<tr><td>(<G>a</G>) Character of the work; wide range</td><td align=right>357-358</td></tr> +<tr><td>(<G>b</G>) List of authors mentioned</td><td align=right>358-360</td></tr> +<tr><td>(<G>g</G>) Translations and editions</td><td align=right>360-361</td></tr> +<tr><td>(<G>d</G>) Summary of contents</td><td align=right>361-439</td></tr> +<tr><td>Book III. Section (1). On the problem of the two +mean proportionals</td><td align=right>361-362</td></tr> +<tr><td>Section (2). The theory of means</td><td align=right>363-365</td></tr> +<tr><td>Section (3). The ‘Paradoxes’ of Erycinus</td><td align=right>365-368</td></tr> +<tr><td>Section (4). The inscribing of the five regular +solids in a sphere</td><td align=right>368-369</td></tr> +<tr><td>Book IV. Section (1). Extension of theorem of +Pythagoras</td><td align=right>369-371</td></tr> +<tr><td>Section (2). On circles inscribed in the <G>a)/rbhlos</G> +(‘shoemaker's knife’)</td><td align=right>371-377</td></tr> +<tr><td>Sections (3), (4). Methods of squaring the circle +and trisecting any angle</td><td align=right>377-386</td></tr> +<tr><td>(<G>a</G>) The Archimedean spiral</td><td align=right>377-379</td></tr> +<tr><td>(<G>b</G>) The conchoid of Nicomedes</td><td align=right>379</td></tr> +<tr><td>(<G>g</G>) The <I>Quadratrix</I></td><td align=right>379-382</td></tr> +<tr><td>(<G>d</G>) Digression: a spiral on a sphere</td><td align=right>382-385</td></tr> +<tr><td>Trisection (or division in any ratio) of any angle</td><td align=right>385-386</td></tr> +<tr><td>Section (5). Solution of the <G>neu=sis</G> of Archimedes, +<I>On Spirals</I>, Prop. 8, by means of conics</td><td align=right>386-388</td></tr> +<tr><td>Book V. Preface on the sagacity of Bees</td><td align=right>389-390</td></tr> +<tr><td>Section (1). Isoperimetry after Zenodorus</td><td align=right>390-393</td></tr> +<tr><td>Section (2). Comparison of volumes of solids having +their surfaces equal. Case of sphere</td><td align=right>393-394</td></tr> +<tr><td>Section (3). Digression on semi-regular solids of +Archimedes</td><td align=right>394</td></tr> +<tr><td>Section (4). Propositions on the lines of Archimedes, +<I>On the Sphere and Cylinder</I></td><td align=right>394-395</td></tr> +<tr><td>Section (5). Of regular solids with surfaces equal, +that is greater which has more faces</td><td align=right>395-396</td></tr> +<tr><td>Book VI.</td><td align=right>396-399</td></tr> +<tr><td>Problem arising out of Euclid's <I>Optics</I></td><td align=right>397-399</td></tr> +<tr><td>Book VII. On the ‘Treasury of Analysis’</td><td align=right>399-427</td></tr> +<tr><td>Definition of Analysis and Synthesis</td><td align=right>400-401</td></tr> +<tr><td>List of works in the ‘Treasury of Analysis’</td><td align=right>401</td></tr> +<tr><td>Description of the treatises</td><td align=right>401-404</td></tr> +<tr><td>Anticipation of Guldin's Theorem</td><td align=right>403</td></tr> +<tr><td>Lemmas to the different treatises</td><td align=right>404-426</td></tr> +<tr><td>(<G>a</G>) Lemmas to the <I>Sectio rationis</I> and <I>Sectio +spatii</I> of Apollonius</td><td align=right>404-405</td></tr> +<tr><td>(<G>b</G>) Lemmas to the <I>Determinate Section</I> of +Apollonius</td><td align=right>405-412</td></tr> +<tr><td>(<G>g</G>) Lemmas on the N<G>eu/seis</G> of Apollonius</td><td align=right>412-416</td></tr> +<tr><td>(<G>d</G>) Lemmas on the <I>On Contacts</I> of Apollonius</td><td align=right>416-417</td></tr> +<tr><td>(<G>e</G>) Lemmas to the <I>Plane Loci</I> of Apollonius</td><td align=right>417-419</td></tr> +<tr><td>(<G>z</G>) Lemmas to the <I>Porisms</I> of Euclid</td><td align=right>419-424</td></tr> +<tr><td>(<G>h</G>) Lemmas to the <I>Conics</I> of Apollonius</td><td align=right>424-425</td></tr> +<tr><td>(<G>q</G>) Lemmas to the <I>Surface Loci</I> of Euclid</td><td align=right>425-426</td></tr> +<tr><td>(<G>i</G>) An unallocated lemma</td><td align=right>426-427</td></tr> +<tr><td>Book VIII. Historical preface</td><td align=right>427-429</td></tr> +<tr><td>The object of the Book</td><td align=right>429-430</td></tr> +<tr><td>On the centre of gravity</td><td align=right>430-433</td></tr> +<pb n=x><head>CONTENTS</head> +<tr><td>XIX. CONTINUED.</td></tr> +<tr><td>Book VIII (<I>continued</I>)</td></tr> +<tr><td>The inclined plane</td><td align=right>PAGES 433-434</td></tr> +<tr><td>Construction of a conic through five points</td><td align=right>434-437</td></tr> +<tr><td>Given two conjugate diameters of an ellipse, to find +the axes</td><td align=right>437-438</td></tr> +<tr><td>Problem of seven hexagons in a circle</td><td align=right>438-439</td></tr> +<tr><td>Construction of toothed wheels and indented screws</td><td align=right>439</td></tr> +<tr><td>XX. ALGEBRA: DIOPHANTUS OF ALEXANDRIA</td><td align=right>440-517</td></tr> +<tr><td>Beginnings learnt from Egypt</td><td align=right>440</td></tr> +<tr><td>‘Hau’-calculations</td><td align=right>440-441</td></tr> +<tr><td>Arithmetical epigrams in the Greek Anthology</td><td align=right>441-443</td></tr> +<tr><td>Indeterminate equations of first degree</td><td align=right>443</td></tr> +<tr><td>Indeterminate equations of second degree before Dio- +phantus</td><td align=right>443-444</td></tr> +<tr><td>Indeterminate equations in Heronian collections</td><td align=right>444-447</td></tr> +<tr><td>Numerical solution of quadratic equations</td><td align=right>448</td></tr> +<tr><td>Works of Diophantus</td><td align=right>448-450</td></tr> +<tr><td>The <I>Arithmetica</I></td><td align=right>449-514</td></tr> +<tr><td>The seven lost Books and their place</td><td align=right>449-450</td></tr> +<tr><td>Relation of ‘Porisms’ to <I>Arithmetica</I></td><td align=right>451-452</td></tr> +<tr><td>Commentators from Hypatia downwards</td><td align=right>453</td></tr> +<tr><td>Translations and editions</td><td align=right>453-455</td></tr> +<tr><td>Notation and definitions</td><td align=right>455-461</td></tr> +<tr><td>Sign for unknown (= <I>x</I>) and its origin</td><td align=right>456-457</td></tr> +<tr><td>Signs for powers of unknown &c.</td><td align=right>458-459</td></tr> +<tr><td>The sign (<FIG>) for <I>minus</I> and its meaning</td><td align=right>459-460</td></tr> +<tr><td>The methods of Diophantus</td><td align=right>462-479</td></tr> +<tr><td>I. Diophantus's treatment of equations</td><td align=right>462-476</td></tr> +<tr><td>(A) Determinate equations</td></tr> +<tr><td>(1) Pure determinate equations</td><td align=right>462-463</td></tr> +<tr><td>(2) Mixed quadratic equations</td><td align=right>463-465</td></tr> +<tr><td>(3) Simultaneous equationsinvolving quadratics</td><td align=right>465</td></tr> +<tr><td>(4) Cubic equation</td><td align=right>465</td></tr> +<tr><td>(B) Indeterminate equations</td></tr> +<tr><td>(<I>a</I>) Indeterminate equations of the second degree</td><td align=right>466-473</td></tr> +<tr><td>(1) Single equation</td><td align=right>466-468</td></tr> +<tr><td>(2) Double equation</td><td align=right>468-473</td></tr> +<tr><td>1. Double equations of first degree</td><td align=right>469-472</td></tr> +<tr><td>2. Double equations of second degree</td><td align=right>472-473</td></tr> +<tr><td>(<I>b</I>) Indeterminate equations of degree higher +than second</td><td align=right>473-476</td></tr> +<tr><td>(1) Single equations</td><td align=right>473-475</td></tr> +<tr><td>(2) Double equations</td><td align=right>475-476</td></tr> +<tr><td>II. Method of limits</td><td align=right>476-477</td></tr> +<tr><td>III. Method of approximation to limits</td><td align=right>477-479</td></tr> +<tr><td>Porisms and propositions in the Theory of Numbers</td><td align=right>479-484</td></tr> +<tr><td>(<G>a</G>) Theorems on the composition of numbers as the +sum of two squares</td><td align=right>481-483</td></tr> +<tr><td>(<G>b</G>) On numbers which are the sum of three squares</td><td align=right>483</td></tr> +<tr><td>(<G>g</G>) Composition of numbers as the sum of four squares</td><td align=right>483-484</td></tr> +<tr><td>Conspectus of <I>Arithmetica</I>, with typical solutions</td><td align=right>484-514</td></tr> +<tr><td>The treatise on Polygonal Numbers</td><td align=right>514-517</td></tr> +<pb n=xi><head>CONTENTS</head> +<tr><td>XXI. COMMENTATORS AND BYZANTINES</td><td align=right>PAGES 518-555</td></tr> +<tr><td>Serenus</td><td align=right>519-526</td></tr> +<tr><td>(<G>a</G>) <I>On the Section of a Cylinder</I></td><td align=right>519-522</td></tr> +<tr><td>(<G>b</G>) <I>On the Section of a Cone</I></td><td align=right>522-526</td></tr> +<tr><td>Theon of Alexandria</td><td align=right>526-528</td></tr> +<tr><td>Commentary on the <I>Syntaxis</I></td><td align=right>526-527</td></tr> +<tr><td>Edition of Euclid's <I>Elements</I></td><td align=right>527-528</td></tr> +<tr><td>Edition of the <I>Optics</I> of Euclid</td><td align=right>528</td></tr> +<tr><td>Hypatia</td><td align=right>528-529</td></tr> +<tr><td>Porphyry. Iamblichus</td><td align=right>529</td></tr> +<tr><td>Proclus</td><td align=right>529-537</td></tr> +<tr><td>Commentary on Euclid, Book I</td><td align=right>530-535</td></tr> +<tr><td>(<G>a</G>) Sources of the Commentary</td><td align=right>530-532</td></tr> +<tr><td>(<G>b</G>) Character of the Commentary</td><td align=right>532-535</td></tr> +<tr><td><I>Hypotyposis of Astronomical Hypotheses</I></td><td align=right>535-536</td></tr> +<tr><td>Commentary on the <I>Republic</I></td><td align=right>536-537</td></tr> +<tr><td>Marinus of Neapolis</td><td align=right>537-538</td></tr> +<tr><td>Domninus of Larissa</td><td align=right>538</td></tr> +<tr><td>Simplicius</td><td align=right>538-540</td></tr> +<tr><td>Extracts from Eudemus</td><td align=right>539</td></tr> +<tr><td>Eutocius</td><td align=right>540-541</td></tr> +<tr><td>Anthemius of Tralles</td><td align=right>541-543</td></tr> +<tr><td><I>On burning-mirrors</I></td><td align=right>541-543</td></tr> +<tr><td>The Papyrus of Akhmīm</td><td align=right>543-545</td></tr> +<tr><td><I>Geodaesia</I> of ‘Heron the Younger’</td><td align=right>545</td></tr> +<tr><td>Michael Psellus</td><td align=right>545-546</td></tr> +<tr><td>Georgius Pachymeres</td><td align=right>546</td></tr> +<tr><td>Maximus Planudes</td><td align=right>546-549</td></tr> +<tr><td>Extraction of the square root</td><td align=right>547-549</td></tr> +<tr><td>Two problems</td><td align=right>549</td></tr> +<tr><td>Manuel Moschopoulos</td><td align=right>549-550</td></tr> +<tr><td>Nicolas Rhabdas</td><td align=right>550-554</td></tr> +<tr><td>Rule for approximating to square root of a non-square +number</td><td align=right>553-554</td></tr> +<tr><td>Ioannes Pediasimus</td><td align=right>554</td></tr> +<tr><td>Barlaam</td><td align=right>554-555</td></tr> +<tr><td>Isaac Argyrus</td><td align=right>555</td></tr> +<tr><td>APPENDIX. On Archimedes's proof of the subtangent-property +of a spiral</td><td align=right>556-561</td></tr> +<tr><td>INDEX OF GREEK WORDS</td><td align=right>563-569</td></tr> +<tr><td>ENGLISH INDEX</td><td align=right>570-586</td></tr> +</table> +<pb><C>XII</C> +<C>ARISTARCHUS OF SAMOS</C> +<p>HISTORIANS of mathematics have, as a rule, given too little +attention to Aristarchus of Samos. The reason is no doubt +that he was an astronomer, and therefore it might be supposed +that his work would have no sufficient interest for the mathe- +matician. The Greeks knew better; they called him Aristar- +chus ‘the mathematician’, to distinguish him from the host +of other Aristarchuses; he is also included by Vitruvius +among the few great men who possessed an equally profound +knowledge of all branches of science, geometry, astronomy, +music, &c. +<p>‘Men of this type are rare, men such as were, in times past, +Aristarchus of Samos, Philolaus and Archytas of Tarentum, +Apollonius of Perga, Eratosthenes of Cyrene, Archimedes and +Scopinas of Syracuse, who left to posterity many mechanical +and gnomonic appliances which they invented and explained +on mathematical (lit. ‘numerical’) principles.’<note>Vitruvius, <I>De architectura,</I> i. 1. 16.</note> +<p>That Aristarchus was a very capable geometer is proved by +his extant work <I>On the sizes and distances of the Sun and +Moon</I> which will be noticed later in this chapter: in the +mechanical line he is credited with the discovery of an im- +proved sun-dial, the so-called <G>ska/fh</G>, which had, not a plane, +but a concave hemispherical surface, with a pointer erected +vertically in the middle throwing shadows and so enabling +the direction and the height of the sun to be read off by means +of lines marked on the surface of the hemisphere. He also +wrote on vision, light and colours. His views on the latter +subjects were no doubt largely influenced by his master, Strato +of Lampsacus; thus Strato held that colours were emanations +from bodies, material molecules, as it were, which imparted to +the intervening air the same colour as that possessed by the +body, while Aristarchus said that colours are ‘shapes or forms +<pb n=2><head>ARISTARCHUS OF SAMOS</head> +stamping the air with impressions like themselves, as it were’, +that ‘colours in darkness have no colouring’, and that ‘light +is the colour impinging on a substratum’. +<p>Two facts enable us to fix Aristarchus's date approximately. +In 281/280 B.C. he made an observation of the summer +solstice; and a book of his, presently to be mentioned, was +published before the date of Archimedes's <I>Psammites</I> or <I>Sand- +reckoner,</I> a work written before 216 B.C. Aristarchus, there- +fore, probably lived <I>circa</I> 310-230 B.C., that is, he was older +than Archimedes by about 25 years. +<p>To Aristarchus belongs the high honour of having been +the first to formulate the Copernican hypothesis, which was +then abandoned again until it was revived by Copernicus +himself. His claim to the title of ‘the ancient Copernicus’ is +still, in my opinion, quite unshaken, notwithstanding the in- +genious and elaborate arguments brought forward by Schia- +parelli to prove that it was Heraclides of Pontus who first +conceived the heliocentric idea. Heraclides is (along with one +Ecphantus, a Pythagorean) credited with having been the first +to hold that the earth revolves about its own axis every 24 +hours, and he was the first to discover that Mercury and Venus +revolve, like satellites, about the sun. But though this proves +that Heraclides came near, if he did not actually reach, the +hypothesis of Tycho Brahe, according to which the earth was +in the centre and the rest of the system, the sun with the +planets revolving round it, revolved round the earth, it does +not suggest that he moved the earth away from the centre. +The contrary is indeed stated by Aëtius, who says that ‘Hera- +clides and Ecphantus make the earth move, <I>not in the sense of +translation,</I> but by way of turning on an axle, like a wheel, +from west to east, about its own centre’.<note>Aët. iii. 13. 3, <I>Vors.</I> i<SUP>3</SUP>, p. 341. 8.</note> None of the +champions of Heraclides have been able to meet this positive +statement. But we have conclusive evidence in favour of the +claim of Aristarchus; indeed, ancient testimony is unanimous +on the point. Not only does Plutarch tell us that Cleanthes +held that Aristarchus ought to be indicted for the impiety of +‘putting the Hearth of the Universe in motion’<note>Plutarch, <I>De faciè in orbe lunae,</I> c. 6, pp. 922 F-923 A.</note>; we have the +best possible testimony in the precise statement of a great +<pb n=3><head>ARISTARCHUS OF SAMOS</head> +contemporary, Archimedes. In the <I>Sand-reckoner</I> Archi- +medes has this passage. +<p>‘You [King Gelon] are aware that “universe” is the name +given by most astronomers to the sphere the centre of which +is the centre of the earth, while its radius is equal to the +straight line between the centre of the sun and the centre of +the earth. This is the common account, as you have heard +from astronomers. But Aristarchus brought out <I>a book con- +sisting of certain hypotheses,</I> wherein it appears, as a conse- +quence of the assumptions made, that the universe is many +times greater than the “universe” just mentioned. His hypo- +theses are that <I>the fixed stars and the sun remain unmoved, +that the earth revolves about the sun in the circumference of a +circle, the sun lying in the middle of the orbit,</I> and that the +sphere of the fixed stars, situated about the same centre as the +sun, is so great that the circle in which he supposes the earth +to revolve bears such a proportion to the distance of the fixed +stars as the centre of the sphere bears to its surface.’ +<p>(The last statement is a variation of a traditional phrase, for +which there are many parallels (cf. Aristarchus's Hypothesis 2 +‘that the earth is in the relation of a point and centre to the +sphere in which the moon moves’), and is a method of saying +that the ‘universe’ is infinitely great in relation not merely to +the size of the sun but even to the orbit of the earth in its +revolution about it; the assumption was necessary to Aris- +tarchus in order that he might not have to take account of +parallax.) +<p>Plutarch, in the passage referred to above, also makes it +clear that Aristarchus followed Heraclides in attributing to +the earth the daily rotation about its axis. The bold hypo- +thesis of Aristarchus found few adherents. Seleucus, of +Seleucia on the Tigris, is the only convinced supporter of it of +whom we hear, and it was speedily abandoned altogether, +mainly owing to the great authority of Hipparchus. Nor do +we find any trace of the heliocentric hypothesis in Aris- +tarchus's extant work <I>On the sizes and distances of the +Sun and Moon.</I> This is presumably because that work was +written before the hypothesis was formulated in the book +referred to by Archimedes. The geometry of the treatise +is, however, unaffected by the difference between the hypo- +theses. +<pb n=4><head>ARISTARCHUS OF SAMOS</head> +<p>Archimedes also says that it was Aristarchus who dis- +covered that the apparent angular diameter of the sun is about +1/720th part of the zodiac circle, that is to say, half a degree. +We do not know how he arrived at this pretty accurate figure: +but, as he is credited with the invention of the <G>ska/fh</G>, he may +have used this instrument for the purpose. But here again +the discovery must apparently have been later than the trea- +tise <I>On sizes and distances,</I> for the value of the subtended +angle is there assumed to be 2° (Hypothesis 6). How Aris- +tarchus came to assume a value so excessive is uncertain. As +the mathematics of his treatise is not dependent on the actual +value taken, 2° may have been assumed merely by way of +illustration; or it may have been a guess at the apparent +diameter made before he had thought of attempting to mea- +sure it. Aristarchus assumed that the angular diameters of +the sun and moon at the centre of the earth are equal. +<p>The method of the treatise depends on the just observation, +which is Aristarchus's third ‘hypothesis’, that ‘when the moon +appears to us halved, the great circle which divides the dark +and the bright portions of the moon is in the direction of our +eye’; the effect of this (since the moon receives its light from +the sun), is that at the time of the dichotomy the centres of +the sun, moon and earth form a triangle right-angled at the +centre of the moon. Two other assumptions were necessary: +first, an estimate of the size of the angle of the latter triangle +at the centre of the earth at the moment of dichotomy: this +Aristarchus assumed (Hypothesis 4) to be ‘less than a quad- +rant by one-thirtieth of a quadrant’, i.e. 87°, again an inaccu- +rate estimate, the true value being 89° 50′; secondly, an esti- +mate of the breadth of the earth's shadow where the moon +traverses it: this he assumed to be ‘the breadth of two +moons’ (Hypothesis 5). +<p>The inaccuracy of the assumptions does not, however, detract +from the mathematical interest of the succeeding investigation. +Here we find the logical sequence of propositions and the abso- +lute rigour of demonstration characteristic of Greek geometry; +the only remaining drawback would be the practical difficulty +of determining the exact moment when the moon ‘appears to +us halved’. The form and style of the book are thoroughly +classical, as befits the period between Euclid and Archimedes; +<pb n=5><head>ARISTARCHUS OF SAMOS</head> +the Greek is even remarkably attractive. The content from +the mathematical point of view is no less interesting, for we +have here the first specimen extant of pure geometry used +with a <I>trigonometrical</I> object, in which respect it is a sort of +forerunner of Archimedes's <I>Measurement of a Circle.</I> Aristar- +chus does not actually evaluate the trigonometrical ratios +on which the ratios of the sizes and distances to be obtained +depend; he finds limits between which they lie, and that by +means of certain propositions which he assumes without proof, +and which therefore must have been generally known to +mathematicians of his day. These propositions are the equi- +valents of the statements that, +<p>(1) if <G>a</G> is what we call the circular measure of an angle +and <G>a</G> is less than 1/2 <G>p</G>, then the ratio sin <G>a/a</G> <I>decreases,</I> and the +ratio tan <G>a/a</G> <I>increases,</I> as <G>a</G> increases from 0 to 1/2 <G>p</G>; +<p>(2) if <G>b</G> be the circular measure of another angle less than +1/2 <G>p</G>, and <G>a</G> > <G>b</G>, then +<MATH>(sin<G>a</G>)/(sin<G>b</G>)<<G>a/b</G><(tan<G>a</G>)/(tan<G>b</G>)</MATH>. +<p>Aristarchus of course deals, not with actual circular measures, +sines and tangents, but with angles (expressed not in degrees +but as fractions of right angles), arcs of circles and their +chords. Particular results obtained by Aristarchus are the +equivalent of the following: +<MATH>1/18>sin3°>1/20</MATH>, [Prop. 7] +<MATH>1/45>sin1°>1/60</MATH>, [Prop. 11] +<MATH>1>cos1°>89/90</MATH>, [Prop. 12] +<MATH>1>cos<SUP>2</SUP>1°>44/45</MATH>. [Prop. 13] +<p>The book consists of eighteen propositions. Beginning with +six hypotheses to the effect already indicated, Aristarchus +declares that he is now in a position to prove +<p>(1) that the distance of the sun from the earth is greater than +eighteen times, but less than twenty times, the distance of the +moon from the earth; +<p>(2) that the diameter of the sun has the same ratio as afore- +said to the diameter of the moon; +<pb n=6><head>ARISTARCHUS OF SAMOS</head> +<p>(3) that the diameter of the sun has to the diameter of the +earth a ratio greater than 19:3, but less than 43:6. +<p>The propositions containing these results are Props. 7, 9 +and 15. +<p>Prop. 1 is preliminary, proving that two equal spheres are +comprehended by one cylinder, and two unequal spheres by +one cone with its vertex in the direction of the lesser sphere, +and the cylinder or cone touches the spheres in circles at +right angles to the line of centres. Prop. 2 proves that, if +a sphere be illuminated by another sphere larger than itself, +the illuminated portion is greater than a hemisphere. Prop. 3 +shows that the circle in the moon which divides the dark from +the bright portion is least when the cone comprehending the +sun and the moon has its vertex at our eye. The ‘dividing +circle’, as we shall call it for short, which was in Hypothesis 3 +spoken of as a great circle, is proved in Prop. 4 to be, not +a great circle, but a small circle not perceptibly different +from a great circle. The proof is typical and is worth giving +along with that of some connected propositions (11 and 12). +<p><I>B</I> is the centre of the moon, <I>A</I> that of the earth, <I>CD</I> the +diameter of the ‘dividing circle in the moon’, <I>EF</I> the parallel +diameter in the moon. <I>BA</I> meets the circular section of the +moon through <I>A</I> and <I>EF</I> in <I>G,</I> and <I>CD</I> in <I>L. GH, GK</I> +are arcs each of which is equal to half the arc <I>CE.</I> By +Hypothesis 6 the angle <I>CAD</I> is ‘one-fifteenth of a sign’ = 2°, +and the angle <I>BAC</I> = 1°. +<p>Now, says Aristarchus, +<MATH>1°:45°[>tan 1°:tan 45°] +><I>BC</I>:<I>CA</I></MATH>, +and, <I>a fortiori,</I> +<MATH><I>BC</I>:<I>BA</I> or <I>BG</I>:<I>BA</I> +<1:45</MATH>; +that is, <MATH><I>BG</I><1/45<I>BA</I> +<1/44<I>GA</I></MATH>; +therefore, <I>a fortiori,</I> +<MATH><I>BH</I><1/44<I>HA</I></MATH>. +<pb n=7><head>ARISTARCHUS OF SAMOS</head> +<p>Now <MATH><I>BH</I>:<I>HA</I>[=sin<I>HAB</I>:sin<I>HBA</I>] +>∠<I>HAB</I>:∠<I>HBA</I></MATH>, +whence <MATH>∠<I>HAB</I><1/44∠<I>HBA</I></MATH>, +<FIG> +and (taking the doubles) <MATH>∠<I>HAK</I><1/44∠<I>HBK</I></MATH>. +<p>But <MATH>∠<I>HBK</I>=∠<I>EBC</I>=1/90<I>R</I> (where <I>R</I> is a right angle)</MATH>; +therefore <MATH>∠<I>HAK</I><1/3960<I>R</I></MATH>. +<p>But ‘a magnitude (arc <I>HK</I>) seen under such an angle is +imperceptible to our eye’; +therefore, <I>a fortiori,</I> the arcs <I>CE, DF</I> are severally imper- +ceptible to our eye. Q.E.D. +<p>An easy deduction from the same figure is Prop. 12, which +shows that the ratio of <I>CD,</I> the diameter of the ‘dividing +circle’, to <I>EF,</I> the diameter of the moon, is < 1 but > 89/90. +<p>We have <MATH>∠<I>EBC</I>=∠<I>BAC</I>=1°</MATH>; +therefore <MATH>(arc <I>EC</I>)=1/90 (arc <I>EG</I>)</MATH>, +and accordingly <MATH>(arc <I>CG</I>):(arc <I>GE</I>)=89:90</MATH>. +<p>Doubling the arcs, we have +<MATH>(arc <I>CGD</I>):(arc <I>EGF</I>)=89:90</MATH>. +<p>But <MATH><I>CD</I>:<I>EF</I>>(arc <I>CGD</I>):(arc <I>EGF</I>) +[equivalent to sin<G>a</G>/sin<G>b</G>><G>a/b</G>, where ∠<I>CBD</I>=2<G>a</G>, +and 2<G>b</G>=<G>p</G></MATH>]; +therefore <MATH><I>CD</I>:<I>EF</I> [=cos 1°]>89:90</MATH>, +while obviously <MATH><I>CD</I>:<I>EF</I><1</MATH>. +<p>Prop. 11 finds limits to the ratio <I>EF</I>:<I>BA</I> (the ratio of the +diameter of the moon to the distance of its centre from +the centre of the earth); the ratio is < 2:45 but > 1:30. +<pb n=8><head>ARISTARCHUS OF SAMOS</head> +<p>The first part follows from the relation found in Prop. 4, +namely <MATH><I>BC</I>:<I>BA</I><1:45</MATH>, +for <MATH><I>EF</I>=2<I>BC</I></MATH>. +<p>The second part requires the use of the circle drawn with +centre <I>A</I> and radius <I>AC.</I> This circle is that on which the +ends of the diameter of the ‘dividing circle’ move as the moon +moves in a circle about the earth. If <I>r</I> is the radius <I>AC</I> +of this circle, a chord in it equal to <I>r</I> subtends at the centre +<I>A</I> an angle of 2/3<I>R</I> or 60°; and the arc <I>CD</I> subtends at <I>A</I> +an angle of 2°. +<p>But <MATH>(arc subtended by <I>CD</I>):(arc subtended by <I>r</I>) +<<I>CD</I>:<I>r</I></MATH>, +or <MATH>2:60<<I>CD</I>:<I>r</I></MATH>; +that is, <MATH><I>CD</I>:<I>CA</I>>1:30</MATH>. +<p>And, by similar triangles, +<MATH><I>CL</I>:<I>CA</I>=<I>CB</I>:<I>BA</I>, or <I>CD</I>:<I>CA</I>=2<I>CB</I>:<I>BA</I>=<I>FE</I>:<I>BA</I></MATH>. +<p>Therefore <MATH><I>FE</I>:<I>BA</I>>1:30</MATH>. +<p>The proposition which is of the greatest interest on the +whole is Prop. 7, to the effect that <I>the distance of the sun +from the earth is greater than 18 times, but less than 20 +times, the distance of the moon from the earth.</I> This result +represents a great improvement on all previous attempts to +estimate the relative distances. The first speculation on the +subject was that of Anaximander (<I>circa</I> 611-545 B.C.), who +seems to have made the distances of the sun and moon from +the earth to be in the ratio 3:2. Eudoxus, according to +Archimedes, made the diameter of the sun 9 times that of +the moon, and Phidias, Archimedes's father, 12 times; and, +assuming that the angular diameters of the two bodies are +equal, the ratio of their distances would be the same. +<p>Aristarchus's proof is shortly as follows. <I>A</I> is the centre of +the sun, <I>B</I> that of the earth, and <I>C</I> that of the moon at the +moment of dichotomy, so that the angle <I>ACB</I> is right. <I>ABEF</I> +is a square, and <I>AE</I> is a quadrant of the sun's circular orbit. +Join <I>BF,</I> and bisect the angle <I>FBE</I> by <I>BG,</I> so that +<MATH>∠<I>GBE</I>=1/4<I>R</I> or 22 1/2°</MATH>. +<pb n=9><head>ARISTARCHUS OF SAMOS</head> +<p>I. Now, by Hypothesis 4, <MATH>∠<I>ABC</I>=87°</MATH>, +so that <MATH>∠<I>HBE</I>=∠<I>BAC</I>=3°</MATH>; +therefore <MATH>∠<I>GBE</I>:∠<I>HBE</I>=(1/4)<I>R</I>:(1/30)<I>R</I> +=15:2</MATH>, +<FIG> +so that <MATH><I>GE</I>:<I>HE</I>[=tan <I>GBE</I>:tan <I>HBE</I>]>∠<I>GBE</I>:∠<I>HBE</I> +>15:2</MATH>. (1) +<p>The ratio which has to be proved > 18:1 is <I>AB</I>:<I>BC</I> or +<I>FE</I>:<I>EH.</I> +<p>Now <MATH><I>FG</I>:<I>GE</I>=<I>FB</I>:<I>BE</I></MATH>, +whence <MATH><I>FG</I><SUP>2</SUP>:<I>GE</I><SUP>2</SUP>=<I>FB</I><SUP>2</SUP>:<I>BE</I><SUP>2</SUP>=2:1</MATH>, +and <MATH><I>FG</I>:<I>GE</I>=√2:1 +>7:5</MATH> +(this is the approximation to √2 mentioned by Plato and +known to the Pythagoreans). +<pb n=10><head>ARISTARCHUS OF SAMOS</head> +<p>Therefore <MATH><I>FE</I>:<I>EG</I>>12:5 or 36:15</MATH>. +<p>Compounding this with (1) above, we have +<MATH><I>FE</I>:<I>EH</I>>36:2 or 18:1</MATH>. +<p>II. To prove <MATH><I>BA</I><20 <I>BC</I></MATH>. +<p>Let <I>BH</I> meet the circle <I>AE</I> in <I>D,</I> and draw <I>DK</I> parallel +to <I>EB.</I> Circumscribe a circle about the triangle <I>BKD,</I> and +let the chord <I>BL</I> be equal to the radius (<I>r</I>) of the circle. +<p>Now <MATH>∠<I>BDK</I>=∠<I>DBE</I>=1/30<I>R</I></MATH>, +so that arc <MATH><I>BK</I>=1/60 (circumference of circle)</MATH>. +<p>Thus <MATH>(arc <I>BK</I>):(arc <I>BL</I>)=(1/60):(1/6), +=1:10</MATH>. +<p>And <MATH>(arc <I>BK</I>):(arc <I>BL</I>)<<I>BK</I>:<I>r</I></MATH> +[this is equivalent to <G>a/b</G><sin<G>a</G>/sin<G>b</G>, where <G>a</G><<G>b</G><1/2<G>p</G>], +so that <MATH><I>r</I><10 <I>BK</I></MATH>, +and <MATH><I>BD</I><20 <I>BK</I></MATH>. +<p>But <MATH><I>BD</I>:<I>BK</I>=<I>AB</I>:<I>BC</I></MATH>; +therefore <MATH><I>AB</I><20<I>BC</I></MATH>. Q.E.D. +<p>The remaining results obtained in the treatise can be +visualized by means of the three figures annexed, which have +reference to the positions of the sun (centre A), the earth +(centre B) and the moon (centre C) during an eclipse. Fig. 1 +shows the middle position of the moon relatively to the earth's +shadow which is bounded by the cone comprehending the sun +and the earth. <I>ON</I> is the arc with centre <I>B</I> along which +move the extremities of the diameter of the dividing circle in +the moon. Fig. 3 shows the same position of the moon in the +middle of the shadow, but on a larger scale. Fig. 2 shows +the moon at the moment when it has just entered the shadow; +and, as the breadth of the earth's shadow is that of two moons +(Hypothesis 5), the moon in the position shown touches <I>BN</I> at +<I>N</I> and <I>BL</I> at <I>L,</I> where <I>L</I> is the middle point of the arc <I>ON.</I> +It should be added that, in Fig. 1, <I>UV</I> is the diameter of the +circle in which the sun is touched by the double cone with <I>B</I> +as vertex, which comprehends both the sun and the moon, +<pb n=11><head>ARISTARCHUS OF SAMOS</head> +while <I>Y, Z</I> are the points in which the perpendicular through +<I>A,</I> the centre of the sun, to <I>BA</I> meets the cone enveloping the +sun and the earth. +<FIG> +<CAP>FIG. 1.</CAP> +<p>This being premised, the main results obtained are as +follows: +Prop. 13. +<p>(1) <MATH><I>ON</I>:(diam. of moon)<2:1</MATH> +but <MATH>>88:45</MATH>. +<pb n=12><head>ARISTARCHUS OF SAMOS</head> +<p>(2) <MATH><I>ON</I>:(diam. of sun)<1:9</MATH> +but <MATH>>22:225</MATH>. +<p>(3) <MATH><I>ON</I>:<I>YZ</I>>979:10125</MATH>. +Prop. 14 (Fig. 3). +<MATH><I>BC</I>:<I>CS</I>>675:1</MATH>. +Prop. 15. +<MATH>(Diam. of sun):(diam. of earth)>19:3</MATH> +but <MATH><43:6</MATH>. +<FIG> +<CAP>FIG. 2.</CAP> +<FIG> +<CAP>FIG. 3.</CAP> +Prop. 17. +<MATH>(Diam. of earth):(diam. of moon)>108:43</MATH> +but <MATH><60:19</MATH>. +<p>It is worth while to show how these results are proved. +Prop. 13. +<p>(1) In Fig. 2 it is clear that +<MATH><I>ON</I><2<I>LN</I> and, <I>a fortiori,</I> <2<I>LP</I></MATH>. +<p>The triangles <I>LON, CLN</I> being similar, +<MATH><I>ON</I>:<I>NL</I>=<I>NL</I>:<I>LC</I></MATH>; +therefore <MATH><I>ON</I>:<I>NL</I>=<I>NL</I>:1/2<I>LP</I> +>89:45</MATH>. (by Prop. 12) +<pb n=13><head>ARISTARCHUS OF SAMOS</head> +<p>Hence <MATH><I>ON</I>:<I>LC</I>=<I>ON</I><SUP>2</SUP>:<I>NL</I><SUP>2</SUP> +>89<SUP>2</SUP>:45<SUP>2</SUP></MATH>; +therefore <MATH><I>ON</I>:<I>LP</I>>7921:4050 +>88:45, says Aristarchus</MATH>. +<p>[If 7921/4050 be developed as a continued fraction, we easily +obtain 1+1/(1+1/(21+1/2)), which is in fact 88/45.] +<p>(2) <MATH><I>ON</I><2(diam. of moon)</MATH>. +<p>But <MATH>(diam. of moon)<1/18(diam. of sun)</MATH>; (Prop. 7) +therefore <MATH><I>ON</I><1/9(diam. of sun)</MATH>. +<p>Again <MATH><I>ON</I>:(diam. of moon)>88:45</MATH>, from above, +and <MATH>(diam. of moon):(diam. of sun)>1:20</MATH>; (Prop. 7) +therefore, <I>ex aequali,</I> +<MATH><I>ON</I>:(diam. of sun)>88:900 +>22:225</MATH>. +<p>(3) Since the same cone comprehends the sun and the moon, +the triangle <I>BUV</I> (Fig. 1) and the triangle <I>BLN</I> (Fig. 2) are +similar, and +<MATH><I>LN</I>:<I>LP</I>=<I>UV</I>:(diam. of sun) +=<I>WU</I>:<I>UA</I> +=<I>UA</I>:<I>AS</I> +<<I>UA</I>:<I>AY</I></MATH>. +<p>But <MATH><I>LN</I>:<I>LP</I>>89:90</MATH>; (Prop. 12) +therefore, <I>a fortiori,</I> <MATH><I>UA</I>:<I>AY</I>>89:90</MATH>. +<p>And <MATH><I>UA</I>:<I>AY</I>=2<I>UA</I>:<I>YZ</I> +=(diam. of sun):<I>YZ</I></MATH>. +<p>But <MATH><I>ON</I>:(diam. of sun)>22:225</MATH>; (Prop. 13) +therefore, <I>ex aequali,</I> +<MATH><I>ON</I>:<I>YZ</I>>89X22:90X225 +>979:10125</MATH>. +<pb n=14><head>ARISTARCHUS OF SAMOS</head> +Prop. 14 (Fig. 3). +<p>The arcs <I>OM, ML, LP, PN</I> are all equal; therefore so are +the chords. <I>BM, BP</I> are tangents to the circle <I>MQP,</I> so that +<I>CM</I> is perpendicular to <I>BM,</I> while <I>BM</I> is perpendicular to <I>OL.</I> +Therefore the triangles <I>LOS, CMR</I> are similar. +<p>Therefore <MATH><I>SO</I>:<I>MR</I>=<I>SL</I>:<I>RC</I></MATH>. +<p>But <MATH><I>SO</I><2<I>MR,</I> since <I>ON</I><2<I>MP</I></MATH>; (Prop. 13) +therefore <MATH><I>SL</I><2<I>RC</I></MATH>, +and, <I>a fortiori,</I> <MATH><I>SR</I><2<I>RC,</I> or <I>SC</I><3<I>RC</I></MATH>, +that is, <MATH><I>CR</I>:<I>CS</I>>1:3</MATH>. +<p>Again, <MATH><I>MC</I>:<I>CR</I>=<I>BC</I>:<I>CM</I> +>45:1</MATH>; (see Prop. 11) +therefore, <I>ex aequali,</I> +<MATH><I>CM</I>:<I>CS</I>>15:1</MATH>. +<p>And <MATH><I>BC</I>:<I>CM</I>>45:1</MATH>; +therefore <MATH><I>BC</I>:<I>CS</I>>675:1</MATH>. +Prop. 15 (Fig. 1). +<p>We have <MATH><I>NO</I>:(diam. of sun)<1:9</MATH>, (Prop. 13) +and, <I>a fortiori,</I> <MATH><I>YZ</I>:<I>NO</I>>9:1</MATH>; +therefore, by similar triangles, if <I>YO, ZN</I> meet in <I>X,</I> +<MATH><I>AX</I>:<I>XR</I>>9:1</MATH>, +and <I>convertendo,</I> <MATH><I>XA</I>:<I>AR</I><9:8</MATH>. +<p>But <MATH><I>AB</I>>18<I>BC,</I> and, <I>a fortiori,</I> >18<I>BR</I></MATH>, +whence <MATH><I>AB</I>>18(<I>AR-AB</I>), or 19<I>AB</I>>18<I>AR</I></MATH>; +that is, <MATH><I>AR</I>:<I>AB</I><19:18</MATH>. +<p>Therefore, <I>ex aequali,</I> +<MATH><I>XA</I>:<I>AB</I><19:16</MATH>, +and, <I>convertendo,</I> <MATH><I>AX</I>:<I>XB</I>>19:3</MATH>; +therefore <MATH>(diam. of sun):(diam. of earth)>19:3</MATH>. +<p>Lastly, since <MATH><I>CB</I>:<I>CR</I>>675:1</MATH>, (Prop. 14) +<MATH><I>CB</I>:<I>BR</I><675:674</MATH>. +<pb n=15><head>ARISTARCHUS OF SAMOS</head> +<p>But <MATH><I>AB</I>:<I>BC</I><20:1</MATH>; +therefore, <I>ex aequali,</I> +<MATH><I>AB</I>:<I>BR</I><13500:674 +<6750:337</MATH>, +whence, by inversion and <I>componendo,</I> +<MATH><I>RA</I>:<I>AB</I>>7087:6750</MATH>. (1) +<p>But <MATH><I>AX</I>:<I>XR</I>=<I>YZ</I>:<I>NO</I> +<10125:979</MATH>; (Prop. 13) +therefore, <I>convertendo,</I> +<MATH><I>XA</I>:<I>AR</I>>10125:9146</MATH>. +<p>From this and (1) we have, <I>ex aequali,</I> +<MATH><I>XA</I>:<I>AB</I>>10125X7087:9146X6750 +>71755875:61735500 +>43:37, <I>a fortiori</I></MATH>. +<p>[It is difficult not to see in 43:37 the expression 1+1/(6+1/6), +which suggests that 43:37 was obtained by developing the +ratio as a continued fraction.] +<p>Therefore, <I>convertendo,</I> +<MATH><I>XA</I>:<I>XB</I><43:6</MATH>, +whence <MATH>(diam. of sun):(diam. of earth)<43:6</MATH>. Q.E.D. +<pb><C>XIII</C> +<C>ARCHIMEDES</C> +<p>THE siege and capture of Syracuse by Marcellus during the +second Punic war furnished the occasion for the appearance of +Archimedes as a personage in history; it is with this histori- +cal event that most of the detailed stories of him are con- +nected; and the fact that he was killed in the sack of the city +in 212 B. C., when he is supposed to have been 75 years of age, +enables us to fix his date at about 287-212 B.C. He was the +son of Phidias, the astronomer, and was on intimate terms +with, if not related to, King Hieron and his son Gelon. It +appears from a passage of Diodorus that he spent some time +in Egypt, which visit was the occasion of his discovery of the +so-called Archimedean screw as a means of pumping water.<note>Diodorus, v. 37. 3.</note> +It may be inferred that he studied at Alexandria with the +successors of Euclid. It was probably at Alexandria that he +made the acquaintance of Conon of Samos (for whom he had +the highest regard both as a mathematician and a friend) and +of Eratosthenes of Cyrene. To the former he was in the habit +of communicating his discourses before their publication; +while it was to Eratosthenes that he sent <I>The Method,</I> with an +introductory letter which is of the highest interest, as well as +(if we may judge by its heading) the famous Cattle-Problem. +<C>Traditions.</C> +<p>It is natural that history or legend should say more of his +mechanical inventions than of his mathematical achievements, +which would appeal less to the average mind. His machines +were used with great effect against the Romans in the siege +of Syracuse. Thus he contrived (so we are told) catapults so +ingeniously constructed as to be equally serviceable at long or +<pb n=17><head>TRADITIONS</head> +short range, machines for discharging showers of missiles +through holes made in the walls, and others consisting of +long movable poles projecting beyond the walls which either +dropped heavy weights on the enemy's ships, or grappled +their prows by means of an iron hand or a beak like that of +a crane, then lifted them into the air and let them fall again.<note>Polybius, <I>Hist.</I> viii. 7, 8; Livy xxiv. 34; Plutarch, <I>Marcellus,</I> cc. 15-17.</note> +Marcellus is said to have derided his own engineers with the +words, ‘Shall we not make an end of fighting against this +geometrical Briareus who uses our ships like cups to ladle +water from the sea, drives off our <I>sambuca</I> ignominiously +with cudgel-blows, and by the multitude of missiles that he +hurls at us all at once outdoes the hundred-handed giants of +mythology?’; but all to no purpose, for the Romans were in +such abject terror that, ‘if they did but see a piece of rope +or wood projecting above the wall, they would cry “there it +is”, declaring that Archimedes was setting some engine in +motion against them, and would turn their backs and run +away’.<note><I>Ib.,</I> c. 17.</note> These things, however, were merely the ‘diversions +of geometry at play’,<note><I>Ib.,</I> c. 14.</note> and Archimedes himself attached no +importance to them. According to Plutarch, +‘though these inventions had obtained for him the renown of +more than human sagacity, he yet would not even deign to +leave behind him any written work on such subjects, but, +regarding as ignoble and sordid the business of mechanics and +every sort of art which is directed to use and profit, he placed +his whole ambition in those speculations the beauty and +subtlety of which is untainted by any admixture of the com- +mon needs of life.’<note><I>Ib.,</I> c. 17.</note> +<C>(<G>a</G>) <I>Astronomy.</I></C> +<p>Archimedes did indeed write one mechanical book, <I>On +Sphere-making,</I> which is lost; this described the construction +of a sphere to imitate the motions of the sun, moon and +planets.<note>Carpus in Pappus, viii, p. 1026. 9; Proclus on Eucl. I, p. 41. 16.</note> Cicero saw this contrivance and gives a description +of it; he says that it represented the periods of the moon +and the apparent motion of the sun with such accuracy that +it would even (over a short period) show the eclipses of the +sun and moon.<note>Cicero, <I>De rep.</I> i. 21, 22, <I>Tusc.</I> i. 63, <I>De nat. deor.</I> ii. 88.</note> As Pappus speaks of ‘those who understand +<pb n=18><head>ARCHIMEDES</head> +the making of spheres and produce a model of the heavens by +means of the circular motion of water’, it is possible that +Archimedes's sphere was moved by water. In any case Archi- +medes was much occupied with astronomy. Livy calls him +‘unicus spectator caeli siderumque’.<note>Livy xxiv. 34. 2.</note> Hipparchus says, ‘From +these observations it is clear that the differences in the years +are altogether small, but, as to the solstices, I almost think +that Archimedes and I have both erred to the extent of a +quarter of a day both in the observation and in the deduction +therefrom’.<note>Ptolemy, <I>Syntaxis,</I> III. 1, vol. i, p. 194. 23.</note> Archimedes then had evidently considered the +length of the year. Macrobius says he discovered the dis- +tances of the planets,<note>Macrobius, <I>In Somn. Scip.</I> ii. 3; cf. the figures in Hippolytus, <I>Refut.,</I> +p. 66. 52 sq., ed. Duncker.</note> and he himself describes in his <I>Sand- +reckoner</I> the apparatus by which he measured the apparent +angular diameter of the sun. +<C>(<G>b</G>) <I>Mechanics.</I></C> +<p>Archimedes wrote, as we shall see, on theoretical mechanics, +and it was by theory that he solved the problem <I>To move a +given weight by a given force,</I> for it was in reliance ‘on the +irresistible cogency of his proof’ that he declared to Hieron +that any given weight could be moved by any given force +(however small), and boasted that, ‘if he were given a place to +stand on, he could move the earth’ (<G>pa= bw=, kai\ kinw= ta\n ga=n</G>, +as he said in his Doric dialect). The story, told by Plutarch, +is that, ‘when Hieron was struck with amazement and asked +Archimedes to reduce the problem to practice and to give an +illustration of some great weight moved by a small force, he +fixed upon a ship of burden with three masts from the king's +arsenal which had only been drawn up with great labour by +many men, and loading her with many passengers and a full +freight, himself the while sitting far off, with no great effort +but only holding the end of a compound pulley (<G>polu/spastos</G>) +quietly in his hand and pulling at it, he drew the ship along +smoothly and safely as if she were moving through the sea.’<note>Plutarch, <I>Marcellus,</I> c. 14.</note> +<p>The story that Archimedes set the Roman ships on fire by +an arrangement of burning-glasses or concave mirrors is not +found in any authority earlier than Lucian; but it is quite +<pb n=19><head>MECHANICS</head> +likely that he discovered some form of burning-mirror, e.g. a +paraboloid of revolution, which would reflect to one point all +rays falling on its concave surface in a direction parallel to +its axis. +<p>Archimedes's own view of the relative importance of his +many discoveries is well shown by his request to his friends +and relatives that they should place upon his tomb a represen- +tation of a cylinder circumscribing a sphere, with an inscrip- +tion giving the ratio which the cylinder bears to the sphere; +from which we may infer that he regarded the discovery of +this ratio as his greatest achievement. Cicero, when quaestor +in Sicily, found the tomb in a neglected state and repaired it<note>Cicero, <I>Tusc.</I> v. 64 sq.</note>; +but it has now disappeared, and no one knows where he was +buried. +<p>Archimedes's entire preoccupation by his abstract studies is +illustrated by a number of stories. We are told that he would +forget all about his food and such necessities of life, and would +be drawing geometrical figures in the ashes of the fire or, when +anointing himself, in the oil on his body.<note>Plutarch, <I>Marcellus,</I> c. 17.</note> Of the same sort +is the tale that, when he discovered in a bath the solution of +the question referred to him by Hieron, as to whether a certain +crown supposed to have been made of gold did not in fact con- +tain a certain proportion of silver, he ran naked through the +street to his home shouting <G>eu(/rhka, eu(/rhka</G>.<note>Vitruvius, <I>De architectura,</I> ix. 1. 9, 10.</note> He was killed +in the sack of Syracuse by a Roman soldier. The story is +told in various forms; the most picturesque is that found in +Tzetzes, which represents him as saying to a Roman soldier +who found him intent on some diagrams which he had drawn +in the dust and came too close, ‘Stand away, fellow, from my +diagram’, whereat the man was so enraged that he killed +him.<note>Tzetzes, <I>Chiliad.</I> ii. 35. 135.</note> +<C>Summary of main achievements.</C> +<p>In geometry Archimedes's work consists in the main of +original investigations into the quadrature of curvilinear +plane figures and the quadrature and cubature of curved +surfaces. These investigations, beginning where Euclid's +Book XII left off, actually (in the words of Chasles) ‘gave +<pb n=20><head>ARCHIMEDES</head> +birth to the calculus of the infinite conceived and brought to +perfection successively by Kepler, Cavalieri, Fermat, Leibniz +and Newton’. He performed in fact what is equivalent to +<I>integration</I> in finding the area of a parabolic segment, and of +a spiral, the surface and volume of a sphere and a segment of +a sphere, and the volumes of any segments of the solids of +revolution of the second degree. In arithmetic he calculated +approximations to the value of <G>p</G>, in the course of which cal- +culation he shows that he could approximate to the value of +square roots of large or small non-square numbers; he further +invented a system of arithmetical terminology by which he +could express in language any number up to that which we +should write down with 1 followed by 80,000 million million +ciphers. In mechanics he not only worked out the principles of +the subject but advanced so far as to find the centre of gravity +of a segment of a parabola, a semicircle, a cone, a hemisphere, +a segment of a sphere, a right segment of a paraboloid and +a spheroid of revolution. His mechanics, as we shall see, has +become more important in relation to his geometry since the +discovery of the treatise called <I>The Method</I> which was formerly +supposed to be lost. Lastly, he invented the whole science of +hydrostatics, which again he carried so far as to give a most +complete investigation of the positions of rest and stability of +a right segment of a paraboloid of revolution floating in a +fluid with its base either upwards or downwards, but so that +the base is either wholly above or wholly below the surface of +the fluid. This represents a sum of mathematical achieve- +ment unsurpassed by any one man in the world's history. +<C>Character of treatises.</C> +<p>The treatises are, without exception, monuments of mathe- +matical exposition; the gradual revelation of the plan of +attack, the masterly ordering of the propositions, the stern +elimination of everything not immediately relevant to the +purpose, the finish of the whole, are so impressive in their +perfection as to create a feeling akin to awe in the mind of +the reader. As Plutarch said, ‘It is not possible to find in +geometry more difficult and troublesome questions or proofs +set out in simpler and clearer propositions’.<note>Plutarch, <I>Marcellus,</I> c. 17.</note> There is at the +<pb n=21><head>CHARACTER OF TREATISES</head> +same time a certain mystery veiling the way in which he +arrived at his results. For it is clear that they were not +<I>discovered</I> by the steps which lead up to them in the finished +treatises. If the geometrical treatises stood alone, Archi- +medes might seem, as Wallis said, ‘as it were of set purpose +to have covered up the traces of his investigation, as if he had +grudged posterity the secret of his method of inquiry, while +he wished to extort from them assent to his results’. And +indeed (again in the words of Wallis) ‘not only Archimedes +but nearly all the ancients so hid from posterity their method +of Analysis (though it is clear that they had one) that more +modern mathematicians found it easier to invent a new +Analysis than to seek out the old’. A partial exception is +now furnished by <I>The Method</I> of Archimedes, so happily dis- +covered by Heiberg. In this book Archimedes tells us how +he discovered certain theorems in quadrature and cubature, +namely by the use of mechanics, weighing elements of a +figure against elements of another simpler figure the mensura- +tion of which was already known. At the same time he is +careful to insist on the difference between (1) the means +which may be sufficient to suggest the truth of theorems, +although not furnishing scientific proofs of them, and (2) the +rigorous demonstrations of them by orthodox geometrical +methods which must follow before they can be finally accepted +as established: +<p>‘certain things’, he says, ‘first became clear to me by a +mechanical method, although they had to be demonstrated by +geometry afterwards because their investigation by the said +method did not furnish an actual demonstration. But it is +of course easier, when we have previously acquired, by the +method, some knowledge of the questions, to supply the proof +than it is to find it without any previous knowledge.’ ‘This’, +he adds, ‘is a reason why, in the case of the theorems that +the volumes of a cone and a pyramid are one-third of the +volumes of the cylinder and prism respectively having the +same base and equal height, the proofs of which Eudoxus was +the first to discover, no small share of the credit should be +given to Democritus who was the first to state the fact, +though without proof.’ +<p>Finally, he says that the very first theorem which he found +out by means of mechanics was that of the separate treatise +<pb n=22><head>ARCHIMEDES</head> +on the <I>Quadrature of the parabola,</I> namely that <I>the area of any +segment of a section of a right-angled cone</I> (<I>i. e. a parabola</I>) <I>is +four-thirds of that of the triangle which has the same base and +height.</I> The mechanical proof, however, of this theorem in the +<I>Quadrature of the Parabola</I> is different from that in the +<I>Method,</I> and is more complete in that the argument is clinched +by formally applying the method of exhaustion. +<C>List of works still extant.</C> +<p>The extant works of Archimedes in the order in which they +appear in Heiberg's second edition, following the order of the +manuscripts so far as the first seven treatises are concerned, +are as follows: +<p>(5) <I>On the Sphere and Cylinder</I>: two Books. +<p>(9) <I>Measurement of a Circle.</I> +<p>(7) <I>On Conoids and Spheroids.</I> +<p>(6) <I>On Spirals.</I> +<p>(1) <I>On Plane Equilibriums,</I> Book I. +<p>(3) ” ” ” Book II. +<p>(10) <I>The Sand-reckoner</I> (<I>Psammites</I>). +<p>(2) <I>Quadrature of the Parabola.</I> +<p>(8) <I>On Floating Bodies</I>: two Books. +<p>? <I>Stomachion</I> (a fragment). +<p>(4) <I>The Method.</I> +<p>This, however, was not the order of composition; and, +judging (<I>a</I>) by statements in Archimedes's own prefaces to +certain of the treatises and (<I>b</I>) by the use in certain treatises +of results obtained in others, we can make out an approxi- +mate chronological order, which I have indicated in the above +list by figures in brackets. The treatise <I>On Floating Bodies</I> +was formerly only known in the Latin translation by William +of Moerbeke, but the Greek text of it has now been in great +part restored by Heiberg from the Constantinople manuscript +which also contains <I>The Method</I> and the fragment of the +<I>Stomachion.</I> +<p>Besides these works we have a collection of propositions +(<I>Liber assumptorum</I>) which has reached us through the +Arabic. Although in the title of the translation by Thãbit b. +<pb n=23><head>LIST OF EXTANT WORKS</head> +Qurra the book is attributed to Archimedes, the propositions +cannot be his in their present form, since his name is several +times mentioned in them; but it is quite likely that some +of them are of Archimedean origin, notably those about the +geometrical figures called <G>a)/rbhlos</G> (‘shoemaker's knife’) and +<G>sa/linon</G> (probably ‘salt-cellar’) respectively and Prop. 8 bear- +ing on the trisection of an angle. +<p>There is also the <I>Cattle-Problem</I> in epigrammatic form, +which purports by its heading to have been communicated by +Archimedes to the mathematicians at Alexandria in a letter +to Eratosthenes. Whether the epigrammatic form is due to +Archimedes himself or not, there is no sufficient reason for +doubting the possibility that the substance of it was set as a +problem by Archimedes. +<C>Traces of lost works.</C> +<p>Of works which are lost we have the following traces. +<p>1. Investigations relating to <I>polyhedra</I> are referred to by +Pappus who, after alluding to the five regular polyhedra, +describes thirteen others discovered by Archimedes which are +semi-regular, being contained by polygons equilateral and +equiangular but not all similar.<note>Pappus, v, pp. 352-8.</note> +<p>2. There was a book of arithmetical content dedicated to +Zeuxippus. We learn from Archimedes himself that it dealt +with the <I>naming of numbers</I> (<G>katono/maxis tw=n a)riqmw=n</G>)<note>Archimedes, vol. ii, pp. 216. 18, 236. 17-22; cf. p. 220. 4.</note> and +expounded the system, which we find in the <I>Sand-reckoner,</I> of +expressing numbers higher than those which could be written +in the ordinary Greek notation, numbers in fact (as we have +said) up to the enormous figure represented by 1 followed by +80,000 million million ciphers. +<p>3. One or more works on mechanics are alluded to contain- +ing propositions not included in the extant treatise <I>On Plane +Equilibriums.</I> Pappus mentions a work <I>On Balances</I> or <I>Levers</I> +(<G>peri\ zugw=n</G>) in which it was proved (as it also was in Philon's +and Heron's <I>Mechanics</I>) that ‘greater circles overpower lesser +circles when they revolve about the same centre’.<note>Pappus, viii, p. 1068.</note> Heron, too, +speaks of writings of Archimedes ‘which bear the title of +<pb n=24><head>ARCHIMEDES</head> +“works on the lever”’.<note>Heron, <I>Mechanics,</I> i. 32.</note> Simplicius refers to <I>problems on the +centre of gravity,</I> <G>kentrobarika/</G>, such as the many elegant +problems solved by Archimedes and others, the object of which +is to show how to find the centre of gravity, that is, the point +in a body such that if the body is hung up from it, the body +will remain at rest in any position.<note>Simpl. on Arist. <I>De caelo,</I> ii, p. 508 a 30, Brandis; p. 543. 24, Heib.</note> This recalls the assump- +tion in the <I>Quadrature of the Parabola</I> (6) that, if a body hangs +at rest from a point, the centre of gravity of the body and the +point of suspension are in the same vertical line. Pappus has +a similar remark with reference to a point of <I>support,</I> adding +that the centre of gravity is determined as the intersection of +two straight lines in the body, through two points of support, +which straight lines are vertical when the body is in equilibrium +so supported. Pappus also gives the characteristic of the centre +of gravity mentioned by Simplicius, observing that this is +the most fundamental principle of the theory of the centre of +gravity, the elementary propositions of which are found in +Archimedes's <I>On Equilibriums</I> (<G>peri\ i)sorropiw=n</G>) and Heron's +<I>Mechanics.</I> Archimedes himself cites propositions which must +have been proved elsewhere, e.g. that the centre of gravity +of a cone divides the axis in the ratio 3:1, the longer segment +being that adjacent to the vertex<note><I>Method,</I> Lemma 10.</note>; he also says that ‘it is +proved in the <I>Equilibriums</I>’ that the centre of gravity of any +segment of a right-angled conoid (i. e. paraboloid of revolution) +divides the axis in such a way that the portion towards the +vertex is double of the remainder.<note><I>On Floating Bodies,</I> ii. 2.</note> It is possible that there +was originally a larger work by Archimedes <I>On Equilibriums</I> +of which the surviving books <I>On Plane Equilibriums</I> formed +only a part; in that case <G>peri\ zugw=n</G> and <G>kentrobarika/</G> may +only be alternative titles. Finally, Heron says that Archi- +medes laid down a certain procedure in a book bearing the +title ‘Book on Supports’.<note>Heron, <I>Mechanics,</I> i. 25.</note> +<p>4. Theon of Alexandria quotes a proposition from a work +of Archimedes called <I>Catoptrica</I> (properties of mirrors) to the +effect that things thrown into water look larger and still +larger the farther they sink.<note>Theon on Ptolemy's <I>Syntaxis,</I> i, p. 29, Halma.</note> Olympiodorus, too, mentions +<pb n=25><head>TRACES OF LOST WORKS</head> +that Archimedes proved the phenomenon of refraction ‘by +means of the ring placed in the vessel (of water)’.<note>Olympiodorus on Arist. <I>Meteorologica,</I> ii, p. 94, Ideler; p. 211. 18, +Busse.</note> A scholiast +to the Pseudo-Euclid's <I>Catoptrica</I> quotes a proof, which he +attributes to Archimedes, of the equality of the angles of +incidence and of reflection in a mirror. +<C>The text of Archimedes.</C> +<p>Heron, Pappus and Theon all cite works of Archimedes +which no longer survive, a fact which shows that such works +were still extant at Alexandria as late as the third and fourth +centuries A.D. But it is evident that attention came to be +concentrated on two works only, the <I>Measurement of a Circle</I> +and <I>On the Sphere and Cylinder.</I> Eutocius (<I>fl.</I> about A. D. 500) +only wrote commentaries on these works and on the <I>Plane +Equilibriums,</I> and he does not seem even to have been +acquainted with the <I>Quadrature of the Parabola</I> or the work +<I>On Spirals,</I> although these have survived. Isidorus of Miletus +revised the commentaries of Eutocius on the <I>Measurement +of a Circle</I> and the two Books <I>On the Sphere and Cylinder,</I> +and it would seem to have been in the school of Isidorus +that these treatises were turned from their original Doric +into the ordinary language, with alterations designed to make +them more intelligible to elementary pupils. But neither in +Isidorus's time nor earlier was there any collected edition +of Archimedes's works, so that those which were less read +tended to disappear. +<p>In the ninth century Leon, who restored the University +of Constantinople, collected together all the works that he +could find at Constantinople, and had the manuscript written +(the archetype, Heiberg's A) which, through its derivatives, +was, up to the discovery of the Constantinople manuscript (C) +containing <I>The Method,</I> the only source for the Greek text. +Leon's manuscript came, in the twelfth century, to the +Norman Court at Palermo, and thence passed to the House +of Hohenstaufen. Then, with all the library of Manfred, it +was given to the Pope by Charles of Anjou after the battle +of Benevento in 1266. It was in the Papal Library in the +years 1269 and 1311, but, some time after 1368, passed into +<pb n=26><head>ARCHIMEDES</head> +private hands. In 1491 it belonged to Georgius Valla, who +translated from it the portions published in his posthumous +work <I>De expetendis et fugiendis rebus</I> (1501), and intended to +publish the whole of Archimedes with Eutocius's commen- +taries. On Valla's death in 1500 it was bought by Albertus +Pius, Prince of Carpi, passing in 1530 to his nephew, Rodolphus +Pius, in whose possession it remained till 1544. At some +time between 1544 and 1564 it disappeared, leaving no +trace. +<p>The greater part of A was translated into Latin in 1269 +by William of Moerbeke at the Papal Court at Viterbo. This +translation, in William's own hand, exists at Rome (Cod. +Ottobon. lat. 1850, Heiberg's B), and is one of our prime +sources, for, although the translation was hastily done and +the translator sometimes misunderstood the Greek, he followed +its wording so closely that his version is, for purposes of +collation, as good as a Greek manuscript. William used also, +for his translation, another manuscript from the same library +which contained works not included in A. This manuscript +was a collection of works on mechanics and optics; William +translated from it the two Books <I>On Floating Bodies,</I> and it +also contained the <I>Plane Equilibriums</I> and the <I>Quadrature +of the Parabola,</I> for which books William used both manu- +scripts. +<p>The four most important extant Greek manuscripts (except +C, the Constantinople manuscript discovered in 1906) were +copied from A. The earliest is E, the Venice manuscript +(Marcianus 305), which was written between the years 1449 +and 1472. The next is D, the Florence manuscript (Laurent. +XXVIII. 4), which was copied in 1491 for Angelo Poliziano, +permission having been obtained with some difficulty in con- +sequence of the jealousy with which Valla guarded his treasure. +The other two are G (Paris. 2360) copied from A after it had +passed to Albertus Pius, and H (Paris. 2361) copied in 1544 +by Christopherus Auverus for Georges d'Armagnac, Bishop +of Rodez. These four manuscripts, with the translation of +William of Moerbeke (B), enable the readings of A to be +inferred. +<p>A Latin translation was made at the instance of Pope +Nicholas V about the year 1450 by Jacobus Cremonensis. +<pb n=27><head>THE TEXT OF ARCHIMEDES</head> +It was made from A, which was therefore accessible to Pope +Nicholas though it does not seem to have belonged to him. +Regiomontanus made a copy of this translation about 1468 +and revised it with the help of E (the Venice manuscript of +the Greek text) and a copy of the same translation belonging +to Cardinal Bessarion, as well as another ‘old copy’ which +seems to have been B. +<p>The <I>editio princeps</I> was published at Basel (<I>apud Herva- +gium</I>) by Thomas Gechauff Venatorius in 1544. The Greek +text was based on a Nürnberg MS. (Norimberg. Cent. V, +app. 12) which was copied in the sixteenth century from A +but with interpolations derived from B; the Latin transla- +tion was Regiomontanus's revision of Jacobus Cremonensis +(Norimb. Cent. V, 15). +<p>A translation by F. Commandinus published at Venice in +1558 contained the <I>Measurement of a Circle, On Spirals,</I> the +<I>Quadrature of the Parabola, On Conoids and Spheroids,</I> and +the <I>Sand-reckoner.</I> This translation was based on the Basel +edition, but Commandinus also consulted E and other Greek +manuscripts. +<p>Torelli's edition (Oxford, 1792) also followed the <I>editio +princeps</I> in the main, but Torelli also collated E. The book +was brought out after Torelli's death by Abram Robertson, +who also collated five more manuscripts, including D, G +and H. The collation, however, was not well done, and the +edition was not properly corrected when in the press. +<p>The second edition of Heiberg's text of all the works of +Archimedes with Eutocius's commentaries, Latin translation, +apparatus criticus, &c., is now available (1910-15) and, of +course, supersedes the first edition (1880-1) and all others. +It naturally includes <I>The Method,</I> the fragment of the <I>Stoma- +chion,</I> and so much of the Greek text of the two Books <I>On +Floating Bodies</I> as could be restored from the newly dis- +covered Constantinople manuscript.<note><I>The Works of Archimedes,</I> edited in modern notation by the present +writer in 1897, was based on Heiberg's first edition, and the Supplement +(1912) containing <I>The Method,</I> on the original edition of Heiberg (in +<I>Hermes,</I> xlii, 1907) with the translation by Zeuthen (<I>Bibliotheca Mathe- +matica,</I> vii<SUB>3</SUB>. 1906/7).</note> +<C>Contents of <I>The Method.</I></C> +<p>Our description of the extant works of Archimedes +may suitably begin with <I>The Method</I> (the full title is <I>On</I> +<pb n=28><head>ARCHIMEDES</head> +<I>Mechanical Theorems, Method</I> (communicated) <I>to Eratosthenes</I>). +Premising certain propositions in mechanics mostly taken +from the <I>Plane Equilibriums,</I> and a lemma which forms +Prop. 1 of <I>On Conoids and Spheroids,</I> Archimedes obtains by +his mechanical method the following results. The area of any +segment of a section of a right-angled cone (parabola) is 4/3 of +the triangle with the same base and height (Prop. 1). The +right cylinder circumscribing a sphere or a spheroid of revolu- +tion and with axis equal to the diameter or axis of revolution +of the sphere or spheroid is 1 1/2 times the sphere or spheroid +respectively (Props. 2, 3). Props. 4, 7, 8, 11 find the volume of +any segment cut off, by a plane at right angles to the axis, +from any right-angled conoid (paraboloid of revolution), +sphere, spheroid, and obtuse-angled conoid (hyperboloid) in +terms of the cone which has the same base as the segment and +equal height. In Props. 5, 6, 9, 10 Archimedes uses his method +to find the centre of gravity of a segment of a paraboloid of +revolution, a sphere, and a spheroid respectively. Props. +12-15 and Prop. 16 are concerned with the cubature of two +special solid figures. (1) Suppose a prism with a square base +to have a cylinder inscribed in it, the circular bases of the +cylinder being circles inscribed in the squares which are +the bases of the prism, and suppose a plane drawn through +one side of one base of the prism and through that diameter of +the circle in the opposite base which is parallel to the said +side. This plane cuts off a solid bounded by two planes and +by part of the curved surface of the cylinder (a solid shaped +like a hoof cut off by a plane); and Props. 12-15 prove that +its volume is one-sixth of the volume of the prism. (2) Sup- +pose a cylinder inscribed in a cube, so that the circular bases +of the cylinder are circles inscribed in two opposite faces of +the cube, and suppose another cylinder similarly inscribed +with reference to two other opposite faces. The two cylinders +enclose a certain solid which is actually made up of eight +‘hoofs’ like that of Prop. 12. Prop. 16 proves that the +volume of this solid is two-thirds of that of the cube. Archi- +medes observes in his preface that a remarkable fact about +<pb n=29><head><I>THE METHOD</I></head> +these solids respectively is that each of them is equal to a +solid enclosed by <I>planes,</I> whereas the volume of curvilinear +solids (spheres, spheroids, &c.) is generally only expressible in +terms of other curvilinear solids (cones and cylinders). In +accordance with his dictum that the results obtained by the +mechanical method are merely indicated, but not actually +proved, unless confirmed by the rigorous methods of pure +geometry, Archimedes proved the facts about the two last- +named solids by the orthodox method of exhaustion as +regularly used by him in his other geometrical treatises; the +proofs, partly lost, were given in Props. 15 and 16. +<p>We will first illustrate the method by giving the argument +of Prop. 1 about the area of a parabolic segment. +<p>Let <I>ABC</I> be the segment, <I>BD</I> its diameter, <I>CF</I> the tangent +at <I>C.</I> Let <I>P</I> be any point on the segment, and let <I>AKF,</I> +<FIG> +<I>OPNM</I> be drawn parallel to <I>BD.</I> Join <I>CB</I> and produce it to +meet <I>MO</I> in <I>N</I> and <I>FA</I> in <I>K,</I> and let <I>KH</I> be made equal to +<I>KC.</I> +<p>Now, by a proposition ‘proved in a lemma’ (cf. <I>Quadrature +of the Parabola,</I> Prop. 5) +<MATH><I>MO</I>:<I>OP</I>=<I>CA</I>:<I>AO</I> +=<I>CK</I>:<I>KN</I> +=<I>HK</I>:<I>KN</I></MATH>. +<p>Also, by the property of the parabola, <MATH><I>EB</I>=<I>BD</I></MATH>, so that +<MATH><I>MN</I>=<I>NO</I></MATH> and <MATH><I>FK</I>=<I>KA</I></MATH>. +<p>It follows that, if <I>HC</I> be regarded as the bar of a balance, +a line <I>TG</I> equal to <I>PO</I> and placed with its middle point at <I>H</I> +balances, about <I>K,</I> the straight line <I>MO</I> placed where it is, +i. e. with its middle point at <I>N.</I> +<p>Similarly with <I>all</I> lines, as <I>MO, PO,</I> in the triangle <I>CFA</I> +and the segment <I>CBA</I> respectively. +<p>And there are the same number of these lines. Therefore +<pb n=30><head>ARCHIMEDES</head> +the whole segment of the parabola acting at <I>H</I> balances the +triangle <I>CFA</I> placed where it is. +<p>But the centre of gravity of the triangle <I>CFA</I> is at <I>W,</I> +where <MATH><I>CW</I>=2<I>WK</I></MATH> [and the whole triangle may be taken as +acting at <I>W</I>]. +<p>Therefore <MATH>(segment <I>ABC</I>):▵<I>CFA</I>=<I>WK</I>:<I>KH</I> +=1:3</MATH>, +so that <MATH>(segment <I>ABC</I>)=1/3▵<I>CFA</I> +=4/3▵<I>ABC</I></MATH>. Q.E.D. +<p>It will be observed that Archimedes takes the segment and +the triangle to be <I>made up</I> of parallel lines indefinitely close +together. In reality they are made up of indefinitely narrow +strips, but the width (<I>dx,</I> as we might say) being the same +for the elements of the triangle and segment respectively, +divides out. And of course the weight of each element in +both is proportional to the area. Archimedes also, without +mentioning <I>moments,</I> in effect assumes that the sum of the +moments of each particle of a figure, acting where it is, is +equal to the moment of the whole figure applied as one mass +at its centre of gravity. +<p>We will now take the case of any segment of a spheroid +of revolution, because that will cover several propositions of +Archimedes as particular cases. +<p>The ellipse with axes <I>AA</I>′, <I>BB</I>′ is a section made by the +plane of the paper in a spheroid with axis <I>AA</I>′. It is required +to find the volume of any right segment <I>ADC</I> of the spheroid +in terms of the right cone with the same base and height. +<p>Let <I>DC</I> be the diameter of the circular base of the segment. +Join <I>AB, AB</I>′, and produce them to meet the tangent at <I>A</I>′ to +the ellipse in <I>K, K</I>′, and <I>DC</I> produced in <I>E, F.</I> +<p>Conceive a cylinder described with axis <I>AA</I>′ and base the +circle on <I>KK</I>′ as diameter, and cones described with <I>AG</I> as +axis and bases the circles on <I>EF, DC</I> as diameters. +<p>Let <I>N</I> be any point on <I>AG,</I> and let <I>MOPQNQ</I>′<I>P</I>′<I>O</I>′<I>M</I>′ be +drawn through <I>N</I> parallel to <I>BB</I>′ or <I>DC</I> as shown in the +figure. +<p>Produce <I>A</I>′<I>A</I> to <I>H</I> so that <MATH><I>HA</I>=<I>AA</I>′</MATH>. +<pb n=31><head><I>THE METHOD</I></head> +<p>Now <MATH><I>HA</I>:<I>AN</I>=<I>A</I>′<I>A</I>:<I>AN</I> +=<I>KA</I>:<I>AQ</I> +=<I>MN</I>:<I>NQ</I> +=<I>MN</I><SUP>2</SUP>:<I>MN.NQ.</I></MATH> +<p>It is now necessary to prove that <MATH><I>MN.NQ</I>=<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP></MATH>. +<FIG> +<p>By the property of the ellipse, +<MATH><I>AN.NA</I>′:<I>NP</I><SUP>2</SUP>=(1/2<I>AA</I>′)<SUP>2</SUP>:(1/2<I>BB</I>′)<SUP>2</SUP> +=<I>AN</I><SUP>2</SUP>:<I>NQ</I><SUP>2</SUP></MATH>; +therefore <MATH><I>NQ</I><SUP>2</SUP>:<I>NP</I><SUP>2</SUP>=<I>AN</I><SUP>2</SUP>:<I>AN.NA</I>′ +=<I>NQ</I><SUP>2</SUP>:<I>NQ.QM</I></MATH>, +whence <MATH><I>NP</I><SUP>2</SUP>=<I>MQ.QN.</I></MATH> +<p>Add <I>NQ</I><SUP>2</SUP> to each side, and we have +<MATH><I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>=<I>MN.NQ.</I></MATH> +<p>Therefore, from above, +<MATH><I>HA</I>:<I>AN</I>=<I>MN</I><SUP>2</SUP>:(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>).</MATH> (1) +<p>But <I>MN</I><SUP>2</SUP>, <I>NP</I><SUP>2</SUP>, <I>NQ</I><SUP>2</SUP> are to one another as the areas of the +circles with <I>MM</I>′, <I>PP</I>′, <I>QQ</I>′ respectively as diameters, and these +<pb n=32><head>ARCHIMEDES</head> +circles are sections made by the plane though <I>N</I> at right +angles to <I>AA</I>′ in the cylinder, the spheroid and the cone <I>AEF</I> +respectively. +<p>Therefore, if <I>HAA</I>′ be a lever, and the sections of the +spheroid and cone be both placed with their centres of gravity +at <I>H</I>, these sections placed at <I>H</I> balance, about <I>A,</I> the section +<I>MM</I>′ of the cylinder where it is. +<p>Treating all the corresponding sections of the segment of +the spheroid, the cone and the cylinder in the same way, +we find that the cylinder with axis <I>AG,</I> where it is, balances, +about <I>A,</I> the cone <I>AEF</I> and the segment <I>ADC</I> together, when +both are placed with their centres of gravity at <I>H</I>; and, +if <I>W</I> be the centre of gravity of the cylinder, i.e. the middle +point of <I>AG,</I> +<MATH><I>HA</I>:<I>AW</I>=(cylinder, axis <I>AG</I>):(cone <I>AEF</I>+segmt. <I>ADC</I>).</MATH> +<p>If we call <I>V</I> the volume of the cone <I>AEF,</I> and <I>S</I> that of the +segment of the spheroid, we have +<MATH>(cylinder):(<I>V</I>+<I>S</I>)=3<I>V.</I>(<I>AA</I>′<SUP>2</SUP>)/(<I>AG</I><SUP>2</SUP>):(<I>V</I>+<I>S</I>)</MATH>, +while <MATH><I>HA</I>:<I>AW</I>=<I>AA</I>′:1/2<I>AG.</I></MATH> +<p>Therefore <MATH><I>AA</I>′:1/2<I>AG</I>=3<I>V.</I>(<I>AA</I>′<SUP>2</SUP>)/(<I>AG</I><SUP>2</SUP>):(<I>V</I>+<I>S</I>)</MATH>, +and <MATH>(<I>V</I>+<I>S</I>)=3/2<I>V.</I>(<I>AA</I>′)/(<I>AG</I>)</MATH>, +whence <MATH><I>S</I>=<I>V</I>((3<I>AA</I>′)/(2<I>AG</I>)-1)</MATH>. +<p>Again, let <I>V</I>′ be the volume of the cone <I>ADC.</I> +<p>Then <MATH><I>V</I>:<I>V</I>′=<I>EG</I><SUP>2</SUP>:<I>DG</I><SUP>2</SUP> +=(<I>BB</I>′<SUP>2</SUP>)/(<I>AA</I>′<SUP>2</SUP>).<I>AG</I><SUP>2</SUP>:<I>DG</I><SUP>2</SUP></MATH>. +<p>But <MATH><I>DG</I><SUP>2</SUP>:<I>AG.GA</I>′=<I>BB</I>′<SUP>2</SUP>:<I>AA</I>′<SUP>2</SUP></MATH>. +<p>Therefore <MATH><I>V</I>:<I>V</I>′=<I>AG</I><SUP>2</SUP>:<I>AG.GA</I>′ +=<I>AG</I>:<I>GA</I>′</MATH>. +<pb n=33><head><I>THE METHOD</I></head> +<p>It follows that <MATH><I>S</I>=<I>V</I>′.<I>AG/GA</I>′(3<I>AA</I>′)/(2<I>AG</I>)-1) +=<I>V</I>′.(3/2<I>AA</I>′-<I>AG</I>)/(<I>A</I>′<I>G</I>) +=<I>V</I>′.(1/2<I>AA</I>′+<I>A</I>′<I>G</I>)/(<I>A</I>′<I>G</I>)</MATH>, +which is the result stated by Archimedes in Prop. 8. +<p>The result is the same for the segment of a sphere. The +proof, of course slightly simpler, is given in Prop. 7. +<p>In the particular case where the segment is half the sphere +or spheroid, the relation becomes +<MATH><I>S</I>=2<I>V</I>′</MATH>, (Props. 2, 3) +and it follows that the volume of the whole sphere or spheroid +is 4<I>V</I>′, where <I>V</I>′ is the volume of the cone <I>ABB</I>′; i.e. the +volume of the sphere or spheroid is two-thirds of that of the +circumscribing cylinder. +<p>In order now to find the centre of gravity of the segment +of a spheroid, we must have the segment acting <I>where it is,</I> +not at <I>H.</I> +<p>Therefore formula (1) above will not serve. But we found +that <MATH><I>MN.NQ</I>=(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>)</MATH>, +whence <MATH><I>MN</I><SUP>2</SUP>:(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>)=(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>):<I>NQ</I><SUP>2</SUP></MATH>; +therefore <MATH><I>HA</I>:<I>AN</I>=(<I>NP</I><SUP>2</SUP>+<I>NQ</I><SUP>2</SUP>):<I>NQ</I><SUP>2</SUP></MATH>. +<p>(This is separately proved by Archimedes for the sphere +in Prop. 9.) +<p>From this we derive, as usual, that the cone <I>AEF</I> and the +segment <I>ADC</I> both acting <I>where they are</I> balance a volume +equal to the cone <I>AEF</I> placed with its centre of gravity at <I>H.</I> +<p>Now the centre of gravity of the cone <I>AEF</I> is on the line +<I>AG</I> at a distance 3/4<I>AG</I> from <I>A.</I> Let <I>X</I> be the required centre +of gravity of the segment. Then, taking moments about <I>A,</I> +we have <MATH><I>V.HA</I>=<I>S.AX</I>+<I>V.</I>3/4<I>AG</I></MATH>, +or <MATH><I>V</I>(<I>AA</I>′-3/4<I>AG</I>)=<I>S.AX</I> +=<I>V</I>(3/2<I>AA</I>′)/<I>AG</I>-1)<I>AX</I></MATH>, from above. +<pb n=34><head>ARCHIMEDES</head> +<p>Therefore <MATH><I>AX</I>:<I>AG</I>=(<I>AA</I>′-3/4<I>AG</I>):(3/2<I>AA</I>′-<I>AG</I>) +=(4<I>AA</I>′-3<I>AG</I>):(6<I>AA</I>′-4<I>AG</I>)</MATH>; +whence <MATH><I>AX</I>:<I>XG</I>=(4<I>AA</I>′-3<I>AG</I>):(2<I>AA</I>′-<I>AG</I>) +=(<I>AG</I>+4<I>A</I>′<I>G</I>):(<I>AG</I>+2<I>A</I>′<I>G</I>)</MATH>, +which is the result obtained by Archimedes in Prop. 9 for the +sphere and in Prop. 10 for the spheroid. +<p>In the case of the hemi-spheroid or hemisphere the ratio +<I>AX</I>:<I>XG</I> becomes 5:3, a result obtained for the hemisphere in +Prop. 6. +<p>The cases of the paraboloid of revolution (Props. 4, 5) and +the hyperboloid of revolution (Prop. 11) follow the same course, +and it is unnecessary to reproduce them. +<p>For the cases of the two solids dealt with at the end of the +treatise the reader must be referred to the propositions them- +selves. Incidentally, in Prop. 13, Archimedes finds the centre +of gravity of the half of a cylinder cut by a plane through +the axis, or, in other words, the centre of gravity of a semi- +circle. +<p>We will now take the other treatises in the order in which +they appear in the editions. +<C>On the Sphere and Cylinder, I, II.</C> +<p>The main results obtained in Book I are shortly stated in +a prefatory letter to Dositheus. Archimedes tells us that +they are new, and that he is now publishing them for the +first time, in order that mathematicians may be able to ex- +amine the proofs and judge of their value. The results are +(1) that the surface of a sphere is four times that of a great +circle of the sphere, (2) that the surface of any segment of a +sphere is equal to a circle the radius of which is equal to the +straight line drawn from the vertex of the segment to a point +on the circumference of the base, (3) that the volume of a +cylinder circumscribing a sphere and with height equal to the +diameter of the sphere is 3/2 of the volume of the sphere, +(4) that the surface of the circumscribing cylinder including +its bases is also 3/2 of the surface of the sphere. It is worthy +of note that, while the first and third of these propositions +appear in the book in this order (Props. 33 and 34 respec- +<pb n=35><head>ON THE SPHERE AND CYLINDER, I</head> +tively), this was not the order of their discovery; for Archi- +medes tells us in <I>The Method</I> that +<p>‘from the theorem that a sphere is four times as great as the +cone with a great circle of the sphere as base and with height +equal to the radius of the sphere I conceived the notion that +the surface of any sphere is four times as great as a great +circle in it; for, judging from the fact that any circle is equal +to a triangle with base equal to the circumference and height +equal to the radius of the circle, I apprehended that, in like +manner, any sphere is equal to a cone with base equal to the +surface of the sphere and height equal to the radius’. +<p>Book I begins with definitions (of ‘concave in the same +direction’ as applied to curves or broken lines and surfaces, of +a ‘solid sector’ and a ‘solid rhombus’) followed by five +Assumptions, all of importance. <I>Of all lines which have the +same extremities the straight line is the least</I>, and, if there are +two curved or bent lines in a plane having the same extremi- +ties and concave in the same direction, but one is wholly +included by, or partly included by and partly common with, +the other, then that which is included is the lesser of the two. +Similarly with plane surfaces and surfaces concave in the +same direction. Lastly, Assumption 5 is the famous ‘Axiom +of Archimedes’, which however was, according to Archimedes +himself, used by earlier geometers (Eudoxus in particular), to +the effect that <I>Of unequal magnitudes the greater exceeds +the less by such a magnitude as, when added to itself, can be +made to exceed any assigned magnitude of the same kind</I>; +the axiom is of course practically equivalent to Eucl. V, Def. 4, +and is closely connected with the theorem of Eucl. X. 1. +<p>As, in applying the method of exhaustion, Archimedes uses +both circumscribed and inscribed figures with a view to <I>com- +pressing</I> them into coalescence with the curvilinear figure to +be measured, he has to begin with propositions showing that, +given two unequal magnitudes, then, however near the ratio +of the greater to the less is to 1, it is possible to find two +straight lines such that the greater is to the less in a still less +ratio (>1), and to circumscribe and inscribe similar polygons to +a circle or sector such that the perimeter or the area of the +circumscribed polygon is to that of the inner in a ratio less +than the given ratio (Props. 2-6): also, just as Euclid proves +<pb n=36><head>ARCHIMEDES</head> +that, if we continually double the number of the sides of the +regular polygon inscribed in a circle, segments will ultimately be +left which are together less than any assigned area, Archimedes +has to supplement this (Prop. 6) by proving that, if we increase +the number of the sides of a <I>circumscribed</I> regular polygon +sufficiently, we can make the excess of the area of the polygon +over that of the circle less than any given area. Archimedes +then addresses himself to the problems of finding the <I>surface</I> of +any right cone or cylinder, problems finally solved in Props. 13 +(the cylinder) and 14 (the cone). Circumscribing and inscrib- +ing regular polygons to the bases of the cone and cylinder, he +erects pyramids and prisms respectively on the polygons as +bases and circumscribed or inscribed to the cone and cylinder +respectively. In Props. 7 and 8 he finds the surface of the +pyramids inscribed and circumscribed to the cone, and in +Props. 9 and 10 he proves that the surfaces of the inscribed +and circumscribed pyramids respectively (excluding the base) +are less and greater than the surface of the cone (excluding +the base). Props. 11 and 12 prove the same thing of the +prisms inscribed and circumscribed to the cylinder, and finally +Props. 13 and 14 prove, by the method of exhaustion, that the +surface of the cone or cylinder (excluding the bases) is equal +to the circle the radius of which is a mean proportional +between the ‘side’ (i.e. generator) of the cone or cylinder and +the radius or diameter of the base (i.e. is equal to <G>p</G><I>rs</I> in the +case of the cone and 2<G>p</G><I>rs</I> in the case of the cylinder, where +<I>r</I> is the radius of the base and <I>s</I> a generator). As Archimedes +here applies the method of exhaustion for the first time, we +will illustrate by the case of the cone (Prop. 14). +<p>Let <I>A</I> be the base of the cone, <I>C</I> a straight line equal to its +<FIG> +radius, <I>D</I> a line equal to a generator of the cone, <I>E</I> a mean +proportional to <I>C, D</I>, and <I>B</I> a circle with radius equal to <I>E</I>. +<pb n=37><head>ON THE SPHERE AND CYLINDER, I</head> +<p>If <I>S</I> is the surface of the cone, we have to prove that <I>S</I>=<I>B</I>. +For, if <I>S</I> is not equal to <I>B</I>, it must be either greater or less. +<p>I. Suppose <I>B</I><<I>S</I>. +<p>Circumscribe a regular polygon about <I>B</I>, and inscribe a similar +polygon in it, such that the former has to the latter a ratio less +than <I>S</I>:<I>B</I> (Prop. 5). Describe about <I>A</I> a similar polygon and +set up from it a pyramid circumscribing the cone. +<p>Then <MATH>(polygon about <I>A</I>):(polygon about <I>B</I>) +=<I>C</I><SUP>2</SUP>:<I>E</I><SUP>2</SUP> +=<I>C</I>:<I>D</I> +=(polygon about <I>A</I>):(surface of pyramid)</MATH>. +<p>Therefore (surface of pyramid)=(polygon about <I>B</I>). +<p>But (polygon about <I>B</I>):(polygon in <I>B</I>)<<I>S</I>:<I>B</I>; +therefore (surface of pyramid):(polygon in <I>B</I>)<<I>S</I>:<I>B</I>. +<p>But this is impossible, since (surface of pyramid)><I>S</I>, while +(polygon in <I>B</I>)<<I>B</I>; +therefore <I>B</I> is not less than <I>S</I>. +<p>II. Suppose <I>B</I>><I>S</I>. +<p>Circumscribe and inscribe similar regular polygons to <I>B</I> +such that the former has to the latter a ratio < <I>B</I>:<I>S</I>. Inscribe +in <I>A</I> a similar polygon, and erect on <I>A</I> the inscribed pyramid. +<p>Then <MATH>(polygon in <I>A</I>):(polygon in <I>B</I>)=<I>C</I><SUP>2</SUP>:<I>E</I><SUP>2</SUP> +=<I>C</I>:<I>D</I> +>(polygon in <I>A</I>):(surface of pyramid)</MATH>. +<p>(The latter inference is clear, because the ratio of <I>C</I>:<I>D</I> is +greater than the ratio of the perpendiculars from the centre of +<I>A</I> and from the vertex of the pyramid respectively on any +side of the polygon in <I>A</I>; in other words, if <MATH><G>b</G><<G>a</G><1/2<G>p</G>, +sin<G>a</G>>sin<G>b</G></MATH>.) +<p>Therefore (surface of pyramid)>(polygon in <I>B</I>). +<p>But (polygon about <I>B</I>):(polygon in <I>B</I>)<<I>B</I>:<I>S</I>, +whence (<I>a fortiori</I>) +<p>(polygon about <I>B</I>):(surface of pyramid)<<I>B</I>:<I>S</I>, +which is impossible, for (polygon about <I>B</I>)><I>B</I>, while (surface +of pyramid) < <I>S</I>. +<pb n=38><head>ARCHIMEDES</head> +<p>Therefore <I>B</I> is not greater than <I>S</I>. +<p>Hence <I>S</I>, being neither greater nor less than <I>B</I>, is equal to <I>B</I>. +<p>Archimedes next addresses himself to the problem of finding +the surface and volume of a sphere or a segment thereof, but +has to interpolate some propositions about ‘solid rhombi’ +(figures made up of two right cones, unequal or equal, with +bases coincident and vertices in opposite directions) the neces- +sity of which will shortly appear. +<p>Taking a great circle of the sphere or a segment of it, he +inscribes a regular polygon of an even number of sides bisected +<FIG> +<CAP>FIG. 1.</CAP> +<FIG> +<CAP>FIG. 2.</CAP> +by the diameter <I>AA</I>′, and approximates to the surface and +volume of the sphere or segment by making the polygon +revolve about <I>AA</I>′ and measuring the surface and volume of +solid so inscribed (Props. 21-7). He then does the same for the +a circumscribed solid (Props. 28-32). Construct the inscribed +polygons as shown in the above figures. Joining <I>BB</I>′, <I>CC</I>′, ..., +<I>CB</I>′, <I>DC</I>′ ... we see that <I>BB</I>′, <I>CC</I>′ ... are all parallel, and so are +<I>AB, CB</I>′, <I>DC</I>′ .... +<p>Therefore, by similar triangles, <MATH><I>BF</I>:<I>FA</I>=<I>A</I>′<I>B</I>:<I>BA</I></MATH>, and +<MATH><I>BF</I>:<I>FA</I>=<I>B</I>′<I>F</I>:<I>FK</I> +=<I>CG</I>:<I>GK</I> +=<I>C</I>′<I>G</I>:<I>GL</I> +. . . . . . . +=<I>E</I>′<I>I</I>:<I>IA</I>′ in Fig. 1 +(=<I>PM</I>:<I>MN</I> in Fig. 2)</MATH>, +<pb n=39><head>ON THE SPHERE AND CYLINDER, I</head> +whence, adding antecedents and consequents, we have +(Fig. 1) <MATH>(<I>BB</I>′+<I>CC</I>′+ ... +<I>EE</I>′):<I>AA</I>′=<I>A</I>′<I>B</I>:<I>BA</I></MATH>, (Prop. 21) +(Fig. 2) <MATH>(<I>BB</I>′+<I>CC</I>′+ ... +1/2<I>PP</I>′):<I>AM</I>=<I>A</I>′<I>B</I>:<I>BA</I></MATH>. (Prop. 22) +<p>When we make the polygon revolve about <I>AA</I>′, the surface +of the inscribed figure so obtained is made up of the surfaces +of cones and frusta of cones; Prop. 14 has proved that the +surface of the cone <I>ABB</I>′ is what we should write <G>p</G>.<I>AB.BF</I>, +and Prop. 16 has proved that the surface of the frustum +<I>BCC</I>′<I>B</I>′ is <MATH><G>p</G>.<I>BC</I> (<I>BF</I>+<I>CG</I>)</MATH>. It follows that, since <I>AB</I>= +<I>BC</I>= ..., the surface of the inscribed solid is +<MATH><G>p</G>.<I>AB</I> {1/2<I>BB</I>′+1/2(<I>BB</I>′+<I>CC</I>′)+ ...}</MATH>, +that is, <MATH><G>p</G>.<I>AB</I>(<I>BB</I>′+<I>CC</I>′+ ... +<I>EE</I>′)</MATH> (Fig. 1), (Prop. 24) +or <MATH><G>p</G>.<I>AB</I>(<I>BB</I>′+<I>CC</I>′+ ... +1/2<I>PP</I>′)</MATH> (Fig. 2). (Prop. 35) +<p>Hence, from above, the surface of the inscribed solid is +<G>p</G>.<I>A</I>′<I>B.AA</I>′ or <G>p</G>.<I>A</I>′<I>B.AM</I>, and is therefore less than +<G>p</G>.<I>AA</I>′<SUP>2</SUP> (Prop. 25) or <G>p</G>.<I>A</I>′<I>A.AM</I>, that is, <G>p</G>.<I>AP</I><SUP>2</SUP> (Prop. 37). +<p>Similar propositions with regard to surfaces formed by the +revolution about <I>AA</I>′ of regular circumscribed solids prove +that their surfaces are greater than <G>p</G>.<I>AA</I>′<SUP>2</SUP> and <G>p</G>.<I>AP</I><SUP>2</SUP> +respectively (Props. 28-30 and Props. 39-40). The case of the +segment is more complicated because the circumscribed poly- +gon with its sides parallel to <I>AB, BC ... DP</I> circumscribes +the <I>sector POP</I>′. Consequently, if the segment is less than a +semicircle, as <I>CAC</I>′, the base of the circumscribed polygon +(<I>cc</I>′) is on the side of <I>CC</I>′ towards <I>A</I>, and therefore the circum- +scribed polygon leaves over a small strip of the inscribed. This +complication is dealt with in Props. 39-40. Having then +arrived at circumscribed and inscribed figures with surfaces +greater and less than <G>p</G>.<I>AA</I>′<SUP>2</SUP> and <G>p</G>.<I>AP</I><SUP>2</SUP> respectively, and +having proved (Props. 32, 41) that the surfaces of the circum- +scribed and inscribed figures are to one another in the duplicate +ratio of their sides, Archimedes proceeds to prove formally, by +the method of exhaustion, that the surfaces of the sphere and +segment are equal to these circles respectively (Props. 33 and +42); <G>p</G>.<I>AA</I>′<SUP>2</SUP> is of course equal to four times the great circle +of the sphere. The segment is, for convenience, taken to be +<pb n=40><head>ARCHIMEDES</head> +less than a hemisphere, and Prop. 43 proves that the same +formula applies also to a segment greater than a hemisphere. +<p>As regards the volumes different considerations involving +‘solid rhombi’ come in. For convenience Archimedes takes, +in the case of the whole sphere, an inscribed polygon of 4<I>n</I> +sides (Fig. 1). It is easily seen that the solid figure formed +by its revolution is made up of the following: first, the solid +rhombus formed by the revolution of the quadrilateral <I>ABOB</I>′ +(the volume of this is shown to be equal to the cone with base +equal to the surface of the cone <I>ABB</I>′ and height equal to <I>p</I>, +the perpendicular from <I>O</I> on <I>AB</I>, Prop. 18); secondly, the +extinguisher-shaped figure formed by the revolution of the +triangle <I>BOC</I> about <I>AA</I>′ (this figure is equal to the difference +between two solid rhombi formed by the revolution of <I>TBOB</I>′ +and <I>TCOC</I>′ respectively about <I>AA</I>′, where <I>T</I> is the point of +intersection of <I>CB, C</I>′<I>B</I>′ produced with <I>A</I>′<I>A</I> produced, and +this difference is proved to be equal to a cone with base equal +to the surface of the frustum of a cone described by <I>BC</I> in its +revolution and height equal to <I>p</I> the perpendicular from <I>O</I> on +<I>BC</I>, Prop. 20); and so on; finally, the figure formed by the +revolution of the triangle <I>COD</I> about <I>AA</I>′ is the difference +between a cone and a solid rhombus, which is proved equal to +a cone with base equal to the surface of the frustum of a cone +described by <I>CD</I> in its revolution and height <I>p</I> (Prop. 19). +Consequently, by addition, the volume of the whole solid of +revolution is equal to the cone with base equal to its whole +surface and height <I>p</I> (Prop. 26). But the whole of the surface +of the solid is less than 4<G>p</G><I>r</I><SUP>2</SUP>, and <I>p</I> < <I>r</I>; therefore the volume +of the inscribed solid is less than four times the cone with +base <G>p</G><I>r</I><SUP>2</SUP> and height <I>r</I> (Prop. 27). +<p>It is then proved in a similar way that the revolution of +the similar circumscribed polygon of 4<I>n</I> sides gives a solid +the volume of which is <I>greater</I> than four times the same cone +(Props. 28-31 Cor.). Lastly, the volumes of the circumscribed +and inscribed figures are to one another in the triplicate ratio of +their sides (Prop. 32); and Archimedes is now in a position to +apply the method of exhaustion to prove that the volume of +the sphere is 4 times the cone with base <G>p</G><I>r</I><SUP>2</SUP> and height <I>r</I> +(Prop. 34). +<p>Dealing with the segment of a sphere, Archimedes takes, for +<pb n=41><head>ON THE SPHERE AND CYLINDER, I</head> +convenience, a segment less than a hemisphere and, by the +same chain of argument (Props. 38, 40 Corr., 41 and 42), proves +(Prop. 44) that the volume of the <I>sector</I> of the sphere bounded +by the surface of the segment is equal to a cone with base +equal to the surface of the segment and height equal to the +radius, i.e. the cone with base <G>p</G>.<I>AP</I><SUP>2</SUP> and height <I>r</I> (Fig. 2). +<p>It is noteworthy that the proportions obtained in Props. 21, +22 (see p. 39 above) can be expressed in trigonometrical form. +If 4<I>n</I> is the number of the sides of the polygon inscribed in +the circle, and 2<I>n</I> the number of the sides of the polygon +inscribed in the segment, and if the angle <I>AOP</I> is denoted +by <G>a</G>, the trigonometrical equivalents of the proportions are +respectively +(1) <MATH>sin<G>p</G>/(2<I>n</I>)+sin(2<G>p</G>)/(2<I>n</I>)+ ... +sin(2<I>n</I>-1) <G>p</G>/(2<I>n</I>)=cot<G>p</G>/(4<I>n</I>)</MATH>; +(2) <MATH>2 {sin<G>a</G>/<I>n</I>+sin(2<G>a</G>)/<I>n</I>+ ... +sin(<I>n</I>-1)<G>a</G>/<I>n</I>} +sin<G>a</G> +=(1-cos<G>a</G>) cot<G>a</G>/(2<I>n</I>)</MATH>. +Thus the two proportions give in effect a summation of the +series +<MATH>sin<G>q</G>+sin2<G>q</G>+ ... +sin(<I>n</I>-1)<G>q</G></MATH>, +both generally where <I>n</I><G>q</G> is equal to any angle <G>a</G> less than <G>p</G> +and in the particular case where <I>n</I> is even and <MATH><G>q</G>=<G>p</G>/<I>n</I></MATH>. +Props. 24 and 35 prove that the areas of the circles equal to +the surfaces of the solids of revolution described by the +polygons inscribed in the sphere and segment are the above +series multiplied by 4<G>p</G><I>r</I><SUP>2</SUP>sin<G>p</G>/(4<I>n</I>) and <G>p</G><I>r</I><SUP>2</SUP>.2 sin<G>a</G>/(2<I>n</I>) respectively +and are therefore 4<G>p</G><I>r</I><SUP>2</SUP>cos <G>p</G>/(4<I>n</I>) and <G>p</G><I>r</I><SUP>2</SUP>.2 cos<G>a</G>/(2<I>n</I>) (1-cos<G>a</G>) +respectively. Archimedes's results for the surfaces of the +sphere and segment, 4<G>p</G><I>r</I><SUP>2</SUP> and 2<G>p</G><I>r</I><SUP>2</SUP>(1-cos<G>a</G>), are the +limiting values of these expressions when <I>n</I> is indefinitely +increased and when therefore cos<G>p</G>/(4<I>n</I>) and cos<G>a</G>/(2<I>n</I>) become +unity. And the two series multiplied by 4<G>p</G><I>r</I><SUP>2</SUP>sin<G>p</G>/(4<I>n</I>) and +<pb n=42><head>ARCHIMEDES</head> +<G>p</G><I>r</I><SUP>2</SUP>.2 sin <G>a</G>/(2<I>n</I>) respectively are (when <I>n</I> is indefinitely increased) +precisely what we should represent by the integrals +<MATH>4<G>p</G><I>r</I><SUP>2</SUP>.(1/2)&int,<<SUP><G>p</G></SUP><SUB>0</SUB>>sin<G>q</G><I>d</I><G>q</G>, or 4<G>p</G><I>r</I><SUP>2</SUP></MATH>, +and <MATH><G>p</G><I>r</I><SUP>2</SUP>.&int,<<SUP><G>a</G></SUP><SUB>0</SUB>> 2 sin<G>q</G><I>d</I><G>q</G>, or 2<G>p</G><I>r</I><SUP>2</SUP>(1-cos<G>a</G>)</MATH>. +<p>Book II contains six problems and three theorems. Of the +theorems Prop. 2 completes the investigation of the volume of +any segment of a sphere, Prop. 44 of Book I having only +brought us to the volume of the corresponding sector. If +<I>ABB</I>′ be a segment of a sphere cut off by a plane at right +angles to <I>AA</I>′, we learnt in I. 44 that the volume of the <I>sector</I> +<FIG> +<I>OBAB</I>′ is equal to the cone with base equal to the surface +of the segment and height equal to the radius, i.e. 1/3<G>p</G>.<I>AB</I><SUP>2</SUP>.<I>r</I>, +where <I>r</I> is the radius. The volume of the segment is therefore +<MATH>1/3<G>p</G>.<I>AB</I><SUP>2</SUP>.<I>r</I>-1/3<G>p</G>.<I>BM</I><SUP>2</SUP>.<I>OM</I></MATH>. +<p>Archimedes wishes to express this as a cone with base the +same as that of the segment. Let <I>AM</I>, the height of the seg- +ment, =<I>h</I>. +<p>Now <MATH><I>AB</I><SUP>2</SUP>:<I>BM</I><SUP>2</SUP>=<I>A</I>′<I>A</I>:<I>A</I>′<I>M</I>=2<I>r</I>:(2<I>r</I>-<I>h</I>)</MATH>. +<p>Therefore +<MATH>1/3<G>p</G>(<I>AB</I><SUP>2</SUP>.<I>r</I>-<I>BM</I><SUP>2</SUP>.<I>OM</I>)=1/3<G>p</G>.<I>BM</I><SUP>2</SUP>{(2<I>r</I><SUP>2</SUP>)/(2<I>r</I>-<I>h</I>)-(<I>r</I>-<I>h</I>)} +=1/3<G>p</G>.<I>BM</I><SUP>2</SUP>.<I>h</I>((3<I>r</I>-<I>h</I>)/(2<I>r</I>-<I>h</I>))</MATH>. +<p>That is, the segment is equal to the cone with the same +base as that of the segment and height <MATH><I>h</I>(3<I>r</I>-<I>h</I>)/(2<I>r</I>-<I>h</I>)</MATH>. +<pb n=43><head>ON THE SPHERE AND CYLINDER, II</head> +This is expressed by Archimedes thus. If <I>HM</I> is the height +of the required cone, +<MATH><I>HM</I>:<I>AM</I>=(<I>OA</I>′+<I>A</I>′<I>M</I>):<I>A</I>′<I>M</I></MATH>, (1) +and similarly the cone equal to the segment <I>A</I>′<I>BB</I>′ has the +height <I>H</I>′<I>M</I>, where +<MATH><I>H</I>′<I>M</I>:<I>A</I>′<I>M</I>=(<I>OA</I>+<I>AM</I>):<I>AM</I></MATH>. (2) +His proof is, of course, not in the above form but purely +geometrical. +<p>This proposition leads to the most important proposition in +the Book, Prop. 4, which solves the problem <I>To cut a given +sphere by a plane in such a way that the volumes of the +segments are to one another in a given ratio</I>. +<C><I>Cubic equation arising out of II. 4</I>.</C> +<p>If <I>m</I>:<I>n</I> be the given ratio of the cones which are equal to +the segments and the heights of which are <I>h, h</I>′, we have +<MATH><I>h</I>((3<I>r</I>-<I>h</I>)/(2<I>r</I>-<I>h</I>))=<I>(m/n)h</I>′((3<I>r</I>-<I>h</I>′)/(2<I>r</I>-<I>h</I>′))</MATH>, +and, if we eliminate <I>h</I>′ by means of the relation <I>h</I>+<I>h</I>′=2<I>r</I>, +we easily obtain the following cubic equation in <I>h</I>, +<MATH><I>h</I><SUP>3</SUP>-3<I>h</I><SUP>2</SUP><I>r</I>+(4<I>m</I>)/(<I>m</I>+<I>n</I>)<I>r</I><SUP>3</SUP>=0</MATH>. +<p>Archimedes in effect reduces the problem to this equation, +which, however, he treats as a particular case of the more +general problem corresponding to the equation +<MATH>(<I>r</I>+<I>h</I>):<I>b</I>=<I>c</I><SUP>2</SUP>:(2<I>r</I>-<I>h</I>)<SUP>2</SUP></MATH>, +where <I>b</I> is a given length and <I>c</I><SUP>2</SUP> any given area, +or <MATH><I>x</I><SUP>2</SUP>(<I>a</I>-<I>x</I>)=<I>bc</I><SUP>2</SUP></MATH>, where <MATH><I>x</I>=2<I>r</I>-<I>h</I></MATH> and <MATH>3<I>r</I>=<I>a</I></MATH>. +<p>Archimedes obtains his cubic equation with one unknown +by means of a <I>geometrical</I> elimination of <I>H, H</I>′ from the +equation <MATH><I>HM</I>=<I>(m/n).H</I>′<I>M</I></MATH>, where <I>HM, H</I>′<I>M</I> have the values +determined by the proportions (1) and (2) above, after which +the one variable point <I>M</I> remaining corresponds to the one +unknown of the cubic equation. His method is, first, to find +<pb n=44><head>ARCHIMEDES</head> +values for each of the ratios <I>A</I>′<I>H</I>′:<I>H</I>′<I>M</I> and <I>H</I>′<I>H</I>:<I>A</I>′<I>H</I>′ which +are alike independent of <I>H, H</I>′ and then, secondly, to equate +the ratio compounded of these two to the known value of the +ratio <I>HH</I>′:<I>H</I>′<I>M</I>. +(<G>a</G>) We have, from (2), +<MATH><I>A</I>′<I>H</I>′:<I>H</I>′<I>M</I>=<I>OA</I>:(<I>OA</I>+<I>AM</I>)</MATH>. (3) +(<G>b</G>) From (1) and (2), <I>separando</I>, +<MATH><I>AH</I>:<I>AM</I>=<I>OA</I>′:<I>A</I>′<I>M</I></MATH>, (4) +<MATH><I>A</I>′<I>H</I>′:<I>A</I>′<I>M</I>=<I>OA</I>:<I>AM</I></MATH>. (5) +<p>Equating the values of the ratio <I>A</I>′<I>M</I>:<I>AM</I> given by (4), (5), +we have <MATH><I>OA</I>′:<I>AH</I>=<I>A</I>′<I>H</I>′:<I>OA</I> +=<I>OH</I>′:<I>OH</I></MATH>, +whence <MATH><I>HH</I>′:<I>OH</I>′=<I>OH</I>′:<I>A</I>′<I>H</I>′</MATH>, (since <MATH><I>OA</I>=<I>OA</I>′</MATH>) +or <MATH><I>HH</I>′.<I>A</I>′<I>H</I>′=<I>OH</I>′<SUP>2</SUP></MATH>, +so that <MATH><I>HH</I>′:<I>A</I>′<I>H</I>′=<I>OH</I>′<SUP>2</SUP>:<I>A</I>′<I>H</I>′<SUP>2</SUP></MATH>. (6) +<p>But, by (5), <MATH><I>OA</I>′:<I>A</I>′<I>H</I>′=<I>AM</I>:<I>A</I>′<I>M</I></MATH>, +and, <I>componendo</I>, <MATH><I>OH</I>′:<I>A</I>′<I>H</I>′=<I>AA</I>′:<I>A</I>′<I>M</I></MATH>. +<p>By substitution in (6), +<MATH><I>HH</I>′:<I>A</I>′<I>H</I>′=<I>AA</I>′<SUP>2</SUP>:<I>A</I>′<I>M</I><SUP>2</SUP></MATH>. (7) +<p>Compounding with (3), we obtain +<MATH><I>HH</I>′:<I>H</I>′<I>M</I>=(<I>AA</I>′<SUP>2</SUP>:<I>A</I>′<I>M</I><SUP>2</SUP>).(<I>OA</I>:<I>OA</I>+<I>AM</I>)</MATH>. (8) +<p>[The algebraical equivalent of this is +<MATH>(<I>m</I>+<I>n</I>)/<I>n</I>=(4<I>r</I><SUP>3</SUP>)/((2<I>r</I>-<I>h</I>)<SUP>2</SUP> (<I>r</I>+<I>h</I>))</MATH>, +which reduces to <MATH>(<I>m</I>+<I>n</I>)/<I>m</I>=(4<I>r</I><SUP>3</SUP>)/(3<I>h</I><SUP>2</SUP><I>r</I>-<I>h</I><SUP>3</SUP>)</MATH>, +or <MATH><I>h</I><SUP>3</SUP>-3<I>h</I><SUP>2</SUP><I>r</I>+(4<I>m</I>)/(<I>m</I>+<I>n</I>)<I>r</I><SUP>3</SUP>=0, as above.]</MATH> +<p>Archimedes expresses the result (8) more simply by pro- +ducing <I>OA</I> to <I>D</I> so that <MATH><I>OA</I>=<I>AD</I></MATH>, and then dividing <I>AD</I> at +<pb n=45><head>ON THE SPHERE AND CYLINDER, II</head> +<I>E</I> so that <MATH><I>AD</I>:<I>DE</I>=<I>HH</I>′:<I>H</I>′<I>M</I></MATH> or (<I>m</I>+<I>n</I>):<I>n</I>. We have +then <MATH><I>OA</I>=<I>AD</I></MATH> and <MATH><I>OA</I>+<I>AM</I>=<I>MD</I></MATH>, so that (8) reduces to +<MATH><I>AD</I>:<I>DE</I>=(<I>AA</I>′<SUP>2</SUP>:<I>A</I>′<I>M</I><SUP>2</SUP>).(<I>AD</I>:<I>MD</I>)</MATH>, +or <MATH><I>MD</I>:<I>DE</I>=<I>AA</I>′<SUP>2</SUP>:<I>A</I>′<I>M</I><SUP>2</SUP></MATH>. +<p>Now, says Archimedes, <I>D</I> is given, since <MATH><I>AD</I>=<I>OA</I></MATH>. Also, +<I>AD</I>:<I>DE</I> being a given ratio, <I>DE</I> is given. Hence the pro- +blem reduces itself to that of dividing <I>A</I>′<I>D</I> into two parts at +<I>M</I> such that +<MATH><I>MD</I>:(a given length)=(a given area):<I>A</I>′<I>M</I><SUP>2</SUP></MATH>. +<p>That is, the generalized equation is of the form +<MATH><I>x</I><SUP>2</SUP>(<I>a</I>-<I>x</I>)=<I>bc</I><SUP>2</SUP></MATH>, as above. +<C>(i) Archimedes's own solution of the cubic.</C> +<p>Archimedes adds that, ‘if the problem is propounded in this +general form, it requires a <G>diorismo/s</G> [i.e. it is necessary to +investigate the limits of possibility], but if the conditions are +added which exist in the present case [i.e. in the actual +problem of Prop. 4], it does not require a <G>diorismo/s</G>’ (in other +words, a solution is always possible). He then promises to +give ‘at the end’ an analysis and synthesis of both problems +[i.e. the <G>diorismo/s</G> and the problem itself]. The promised +solutions do not appear in the treatise as we have it, but +Eutocius gives solutions taken from ‘an old book’ which he +managed to discover after laborious search, and which, since it +was partly written in Archimedes's favourite Doric, he with +fair reason assumed to contain the missing <I>addendum</I> by +Archimedes. +<p>In the Archimedean fragment preserved by Eutocius the +above equation, <MATH><I>x</I><SUP>2</SUP>(<I>a</I>-<I>x</I>)=<I>bc</I><SUP>2</SUP></MATH>, is solved by means of the inter- +section of a parabola and a rectangular hyperbola, the equations +of which may be written thus +<MATH><I>x</I><SUP>2</SUP>=(<I>c</I><SUP>2</SUP>/<I>a</I>)<I>y</I>, (<I>a</I>-<I>x</I>)<I>y</I>=<I>ab</I></MATH>. +<p>The <G>diorismo/s</G> takes the form of investigating the maximum +possible value of <MATH><I>x</I><SUP>2</SUP>(<I>a</I>-<I>x</I>)</MATH>, and it is proved that this maximum +value for a real solution is that corresponding to the value +<MATH><I>x</I>=2/3<I>a</I></MATH>. This is established by showing that, if <MATH><I>bc</I><SUP>2</SUP>=(4/27)<I>a</I><SUP>3</SUP></MATH>, +<pb n=46><head>ARCHIMEDES</head> +the curves touch at the point for which <MATH><I>x</I>=2/3<I>a</I></MATH>. If on the +other hand <MATH><I>bc</I><SUP>2</SUP><(4/27)<I>a</I><SUP>3</SUP></MATH>, it is proved that there are two real +solutions. In the particular case arising in Prop. 4 it is clear +that the condition for a real solution is satisfied, for the +expression corresponding to <I>bc</I><SUP>2</SUP> is <MATH><I>m</I>/(<I>m</I>+<I>n</I>)4<I>r</I><SUP>3</SUP></MATH>, and it is only +necessary that <MATH><I>m</I>/(<I>m</I>+<I>n</I>)4<I>r</I><SUP>3</SUP></MATH> should be not greater than <MATH>(4/27)<I>a</I><SUP>3</SUP></MATH> or +4<I>r</I><SUP>3</SUP>, which is obviously the case. +<C>(ii) Solution of the cubic by Dionysodorus.</C> +<p>It is convenient to add here that Eutocius gives, in addition +to the solution by Archimedes, two other solutions of our +problem. One, by Dionysodorus, solves the cubic equation in +the less general form in which it is required for Archimedes's +proposition. This form, obtained from (8) above, by putting +<MATH><I>A</I>′<I>M</I>=<I>x</I></MATH>, is +<MATH>4<I>r</I><SUP>2</SUP>:<I>x</I><SUP>2</SUP>=(3<I>r</I>-<I>x</I>):<I>n</I>/(<I>m</I>+<I>n</I>)<I>r</I></MATH>, +and the solution is obtained by drawing the parabola and +<FIG> +the rectangular hyperbola which we should represent by the +equations +<MATH><I>n</I>/(<I>m</I>+<I>n</I>)<I>r</I>(3<I>r</I>-<I>x</I>)=<I>y</I><SUP>2</SUP> and <I>n</I>/(<I>m</I>+<I>n</I>)2<I>r</I><SUP>2</SUP>=<I>xy</I></MATH>, +referred to <I>A</I>′<I>A</I> and the perpendicular to it through <I>A</I> as axes +of <I>x, y</I> respectively. +<p>(We make <I>FA</I> equal to <I>OA</I>, and draw the perpendicular +<I>AH</I> of such a length that +<MATH><I>FA</I>:<I>AH</I>=<I>CE</I>:<I>ED</I>=(<I>m</I>+<I>n</I>):<I>n</I></MATH>.) +<pb n=47><head>ON THE SPHERE AND CYLINDER, II</head> +<C>(iii) Solution of the original problem of II. 4 by Diocles.</C> +<p>Diocles proceeded in a different manner, satisfying, by +a geometrical construction, not the derivative cubic equation, +but the three simultaneous relations which hold in Archi- +medes's proposition, namely +<MATH><BRACE><I>HM</I>:<I>H</I>′<I>M</I>=<I>m</I>:<I>n</I> +<I>HA</I>:<I>h</I>=<I>r</I>:<I>h</I>′ +<I>H</I>′<I>A</I>′:<I>h</I>′=<I>r</I>:<I>h</I></BRACE></MATH>, +with the slight generalization that he substitutes for <I>r</I> in +these equations another length <I>a</I>. +<FIG> +The problem is, given a straight line <I>AA</I>′, a ratio <I>m</I>:<I>n</I>, and +another straight line <I>AK</I> (= <I>a</I>), to divide <I>AA</I>′ at a point <I>M</I> +and at the same time to find two points <I>H, H</I>′ on <I>AA</I>′ +produced such that the above relations (with <I>a</I> in place +of <I>r</I>) hold. +<p>The analysis leading to the construction is very ingenious. +Place <I>AK</I> (= <I>a</I>) at right angles to <I>AA</I>′, and draw <I>A</I>′<I>K</I>′ equal +and parallel to it. +<p>Suppose the problem solved, and the points <I>M, H, H</I>′ all +found. +<p>Join <I>KM</I>, produce it, and complete the rectangle <I>KGEK</I>′. +<pb n=48><head>ARCHIMEDES</head> +Draw <I>QMN</I> through <I>M</I> parallel to <I>AK</I>. Produce <I>K</I>′<I>M</I> to +meet <I>KG</I> produced in <I>F</I>. +<p>By similar triangles, +<MATH><I>FA</I>:<I>AM</I>=<I>K</I>′<I>A</I>′:<I>A</I>′<I>M</I>, or <I>FA</I>:<I>h</I>=<I>a</I>:<I>h</I>′</MATH>, +whence <MATH><I>FA</I>=<I>AH</I> (<I>k</I>, suppose)</MATH>. +Similarly <MATH><I>A</I>′<I>E</I>=<I>A</I>′<I>H</I>′ (<I>k</I>′, suppose)</MATH>. +<p>Again, by similar triangles, +<MATH>(<I>FA</I>+<I>AM</I>):(<I>A</I>′<I>K</I>′+<I>A</I>′<I>M</I>)=<I>AM</I>:<I>A</I>′<I>M</I> +=(<I>AK</I>+<I>AM</I>):(<I>EA</I>′+<I>A</I>′<I>M</I>)</MATH>, +or <MATH>(<I>k</I>+<I>h</I>):(<I>a</I>+<I>h</I>′)=(<I>a</I>+<I>h</I>):(<I>k</I>′+<I>h</I>′)</MATH>, +i.e. <MATH>(<I>k</I>+<I>h</I>) (<I>k</I>′+<I>h</I>′)=(<I>a</I>+<I>h</I>) (<I>a</I>+<I>h</I>′)</MATH>. (1) +<p>Now, by hypothesis, +<MATH><I>m</I>:<I>n</I>=(<I>k</I>+<I>h</I>):(<I>k</I>′+<I>h</I>′) +=(<I>k</I>+<I>h</I>) (<I>k</I>′+<I>h</I>′):(<I>k</I>′+<I>h</I>′)<SUP>2</SUP> +=(<I>a</I>+<I>h</I>) (<I>a</I>+<I>h</I>′):(<I>k</I>′+<I>h</I>′)<SUP>2</SUP> [by (1)]</MATH>. (2) +<p>Measure <I>AR, A</I>′<I>R</I>′ on <I>AA</I>′ produced both ways equal to <I>a</I>. +Draw <I>RP, R</I>′<I>P</I>′ at right angles to <I>RR</I>′ as shown in the figure. +Measure along <I>MN</I> the length <I>MV</I> equal to <I>MA</I>′ or <I>h</I>′, and +draw <I>PP</I>′ through <I>V, A</I>′ to meet <I>RP, R</I>′<I>P</I>′. +<p>Then <MATH><I>QV</I>=<I>k</I>′+<I>h</I>′, <I>P</I>′<I>V</I>=√2(<I>a</I>+<I>h</I>′), +<I>PV</I>=√2(<I>a</I>+<I>h</I>)</MATH>, +whence <MATH><I>PV.P</I>′<I>V</I>=2(<I>a</I>+<I>h</I>) (<I>a</I>+<I>h</I>′)</MATH>; +and, from (2) above, +<MATH>2<I>m</I>:<I>n</I>=2(<I>a</I>+<I>h</I>) (<I>a</I>+<I>h</I>′):(<I>k</I>′+<I>h</I>′)<SUP>2</SUP> +=<I>PV.P</I>′<I>V</I>:<I>QV</I><SUP>2</SUP></MATH>. (3) +<p>Therefore <I>Q</I> is on an ellipse in which <I>PP</I>′ is a diameter, and +<I>QV</I> is an ordinate to it. +<p>Again, ▭ <I>GQNK</I> is equal to ▭ <I>AA</I>′<I>K</I>′<I>K</I>, whence +<MATH><I>GQ.QN</I>=<I>AA</I>′.<I>A</I>′<I>K</I>′=(<I>h</I>+<I>h</I>′)<I>a</I>=2<I>ra</I></MATH>, (4) +and therefore <I>Q</I> is on the rectangular hyperbola with <I>KF</I>, +<I>KK</I>′ as asymptotes and passing through <I>A</I>′. +<pb n=49><head>ON THE SPHERE AND CYLINDER, II</head> +<p>How this ingenious analysis was suggested it is not possible +to say. It is the equivalent of reducing the four unknowns +<I>h, h</I>′, <I>k, k</I>′ to two, by putting <I>h</I>=<I>r</I>+<I>x, h</I>′=<I>r</I>-<I>x</I> and <I>k</I>′=<I>y</I>, +and then reducing the given relations to two equations in <I>x, y</I>, +which are coordinates of a point in relation to <I>Ox, Oy</I> as axes, +where <I>O</I> is the middle point of <I>AA</I>′, and <I>Ox</I> lies along <I>OA</I>′, +while <I>Oy</I> is perpendicular to it. +<p>Our original relations (p. 47) give +<MATH><I>y</I>=<I>k</I>′=(<I>ah</I>′)/<I>h</I>=<I>a</I>(<I>r</I>-<I>x</I>)/(<I>r</I>+<I>x</I>), <I>k</I>=(<I>ah</I>)/<I>h</I>′=<I>a</I>(<I>r</I>+<I>x</I>)/(<I>r</I>-<I>x</I>)</MATH>, and +<MATH><I>m</I>/<I>n</I>=(<I>h</I>+<I>k</I>)/(<I>h</I>′+<I>k</I>′)</MATH>. +<p>We have at once, from the first two equations, +<MATH><I>ky</I>=<I>a</I>(<I>r</I>+<I>x</I>)/(<I>r</I>-<I>x</I>)<I>y</I>=<I>a</I><SUP>2</SUP></MATH>, +whence <MATH>(<I>r</I>+<I>x</I>)<I>y</I>=<I>a</I>(<I>r</I>-<I>x</I>)</MATH>, +and <MATH>(<I>x</I>+<I>r</I>) (<I>y</I>+<I>a</I>)=2<I>ra</I></MATH>, +which is the rectangular hyperbola (4) above. +<p>Again, <MATH><I>m</I>/<I>n</I>=(<I>h</I>+<I>k</I>)/(<I>h</I>′+<I>k</I>′)=((<I>r</I>+<I>x</I>)(1+<I>a</I>/(<I>r</I>-<I>x</I>)))/((<I>r</I>-<I>x</I>) (1+<I>a</I>/(<I>r</I>+<I>x</I>)))</MATH>, +whence we obtain a cubic equation in <I>x</I>, +<MATH>(<I>r</I>+<I>x</I>)<SUP>2</SUP>(<I>r</I>+<I>a</I>-<I>x</I>)=(<I>m/n</I>)(<I>r</I>-<I>x</I>)<SUP>2</SUP>(<I>r</I>+<I>a</I>+<I>x</I>)</MATH>, +which gives +<MATH>(<I>m/n</I>)(<I>r</I>-<I>x</I>)<SUP>2</SUP>((<I>r</I>+<I>a</I>+<I>x</I>)/(<I>r</I>+<I>x</I>))<SUP>2</SUP>=(<I>r</I>+<I>a</I>)<SUP>2</SUP>-<I>x</I><SUP>2</SUP></MATH>. +<p>But <MATH><I>y</I>/(<I>r</I>-<I>x</I>)=<I>a</I>/(<I>r</I>+<I>x</I>)</MATH>, whence <MATH>(<I>y</I>+<I>r</I>-<I>x</I>)/(<I>r</I>-<I>x</I>)=(<I>r</I>+<I>a</I>+<I>x</I>)/(<I>r</I>+<I>x</I>)</MATH>, +and the equation becomes +<MATH>(<I>m/n</I>)(<I>y</I>+<I>r</I>-<I>x</I>)<SUP>2</SUP>=(<I>r</I>+<I>a</I>)<SUP>2</SUP>-<I>x</I><SUP>2</SUP></MATH>, +which is the ellipse (3) above. +<pb n=50><head>ARCHIMEDES</head> +<p>To return to Archimedes. Book II of our treatise contains +further problems: To find a sphere equal to a given cone or +cylinder (Prop. 1), solved by reduction to the finding of two +mean proportionals; to cut a sphere by a plane into two +segments having their surfaces in a given ratio (Prop. 3), +which is easy (by means of I. 42, 43); given two segments of +spheres, to find a third segment of a sphere similar to one +of the given segments and having its surface equal to that of +the other (Prop. 6); the same problem with volume substituted +for surface (Prop. 5), which is again reduced to the finding +of two mean proportionals; from a given sphere to cut off +a segment having a given ratio to the cone with the same +base and equal height (Prop. 7). The Book concludes with +two interesting theorems. If a sphere be cut by a plane into +two segments, the greater of which has its surface equal to <I>S</I> +and its volume equal to <I>V,</I> while <I>S</I>′, <I>V</I>′ are the surface and +volume of the lesser, then <I>V</I>:<I>V</I>′<<I>S</I><SUP>2</SUP>:<I>S</I>′<SUP>2</SUP> but > <I>S</I><SUP>3/2</SUP>:<I>S</I>′<SUP>3/2</SUP> +(Prop. 8): and, of all segments of spheres which have their +surfaces equal, the hemisphere is the greatest in volume +(Prop. 9). +<C>Measurement of a Circle.</C> +<p>The book on the <I>Measurement of a Circle</I> consists of three +propositions only, and is not in its original form, having lost +(as the treatise <I>On the Sphere and Cylinder</I> also has) prac- +tically all trace of the Doric dialect in which Archimedes +wrote; it may be only a fragment of a larger treatise. The +three propositions which survive prove (1) that the area of +a circle is equal to that of a right-angled triangle in which +the perpendicular is equal to the radius, and the base to the +circumference, of the circle, (2) that the area of a circle is to +the square on its diameter as 11 to 14 (the text of this proposition +is, however, unsatisfactory, and it cannot have been +placed by Archimedes before Prop. 3, on which it depends), +(3) <I>that the ratio of the circumference of any circle to its +diameter</I> (i. e. <G>p</G>) <I>is</I> < 3 1/7 <I>but</I> > 3 10/71. Prop. 1 is proved by +the method of exhaustion in Archimedes's usual form: he +approximates to the area of the circle in both directions +(<I>a</I>) by inscribing successive regular polygons with a number of +<pb n=51><head>MEASUREMENT OF A CIRCLE</head> +sides continually doubled, beginning from a square, (<I>b</I>) by +circumscribing a similar set of regular polygons beginning +from a square, it being shown that, if the number of the +sides of these polygons be continually doubled, more than half +of the portion of the polygon outside the circle will be taken +away each time, so that we shall ultimately arrive at a circum- +scribed polygon greater than the circle by a space less than +any assigned area. +<p>Prop. 3, containing the arithmetical approximation to <G>p</G>, is +the most interesting. The method amounts to calculating +approximately the perimeter of two regular polygons of 96 +sides, one of which is circumscribed, and the other inscribed, +to the circle; and the calculation starts from a greater and +a lesser limit to the value of √3, which Archimedes assumes +without remark as known, namely +<MATH>265/153<√3<1351/780</MATH>. +<p>How did Archimedes arrive at these particular approximations? +No puzzle has exercised more fascination upon +writers interested in the history of mathematics. De Lagny, +Mollweide, Buzengeiger, Hauber, Zeuthen, P. Tannery, Heilermann, +Hultsch, Hunrath, Wertheim, Bobynin: these are the +names of some of the authors of different conjectures. The +simplest supposition is certainly that of Hunrath and Hultsch, +who suggested that the formula used was +<MATH><I>a</I>±<I>b</I>/2<I>a</I>>√(<I>a</I><SUP>2</SUP>±<I>b</I>)><I>a</I>±<I>b</I>/(2<I>a</I>±1)</MATH>, +where <I>a</I><SUP>2</SUP> is the nearest square number above or below <I>a</I><SUP>2</SUP>±<I>b</I>, +as the case may be. The use of the first part of this formula +by Heron, who made a number of such approximations, is +proved by a passage in his <I>Metrica</I><note>Heron, <I>Metrica,</I> i. 8.</note>, where a rule equivalent +to this is applied to √720; the second part of the formula is +used by the Arabian Alkarkhī (eleventh century) who drew +from Greek sources, and one approximation in Heron may be +obtained in this way.<note><I>Stereom.</I> ii, p. 184. 19, Hultsch; p. 154. 19, Heib. <MATH>√54=7 1/3=7 5/15</MATH> instead of 7 5/14.</note> Another suggestion (that of Tannery +<pb n=52><head>ARCHIMEDES</head> +and Zeuthen) is that the successive solutions in integers of +the equations +<MATH><BRACE><I>x</I><SUP>2</SUP>-3<I>y</I><SUP>2</SUP>=1 +<I>x</I><SUP>2</SUP>-3<I>y</I><SUP>2</SUP>=-2</BRACE></MATH> +may have been found in a similar way to those of the +equations <MATH><I>x</I><SUP>2</SUP>-2<I>y</I><SUP>2</SUP>=±1</MATH> given by Theon of Smyrna after +the Pythagoreans. The rest of the suggestions amount for the +most part to the use of the method of continued fractions +more or less disguised. +<p>Applying the above formula, we easily find +<MATH>2-1/4>√3>2-1/3</MATH>, +or <MATH>7/4>√3>5/3</MATH>. +<p>Next, clearing of fractions, we consider 5 as an approxi- +mation to √(3.3<SUP>2</SUP>) or √27, and we have +<MATH>5+2/10>3√3>5+2/11</MATH>, +whence <MATH>26/15>√3>19/11</MATH>. +<p>Clearing of fractions again, and taking 26 as an approxi- +mation to √(3.15<SUP>2</SUP>) or √675, we have +<MATH>26-1/52>15√3>26-1/51</MATH>, +which reduces to +<MATH>1351/780>√3>265/153</MATH>. +<p>Archimedes first takes the case of the circumscribed polygon. +Let <I>CA</I> be the tangent at <I>A</I> to a circular arc with centre <I>O.</I> +Make the angle <I>AOC</I> equal to one-third of a right angle. +Bisect the angle <I>AOC</I> by <I>OD,</I> the angle <I>AOD</I> by <I>OE,</I> the +angle <I>AOE</I> by <I>OF,</I> and the angle <I>AOF</I> by <I>OG.</I> Produce <I>GA</I> +to <I>AH,</I> making <I>AH</I> equal to <I>AG.</I> The angle <I>GOH</I> is then +equal to the angle <I>FOA</I> which is 1/24th of a right angle, so +that <I>GH</I> is the side of a circumscribed regular polygon with +96 sides. +<p>Now <MATH><I>OA</I>:<I>AC</I>[=√3:1]>265:153</MATH>, (1) +and <MATH><I>OC</I>:<I>CA</I>=2:1=306:153</MATH>. (2) +<pb n=53><head>MEASUREMENT OF A CIRCLE</head> +<p>And, since <I>OD</I> bisects the angle <I>COA,</I> +<MATH><I>CO</I>:<I>OA</I>=<I>CD</I>:<I>DA</I></MATH>, +so that <MATH>(<I>CO</I>+<I>OA</I>):<I>OA</I>=<I>CA</I>:<I>DA</I></MATH>, +or <MATH>(<I>CO</I>+<I>OA</I>):<I>CA</I>=<I>OA</I>:<I>AD</I></MATH>. +<p>Hence <MATH><I>OA</I>:<I>AD</I>>571:153</MATH>, by (1) and (2). +<FIG> +<p>And <MATH><I>OD</I><SUP>2</SUP>:<I>AD</I><SUP>2</SUP>=(<I>OA</I><SUP>2</SUP>+<I>AD</I><SUP>2</SUP>):<I>AD</I><SUP>2</SUP> +>(571<SUP>2</SUP>+153<SUP>2</SUP>):153<SUP>2</SUP> +>349450:23409</MATH>. +<p>Therefore, says Archimedes, +<MATH><I>OD</I>:<I>DA</I>>591 1/8:153</MATH>. +<p>Next, just as we have found the limit of <I>OD</I>:<I>AD</I> +from <I>OC</I>:<I>CA</I> and the limit of <I>OA</I>:<I>AC</I>, we find the limits +of <I>OA</I>:<I>AE</I> and <I>OE</I>:<I>AE</I> from the limits of <I>OD</I>:<I>DA</I> and +<I>OA</I>:<I>AD</I>, and so on. This gives ultimately the limit of +<I>OA</I>:<I>AG.</I> +<p>Dealing with the inscribed polygon, Archimedes gets a +similar series of approximations. <I>ABC</I> being a semicircle, the +angle <I>BAC</I> is made equal to one-third of a right angle. Then, +if the angle <I>BAC</I> is bisected by <I>AD,</I> the angle <I>BAD</I> by <I>AE,</I> +the angle <I>BAE</I> by <I>AF,</I> and the angle <I>BAF</I> by <I>AG,</I> the +straight line <I>BG</I> is the side of an inscribed polygon with +96 sides. +<pb n=54><head>ARCHIMEDES</head> +<p>Now the triangles <I>ADB, BDd, ACd</I> are similar; +therefore <MATH><I>AD</I>:<I>DB</I>=<I>BD</I>:<I>Dd</I>=<I>AC</I>:<I>Cd</I> +=<I>AB</I>:<I>Bd,</I> since <I>AD</I> bisects ∠<I>BAC,</I> +=(<I>AB</I>+<I>AC</I>):(<I>Bd</I>+<I>Cd</I>) +=(<I>AB</I>+<I>AC</I>):<I>BC</I></MATH>. +<p>But <MATH><I>AC</I>:<I>CB</I><1351:780</MATH>, +while <MATH><I>BA</I>:<I>BC</I>=2:1=1560:780</MATH>. +<p>Therefore <MATH><I>AD</I>:<I>DB</I><2911:780</MATH>. +<FIG> +<p>Hence <MATH><I>AB</I><SUP>2</SUP>:<I>BD</I><SUP>2</SUP><(2911<SUP>2</SUP>+780<SUP>2</SUP>):780<SUP>2</SUP> +<9082321:608400</MATH>, +and, says Archimedes, +<MATH><I>AB</I>:<I>BD</I><3013 3/4:780</MATH>. +<p>Next, just as a limit is found for <I>AD</I>:<I>DB</I> and <I>AB</I>:<I>BD</I> +from <I>AB</I>:<I>BC</I> and the limit of <I>AC</I>:<I>CB,</I> so we find limits for +<I>AE</I>:<I>EB</I> and <I>AB</I>:<I>BE</I> from the limits of <I>AB</I>:<I>BD</I> and <I>AD</I>:<I>DB,</I> +and so on, and finally we obtain the limit of <I>AB</I>:<I>BG.</I> +<p>We have therefore in both cases two series of terms <I>a</I><SUB>0</SUB>, <I>a</I><SUB>1</SUB>, +<I>a</I><SUB>2</SUB> ... <I>a</I><SUB>n</SUB> and <I>b</I><SUB>0</SUB>, <I>b</I><SUB>1</SUB>, <I>b</I><SUB>2</SUB> ... <I>b</I><SUB>n</SUB>, for which the rule of formation is +<MATH><I>a</I><SUB>1</SUB>=<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>, <I>a</I><SUB>2</SUB>=<I>a</I><SUB>1</SUB>+<I>b</I><SUB>1</SUB>,...</MATH>, +where <MATH><I>b</I><SUB>1</SUB>=√(<I>a</I><SUB>1</SUB><SUP>2</SUP>+<I>c</I><SUP>2</SUP>), <I>b</I><SUB>2</SUB>=√(<I>a</I><SUB>2</SUB><SUP>2</SUP>+<I>c</I><SUP>2</SUP>) ...;</MATH> +and in the first case +<MATH><I>a</I><SUB>0</SUB>=265, <I>b</I><SUB>0</SUB>=306, <I>c</I>=153</MATH>, +while in the second case +<MATH><I>a</I><SUB>0</SUB>=1351, <I>b</I><SUB>0</SUB>1560, <I>c</I>=780</MATH>. +<pb n=55><head>MEASUREMENT OF A CIRCLE</head> +<p>The series of values found by Archimedes are shown in the +following table: +<MATH></MATH><note>Here the ratios of <I>a</I> to <I>c</I> are in the first instance reduced to lower terms.</note> +and, bearing in mind that in the first case the final ratio +<MATH><I>a</I><SUB>4</SUB>:<I>c</I></MATH> is the ratio <MATH><I>OA</I>:<I>AG</I>=2<I>OA</I>:<I>GH</I></MATH>, and in the second case +the final ratio <I>b</I><SUB>4</SUB>:<I>c</I> is the ratio <I>AB</I>:<I>BG</I>, while <I>GH</I> in the first +figure and <I>BG</I> in the second are the sides of regular polygons +of 96 sides circumscribed and inscribed respectively, we have +finally +<MATH>96X153/(4673 1/2)><G>p</G>>96X66/(2017 1/4)</MATH>. +<p>Archimedes simply infers from this that +<MATH>3 1/7><G>p</G>>3 10/71</MATH>. +<p>As a matter of fact <MATH>96X153/(4673 1/2)=3 (667 1/2)/(4673 1/2)</MATH>, and <MATH>(667 1/2)/(4672 1/2)=1/7</MATH>. +It is also to be observed that <MATH>3 10/71=3+1/(7+1/10)</MATH>, and it may +have been arrived at by a method equivalent to developing +the fraction 6336/(2017 1/4) in the form of a continued fraction. +<p>It should be noted that, in the text as we have it, the values +of <I>b</I><SUB>1</SUB>, <I>b</I><SUB>2</SUB>, <I>b</I><SUB>3</SUB>, <I>b</I><SUB>4</SUB> are simply stated in their final form without +the intermediate step containing the radical except in the first +<pb n=56><head>ARCHIMEDES</head> +case of all, where we are told that <MATH><I>OD</I><SUP>2</SUP>:<I>AD</I><SUP>2</SUP>>349450:23409</MATH> +and then that <MATH><I>OD</I>:<I>DA</I>>591 1/8:153</MATH>. At the points marked +* and <FIG> in the table Archimedes simplifies the ratio <I>a</I><SUB>2</SUB>:<I>c</I> and +<I>a</I><SUB>3</SUB>:<I>c</I> before calculating <I>b</I><SUB>2</SUB>, <I>b</I><SUB>3</SUB> respectively, by multiplying each +term in the first case by 4/13 and in the second case by 11/40. +He gives no explanation of the exact figure taken as the +approximation to the square root in each case, or of the +method by which he obtained it. We may, however, be sure +that the method amounted to the use of the formula <MATH>(<I>a</I>±<I>b</I>)<SUP>2</SUP> +=<I>a</I><SUP>2</SUP>±2<I>ab</I>+<I>b</I><SUP>2</SUP></MATH>, much as our method of extracting the square +root also depends upon it. +<p>We have already seen (vol. i, p. 232) that, according to +Heron, Archimedes made a still closer approximation to the +value of <G>p</G>. +<C>On Conoids and Spheroids.</C> +<p>The main problems attacked in this treatise are, in Archi- +medes's manner, stated in his preface addressed to Dositheus, +which also sets out the premisses with regard to the solid +figures in question. These premisses consist of definitions and +obvious inferences from them. The figures are (1) the <I>right- +angled conoid</I> (paraboloid of revolution), (2) the <I>obtuse-angled +conoid</I> (hyperboloid of revolution), and (3) the <I>spheroids</I> +(<I>a</I>) the <I>oblong,</I> described by the revolution of an ellipse about +its ‘greater diameter’ (major axis), (<I>b</I>) the <I>flat,</I> described by +the revolution of an ellipse about its ‘lesser diameter’ (minor +axis). Other definitions are those of the <I>vertex</I> and <I>axis</I> of the +figures or segments thereof, the vertex of a segment being +the point of contact of the tangent plane to the solid which +is parallel to the base of the segment. The <I>centre</I> is only +recognized in the case of the spheroid; what corresponds to +the centre in the case of the hyperboloid is the ‘vertex of +the enveloping cone’ (described by the revolution of what +Archimedes calls the ‘nearest lines to the section of the +obtuse-angled cone’, i.e. the asymptotes of the hyperbola), +and the line between this point and the vertex of the hyper- +boloid or segment is called, not the axis or diameter, but (the +line) ‘adjacent to the axis’. The axis of the segment is in +the case of the paraboloid the line through the vertex of the +segment parallel to the axis of the paraboloid, in the case +<pb n=57><head>ON CONOIDS AND SPHEROIDS</head> +of the hyperboloid the portion within the solid of the line +joining the vertex of the enveloping cone to the vertex of +the segment and produced, and in the case of the spheroids the +line joining the points of contact of the two tangent planes +parallel to the base of the segment. Definitions are added of +a ‘segment of a cone’ (the figure cut off towards the vertex by +an elliptical, not circular, section of the cone) and a ‘frustum +of a cylinder’ (cut off by two parallel elliptical sections). +<p>Props. 1 to 18 with a Lemma at the beginning are preliminary +to the main subject of the treatise. The Lemma and Props. 1, 2 +are general propositions needed afterwards. They include +propositions in summation, +<MATH>2{<I>a</I>+2<I>a</I>+3<I>a</I>+...+<I>na</I>}><I>n.na</I>>2{<I>a</I>+2<I>a</I>+...+(<I>n</I>-1)<I>a</I>}</MATH> +(Lemma) +(this is clear from <MATH><I>S</I><SUB><I>n</I></SUB>=1/2<I>n</I>(<I>n</I>+1)<I>a</I>)</MATH>; +<MATH>(<I>n</I>+1)(<I>na</I>)<SUP>2</SUP>+<I>a</I>(<I>a</I>+2<I>a</I>+3<I>a</I>+...+<I>na</I>) +=3{<I>a</I><SUP>2</SUP>+(2<I>a</I>)<SUP>2</SUP>+(3<I>a</I>)<SUP>2</SUP>+...+(<I>na</I>)<SUP>2</SUP>};</MATH> +(Lemma to Prop. 2) +whence (Cor.) +<MATH>3{<I>a</I><SUP>2</SUP>+(2<I>a</I>)<SUP>2</SUP>+(3<I>a</I>)<SUP>2</SUP>+...+(<I>na</I>)<SUP>2</SUP>}><I>n</I>(<I>na</I>)<SUP>2</SUP> +>3{<I>a</I><SUP>2</SUP>+(2<I>a</I>)<SUP>2</SUP>+...+(<I>n</I>-1<I>a</I>)<SUP>2</SUP>};</MATH> +lastly, Prop. 2 gives limits for the sum of <I>n</I> terms of the +series <MATH><I>ax</I>+<I>x</I><SUP>2</SUP>, <I>a.</I>2<I>x</I>+(2<I>x</I>)<SUP>2</SUP>, <I>a.</I>3<I>x</I>+(3<I>x</I>)<SUP>2</SUP>,...</MATH>, in the form of +inequalities of ratios, thus: +<MATH><I>n</I>{<I>a.nx</I>+(<I>nx</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>a.rx</I>+(<I>rx</I>)<SUP>2</SUP>} +>(<I>a</I>+<I>nx</I>):(1/2<I>a</I>+1/3<I>nx</I>) +><I>n</I>{<I>a.nx</I>+(<I>nx</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>a.rx</I>+(<I>rx</I>)<SUP>2</SUP>}</MATH>. +Prop. 3 proves that, if <I>QQ</I>′ be a chord of a parabola bisected +at <I>V</I> by the diameter <I>PV,</I> then, if <I>PV</I> be of constant length, +the areas of the triangle <I>PQQ</I>′ and of the segment <I>PQQ</I>′ are +also constant, whatever be the direction of <I>QQ</I>′; to prove it +Archimedes assumes a proposition ‘proved in the conics’ and +by no means easy, namely that, if <I>QD</I> be perpendicular to <I>PV,</I> +and if <I>p, p<SUB>a</SUB></I> be the parameters corresponding to the ordinates +parallel to <I>QQ</I>′ and the principal ordinates respectively, then +<MATH><I>QV</I><SUP>2</SUP>:<I>QD</I><SUP>2</SUP>=<I>p</I>:<I>p<SUB>a</SUB></I></MATH>. +Props. 4-6 deal with the area of an ellipse, which is, in the +<pb n=58><head>ARCHIMEDES</head> +first of the three propositions, proved to be to the area of +the auxiliary circle as the minor axis to the major; equilateral +polygons of 4<I>n</I> sides are inscribed in the circle and compared +with corresponding polygons inscribed in the ellipse, which are +determined by the intersections with the ellipse of the double +ordinates passing through the angular points of the polygons +inscribed in the circle, and the method of exhaustion is then +applied in the usual way. Props. 7, 8 show how, given an ellipse +with centre <I>C</I> and a straight line <I>CO</I> in a plane perpendicular to +that of the ellipse and passing through an axis of it, (1) in the +case where <I>OC</I> is perpendicular to that axis, (2) in the case +where it is not, we can find an (in general oblique) circular +cone with vertex <I>O</I> such that the given ellipse is a section of it, +or, in other words, how we can find the circular sections of the +cone with vertex <I>O</I> which passes through the circumference of +the ellipse; similarly Prop. 9 shows how to find the circular +sections of a cylinder with <I>CO</I> as axis and with surface passing +through the circumference of an ellipse with centre <I>C,</I> where +<I>CO</I> is in the plane through an axis of the ellipse and perpen- +dicular to its plane, but is not itself perpendicular to that +axis. Props. 11-18 give simple properties of the conoids and +spheroids, easily derivable from the properties of the respective +conics; they explain the nature and relation of the sections +made by planes cutting the solids respectively in different ways +(planes through the axis, parallel to the axis, through the centre +or the vertex of the enveloping cone, perpendicular to the axis, +or cutting it obliquely, respectively), with especial reference to +the elliptical sections of each solid, the similarity of parallel +elliptical sections, &c. Then with Prop. 19 the real business +of the treatise begins, namely the investigation of the volume +of segments (right or oblique) of the two conoids and the +spheroids respectively. +<p>The method is, in all cases, to circumscribe and inscribe to +the segment solid figures made up of cylinders or ‘frusta of +cylinders’, which can be made to differ as little as we please +from one another, so that the circumscribed and inscribed +figures are, as it were, compressed together and into coincidence +with the segment which is intermediate between them. +<p>In each diagram the plane of the paper is a plane through +the axis of the conoid or spheroid at right angles to the plane +<pb n=59><head>ON CONOIDS AND SPHEROIDS</head> +of the section which is the base of the segment, and which +is a circle or an ellipse according as the said base is or is not +at right angles to the axis; the plane of the paper cuts the +base in a diameter of the circle or an axis of the ellipse as +the case may be. +<FIG> +<p>The nature of the inscribed and circumscribed figures will +be seen from the above figures showing segments of a para- +boloid, a hyperboloid and a spheroid respectively, cut off +<pb n=60><head>ARCHIMEDES</head> +by planes obliquely inclined to the axis. The base of the +segment is an ellipse in which <I>BB</I>′ is an axis, and its plane is +at right angles to the plane of the paper, which passes through +the axis of the solid and cuts it in a parabola, a hyperbola, or +an ellipse respectively. The axis of the segment is cut into a +number of equal parts in each case, and planes are drawn +through each point of section parallel to the base, cutting the +solid in ellipses, similar to the base, in which <I>PP</I>′, <I>QQ</I>′, &c., are +axes. Describing frusta of cylinders with axis <I>AD</I> and passing +through these elliptical sections respectively, we draw the +circumscribed and inscribed solids consisting of these frusta. +It is evident that, beginning from <I>A,</I> the first inscribed frustum +is equal to the first circumscribed frustum, the second to the +second, and so on, but there is one more circumscribed frustum +than inscribed, and the difference between the circumscribed +and inscribed solids is equal to the <I>last frustum</I> of which <I>BB</I>′ +is the base, and <I>ND</I> is the axis. Since <I>ND</I> can be made as +small as we please, the difference between the circumscribed +and inscribed solids can be made less than any assigned solid +whatever. Hence we have the requirements for applying the +method of exhaustion. +<p>Consider now separately the cases of the paraboloid, the +hyperboloid and the spheroid. +<p>I. The <I>paraboloid</I> (Props. 20-22). +<p>The frustum the base of which is the ellipse in which <I>PP</I>′ is +an axis is proportional to <I>PP</I>′<SUP>2</SUP> or <I>PN</I><SUP>2</SUP>, i.e. proportional to +<I>AN.</I> Suppose that the axis <I>AD</I> (=<I>c</I>) is divided into <I>n</I> equal +parts. Archimedes compares each frustum in the inscribed +and circumscribed figure with the frustum of the whole cylinder +<I>BF</I> cut off by the same planes. Thus +<MATH>(first frustum in <I>BF</I>):(first frustum in inscribed figure) +=<I>BD</I><SUP>2</SUP>:<I>PN</I><SUP>2</SUP> +=<I>AD</I>:<I>AN</I> +=<I>BD</I>:<I>TN</I></MATH>. +Similarly +<MATH>(second frustum in <I>BF</I>):(second in inscribed figure) +=<I>HN</I>:<I>SM</I></MATH>, +and so on. The last frustum in the cylinder <I>BF</I> has none to +<pb n=61><head>ON CONOIDS AND SPHEROIDS</head> +correspond to it in the inscribed figure, and we should write +the ratio as (<I>BD</I>:zero). +<p>Archimedes concludes, by means of a lemma in proportions +forming Prop. 1, that +<MATH>(frustum <I>BF</I>):(inscribed figure) +=(<I>BD</I>+<I>HN</I>+...):(<I>TN</I>+<I>SM</I>+...+<I>XO</I>) +=<I>n</I><SUP>2</SUP><I>k</I>:(<I>k</I>+2<I>k</I>+3<I>k</I>+...+―(<I>n</I>-1)<I>k</I>)</MATH>, +where <MATH><I>XO</I>=<I>k,</I> so that <I>BD</I>=<I>nk</I></MATH>. +<p>In like manner, he concludes that +<MATH>(frustum <I>BF</I>):(circumscribed figure) +=<I>n</I><SUP>2</SUP><I>k</I>:(<I>k</I>+2<I>k</I>+3<I>k</I>+...+<I>nk</I>)</MATH>. +<p>But, by the Lemma preceding Prop. 1, +<MATH><I>k</I>+2<I>k</I>+3<I>k</I>+...+―(<I>n</I>-1)<I>k</I><1/2<I>n</I><SUP>2</SUP><I>k</I><<I>k</I>+2<I>k</I>+3<I>k</I>+...+<I>nk</I></MATH>, +whence +<MATH>(frustum <I>BF</I>):(inscr.fig.)>2>(frustum <I>BF</I>):(circumscr. fig.)</MATH>. +<p>This indicates the desired result, which is then confirmed by +the method of exhaustion, namely that +<MATH>(frustum <I>BF</I>)=2(segment of paraboloid)</MATH>, +or, if <I>V</I> be the volume of the ‘segment of a cone’, with vertex +<I>A</I> and base the same as that of the segment, +<MATH>(volume of segment)=3/2<I>V</I></MATH>. +<p>Archimedes, it will be seen, proves in effect that, if <I>k</I> be +indefinitely diminished, and <I>n</I> indefinitely increased, while <I>nk</I> +remains equal to <I>c,</I> then +limit of <MATH><I>k</I>{<I>k</I>+2<I>k</I>+3<I>k</I>+...+(<I>n</I>-1)<I>k</I>}=1/2<I>c</I><SUP>2</SUP></MATH>, +that is, in our notation, +<MATH>∫<SUP><I>c</I></SUP><SUB>0</SUB><I>xdx</I>=1/2<I>c</I><SUP>2</SUP></MATH>. +<p>Prop. 23 proves that the volume is constant for a given +length of axis <I>AD,</I> whether the segment is cut off by a plane +perpendicular or not perpendicular to the axis, and Prop. 24 +shows that the volumes of two segments are as the squares on +their axes. +<pb n=62><head>ARCHIMEDES</head> +<p>II. In the case of the <I>hyperboloid</I> (Props. 25, 26) let the axis +<I>AD</I> be divided into <I>n</I> parts, each of length <I>h</I>, and let <MATH><I>AA</I>′=<I>a</I></MATH>. +Then the ratio of the volume of the frustum of a cylinder on +the ellipse of which any double ordinate <I>QQ</I>′ is an axis to the +volume of the corresponding portion of the whole frustum <I>BF</I> +takes a different form; for, if <I>AM</I>=<I>rh,</I> we have +<MATH>(frustum in <I>BF</I>):(frustum on base <I>QQ</I>′ +=<I>BD</I><SUP>2</SUP>:<I>QM</I><SUP>2</SUP> +=<I>AD.A</I>′<I>D</I>:<I>AM.A</I>′<I>M</I> +={<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>. +By means of this relation Archimedes proves that +<MATH>(frustum <I>BF</I>):(inscribed figure) +=<I>n</I>{<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>, +and +<MATH>(frustum <I>BF</I>):(circumscribed figure) +=<I>n</I>{<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>. +But, by Prop. 2, +<MATH><I>n</I>{<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}>(<I>a</I>+<I>nh</I>):(1/2<I>a</I>+1/3<I>nh</I>) +><I>n</I>{<I>a.nh</I>+(<I>nh</I>)<SUP>2</SUP>}:<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>a.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>. +<p>From these relations it is inferred that +<MATH>(frustum <I>BF</I>):(volume of segment)=(<I>a</I>+<I>nh</I>):(1/2<I>a</I>+1/3<I>nh</I>)</MATH>, +or <MATH>(volume of segment):(volume of cone <I>ABB</I>′) +=(<I>AD</I>+3<I>CA</I>):(<I>AD</I>+2<I>CA</I>);</MATH> +and this is confirmed by the method of exhaustion. +<p>The result obtained by Archimedes is equivalent to proving +that, if <I>h</I> be indefinitely diminished while <I>n</I> is indefinitely +increased but <I>nh</I> remains always equal to <I>b,</I> then +limit of <MATH><I>n</I>(<I>ab</I>+<I>b</I><SUP>2</SUP>)/<I>S</I><SUB><I>n</I></SUB>=(<I>a</I>+<I>b</I>)/(1/2<I>a</I>+1/3<I>b</I>)</MATH>, +or limit of <MATH><I>b/n S<SUB>n</SUB></I>=<I>b</I><SUP>2</SUP>(1/2<I>a</I>+1/3<I>b</I>)</MATH>, +where +<MATH><I>S<SUB>n</SUB></I>=<I>a</I>(<I>h</I>+2<I>h</I>+3<I>h</I>+...+<I>nh</I>)+{<I>h</I><SUP>2</SUP>+(2<I>h</I>)<SUP>2</SUP>+(3<I>h</I>)<SUP>2</SUP>+...+(<I>nh</I>)<SUP>2</SUP>}</MATH> +<pb n=63><head>ON CONOIDS AND SPHEROIDS</head> +so that +<MATH><I>hS<SUB>n</SUB></I>=<I>ah</I>(<I>h</I>+2<I>h</I>+...+<I>nh</I>)+<I>h</I>{<I>h</I><SUP>2</SUP>+(2<I>h</I>)<SUP>2</SUP>+...+(<I>nh</I>)<SUP>2</SUP>}</MATH>. +<p>The limit of this latter expression is what we should write +<MATH>∫<SUP><I>b</I></SUP><SUB>0</SUB>(<I>ax</I>+<I>x</I><SUP>2</SUP>)<I>dx</I>=<I>b</I><SUP>2</SUP>(1/2<I>a</I>+1/3<I>b</I>)</MATH>, +and Archimedes's procedure is the equivalent of this integration. +<p>III. In the case of the <I>spheroid</I> (Props. 29, 30) we take +a segment less than half the spheroid. +<p>As in the case of the hyperboloid, +<MATH>(frustum in <I>BF</I>):(frustum on base <I>QQ</I>′) +=<I>BD</I><SUP>2</SUP>:<I>QM</I><SUP>2</SUP> +=<I>AD.A</I>′<I>D</I>:<I>AM.A</I>′<I>M</I></MATH>; +but, in order to reduce the summation to the same as that in +Prop. 2, Archimedes expresses <I>AM.A</I>′<I>M</I> in a different form +equivalent to the following. +<p>Let <I>AD</I> (=<I>b</I>) be divided into <I>n</I> equal parts of length <I>h,</I> +and suppose that <MATH><I>AA</I>′=<I>a, CD</I>=1/2<I>c.</I></MATH> +<p>Then <MATH><I>AD.A</I>′<I>D</I>=1/4<I>a</I><SUP>2</SUP>-1/4<I>c</I><SUP>2</SUP></MATH>, +and <MATH><I>AM.A</I>′<I>M</I>=1/4<I>a</I><SUP>2</SUP>-(1/2<I>c</I>+<I>rh</I>)<SUP>2</SUP> (<I>DM</I>=<I>rh</I>) +=<I>AD.A</I>′<I>D</I>-{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>} +=<I>cb</I>+<I>b</I><SUP>2</SUP>-{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}</MATH>. +<p>Thus in this case we have +<MATH>(frustum <I>BF</I>):(inscribed figure) +=<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>):[<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>)-<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}]</MATH> +and +<MATH>(frustum <I>BF</I>):(circumscribed figure) +=<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>):[<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>)-<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}]</MATH>. +<p>And, since <I>b</I>=<I>nh,</I> we have, by means of Prop. 2, +<MATH><I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>):[<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>)-<G>*s</G><SUB>1</SUB><SUP><I>n</I></SUP>{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}] +>(<I>c</I>+<I>b</I>):{<I>c</I>+<I>b</I>-(1/2<I>c</I>+1/3<I>b</I>)} +><I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>):[<I>n</I>(<I>cb</I>+<I>b</I><SUP>2</SUP>)-<G>*s</G><SUB>1</SUB><SUP><I>n</I>-1</SUP>{<I>c.rh</I>+(<I>rh</I>)<SUP>2</SUP>}]</MATH>. +<pb n=64><head>ARCHIMEDES</head> +<p>The conclusion, confirmed as usual by the method of ex- +haustion, is that +<MATH>(frustum <I>BF</I>):(segment of spheroid)=(<I>c</I>+<I>b</I>):{<I>c</I>+<I>b</I>-(1/2<I>c</I>+1/3<I>b</I>)} +=(<I>c</I>+<I>b</I>):(1/2<I>c</I>+2/3<I>b</I>), +whence (volume of segment):(volume of cone <I>ABB</I>′) +=(3/2<I>c</I>+2<I>b</I>):(<I>c</I>+<I>b</I>) +=(3<I>CA</I>-<I>AD</I>):(2<I>CA</I>-<I>AD</I>), since <I>CA</I>=1/2<I>c</I>+<I>b</I></MATH>. +<p>As a particular case (Props. 27, 28), half the spheroid is +double of the corresponding cone. +<p>Props. 31, 32, concluding the treatise, deduce the similar +formula for the volume of the greater segment, namely, in our +figure, +<MATH>(greater segmt.):(cone or segmt.of cone with same base and axis) +=(<I>CA</I>+<I>AD</I>):<I>AD</I></MATH>. +<C>On Spirals.</C> +<p>The treatise <I>On Spirals</I> begins with a preface addressed to +Dositheus in which Archimedes mentions the death of Conon +as a grievous loss to mathematics, and then summarizes the +main results of the treatises <I>On the Sphere and Cylinder</I> and +<I>On Conoids and Spheroids,</I> observing that the last two pro- +positions of Book II of the former treatise took the place +of two which, as originally enunciated to Dositheus, were +wrong; lastly, he states the main results of the treatise +<I>On Spirals,</I> premising the definition of a spiral which is as +follows: +<p>‘If a straight line one extremity of which remains fixed be +made to revolve at a uniform rate in a plane until it returns +to the position from which it started, and if, at the same time +as the straight line is revolving, a point move at a uniform +rate along the straight line, starting from the fixed extremity, +the point will describe a spiral in the plane.’ +<p>As usual, we have a series of propositions preliminary to +the main subject, first two propositions about uniform motion, +<pb n=65><head>ON SPIRALS</head> +then two simple geometrical propositions, followed by pro- +positions (5-9) which are all of one type. Prop. 5 states that, +given a circle with centre <I>O,</I> a tangent to it at <I>A,</I> and <I>c,</I> the +<FIG> +<CAP>FIG. 1.</CAP> +circumference of any circle whatever, it is possible to draw +a straight line <I>OPF</I> meeting the circle in <I>P</I> and the tangent +in <I>F</I> such that +<MATH><I>FP</I>:<I>OP</I><(arc <I>AP</I>):<I>c.</I></MATH> +<p>Archimedes takes <I>D</I> a straight line greater than <I>c,</I> draws +<I>OH</I> parallel to the tangent at <I>A</I> and then says ‘let <I>PH</I> be +placed equal to <I>D verging</I> (<G>neu/ousa</G>) towards <I>A</I>’. This is the +usual phraseology of the type of problem known as <G>neu=sis</G> +where a straight line of given length has to be placed between +two lines or curves in such a position that, if produced, it +passes through a given point (this is the meaning of <I>verging</I>). +Each of the propositions 5-9 depends on a <G>neu=sis</G> of this kind, +<FIG> +<CAP>FIG. 2.</CAP> +which Archimedes assumes as ‘possible’ without showing how +it is effected. Except in the case of Prop. 5, the theoretical +solution cannot be effected by means of the straight line and +circle; it depends in general on the solution of an equation +of the fourth degree, which can be solved by means of the +<pb n=66><head>ARCHIMEDES</head> +points of intersection of a certain rectangular hyperbola +and a certain parabola. It is quite possible, however, that +such problems were in practice often solved by a mechanical +method, namely by placing a ruler, by trial, in the position of +the required line: for it is only necessary to place the ruler +so that it passes through the given point and then turn it +round that point as a pivot till the intercept becomes of the +given length. In Props. 6-9 we have a circle with centre <I>O,</I> +a chord <I>AB</I> less than the diameter in it, <I>OM</I> the perpendicular +from <I>O</I> on <I>AB, BT</I> the tangent at <I>B, OT</I> the straight line +through <I>O</I> parallel to <I>AB; D</I>:<I>E</I> is any ratio less or greater, +as the case may be, than the ratio <I>BM</I>:<I>MO</I>. Props. 6, 7 +(Fig. 2) show that it is possible to draw a straight line <I>OFP</I> +<FIG> +<CAP>FIG. 3.</CAP> +meeting <I>AB</I> in <I>F</I> and the circle in <I>P</I> such that <I>FP</I>:<I>PB</I>=<I>D</I>:<I>E</I> +(<I>OP</I> meeting <I>AB</I> in the case where <I>D</I>:<I>E</I><<I>BM</I>:<I>MO,</I> and +meeting <I>AB</I> produced when <I>D</I>:<I>E</I>><I>BM</I>:<I>MO</I>). In Props. 8, 9 +(Fig. 3) it is proved that it is possible to draw a straight line +<I>OFP</I> meeting <I>AB</I> in <I>F,</I> the circle in <I>P</I> and the tangent at <I>B</I> in +<I>G,</I> such that <I>FP</I>:<I>BG</I>=<I>D</I>:<I>E</I> (<I>OP</I> meeting <I>AB</I> itself in the case +where <I>D</I>:<I>E</I><<I>BM</I>:<I>MO,</I> and meeting <I>AB</I> produced in the +case where <I>D</I>:<I>E</I>><I>BM</I>:<I>MO</I>). +<p>We will illustrate by the constructions in Props. 7, 8, +as it is these propositions which are actually cited later. +Prop. 7. If <I>D</I>:<I>E</I> is any ratio><I>BM</I>:<I>MO,</I> it is required (Fig. 2) +to draw <I>OP′F</I>′ meeting the circle in <I>P</I>′ and <I>AB</I> produced in +<I>F</I>′ so that +<MATH><I>F</I>′<I>P</I>′:<I>P</I>′<I>B</I>=<I>D</I>:<I>E</I></MATH>. +<p>Draw <I>OT</I> parallel to <I>AB,</I> and let the tangent to the circle at +<I>B</I> meet <I>OT</I> in <I>T.</I> +<pb n=67><head>ON SPIRALS</head> +<p>Then <MATH><I>D</I>:<I>E</I>><I>BM</I>:<I>MO</I></MATH>, by hypothesis, +<MATH>><I>OB</I>:<I>BT</I></MATH>, by similar triangles. +<p>Take a straight line <I>P</I>′<I>H</I>′ (less than <I>BT</I>) such that <MATH><I>D</I>:<I>E</I> +=<I>OB</I>:<I>P</I>′<I>H</I>′</MATH>, and place <I>P</I>′<I>H</I>′ between the circle and <I>OT</I> +‘verging towards <I>B</I>’ (construction assumed). +<p>Then <MATH><I>F</I>′<I>P</I>′:<I>P</I>′<I>B</I>=<I>OP</I>′:<I>P</I>′<I>H</I>′ +=<I>OB</I>:<I>P</I>′<I>H</I>′ +=<I>D</I>:<I>E</I></MATH>. +<p>Prop. 8. If <I>D</I>:<I>E</I> is any given ratio < <I>BM</I>:<I>MO</I>, it is required +to draw <I>OFPG</I> meeting <I>AB</I> in <I>F</I>, the circle in <I>P</I>, and the +tangent at <I>B</I> to the circle in <I>G</I> so that +<MATH><I>FP</I>:<I>BG</I>=<I>D</I>:<I>E</I></MATH>. +<FIG> +<p>If <I>OT</I> is parallel to <I>AB</I> and meets the tangent at <I>B</I> in <I>T</I>, +<MATH><I>BM</I>:<I>MO</I>=<I>OB</I>:<I>BT</I></MATH>, by similar triangles, +whence <MATH><I>D</I>:<I>E</I><<I>OB</I>:<I>BT</I></MATH>. +<p>Produce <I>TB</I> to <I>C,</I> making <I>BC</I> of such length that +<MATH><I>D</I>:<I>E</I>=<I>OB</I>:<I>BC</I></MATH>, +so that <I>BC</I>><I>BT.</I> +<p>Describe a circle through the three points <I>O, T, C</I> and let <I>OB</I> +produced meet this circle in <I>K</I>. +<p>‘Then, since <I>BC</I>><I>BT</I>, and <I>OK</I> is perpendicular to <I>CT</I>, it is +possible to place <I>QG</I> [between the circle <I>TKC</I> and <I>BC</I>] equal to +<I>BK</I> and verging towards <I>O</I>’ (construction assumed). +<pb n=68><head>ARCHIMEDES</head> +<p>Let <I>QGO</I> meet the original circle in <I>P</I> and <I>AB</I> in <I>F</I>. Then +<I>OFPG</I> is the straight line required. +<p>For <MATH><I>CG.GT</I> = <I>OG.GQ</I> = <I>OG.BK</I></MATH>. +<p>But <MATH><I>OF</I>:<I>OG</I> = <I>BT</I>:<I>GT</I></MATH>, by parallels, +whence <MATH><I>OF.GT</I> = <I>OG.BT</I></MATH>. +<p>Therefore <MATH><I>CG.GT</I>:<I>OF.GT</I> = <I>OG.BK</I>:<I>OG.BT</I></MATH>, +whence <MATH><I>CG</I>:<I>OF</I> = <I>BK</I>:<I>BT</I> += <I>BC</I>:<I>OB</I> += <I>BC</I>:<I>OP</I></MATH>. +<p>Therefore <MATH><I>OP</I>:<I>OF</I> = <I>BC</I>:<I>CG</I></MATH>, +and hence <MATH><I>PF</I>:<I>OP</I> = <I>BG</I>:<I>BC</I></MATH>, +or <MATH><I>PF</I>:<I>BG</I> = <I>OB</I>:<I>BC</I> = <I>D</I>:<I>E</I></MATH>. +<p>Pappus objects to Archimedes's use of the <G>neu=sis</G> assumed in +Prop. 8, 9 in these words: +<p>‘it seems to be a grave error into which geometers fall +whenever any one discovers the solution of a plane problem +by means of conics or linear (higher) curves, or generally +solves it by means of a foreign kind, as is the case e.g. (1) with +the problem in the fifth Book of the Conics of Apollonius +relating to the parabola, and (2) when Archimedes assumes in +his work on the spiral a <G>neu=sis</G> of a “solid” character with +reference to a circle; for it is possible without calling in the +aid of anything solid to find the proof of the theorem given by +Archimedes, that is, to prove that the circumference of the +circle arrived at in the first revolution is equal to the straight +line drawn at right angles to the initial line to meet the tangent +to the spiral (i.e. the subtangent).’ +<p>There is, however, this excuse for Archimedes, that he only +assumes that the problem <I>can</I> be solved and does not assume +the actual solution. Pappus<note>Pappus, iv, pp. 298-302.</note> himself gives a solution of the +particular <G>neu=sis</G> by means of conics. Apollonius wrote two +Books of <G>neu/seis</G>, and it is quite possible that by Archimedes's +time there may already have been a collection of such problems +to which tacit reference was permissible. +<p>Prop. 10 repeats the result of the Lemma to Prop. 2 of <I>On</I> +<pb n=69><head>ON SPIRALS</head> +<I>Conoids and Spheroids</I> involving the summation of the series +<MATH>1<SUP>2</SUP> + 2<SUP>2</SUP> + 3<SUP>2</SUP> + ... + <I>n</I><SUP>2</SUP></MATH>. Prop 11 proves another proposition in +summation, namely that +<MATH>(<I>n</I>-1)(<I>na</I>)<SUP>2</SUP>:{<I>a</I><SUP>2</SUP>+(2<I>a</I>)<SUP>2</SUP>+(3<I>a</I>)<SUP>2</SUP>+...+(―(<I>n</I>-1))<I>a</I>)<SUP>2</SUP>} +>(<I>na</I>)<SUP>2</SUP>:{<I>na.a</I>+1/3(<I>na</I>-<I>a</I>)<SUP>2</SUP>} +>(<I>n</I>-1)(<I>na</I>)<SUP>2</SUP>:{(2<I>a</I>)<SUP>2</SUP>+(3<I>a</I>)<SUP>2</SUP>+...+(<I>na</I>)<SUP>2</SUP>}</MATH>. +The same proposition is also true if the terms of the series +are <MATH><I>a</I><SUP>2</SUP>, (<I>a</I>+<I>b</I>)<SUP>2</SUP>, (<I>a</I>+2<I>b</I>)<SUP>2</SUP> ... (<I>a</I>+―(<I>n</I>-1)<I>b</I>)<SUP>2</SUP></MATH>, and it is assumed in +the more general form in Props. 25, 26. +<p>Archimedes now introduces his Definitions, of the <I>spiral</I> +itself, the <I>origin</I>, the <I>initial line</I>, the <I>first distance</I> (= the +radius vector at the end of one revolution), the <I>second distance</I> +(= the equal length added to the radius vector during the +second complete revolution), and so on; the <I>first area</I> (the area +bounded by the spiral described in the first revolution and +the ‘first distance’), the <I>second area</I> (that bounded by the spiral +described in the second revolution and the ‘second distance’), +and so on; the <I>first circle</I> (the circle with the ‘first distance’ +as radius), the <I>second circle</I> (the circle with radius equal to the +sum of the ‘first’ and ‘second distances’, or twice the first +distance), and so on. +<p>Props. 12, 14, 15 give the fundamental property of the +spiral connecting the length of the radius vector with the angle +through which the initial line has revolved from its original +position, and corresponding to the equation in polar coordinates +<MATH><I>r</I> = <I>a</I><G>q</G></MATH>. As Archimedes does not speak of angles greater +than <G>p</G>, or 2<G>p</G>, he has, in the case of points on any turn after +the first, to use multiples of the circumference +of a circle as well as arcs of it. He uses the +‘first circle’ for this purpose. Thus, if <I>P, Q</I> +are two points on the first turn, +<FIG> +<MATH><I>OP</I>:<I>OQ</I> = (arc <I>AKP</I>′):(arc <I>AKQ</I>′)</MATH>; +if <I>P, Q</I> are points on the <I>n</I>th turn of the +spiral, and <I>c</I> is the circumference of the first circle, +<MATH><I>OP</I>:<I>OQ</I> = {(<I>n</I> - 1)<I>c</I> + arc <I>AKP</I>′}:{(<I>n</I> - 1)<I>c</I> + arc <I>AKQ</I>′}</MATH>. +<p>Prop. 13 proves that, if a straight line touches the spiral, it +<pb n=70><head>ARCHIMEDES</head> +touches it at one point only. For, if possible, let the tangent +at <I>P</I> touch the spiral at another point <I>Q</I>. Then, if we bisect +the angle <I>POQ</I> by <I>OL</I> meeting <I>PQ</I> in <I>L</I> and the spiral in <I>R</I>, +<MATH><I>OP</I> + <I>OQ</I> = 2<I>OR</I></MATH> by the property of the spiral. But by +the property of the triangle (assumed, but easily proved) +<MATH><I>OP</I> + <I>OQ</I> > 2<I>OL</I></MATH>, so that <MATH><I>OL</I> < <I>OR</I></MATH>, and some point of <I>PQ</I> +lies within the spiral. Hence <I>PQ</I> cuts the spiral, which is +contrary to the hypothesis. +<p>Props. 16, 17 prove that the angle made by the tangent +at a point with the radius vector to that point is obtuse on the +‘forward’ side, and acute on the ‘backward’ side, of the radius +vector. +<p>Props. 18-20 give the fundamental proposition about the +tangent, that is to say, they give the length of the <I>subtangent</I> +at any point <I>P</I> (the distance between <I>O</I> and the point of inter- +section of the tangent with the perpendicular from <I>O</I> to <I>OP</I>). +Archimedes always deals first with the first turn and then +with any subsequent turn, and with each complete turn before +parts or points of any particular turn. Thus he deals with +tangents in this order, (1) the tangent at <I>A</I> the end of the first +turn, (2) the tangent at the end of the second and any subse- +quent turn, (3) the tangent at any intermediate point of the +first or any subsequent turn. We will take as illustrative +the case of the tangent at any intermediate point <I>P</I> of the first +turn (Prop. 20). +<p>If <I>OA</I> be the initial line, <I>P</I> any point on the first turn, <I>PT</I> +the tangent at <I>P</I> and <I>OT</I> perpendicular to <I>OP</I>, then it is to be +proved that, if <I>ASP</I> be the circle through <I>P</I> with centre <I>O</I>, +meeting <I>PT</I> in <I>S</I>, then +<MATH>(subtangent <I>OT</I>) = (arc <I>ASP</I>)</MATH>. +<p>I. If possible, let <I>OT</I> be greater than the arc <I>ASP</I>. +<p>Measure off <I>OU</I> such that <I>OU</I> > arc <I>ASP</I> but < <I>OT</I>. +<p>Then the ratio <I>PO</I>:<I>OU</I> is greater than the ratio <I>PO</I>:<I>OT</I>, +i.e. greater than the ratio of 1/2<I>PS</I> to the perpendicular from <I>O</I> +on <I>PS</I>. +<p>Therefore (Prop. 7) we can draw a straight line <I>OQF</I> meeting +<I>TP</I> produced in <I>F</I>, and the circle in <I>Q</I>, such that +<MATH><I>FQ</I>:<I>PQ</I> = <I>PO</I>:<I>OU</I></MATH>. +<pb n=71><head>ON SPIRALS</head> +<p>Let <I>OF</I> meet the spiral in <I>Q</I>′. +<p>Then we have, <I>alternando</I>, since <MATH><I>PO</I> = <I>QO</I>, +<I>FQ</I>:<I>QO</I> = <I>PQ</I>:<I>OU</I> +< (arc <I>PQ</I>):(arc <I>ASP</I>)</MATH>, by hypothesis and <I>a fortiori. +Componendo</I>, <MATH><I>FO</I>:<I>QO</I> < (arc <I>ASQ</I>):(arc <I>ASP</I>) +< <I>OQ</I>′:<I>OP</I></MATH>. +<p>But <MATH><I>QO</I> = <I>OP</I></MATH>; therefore <MATH><I>FO</I> < <I>OQ</I>′</MATH>; which is impossible. +<p>Therefore <I>OT</I> is not greater than the arc <I>ASP</I>. +<FIG> +<p>II. Next suppose, if possible, that <I>OT</I> < arc <I>ASP</I>. +<p>Measure <I>OV</I> along <I>OT</I> such that <I>OV</I> is greater than <I>OT</I> but +less than the arc <I>ASP</I>. +<p>Then the ratio <I>PO</I>:<I>OV</I> is less than the ratio <I>PO</I>:<I>OT</I>, i.e. +than the ratio of 1/2<I>PS</I> to the perpendicular from <I>O</I> on <I>PS</I>; +therefore it is possible (Prop. 8) to draw a straight line <I>OF</I>′<I>RG</I> +meeting <I>PS</I>, the circle <I>PSA</I>, and the tangent to the circle at <I>P</I> +in <I>F</I>′, <I>R, G</I> respectively, and such that +<MATH><I>F</I>′<I>R</I>:<I>GP</I> = <I>PO</I>:<I>OV</I></MATH>. +<pb n=72><head>ARCHIMEDES</head> +<p>Let <I>OF</I>′<I>G</I> meet the spiral in <I>R</I>′. +<p>Then, since <I>PO</I> = <I>RO</I>, we have, <I>alternando</I>, +<MATH><I>F</I>′<I>R</I>:<I>RO</I> = <I>GP</I>:<I>OV</I> +> (arc <I>PR</I>):(arc <I>ASP</I>)</MATH>, <I>a fortiori</I>, +whence <MATH><I>F</I>′<I>O</I>:<I>RO</I> < (arc <I>ASR</I>):(arc <I>ASP</I>) +< <I>OR</I>′:<I>OP</I></MATH>, +so that <I>F</I>′<I>O</I> < <I>OR</I>′; which is impossible. +<p>Therefore <I>OT</I> is not less than the arc <I>ASP</I>. And it was +proved not greater than the same arc. Therefore +<MATH><I>OT</I> = (arc <I>ASP</I>)</MATH>. +<p>As particular cases (separately proved by Archimedes), if +<I>P</I> be the extremity of the first turn and <I>c</I><SUB>1</SUB> the circumference +of the first circle, the subtangent = <I>c</I><SUB>1</SUB>; if <I>P</I> be the extremity +of the second turn and <I>c</I><SUB>2</SUB> the circumference of the ‘second +circle’, the subtangent = 2<I>c</I><SUB>2</SUB>; and generally, if <I>c<SUB>n</SUB></I> be the +circumference of the <I>n</I>th circle (the circle with the radius +vector to the extremity of the <I>n</I>th turn as radius), the sub- +tangent to the tangent at the extremity of the <I>n</I>th turn = <I>nc<SUB>n</SUB></I>. +<p>If <I>P</I> is a point on the <I>n</I>th turn, not the extremity, and the +circle with <I>O</I> as centre and <I>OP</I> as radius cuts the initial line +in <I>K</I>, while <I>p</I> is the circumference of the circle, the sub- +tangent to the tangent at <MATH><I>P</I> = (<I>n</I> - 1)<I>p</I> + arc <I>KP</I></MATH> (measured +‘forward’).<note>On the whole course of Archimedes's proof of the property of the +subtangent, see note in the Appendix.</note> +<p>The remainder of the book (Props. 21-8) is devoted to +finding the areas of portions of the spiral and its several +turns cut off by the initial line or any two radii vectores. +We will illustrate by the general case (Prop. 26). Take +<I>OB, OC</I>, two bounding radii vectores, including an arc <I>BC</I> +of the spiral. With centre <I>O</I> and radius <I>OC</I> describe a circle. +Divide the angle <I>BOC</I> into any number of equal parts by +radii of this circle. The spiral meets these radii in points +<I>P, Q ... Y, Z</I> such that the radii vectores <I>OB, OP, OQ ... OZ, OC</I> +<pb n=73><head>ON SPIRALS</head> +are in arithmetical progression. Draw arcs of circles with +radii <I>OB, OP, OQ</I> ... as shown; this produces a figure circum- +scribed to the spiral and consisting of the sum of small sectors +of circles, and an inscribed figure of the same kind. As the +first sector in the circumscribed figure is equal to the second +sector in the inscribed, it is easily seen that the areas of the +circumscribed and inscribed figures differ by the difference +between the sectors <I>OzC</I> and <I>OBp</I>′; therefore, by increasing +the number of divisions of the angle <I>BOC</I>, we can make the +<FIG> +difference between the areas of the circumscribed and in- +scribed figures as small as we please; we have, therefore, the +elements necessary for the application of the method of +exhaustion. +<p>If there are <I>n</I> radii <I>OB, OP ... OC</I>, there are (<I>n</I> - 1) parts of +the angle <I>BOC</I>. Since the angles of all the small sectors are +equal, the sectors are as the square on their radii. +<p>Thus <MATH>(whole sector <I>Ob</I>′<I>C</I>):(circumscribed figure) += (<I>n</I> - 1)<I>OC</I><SUP>2</SUP>:(<I>OP</I><SUP>2</SUP> + <I>OQ</I><SUP>2</SUP> + ... + <I>OC</I><SUP>2</SUP>)</MATH>, +and <MATH>(whole sector <I>Ob</I>′<I>C</I>):(inscribed figure) += (<I>n</I> - 1)<I>OC</I><SUP>2</SUP>:(<I>OB</I><SUP>2</SUP> + <I>OP</I><SUP>2</SUP> + <I>OQ</I><SUP>2</SUP> + ... + <I>OZ</I><SUP>2</SUP>)</MATH>. +<pb n=74><head>ARCHIMEDES</head> +<p>And <I>OB, OP, OQ, ... OZ, OC</I> is an arithmetical progression +of <I>n</I> terms; therefore (cf. Prop. 11 and Cor.), +<MATH>(<I>n</I> - 1)<I>OC</I><SUP>2</SUP>:(<I>OP</I><SUP>2</SUP> + <I>OQ</I><SUP>2</SUP> + ... + <I>OC</I><SUP>2</SUP>) +< <I>OC</I><SUP>2</SUP>:{<I>OC.OB</I> + 1/3(<I>OC</I> - <I>OB</I>)<SUP>2</SUP>} +< (<I>n</I> - 1)<I>OC</I><SUP>2</SUP>:(<I>OB</I><SUP>2</SUP> + <I>OP</I><SUP>2</SUP> + ... + <I>OZ</I><SUP>2</SUP>)</MATH>. +<p>Compressing the circumscribed and inscribed figures together +in the usual way, Archimedes proves by exhaustion that +<MATH>(sector <I>Ob</I>′<I>C</I>):(area of spiral <I>OBC</I>) += <I>OC</I><SUP>2</SUP>:{<I>OC.OB</I> + 1/3(<I>OC</I> - <I>OB</I>)<SUP>2</SUP>}</MATH>. +<p>If <MATH><I>OB</I> = <I>b, OC</I> = <I>c</I>, and (<I>c</I> - <I>b</I>) = (<I>n</I> - 1)<I>h</I></MATH>, Archimedes's +result is the equivalent of saying that, when <I>h</I> diminishes and +<I>n</I> increases indefinitely, while <I>c</I> - <I>b</I> remains constant, +limit of <MATH><I>h</I>{<I>b</I><SUP>2</SUP> + (<I>b</I> + <I>h</I>)<SUP>2</SUP> + (<I>b</I> + 2<I>h</I>)<SUP>2</SUP> + ... + (<I>b</I> + ―(<I>n</I> - 2)<I>h</I>)<SUP>2</SUP>} += (<I>c</I> - <I>b</I>) {<I>cb</I> + 1/3(<I>c</I> - <I>b</I>)<SUP>2</SUP>} += 1/3(<I>c</I><SUP>3</SUP> - <I>b</I><SUP>3</SUP>)</MATH>; +that is, with our notation, +<MATH>∫<SUP>c</SUP><SUB>b</SUB><I>x</I><SUP>2</SUP><I>dx</I> = 1/3(<I>c</I><SUP>3</SUP> - <I>b</I><SUP>3</SUP>)</MATH>. +<p>In particular, the area included by the first turn and the +initial line is bounded by the radii vectores 0 and 2<G>p</G><I>a</I>; +the area, therefore, is to the circle with radius 2<G>p</G><I>a</I> as 1/3(2<G>p</G><I>a</I>)<SUP>2</SUP> +to (2<G>p</G><I>a</I>)<SUP>2</SUP>, that is to say, it is 1/3 of the circle or 1/3<G>p</G>(2<G>p</G><I>a</I>)<SUP>2</SUP>. +This is separately proved in Prop. 24 by means of Prop. 10 +and Corr. 1, 2. +<p>The area of the ring added while the radius vector describes +the second turn is the area bounded by the radii vectores 2<G>p</G><I>a</I> +and 4<G>p</G><I>a</I>, and is to the circle with radius 4<G>p</G><I>a</I> in the ratio +of <MATH>{<I>r</I><SUB>2</SUB><I>r</I><SUB>1</SUB> + 1/3(<I>r</I><SUB>2</SUB> - <I>r</I><SUB>1</SUB>)<SUP>2</SUP>} to <I>r</I><SUP>2</SUP><SUB>2</SUB>, where <I>r</I><SUB>1</SUB> = 2<G>p</G><I>a</I> and <I>r</I><SUB>2</SUB> = +4<G>p</G><I>a</I></MATH>; the ratio is 7:12 (Prop. 25). +<p>If <I>R</I><SUB>1</SUB> be the area of the first turn of the spiral bounded by +the initial line, <I>R</I><SUB>2</SUB> the area of the ring added by the second +complete turn, <I>R</I><SUB>3</SUB> that of the ring added by the third turn, +and so on, then (Prop. 27) +<MATH><I>R</I><SUB>3</SUB> = 2<I>R</I><SUB>2</SUB>, <I>R</I><SUB>4</SUB> = 3<I>R</I><SUB>2</SUB>, <I>R</I><SUB>5</SUB> = 4<I>R</I><SUB>2</SUB>, ... <I>R<SUB>n</SUB></I> = (<I>n</I> - 1)<I>R</I><SUB>2</SUB></MATH>. +Also <MATH><I>R</I><SUB>2</SUB> = 6<I>R</I><SUB>1</SUB></MATH>. +<pb n=75><head>ON SPIRALS</head> +<p>Lastly, if <I>E</I> be the portion of the sector <I>b</I>′<I>OC</I> bounded by +<I>b</I>′<I>B</I>, the arc <I>b</I>′<I>zC</I> of the circle and the arc <I>BC</I> of the spiral, and +<I>F</I> the portion cut off between the arc <I>BC</I> of the spiral, the +radius <I>OC</I> and the arc intercepted between <I>OB</I> and <I>OC</I> of +the circle with centre <I>O</I> and radius <I>OB</I>, it is proved that +<MATH><I>E</I>:<I>F</I> = {<I>OB</I> + 2/3(<I>OC</I> - <I>OB</I>)}:{<I>OB</I> + 1/3(<I>OC</I> - <I>OB</I>)}</MATH> (Prop. 28). +<C>On Plane Equilibriums, I, II.</C> +<p>In this treatise we have the fundamental principles of +mechanics established by the methods of geometry in its +strictest sense. There were doubtless earlier treatises on +mechanics, but it may be assumed that none of them had +been worked out with such geometrical rigour. Archimedes +begins with seven Postulates including the following prin- +ciples. Equal weights at equal distances balance; if unequal +weights operate at equal distances, the larger weighs down +the smaller. If when equal weights are in equilibrium some- +thing be added to, or subtracted from, one of them, equilibrium +is not maintained but the weight which is increased or is not +diminished prevails. When equal and similar plane figures +coincide if applied to one another, their centres of gravity +similarly coincide; and in figures which are unequal but +similar the centres of gravity will be ‘similarly situated’. +In any figure the contour of which is concave in one and the +same direction the centre of gravity must be within the figure. +Simple propositions (1-5) follow, deduced by <I>reductio ad +absurdum</I>; these lead to the fundamental theorem, proved +first for commensurable and then by <I>reductio ad absurdum</I> +for incommensurable magnitudes, that <I>Two magnitudes, +whether commensurable or incommensurable, balance at dis- +tances reciprocally proportional to the magnitudes</I> (Props. +6, 7). Prop. 8 shows how to find the centre of gravity of +a part of a magnitude when the centres of gravity of the +other part and of the whole magnitude are given. Archimedes +then addresses himself to the main problems of Book I, namely +to find the centres of gravity of (1) a parallelogram (Props. +9, 10), (2) a triangle (Props. 13, 14), and (3) a parallel- +trapezium (Prop. 15), and here we have an illustration of the +extraordinary rigour which he requires in his geometrical +<pb n=76><head>ARCHIMEDES</head> +proofs. We do not find him here assuming, as in <I>The Method</I>, +that, if all the lines that can be drawn in a figure parallel to +(and including) one side have their middle points in a straight +line, the centre of gravity must lie somewhere on that straight +line; he is not content to regard the figure as <I>made up</I> of an +infinity of such parallel lines; pure geometry realizes that +the parallelogram is made up of elementary parallelograms, +indefinitely narrow if you please, but still parallelograms, and +the triangle of elementary <I>trapezia</I>, not straight lines, so +that to assume directly that the centre of gravity lies on the +straight line bisecting the parallelograms would really be +a <I>petitio principii</I>. Accordingly the result, no doubt dis- +covered in the informal way, is clinched by a proof by <I>reductio +ad absurdum</I> in each case. In the case of the parallelogram +<I>ABCD</I> (Prop. 9), if the centre of gravity is not on the straight +line <I>EF</I> bisecting two opposite sides, let it be at <I>H</I>. Draw +<I>HK</I> parallel to <I>AD</I>. Then it is possible by bisecting <I>AE, ED</I>, +then bisecting the halves, and so on, ultimately to reach +a length less than <I>KH</I>. Let this be done, and through the +<FIG> +points of division of <I>AD</I> draw parallels to <I>AB</I> or <I>DC</I> making +a number of equal and similar parallelograms as in the figure. +The centre of gravity of each of these parallelograms is +similarly situated with regard to it. Hence we have a number +of equal magnitudes with their centres of gravity at equal +distances along a straight line. Therefore the centre of +gravity of the whole is on the line joining the centres of gravity +of the two middle parallelograms (Prop. 5, Cor. 2). But this +is impossible, because <I>H</I> is outside those parallelograms. +Therefore the centre of gravity cannot but lie on <I>EF</I>. +<p>Similarly the centre of gravity lies on the straight line +bisecting the other opposite sides <I>AB, CD</I>; therefore it lies at +the intersection of this line with <I>EF</I>, i.e. at the point of +intersection of the diagonals. +<pb n=77><head>ON PLANE EQUILIBRIUMS, I</head> +<p>The proof in the case of the triangle is similar (Prop. 13). +Let <I>AD</I> be the median through <I>A</I>. The centre of gravity +must lie on <I>AD</I>. +<p>For, if not, let it be at <I>H</I>, and draw <I>HI</I> parallel to <I>BC</I>. +Then, if we bisect <I>DC</I>, then bisect the halves, and so on, +we shall arrive at a length <I>DE</I> less than <I>IH</I>. Divide <I>BC</I> into +lengths equal to <I>DE</I>, draw parallels to <I>DA</I> through the points +of division, and complete the small parallelograms as shown in +the figure. +<p>The centres of gravity of the whole parallelograms <I>SN, TP, +FQ</I> lie on <I>AD</I> (Prop. 9); therefore the centre of gravity of the +<FIG> +figure formed by them all lies on <I>AD</I>; let it be <I>O</I>. Join <I>OH</I>, +and produce it to meet in <I>V</I> the parallel through <I>C</I> to <I>AD</I>. +<p>Now it is easy to see that, if <I>n</I> be the number of parts into +which <I>DC, AC</I> are divided respectively, +<MATH>(sum of small ▵s <I>AMR, MLS ... ARN, NUP</I> ...):(▵<I>ABC</I>) += <I>n.AN</I><SUP>2</SUP>:<I>AC</I><SUP>2</SUP> += 1:<I>n</I></MATH>; +whence +<MATH>(sum of small ▵s):(sum of parallelograms) = 1:(<I>n</I> - 1)</MATH>. +<p>Therefore the centre of gravity of the figure made up of all +the small triangles is at a point <I>X</I> on <I>OH</I> produced such that +<MATH><I>XH</I> = (<I>n</I>-1)<I>OH</I></MATH>. +<p>But <MATH><I>VH</I>:<I>HO</I> < <I>CE</I>:<I>ED</I> or (<I>n</I>-1):1</MATH>; therefore <I>XH</I> > <I>VH</I>. +It follows that the centre of gravity of all the small +triangles taken together lies at <I>X</I> notwithstanding that all +the triangles lie on one side of the parallel to <I>AD</I> drawn +through <I>X</I>: which is impossible. +<pb n=78><head>ARCHIMEDES</head> +<p>Hence the centre of gravity of the whole triangle cannot +but lie on <I>AD</I>. +<p>It lies, similarly, on either of the other two medians; so +that it is at the intersection of any two medians (Prop. 14). +<p>Archimedes gives alternative proofs of a direct character, +both for the parallelogram and the triangle, depending on the +postulate that the centres of gravity of similar figures are +‘similarly situated’ in regard to them (Prop. 10 for the +parallelogram, Props. 11, 12 and part 2 of Prop. 13 for the +triangle). +<p>The geometry of Prop. 15 deducing the centre of gravity of +a trapezium is also interesting. It is proved that, if <I>AD, BC</I> +are the parallel sides (<I>AD</I> being the smaller), and <I>EF</I> is the +straight line joining their middle points, the centre of gravity +is at a point <I>G</I> on <I>EF</I> such that +<MATH><I>GE</I>:<I>GF</I> = (2<I>BC</I> + <I>AD</I>):(2<I>AD</I> + <I>BC</I>)</MATH>. +<p>Book II of the treatise is entirely devoted to finding the +centres of gravity of a parabolic segment (Props. 1-8) and +of a portion of it cut off by a parallel to the base (Props. 9, 10). +Prop. 1 (really a particular case of I. 6, 7) proves that, if <I>P, P</I>′ +<FIG> +be the areas of two parabolic segments and <I>D, E</I> their centres +of gravity, the centre of gravity of both taken together is +at a point <I>C</I> on <I>DE</I> such that +<MATH><I>P</I>:<I>P</I>′ = <I>CE</I>:<I>CD</I></MATH>. +<pb n=79><head>ON PLANE EQUILIBRIUMS, I, II</head> +This is merely preliminary. Then begins the real argument, +the course of which is characteristic and deserves to be set out. +Archimedes uses a series of figures inscribed to the segment, +as he says, ‘in the recognized manner’ (<G>gnwri/mws</G>). The rule +is as follows. Inscribe in the segment the triangle <I>ABB</I>′ with +the same base and height; the vertex <I>A</I> is then the point +of contact of the tangent parallel to <I>BB</I>′. Do the same with +the remaining segments cut off by <I>AB, AB</I>′, then with the +segments remaining, and so on. If <I>BRQPAP</I>′<I>Q</I>′<I>R</I>′<I>B</I>′ is such +a figure, the diameters through <I>Q, Q</I>′, <I>P, P</I>′, <I>R, R</I>′ bisect the +straight lines <I>AB, AB</I>′, <I>AQ, AQ</I>′, <I>QB, Q</I>′<I>B</I>′ respectively, and +<I>BB</I>′ is divided by the diameters into parts which are all +equal. It is easy to prove also that <I>PP</I>′, <I>QQ</I>′, <I>RR</I>′ are all +parallel to <I>BB</I>′, and that <MATH><I>AL</I>:<I>LM</I>:<I>MN</I>:<I>NO</I> = 1:3:5:7</MATH>, the +same relation holding if the number of sides of the polygon +is increased; i.e. the segments of <I>AO</I> are always in the ratio +of the successive odd numbers (Lemmas to Prop. 2). The +centre of gravity of the inscribed figure lies on <I>AO</I> (Prop. 2). +If there be two parabolic segments, and two figures inscribed +in them ‘in the recognized manner’ with an equal number of +sides, the centres of gravity divide the respective axes in the +same proportion, for the ratio depends on the same ratio of odd +numbers 1:3:5:7 ... (Prop. 3). The centre of gravity of the +parabolic segment itself lies on the diameter <I>AO</I> (this is proved +in Prop. 4 by <I>reductio ad absurdum</I> in exactly the same way +as for the triangle in I. 13). It is next proved (Prop. 5) that +the centre of gravity of the segment is nearer to the vertex <I>A</I> +than the centre of gravity of the inscribed figure is; but that +it is possible to inscribe in the segment in the recognized +manner a figure such that the distance between the centres of +gravity of the segment and of the inscribed figure is less than +any assigned length, for we have only to increase the number +of sides sufficiently (Prop. 6). Incidentally, it is observed in +Prop. 4 that, if in any segment the triangle with the same +base and equal height is inscribed, the triangle is greater than +half the segment, whence it follows that, each time we increase +the number of sides in the inscribed figure, we take away +more than half of the segments remaining over; and in Prop. 5 +that corresponding segments on opposite sides of the axis, e.g. +<I>QRB, Q</I>′<I>R</I>′<I>B</I>′ have their axes equal and therefore are equal in +<pb n=80><head>ARCHIMEDES</head> +area. Lastly (Prop. 7), if there be two parabolic segments, +their centres of gravity divide their diameters in the same +ratio (Archimedes enunciates this of similar segments only, +but it is true of any two segments and is required of any two +segments in Prop. 8). Prop. 8 now finds the centre of gravity +of any segment by using the last proposition. It is the +geometrical equivalent of the solution of a simple equation in +the ratio (<I>m</I>, say) of <I>AG</I> to <I>AO</I>, where <I>G</I> is the centre of +gravity of the segment. +<p>Since the segment = 4/3(▵<I>ABB</I>′), the sum of the two seg- +ments <MATH><I>AQB, AQ</I>′<I>B</I>′ = 1/3(▵<I>ABB</I>′)</MATH>. +<p>Further, if <I>QD, Q</I>′<I>D</I>′ are the diameters of these segments, +<I>QD, Q</I>′<I>D</I>′ are equal, and, since the centres +of gravity <I>H, H</I>′ of the segments divide +<I>QD, Q</I>′<I>D</I>′ proportionally, <I>HH</I>′ is parallel +to <I>QQ</I>′, and the centre of gravity of the +two segments together is at <I>K</I>, the point +where <I>HH</I>′ meets <I>AO</I>. +<FIG> +<p>Now <MATH><I>AO</I> = 4<I>AV</I></MATH> (Lemma 3 to Prop. +2), and <MATH><I>QD</I> = 1/2<I>AO</I> - <I>AV</I> = <I>AV</I></MATH>. But +<I>H</I> divides <I>QD</I> in the same ratio as <I>G</I> +divides <I>AO</I> (Prop. 7); therefore +<MATH><I>VK</I> = <I>QH</I> = <I>m.QD</I> = <I>m.AV</I></MATH>. +<p>Taking moments about <I>A</I> of the segment, the triangle <I>ABB</I>′ +and the sum of the small segments, we have (dividing out by +<I>AV</I> and ▵<I>ABB</I>′) +<MATH>1/3(1 + <I>m</I>) + 2/3.4 = 4/3.4<I>m</I></MATH>, +or <MATH>15<I>m</I> = 9</MATH>, +and <MATH><I>m</I> = 3/5</MATH>. +<p>That is, <MATH><I>AG</I> = 3/5<I>AO</I>, or <I>AG</I>:<I>GO</I> = 3:2</MATH>. +<p>The final proposition (10) finds the centre of gravity of the +portion of a parabola cut off between two parallel chords <I>PP</I>′, +<I>BB</I>′. If <I>PP</I>′ is the shorter of the chords and the diameter +bisecting <I>PP</I>′, <I>BB</I>′ meets them in <I>N, O</I> respectively, Archi- +medes proves that, if <I>NO</I> be divided into five equal parts of +which <I>LM</I> is the middle one (<I>L</I> being nearer to <I>N</I> than <I>M</I> is), +<pb n=81><head>ON PLANE EQUILIBRIUMS, II</head> +the centre of gravity <I>G</I> of the portion of the parabola between +<I>PP</I>′ and <I>BB</I>′ divides <I>LM</I> in such a way that +<MATH><I>LG</I>:<I>GM</I> = <I>BO</I><SUP>2</SUP>.(2<I>PN</I> + <I>BO</I>):<I>PN</I><SUP>2</SUP>.(2<I>BO</I> + <I>PN</I>)</MATH>. +<p>The geometrical proof is somewhat difficult, and uses a very +remarkable Lemma which forms Prop. 9. If <I>a, b, c, d, x, y</I> are +straight lines satisfying the conditions +<MATH><BRACE><I>a</I>/<I>b</I> = <I>b</I>/<I>c</I> = <I>c</I>/<I>d</I>(<I>a</I> > <I>b</I> > <I>c</I> > <I>d</I>), +<I>d</I>/(<I>a</I> - <I>d</I>) = <I>x</I>/(3/5(<I>a</I> - <I>c</I>)), +and (2<I>a</I> + 4<I>b</I> + 6<I>c</I> + 3<I>d</I>)/(5<I>a</I> + 10<I>b</I> + 10<I>c</I> + 5<I>d</I>) = <I>y</I>/(<I>a</I> - <I>c</I>), +then must <I>x</I> + <I>y</I> = 2/5<I>a</I></BRACE></MATH>. +<p>The proof is entirely geometrical, but amounts of course to +the elimination of three quantities <I>b, c, d</I> from the above four +equations. +<C>The Sand-reckoner (<I>Psammites</I> or <I>Arenarius</I>).</C> +<p>I have already described in a previous chapter the remark- +able system, explained in this treatise and in a lost work, +’<I>A<G>rxai/</G>, Principles</I>, addressed to Zeuxippus, for expressing very +large numbers which were beyond the range of the ordinary +Greek arithmetical notation. Archimedes showed that his +system would enable any number to be expressed up to that +which in our notation would require 80,000 million million +ciphers and then proceeded to prove that this system more +than sufficed to express the number of grains of sand which +it would take to fill the universe, on a reasonable view (as it +seemed to him) of the size to be attributed to the universe. +Interesting as the book is for the course of the argument by +which Archimedes establishes this, it is, in addition, a docu- +ment of the first importance historically. It is here that we +learn that Aristarchus put forward the Copernican theory of +the universe, with the sun in the centre and the planets +including the earth revolving round it, and that Aristarchus +further discovered the angular diameter of the sun to be 1/(720)th +of the circle of the zodiac or half a degree. Since Archimedes, +in order to calculate a safe figure (not too small) for the size +<pb n=82><head>ARCHIMEDES</head> +of the universe, has to make certain assumptions as to the +sizes and distances of the sun and moon and their relation +to the size of the universe, he takes the opportunity of +quoting earlier views. Some have tried, he says, to prove +that the perimeter of the earth is about 300,000 stades; in +order to be quite safe he will take it to be about ten times +this, or 3,000,000 stades, and not greater. The diameter of +the earth, like most earlier astronomers, he takes to be +greater than that of the moon but less than that of the sun. +Eudoxus, he says, declared the diameter of the sun to be nine +times that of the moon, Phidias, his own father, twelve times, +while Aristarchus tried to prove that it is greater than 18 but +less than 20 times the diameter of the moon; he will again be +on the safe side and take it to be 30 times, but not more. The +position is rather more difficult as regards the ratio of the +distance of the sun to the size of the universe. Here he seizes +upon a dictum of Aristarchus that the sphere of the fixed +stars is so great that the circle in which he supposes the earth +to revolve (round the sun) ‘bears such a proportion to the +distance of the fixed stars as the centre of the sphere bears to +its surface’. If this is taken in a strictly mathematical sense, +it means that the sphere of the fixed stars is infinite in size, +which would not suit Archimedes's purpose; to get another +meaning out of it he presses the point that Aristarchus's +words cannot be taken quite literally because the centre, being +without magnitude, cannot be in any ratio to any other mag- +nitude; hence he suggests that a reasonable interpretation of +the statement would be to suppose that, if we conceive a +sphere with radius equal to the distance between the centre +of the sun and the centre of the earth, then +(diam. of earth):(diam. of said sphere) += (diam. of said sphere):(diam. of sphere of fixed stars). +This is, of course, an arbitrary interpretation; Aristarchus +presumably meant no such thing, but merely that the size of +the earth is negligible in comparison with that of the sphere +of the fixed stars. However, the solution of Archimedes's +problem demands some assumption of the kind, and, in making +this assumption, he was no doubt aware that he was taking +a liberty with Aristarchus for the sake of giving his hypo- +thesis an air of authority. +<pb n=83><head>THE <I>SAND-RECKONER</I></head> +<p>Archimedes has, lastly, to compare the diameter of the sun +with the circumference of the circle described by its centre. +Aristarchus had made the apparent diameter of the sun 1/(720)th +of the said circumference; Archimedes will prove that the +said circumference cannot contain as many as 1,000 sun's +diameters, or that the diameter of the sun is greater than the +side of a regular chiliagon inscribed in the circle. First he +made an experiment of his own to determine the apparent +diameter of the sun. With a small cylinder or disc in a plane +at right angles to a long straight stick and moveable along it, +he observed the sun at the moment when it cleared the +horizon in rising, moving the disc till it just covered and just +failed to cover the sun as he looked along the straight stick. +He thus found the angular diameter to lie between 1/(164)<I>R</I> and +1/(200)<I>R</I>, where <I>R</I> is a right angle. But as, under his assump- +tions, the size of the earth is not negligible in comparison with +the sun's circle, he had to allow for parallax and find limits +for the angle subtended by the sun at the centre of the earth. +This he does by a geometrical argument very much in the +manner of Aristarchus. +<FIG> +<p>Let the circles with centres <I>O, C</I> represent sections of the sun +and earth respectively, <I>E</I> the position of the observer observing +<pb n=84><head>ARCHIMEDES</head> +the sun when it has just cleared the horizon. Draw from <I>E</I> +two tangents <I>EP, EQ</I> to the circle with centre <I>O</I>, and from +<I>C</I> let <I>CF, CG</I> be drawn touching the same circle. With centre +<I>C</I> and radius <I>CO</I> describe a circle: this will represent the path +of the centre of the sun round the earth. Let this circle meet +the tangents from <I>C</I> in <I>A, B</I>, and join <I>AB</I> meeting <I>CO</I> in <I>M</I>. +<p>Archimedes's observation has shown that +<MATH>(1/164)<I>R</I> > ∠<I>PEQ</I> > 1/200 <I>R</I></MATH>; +and he proceeds to prove that <I>AB</I> is less than the side of a +regular polygon of 656 sides inscribed in the circle <I>AOB</I>, +but greater than the side of an inscribed regular polygon of +1,000 sides, in other words, that +<MATH>(1/164)<I>R</I> > ∠<I>FCG</I> > (1/250)<I>R</I></MATH>. +The first relation is obvious, for, since <I>CO</I> > <I>EO</I>, +<MATH>∠<I>PEQ</I> > ∠<I>FCG</I></MATH>. +<p>Next, the perimeter of any polygon inscribed in the circle +<I>AOB</I> is less than 44/7 <I>CO</I> (i.e. 22/7 times the diameter); +Therefore <MATH><I>AB</I> < (1/656).(44/7) <I>CO</I> or (11/1148)<I>CO</I></MATH>, +and, <I>a fortiori</I>, <MATH><I>AB</I> < (1/100)<I>CO</I></MATH>. +<p>Now, the triangles <I>CAM, COF</I> being equal in all respects, +<MATH><I>AM</I> = <I>OF</I>, so that <I>AB</I> = 2<I>OF</I> = (diameter of sun) > <I>CH</I> + <I>OK</I></MATH>, +since the diameter of the sun is greater than that of the earth; +therefore <MATH><I>CH</I> + <I>OK</I> < (1/100)<I>CO</I>, and <I>HK</I> > (99/100)<I>CO</I></MATH>. +<p>And <MATH><I>CO</I> > <I>CF</I>, while <I>HK</I> < <I>EQ</I>, so that <I>EQ</I> > (99/100)<I>CF</I></MATH>. +<p>We can now compare the angles <I>OCF, OEQ</I>; +for <MATH>(∠<I>OCF</I>)/(∠<I>OEQ</I>)[ > (tan <I>OCF</I>)/(tan <I>OEQ</I>)] +> <I>EQ</I>/<I>CF</I> +> 99/100</MATH>, <I>a fortiori</I>. +<p>Doubling the angles, we have +<MATH>∠<I>FCG</I> > (99/100).∠<I>PEQ</I> +> (99/20000)<I>R</I>, since ∠<I>PEQ</I> > (1/200)<I>R</I>, +> (1/203)<I>R</I></MATH>. +<pb n=85><head>THE <I>SAND-RECKONER</I></head> +<p>Hence <I>AB</I> is greater than the side of a regular polygon of +812 sides, and <I>a fortiori</I> greater than the side of a regular +polygon of 1,000 sides, inscribed in the circle <I>AOB</I>. +<p>The perimeter of the chiliagon, as of any regular polygon +with more sides than six, inscribed in the circle <I>AOB</I> is greater +than 3 times the diameter of the sun's orbit, but is less than +1,000 times the diameter of the sun, and <I>a fortiori</I> less than +30,000 times the diameter of the earth; +<MATH>therefore (diameter of sun's orbit) < 10,000 (diam. of earth) +< 10,000,000,000 stades.</MATH> +<p>But <MATH>(diam. of earth):(diam. of sun's orbit) += (diam. of sun's orbit):(diam. of universe);</MATH> +therefore the universe, or the sphere of the fixed stars, is less +than 10,000<SUP>3</SUP> times the sphere in which the sun's orbit is a +great circle. +<p>Archimedes takes a quantity of sand not greater than +a poppy-seed and assumes that it contains not more than 10,000 +grains; the diameter of a poppy-seed he takes to be not less +than 1/(40)th of a finger-breadth; thus a sphere of diameter +1 finger-breadth is not greater than 64,000 poppy-seeds and +therefore contains not more than 640,000,000 grains of sand +(‘6 units of <I>second order</I> + 40,000,000 units of <I>first order</I>’) +and <I>a fortiori</I> not more than 1,000,000,000 (‘10 units of +<I>second order</I> of numbers’). Gradually increasing the diameter +of the sphere by multiplying it each time by 100 (making the +sphere 1,000,000 times larger each time) and substituting for +10,000 finger-breadths a stadium (< 10,000 finger-breadths), +he finds the number of grains of sand in a sphere of diameter +10,000,000,000 stadia to be less than ‘1,000 units of <I>seventh +order</I> of numbers’ or 10<SUP>51</SUP>, and the number in a sphere 10,000<SUP>3</SUP> +times this size to be less than ‘10,000,000 units of the <I>eighth +order</I> of numbers’ or 10<SUP>63</SUP>. +<C>The Quadrature of the Parabola.</C> +<p>In the preface, addressed to Dositheus after the death of +Conon, Archimedes claims originality for the solution of the +problem of finding the area of a segment of a parabola cut off +by any chord, which he says he first discovered by means of +mechanics and then confirmed by means of geometry, using +the lemma that, if there are two unequal areas (or magnitudes +<pb n=86><head>ARCHIMEDES</head> +generally), then however small the excess of the greater over +the lesser, it can by being continually added to itself be made +to exceed the greater; in other words, he confirmed the solution +by the method of exhaustion. One solution by means of +mechanics is, as we have seen, given in <I>The Method</I>; the +present treatise contains a solution by means of mechanics +confirmed by the method of exhaustion (Props. 1-17), and +then gives an entirely independent solution by means of pure +geometry, also confirmed by exhaustion (Props. 18-24). +<p>I. The mechanical solution depends upon two properties of +the parabola proved in Props. 4, 5. If <I>Qq</I> be the base, and <I>P</I> +<FIG> +the vertex, of a parabolic segment, <I>P</I> is the point of contact +of the tangent parallel to <I>Qq</I>, the diameter <I>PV</I> through <I>P</I> +bisects <I>Qq</I> in <I>V</I>, and, if <I>VP</I> produced meets the tangent at <I>Q</I> +in <I>T</I>, then <I>TP</I> = <I>PV</I>. These properties, along with the funda- +mental property that <I>QV</I><SUP>2</SUP> varies as <I>PV</I>, Archimedes uses to +prove that, if <I>EO</I> be any parallel to <I>TV</I> meeting <I>QT, QP</I> +(produced, if necessary), the curve, and <I>Qq</I> in <I>E, F, R, O</I> +respectively, then +<MATH><I>QV</I>:<I>VO</I> = <I>OF</I>:<I>FR</I></MATH>, +and <MATH><I>QO</I>:<I>Oq</I> = <I>ER</I>:<I>RO</I></MATH>. (Props. 4, 5.) +<p>Now suppose a parabolic segment <I>QR</I><SUB>1</SUB><I>q</I> so placed in relation +to a horizontal straight line <I>QA</I> through <I>Q</I> that the diameter +bisecting <I>Qq</I> is at right angles to <I>QA</I>, i.e. vertical, and let the +tangent at <I>Q</I> meet the diameter <I>qO</I> through <I>q</I> in <I>E</I>. Produce +<I>QO</I> to <I>A</I>, making <I>OA</I> equal to <I>OQ</I>. +<p>Divide <I>Qq</I> into any number of equal parts at <I>O</I><SUB>1</SUB>, <I>O</I><SUB>2</SUB> ... <I>O<SUB>n</SUB></I>, +and through these points draw parallels to <I>OE</I>, i.e. vertical +lines meeting <I>OQ</I> in <I>H</I><SUB>1</SUB>, <I>H</I><SUB>2</SUB>, ..., <I>EQ</I> in <I>E</I><SUB>1</SUB>, <I>E</I><SUB>2</SUB>, ..., and the +<pb n=87><head>THE QUADRATURE OF THE PARABOLA</head> +curve in <I>R</I><SUB>1</SUB>, <I>R</I><SUB>2</SUB>, .... Join <I>QR</I><SUB>1</SUB>, and produce it to meet <I>OE</I> in +<I>F, QR</I><SUB>2</SUB> meeting <I>O</I><SUB>1</SUB><I>E</I><SUB>1</SUB> in <I>F</I><SUB>1</SUB>, and so on. +<FIG> +<p>Now Archimedes has proved in a series of propositions +(6-13) that, if a trapezium such as <I>O</I><SUB>1</SUB><I>E</I><SUB>1</SUB><I>E</I><SUB>2</SUB><I>O</I><SUB>2</SUB> is suspended +from <I>H</I><SUB>1</SUB><I>H</I><SUB>2</SUB>, and an area <I>P</I> suspended at <I>A</I> balances <I>O</I><SUB>1</SUB><I>E</I><SUB>1</SUB><I>E</I><SUB>2</SUB><I>O</I><SUB>2</SUB> +so suspended, it will take a greater area than <I>P</I> suspended at +<I>A</I> to balance the same trapezium suspended from <I>H</I><SUB>2</SUB> and +a less area than <I>P</I> to balance the same trapezium suspended +from <I>H</I><SUB>1</SUB>. A similar proposition holds with regard to a triangle +such as <I>E<SUB>n</SUB>H<SUB>n</SUB>Q</I> suspended where it is and suspended at <I>Q</I> and +<I>H<SUB>n</SUB></I> respectively. +<p>Suppose (Props. 14, 15) the triangle <I>QqE</I> suspended where +it is from <I>OQ</I>, and suppose that the trapezium <I>EO</I><SUB>1</SUB>, suspended +where it is, is balanced by an area <I>P</I><SUB>1</SUB> suspended at <I>A</I>, the +trapezium <I>E</I><SUB>1</SUB><I>O</I><SUB>2</SUB>, suspended where it is, is balanced by <I>P</I><SUB>2</SUB> +suspended at <I>A</I>, and so on, and finally the triangle <I>E<SUB>n</SUB>O<SUB>n</SUB>Q</I>, +suspended where it is, is balanced by <I>P</I><SUB><I>n</I> + 1</SUB> suspended at <I>A</I>; +then <MATH><I>P</I><SUB>1</SUB> + <I>P</I><SUB>2</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB></MATH> at <I>A</I> balances the whole triangle, so that +<MATH><I>P</I><SUB>1</SUB> + <I>P</I><SUB>2</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB> = 1/3▵<I>EqQ</I></MATH>, +since the whole triangle may be regarded as suspended from +the point on <I>OQ</I> vertically above its centre of gravity. +<p>Now <MATH><I>AO</I>:<I>OH</I><SUB>1</SUB> = <I>QO</I>:<I>OH</I><SUB>1</SUB> += <I>Qq</I>:<I>qQ</I><SUB>1</SUB> += <I>E</I><SUB>1</SUB><I>O</I><SUB>1</SUB>:<I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB>, by Prop. 5, += (trapezium <I>EO</I><SUB>1</SUB>):(trapezium <I>FO</I><SUB>1</SUB>)</MATH>, +<pb n=88><head>ARCHIMEDES</head> +that is, it takes the trapezium <I>FO</I><SUB>1</SUB> suspended at <I>A</I> to balance +the trapezium <I>EO</I><SUB>1</SUB> suspended at <I>H</I><SUB>1</SUB>. And <I>P</I><SUB>1</SUB> balances <I>EO</I><SUB>1</SUB> +where it is. +<p>Therefore <MATH>(<I>FO</I><SUB>1</SUB>) > <I>P</I><SUB>1</SUB></MATH>. +<p>Similarly <MATH>(<I>F</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) > <I>P</I><SUB>2</SUB></MATH>, and so on. +<p>Again <MATH><I>AO</I>:<I>OH</I><SUB>1</SUB> = <I>E</I><SUB>1</SUB><I>O</I><SUB>1</SUB>:<I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB> += (trapezium <I>E</I><SUB>1</SUB><I>O</I><SUB>2</SUB>):(trapezium <I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>)</MATH>, +that is, (<I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) at <I>A</I> will balance (<I>E</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) suspended at <I>H</I><SUB>1</SUB>, +while <I>P</I><SUB>2</SUB> at <I>A</I> balances (<I>E</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) suspended where it is, +whence <MATH><I>P</I><SUB>2</SUB> > <I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB></MATH>. +<p>Therefore <MATH>(<I>F</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) > <I>P</I><SUB>2</SUB> > (<I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>), +(<I>F</I><SUB>2</SUB><I>O</I><SUB>3</SUB>) > <I>P</I><SUB>3</SUB> > <I>R</I><SUB>2</SUB><I>O</I><SUB>3</SUB></MATH>, and so on; +and finally, <MATH>▵<I>E<SUB>n</SUB>O<SUB>n</SUB>Q</I> > <I>P</I><SUB><I>n</I> + 1</SUB> > ▵<I>R<SUB>n</SUB>O<SUB>n</SUB>Q</I></MATH>. +<p>By addition, +<MATH>(<I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) + (<I>R</I><SUB>2</SUB><I>O</I><SUB>3</SUB>) + ... + (▵<I>R<SUB>n</SUB>O<SUB>n</SUB>Q</I>) < <I>P</I><SUB>2</SUB> + <I>P</I><SUB>3</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB></MATH>; +therefore, <I>a fortiori</I>, +<MATH>(<I>R</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) + (<I>R</I><SUB>2</SUB><I>O</I><SUB>3</SUB>) + ... + ▵<I>R<SUB>n</SUB>O<SUB>n</SUB>Q</I> < <I>P</I><SUB>1</SUB> + <I>P</I><SUB>2</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB> +< (<I>FO</I><SUB>1</SUB>) + (<I>F</I><SUB>1</SUB><I>O</I><SUB>2</SUB>) + ... + ▵<I>E<SUB>n</SUB>O<SUB>n</SUB>Q</I></MATH>. +<p>That is to say, we have an inscribed figure consisting of +trapezia and a triangle which is less, and a circumscribed +figure composed in the same way which is greater, than +<MATH><I>P</I><SUB>1</SUB> + <I>P</I><SUB>2</SUB> + ... + <I>P</I><SUB><I>n</I> + 1</SUB>, i.e. 1/3▵<I>EqQ</I></MATH>. +<p>It is therefore inferred, and proved by the method of ex- +haustion, that the segment itself is <I>equal</I> to 1/3▵<I>EqQ</I> (Prop. 16). +<p>In order to enable the method to be applied, it has only +to be proved that, by increasing the number of parts in <I>Qq</I> +sufficiently, the difference between the circumscribed and +inscribed figures can be made as small as we please. This +can be seen thus. We have first to show that all the parts, as +<I>qF</I>, into which <I>qE</I> is divided are equal. +<p>We have <MATH><I>E</I><SUB>1</SUB><I>O</I><SUB>1</SUB>:<I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB> = <I>QO</I>:<I>OH</I><SUB>1</SUB> = (<I>n</I> + 1):1</MATH>, +or <MATH><I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB> = 1/(<I>n</I> + 1).<I>E</I><SUB>1</SUB><I>O</I><SUB>1</SUB></MATH>, whence also <MATH><I>O</I><SUB>2</SUB><I>S</I> = 1/(<I>n</I> + 1).<I>O</I><SUB>2</SUB><I>E</I><SUB>2</SUB></MATH>. +<pb n=89><head>THE QUADRATURE OF THE PARABOLA</head> +<p>And <MATH><I>E</I><SUB>2</SUB><I>O</I><SUB>2</SUB>:<I>O</I><SUB>2</SUB><I>R</I><SUB>2</SUB> = <I>QO</I>:<I>OH</I><SUB>2</SUB> = (<I>n</I> + 1):2</MATH>, +or <MATH><I>O</I><SUB>2</SUB><I>R</I><SUB>2</SUB> = 2/(<I>n</I> + 1).<I>O</I><SUB>2</SUB><I>E</I><SUB>2</SUB></MATH>. +<p>It follows that <I>O</I><SUB>2</SUB><I>S</I> = <I>SR</I><SUB>2</SUB>, and so on. +<p>Consequently <I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB>, <I>O</I><SUB>2</SUB><I>R</I><SUB>2</SUB>, <I>O</I><SUB>3</SUB><I>R</I><SUB>3</SUB> ... are divided into 1, 2, 3 ... +equal parts respectively by the lines from <I>Q</I> meeting <I>qE</I>. +<p>It follows that the difference between the circumscribed and +inscribed figures is equal to the triangle <I>FqQ</I>, which can be +made as small as we please by increasing the number of +divisions in <I>Qq</I>, i.e. in <I>qE</I>. +<p>Since the area of the segment is equal to 1/3▵<I>EqQ</I>, and it is +easily proved (Prop. 17) that ▵<I>EqQ</I> = 4 (triangle with same +base and equal height with segment), it follows that the area +of the segment = 4/3 times the latter triangle. +<p>It is easy to see that this solution is essentially the same as +that given in <I>The Method</I> (see pp. 29-30, above), only in a more +orthodox form (geometrically speaking). For there Archi- +medes took the sum of all the <I>straight lines</I>, as <I>O</I><SUB>1</SUB><I>R</I><SUB>1</SUB>, <I>O</I><SUB>2</SUB><I>R</I><SUB>2</SUB> ..., +as making up the segment notwithstanding that there are an +infinite number of them and straight lines have no breadth. +Here he takes inscribed and circumscribed trapezia propor- +tional to the straight lines and having finite breadth, and then +compresses the figures together into the segment itself by +increasing indefinitely the number of trapezia in each figure, +i.e. diminishing their breadth indefinitely. +<p>The procedure is equivalent to an integration, thus: +<p>If <I>X</I> denote the area of the triangle <I>FqQ</I>, we have, if <I>n</I> be +the number of parts in <I>Qq</I>, +<MATH>(circumscribed figure) += sum of ▵s <I>QqF, QR</I><SUB>1</SUB><I>F</I><SUB>1</SUB>, <I>QR</I><SUB>2</SUB><I>F</I><SUB>2</SUB>, ... += sum of ▵s <I>QqF, QO</I><SUB>1</SUB><I>R</I><SUB>1</SUB>, <I>QO</I><SUB>2</SUB><I>S</I>, ... += <I>X</I> {1 + ((<I>n</I> - 1)<SUP>2</SUP>)/(<I>n</I><SUP>2</SUP>) + ((<I>n</I> - 2)<SUP>2</SUP>)/(<I>n</I><SUP>2</SUP>) + ... + 1/(<I>n</I><SUP>2</SUP>)} += 1/(<I>n</I><SUP>2</SUP><I>X</I><SUP>2</SUP>).<I>X</I> (<I>X</I><SUP>2</SUP> + 2<SUP>2</SUP><I>X</I><SUP>2</SUP> + 3<SUP>2</SUP><I>X</I><SUP>2</SUP> + ... + <I>n</I><SUP>2</SUP><I>X</I><SUP>2</SUP>)</MATH>. +Similarly, we find that +<MATH>(inscribed figure) = 1/(<I>n</I><SUP>2</SUP><I>X</I><SUP>2</SUP>).<I>X</I> {<I>X</I><SUP>2</SUP> + 2<SUP>2</SUP><I>X</I><SUP>2</SUP> + ... + (<I>n</I> - 1)<SUP>2</SUP><I>X</I><SUP>2</SUP>}</MATH>. +<pb n=90><head>ARCHIMEDES</head> +<p>Taking the limit, we have, if <I>A</I> denote the area of the +triangle <I>EqQ,</I> so that <MATH><I>A</I>=<I>nX,</I></MATH> +<MATH>area of segment=1/(<I>A</I><SUP>2</SUP>)∫<SUP>A</SUP><SUB>0</SUB><I>X</I><SUP>2</SUP><I>dX</I> +=1/3<I>A.</I></MATH> +<p>II. The purely geometrical method simply <I>exhausts</I> the +parabolic segment by inscribing successive figures ‘in the +recognized manner’ (see p. 79, above). For this purpose +it is necessary to find, in terms of the triangle with the same +<FIG> +base and height, the area added to the +inscribed figure by doubling the number of +sides other than the base of the segment. +<p>Let <I>QPq</I> be the triangle inscribed ‘in the +recognized manner’, <I>P</I> being the point of +contact of the tangent parallel to <I>Qq,</I> and +<I>PV</I> the diameter bisecting <I>Qq.</I> If <I>QV, Vq</I> +be bisected in <I>M, m,</I> and <I>RM, rm</I> be drawn +parallel to <I>PV</I> meeting the curve in <I>R, r,</I> +the latter points are vertices of the next +figure inscribed ‘in the recognized manner’, +for <I>RY, ry</I> are diameters bisecting <I>PQ, Pq</I> +respectively. +<p>Now <MATH><I>QV</I><SUP>2</SUP>=4<I>RW</I><SUP>2</SUP></MATH>, so that <MATH><I>PV</I>=4<I>PW,</I></MATH> or <MATH><I>RM</I>=3<I>PW.</I></MATH> +<p>But <MATH><I>YM</I>=1/2<I>PV</I>=2<I>PW,</I></MATH> so that <MATH><I>YM</I>=2<I>RY.</I></MATH> +<p>Therefore <MATH>▵<I>PRQ</I>=1/2▵<I>PQM</I>=1/4▵<I>PQV.</I></MATH> +<p>Similarly +<MATH>▵<I>Prq</I>=1/4▵<I>PVq</I></MATH>; whence <MATH>(▵<I>PRQ</I>+▵<I>Prq</I>)=1/4<I>PQq.</I></MATH> (Prop. 21.) +<p>In like manner it can be proved that the next addition +to the inscribed figure adds 1/4 of the sum of ▵s<I>PRQ, Prq,</I> +and so on. +<p><MATH>Therefore the area of the inscribed figure +={1+1/4+(1/4)<SUP>2</SUP>+ ...}.▵<I>PQq.</I></MATH> (Prop. 22.) +<p>Further, each addition to the inscribed figure is greater +than half the segments of the parabola left over before the +addition is made. For, if we draw the tangent at <I>P</I> and +complete the parallelogram <I>EQqe</I> with side <I>EQ</I> parallel to <I>PV,</I> +<pb n=91><head>THE QUADRATURE OF THE PARABOLA</head> +the triangle <I>PQq</I> is half of the parallelogram and therefore +more than half the segment. And so on (Prop. 20). +<p>We now have to sum <I>n</I> terms of the above geometrical +series. Archimedes enunciates the problem in the form, Given +a series of areas <I>A, B, C, D</I> ... <I>Z,</I> of which <I>A</I> is the greatest, and +each is equal to four times the next in order, then (Prop. 23) +<MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>Z</I>+1/3<I>Z</I>=4/3<I>A.</I></MATH> +<p>The algebraical equivalent of this is of course +<MATH>1+1/4+(1/4)<SUP>2</SUP>+ ... +(1/4)<SUP><I>n</I>-1</SUP>=4/3-1/3(1/4)<SUP><I>n</I>-1</SUP>=(1-(1/4)<SUP><I>n</I></SUP>)/(1-1/4)</MATH>. +<p>To find the area of the segment, Archimedes, instead of +taking the limit, as we should, uses the method of <I>reductio ad +absurdum.</I> +<p>Suppose <MATH><I>K</I>=4/3.▵<I>PQq.</I></MATH> +<p>(1) If possible, let the area of the segment be greater than <I>K.</I> +<p>We then inscribe a figure ‘in the recognized manner’ such +that the segment exceeds it by an area less than the excess of +the segment over <I>K.</I> Therefore the inscribed figure must be +greater than <I>K,</I> which is impossible since +<MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>Z</I><4/3<I>A,</I></MATH> +where <MATH><I>A</I>=▵<I>PQq</I></MATH> (Prop. 23). +<p>(2) If possible, let the area of the segment be less than <I>K.</I> +<p>If then <MATH>▵<I>PQq</I>=<I>A, B</I>=1/4<I>A, C</I>=1/4<I>B,</I></MATH> and so on, until we +arrive at an area <I>X</I> less than the excess of <I>K</I> over the area of +the segment, we have +<MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>X</I>+1/3<I>X</I>=4/3<I>A</I>=<I>K.</I></MATH> +<p>Thus <I>K</I> exceeds <MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>X</I></MATH> by an area less than <I>X,</I> +and exceeds the segment by an area greater than <I>X.</I> +<p>It follows that <MATH><I>A</I>+<I>B</I>+<I>C</I>+ ... +<I>X</I> > (the segment)</MATH>; which +is impossible (Prop. 22). +<p>Therefore the area of the segment, being neither greater nor +less than <I>K,</I> is equal to <I>K</I> or 4/3▵<I>PQq.</I> +<C>On Floating Bodies, I, II.</C> +<p>In Book I of this treatise Archimedes lays down the funda- +mental principles of the science of hydrostatics. These are +<pb n=92><head>ARCHIMEDES</head> +deduced from Postulates which are only two in number. The +first which begins Book I is this: +<p>‘let it be assumed that a fluid is of such a nature that, of the +parts of it which lie evenly and are continuous, that which is +pressed the less is driven along by that which is pressed the +more; and each of its parts is pressed by the fluid which is +perpendicularly above it except when the fluid is shut up in +anything and pressed by something else’; +<p>the second, placed after Prop. 7, says +<p>‘let it be assumed that, of bodies which are borne upwards in +a fluid, each is borne upwards along the perpendicular drawn +through its centre of gravity’. +<p>Prop. 1 is a preliminary proposition about a sphere, and +then Archimedes plunges <I>in medias res</I> with the theorem +(Prop. 2) that ‘<I>the surface of any fluid at rest is a sphere the +centre of which is the same as that of the earth</I>’, and in the +whole of Book I the surface of the fluid is always shown in +the diagrams as spherical. The method of proof is similar to +what we should expect in a modern elementary textbook, the +main propositions established being the following. A solid +which, size for size, is of equal weight with a fluid will, if let +down into the fluid, sink till it is just covered but not lower +(Prop. 3); a solid lighter than a fluid will, if let down into it, +be only partly immersed, in fact just so far that the weight +of the solid is equal to the weight of the fluid displaced +(Props. 4, 5), and, if it is forcibly immersed, it will be driven +upwards by a force equal to the difference between its weight +and the weight of the fluid displaced (Prop. 6). +<p>The important proposition follows (Prop. 7) that a solid +heavier than a fluid will, if placed in it, sink to the bottom of +the fluid, and the solid will, when weighed in the fluid, be +lighter than its true weight by the weight of the fluid +displaced. +<C><I>The problem of the Crown.</I></C> +<p>This proposition gives a method of solving the famous +problem the discovery of which in his bath sent Archimedes +home naked crying <G>en(/rhka, en(/rhka</G>, namely the problem of +<pb n=93><head>ON FLOATING BODIES, I</head> +determining the proportions of gold and silver in a certain +crown. +<p>Let <I>W</I> be the weight of the crown, <I>w</I><SUB>1</SUB> and <I>w</I><SUB>2</SUB> the weights of +the gold and silver in it respectively, so that <MATH><I>W</I>=<I>w</I><SUB>1</SUB>+<I>w</I><SUB>2</SUB></MATH>. +<p>(1) Take a weight <I>W</I> of pure gold and weigh it in the fluid. +The apparent loss of weight is then equal to the weight of the +fluid displaced; this is ascertained by weighing. Let it be <I>F</I><SUB>1</SUB>. +<p>It follows that the weight of the fluid displaced by a weight +<I>w</I><SUB>1</SUB> of gold is <MATH>(<I>w</I><SUB>1</SUB>/<I>W</I>).<I>F</I><SUB>1</SUB></MATH>. +<p>(2) Take a weight <I>W</I> of silver, and perform the same +operation. Let the weight of the fluid displaced be <I>F</I><SUB>2</SUB>. +Then the weight of the fluid displaced by a weight <I>w</I><SUB>2</SUB> of +silver is <MATH><I>w</I><SUB>2</SUB>/<I>W</I>.<I>F</I><SUB>2</SUB></MATH>. +<p>(3) Lastly weigh the crown itself in the fluid, and let <I>F</I> be +loss of weight or the weight of the fluid displaced. +<p>We have then <MATH>(<I>w</I><SUB>1</SUB>/<I>W</I>).<I>F</I><SUB>1</SUB>+(<I>w</I><SUB>2</SUB>/<I>W</I>).<I>F</I><SUB>2</SUB>=<I>F,</I></MATH> +that is, <MATH><I>w</I><SUB>1</SUB><I>F</I><SUB>1</SUB>+<I>w</I><SUB>2</SUB><I>F</I><SUB>2</SUB>=(<I>w</I><SUB>1</SUB>+<I>w</I><SUB>2</SUB>)<I>F,</I></MATH> +whence <MATH><I>w</I><SUB>1</SUB>/<I>w</I><SUB>2</SUB>=(<I>F</I><SUB>2</SUB>-<I>F</I>)/(<I>F</I>-<I>F</I><SUB>1</SUB>)</MATH>. +<p>According to the author of the poem <I>de ponderibus et men- +suris</I> (written probably about A.D. 500) Archimedes actually +used a method of this kind. We first take, says our authority, +two equal weights of gold and silver respectively and weigh +them against each other when both are immersed in water; +this gives the relation between their weights in water, and +therefore between their losses of weight in water. Next we +take the mixture of gold and silver and an equal weight of +silver, and weigh them against each other in water in the +same way. +<p>Nevertheless I do not think it probable that this was the +way in which the solution of the problem was <I>discovered.</I> As +we are told that Archimedes discovered it in his bath, and +that he noticed that, if the bath was full when he entered it, +so much water overflowed as was displaced by his body, he is +more likely to have discovered the solution by the alternative +<pb n=94><head>ARCHIMEDES</head> +method attributed to him by Vitruvius,<note><I>De architectura,</I> ix. 3.</note> namely by measuring +successively the <I>volumes</I> of fluid displaced by three equal +weights, (1) the crown, (2) an equal weight of gold, (3) an +equal weight of silver respectively. Suppose, as before, that +the weight of the crown is <I>W</I> and that it contains weights +<I>w</I><SUB>1</SUB> and <I>w</I><SUB>2</SUB> of gold and silver respectively. Then +<p>(1) the crown displaces a certain volume of the fluid, <I>V,</I> say; +<p>(2) the weight <I>W</I> of gold displaces a volume <I>V</I><SUB>1</SUB>, say, of the +fluid; +<p>therefore a weight <I>w</I><SUB>1</SUB> of gold displaces a volume <MATH>(<I>w</I><SUB>1</SUB>/<I>W</I>).<I>V</I><SUB>1</SUB></MATH> of +the fluid; +<p>(3) the weight <I>W</I> of silver displaces <I>V</I><SUB>2</SUB>, say, of the fluid; +therefore a weight <I>w</I><SUB>2</SUB> of silver displaces <MATH>(<I>w</I><SUB>2</SUB>/<I>W</I>).<I>V</I><SUB>2</SUB></MATH>. +<p>It follows that <MATH><I>V</I>=(<I>w</I><SUB>1</SUB>/<I>W</I>).<I>V</I><SUB>1</SUB>+(<I>w</I><SUB>2</SUB>/<I>W</I>).<I>V</I><SUB>2</SUB></MATH>, +whence we derive (since <MATH><I>W</I>=<I>w</I><SUB>1</SUB>+<I>w</I><SUB>2</SUB>) +<I>w</I><SUB>1</SUB>/<I>w</I><SUB>2</SUB>=(<I>V</I><SUB>2</SUB>-<I>V</I>)/(<I>V</I>-<I>V</I><SUB>1</SUB>)</MATH>, +the latter ratio being obviously equal to that obtained by the +other method. +<p>The last propositions (8 and 9) of Book I deal with the case +of any segment of a sphere lighter than a fluid and immersed +in it in such a way that either (1) the curved surface is down- +wards and the base is entirely outside the fluid, or (2) the +curved surface is upwards and the base is entirely submerged, +and it is proved that in either case the segment is in stable +equilibrium when the axis is vertical. This is expressed here +and in the corresponding propositions of Book II by saying +that, ‘if the figure be forced into such a position that the base +of the segment touches the fluid (at one point), the figure will +not remain inclined but will return to the upright position’. +<p>Book II, which investigates fully the conditions of stability +of a right segment of a paraboloid of revolution floating in +a fluid for different values of the specific gravity and different +ratios between the axis or height of the segment and the +<pb n=95><head>ON FLOATING BODIES, I, II</head> +principal parameter of the generating parabola, is a veritable +<I>tour de force</I> which must be read in full to be appreciated. +Prop. 1 is preliminary, to the effect that, if a solid lighter than +a fluid be at rest in it, the weight of the solid will be to that +of the same volume of the fluid as the immersed portion of +the solid is to the whole. The results of the propositions +about the segment of a paraboloid may be thus summarized. +Let <I>h</I> be the axis or height of the segment, <I>p</I> the principal +parameter of the generating parabola, <I>s</I> the ratio of the +specific gravity of the solid to that of the fluid (<I>s</I> always<1). +The segment is supposed to be always placed so that its base +is either entirely above, or entirely below, the surface of the +fluid, and what Archimedes proves in each case is that, if +the segment is so placed with its axis inclined to the vertical +at any angle, it will not rest there but will return to the +position of stability. +<p>I. If <I>h</I> is not greater than 3/4<I>p,</I> the position of stability is with +the axis vertical, whether the curved surface is downwards or +upwards (Props. 2, 3). +<p>II. If <I>h</I> is greater than 3/4<I>p,</I> then, in order that the position of +stability may be with the axis vertical, <I>s</I> must be not less +than <MATH>(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>/<I>h</I><SUP>2</SUP></MATH> if the curved surface is downwards, and not +greater than <MATH>{<I>h</I><SUP>2</SUP>-(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>}/<I>h</I><SUP>2</SUP></MATH> if the curved surface is +upwards (Props. 4, 5). +<p>III. If <MATH><I>h</I>>3/4<I>p,</I></MATH> but <MATH><I>h</I>/(1/2<I>p</I>)<15/4</MATH>, the segment, if placed with +one point of the base touching the surface, will never remain +there whether the curved surface be downwards or upwards +(Props. 6, 7). (The segment will move in the direction of +bringing the axis nearer to the vertical position.) +<p>IV. If <MATH><I>h</I>>3/4<I>p,</I></MATH> but <MATH><I>h</I>/(1/2<I>p</I>)<15/4</MATH>, and if <I>s</I> is less than +<MATH>(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>/<I>h</I><SUP>2</SUP></MATH> in the case where the curved surface is down- +wards, but greater than <MATH>{<I>h</I><SUP>2</SUP>-(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>}/<I>h</I><SUP>2</SUP></MATH> in the case where +the curved surface is upwards, then the position of stability is +one in which the axis is not vertical but inclined to the surface +of the fluid at a certain angle (Props. 8, 9). (The angle is drawn +in an auxiliary figure. The construction for it in Prop. 8 is +equivalent to the solution of the following equation in <G>q</G>, +<MATH>1/4<I>p</I>cot<SUP>2</SUP><G>q</G>=2/3(<I>h</I>-<I>k</I>)-1/2<I>p,</I></MATH> +<pb n=96><head>ARCHIMEDES</head> +where <I>k</I> is the axis of the segment of the paraboloid cut off by +the surface of the fluid.) +<p>V. Prop. 10 investigates the positions of stability in the cases +where <MATH><I>h</I>/(1/2<I>p</I>)>15/4</MATH>, the base is entirely above the surface, and +<I>s</I> has values lying between five pairs of ratios respectively. +Only in the case where <I>s</I> is not less than <MATH>(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>/<I>h</I><SUP>2</SUP></MATH> is the +position of stability that in which the axis is vertical. +<p><I>BAB</I><SUB>1</SUB> is a section of the paraboloid through the axis <I>AM.</I> +<I>C</I> is a point on <I>AM</I> such that <MATH><I>AC</I>=2<I>CM, K</I></MATH> is a point on <I>CA</I> +such that <MATH><I>AM</I>:<I>CK</I>=15:4</MATH>. <I>CO</I> is measured along <I>CA</I> such +that <MATH><I>CO</I>=1/2<I>p</I></MATH>, and <I>R</I> is a point on <I>AM</I> such that <MATH><I>MR</I>=3/2<I>CO.</I></MATH> +<I>A</I><SUB>2</SUB> is the point in which the perpendicular to <I>AM</I> from <I>K</I> +meets <I>AB,</I> and <I>A</I><SUB>3</SUB> is the middle point of <I>AB.</I> <I>BA</I><SUB>2</SUB><I>B</I><SUB>2</SUB>, <I>BA</I><SUB>3</SUB><I>M</I> +are parabolic segments on <I>A</I><SUB>2</SUB><I>M</I><SUB>2</SUB>, <I>A</I><SUB>3</SUB><I>M</I><SUB>3</SUB> (parallel to <I>AM</I>) as axes +<FIG> +and similar to the original segment. (The parabola <I>BA</I><SUB>2</SUB><I>B</I><SUB>2</SUB> +is proved to pass through <I>C</I> by using the above relation +<MATH><I>AM</I>:<I>CK</I>=15:4</MATH> and applying Prop. 4 of the <I>Quadrature of +the Parabola.</I>) The perpendicular to <I>AM</I> from <I>O</I> meets the +parabola <I>BA</I><SUB>2</SUB><I>B</I><SUB>2</SUB> in two points <I>P</I><SUB>2</SUB>, <I>Q</I><SUB>2</SUB>, and straight lines +through these points parallel to <I>AM</I> meet the other para- +bolas in <I>P</I><SUB>1</SUB>, <I>Q</I><SUB>1</SUB> and <I>P</I><SUB>3</SUB>, <I>Q</I><SUB>3</SUB> respectively. <I>P</I><SUB>1</SUB><I>T</I> and <I>Q</I><SUB>1</SUB><I>U</I> are +tangents to the original parabola meeting the axis <I>MA</I> pro- +duced in <I>T, U.</I> Then +<p>(i) if <I>s</I> is not less than <MATH><I>AR</I><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH> or <MATH>(<I>h</I>-3/4<I>p</I>)<SUP>2</SUP>:<I>h</I><SUP>2</SUP></MATH>, there is +stable equilibrium when <I>AM</I> is vertical; +<pb n=97><head>THE CATTLE-PROBLEM</head> +<p>(ii) if <MATH><I>s</I><<I>AR</I><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH> but <MATH>><I>Q</I><SUB>1</SUB><I>Q</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, the solid will not rest +with its base touching the surface of the fluid in one point +only, but in a position with the base entirely out of the fluid +and the axis making with the surface an angle greater +than <I>U</I>; +<p>(iiia) if <MATH><I>s</I>=<I>Q</I><SUB>1</SUB><I>Q</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, there is stable equilibrium with one +point of the base touching the surface and <I>AM</I> inclined to it +at an angle equal to <I>U</I>; +<p>(iiib) if <MATH><I>s</I>=<I>P</I><SUB>1</SUB><I>P</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, there is stable equilibrium with one +point of the base touching the surface and with <I>AM</I> inclined +to it at an angle equal to <I>T</I>; +<p>(iv) if <MATH><I>s</I>><I>P</I><SUB>1</SUB><I>P</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH> but <MATH><<I>Q</I><SUB>1</SUB><I>Q</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, there will be stable +equilibrium in a position in which the base is more submerged; +<p>(v) if <MATH><I>s</I><<I>P</I><SUB>1</SUB><I>P</I><SUB>3</SUB><SUP>2</SUP>:<I>AM</I><SUP>2</SUP></MATH>, there will be stable equilibrium with +the base entirely out of the fluid and with the axis <I>AM</I> +inclined to the surface at an angle less than <I>T.</I> +<p>It remains to mention the traditions regarding other in- +vestigations by Archimedes which have reached us in Greek +or through the Arabic. +<C>(<G>a</G>) <I>The Cattle-Problem.</I></C> +<p>This is a difficult problem in indeterminate analysis. It is +required to find the number of bulls and cows of each of four +colours, or to find 8 unknown quantities. The first part of +the problem connects the unknowns by seven simple equations; +and the second part adds two more conditions to which the +unknowns must be subject. If <I>W, w</I> be the numbers of white +bulls and cows respectively and (<I>X, x</I>), (<I>Y, y</I>), (<I>Z, z</I>) represent +the numbers of the other three colours, we have first the +following equations: +<MATH>(I) <I>W</I>=(1/2+1/3)<I>X</I>+<I>Y,</I> (<G>a</G>) +<I>X</I>=(1/4+1/5)<I>Z</I>+<I>Y,</I> (<G>b</G>) +<I>Z</I>=(1/6+1/7)<I>W</I>+<I>Y</I>, (<G>g</G>) +(II) <I>w</I>=(1/3+1/4)(<I>X</I>+<I>x</I>), (<G>d</G>) +<I>x</I>=(1/4+1/5)(<I>Z</I>+<I>z</I>), (<G>e</G>) +<I>z</I>=(1/5+1/6)(<I>Y</I>+<I>y</I>), (<G>z</G>) +<I>y</I>=(1/6+1/7)(<I>W</I>+<I>w</I>). (<G>h</G>)</MATH> +<pb n=98><head>ARCHIMEDES</head> +<p>Secondly, it is required that +<MATH><I>W</I>+<I>X</I>=a square, (<G>q</G>) +<I>Y</I>+<I>Z</I>=a triangular number. (<G>i</G>)</MATH> +There is an ambiguity in the text which makes it just possible +that <MATH><I>W</I>+<I>X</I></MATH> need only be the product of two whole numbers +instead of a square as in (<G>q</G>). Jul. Fr. Wurm solved the problem +in the simpler form to which this change reduces it. The +complete problem is discussed and partly solved by Amthor.<note><I>Zeitschrift für Math. u. Physik</I> (Hist.-litt. Abt.) xxv. (1880), pp. +156 sqq.</note> +<p>The general solution of the first seven equations is +<MATH><I>W</I>=2 . 3 . 7 . 53 . 4657<I>n</I>=10366482<I>n,</I> +<I>X</I>=2 . 3<SUP>2</SUP> . 89 . 4657<I>n</I>=7460514<I>n,</I> +<I>Y</I>=3<SUP>4</SUP> . 11 . 4657<I>n</I>=4149387<I>n,</I> +<I>Z</I>=2<SUP>2</SUP> . 5 . 79 . 4657<I>n</I>=7358060<I>n,</I> +<I>w</I>=2<SUP>3</SUP> . 3 . 5 . 7 . 23 . 373<I>n</I>=7206360<I>n,</I> +<I>x</I>=2.3<SUP>2</SUP> . 17 . 15991<I>n</I>=4893246<I>n,</I> +<I>y</I>=3<SUP>2</SUP> . 13 . 46489<I>n</I>=5439213<I>n,</I> +<I>z</I>=2<SUP>2</SUP> . 3 . 5 . 7 . 11 . 761<I>n</I>=3515820<I>n.</I></MATH> +<p>It is not difficult to find such a value of <I>n</I> that <MATH><I>W</I>+<I>X</I>=a</MATH> +square number; it is <MATH><I>n</I>=3 . 11 . 29 . 4657<G>x</G><SUP>2</SUP>=4456749<G>x</G><SUP>2</SUP></MATH>, +where <G>x</G> is any integer. We then have to make <MATH><I>Y</I>+<I>Z</I></MATH> +a triangular number, i.e. a number of the form <MATH>1/2<I>q</I>(<I>q</I>+1)</MATH>. +This reduces itself to the solution of the ‘Pellian’ equation +<MATH><I>t</I><SUP>2</SUP>-4729494<I>u</I><SUP>2</SUP>=1</MATH>, +which leads to prodigious figures; one of the eight unknown +quantities alone would have more than 206,500 digits! +<C>(<G>b</G>) <I>On semi-regular polyhedra.</I></C> +<p>In addition, Archimedes investigated polyhedra of a certain +type. This we learn from Pappus.<note>Pappus, v, pp. 352-8.</note> The polyhedra in question +are semi-regular, being contained by equilateral and equi- +<pb n=99><head>ON SEMI-REGULAR POLYHEDRA</head> +angular, but not similar, polygons; those discovered by +Archimedes were 13 in number. If we for convenience +designate a polyhedron contained by <I>m</I> regular polygons +of <G>a</G> sides, <I>n</I> regular polygons of <G>b</G> sides, &c., by (<I>m</I><SUB><G>a</G></SUB>, <I>n</I><SUB><G>b</G></SUB> ...), +the thirteen Archimedean polyhedra, which we will denote by +<I>P</I><SUB>1</SUB>, <I>P</I><SUB>2</SUB> ... <I>P</I><SUB>13</SUB>, are as follows: +<MATH>Figure with 8 faces: <I>P</I><SUB>1</SUB>&equalse3;(4<SUB>3</SUB>, 4<SUB>6</SUB>). +Figures with 14 faces: <I>P</I><SUB>2</SUB>&equalse3;(8<SUB>3</SUB>, 6<SUB>4</SUB>), <I>P</I><SUB>3</SUB>&equalse3;(6<SUB>4</SUB>, 8<SUB>6</SUB>), +<I>P</I><SUB>4</SUB>&equalse3;(8<SUB>3</SUB>, 6<SUB>8</SUB>). +Figures with 26 faces: <I>P</I><SUB>5</SUB>&equalse3;(8<SUB>3</SUB>, 18<SUB>4</SUB>), <I>P</I><SUB>6</SUB>&equalse3;(12<SUB>4</SUB>, 8<SUB>6</SUB>, 6<SUB>8</SUB>). +Figures with 32 faces: <I>P</I><SUB>7</SUB>&equalse3;(20<SUB>3</SUB>, 12<SUB>5</SUB>), <I>P</I><SUB>8</SUB>&equalse3;(12<SUB>5</SUB>, 20<SUB>6</SUB>), +<I>P</I><SUB>9</SUB>&equalse3;(20<SUB>3</SUB>, 12<SUB>10</SUB>). +Figure with 38 faces: <I>P</I><SUB>10</SUB>&equalse3;(32<SUB>3</SUB>, 6<SUB>4</SUB>). +Figures with 62 faces: <I>P</I><SUB>11</SUB>&equalse3;(20<SUB>3</SUB>, 30<SUB>4</SUB>, 12<SUB>5</SUB>), +<I>P</I><SUB>12</SUB>&equalse3;(30<SUB>4</SUB>, 20<SUB>6</SUB>, 12<SUB>10</SUB>). +Figure with 92 faces: <I>P</I><SUB>13</SUB>&equalse3;(80<SUB>3</SUB>, 12<SUB>5</SUB>)</MATH>. +<p>Kepler<note>Kepler, <I>Harmonice mundi</I> in <I>Opera</I> (1864), v, pp. 123-6.</note> showed how these figures can be obtained. A +method of obtaining some of them is indicated in a fragment +of a scholium to the Vatican MS. of Pappus. If a solid +angle of one of the regular solids be cut off symmetrically by +a plane, i.e. in such a way that the plane cuts off the same +length from each of the edges meeting at the angle, the +section is a regular polygon which is a triangle, square or +pentagon according as the solid angle is formed of three, four, +or five plane angles. If certain equal portions be so cut off +from all the solid angles respectively, they will leave regular +polygons inscribed in the faces of the solid; this happens +(A) when the cutting planes bisect the sides of the faces and +so leave in each face a polygon of the same kind, and (B) when +the cutting planes cut off a smaller portion from each angle in +such a way that a regular polygon is left in each face which +has double the number of sides (as when we make, say, an +octagon out of a square by cutting off the necessary portions, +<pb n=100><head>ARCHIMEDES</head> +symmetrically, from the corners). We have seen that, accord- +ing to Heron, two of the semi-regular solids had already been +discovered by Plato, and this would doubtless be his method. +The methods (A) and (B) applied to the five regular solids +give the following out of the 13 semi-regular solids. We +obtain (1) from the tetrahedron, <I>P</I><SUB>1</SUB> by cutting off angles +so as to leave hexagons in the faces; (2) from the cube, <I>P</I><SUB>2</SUB> by +leaving squares, and <I>P</I><SUB>4</SUB> by leaving octagons, in the faces; +(3) from the octahedron, <I>P</I><SUB>2</SUB> by leaving triangles, and <I>P</I><SUB>3</SUB> by +leaving hexagons, in the faces; (4) from the icosahedron, +<I>P</I><SUB>7</SUB> by leaving triangles, and <I>P</I><SUB>8</SUB> by leaving hexagons, in the +faces; (5) from the dodecahedron, <I>P</I><SUB>7</SUB> by leaving pentagons, +and <I>P</I><SUB>9</SUB> by leaving decagons in the faces. +<p>Of the remaining six, four are obtained by cutting off all +the edges symmetrically and equally by planes parallel to the +edges, and then cutting off angles. Take first the cube. +(1) Cut off from each four parallel edges portions which leave +an octagon as the section of the figure perpendicular to the +edges; then cut off equilateral triangles from the corners +(see Fig. 1); this gives <I>P</I><SUB>5</SUB> containing 8 equilateral triangles +and 18 squares. (<I>P</I><SUB>5</SUB> is also obtained by bisecting all the +edges of <I>P</I><SUB>2</SUB> and cutting off corners.) (2) Cut off from the +edges of the cube a smaller portion so as to leave in each +face a square such that the octagon described in it has its +side equal to the breadth of the section in which each edge is +cut; then cut off hexagons from each angle (see Fig. 2); this +<FIG> +<CAP>FIG. 1.</CAP> +<FIG> +<CAP>FIG. 2.</CAP> +gives 6 octagons in the faces, 12 squares under the edges and +8 hexagons at the corners; that is, we have <I>P</I><SUB>6</SUB>. An exactly +<pb n=101><head>ON SEMI-REGULAR POLYHEDRA</head> +similar procedure with the icosahedron and dodecahedron +produces <I>P</I><SUB>11</SUB> and <I>P</I><SUB>12</SUB> (see Figs. 3, 4 for the case of the icosa- +hedron). +<FIG> +<CAP>FIG. 3.</CAP> +<FIG> +<CAP>FIG. 4.</CAP> +<p>The two remaining solids <I>P</I><SUB>10</SUB>, <I>P</I><SUB>13</SUB> cannot be so simply pro- +duced. They are represented in Figs. 5, 6, which I have +<FIG> +<CAP>FIG. 5.</CAP> +<FIG> +<CAP>FIG. 6.</CAP> +taken from Kepler. <I>P</I><SUB>10</SUB> is the <I>snub cube</I> in which each +solid angle is formed by the angles of four equilateral triangles +and one square; <I>P</I><SUB>13</SUB> is the <I>snub dodecahedron,</I> each solid +angle of which is formed by the angles of four equilateral +triangles and <I>one regular pentagon.</I> +<p>We are indebted to Arabian tradition for +<C>(<G>g</G>) <I>The Liber Assumptorum.</I></C> +<p>Of the theorems contained in this collection many are +so elegant as to afford a presumption that they may really +be due to Archimedes. In three of them the figure appears +which was called <G>a)/rbhlos</G>, a shoemaker's knife, consisting of +three semicircles with a common diameter as shown in the +annexed figure. If <I>N</I> be the point at which the diameters +<pb n=102><head>ARCHIMEDES</head> +of the two smaller semicircles adjoin, and <I>NP</I> be drawn at +right angles to <I>AB</I> meeting the external semicircle in <I>P,</I> the +area of the <G>a)/rbhlos</G> (included between the three semicircular +arcs) is equal to the circle on <I>PN</I> as diameter (Prop. 4). In +Prop. 5 it is shown that, if a circle be described in the space +between the arcs <I>AP, AN</I> and the straight line <I>PN</I> touching +<FIG> +all three, and if a circle be similarly described in the space +between the arcs <I>PB, NB</I> and the straight line <I>PN</I> touching +all three, the two circles are equal. If one circle be described +in the <G>a)/rbhlos</G> touching all three semicircles, Prop. 6 shows +that, if the ratio of <I>AN</I> to <I>NB</I> be given, we can find the +relation between the diameter of the circle inscribed to the +<G>a)/rbhlos</G> and the straight line <I>AB</I>; the proof is for the parti- +cular case <MATH><I>AN</I>=3/2<I>BN</I></MATH>, and shows that the diameter of the +inscribed <MATH>circle=6/19<I>AB</I></MATH>. +<p>Prop. 8 is of interest in connexion with the problem of +<FIG> +trisecting any angle. If <I>AB</I> be any chord of a circle with +centre <I>O,</I> and <I>BC</I> on <I>AB</I> produced be made equal to the radius, +draw <I>CO</I> meeting the circle in <I>D, E</I>; then will the arc <I>BD</I> be +one-third of the arc <I>AE</I> (or <I>BF,</I> if <I>EF</I> be the chord through <I>E</I> +parallel to <I>AB</I>). The problem is by this theorem reduced to +a <G>neu=sis</G> (cf. vol. i, p. 241). +<pb n=103><head>THE <I>LIBER ASSUMPTORUM</I></head> +<p>Lastly, we may mention the elegant theorem about the +area of the <I>Salinon</I> (presumably ‘salt-cellar’) in Prop. 14. +<I>ACB</I> is a semicircle on <I>AB</I> as diameter, <I>AD, EB</I> are equal +lengths measured from <I>A</I> and <I>B</I> on <I>AB.</I> Semicircles are +drawn with <I>AD, EB</I> as diameters on the side towards <I>C,</I> and +<FIG> +a semicircle with <I>DE</I> as diameter is drawn on the other side of +<I>AB. CF</I> is the perpendicular to <I>AB</I> through <I>O,</I> the centre +of the semicircles <I>ACB, DFE.</I> Then is the area bounded by +all the semicircles (the <I>Salinon</I>) equal to the circle on <I>CF</I> +as diameter. +<p>The Arabians, through whom the Book of Lemmas has +reached us, attributed to Archimedes other works (1) on the +Circle, (2) on the Heptagon in a Circle, (3) on Circles touch- +ing one another, (4) on Parallel Lines, (5) on Triangles, (6) on +the properties of right-angled triangles, (7) a book of Data, +(8) De clepsydris: statements which we are not in a position +to check. But the author of a book on the finding of chords +in a circle<note>See <I>Bibliotheca mathematica,</I> xi<SUB>3</SUB>, pp. 11-78.</note> Abū'l Raihān Muh. al-Bīrūnī, quotes some alterna- +tive proofs as coming from the first of these works. +<C>(<G>d</G>) <I>Formula for area of triangle.</I></C> +<p>More important, however, is the mention in this same work +of Archimedes as the discoverer of two propositions hitherto +attributed to Heron, the first being the problem of finding +the perpendiculars of a triangle when the sides are given, and +the second the famous formula for the area of a triangle in +terms of the sides, +<MATH>√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)}</MATH>. +<pb n=104><head>ERATOSTHENES</head> +<p>Long as the present chapter is, it is nevertheless the most +appropriate place for ERATOSTHENES of Cyrene. It was to him +that Archimedes dedicated <I>The Method,</I> and the <I>Cattle-Problem</I> +purports, by its heading, to have been sent through him to +the mathematicians of Alexandria. It is evident from the +preface to <I>The Method</I> that Archimedes thought highly of his +mathematical ability. He was, indeed, recognized by his con- +temporaries as a man of great distinction in all branches of +knowledge, though in each subject he just fell short of the +highest place. On the latter ground he was called Beta, and +another nickname applied to him, <I>Pentathlos,</I> has the same +implication, representing as it does an all-round athlete who +was not the first runner or wrestler but took the second prize +in these contests as well as in others. He was very little +younger than Archimedes; the date of his birth was probably +284 B.C. or thereabouts. He was a pupil of the philosopher +Ariston of Chios, the grammarian Lysanias of Cyrene, and +the poet Callimachus; he is said also to have been a pupil of +Zeno the Stoic, and he may have come under the influence of +Arcesilaus at Athens, where he spent a considerable time. +Invited, when about 40 years of age, by Ptolemy Euergetes +to be tutor to his son (Philopator), he became librarian at +Alexandria; his obligation to Ptolemy he recognized by the +column which he erected with a graceful epigram inscribed on +it. This is the epigram, with which we are already acquainted +(vol. i, p. 260), relating to the solutions, discovered up to date, +of the problem of the duplication of the cube, and commend- +ing his own method by means of an appliance called <G>meso/labon</G>, +itself represented in bronze on the column. +<p>Eratosthenes wrote a book with the title <G>*platwniko/s</G>, and, +whether it was a sort of commentary on the <I>Timaeus</I> of +Plato, or a dialogue in which the principal part was played by +Plato, it evidently dealt with the fundamental notions of +mathematics in connexion with Plato's philosophy. It was +naturally one of the important sources of Theon of Smyrna's +work on the mathematical matters which it was necessary for +the student of Plato to know; and Theon cites the work +twice by name. It seems to have begun with the famous +problem of Delos, telling the story quoted by Theon how the +god required, as a means of stopping a plague, that the altar +<pb n=105><head><I>PLATONICUS</I> AND <I>ON MEANS</I></head> +there, which was cubical in form, should be doubled in size. +The book evidently contained a disquisition on <I>proportion</I> +(<G>a)nalogi/a</G>); a quotation by Theon on this subject shows that +Eratosthenes incidentally dealt with the fundamental defini- +tions of geometry and arithmetic. The principles of music +were discussed in the same work. +<p>We have already described Eratosthenes's solution of the +problem of Delos, and his contribution to the theory of arith- +metic by means of his <I>sieve</I> (<G>ko/skinon</G>) for finding successive +prime numbers. +<p>He wrote also an independent work <I>On means.</I> This was in +two Books, and was important enough to be mentioned by +Pappus along with works by Euclid, Aristaeus and Apol- +lonius as forming part of the <I>Treasury of Analysis</I><note>Pappus, vii, p. 636. 24.</note>; this +proves that it was a systematic geometrical treatise. Another +passage of Pappus speaks of certain loci which Eratosthenes +called ‘loci with reference to means’ (<G>to/poi pro\s meso/thtas</G>)<note><I>Ib.,</I> p. 662. 15 sq.</note>; +these were presumably discussed in the treatise in question. +What kind of loci these were is quite uncertain; Pappus (if it +is not an interpolator who speaks) merely says that these loci +‘belong to the aforesaid classes of loci’, but as the classes are +numerous (including ‘plane’, ‘solid’, ‘linear’, ‘loci on surfaces’, +&c.), we are none the wiser. Tannery conjectured that they +were loci of points such that their distances from three fixed +straight lines furnished a ‘médiété’, i.e. loci (straight lines +and conics) which we should represent in trilinear coordinates +by such equations as <MATH>2<I>y</I>=<I>x</I>+<I>z, y</I><SUP>2</SUP>=<I>xz, y</I>(<I>x</I>+<I>z</I>)=2<I>xz,</I> +<I>x</I>(<I>x</I>-<I>y</I>)=<I>z</I>(<I>y</I>-<I>z</I>), <I>x</I>(<I>x</I>-<I>y</I>)=<I>y</I>(<I>y</I>-<I>z</I>)</MATH>, the first three equations +representing the arithmetic, geometric and harmonic means, +while the last two represent the ‘subcontraries’ to the +harmonic and geometric means respectively. Zeuthen has +a different conjecture.<note>Zeuthen, <I>Die Lehre von den Kegelschnitten im Altertum,</I> 1886, pp. +320, 321.</note> He points out that, if <I>QQ</I>′ be the +polar of a given point <I>C</I> with reference to a conic, and <I>CPOP</I>′ +be drawn through <I>C</I> meeting <I>QQ</I>′ in <I>O</I> and the conic in <I>P, P</I>′, +then <I>CO</I> is the harmonic mean to <I>CP, CP</I>′; the locus of <I>O</I> for +all transversals <I>CPP</I>′ is then the straight line <I>QQ</I>′. If <I>A, G</I> +are points on <I>PP</I>′ such that <I>CA</I> is the arithmetic, and <I>CG</I> the +<pb n=106><head>ERATOSTHENES</head> +geometric mean between <I>CP, CP</I>′, the loci of <I>A, G</I> respectively +are conics. Zeuthen therefore suggests that these loci and +the corresponding loci of the points on <I>CPP</I>′ at a distance +from <I>C</I> equal to the subcontraries of the geometric and +harmonic means between <I>CP</I> and <I>CP</I>′ are the ‘loci with +reference to means’ of Eratosthenes; the latter two loci are +‘linear’, i.e. higher curves than conics. Needless to say, we +have no confirmation of this conjecture. +<C><I>Eratosthenes's measurement of the Earth.</I></C> +<p>But the most famous scientific achievement of Eratosthenes +was his measurement of the earth. Archimedes mentions, as +we have seen, that some had tried to prove that the circum- +ference of the earth is about 300,000 stades. This was +evidently the measurement based on observations made at +Lysimachia (on the Hellespont) and Syene. It was observed +that, while both these places were on one meridian, the head +of Draco was in the zenith at Lysimachia, and Cancer in the +zenith at Syene; the arc of the meridian separating the two +in the heavens was taken to be 1/15th of the complete circle. +<FIG> +The distance between the two towns +was estimated at 20,000 stades, and +accordingly the whole circumference of +the earth was reckoned at 300,000 +stades. Eratosthenes improved on this. +He observed (1) that at Syene, at +noon, at the summer solstice, the +sun cast no shadow from an upright +gnomon (this was confirmed by the +observation that a well dug at the +same place was entirely lighted up at +the same time), while (2) at the same moment the gnomon fixed +upright at Alexandria (taken to be on the same meridian with +Syene) cast a shadow corresponding to an angle between the +gnomon and the sun's rays of 1/50th of a complete circle or +four right angles. The sun's rays are of course assumed to be +parallel at the two places represented by <I>S</I> and <I>A</I> in the +annexed figure. If <G>a</G> be the angle made at <I>A</I> by the sun's rays +with the gnomon (<I>OA</I> produced), the angle <I>SOA</I> is also equal to +<pb n=107><head>MEASUREMENT OF THE EARTH</head> +<G>a</G>, or 1/50th of four right angles. Now the distance from <I>S</I> +to <I>A</I> was known by measurement to be 5,000 stades; it +followed that the circumference of the earth was 250,000 +stades. This is the figure given by Cleomedes, but Theon of +Smyrna and Strabo both give it as 252,000 stades. The +reason of the discrepancy is not known; it is possible that +Eratosthenes corrected 250,000 to 252,000 for some reason, +perhaps in order to get a figure divisible by 60 and, inci- +dentally, a round number (700) of stades for one degree. If +Pliny is right in saying that Eratosthenes made 40 stades +equal to the Egyptian <G>sxoi=nos</G>, then, taking the <G>sxoi=nos</G> at +12,000 Royal cubits of 0.525 metres, we get 300 such cubits, +or 157.5 metres, i.e. 516.73 feet, as the length of the stade. +On this basis 252,000 stades works out to 24,662 miles, and +the diameter of the earth to about 7,850 miles, only 50 miles +shorter than the true polar diameter, a surprisingly close +approximation, however much it owes to happy accidents +in the calculation. +<p>We learn from Heron's <I>Dioptra</I> that the measurement of +the earth by Eratosthenes was given in a separate work <I>On +the Measurement of the Earth.</I> According to Galen<note>Galen, <I>Instit. Logica,</I> 12 (p. 26 Kalbfleisch).</note> this work +dealt generally with astronomical or mathematical geography, +treating of ‘the size of the equator, the distance of the tropic +and polar circles, the extent of the polar zone, the size and +distance of the sun and moon, total and partial eclipses of +these heavenly bodies, changes in the length of the day +according to the different latitudes and seasons’. Several +details are preserved elsewhere of results obtained by +Eratosthenes, which were doubtless contained in this work. +He is supposed to have estimated the distance between the +tropic circles or twice the obliquity of the ecliptic at 11/83rds +of a complete circle or 47° 42′ 39″; but from Ptolemy's +language on this subject it is not clear that this estimate was +not Ptolemy's own. What Ptolemy says is that he himself +found the distance between the tropic circles to lie always +between 47° 40′ and 47° 45′, ‘from which we obtain <I>about</I> +(<G>sxedo/n</G>) the same ratio as that of Eratosthenes, which +Hipparchus also used. For the distance between the tropics +becomes (or <I>is found to be,</I> <G>gi/netai</G>) very nearly 11 parts +<pb n=108><head>ERATOSTHENES</head> +out of 83 contained in the whole meridian circle’.<note>Ptolemy, <I>Syntaxis,</I> i. 12, pp. 67. 22-68. 6.</note> The +mean of Ptolemy's estimates, 47° 42′ 30″, is of course nearly +11/83rds of 360°. It is consistent with Ptolemy's language +to suppose that Eratosthenes adhered to the value of the +obliquity of the ecliptic discovered before Euclid's time, +namely 24°, and Hipparchus does, in his extant <I>Commentary +on the Phaenomena of Aratus and Eudoxus,</I> say that the +summer tropic is ‘very nearly 24° north of the equator’. +<p>The <I>Doxographi</I> state that Eratosthenes estimated the +distance of the moon from the earth at 780,000 stades and +the distance of the sun from the earth at 804,000,000 stades +(the versions of Stobaeus and Joannes Lydus admit 4,080,000 +as an alternative for the latter figure, but this obviously +cannot be right). Macrobius<note>Macrobius, <I>In Somn. Scip.</I> i. 20. 9.</note> says that Eratosthenes made +the ‘measure’ of the sun to be 27 times that of the earth. +It is not certain whether measure means ‘solid content’ or +‘diameter’ in this case; the other figures on record make the +former more probable, in which case the diameter of the sun +would be three times that of the earth. Macrobius also tells +us that Eratosthenes's estimates of the distances of the sun +and moon were obtained by means of lunar eclipses. +<p>Another observation by Eratosthenes, namely that at Syene +(which is under the summer tropic) and throughout a circle +round it with a radius of 300 stades the upright gnomon +throws no shadow at noon, was afterwards made use of by +Posidonius in his calculation of the size of the sun. Assuming +that the circle in which the sun apparently moves round the +earth is 10,000 times the size of a circular section of the earth +through its centre, and combining with this hypothesis the +datum just mentioned, Posidonius arrived at 3,000,000 stades +as the diameter of the sun. +<p>Eratosthenes wrote a poem called <I>Hermes</I> containing a good +deal of descriptive astronomy; only fragments of this have +survived. The work <I>Catasterismi</I> (literally ‘placings among +the stars’) which is extant can hardly be genuine in the form +in which it has reached us; it goes back, however, to a genuine +work by Eratosthenes which apparently bore the same name; +alternatively it is alluded to as <G>*kata/logoi</G> or by the general +<pb n=109><head>ASTRONOMY, ETC.</head> +word <G>*)astronomi/a</G> (Suidas), which latter word is perhaps a mis- +take for <G>*)astroqesi/a</G> corresponding to the title <G>*)astroqesi/ai +zw|di/wn</G> found in the manuscripts. The work as we have it +contains the story, mythological and descriptive, of the con- +stellations, &c., under forty-four heads; there is little or +nothing belonging to astronomy proper. +<p>Eratosthenes is also famous as the first to attempt a scientific +chronology beginning from the siege of Troy; this was the +subject of his <G>*xronografi/ai</G>, with which must be connected +the separate <G>*)olumpioni=kai</G> in several books. Clement of +Alexandria gives a short <I>résumé</I> of the main results of the +former work, and both works were largely used by Apollo- +dorus. Another lost work was on the Octaëteris (or eight- +years' period), which is twice mentioned, by Geminus and +Achilles; from the latter we learn that Eratosthenes re- +garded the work on the same subject attributed to Eudoxus +as not genuine. His <I>Geographica</I> in three books is mainly +known to us through Suidas's criticism of it. It began with +a history of geography down to his own time; Eratosthenes +then proceeded to mathematical geography, the spherical form +of the earth, the negligibility in comparison with this of the +unevennesses caused by mountains and valleys, the changes of +features due to floods, earthquakes and the like. It would +appear from Theon of Smyrna's allusions that Eratosthenes +estimated the height of the highest mountain to be 10 stades +or about 1/8000th part of the diameter of the earth. +<pb><C>XIV</C> +<C>CONIC SECTIONS. APOLLONIUS OF PERGA</C> +<C>A. HISTORY OF CONICS UP TO APOLLONIUS</C> +<C>Discovery of the conic sections by Menaechmus.</C> +<p>WE have seen that Menaechmus solved the problem of the +two mean proportionals (and therefore the duplication of +the cube) by means of conic sections, and that he is credited +with the discovery of the three curves; for the epigram of +Eratosthenes speaks of ‘the <I>triads</I> of Menaechmus’, whereas +of course only two conics, the parabola and the rectangular +hyperbola, actually appear in Menaechmus's solutions. The +question arises, how did Menaechmus come to think of obtain- +ing curves by cutting a cone? On this we have no informa- +tion whatever. Democritus had indeed spoken of a section of +a cone parallel and very near to the base, which of course +would be a circle, since the cone would certainly be the right +circular cone. But it is probable enough that the attention +of the Greeks, whose observation nothing escaped, would be +attracted to the shape of a section of a cone or a cylinder by +a plane obliquely inclined to the axis when it occurred, as it +often would, in real life; the case where the solid was cut +right through, which would show an ellipse, would presum- +ably be noticed first, and some attempt would be made to +investigate the nature and geometrical measure of the elonga- +tion of the figure in relation to the circular sections of the +same solid; these would in the first instance be most easily +ascertained when the solid was a right cylinder; it would +then be a natural question to investigate whether the curve +arrived at by cutting the cone had the same property as that +obtained by cutting the cylinder. As we have seen, the +<pb n=111><head>DISCOVERY OF THE CONIC SECTIONS</head> +observation that an ellipse can be obtained from a cylinder +as well as a cone is actually made by Euclid in his <I>Phaeno- +mena</I>: ‘if’, says Euclid, ‘a cone or a cylinder be cut by +a plane not parallel to the base, the resulting section is a +section of an acute-angled cone which is similar to a <G>qureo/s</G> +(shield).’ After this would doubtless follow the question +what sort of curves they are which are produced if we +cut a cone by a plane which does not cut through the +cone completely, but is either parallel or not parallel to +a generator of the cone, whether these curves have the +same property with the ellipse and with one another, and, +if not, what exactly are their fundamental properties respec- +tively. +<p>As it is, however, we are only told how the first writers on +conics obtained them in actual practice. We learn on the +authority of Geminus<note>Eutocius, <I>Comm. on Conics</I> of Apollonius.</note> that the ancients defined a cone as the +surface described by the revolution of a right-angled triangle +about one of the sides containing the right angle, and that +they knew no cones other than right cones. Of these they +distinguished three kinds; according as the vertical angle of +the cone was less than, equal to, or greater than a right angle, +they called the cone acute-angled, right-angled, or obtuse- +angled, and from each of these kinds of cone they produced +one and only one of the three sections, the section being +always made perpendicular to one of the generating lines of +the cone; the curves were, on this basis, called ‘section of an +acute-angled cone’ (= an ellipse), ‘section of a right-angled +cone’ (= a parabola), and ‘section of an obtuse-angled cone’ +(= a hyperbola) respectively. These names were still used +by Euclid and Archimedes. +<C><I>Menaechmus's probable procedure.</I></C> +<p>Menaechmus's constructions for his curves would presum- +ably be the simplest and the most direct that would show the +desired properties, and for the parabola nothing could be +simpler than a section of a right-angled cone by a plane at right +angles to one of its generators. Let <I>OBC</I> (Fig. 1) represent +<pb n=112><head>CONIC SECTIONS</head> +a section through the axis <I>OL</I> of a right-angled cone, and +conceive a section through <I>AG</I> (perpendicular to <I>OA</I>) and at +right angles to the plane of the paper. +<FIG> +<CAP>FIG. 1.</CAP> +<p>If <I>P</I> is any point on the curve, and <I>PN</I> perpendicular to +<I>AG</I>, let <I>BC</I> be drawn through <I>N</I> perpendicular to the axis of +the cone. Then <I>P</I> is on the circular section of the cone about +<I>BC</I> as diameter. +<p>Draw <I>AD</I> parallel to <I>BC</I>, and <I>DF, CG</I> parallel to <I>OL</I> meet- +ing <I>AL</I> produced in <I>F, G.</I> Then <I>AD, AF</I> are both bisected +by <I>OL.</I> +<p>If now <MATH><I>PN</I>=<I>y</I>, <I>AN</I>=<I>x</I>, +<I>y</I><SUP>2</SUP>=<I>PN</I><SUP>2</SUP>=<I>BN.NC</I></MATH>. +<p>But <I>B, A, C, G</I> are concyclic, so that +<MATH><I>BN.NC</I>=<I>AN.NG</I> +=<I>AN.AF</I> +=<I>AN</I>.2<I>AL</I></MATH>. +<p>Therefore <MATH><I>y</I><SUP>2</SUP>=<I>AN</I>.2<I>AL</I> +=2<I>AL.x</I></MATH>, +and 2<I>AL</I> is the ‘parameter’ of the principal ordinates <I>y.</I> +<p>In the case of the hyperbola Menaechmus had to obtain the +<pb n=113><head>MENAECHMUS'S PROCEDURE</head> +particular hyperbola which we call rectangular or equilateral, +and also to obtain its property with reference to its asymp- +totes, a considerable advance on what was necessary in the +case of the parabola. Two methods of obtaining the particular +hyperbola were possible, namely (1) to obtain the hyperbola +arising from the section of any obtuse-angled cone by a plane +at right angles to a generator, and then to show how a +rectangular hyperbola can be obtained as a particular case +by finding the vertical angle which the cone must have to +give a rectangular hyperbola when cut in the particular way, +or (2) to obtain the rectangular hyperbola direct by cutting +another kind of cone by a section not necessarily perpen- +dicular to a generator. +<p>(1) Taking the first method, we draw (Fig. 2) a cone with its +vertical angle <I>BOC</I> obtuse. Imagine a section perpendicular +to the plane of the paper and passing through <I>AG</I> which is +perpendicular to <I>OB</I>. Let <I>GA</I> produced meet <I>CO</I> produced in +<I>A</I>′, and complete the same construction as in the case of +the parabola. +<FIG> +<CAP>FIG. 2.</CAP> +<p>In this case we have +<MATH><I>PN</I><SUP>2</SUP>=<I>BN.NC</I>=<I>AN.NG</I></MATH>. +<pb n=114><head>CONIC SECTIONS</head> +<p>But, by similar triangles, +<MATH><I>NG</I>:<I>AF</I>=<I>NC</I>:<I>AD</I> +=<I>A</I>′<I>N</I>:<I>AA</I>′</MATH>. +<p>Hence <MATH><I>PN</I><SUP>2</SUP>=<I>AN.A</I>′<I>N.AF</I>/<I>AA</I>′ +=(2<I>AL</I>)/<I>AA</I>′.<I>AN.A</I>′<I>N</I></MATH>, +which is the property of the hyperbola, <I>AA</I>′ being what we +call the transverse axis, and 2 <I>AL</I> the parameter of the principal +ordinates. +<p>Now, in order that the hyperbola may be rectangular, we +must have 2 <I>AL</I>:<I>AA</I>′ equal to 1. The problem therefore now +is: given a straight line <I>AA</I>′, and <I>AL</I> along <I>A</I>′<I>A</I> produced +equal to 1/2<I>AA</I>′, to find a cone such that <I>L</I> is on its axis and +the section through <I>AL</I> perpendicular to the generator through +<I>A</I> is a rectangular hyperbola with <I>A</I>′<I>A</I> as transverse axis. In +other words, we have to find a point <I>O</I> on the straight line +through <I>A</I> perpendicular to <I>AA</I>′ such that <I>OL</I> bisects the +angle which is the supplement of the angle <I>A</I>′<I>OA.</I> +<p>This is the case if <MATH><I>A</I>′<I>O</I>:<I>OA</I>=<I>A</I>′<I>L</I>:<I>LA</I>=3:1</MATH>; +therefore <I>O</I> is on the circle which is the locus of all points +such that their distances from the two fixed points <I>A</I>′, <I>A</I> +are in the ratio 3:1. This circle is the circle on <I>KL</I> as +diameter, where <MATH><I>A</I>′<I>K</I>:<I>KA</I>=<I>A</I>′<I>L</I>:<I>LA</I>=3:1</MATH>. Draw this +circle, and <I>O</I> is then determined as the point in which <I>AO</I> +drawn perpendicular to <I>AA</I>′ intersects the circle. +<p>It is to be observed, however, that this deduction of a +particular from a more general case is not usual in early +Greek mathematics; on the contrary, the particular usually +led to the more general. Notwithstanding, therefore, that the +orthodox method of producing conic sections is said to have +been by cutting the generator of each cone perpendicularly, +I am inclined to think that Menaechmus would get his rect- +angular hyperbola directly, and in an easier way, by means of +a different cone differently cut. Taking the right-angled cone, +already used for obtaining a parabola, we have only to make +a section parallel to the axis (instead of perpendicular to a +generator) to get a rectangular hyperbola. +<pb n=115><head>MENAECHMUS'S PROCEDURE</head> +<p>For, let the right-angled cone <I>HOK</I> (Fig. 3) be cut by a +plane through <I>A</I>′<I>AN</I> parallel +to the axis <I>OM</I> and cutting the +sides of the axial triangle <I>HOK</I> +in <I>A</I>′, <I>A, N</I> respectively. Let +<I>P</I> be the point on the curve +for which <I>PN</I> is the principal +ordinate. Draw <I>OC</I> parallel +to <I>HK.</I> We have at once +<FIG> +<CAP>FIG. 3.</CAP> +<MATH><I>PN</I><SUP>2</SUP>=<I>HN.NK</I> +=<I>MK</I><SUP>2</SUP>-<I>MN</I><SUP>2</SUP> +=<I>CN</I><SUP>2</SUP>-<I>CA</I><SUP>2</SUP></MATH>, since <MATH><I>MK</I>=<I>OM</I></MATH>, and <MATH><I>MN</I>=<I>OC</I>=<I>CA</I></MATH>. +This is the property of the rectangular hyperbola having <I>A</I>′<I>A</I> +as axis. To obtain a particular rectangular hyperbola with +axis of given length we have only to choose the cutting plane +so that the intercept <I>A</I>′<I>A</I> may have the given length. +<p>But Menaechmus had to prove the asymptote-property of +his rectangular hyperbola. As he can hardly be supposed to +have got as far as Apollonius in investigating the relations of +the hyperbola to its asymptotes, it is probably safe to assume +that he obtained the particular property in the simplest way, +i.e. directly from the property of the curve in relation to +its axes. +<FIG> +<CAP>FIG. 4.</CAP> +<p>If (Fig. 4) <I>CR, CR</I>′ be the asymptotes (which are therefore +<pb n=116><head>CONIC SECTIONS</head> +at right angles) and <I>A</I>′<I>A</I> the axis of a rectangular hyperbola, +<I>P</I> any point on the curve, <I>PN</I> the principal ordinate, draw +<I>PK, PK</I>′ perpendicular to the asymptotes respectively. Let +<I>PN</I> produced meet the asymptotes in <I>R, R</I>′. +<p>Now, by the axial property, +<MATH><I>CA</I><SUP>2</SUP>=<I>CN</I><SUP>2</SUP>-<I>PN</I><SUP>2</SUP> +=<I>RN</I><SUP>2</SUP>-<I>PN</I><SUP>2</SUP> +=<I>RP.PR</I>′ +=2<I>PK.PK</I>′</MATH>, since ∠<I>PRK</I> is half a right angle; +therefore <MATH><I>PK.PK</I>′=1/2<I>CA</I><SUP>2</SUP></MATH>. +<C>Works by Aristaeus and Euclid.</C> +<p>If Menaechmus was really the discoverer of the three conic +sections at a date which we must put at about 360 or 350 B.C., +the subject must have been developed very rapidly, for by the +end of the century there were two considerable works on +conics in existence, works which, as we learn from Pappus, +were considered worthy of a place, alongside the <I>Conics</I> of +Apollonius, in the <I>Treasury of Analysis.</I> Euclid flourished +about 300 B.C., or perhaps 10 or 20 years earlier; but his +<I>Conics</I> in four books was preceded by a work of Aristaeus +which was still extant in the time of Pappus, who describes it +as ‘five books of <I>Solid Loci</I> connected (or continuous, <G>sunexh=</G>) +with the conics’. Speaking of the relation of Euclid's <I>Conics</I> +in four books to this work, Pappus says (if the passage is +genuine) that Euclid gave credit to Aristaeus for his dis- +coveries in conics and did not attempt to anticipate him or +wish to construct anew the same system. In particular, +Euclid, when dealing with what Apollonius calls the three- +and four-line locus, ‘wrote so much about the locus as was +possible by means of the conics of Aristaeus, without claiming +completeness for his demonstrations’.<note>Pappus, vii, p. 678. 4.</note> We gather from these +remarks that Euclid's <I>Conics</I> was a compilation and rearrange- +ment of the geometry of the conics so far as known in his +<pb n=117><head>WORKS BY ARISTAEUS AND EUCLID</head> +time, whereas the work of Aristaeus was more specialized and +more original. +<C>‘<I>Solid loci</I>’ <I>and</I> ‘<I>solid problems</I>’.</C> +<p>‘Solid loci’ are of course simply conics, but the use of the +title ‘Solid loci’ instead of ‘conics’ seems to indicate that +the work was in the main devoted to conics regarded as loci. +As we have seen, ‘solid loci’ which are conics are distinguished +from ‘plane loci’, on the one hand, which are straight lines +and circles, and from ‘linear loci’ on the other, which are +curves higher than conics. There is some doubt as to the +real reason why the term ‘solid loci’ was applied to the conic +sections. We are told that ‘plane’ loci are so called because +they are generated in a plane (but so are some of the higher +curves, such as the <I>quadratrix</I> and the spiral of Archimedes), +and that ‘solid loci’ derived their name from the fact that +they arise as sections of solid figures (but so do some higher +curves, e.g. the spiric curves which are sections of the <G>spei=ra</G> +or <I>tore</I>). But some light is thrown on the subject by the corre- +sponding distinction which Pappus draws between ‘plane’, +‘solid’ and ‘linear’ <I>problems.</I> +<p>‘Those problems’, he says, ‘which can be solved by means +of a straight line and a circumference of a circle may pro- +perly be called <I>plane</I>; for the lines by means of which such +problems are solved have their origin in a plane. Those, +however, which are solved by using for their discovery one or +more of the sections of the cone have been called <I>solid</I>; for +their construction requires the use of surfaces of solid figures, +namely those of cones. There remains a third kind of pro- +blem, that which is called <I>linear</I>; for other lines (curves) +besides those mentioned are assumed for the construction, the +origin of which is more complicated and less natural, as they +are generated from more irregular surfaces and intricate +movements.’<note>Pappus, iv, p. 270. 5-17.</note> +<p>The true significance of the word ‘plane’ as applied to +problems is evidently, not that straight lines and circles have +their origin in a plane, but that the problems in question can +be solved by the ordinary plane methods of transformation of +<pb n=118><head>CONIC SECTIONS</head> +areas, manipulation of simple equations between areas and, in +particular, the application of areas; in other words, plane +problems were those which, if expressed algebraically, depend +on equations of a degree not higher than the second. +Problems, however, soon arose which did not yield to ‘plane’ +methods. One of the first was that of the duplication of the +cube, which was a problem of geometry in three dimensions or +solid geometry. Consequently, when it was found that this +problem could be solved by means of conics, and that no +higher curves were necessary, it would be natural to speak of +them as ‘solid’ loci, especially as they were in fact produced +from sections of a solid figure, the cone. The propriety of the +term would be only confirmed when it was found that, just as +the duplication of the cube depended on the solution of a pure +cubic equation, other problems such as the trisection of an +angle, or the cutting of a sphere into two segments bearing +a given ratio to one another, led to an equation between +volumes in one form or another, i. e. a mixed cubic equation, +and that this equation, which was also a solid problem, could +likewise be solved by means of conics. +<C>Aristaeus's <I>Solid Loci.</I></C> +<p>The <I>Solid Loci</I> of Aristaeus, then, presumably dealt with +loci which proved to be conic sections. In particular, he must +have discussed, however imperfectly, the locus with respect to +three or four lines the synthesis of which Apollonius says that +he found inadequately worked out in Euclid's <I>Conics.</I> The +theorems relating to this locus are enunciated by Pappus in +this way: +<p>‘If three straight lines be given in position and from one and +the same point straight lines be drawn to meet the three +straight lines at given angles, and if the ratio of the rectangle +contained by two of the straight lines so drawn to the square +on the remaining one be given, then the point will lie on a +solid locus given in position, that is, on one of the three conic +sections. And if straight lines be so drawn to meet, at given +angles, four straight lines given in position, and the ratio of +the rectangle contained by two of the lines so drawn to the +rectangle contained by the remaining two be given, then in +<pb n=119><head>ARISTAEUS'S <I>SOLID LOCI</I></head> +the same way the point will lie on a conic section given in +position.’<note>Pappus, vii, p. 678. 15-24.</note> +<p>The reason why Apollonius referred in this connexion to +Euclid and not to Aristaeus was probably that it was Euclid's +work that was on the same lines as his own. +<p>A very large proportion of the standard properties of conics +admit of being stated in the form of locus-theorems; if a +certain property holds with regard to a certain point, then +that point lies on a conic section. But it may be assumed +that Aristaeus's work was not merely a collection of the +ordinary propositions transformed in this way; it would deal +with new locus-theorems not implied in the fundamental +definitions and properties of the conics, such as those just +mentioned, the theorems of the three- and four-line locus. +But one (to us) ordinary property, the focus-directrix property, +was, as it seems to me, in all probability included. +<C>Focus-directrix property known to Euclid.</C> +<p>It is remarkable that the directrix does not appear at all in +Apollonius's great treatise on conics. The focal properties of +the central conics are given by Apollonius, but the foci are +obtained in a different way, without any reference to the +directrix; the focus of the parabola does not appear at all. +We may perhaps conclude that neither did Euclid's <I>Conics</I> +contain the focus-directrix property; for, according to Pappus, +Apollonius based his first four books on Euclid's four books, +while filling them out and adding to them. Yet Pappus gives +the proposition as a lemma to Euclid's <I>Surface-Loci</I>, from +which we cannot but infer that it was assumed in that +treatise without proof. If, then, Euclid did not take it from +his own <I>Conics</I>, what more likely than that it was contained +in Aristaeus's <I>Solid Loci</I>? +<p>Pappus's enunciation of the theorem is to the effect that the +locus of a point such that its distance from a given point is in +a given ratio to its distance from a fixed straight line is a conic +section, and is an ellipse, a parabola, or a hyperbola according +as the given ratio is less than, equal to, or greater than unity. +<pb n=120><head>CONIC SECTIONS</head> +<C><I>Proof from Pappus.</I></C> +<p>The proof in the case where the given ratio is different from +unity is shortly as follows. +<p>Let <I>S</I> be the fixed point, <I>SX</I> the perpendicular from <I>S</I> on +the fixed line. Let <I>P</I> be any point on the locus and <I>PN</I> +<FIG> +perpendicular to <I>SX</I>, so that <I>SP</I> is to <I>NX</I> in the given +ratio (<I>e</I>); +thus <MATH><I>e</I><SUP>2</SUP>=(<I>PN</I><SUP>2</SUP>+<I>SN</I><SUP>2</SUP>):<I>NX</I><SUP>2</SUP></MATH>. +<p>Take <I>K</I> on <I>SX</I> such that +<MATH><I>e</I><SUP>2</SUP>=<I>SN</I><SUP>2</SUP>:<I>NK</I><SUP>2</SUP></MATH>; +then, if <I>K</I>′ be another point on <I>SN</I>, produced if necessary, +such that <MATH><I>NK</I>=<I>NK</I>′</MATH>, +<MATH><I>e</I><SUP>2</SUP>:1=(<I>PN</I><SUP>2</SUP>+<I>SN</I><SUP>2</SUP>):<I>NX</I><SUP>2</SUP>=<I>SN</I><SUP>2</SUP>:<I>NK</I><SUP>2</SUP> +=<I>PN</I><SUP>2</SUP>:(<I>NX</I><SUP>2</SUP>-<I>NK</I><SUP>2</SUP>) +=<I>PN</I><SUP>2</SUP>:<I>XK.XK</I>′</MATH>. +<p>The positions of <I>N, K, K</I>′ change with the position of <I>P.</I> +If <I>A, A</I>′ be the points on which <I>N</I> falls when <I>K, K</I>′ coincide +with <I>X</I> respectively, we have +<MATH><I>SA</I>:<I>AX</I>=<I>SN</I>:<I>NK</I>=<I>e</I>:1=<I>SN</I>:<I>NK</I>′=<I>SA</I>′:<I>A</I>′<I>X</I></MATH>. +<p>Therefore <MATH><I>SX</I>:<I>SA</I>=<I>SK</I>:<I>SN</I>=(1+<I>e</I>):<I>e</I></MATH>, +whence <MATH>(1+<I>e</I>):<I>e</I>=(<I>SX</I>-<I>SK</I>):(<I>SA</I>-<I>SN</I>) +=<I>XK</I>:<I>AN</I></MATH>. +<pb n=121><head>FOCUS-DIRECTRIX PROPERTY</head> +<p>Similarly it can be shown that +<MATH>(1-<I>e</I>):<I>e</I>=<I>XK</I>′:<I>A</I>′<I>N</I></MATH>. +<p>By multiplication, <MATH><I>XK.XK</I>′:<I>AN.A</I>′<I>N</I>=(1-<I>e</I><SUP>2</SUP>):<I>e</I><SUP>2</SUP></MATH>; +and it follows from above, <I>ex aequali</I>, that +<MATH><I>PN</I><SUP>2</SUP>:<I>AN.A</I>′<I>N</I>=(1-<I>e</I><SUP>2</SUP>):1</MATH>, +which is the property of a central conic. +<p>When <I>e</I><1, <I>A</I> and <I>A</I>′ lie on the same side of <I>X</I>, while +<I>N</I> lies on <I>AA</I>′, and the conic is an ellipse; when <I>e</I>>1, <I>A</I> and +<I>A</I>′ lie on opposite sides of <I>X</I>, while <I>N</I> lies on <I>A</I>′<I>A</I> produced, +and the conic is a hyperbola. +<p>The case where <I>e</I>=1 and the curve is a parabola is easy +and need not be reproduced here. +<p>The treatise would doubtless contain other loci of types +similar to that which, as Pappus says, was used for the +trisection of an angle: I refer to the proposition already +quoted (vol. i, p. 243) that, if <I>A, B</I> are the base angles of +a triangle with vertex <I>P</I>, and <MATH>∠<I>B</I>=2∠<I>A</I></MATH>, the locus of <I>P</I> +is a hyperbola with eccentricity 2. +<C>Propositions included in Euclid's <I>Conics.</I></C> +<p>That Euclid's <I>Conics</I> covered much of the same ground as +the first three Books of Apollonius is clear from the language +of Apollonius himself. Confirmation is forthcoming in the +quotations by Archimedes of propositions (1) ‘proved in +the elements of conics’, or (2) assumed without remark as +already known. The former class include the fundamental +ordinate properties of the conics in the following forms: +<p>(1) for the ellipse, +<MATH><I>PN</I><SUP>2</SUP>:<I>AN.A</I>′<I>N</I>=<I>P</I>′<I>N</I>′<SUP>2</SUP>:<I>AN</I>′.<I>A</I>′<I>N</I>′=<I>BC</I><SUP>2</SUP>:<I>AC</I><SUP>2</SUP></MATH>; +<p>(2) for the hyperbola, +<MATH><I>PN</I><SUP>2</SUP>:<I>AN.A</I>′<I>N</I>=<I>P</I>′<I>N</I>′<SUP>2</SUP>:<I>AN</I>′.<I>A</I>′<I>N</I>′</MATH>; +<p>(3) for the parabola, <MATH><I>PN</I><SUP>2</SUP>=<I>p<SUB>a</SUB>.AN</I></MATH>; +the principal tangent properties of the parabola; +the property that, if there are two tangents drawn from one +point to any conic section whatever, and two intersecting +<pb n=122><head>CONIC SECTIONS</head> +chords drawn parallel to the tangents respectively, the rect- +angles contained by the segments of the chords respectively +are to one another as the squares of the parallel tangents; +the by no means easy proposition that, if in a parabola the +diameter through <I>P</I> bisects the chord <I>QQ</I>′ in <I>V</I>, and <I>QD</I> is +drawn perpendicular to <I>PV</I>, then +<MATH><I>QV</I><SUP>2</SUP>:<I>QD</I><SUP>2</SUP>=<I>p</I>:<I>p<SUB>a</SUB></I></MATH>, +where <I>p<SUB>a</SUB></I> is the parameter of the principal ordinates and <I>p</I> is +the parameter of the ordinates to the diameter <I>PV.</I> +<C>Conic sections in Archimedes.</C> +<p>But we must equally regard Euclid's <I>Conics</I> as the source +from which Archimedes took most of the other ordinary +properties of conics which he assumes without proof. Before +summarizing these it will be convenient to refer to Archi- +medes's terminology. We have seen that the axes of an +ellipse are not called axes but <I>diameters</I>, greater and lesser; +the axis of a parabola is likewise its <I>diameter</I> and the other +diameters are ‘lines parallel to the diameter’, although in +a segment of a parabola the diameter bisecting the base is +the ‘diameter’ of the segment. The two ‘diameters’ (axes) +of an ellipse are <I>conjugate.</I> In the case of the hyperbola the +‘diameter’ (axis) is the portion of it within the (single-branch) +hyperbola; the centre is not called the ‘centre’, but the point +in which the ‘nearest lines to the section of an obtuse-angled +cone’ (the asymptotes) meet; the half of the axis (<I>CA</I>) is +‘the line adjacent to the axis’ (of the hyperboloid of revolution +obtained by making the hyperbola revolve about its ‘diameter’), +and <I>A</I>′<I>A</I> is double of this line. Similarly <I>CP</I> is the line +‘adjacent to the axis’ of a segment of the hyperboloid, and +<I>P</I>′<I>P</I> double of this line. It is clear that Archimedes did not +yet treat the two branches of a hyperbola as forming one +curve; this was reserved for Apollonius. +<p>The main properties of conics assumed by Archimedes in +addition to those above mentioned may be summarized thus. +<C><I>Central Conics.</I></C> +<p>1. The property of the ordinates to any diameter <I>PP</I>′, +<MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I>=<I>Q</I>′<I>V</I>′<SUP>2</SUP>:<I>PV</I>′.<I>P</I>′<I>V</I>′</MATH>. +<pb n=123><head>CONIC SECTIONS IN ARCHIMEDES</head> +<p>In the case of the hyperbola Archimedes does not give +any expression for the constant ratios <MATH><I>PN</I><SUP>2</SUP>:<I>AN.A</I>′<I>N</I></MATH> and +<MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I></MATH> respectively, whence we conclude that he had +no conception of diameters or radii of a hyperbola not meeting +the curve. +<p>2. The straight line drawn from the centre of an ellipse, or +the point of intersection of the asymptotes of a hyperbola, +through the point of contact of any tangent, bisects all chords +parallel to the tangent. +<p>3. In the ellipse the tangents at the extremities of either of two +conjugate diameters are both parallel to the other diameter. +<p>4. If in a hyperbola the tangent at <I>P</I> meets the transverse +axis in <I>T</I>, and <I>PN</I> is the principal ordinate, <I>AN</I>><I>AT.</I> (It +is not easy to see how this could be proved except by means +of the general property that, if <I>PP</I>′ be any diameter of +a hyperbola, <I>QV</I> the ordinate to it from <I>Q</I>, and <I>QT</I> the tangent +at <I>Q</I> meeting <I>P</I>′<I>P</I> in <I>T</I>, then <MATH><I>TP</I>:<I>TP</I>′=<I>PV</I>:<I>P</I>′<I>V</I></MATH>.) +<p>5. If a cone, right or oblique, be cut by a plane meeting all +the generators, the section is either a circle or an ellipse. +<p>6. If a line between the asymptotes meets a hyperbola and +is bisected at the point of concourse, it will touch the +hyperbola. +<p>7. If <I>x, y</I> are straight lines drawn, in fixed directions respec- +tively, from a point on a hyperbola to meet the asymptotes, +the rectangle <I>xy</I> is constant. +<p>8. If <I>PN</I> be the principal ordinate of <I>P</I>, a point on an ellipse, +and if <I>NP</I> be produced to meet the auxiliary circle in <I>p</I>, the +ratio <I>pN</I>:<I>PN</I> is constant. +<p>9. The criteria of similarity of conics and segments of +conics are assumed in practically the same form as Apollonius +gives them. +<C><I>The Parabola.</I></C> +<p>1. The fundamental properties appear in the alternative forms +<MATH><I>PN</I><SUP>2</SUP>:<I>P</I>′<I>N</I>′<SUP>2</SUP>=<I>AN</I>:<I>AN</I>′, or <I>PN</I><SUP>2</SUP>=<I>p<SUB>a</SUB>.AN</I>, +<I>QV</I><SUP>2</SUP>:<I>Q</I>′<I>V</I>′<SUP>2</SUP>=<I>PV</I>:<I>PV</I>′, or <I>QV</I><SUP>2</SUP>=<I>p.PV</I></MATH>. +<p>Archimedes applies the term <I>parameter</I> (<G>a( par) a(\v du/nantai +ai( a)po\ ta=s toma=s</G>) to the parameter of the principal ordinates +<pb n=124><head>CONIC SECTIONS</head> +only: <I>p</I> is simply the line to which the rectangle equal to <I>QV</I><SUP>2</SUP> +and of width equal to <I>PV</I> is applied. +<p>2. Parallel chords are bisected by one straight line parallel to +the axis, which passes through the point of contact of the +tangent parallel to the chords. +<p>3. If the tangent at <I>Q</I> meet the diameter <I>PV</I> in <I>T</I>, and <I>QV</I> be +the ordinate to the diameter, <MATH><I>PV</I>=<I>PT</I></MATH>. +<p>By the aid of this proposition a tangent to the parabola can +be drawn (<I>a</I>) at a point on it, (<I>b</I>) parallel to a given chord. +<p>4. Another proposition assumed is equivalent to the property +of the subnormal, <MATH><I>NG</I>=1/2<I>p<SUB>a</SUB></I></MATH>. +<p>5. If <I>QQ</I>′ be a chord of a parabola perpendicular to the axis +and meeting the axis in <I>M</I>, while <I>QVq</I> another chord parallel +to the tangent at <I>P</I> meets the diameter through <I>P</I> in <I>V</I>, and +<I>RHK</I> is the principal ordinate of any point <I>R</I> on the curve +meeting <I>PV</I> in <I>H</I> and the axis in <I>K</I>, then <I>PV</I>:<I>PH</I>> or +=<I>MK</I>:<I>KA</I>; ‘for this is proved’ (<I>On Floating Bodies</I>, II. 6). +<p>Where it was proved we do not know; the proof is not +altogether easy.<note>See <I>Apollonius of Perga</I>, ed. Heath, p. liv.</note> +<p>6. All parabolas are similar. +<p>As we have seen, Archimedes had to specialize in the +parabola for the purpose of his treatises on the <I>Quadrature +of the Parabola, Conoids and Spheroids, Floating Bodies</I>, +Book II, and <I>Plane Equilibriums</I>, Book II; consequently he +had to prove for himself a number of special propositions, which +have already been given in their proper places. A few others +are assumed without proof, doubtless as being easy deductions +from the propositions which he does prove. They refer mainly +to similar parabolic segments so placed that their bases are in +one straight line and have one common extremity. +<p>1. If any three similar and similarly situated parabolic +segments <I>BQ</I><SUB>1</SUB>, <I>BQ</I><SUB>2</SUB>, <I>BQ</I><SUB>3</SUB> lying along the same straight line +as bases (<I>BQ</I><SUB>1</SUB><<I>BQ</I><SUB>2</SUB><<I>BQ</I><SUB>3</SUB>), and if <I>E</I> be any point on the +tangent at <I>B</I> to one of the segments, and <I>EO</I> a straight line +through <I>E</I> parallel to the axis of one of the segments and +meeting the segments in <I>R</I><SUB>3</SUB>, <I>R</I><SUB>2</SUB>, <I>R</I><SUB>1</SUB> respectively and <I>BQ</I><SUB>3</SUB> +in <I>O</I>, then +<MATH><I>R</I><SUB>3</SUB><I>R</I><SUB>2</SUB>:<I>R</I><SUB>2</SUB><I>R</I><SUB>1</SUB>=(<I>Q</I><SUB>2</SUB><I>Q</I><SUB>3</SUB>:<I>BQ</I><SUB>3</SUB>).(<I>BQ</I><SUB>1</SUB>:<I>Q</I><SUB>1</SUB><I>Q</I><SUB>2</SUB>)</MATH>. +<pb n=125><head>CONIC SECTIONS IN ARCHIMEDES</head> +<p>2. If two similar parabolic segments with bases <I>BQ</I><SUB>1</SUB>, <I>BQ</I><SUB>2</SUB> be +placed as in the last proposition, and if <I>BR</I><SUB>1</SUB><I>R</I><SUB>2</SUB> be any straight +line through <I>B</I> meeting the segments in <I>R</I><SUB>1</SUB>, <I>R</I><SUB>2</SUB> respectively, +<MATH><I>BQ</I><SUB>1</SUB>:<I>BQ</I><SUB>2</SUB>=<I>BR</I><SUB>1</SUB>:<I>BR</I><SUB>2</SUB></MATH>. +<p>These propositions are easily deduced from the theorem +proved in the <I>Quadrature of the Parabola</I>, that, if through <I>E</I>, +a point on the tangent at <I>B</I>, a straight line <I>ERO</I> be drawn +parallel to the axis and meeting the curve in <I>R</I> and any chord +<I>BQ</I> through <I>B</I> in <I>O</I>, then +<MATH><I>ER</I>:<I>RO</I>=<I>BO</I>:<I>OQ</I></MATH>. +<p>3. On the strength of these propositions Archimedes assumes +the solution of the problem of placing, between two parabolic +segments similar to one another and placed as in the above +propositions, a straight line of a given length and in a direction +parallel to the diameters of either parabola. +<p>Euclid and Archimedes no doubt adhered to the old method +of regarding the three conics as arising from sections of three +kinds of right circular cones (right-angled, obtuse-angled and +acute-angled) by planes drawn in each case at right angles to +a generator of the cone. Yet neither Euclid nor Archimedes +was unaware that the ‘section of an acute-angled cone’, or +ellipse, could be otherwise produced. Euclid actually says in +his <I>Phaenomena</I> that ‘if a cone or cylinder (presumably right) +be cut by a plane not parallel to the base, the resulting section +is a section of an acute-angled cone which is similar to +a <G>qureo/s</G> (shield)’. Archimedes knew that the non-circular +sections even of an oblique circular cone made by planes +cutting all the generators are ellipses; for he shows us how, +given an ellipse, to draw a cone (in general oblique) of which +it is a section and which has its vertex outside the plane +of the ellipse on any straight line through the centre of the +ellipse in a plane at right angles to the ellipse and passing +through one of its axes, whether the straight line is itself +perpendicular or not perpendicular to the plane of the ellipse; +drawing a cone in this case of course means finding the circular +sections of the surface generated by a straight line always +passing through the given vertex and all the several points of +the given ellipse. The method of proof would equally serve +<pb n=126><head>APOLLONIUS OF PERGA</head> +for the other two conics, the hyperbola and parabola, and we +can scarcely avoid the inference that Archimedes was equally +aware that the parabola and the hyperbola could be found +otherwise than by the old method. +<p>The first, however, to base the theory of conics on the +production of all three in the most general way from any +kind of circular cone, right or oblique, was Apollonius, to +whose work we now come. +<C>B. APOLLONIUS OF PERGA</C> +<p>Hardly anything is known of the life of Apollonius except +that he was born at Perga, in Pamphylia, that he went +when quite young to Alexandria, where he studied with the +successors of Euclid and remained a long time, and that +he flourished (<G>ge/gone</G>) in the reign of Ptolemy Euergetes +(247-222 B.C). Ptolemaeus Chennus mentions an astronomer +of the same name, who was famous during the reign of +Ptolemy Philopator (222-205 B.C.), and it is clear that our +Apollonius is meant. As Apollonius dedicated the fourth and +following Books of his <I>Conics</I> to King Attalus I (241-197 B.C.) +we have a confirmation of his approximate date. He was +probably born about 262 B.C., or 25 years after Archimedes. +We hear of a visit to Pergamum, where he made the acquain- +tance of Eudemus of Pergamum, to whom he dedicated the +first two Books of the <I>Conics</I> in the form in which they have +come down to us; they were the first two instalments of a +second edition of the work. +<C>The text of the <I>Conics.</I></C> +<p>The <I>Conics</I> of Apollonius was at once recognized as the +authoritative treatise on the subject, and later writers regu- +larly cited it when quoting propositions in conics. Pappus +wrote a number of lemmas to it; Serenus wrote a commen- +tary, as also, according to Suidas, did Hypatia. Eutocius +(fl. A.D. 500) prepared an edition of the first four Books and +wrote a commentary on them; it is evident that he had before +him slightly differing versions of the completed work, and he +may also have had the first unrevised edition which had got +into premature circulation, as Apollonius himself complains in +the Preface to Book I. +<pb n=127> +<head>THE TEXT OF THE <I>CONICS</I></head> +<p>The edition of Eutocius suffered interpolations which were +probably made in the ninth century when, under the auspices +of Leon, mathematical studies were revived at Constantinople; +for it was at that date that the uncial manuscripts were +written, from which our best manuscripts, V (= Cod. Vat. gr. +206 of the twelfth to thirteenth century) for the <I>Conics</I>, and +W (= Cod. Vat. gr. 204 of the tenth century) for Eutocius, +were copied. +<p>Only the first four Books survive in Greek; the eighth +Book is altogether lost, but the three Books V-VII exist in +Arabic. It was A&hdot;mad and al-&Hdot;asa&ndot;, two sons of Mu&hdot;. b. +Mūsā b. Shākir, who first contemplated translating the <I>Conics</I> +into Arabic. They were at first deterred by the bad state of +their manuscripts; but afterwards A&hdot;mad obtained in Syria +a copy of Eutocius's edition of Books I-IV and had them +translated by Hilāl b. Abī Hilāl al-&Hdot;im⋅ī (died 883/4). +Books V-VII were translated, also for A&hdot;mad, by Thābit +b. Qurra ( 826-901) from another manuscript. Na⋅īraddīn's +recension of this translation of the seven Books, made in 1248, +is represented by two copies in the Bodleian, one of the year +1301 (No. 943) and the other of 1626 containing Books V-VII +only (No. 885). +<p>A Latin translation of Books I-IV was published by +Johannes Baptista Memus at Venice in 1537; but the first +important edition was the translation by Commandinus +(Bologna, 1566), which included the lemmas of Pappus and +the commentary of Eutocius, and was the first attempt to +make the book intelligible by means of explanatory notes. +For the Greek text Commandinus used Cod. Marcianus 518 +and perhaps also Vat. gr. 205, both of which were copies of V, +but not V itself. +<p>The first published version of Books V-VII was a Latin +translation by Abraham Echellensis and Giacomo Alfonso +Borelli (Florence, 1661) of a reproduction of the Books written +in 983 by Abū 'l Fat&hdot; al-I⋅fahānī. +<p>The <I>editio princeps</I> of the Greek text is the monumental +work of Halley (Oxford, 1710). The original intention was +that Gregory should edit the four Books extant in Greek, with +Eutocius's commentary and a Latin translation, and that +Halley should translate Books V-VII from the Arabic into +<pb n=128> +<head>APOLLONIUS OF PERGA</head> +Latin. Gregory, however, died while the work was proceeding, +and Halley then undertook responsibility for the whole. The +Greek manuscripts used were two, one belonging to Savile +and the other lent by D. Baynard; their whereabouts cannot +apparently now be traced, but they were both copies of Paris. +gr. 2356, which was copied in the sixteenth century from Paris. +gr. 2357 of the sixteenth century, itself a copy of V. For the +three Books in Arabic Halley used the Bodleian MS. 885, but +also consulted (<I>a</I>) a compendium of the three Books by ‘Abdel- +melik al-Shīrāzī (twelfth century), also in the Bodleian (913), +(<I>b</I>) Borelli's edition, and (<I>c</I>) Bodl. 943 above mentioned, by means +of which he revised and corrected his translation when com- +pleted. Halley's edition is still, so far as I know, the only +available source for Books V-VII, except for the beginning of +Book V (up to Prop. 7) which was edited by L. Nix (Leipzig, +1889). +<p>The Greek text of Books I-IV is now available, with the +commentaries of Eutocius, the fragments of Apollonius, &c., +in the definitive edition of Heiberg (Teubner, 1891-3). +<C>Apollonius's own account of the <I>Conics.</I></C> +<p>A general account of the contents of the great work which, +according to Geminus, earned for him the title of the ‘great +geometer’ cannot be better given than in the words of the +writer himself. The prefaces to the several Books contain +interesting historical details, and, like the prefaces of Archi- +medes, state quite plainly and simply in what way the +treatise differs from those of his predecessors, and how much +in it is claimed as original. The strictures of Pappus (or +more probably his interpolator), who accuses him of being a +braggart and unfair towards his predecessors, are evidently +unfounded. The prefaces are quoted by v. Wilamowitz- +Moellendorff as specimens of admirable Greek, showing how +perfect the style of the great mathematicians could be +when they were free from the trammels of mathematical +terminology. +<C>Book I. General Preface.</C> +<p>Apollonius to Eudemus, greeting. +<p>If you are in good health and things are in other respects +as you wish, it is well; with me too things are moderately +<pb n=129> +<head>THE <I>CONICS</I></head> +well. During the time I spent with you at Pergamum +I observed your eagerness to become acquainted with my +work in conics; I am therefore sending you the first book, +which I have corrected, and I will forward the remaining +books when I have finished them to my satisfaction. I dare +say you have not forgotten my telling you that I undertook +the investigation of this subject at the request of Naucrates +the geometer, at the time when he came to Alexandria and +stayed with me, and, when I had worked it out in eight +books, I gave them to him at once, too hurriedly, because he +was on the point of sailing; they had therefore not been +thoroughly revised, indeed I had put down everything just as +it occurred to me, postponing revision till the end. Accord- +ingly I now publish, as opportunities serve from time to time, +instalments of the work as they are corrected. In the mean- +time it has happened that some other persons also, among +those whom I have met, have got the first and second books +before they were corrected; do not be surprised therefore if +you come across them in a different shape. +<p>Now of the eight books the first four form an elementary +introduction. The first contains the modes of producing the +three sections and the opposite branches (of the hyperbola), +and the fundamental properties subsisting in them, worked +out more fully and generally than in the writings of others. +The second book contains the properties of the diameters and +the axes of the sections as well as the asymptotes, with other +things generally and necessarily used for determining limits +of possibility (<G>diorismoi/</G>); and what I mean by diameters +and axes respectively you will learn from this book. The +third book contains many remarkable theorems useful for +the syntheses of solid loci and for <I>diorismi</I>; the most and +prettiest of these theorems are new, and it was their discovery +which made me aware that Euclid did not work out the +synthesis of the locus with respect to three and four lines, but +only a chance portion of it, and that not successfully; for it +was not possible for the said synthesis to be completed without +the aid of the additional theorems discovered by me. The +fourth book shows in how many ways the sections of cones +can meet one another and the circumference of a circle; it +contains other things in addition, none of which have been +discussed by earlier writers, namely the questions in how +many points a section of a cone or a circumference of a circle +can meet [a double-branch hyperbola, or two double-branch +hyperbolas can meet one another]. +<p>The rest of the books are more by way of surplusage +(<G>periousiastikw/tera</G>): one of them deals somewhat fully with +<pb n=130> +<head>APOLLONIUS OF PERGA</head> +<I>minima</I> and <I>maxima</I>, another with equal and similar sections +of cones, another with theorems of the nature of determina- +tions of limits, and the last with determinate conic problems. +But of course, when all of them are published, it will be open +to all who read them to form their own judgement about them, +according to their own individual tastes. Farewell. +<p>The preface to Book II merely says that Apollonius is +sending the second Book to Eudemus by his son Apollonius, +and begs Eudemus to communicate it to earnest students of the +subject, and in particular to Philonides the geometer whom +Apollonius had introduced to Eudemus at Ephesus. There is +no preface to Book III as we have it, although the preface to +Book IV records that it also was sent to Eudemus. +<C>Preface to Book IV.</C> +<p>Apollonius to Attalus, greeting. +<p>Some time ago I expounded and sent to Eudemus of Per- +gamum the first three books of my conics which I have +compiled in eight books, but, as he has passed away, I have +resolved to dedicate the remaining books to you because of +your earnest desire to possess my works. I am sending you +on this occasion the fourth book. It contains a discussion of +the question, in how many points at most it is possible for +sections of cones to meet one another and the circumference +of a circle, on the assumption that they do not coincide +throughout, and further in how many points at most a +section of a cone or the circumference of a circle can meet the +hyperbola with two branches, [or two double-branch hyper- +bolas can meet one another]; and, besides these questions, +the book considers a number of others of a similar kind. +Now the first question Conon expounded to Thrasydaeus, with- +out, however, showing proper mastery of the proofs, and on +this ground Nicoteles of Cyrene, not without reason, fell foul +of him. The second matter has merely been mentioned by +Nicoteles, in connexion with his controversy with Conon, +as one capable of demonstration; but I have not found it +demonstrated either by Nicoteles himself or by any one else. +The third question and the others akin to it I have not found +so much as noticed by any one. All the matters referred to, +which I have not found anywhere, required for their solution +many and various novel theorems, most of which I have, +as a matter of fact, set out in the first three books, while the +rest are contained in the present book. These theorems are +of considerable use both for the syntheses of problems and for +<pb n=131> +<head>THE <I>CONICS</I></head> +<I>diorismi.</I> Nicoteles indeed, on account of his controversy +with Conon, will not have it that any use can be made of the +discoveries of Conon for the purpose of <I>diorismi</I>; he is, +however, mistaken in this opinion, for, even if it is possible, +without using them at all, to arrive at results in regard to +limits of possibility, yet they at all events afford a readier +means of observing some things, e.g. that several or so many +solutions are possible, or again that no solution is possible; +and such foreknowledge secures a satisfactory basis for in- +vestigations, while the theorems in question are again useful +for the analyses of <I>diorismi.</I> And, even apart from such +usefulness, they will be found worthy of acceptance for the +sake of the demonstrations themselves, just as we accept +many other things in mathematics for this reason and for +no other. +<p>The prefaces to Books V-VII now to be given are repro- +duced for Book V from the translation of L. Nix and for +Books VI, VII from that of Halley. +<C>Preface to Book V.</C> +<p>Apollonius to Attalus, greeting. +<p>In this fifth book I have laid down propositions relating to +<I>maximum</I> and <I>minimum</I> straight lines. You must know +that my predecessors and contemporaries have only super- +ficially touched upon the investigation of the shortest lines, +and have only proved what straight lines touch the sections +and, conversely, what properties they have in virtue of which +they are tangents. For my part, I have proved these pro- +perties in the first book (without however making any use, in +the proofs, of the doctrine of the shortest lines), inasmuch as +I wished to place them in close connexion with that part +of the subject in which I treat of the production of the three +conic sections, in order to show at the same time that in each +of the three sections countless properties and necessary results +appear, as they do with reference to the original (transverse) +diameter. The propositions in which I discuss the shortest +lines I have separated into classes, and I have dealt with each +individual case by careful demonstration; I have also con- +nected the investigation of them with the investigation of +the greatest lines above mentioned, because I considered that +those who cultivate this science need them for obtaining +a knowledge of the analysis, and determination of limits of +possibility, of problems as well as for their synthesis: in +addition to which, the subject is one of those which seem +worthy of study for their own sake. Farewell. +<pb n=132> +<head>APOLLONIUS OF PERGA</head> +<C>Preface to Book VI.</C> +<p>Apollonius to Attalus, greeting. +<p>I send you the sixth book of the conics, which embraces +propositions about conic sections and segments of conics equal +and unequal, similar and dissimilar, besides some other matters +left out by those who have preceded me. In particular, you +will find in this book how, in a given right cone, a section can +be cut which is equal to a given section, and how a right cone +can be described similar to a given cone but such as to contain +a given conic section. And these matters in truth I have +treated somewhat more fully and clearly than those who wrote +before my time on these subjects. Farewell. +<C>Preface to Book VII.</C> +<p>Apollonius to Attalus, greeting. +<p>I send to you with this letter the seventh book on conic +sections. In it are contained a large number of new proposi- +tions concerning diameters of sections and the figures described +upon them; and all these propositions have their uses in many +kinds of problems, especially in the determination of the +limits of their possibility. Several examples of these occur +in the determinate conic problems solved and demonstrated +by me in the eighth book, which is by way of an appendix, +and which I will make a point of sending to you as soon +as possible. Farewell. +<C><I>Extent of claim to originality.</I></C> +<p>We gather from these prefaces a very good idea of the +plan followed by Apollonius in the arrangement of the sub- +ject and of the extent to which he claims originality. The +first four Books form, as he says, an elementary introduction, +by which he means an exposition of the elements of conics, +that is, the definitions and the fundamental propositions +which are of the most general use and application; the term +‘elements’ is in fact used with reference to conics in exactly +the same sense as Euclid uses it to describe his great work. +The remaining Books beginning with Book V are devoted to +more specialized investigation of particular parts of the sub- +ject. It is only for a very small portion of the <I>content</I> of the +treatise that Apollonius claims originality; in the first three +Books the claim is confined to certain propositions bearing on +the ‘locus with respect to three or four lines’; and in the +fourth Book (on the number of points at which two conics +<pb n=133> +<head>THE <I>CONICS</I></head> +may intersect, touch, or both) the part which is claimed +as new is the extension to the intersections of the parabola, +ellipse, and circle with the double-branch hyperbola, and of +two double-branch hyperbolas with one another, of the in- +vestigations which had theretofore only taken account of the +single-branch hyperbola. Even in Book V, the most remark- +able of all, Apollonius does not say that normals as ‘the shortest +lines’ had not been considered before, but only that they had +been superficially touched upon, doubtless in connexion with +propositions dealing with the tangent properties. He explains +that he found it convenient to treat of the tangent properties, +without any reference to normals, in the first Book in order +to connect them with the chord properties. It is clear, there- +fore, that in treating normals as <I>maxima</I> and <I>minima</I>, and by +themselves, without any reference to tangents, as he does in +Book V, he was making an innovation; and, in view of the +extent to which the theory of normals as maxima and minima +is developed by him (in 77 propositions), there is no wonder +that he should devote a whole Book to the subject. Apart +from the developments in Books III, IV, V, just mentioned, +and the numerous new propositions in Book VII with the +problems thereon which formed the lost Book VIII, Apollonius +only claims to have treated the whole subject more fully and +generally than his predecessors. +<C><I>Great generality of treatment from the beginning.</I></C> +<p>So far from being a braggart and taking undue credit to +himself for the improvements which he made upon his prede- +cessors, Apollonius is, if anything, too modest in his descrip- +tion of his personal contributions to the theory of conic +sections. For the ‘more fully and generally’ of his first +preface scarcely conveys an idea of the extreme generality +with which the whole subject is worked out. This character- +istic generality appears at the very outset. +<C>Analysis of the <I>Conics.</I></C> +<C>Book I.</C> +<p>Apollonius begins by describing a double oblique circular +cone in the most general way. Given a circle and any point +outside the plane of the circle and in general not lying on the +<pb n=134> +<head>APOLLONIUS OF PERGA</head> +straight line through the centre of the circle perpendicular to +its plane, a straight line passing through the point and pro- +duced indefinitely in both directions is made to move, while +always passing through the fixed point, so as to pass succes- +sively through all the points of the circle; the straight line +thus describes a double cone which is in general oblique or, as +Apollonius calls it, <I>scalene.</I> Then, before proceeding to the +geometry of a cone, Apollonius gives a number of definitions +which, though of course only required for conics, are stated as +applicable to any curve. +<p>‘In any curve,’ says Apollonius, ‘I give the name <I>diameter</I> to +any straight line which, drawn from the curve, bisects all the +straight lines drawn in the curve (chords) parallel to any +straight line, and I call the extremity of the straight line +(i.e. the diameter) which is at the curve a <I>vertex</I> of the curve +and each of the parallel straight lines (chords) an ordinate +(lit. drawn ordinate-wise, <G>tetagme/nws kath=xqai</G>) to the +diameter.’ +<p>He then extends these terms to a pair of curves (the primary +reference being to the double-branch hyperbola), giving the +name <I>transverse diameter</I> to any straight line bisecting all the +chords in both curves which are parallel to a given straight +line (this gives two vertices where the diameter meets the +curves respectively), and the name <I>erect diameter</I> (<G>o)rqi/a</G>) to +any straight line which bisects all straight lines drawn +between one curve and the other which are parallel to any +straight line; the <I>ordinates</I> to any diameter are again the +parallel straight lines bisected by it. <I>Conjugate diameters</I> in +any curve or pair of curves are straight lines each of which +bisects chords parallel to the other. <I>Axes</I> are the particular +diameters which cut at right angles the parallel chords which +they bisect; and <I>conjugate axes</I> are related in the same way +as conjugate diameters. Here we have practically our modern +definitions, and there is a great advance on Archimedes's +terminology. +<C><I>The conics obtained in the most general way from an +oblique cone.</I></C> +<p>Having described a cone (in general oblique), Apollonius +defines the <I>axis</I> as the straight line drawn from the vertex to +<pb n=135> +<head>THE <I>CONICS</I>, BOOK I</head> +the centre of the circular base. After proving that all +sections parallel to the base are also circles, and that there +is another set of circular sections subcontrary to these, he +proceeds to consider sections of the cone drawn in any +manner. Taking any triangle through the axis (the base of +the triangle being consequently a diameter of the circle which +is the base of the cone), he is careful to make his section cut +the base in a straight line perpendicular to the particular +diameter which is the base of the axial triangle. (There is +no loss of generality in this, for, if any section is taken, +without reference to any axial triangle, we have only to +select the particular axial triangle the base of which is that +diameter of the circular base which is +<FIG> +at right angles to the straight line in +which the section of the cone cuts the +base.) Let <I>ABC</I> be any axial triangle, +and let any section whatever cut the +base in a straight line <I>DE</I> at right +angles to <I>BC</I>; if then <I>PM</I> be the in- +tersection of the cutting plane and the +axial triangle, and if <I>QQ</I>′ be any chord +in the section parallel to <I>DE</I>, Apollonius +proves that <I>QQ</I>′ is bisected by <I>PM.</I> In +other words, <I>PM</I> is a <I>diameter</I> of the section. Apollonius is +careful to explain that, +<p>‘if the cone is a right cone, the straight line in the base (<I>DE</I>) +will be at right angles to the common section (<I>PM</I>) of the +cutting plane and the triangle through the axis, but, if the +cone is scalene, it will not in general be at right angles to <I>PM</I>, +but will be at right angles to it only when the plane through +the axis (i.e. the axial triangle) is at right angles to the base +of the cone’ (I. 7). +<p>That is to say, Apollonius works out the properties of the +conics in the most general way with reference to a diameter +which is not one of the principal diameters or axes, but in +general has its ordinates obliquely inclined to it. The axes do +not appear in his exposition till much later, after it has been +shown that each conic has the same property with reference +to any diameter as it has with reference to the original +diameter arising out of the construction; the axes then appear +<pb n=136> +<head>APOLLONIUS OF PERGA</head> +as particular cases of the new diameter of reference. The +three sections, the parabola, hyperbola, and ellipse are made +in the manner shown in the figures. In each case they pass +<FIG> +through a straight line <I>DE</I> in the plane of the base which +is at right angles to <I>BC</I>, the base of the axial triangle, or +to <I>BC</I> produced. The diameter <I>PM</I> is in the case of the +<pb n=137> +<head>THE <I>CONICS</I>, BOOK I</head> +parabola parallel to <I>AC</I>; in the case of the hyperbola it meets +the other half of the double cone in <I>P</I>′; and in the case of the +ellipse it meets the cone itself again in <I>P</I>′. We draw, in +<FIG> +the cases of the hyperbola and ellipse, <I>AF</I> parallel to <I>PM</I> +to meet <I>BC</I> or <I>BC</I> produced in <I>F.</I> +<p>Apollonius expresses the properties of the three curves by +means of a certain straight line <I>PL</I> drawn at right angles +to <I>PM</I> in the plane of the section. +<p>In the case of the parabola, <I>PL</I> is taken such that +<MATH><I>PL</I>:<I>PA</I>=<I>BC</I><SUP>2</SUP>:<I>BA.AC</I></MATH>; +and in the case of the hyperbola and ellipse such that +<MATH><I>PL</I>:<I>PP</I>′=<I>BF.FC</I>:<I>AF</I><SUP>2</SUP></MATH>. +<p>In the latter two cases we join <I>P</I>′<I>L</I>, and then draw <I>VR</I> +parallel to <I>PL</I> to meet <I>P</I>′<I>L</I>, produced if necessary, in <I>R.</I> +<p>If <I>HK</I> be drawn through <I>V</I> parallel to <I>BC</I> and meeting +<I>AB, AC</I> in <I>H, K</I> respectively, <I>HK</I> is the diameter of the circular +section of the cone made by a plane parallel to the base. +<p>Therefore <MATH><I>QV</I><SUP>2</SUP>=<I>HV.VK</I></MATH>. +<p>Then (1) for the parabola we have, by parallels and similar +triangles, +<MATH><I>HV</I>:<I>PV</I>=<I>BC</I>:<I>CA</I></MATH>, +and <MATH><I>VK</I>:<I>PA</I>=<I>BC</I>:<I>BA</I></MATH>. +<pb n=138> +<head>APOLLONIUS OF PERGA</head> +<p>Therefore <MATH><I>QV</I><SUP>2</SUP>:<I>PV.PA</I>=<I>HV.VK</I>:<I>PV.PA</I> +=<I>BC</I><SUP>2</SUP>:<I>BA.AC</I> +=<I>PL</I>:<I>PA</I>, by hypothesis, +=<I>PL.PV</I>:<I>PV.PA</I>, +whence <I>QV</I><SUP>2</SUP>=<I>PL.PV</I></MATH>. +<p>(2) In the case of the hyperbola and ellipse, +<MATH><I>HV</I>:<I>PV</I>=<I>BF</I>:<I>FA</I>, +<I>VK</I>:<I>P</I>′<I>V</I>=<I>FC</I>:<I>AF</I></MATH>. +<p>Therefore <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I>=<I>HV.VK</I>:<I>PV.P</I>′<I>V</I> +=<I>BF.FC</I>:<I>AF</I><SUP>2</SUP> +=<I>PL</I>:<I>PP</I>′, by hypothesis, +=<I>RV</I>:<I>P</I>′<I>V</I> +=<I>PV.VR</I>:<I>PV.P</I>′<I>V</I>, +whence <I>QV</I><SUP>2</SUP>=<I>PV.VR</I></MATH>. +<C><I>New names, ‘parabola’, ‘ellipse’, ‘hyperbola’.</I></C> +<p>Accordingly, in the case of the parabola, the square of the +ordinate (<I>QV</I><SUP>2</SUP>) is equal to the rectangle <I>applied</I> to <I>PL</I> and +with width equal to the abscissa (<I>PV</I>); +in the case of the hyperbola the rectangle applied to <I>PL</I> +which is equal to <I>QV</I><SUP>2</SUP> and has its width equal to the abscissa +<I>PV overlaps</I> or <I>exceeds</I> (<G>u(perba/llei</G>) by the small rectangle <I>LR</I> +which is similar and similarly situated to the rectangle con- +tained by <I>PL, PP</I>′; +in the case of the ellipse the corresponding rectangle <I>falls +short</I> (<G>e)llei/pei</G>) by a rectangle similar and similarly situated +to the rectangle contained by <I>PL, PP</I>′. +<p>Here then we have the properties of the three curves +expressed in the precise language of the Pythagorean applica- +tion of areas, and the curves are named accordingly: <I>parabola</I> +(<G>parabolh/</G>) where the rectangle is exactly <I>applied, hyperbola</I> +(<G>u(perbolh/</G>) where it <I>exceeds</I>, and <I>ellipse</I> (<G>e)/lleiyis</G>) where it +<I>falls short.</I> +<pb n=139> +<head>THE <I>CONICS</I>, BOOK I</head> +<p><I>PL</I> is called the <I>latus rectum</I> (<G>o)rqi/a</G>) or the <I>parameter of +the ordinates</I> (<G>par) h(\n du/nantai ai( katago/menai tetagme/nws</G>) in +each case. In the case of the central conics, the diameter <I>PP</I>′ +is the <I>transverse</I> (<G>h( plagi/a</G>) or <I>transverse diameter</I>; while, +even more commonly, Apollonius speaks of the diameter and +the corresponding parameter together, calling the latter the +<I>latus rectum or erect side</I> (<G>o)rqi/a pleura/</G>) and the former +the <I>transverse side</I> of the <I>figure</I> (<G>ei)=dos</G>) <I>on</I>, or <I>applied to</I>, the +diameter. +<C><I>Fundamental properties equivalent to Cartesian equations.</I></C> +<p>If <I>p</I> is the parameter, and <I>d</I> the corresponding diameter, +the properties of the curves are the equivalent of the Cartesian +equations, referred to the diameter and the tangent at its +extremity as axes (in general oblique), +<MATH><I>y</I><SUP>2</SUP>=<I>px</I></MATH> (the parabola), +<MATH><I>y</I><SUP>2</SUP>=<I>px</I>±<I>(p/d)x</I><SUP>2</SUP></MATH> (the hyperbola and ellipse respectively). +<p>Thus Apollonius expresses the fundamental property of the +central conics, like that of the parabola, as an equation +between areas, whereas in Archimedes it appears as a +proportion +<MATH><I>y</I><SUP>2</SUP>:(<I>a</I><SUP>2</SUP>±<I>x</I><SUP>2</SUP>)=<I>b</I><SUP>2</SUP>:<I>a</I><SUP>2</SUP></MATH>, +which, however, is equivalent to the Cartesian equation +referred to axes with the centre as origin. The latter pro- +perty with reference to the original diameter is separately +proved in I. 21, to the effect that <I>QV</I><SUP>2</SUP> varies as <I>PV.P</I>′<I>V</I>, as +is really evident from the fact that <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I>=<I>PL</I>:<I>PP</I>′</MATH>, +seeing that <I>PL</I>:<I>PP</I>′ is constant for any fixed diameter <I>PP</I>′. +<p>Apollonius has a separate proposition (I. 14) to prove that +the opposite branches of a hyperbola have the same diameter +and equal <I>latera recta</I> corresponding thereto. As he was the +first to treat the double-branch hyperbola fully, he generally +discusses the <I>hyperbola</I> (i.e. the single branch) along with +the ellipse, and <I>the opposites</I>, as he calls the double-branch +hyperbola, separately. The properties of the single-branch +hyperbola are, where possible, included in one enunciation +with those of the ellipse and circle, the enunciation beginning, +<pb n=140> +<head>APOLLONIUS OF PERGA</head> +‘If in a hyperbola, an ellipse, or the circumference of a circle’; +sometimes, however, the double-branch hyperbola and the +ellipse come in one proposition, e.g. in I. 30: ‘If in an ellipse +or the opposites (i.e. the double hyperbola) a straight line be +drawn through the centre meeting the curve on both sides of +the centre, it will be bisected at the centre.’ The property of +conjugate diameters in an ellipse is proved in relation to +the original diameter of reference and its conjugate in I. 15, +where it is shown that, if <I>DD</I>′ is the diameter conjugate to +<I>PP</I>′ (i.e. the diameter drawn ordinate-wise to <I>PP</I>′), just as +<I>PP</I>′ bisects all chords parallel to <I>DD</I>′, so <I>DD</I>′ bisects all chords +parallel to <I>PP</I>′; also, if <I>DL</I>′ be drawn at right angles to <I>DD</I>′ +and such that <MATH><I>DL</I>′.<I>DD</I>′=<I>PP</I>′<SUP>2</SUP></MATH> (or <I>DL</I>′ is a third proportional +to <I>DD</I>′, <I>PP</I>′), then the ellipse has the same property in rela- +tion to <I>DD</I>′ as diameter and <I>DL</I>′ as parameter that it has in +relation to <I>PP</I>′ as diameter and <I>PL</I> as the corresponding para- +meter. Incidentally it appears that <MATH><I>PL.PP</I>′=<I>DD</I>′<SUP>2</SUP></MATH>, or <I>PL</I> is +a third proportional to <I>PP</I>′, <I>DD</I>′, as indeed is obvious from the +property of the curve <MATH><I>QV</I><SUP>2</SUP>:<I>PV.PV</I>′=<I>PL</I>:<I>PP</I>′=<I>DD</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP></MATH>. +The next proposition, I. 16, introduces the <I>secondary diameter</I> +of the double-branch hyperbola (i.e. the diameter conjugate to +the transverse diameter of reference), which does not meet the +curve; this diameter is defined as that straight line drawn +through the centre parallel to the ordinates of the transverse +diameter which is bisected at the centre and is of length equal +to the mean proportional between the ‘sides of the figure’, +i.e. the transverse diameter <I>PP</I>′ and the corresponding para- +meter <I>PL.</I> The <I>centre</I> is defined as the middle point of the +diameter of reference, and it is proved that all other diameters +are bisected at it (I. 30). +<p>Props. 17-19, 22-9, 31-40 are propositions leading up to +and containing the tangent properties. On lines exactly like +those of Eucl. III. 16 for the circle, Apollonius proves that, if +a straight line is drawn through the vertex (i.e. the extremity +of the diameter of reference) parallel to the ordinates to the +diameter, it will fall outside the conic, and no other straight +line can fall between the said straight line and the conic; +therefore the said straight line touches the conic (I. 17, 32). +Props. I. 33, 35 contain the property of the tangent at any +point on the parabola, and Props. I. 34, 36 the property of +<pb n=141> +<head>THE <I>CONICS</I>, BOOK I</head> +the tangent at any point of a central conic, in relation +to the original diameter of reference; if <I>Q</I> is the point of +contact, <I>QV</I> the ordinate to the diameter through <I>P</I>, and +if <I>QT</I>, the tangent at <I>Q</I>, meets the diameter produced in <I>T</I>, +then (1) for the parabola <MATH><I>PV</I>=<I>PT</I></MATH>, and (2) for the central +conic <MATH><I>TP</I>:<I>TP</I>′=<I>PV</I>:<I>VP</I>′</MATH>. The method of proof is to take a +point <I>T</I> on the diameter produced satisfying the respective +relations, and to prove that, if <I>TQ</I> be joined and produced, +any point on <I>TQ</I> on either side of <I>Q</I> is outside the curve: the +form of proof is by <I>reductio ad absurdum</I>, and in each +case it is again proved that no other straight line can fall +between <I>TQ</I> and the curve. The fundamental property +<MATH><I>TP</I>:<I>TP</I>′=<I>PV</I>:<I>VP</I>′</MATH> for the central conic is then used to +prove that <MATH><I>CV.CT</I>=<I>CP</I><SUP>2</SUP></MATH> and <MATH><I>QV</I><SUP>2</SUP>:<I>CV.VT</I>=<I>p</I>:<I>PP</I>′</MATH> (or +<I>CD</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP>) and the corresponding properties with reference to +the diameter <I>DD</I>′ conjugate to <I>PP</I>′ and <I>v, t</I>, the points where +<I>DD</I>′ is met by the ordinate to it from <I>Q</I> and by the tangent +at <I>Q</I> respectively (Props. I. 37-40). +<C><I>Transition to new diameter and tangent at its extremity.</I></C> +<p>An important section of the Book follows (I. 41-50), con- +sisting of propositions leading up to what amounts to a trans- +formation of coordinates from the original diameter and the +tangent at its extremity to <I>any</I> diameter and the tangent at +its extremity; what Apollonius proves is of course that, if +<I>any</I> other diameter be taken, the ordinate-property of the +conic with reference to that diameter is of the same form as it +is with reference to the original diameter. It is evident that +this is vital to the exposition. The propositions leading up to +the result in I. 50 are not usually given in our text-books of +geometrical conics, but are useful and interesting. +<p>Suppose that the tangent at any point <I>Q</I> meets the diameter +of reference <I>PV</I> in <I>T</I>, and that the tangent at <I>P</I> meets the +diameter through <I>Q</I> in <I>E.</I> Let <I>R</I> be any third point on +the curve; let the ordinate <I>RW</I> to <I>PV</I> meet the diameter +through <I>Q</I> in <I>F</I>, and let <I>RU</I> parallel to the tangent at <I>Q</I> meet +<I>PV</I> in <I>U.</I> Then +<p>(1) in the parabola, the triangle <I>RUW</I>=the parallelogram +<I>EW</I>, and +<pb n=142> +<head>APOLLONIUS OF PERGA</head> +<FIG> +<pb n=143> +<head>THE <I>CONICS</I>, BOOK I</head> +<p>(2) in the hyperbola or ellipse, ▵<I>RUW</I>=the difference +between the triangles <I>CFW</I> and <I>CPE.</I> +<p>(1) In the parabola <MATH>▵<I>RUW</I>:▵<I>QTV</I>=<I>RW</I><SUP>2</SUP>:<I>QV</I><SUP>2</SUP> +=<I>PW</I>:<I>PV</I> +=▭<I>EW</I>:▭<I>EV</I></MATH>. +<p>But, since <MATH><I>TV</I>=2<I>PV</I>,▵<I>QTV</I>=▭<I>EV</I></MATH>; +therefore <MATH>▵<I>RUW</I>=▭<I>EW</I></MATH>. +<p>(2) The proof of the proposition with reference to the +central conic depends on a Lemma, proved in I. 41, to the effect +that, if <I>PX, VY</I> be similar parallelograms on <I>CP, CV</I> as bases, +and if <I>VZ</I> be an equiangular parallelogram on <I>QV</I> as base and +such that, if the ratio of <I>CP</I> to the other side of <I>PX</I> is <I>m</I>, the +ratio of <I>QV</I> to the other side of <I>VZ</I> is <I>m.p/PP</I>′, then <I>VZ</I> is +equal to the difference between <I>VY</I> and <I>PX.</I> The proof of the +Lemma by Apollonius is difficult, but the truth of it can be +easily seen thus. +<p>By the property of the curve, <MATH><I>QV</I><SUP>2</SUP>:<I>CV</I><SUP>2</SUP>-<I>CP</I><SUP>2</SUP>=<I>p</I>:<I>PP</I>′</MATH>; +therefore <MATH><I>CV</I><SUP>2</SUP>-<I>CP</I><SUP>2</SUP>=<I>PP</I>′/<I>p.QV</I><SUP>2</SUP></MATH>. +<p>Now <MATH>▭<I>PX</I>=<G>m</G>.<I>CP</I><SUP>2</SUP>/<I>m</I></MATH>, where <G>m</G> is a constant depending +on the angle of the parallelogram. +<p>Similarly +<MATH>▭<I>VY</I>=<G>m</G>.<I>CV</I><SUP>2</SUP>/<I>m</I>, and ▭<I>VZ</I>=<G>m</G>.(<I>PP</I>′/<I>p</I>)<I>QV</I><SUP>2</SUP>/<I>m</I></MATH>. +<p>It follows that <MATH>▭<I>VY</I>-▭<I>PX</I>=▭<I>VZ</I></MATH>. +<p>Taking now the triangles <I>CFW, CPE</I> and <I>RUW</I> in the +ellipse or hyperbola, we see that <I>CFW, CPE</I> are similar, and +<I>RUW</I> has one angle (at <I>W</I>) equal or supplementary to the +angles at <I>P</I> and <I>V</I> in the other two triangles, while we have +<MATH><I>QV</I><SUP>2</SUP>:<I>CV.VT</I>=<I>p</I>:<I>PP</I>′</MATH>, +whence <MATH><I>QV</I>:<I>VT</I>=(<I>p</I>:<I>PP</I>′).(<I>CV</I>:<I>QV</I>)</MATH>, +and, by parallels, +<MATH><I>RW</I>:<I>WU</I>=(<I>p</I>:<I>PP</I>′).(<I>CP</I>:<I>PE</I>)</MATH>. +<pb n=144><head>APOLLONIUS OF PERGA</head> +<p>Therefore <I>RUW, CPE, CFW</I> are the halves of parallelograms +related as in the lemma; +therefore <MATH>▵<I>RUW</I>=▵<I>CFW</I>-▵<I>CPE</I></MATH>. +<p>The same property with reference to the diameter <I>secondary</I> +to <I>CPV</I> is proved in I. 45. +<p>It is interesting to note the exact significance of the property +thus proved for the central conic. The proposition, which is +the foundation of Apollonius's method of transformation of +coordinates, amounts to this. If <I>CP, CQ</I> are fixed semi- +diameters and <I>R</I> a variable point, the area of the quadrilateral +<I>CFRU</I> is constant for all positions of <I>R</I> on the conic. Suppose +now that <I>CP, CQ</I> are taken as axes of <I>x</I> and <I>y</I> respectively. +If we draw <I>RX</I> parallel to <I>CQ</I> to meet <I>CP</I> and <I>RY</I> parallel to +<I>CP</I> to meet <I>CQ</I>, the proposition asserts that (subject to the +proper convention as to sign) +<MATH>▵<I>RYF</I>+▭<I>CXRY</I>+▵<I>RXU</I>=(const.)</MATH>. +<p>But since <I>RX, RY, RF, RU</I> are in fixed directions, +<p>▵<I>RYF</I> varies as <I>RY</I><SUP>2</SUP> or <I>x</I><SUP>2</SUP>, ▭<I>CXRY</I> as <I>RX.RY</I> or <I>xy</I>, +and ▵<I>RXU</I> as <I>RX</I><SUP>2</SUP> or <I>y</I><SUP>2</SUP>. +<p>Hence, if <I>x, y</I> are the coordinates of <I>R</I>, +<MATH><G>a</G><I>x</I><SUP>2</SUP>+<G>b</G><I>xy</I>+<G>g</G><I>y</I><SUP>2</SUP>=<I>A</I></MATH>, +which is the Cartesian equation of the conic referred to the +centre as origin and any two diameters as axes. +<p>The properties so obtained are next used to prove that, +if <I>UR</I> meets the curve again in <I>R</I>′ and the diameter through +<I>Q</I> in <I>M</I>, then <I>RR</I>′ is bisected at <I>M.</I> (I. 46-8). +<p>Taking (1) the case of the parabola, we have, +<MATH>▵<I>RUW</I>=▭<I>EW</I></MATH>, +and <MATH>▵<I>R</I>′<I>UW</I>′=▭<I>EW</I>′</MATH>. +<p>By subtraction, <MATH>(<I>RWW</I>′<I>R</I>′)=▭<I>F</I>′<I>W</I></MATH>, +whence <MATH>▵<I>RFM</I>=▵<I>R</I>′<I>F</I>′<I>M</I></MATH>, +and, since the triangles are similar, <MATH><I>RM</I>=<I>R</I>′<I>M</I></MATH>. +<p>The same result is easily obtained for the central conic. +<p>It follows that <I>EQ</I> produced in the case of the parabola, +<pb n=145><head>THE <I>CONICS</I>, BOOK I</head> +or <I>CQ</I> in the case of the central conic, bisects all chords as +<I>RR</I>′ parallel to the tangent at <I>Q.</I> Consequently <I>EQ</I> and <I>CQ</I> +are <I>diameters</I> of the respective conics. +<p>In order to refer the conic to the new diameter and the +corresponding ordinates, we have only to determine the <I>para- +meter</I> of these ordinates and to show that the property of the +conic with reference to the new parameter and diameter is in +the same form as that originally found. +<p>The propositions I. 49, 50 do this, and show that the new +parameter is in all the cases <I>p</I>′, where (if <I>O</I> is the point of +intersection of the tangents at <I>P</I> and <I>Q</I>) +<MATH><I>OQ</I>:<I>QE</I>=<I>p</I>′:2<I>QT</I></MATH>. +<p>(1) In the case of the parabola, we have <MATH><I>TP</I>=<I>PV</I>=<I>EQ</I></MATH>, +whence <MATH>▵<I>EOQ</I>=▵<I>POT</I></MATH>. +<p>Add to each the figure <I>POQF</I>′<I>W</I>′; +therefore <MATH><I>QTW</I>′<I>F</I>′=▭<I>EW</I>′=▵<I>R</I>′<I>UW</I>′</MATH>, +whence, subtracting <I>MUW</I>′<I>F</I>′ from both, we have +<MATH>▵<I>R</I>′<I>MF</I>′=▭<I>QU</I></MATH>. +<p>Therefore <MATH><I>R</I>′<I>M.MF</I>′=2<I>QT.QM</I></MATH>. +<p>But <MATH><I>R</I>′<I>M</I>:<I>MF</I>′=<I>OQ</I>:<I>QE</I>=<I>p</I>′:2<I>QT</I></MATH>, by hypothesis; +therefore <MATH><I>R</I>′<I>M</I><SUP>2</SUP>.<I>R</I>′<I>M.MF</I>′=<I>p</I>′.<I>QM</I>:2<I>QT.QM</I></MATH>. +<p>And <MATH><I>R</I>′<I>M.MF</I>′=2<I>QT.QM</I></MATH>, from above; +therefore <MATH><I>R</I>′<I>M</I><SUP>2</SUP>=<I>p</I>′.<I>QM</I></MATH>, +which is the desired property.<note>The proposition that, in the case of the parabola, if <I>p</I> be the para- +meter of the ordinates to the diameter through <I>Q</I>, then (see the first figure +on p. 142) +<MATH><I>OQ</I>:<I>QE</I>=<I>p</I>:2<I>QT</I></MATH> +has an interesting application; for it enables us to prove the proposition, +assumed without proof by Archimedes (but not easy to prove otherwise), +that, if in a parabola the diameter through <I>P</I> bisects the chord <I>QQ</I>′ in <I>V</I>, +and <I>QD</I> is drawn perpendicular to <I>PV</I>, then +<MATH><I>QV</I><SUP>2</SUP>:<I>QD</I><SUP>2</SUP>=<I>p</I>:<I>p<SUB>a</SUB></I></MATH>, +where <I>p<SUB>a</SUB></I> is the parameter of the principal ordinates and <I>p</I> the para- +meter of the ordinates to the diameter +<I>PV.</I> +<FIG> +<p>If the tangent at the vertex <I>A</I> meets +<I>VP</I> produced in <I>E</I>, and <I>PT</I>, the tangent +at <I>P</I>, in <I>O</I>, the proposition of Apollonius +proves that +<MATH><I>OP</I>:<I>PE</I>=<I>p</I>:2<I>PT</I></MATH>. +<p>But <MATH><I>OP</I>=1/2<I>PT</I></MATH>; +therefore <MATH><I>PT</I><SUP>2</SUP>=<I>p.PE</I> +=<I>p.AN</I></MATH>. +<p>Thus <MATH><I>QV</I><SUP>2</SUP>:<I>QD</I><SUP>2</SUP>=<I>PT</I><SUP>2</SUP>:<I>PN</I><SUP>2</SUP></MATH>, by similar triangles, +<MATH>=<I>p.AN</I>:<I>p<SUB>a</SUB>.AN</I> +=<I>p</I>:<I>p<SUB>a</SUB></I></MATH>.</note> +<pb n=146><head>APOLLONIUS OF PERGA</head> +<p>(2) In the case of the central conic, we have +<MATH>▵<I>R</I>′<I>UW</I>′=▵<I>CF</I>′<I>W</I>′-▵<I>CPE</I></MATH>. +(Apollonius here assumes what he does not prove till III. 1, +namely that <MATH>▵<I>CPE</I>=▵<I>CQT</I></MATH>. This is proved thus. +<p>We have <MATH><I>CV</I>:<I>CT</I>=<I>CV</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP>; (I. 37, 39.)</MATH> +therefore <MATH>▵<I>CQV</I>:▵<I>CQT</I>=▵<I>CQV</I>:▵<I>CPE</I></MATH>, +so that <MATH>▵<I>CQT</I>=▵<I>CPE</I></MATH>.) +<p>Therefore <MATH>▵<I>R</I>′<I>UW</I>′=▵<I>CF</I>′<I>W</I>′-▵<I>CQT</I></MATH>, +and it is easy to prove that in all cases +<MATH>▵<I>R</I>′<I>MF</I>′=<I>QTUM</I></MATH>. +<p>Therefore <MATH><I>R</I>′<I>M.MF</I>′=<I>QM</I>(<I>QT</I>+<I>MU</I>)</MATH>. +<p>Let <I>QL</I> be drawn at right angles to <I>CQ</I> and equal to <I>p</I>′. +Join <I>Q</I>′<I>L</I> and draw <I>MK</I> parallel to <I>QL</I> to meet <I>Q</I>′<I>L</I> in <I>K</I>. +Draw <I>CH</I> parallel to <I>Q</I>′<I>L</I> to meet <I>QL</I> in <I>H</I> and <I>MK</I> in <I>N.</I> +<p>Now <MATH><I>R</I>′<I>M</I>:<I>MF</I>′=<I>OQ</I>:<I>QE</I> +=<I>QL</I>:2<I>QT</I></MATH>, by hypothesis, +<MATH>=<I>QH</I>:<I>QT</I></MATH>. +<p>But <MATH><I>QT</I>:<I>MU</I>=<I>CQ</I>:<I>CM</I>=<I>QH</I>:<I>MN</I></MATH>, +so that <MATH>(<I>QH</I>+<I>MN</I>):(<I>QT</I>+<I>MU</I>)=<I>QH</I>:<I>QT</I> +=<I>R</I>′<I>M</I>:<I>MF</I>′</MATH>, from above. +<pb n=147><head>THE <I>CONICS</I>, BOOK I</head> +<p>It follows that +<MATH><I>QM</I>(<I>QH</I>+<I>MN</I>):<I>QM</I>(<I>QT</I>+<I>MU</I>)=<I>R</I>′<I>M</I><SUP>2</SUP>:<I>R</I>′<I>M.MF</I>′</MATH>; +but, from above, <MATH><I>QM</I>(<I>QT</I>+<I>MU</I>)=<I>R</I>′<I>M.MF</I>′</MATH>; +therefore <MATH><I>R</I>′<I>M</I><SUP>2</SUP>=<I>QM</I>(<I>QH</I>+<I>MN</I>) +=<I>QM.MK</I></MATH>, +which is the desired property. +<p>In the case of the hyperbola, the same property is true for +the opposite branch. +<p>These important propositions show that the ordinate property +of the three conics is of the same form whatever diameter is +taken as the diameter of reference. It is therefore a matter +of indifference to which particular diameter and ordinates the +conic is referred. This is stated by Apollonius in a summary +which follows I. 50. +<C><I>First appearance of principal axes.</I></C> +<p>The <I>axes</I> appear for the first time in the propositions next +following (I. 52-8), where Apollonius shows how to construct +each of the conics, given in each case (1) a diameter, (2) the +length of the corresponding parameter, and (3) the inclination +of the ordinates to the diameter. In each case Apollonius +first assumes the angle between the ordinates and the diameter +to be a right angle; then he reduces the case where the angle +is oblique to the case where it is right by his method of trans- +formation of coordinates; i.e. from the given diameter and +parameter he <I>finds</I> the <I>axis</I> of the conic and the length of the +corresponding parameter, and he then constructs the conic as +in the first case where the ordinates are at right angles to the +diameter. Here then we have a case of the proof of <I>existence</I> +by means of <I>construction.</I> The conic is in each case con- +structed by finding the cone of which the given conic is a +section. The problem of finding the axis of a parabola and +the centre and the axes of a central conic when the conic (and +not merely the elements, as here) is given comes later (in II. +44-7), where it is also proved (II. 48) that no central conic +can have more than two axes. +<pb n=148><head>APOLLONIUS OF PERGA</head> +<p>It has been my object, by means of the above detailed +account of Book I, to show not merely what results are +obtained by Apollonius, but the way in which he went to +work; and it will have been realized how entirely scientific +and general the method is. When the foundation is thus laid, +and the fundamental properties established, Apollonius is able +to develop the rest of the subject on lines more similar to +those followed in our text-books. My description of the rest +of the work can therefore for the most part be confined to a +summary of the contents. +<p>Book II begins with a section devoted to the properties of +the asymptotes. They are constructed in II. 1 in this way. +Beginning, as usual, with <I>any</I> diameter of reference and the +corresponding parameter and inclination of ordinates, Apol- +lonius draws at <I>P</I> the vertex (the extremity of the diameter) +a tangent to the hyperbola and sets off along it lengths <I>PL, PL</I>′ +on either side of <I>P</I> such that <MATH><I>PL</I><SUP>2</SUP>=<I>PL</I>′<SUP>2</SUP>=1/4<I>p.PP</I>′[=<I>CD</I><SUP>2</SUP>]</MATH>, +where <I>p</I> is the parameter. He then proves that <I>CL, CL</I>′ pro- +duced will not meet the curve in any finite point and are there- +fore <I>asymptotes.</I> II. 2 proves further that no straight line +through <I>C</I> within the angle between the asymptotes can itself +be an asymptote. II. 3 proves that the intercept made by the +asymptotes on the tangent at any point <I>P</I> is bisected at <I>P</I>, and +that the square on each half of the intercept is equal to one- +fourth of the ‘figure’ corresponding to the diameter through +<I>P</I> (i.e. one-fourth of the rectangle contained by the ‘erect’ +side, the <I>latus rectum</I> or parameter corresponding to the +diameter, and the diameter itself); this property is used as a +means of drawing a hyperbola when the asymptotes and one +point on the curve are given (II. 4). II. 5-7 are propositions +about a tangent at the extremity of a diameter being parallel +to the chords bisected by it. Apollonius returns to the +asymptotes in II. 8, and II. 8-14 give the other ordinary +properties with reference to the asymptotes (II. 9 is a con- +verse of II. 3), the equality of the intercepts between the +asymptotes and the curve of any chord (II. 8), the equality of +the rectangle contained by the distances between either point +in which the chord meets the curve and the points of inter- +section with the asymptotes to the square on the parallel +semi-diameter (II. 10), the latter property with reference to +<pb n=149><head>THE <I>CONICS</I>, BOOK II</head> +the portions of the asymptotes which include between them +a branch of the conjugate hyperbola (II. 11), the constancy of +the rectangle contained by the straight lines drawn from any +point of the curve in fixed directions to meet the asymptotes +(equivalent to the Cartesian equation with reference to the +asymptotes, <MATH><I>xy</I> = const</MATH>.) (II. 12), and the fact that the curve +and the asymptotes proceed to infinity and approach con- +tinually nearer to one another, so that the distance separating +them can be made smaller than any given length (II. 14). II. 15 +proves that the two opposite branches of a hyperbola have the +same asymptotes and II. 16 proves for the chord connecting +points on two branches the property of II. 8. II. 17 shows that +‘conjugate opposites’ (two conjugate double-branch hyper- +bolas) have the same asymptotes. Propositions follow about +conjugate hyperbolas; any tangent to the conjugate hyper- +bola will meet both branches of the original hyperbola +and will be bisected at the point of contact (II. 19); if <I>Q</I> be +any point on a hyperbola, and <I>CE</I> parallel to the tangent +at <I>Q</I> meets the conjugate hyperbola in <I>E</I>, the tangent at +<I>E</I> will be parallel to <I>CQ</I> and <I>CQ, CE</I> will be conjugate +diameters (II. 20), while the tangents at <I>Q, E</I> will meet on one +of the asymptotes (II. 21); if a chord <I>Qq</I> in one branch of +a hyperbola meet the asymptotes in <I>R, r</I> and the conjugate +hyperbola in <I>Q</I>′, <I>q</I>′, then <MATH><I>Q</I>′<I>Q.Qq</I>′=2<I>CD</I><SUP>2</SUP> (II. 23)</MATH>. Of the +rest of the propositions in this part of the Book the following +may be mentioned: if <I>TQ, TQ</I>′ are two tangents to a conic +and <I>V</I> is the middle point of <I>QQ</I>′, <I>TV</I> is a diameter (II. 29, +30, 38); if <I>tQ, tQ</I>′ be tangents to opposite branches of a hyper- +bola, <I>RR</I>′ the chord through <I>t</I> parallel to <I>QQ</I>′, <I>v</I> the middle +point of <I>QQ</I>′, then <I>vR, vR</I>′ are tangents to the hyperbola +(II. 40); in a conic, or a circle, or in conjugate hyperbolas, if +two chords not passing through the centre intersect, they do not +bisect each other (II. 26, 41, 42). II. 44-7 show how to find +a diameter of a conic and the centre of a central conic, the +axis of a parabola and the axes of a central conic. The Book +concludes with problems of drawing tangents to conics in +certain ways, through any point on or outside the curve +(II. 49), making with the axis an angle equal to a given acute +angle (II. 50), making a given angle with the diameter through +the point of contact (II. 51, 53); II. 52 contains a <G>diorismo/s</G> for +<pb n=150><head>APOLLONIUS OF PERGA</head> +the last problem, proving that, if the tangent to an ellipse at +any point <I>P</I> meets the major axis in <I>T</I>, the angle <I>CPT</I> is not +greater than the angle <I>ABA</I>′, where <I>B</I> is one extremity of the +minor axis. +<p>Book III begins with a series of propositions about the +equality of certain areas, propositions of the same kind as, and +easily derived from, the propositions (I. 41-50) by means of +which, as already shown, the transformation of coordinates is +effected. We have first the proposition that, if the tangents +at any points <I>P, Q</I> of a conic meet in <I>O</I>, and if they meet +the diameters through <I>Q, P</I> respectively in <I>E, T</I>, then +<MATH>▵<I>OPT</I>=▵<I>OQE</I> (III. 1, 4)</MATH>; and, if <I>P, Q</I> be points on adjacent +branches of conjugate hyperbolas, <MATH>▵<I>CPE</I>=▵<I>CQT</I> (III. 13)</MATH>. +With the same notation, if <I>R</I> be any other point on the conic, +and if we draw <I>RU</I> parallel to the tangent at <I>Q</I> meeting the +diameter through <I>P</I> in <I>U</I> and the diameter through <I>Q</I> in <I>M</I>, +and <I>RW</I> parallel to the tangent at <I>P</I> meeting <I>QT</I> in <I>H</I> and +the diameters through <I>Q, P</I> in <I>F, W</I>, then <MATH>▵<I>HQF</I>=quadri- +lateral <I>HTUR</I> (III. 2. 6)</MATH>; this is proved at once from the fact +that <MATH>▵<I>RMF</I>=quadrilateral <I>QTUM</I> (see I. 49, 50, or pp. 145-6 +above)</MATH> by subtracting or adding the area <I>HRMQ</I> on each +side. Next take any other point <I>R</I>′, and draw <I>R</I>′<I>U</I>′, <I>F</I>′<I>H</I>′<I>R</I>′<I>W</I>′ +in the same way as before; it is then proved that, if <I>RU, R</I>′<I>W</I>′ +meet in <I>I</I> and <I>R</I>′<I>U</I>′, <I>RW</I> in <I>J</I>, the quadrilaterals <I>F</I>′<I>IRF, IUU</I>′<I>R</I>′ +are equal, and also the quadrilaterals <I>FJR</I>′<I>F</I>′, <I>JU</I>′<I>UR</I> (III. 3, +7, 9, 10). The proof varies according to the actual positions +of the points in the figures. +<p>In Figs. 1, 2 <MATH>▵<I>HFQ</I>=quadrilateral <I>HTUR</I>, +▵<I>H</I>′<I>F</I>′<I>Q</I>=<I>H</I>′<I>TU</I>′<I>R</I>′</MATH>. +<p>By subtraction, <MATH><I>FHH</I>′<I>F</I>′=<I>IUU</I>′<I>R</I>′∓(<I>IH</I>)</MATH>; +whence, if <I>IH</I> be added or subtracted, <MATH><I>F</I>′<I>IRF</I>=<I>IUU</I>′<I>R</I>′</MATH>, +and again, if <I>IJ</I> be added to both, <MATH><I>FJR</I>′<I>F</I>′=<I>JU</I>′<I>UR</I></MATH>. +<p>In Fig. 3 <MATH>▵<I>R</I>′<I>U</I>′<I>W</I>′=▵<I>CF</I>′<I>W</I>′-▵<I>CQT</I></MATH>, +so that <MATH>▵<I>CQT</I>=<I>CU</I>′<I>R</I>′<I>F</I>′</MATH>. +<pb n=151><head>THE <I>CONICS</I>, BOOK III</head> +<FIG> +<CAP>FIG. 1.</CAP> +<FIG> +<CAP>FIG. 2.</CAP> +<FIG> +<CAP>FIG. 3.</CAP> +<pb n=152><head>APOLLONIUS OF PERGA</head> +<p>Adding the quadrilateral <I>CF</I>′<I>H</I>′<I>T</I>, we have +<MATH>▵<I>H</I>′<I>F</I>′<I>Q</I>=<I>H</I>′<I>TU</I>′<I>R</I>′</MATH>, +and similarly <MATH>▵<I>HFQ</I>=<I>HTUR</I></MATH>. +<p>By subtraction, <MATH><I>F</I>′<I>H</I>′<I>HF</I>=<I>H</I>′<I>TU</I>′<I>R</I>′-<I>HTUR</I></MATH>. +<p>Adding <I>H</I>′<I>IRH</I> to each side, we have +<MATH><I>F</I>′<I>IRF</I>=<I>IUU</I>′<I>R</I>′</MATH>. +<p>If each of these quadrilaterals is subtracted from <I>IJ</I>, +<MATH><I>FJR</I>′<I>F</I>′=<I>JU</I>′<I>UR</I></MATH>. +<p>The corresponding results are proved in III. 5, 11, 12, 14 +for the case where the ordinates through <I>RR</I>′ are drawn to +a <I>secondary</I> diameter, and in III. 15 for the case where <I>P, Q</I> +are on the original hyperbola and <I>R, R</I>′ on the conjugate +hyperbola. +<p>The importance of these propositions lies in the fact that +they are immediately used to prove the well-known theorems +about the rectangles contained by the segments of intersecting +chords and the harmonic properties of the pole and polar. +The former question is dealt with in III. 16-23, which give +a great variety of particular cases. We will give the proof +of one case, to the effect that, if <I>OP, OQ</I> be two tangents +to any conic and <I>Rr, R</I>′<I>r</I>′ be any two chords parallel to +them respectively and intersecting in <I>J</I>, an internal or external +point, +then <MATH><I>RJ.Jr</I>:<I>R</I>′<I>J.Jr</I>′=<I>OP</I><SUP>2</SUP>:<I>OQ</I><SUP>2</SUP>=(const.)</MATH>. +<p>We have +<MATH><I>RJ.Jr</I>=<I>RW</I><SUP>2</SUP>-<I>JW</I><SUP>2</SUP></MATH>, and <MATH><I>RW</I><SUP>2</SUP>:<I>JW</I><SUP>2</SUP>=▵<I>RUW</I>:▵<I>JU</I>′<I>W</I></MATH>; +therefore +<MATH><I>RJ.Jr</I>:<I>RW</I><SUP>2</SUP>=(<I>RW</I><SUP>2</SUP>-<I>JW</I><SUP>2</SUP>):<I>RW</I><SUP>2</SUP>=<I>JU</I>′<I>UR</I>:▵<I>RUW</I></MATH>. +<p>But <MATH><I>RW</I><SUP>2</SUP>:<I>OP</I><SUP>2</SUP>=▵<I>RUW</I>:▵<I>OPT</I></MATH>; +therefore, <I>ex aequali</I>, <MATH><I>RJ.Jr</I>:<I>OP</I><SUP>2</SUP>=<I>JU</I>′<I>UR</I>:▵<I>OPT</I></MATH>. +<pb n=153><head>THE <I>CONICS</I>, BOOK III</head> +<p>Similarly <MATH><I>R</I>′<I>M</I>′<SUP>2</SUP>:<I>JM</I>′<SUP>2</SUP>=▵<I>R</I>′<I>F</I>′<I>M</I>′:▵<I>JFM</I>′</MATH>, +whence <MATH><I>R</I>′<I>J.Jr</I>′:<I>R</I>′<I>M</I>′<SUP>2</SUP>=<I>FJR</I>′<I>F</I>′:▵<I>R</I>′<I>F</I>′<I>M</I>′</MATH>. +<p>But <MATH><I>R</I>′<I>M</I>′<SUP>2</SUP>:<I>OQ</I><SUP>2</SUP>=▵<I>R</I>′<I>F</I>′<I>M</I>′:▵<I>OQE</I></MATH>; +therefore, <I>ex aequali</I>, <MATH><I>R</I>′<I>J.Jr</I>′:<I>OQ</I><SUP>2</SUP>=<I>FJR</I>′<I>F</I>′:▵<I>OQE</I></MATH>. +<p>It follows, since <MATH><I>FJR</I>′<I>F</I>′=<I>JU</I>′<I>UR</I></MATH>, and <MATH>▵<I>OPT</I>=▵<I>OQE</I></MATH>, +that <MATH><I>RJ.Jr</I>:<I>OP</I><SUP>2</SUP>=<I>R</I>′<I>J.Jr</I>′:<I>OQ</I><SUP>2</SUP></MATH>, +or <MATH><I>RJ.Jr</I>:<I>R</I>′<I>J.Jr</I>′=<I>OP</I><SUP>2</SUP>:<I>OQ</I><SUP>2</SUP></MATH>. +<p>If we had taken chords <I>Rr</I><SUB>1</SUB>, <I>R</I>′<I>r</I><SUB>1</SUB>′ parallel respectively to +<I>OQ, OP</I> and intersecting in <I>I</I>, an internal or external point, +we should have in like manner +<MATH><I>RI.Ir</I><SUB>1</SUB>:<I>R</I>′<I>I.Ir</I><SUB>1</SUB>′=<I>OQ</I><SUP>2</SUP>:<I>OP</I><SUP>2</SUP></MATH>. +<p>As a particular case, if <I>PP</I>′ be a diameter, and <I>Rr, R</I>′<I>r</I>′ be +chords parallel respectively to the tangent at <I>P</I> and the +diameter <I>PP</I>′ and intersecting in <I>I</I>, then (as is separately +proved) +<MATH><I>RI.Ir</I>:<I>R</I>′<I>I.Ir</I>′=<I>p</I>:<I>PP</I>′</MATH>. +The corresponding results are proved in the cases where certain +of the points lie on the conjugate hyperbola. +<p>The six following propositions about the segments of inter- +secting chords (III. 24-9) refer to two chords in conjugate +hyperbolas or in an ellipse drawn parallel respectively to two +conjugate diameters <I>PP</I>′, <I>DD</I>′, and the results in modern form +are perhaps worth quoting. If <I>Rr, R</I>′<I>r</I>′ be two chords so +drawn and intersecting in <I>O</I>, then +<p>(<I>a</I>) in the conjugate hyperbolas +<MATH><I>RO.Or</I>/<I>CP</I><SUP>2</SUP>±<I>R</I>′<I>O.Or</I>′/<I>CD</I><SUP>2</SUP>=2</MATH>, +and <MATH>(<I>RO</I><SUP>2</SUP>+<I>Or</I><SUP>2</SUP>):(<I>R</I>′<I>O</I><SUP>2</SUP>+<I>Or</I>′<SUP>2</SUP>)=<I>CP</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP></MATH>; +<p>(<I>b</I>) in the ellipse +<MATH>(<I>RO</I><SUP>2</SUP>+<I>Or</I><SUP>2</SUP>)/<I>CP</I><SUP>2</SUP>+(<I>R</I>′<I>O</I><SUP>2</SUP>+<I>Or</I>′<SUP>2</SUP>)/<I>CD</I><SUP>2</SUP>=4</MATH>. +<pb n=154><head>APOLLONIUS OF PERGA</head> +<p>The general propositions containing the harmonic properties +of the pole and polar of a conic are III. 37-40, which prove +that in any conic, if <I>TQ, Tq</I> be tangents, and if <I>Qq</I> the chord +of contact be bisected in <I>V</I>, then +<p>(1) if any straight line through <I>T</I> meet the conic in <I>R</I>′, <I>R</I> and +<I>Qq</I> in <I>I</I>, then (Fig. 1) <MATH><I>RT</I>:<I>TR</I>′=<I>RI</I>:<I>IR</I>′</MATH>; +<FIG> +<CAP>FIG. 1.</CAP> +<p>(2) if any straight line through <I>V</I> meet the conic in <I>R, R</I>′ and +the parallel through <I>T</I> to <I>Qq</I> in <I>O</I>, then (Fig. 2) +<MATH><I>RO</I>:<I>OR</I>′=<I>RV</I>:<I>VR</I>′</MATH>. +<FIG> +<CAP>FIG. 2.</CAP> +<p>The above figures represent theorem (1) for the parabola and +theorem (2) for the ellipse. +<pb n=155><head>THE <I>CONICS</I>, BOOK III</head> +<p>To prove (1) we have +<MATH><I>R</I>′<I>I</I><SUP>2</SUP>:<I>IR</I><SUP>2</SUP>=<I>H</I>′<I>Q</I><SUP>2</SUP>:<I>QH</I><SUP>2</SUP>=▵<I>H</I>′<I>F</I>′<I>Q</I>:▵<I>HFQ</I>=<I>H</I>′<I>TU</I>′<I>R</I>′:<I>HTUR</I> +(III. 2, 3, &c.)</MATH>. +<p>Also <MATH><I>R</I>′<I>T</I><SUP>2</SUP>:<I>TR</I><SUP>2</SUP>=<I>R</I>′<I>U</I>′<SUP>2</SUP>:<I>UR</I><SUP>2</SUP>=▵<I>R</I>′<I>U</I>′<I>W</I>′:▵<I>RUW</I></MATH>, +and <MATH><I>R</I>′<I>T</I><SUP>2</SUP>:<I>TR</I><SUP>2</SUP>=<I>TW</I>′<SUP>2</SUP>:<I>TW</I><SUP>2</SUP>=▵<I>TH</I>′<I>W</I>′:▵<I>THW</I></MATH>, +so that <MATH><I>R</I>′<I>T</I><SUP>2</SUP>:<I>TR</I><SUP>2</SUP>=▵<I>TH</I>′<I>W</I>′-▵<I>R</I>′<I>U</I>′<I>W</I>′:▵<I>THW</I>-▵<I>RUW</I> +=<I>H</I>′<I>TU</I>′<I>R</I>′:<I>HTUR</I> +=<I>R</I>′<I>I</I><SUP>2</SUP>:<I>IR</I><SUP>2</SUP></MATH>, from above. +<p>To prove (2) we have +<MATH><I>RV</I><SUP>2</SUP>:<I>VR</I>′<SUP>2</SUP>=<I>RU</I><SUP>2</SUP>:<I>R</I>′<I>U</I>′<SUP>2</SUP>=▵<I>RUW</I>:▵<I>R</I>′<I>U</I>′<I>W</I>′</MATH>, +and also +<MATH>=<I>HQ</I><SUP>2</SUP>:<I>QH</I>′<SUP>2</SUP>=▵<I>HFQ</I>:▵<I>H</I>′<I>F</I>′<I>Q</I>=<I>HTUR</I><note>Where a quadrilateral, as <I>HTUR</I> in the figure, is a cross-quadri- +lateral, the area is of course the difference between the two triangles +which it forms, as <I>HTW-RUW.</I></note>:<I>H</I>′<I>TU</I>′<I>R</I>′</MATH>, +so that +<MATH><I>RV</I><SUP>2</SUP>:<I>VR</I>′<SUP>2</SUP>=<I>HTUR</I>±▵<I>RUW</I>:<I>H</I>′<I>TU</I>′<I>R</I>′±▵<I>R</I>′<I>U</I>′<I>W</I>′ +=▵<I>THW</I>:▵<I>TH</I>′<I>W</I>′ +=<I>TW</I><SUP>2</SUP>:<I>TW</I>′<SUP>2</SUP> +=<I>RO</I><SUP>2</SUP>:<I>OR</I>′<SUP>2</SUP></MATH>. +<p>Props. III. 30-6 deal separately with the particular cases +in which (<I>a</I>) the transversal is parallel to an asymptote of the +hyperbola or (<I>b</I>) the chord of contact is parallel to an asymp- +tote, i.e. where one of the tangents is an asymptote, which is +the tangent at infinity. +<p>Next we have propositions about intercepts made by two +tangents on a third: If the tangents at three points of a +parabola form a triangle, all three tangents will be cut by the +points of contact in the same proportion (III. 41); if the tan- +gents at the extremities of a diameter <I>PP</I>′ of a central conic +are cut in <I>r, r</I>′ by any other tangent, <MATH><I>Pr.P</I>′<I>r</I>′=<I>CD</I><SUP>2</SUP> (III. 42)</MATH>; +if the tangents at <I>P, Q</I> to a hyperbola meet the asymptotes in +<pb n=156><head>APOLLONIUS OF PERGA</head> +<I>L, L</I>′ and <I>M, M</I>′ respectively, then <I>L</I>′<I>M, LM</I>′ are both parallel +to <I>PQ</I> (III. 44). +<p>The first of these propositions asserts that, if the tangents at +three points <I>P, Q, R</I> of a parabola form a triangle <I>pqr</I>, then +<MATH><I>Pr</I>:<I>rq</I>=<I>rQ</I>:<I>Qp</I>=<I>qp</I>:<I>pR</I></MATH>. +<p>From this property it is easy to deduce the Cartesian +equation of a parabola referred to two fixed tangents as +coordinate axes. Taking.<I>qR, qP</I> as fixed coordinate axes, we +find the locus of <I>Q</I> thus. Let <I>x, y</I> be the coordinates of <I>Q.</I> +Then, if <MATH><I>qp</I>=<I>x</I><SUB>1</SUB>, <I>qr</I>=<I>y</I><SUB>1</SUB>, <I>qR</I>=<I>h, qP</I>=<I>k</I></MATH>, we have +<MATH><I>x</I>/(<I>x</I><SUB>1</SUB>-<I>x</I>)=<I>rQ</I>/<I>Qp</I>=(<I>y</I><SUB>1</SUB>-<I>y</I>)/<I>y</I>=(<I>k-y</I><SUB>1</SUB>)/<I>y</I><SUB>1</SUB>=<I>x</I><SUB>1</SUB>/(<I>h-x</I><SUB>1</SUB>)</MATH>. +<p>From these equations we derive +<MATH><I>x</I><SUB>1</SUB><SUP>2</SUP>=<I>hx, y</I><SUB>1</SUB><SUP>2</SUP>=<I>ky</I></MATH>; +also, since <MATH><I>x</I><SUB>1</SUB>/<I>x</I>=<I>y</I><SUB>1</SUB>/(<I>y</I><SUB>1</SUB>-<I>y</I>)</MATH>, we have <MATH><I>x</I>/<I>x</I><SUB>1</SUB>+<I>y</I>/<I>y</I><SUB>1</SUB>=1</MATH>. +<p>By substituting for <I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB> the values √(<I>hx</I>), √(<I>ky</I>) we +obtain +<MATH>(<I>x</I>/<I>h</I>)<SUP>1/2</SUP>+(<I>y</I>/<I>k</I>)<SUP>1/2</SUP>=1</MATH>. +<p>The focal properties of central conics are proved in +III. 45-52 without any reference to the directrix; there is +no mention of the focus of a parabola. The foci are called +‘the points arising out of the application’ (<G>ta\ e)k th=s para- +bolh=s gino/mena shmei=a</G>), the meaning being that <I>S,S</I>′ are taken +on the axis <I>AA</I>′ such that <MATH><I>AS.SA</I>′=<I>AS</I>′.<I>S</I>′<I>A</I>′=1/4<I>P<SUB>a</SUB>.AA</I>′</MATH> +or <I>CB</I><SUP>2</SUP>, that is, in the phraseology of application of areas, +a rectangle is applied to <I>AA</I>′ as base equal to one-fourth +part of the ‘figure’, and in the case of the hyperbola ex- +ceeding, but in the case of the ellipse falling short, by a +square figure. The foci being thus found, it is proved that, +if the tangents <I>Ar, A</I>′<I>r</I>′ at the extremities of the axis are met +by the tangent at any point <I>P</I> in <I>r, r</I>′ respectively, <I>rr</I>′ subtends +a right angle at <I>S, S</I>′, and the angles <I>rr</I>′<I>S, A</I>′<I>r</I>′<I>S</I>′ are equal, as +also are the angles <I>r</I>′<I>rS</I>′, <I>ArS</I> (III. 45, 46). It is next shown +that, if <I>O</I> be the intersection of <I>rS</I>′, <I>r</I>′<I>S</I>, then <I>OP</I> is perpen- +dicular to the tangent at <I>P</I> (III. 47). These propositions are +<pb n=157><head>THE <I>CONICS</I>, BOOK III</head> +used to prove that the focal distances of <I>P</I> make equal angles +with the tangent at <I>P</I> (III. 48). In III. 49-52 follow the +other ordinary properties, that, if <I>SY</I> be perpendicular to +the tangent at <I>P</I>, the locus of <I>Y</I> is the circle on <I>AA</I>′ as +diameter, that the lines from <I>C</I> drawn parallel to the focal +distances to meet the tangent at <I>P</I> are equal to <I>CA</I>, and that +the sum or difference of the focal distances of any point is +equal to <I>AA</I>′. +<p>The last propositions of Book III are of use with reference +to the locus with respect to three or four lines. They are as +follows. +<p>1. If <I>PP</I>′ be a diameter of a central conic, and if <I>PQ, P</I>′<I>Q</I> +drawn to any other point <I>Q</I> of the conic meet the tangents at +<I>P</I>′, <I>P</I> in <I>R</I>′, <I>R</I> respectively, then <MATH><I>PR.P</I>′<I>R</I>′=4<I>CD</I><SUP>2</SUP> (III. 53)</MATH>. +<p>2. If <I>TQ, TQ</I>′ be two tangents to a conic, <I>V</I> the middle point +of <I>QQ</I>′, <I>P</I> the point of contact of the tangent parallel to <I>QQ</I>′, +and <I>R</I> any other point on the conic, let <I>Qr</I> parallel to <I>TQ</I>′ +meet <I>Q</I>′<I>R</I> in <I>r</I>, and <I>Q</I>′<I>r</I>′ parallel to <I>TQ</I> meet <I>QR</I> in <I>r</I>′; then +<MATH><I>Qr.Q</I>′<I>r</I>′:<I>QQ</I>′<SUP>2</SUP>=(<I>PV</I><SUP>2</SUP>:<I>PT</I><SUP>2</SUP>).(<I>TQ.TQ</I>′:<I>QV</I><SUP>2</SUP>). (III. 54, 56.)</MATH> +<p>3. If the tangents are tangents to opposite branches of a +hyperbola and meet in <I>t</I>, and if <I>R, r, r</I>′ are taken as before, +while <I>tq</I> is half the chord through <I>t</I> parallel to <I>QQ</I>′, then +<MATH><I>Qr.Q</I>′<I>r</I>′:<I>QQ</I>′<SUP>2</SUP>=<I>tQ.tQ</I>′:<I>tq</I><SUP>2</SUP>. (III. 55.)</MATH> +<p>The second of these propositions leads at once to the three- +line locus, and from this we easily obtain the Cartesian +equation to a conic with reference to two fixed tangents as +axes, where the lengths of the tangents are <I>h, k</I>, viz. +<MATH>(<I>x</I>/<I>h</I>+<I>y</I>/<I>k</I>-1)<SUP>2</SUP>=2<G>l</G>(<I>xy</I>/<I>hk</I>)<SUP>1/2</SUP></MATH>. +<p>Book IV is on the whole dull, and need not be noticed at +length. Props. 1-23 prove the converse of the propositions in +Book III about the harmonic properties of the pole and polar +for a large number of particular cases. One of the proposi- +tions (IV. 9) gives a method of drawing two tangents to +a conic from an external point <I>T.</I> Draw any two straight +lines through <I>T</I> cutting the conic in <I>Q, Q</I>′ and in <I>R, R</I>′ respec- +<pb n=158><head>APOLLONIUS OF PERGA</head> +tively. Take <I>O</I> on <I>QQ</I>′ and <I>O</I>′ on <I>RR</I>′ so that <I>TQ</I>′, <I>TR</I>′ are +harmonically divided. The intersections of <I>OO</I>′ produced with +the conic give the two points of contact required. +<p>The remainder of the Book (IV.24-57) deals with intersecting +conics, and the number of points in which, in particular cases, +they can intersect or touch. IV. 24 proves that no two conics +can meet in such a way that part of one of them is common +to both, while the rest is not. The rest of the propositions +can be divided into five groups, three of which can be brought +under one general enunciation. Group I consists of particular +cases depending on the more elementary considerations affect- +ing conics: e.g. two conics having their concavities in oppo- +site directions will not meet in more than two points (IV. 35); +if a conic meet one branch of a hyperbola, it will not meet +the other branch in more points than two (IV. 37); a conic +touching one branch of a hyperbola with its concave side +will not meet the opposite branch (IV. 39). IV. 36, 41, 42, 45, +54 belong to this group. Group II contains propositions +(IV. 25, 38, 43, 44, 46, 55) showing that no two conics +(including in the term the double-branch hyperbola) can +intersect in more than four points. Group III (IV. 26, 47, 48, +49, 50, 56) are particular cases of the proposition that two +conics which touch at one point cannot intersect at more than +two other points. Group IV (IV. 27, 28, 29, 40, 51, 52, 53, 57) +are cases of the proposition that no two conics which touch +each other at two points can intersect at any other point. +Group V consists of propositions about double contact. A +parabola cannot touch another parabola in more points than +one (IV. 30); this follows from the property <MATH><I>TP</I>=<I>PV</I></MATH>. A +parabola, if it fall outside a hyperbola, cannot have double +contact with it (IV. 31); it is shown that for the hyperbola +<MATH><I>PV</I>><I>PT</I></MATH>, while for the parabola <MATH><I>P</I>′<I>V</I>=<I>P</I>′<I>T</I></MATH>; therefore the +hyperbola would fall outside the parabola, which is impossible. +A parabola cannot have internal double contact with an ellipse +or circle (IV. 32). A hyperbola cannot have double contact +with another hyperbola having the same centre (IV. 33); +proved by means of <MATH><I>CV.CT</I>=<I>CP</I><SUP>2</SUP></MATH>. If an ellipse have double +contact with an ellipse or a circle, the chord of contact will +pass through the centre (IV. 34). +<p>Book V is of an entirely different order, indeed it is the +<pb n=159><head>THE <I>CONICS</I>, BOOKS IV-V</head> +most remarkable of the extant Books. It deals with normals +to conics regarded as <I>maximum</I> and <I>minimum</I> straight lines +drawn from particular points to the curve. Included in it are +a series of propositions which, though worked out by the +purest geometrical methods, actually lead immediately to the +determination of the evolute of each of the three conics; that +is to say, the Cartesian equations to the evolutes can be easily +deduced from the results obtained by Apollonius. There can +be no doubt that the Book is almost wholly original, and it is +a veritable geometrical <I>tour de force.</I> +<p>Apollonius in this Book considers various points and classes +of points with reference to the maximum or minimum straight +lines which it is possible to draw from them to the conics, +i.e. as the feet of normals to the curve. He begins naturally +with points on the axis, and he takes first the point <I>E</I> where +<I>AE</I> measured along the axis from the vertex <I>A</I> is 1/2<I>p, p</I> being +the principal parameter. The first three propositions prove +generally and for certain particular cases that, if in an ellipse +or a hyperbola <I>AM</I> be drawn at right angles to <I>AA</I>′ and equal +to 1/2<I>p</I>, and if <I>CM</I> meet the ordinate <I>PN</I> of any point <I>P</I> of the +curve in <I>H</I>, then <MATH><I>PN</I><SUP>2</SUP>=2 (quadrilateral <I>MANH</I>)</MATH>; this is a +lemma used in the proofs of later propositions, V. 5, 6, &c. +Next, in V. 4, 5, 6, he proves that, if <MATH><I>AE</I>=1/2<I>p</I></MATH>, then <I>AE</I> is the +<I>minimum</I> straight line from <I>E</I> to the curve, and if <I>P</I> be any +other point on it, <I>PE</I> increases as <I>P</I> moves farther away from +<I>A</I> on either side; he proves in fact that, if <I>PN</I> be the ordinate +from <I>P</I>, +<p>(1) in the case of the parabola <MATH><I>PE</I><SUP>2</SUP>=<I>AE</I><SUP>2</SUP>+<I>AN</I><SUP>2</SUP></MATH>, +<p>(2) in the case of the hyperbola or ellipse +<MATH><I>PE</I><SUP>2</SUP>=<I>AE</I><SUP>2</SUP>+<I>AN</I><SUP>2</SUP>.(<I>AA</I>′±<I>p</I>)/<I>AA</I>′</MATH>, +where of course <MATH><I>p</I>=<I>BB</I>′<SUP>2</SUP>/<I>AA</I>′</MATH>, and therefore <MATH>(<I>AA</I>′±<I>p</I>)/<I>AA</I>′</MATH> +is equivalent to what we call <I>e</I><SUP>2</SUP>, the square of the eccentricity. +It is also proved that <I>EA</I>′ is the <I>maximum</I> straight line from +<I>E</I> to the curve. It is next proved that, if <I>O</I> be any point on +the axis between <I>A</I> and <I>E, OA</I> is the minimum straight line +from <I>O</I> to the curve and, if <I>P</I> is any other point on the eurve, +<I>OP</I> increases as <I>P</I> moves farther from <I>A</I> (V. 7). +<pb n=160><head>APOLLONIUS OF PERGA</head> +<p>Next Apollonius takes points <I>G</I> on the axis at a distance +from <I>A</I> greater than 1/2<I>p</I>, and he proves that the <I>minimum</I> +straight line from <I>G</I> to the curve (i.e. the normal) is <I>GP</I>, +where <I>P</I> is such a point that +<p>(1) in the case of the parabola <MATH><I>NG</I>=1/2<I>p</I></MATH>; +<p>(2) in the case of the central conic <MATH><I>NG</I>:<I>CN</I>=<I>p</I>:<I>AA</I>′</MATH>; +and, if <I>P</I>′ is any other point on the conic, <I>P</I>′<I>G</I> increases as <I>P</I>′ +moves away from <I>P</I> on either side; this is proved by show- +ing that +<p>(1) for the parabola <MATH><I>P</I>′<I>G</I><SUP>2</SUP>=<I>PG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP></MATH>; +<p>(2) for the central conic <MATH><I>P</I>′<I>G</I><SUP>2</SUP>=<I>PG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP>.(<I>AA</I>′±<I>p</I>)/<I>AA</I>′</MATH>. +<FIG> +<p>As these propositions contain the fundamental properties of +the subnormals, it is worth while to reproduce Apollonius's +proofs. +<p>(1) In the parabola, if <I>G</I> be any point on the axis such that +<MATH><I>AG</I>>1/2<I>p</I></MATH>, measure <I>GN</I> towards <I>A</I> equal to 1/2<I>p</I>. Let <I>PN</I> be +the ordinate through <I>N, P</I>′ any other point on the curve. +Then shall <I>PG</I> be the minimum line from <I>G</I> to the curve, &c. +<pb n=161><head>THE <I>CONICS</I>, BOOK V</head> +<p>We have <MATH><I>P</I>′<I>N</I>′<SUP>2</SUP>=<I>p.AN</I>′=2<I>NG.AN</I>′</MATH>; +and <MATH><I>N</I>′<I>G</I><SUP>2</SUP>=<I>NN</I>′<SUP>2</SUP>+<I>NG</I><SUP>2</SUP>±2<I>NG.NN</I>′</MATH>, +according to the position of <I>N</I>′. +<p>Therefore <MATH><I>P</I>′<I>G</I><SUP>2</SUP>=2<I>NG.AN</I>+<I>NG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP> +=<I>PN</I><SUP>2</SUP>+<I>NG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP> +=<I>PG</I><SUP>2</SUP>+<I>NN</I>′<SUP>2</SUP></MATH>; +and the proposition is proved. +<p>(2) In the case of the central conic, take <I>G</I> on the axis such +that <MATH><I>AG</I> > 1/2<I>p</I></MATH>, and measure <I>GN</I> towards <I>A</I> such that +<MATH><I>NG</I>:<I>CN</I>=<I>p</I>:<I>AA</I>′</MATH>. +Draw the ordinate <I>PN</I> through <I>N</I>, and also the ordinate <I>P</I>′<I>N</I>′ +from any other point <I>P</I>′. +<p>We have first to prove the lemma (V. 1, 2, 3) that, if <I>AM</I> be +drawn perpendicular to <I>AA</I>′ and equal to 1/2<I>p</I>, and if <I>CM</I>, +produced if necessary, meet <I>PN</I> in <I>H</I>, then +<MATH><I>PN</I><SUP>2</SUP>=2(quadrilateral <I>MANH</I>)</MATH>. +<p>This is easy, for, if <MATH><I>AL</I>(=2<I>AM</I>)</MATH> be the parameter, and <I>A</I>′<I>L</I> +meet <I>PN</I> in <I>R</I>, then, by the property of the curve, +<MATH><I>PN</I><SUP>2</SUP>=<I>AN.NR</I> +=<I>AN</I>(<I>NH</I>+<I>AM</I>) +=2(quadrilateral <I>MANH</I>)</MATH>. +<p>Let <I>GH</I>, produced if necessary, meet <I>P</I>′<I>N</I>′ in <I>H</I>′. From <I>H</I> +draw <I>HI</I> perpendicular to <I>P</I>′<I>H</I>′. +<p>Now, since, by hypothesis, <MATH><I>NG</I>:<I>CN</I>=<I>p</I>:<I>AA</I>′ +=<I>AM</I>:<I>AC</I> +=<I>HN</I>:<I>NC</I></MATH>, +<MATH><I>NH</I>=<I>NG</I></MATH>, whence also <MATH><I>H</I>′<I>N</I>′=<I>N</I>′<I>G</I></MATH>. +<p>Therefore <MATH><I>NG</I><SUP>2</SUP>=2▵<I>HNG, N</I>′<I>G</I><SUP>2</SUP>=2▵<I>H</I>′<I>N</I>′<I>G</I></MATH>. +<p>And <MATH><I>PN</I><SUP>2</SUP>=2(<I>MANH</I>)</MATH>; +therefore <MATH><I>PG</I><SUP>2</SUP>=<I>NG</I><SUP>2</SUP>+<I>PN</I><SUP>2</SUP>=2(<I>AMHG</I>)</MATH>. +<pb n=162><head>APOLLONIUS OF PERGA</head> +<p>Similarly, if <I>CM</I> meets <I>P</I>′<I>N</I>′ in <I>K</I>, +<MATH><I>P</I>′<I>G</I><SUP>2</SUP>=<I>N</I>′<I>G</I><SUP>2</SUP>+<I>P</I>′<I>N</I>′<SUP>2</SUP> +=2▵<I>H</I>′<I>N</I>′<I>G</I>+2(<I>AMKN</I>′) +=2(<I>AMHG</I>)+2▵<I>HH</I>′<I>K</I></MATH>. +<p>Therefore, by subtraction, +<MATH><I>P</I>′<I>G</I><SUP>2</SUP>-<I>PG</I><SUP>2</SUP>=2▵<I>HH</I>′<I>K</I> +=<I>HI</I>.(<I>H</I>′<I>I</I>±<I>IK</I>) +=<I>HI</I>.(<I>HI</I>±<I>IK</I>) +=<I>HI</I><SUP>2</SUP>.(<I>CA</I>±<I>AM</I>)/<I>CA</I> +=<I>NN</I>′<SUP>2</SUP>.(<I>AA</I>′±<I>p</I>)/<I>AA</I>′</MATH>; +which proves the proposition. +<p>If <I>O</I> be any point on <I>PG, OP</I> is the minimum straight line +from <I>O</I> to the eurve, and <I>OP</I>′ increases as <I>P</I>′ moves away from +<I>P</I> on either side; this is proved in V. 12. (Since <MATH><I>P</I>′<I>G</I> > <I>PG</I></MATH>, +<MATH>∠<I>GPP</I>′ > ∠<I>GP</I>′<I>P</I></MATH>; therefore, <I>a fortiori</I>, <MATH>∠<I>OPP</I>′ > ∠<I>OP</I>′<I>P</I></MATH>, +and <MATH><I>OP</I>′ > <I>OP</I></MATH>.) +<p>Apollonius next proves the corresponding propositions with +reference to points on the <I>minor</I> axis of an ellipse. If <I>p</I>′ be +the parameter of the ordinates to the minor axis, <MATH><I>p</I>′=<I>AA</I>′<SUP>2</SUP>/<I>BB</I>′</MATH>, +or <MATH>1/2<I>p</I>′=<I>CA</I><SUP>2</SUP>/<I>CB</I></MATH>. If now <I>E</I>′ be so taken that <MATH><I>BE</I>′=1/2<I>p</I>′</MATH>, +then <I>BE</I>′ is the <I>maximum</I> straight line from <I>E</I>′ to the curve +and, if <I>P</I> be any other point on it, <I>E</I>′<I>P</I> diminishes as <I>P</I> moves +farther from <I>B</I> on either side, and <I>E</I>′<I>B</I>′ is the <I>minimum</I> +straight line from <I>E</I>′ to the curve. It is, in fact, proved that +<MATH><I>E</I>′<I>B</I><SUP>2</SUP>-<I>E</I>′<I>P</I><SUP>2</SUP>=<I>Bn</I><SUP>2</SUP>.(<I>p</I>′-<I>BB</I>′)/<I>BB</I>′</MATH>, where <I>Bn</I> is the abscissa of <I>P</I> +(V. 16-18). If <I>O</I> be any point on the minor axis such that +<MATH><I>BO</I> > <I>BE</I>′</MATH>, then <I>OB</I> is the <I>maximum</I> straight line from <I>O</I> to +the curve, &c. (V. 19). +<p>If <I>g</I> be a point on the minor axis such that <MATH><I>Bg</I> > <I>BC</I></MATH>, but +<MATH><I>Bg</I> < 1/2<I>p</I>′</MATH>, and if <I>Cn</I> be measured towards <I>B</I> so that +<MATH><I>Cn</I>:<I>ng</I>=<I>BB</I>′:<I>p</I>′</MATH>, +then <I>n</I> is the foot of the ordinates of two points <I>P</I> such that +<I>Pg</I> is the <I>maximum</I> straight line from <I>g</I> to the curve. Also, +<pb n=163><head>THE <I>CONICS</I>, BOOK V</head> +if <I>P</I>′ be any other point on it, <I>P</I>′<I>g</I> diminishes as <I>P</I>′ moves +farther from <I>P</I> on either side to <I>B</I> or <I>B</I>′, and +<MATH><I>Pg</I><SUP>2</SUP>-<I>P</I>′<I>g</I><SUP>2</SUP>=<I>nn</I>′<SUP>2</SUP>.(<I>p</I>′-<I>BB</I>′)/<I>BB</I>′</MATH> or +<MATH><I>nn</I>′<SUP>2</SUP>.(<I>CA</I><SUP>2</SUP>-<I>CB</I><SUP>2</SUP>)/<I>CB</I><SUP>2</SUP></MATH>. +If <I>O</I> be any point on <I>Pg</I> produced beyond the minor axis, <I>PO</I> +is the <I>maximum</I> straight line from <I>O</I> to the same part of the +ellipse for which <I>Pg</I> is a maximum, i.e. the semi-ellipse <I>BPB</I>′, +&c. (V. 20-2). +<p>In V. 23 it is proved that, if <I>g</I> is on the minor axis, and <I>gP</I> +a maximum straight line to the curve, and if <I>Pg</I> meets <I>AA</I>′ +in <I>G</I>, then <I>GP</I> is the <I>minimum</I> straight line from <I>G</I> to the +curve; this is proved by similar triangles. Only one normal +can be drawn from any one point on a conic (V. 24-6). The +normal at any point <I>P</I> of a conic, whether regarded as a +minimum straight line from <I>G</I> on the major axis or (in the +case of the ellipse) as a <I>maximum</I> straight line from <I>g</I> on the +minor axis, is perpendicular to the tangent at <I>P</I> (V. 27-30); +in general (1) if <I>O</I> be any point within a conic, and <I>OP</I> be +a maximum or a minimum straight line from <I>O</I> to the conic, +the straight line through <I>P</I> perpendicular to <I>PO</I> touches the +conic, and (2) if <I>O</I>′ be any point on <I>OP</I> produced outside the +conic, <I>O</I>′<I>P</I> is the minimum straight line from <I>O</I>′ to the conic, +&c. (V. 31-4). +<C><I>Number of normals from a point.</I></C> +<p>We now come to propositions about two or more normals +meeting at a point. If the normal at <I>P</I> meet the axis of +a parabola or the axis <I>AA</I>′ of a hyperbola or ellipse in <I>G</I>, the +angle <I>PGA</I> increases as <I>P</I> or <I>G</I> moves farther away from <I>A</I>, +but in the case of the hyperbola the angle will always be less +than the complement of half the angle between the asymptotes. +Two normals at points on the same side of <I>AA</I>′ will meet on +the opposite side of that axis; and two normals at points on +the same quadrant of an ellipse as <I>AB</I> will meet at a point +within the angle <I>ACB</I>′ (V. 35-40). In a parabola or an +ellipse any normal <I>PG</I> will meet the curve again; in the +hyperbola, (1) if <I>AA</I>′ be not greater than <I>p</I>, no normal can +meet the curve at a second point on the same branch, but +<pb n=164><head>APOLLONIUS OF PERGA</head> +(2) if <MATH><I>AA</I>′ > <I>p</I></MATH>, some normals will meet the same branch again +and others not (V. 41-3). +<p>If <I>P</I><SUB>1</SUB><I>G</I><SUB>1</SUB>, <I>P</I><SUB>2</SUB><I>G</I><SUB>2</SUB> be normals at points on one side of the axis of +a conic meeting in <I>O</I>, and if <I>O</I> be joined to any other point <I>P</I> +on the conic (it being further supposed in the case of the +ellipse that all three lines <I>OP</I><SUB>1</SUB>, <I>OP</I><SUB>2</SUB>, <I>OP</I> cut the same half of +the axis), then +<p>(1) <I>OP</I> cannot be a normal to the curve; +<p>(2) if <I>OP</I> meet the axis in <I>K</I>, and <I>PG</I> be the normal at <I>P, AG</I> +is less or greater than <I>AK</I> according as <I>P</I> does or does not lie +between <I>P</I><SUB>1</SUB> and <I>P</I><SUB>2</SUB>. +<p>From this proposition it is proved that (1) three normals at +points on one quadrant of an ellipse cannot meet at one point, +and (2) four normals at points on one semi-ellipse bounded by +the major axis cannot meet at one point (V. 44-8). +<p>In any conic, if <I>M</I> be any point on the axis such that <I>AM</I> +is not greater than 1/2<I>p</I>, and if <I>O</I> be any point on the double +ordinate through <I>M</I>, then no straight line drawn to any point +on the curve on the other side of the axis from <I>O</I> and meeting +the axis between <I>A</I> and <I>M</I> can be a normal (V. 49, 50). +<C><I>Propositions leading immediately to the determination +of the</I> evolute <I>of a conic.</I></C> +<p>These great propositions are V. 51, 52, to the following +effect: +<p>If <I>AM</I> measured along the axis be greater than 1/2<I>p</I> (but in +the case of the ellipse less than <I>AC</I>), and if <I>MO</I> be drawn per- +pendicular to the axis, then a certain length (<I>y</I>, say) can be +assigned such that +<p>(<I>a</I>) if <MATH><I>OM</I> > <I>y</I></MATH>, no normal can be drawn through <I>O</I> which cuts +the axis; but, if <I>OP</I> be any straight line drawn to the curve +cutting the axis in <MATH><I>K, NK</I> < <I>NG</I></MATH>, where <I>PN</I> is the ordinate +and <I>PG</I> the normal at <I>P</I>; +<p>(<I>b</I>) if <MATH><I>OM</I>=<I>y</I></MATH>, only one normal can be so drawn through <I>O</I>, +and, if <I>OP</I> be any other straight line drawn to the curve and +cutting the axis in <MATH><I>K, NK</I> < <I>NG</I></MATH>, as before; +<p>(<I>c</I>) if <MATH><I>OM</I> < <I>y</I></MATH>, two normals can be so drawn through <I>O</I>, and, if +<I>OP</I> be any other straight line drawn to the curve, <I>NK</I> is +<pb n=165><head>THE <I>CONICS</I>, BOOK V</head> +greater or less than <I>NG</I> according as <I>OP</I> is or is not inter- +mediate between the two normals (V. 51, 52). +<p>The proofs are of course long and complicated. The length +<I>y</I> is determined in this way: +<p>(1) In the case of the parabola, measure <I>MH</I> towards the +vertex equal to 1/2<I>p</I>, and divide <I>AH</I> at <I>N</I><SUB>1</SUB> so that <MATH><I>HN</I><SUB>1</SUB>=2<I>N</I><SUB>1</SUB><I>A</I></MATH>. +The length <I>y</I> is then taken such that +<MATH><I>y</I>:<I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB>=<I>N</I><SUB>1</SUB><I>H</I>:<I>HM</I></MATH>, +where <I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB> is the ordinate passing through <I>N</I><SUB>1</SUB>; +<p>(2) In the case of the hyperbola and ellipse, we have +<MATH><I>AM</I> > 1/2<I>p</I></MATH>, so that <MATH><I>CA</I>:<I>AM</I> < <I>AA</I>′:<I>p</I></MATH>; therefore, if <I>H</I> be taken +on <I>AM</I> such that <MATH><I>CH</I>:<I>HM</I>=<I>AA</I>′:<I>p, H</I></MATH> will fall between <I>A</I> +and <I>M.</I> +<p>Take two mean proportionals <I>CN</I><SUB>1</SUB>, <I>CI</I> between <I>CA</I> and <I>CH</I>, +and let <I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB> be the ordinate through <I>N</I><SUB>1</SUB>. +<p>The length <I>y</I> is then taken such that +<MATH><I>y</I>:<I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB>=(<I>CM</I>:<I>MH</I>).(<I>HN</I><SUB>1</SUB>:<I>N</I><SUB>1</SUB><I>C</I>)</MATH>. +<p>In the case (<I>b</I>), where <MATH><I>OM</I>=<I>y, O</I></MATH> is the point of intersection +of consecutive normals, i.e. <I>O</I> is the centre of curvature at the +point <I>P</I>; and, by considering the coordinates of <I>O</I> with reference +to two coordinate axes, we can derive the Cartesian equations +of the evolutes. E. g. (1) in the case of the parabola let the +coordinate axes be the axis and the tangent at the vertex. +Then <MATH><I>AM</I>=<I>x</I></MATH>, <MATH><I>OM</I>=<I>y</I></MATH>. Let <MATH><I>p</I>=4<I>a</I></MATH>; then +<MATH><I>HM</I>=2<I>a</I></MATH>, <MATH><I>N</I><SUB>1</SUB><I>H</I>=2/3(<I>x</I>-2<I>a</I>)</MATH>, and <MATH><I>AN</I><SUB>1</SUB>=1/3(<I>x</I>-2<I>a</I>)</MATH>. +<p>But <MATH><I>y</I><SUP>2</SUP>:<I>P</I><SUB>1</SUB><I>N</I><SUB>1</SUB><SUP>2</SUP>=<I>N</I><SUB>1</SUB><I>H</I><SUP>2</SUP>:<I>HM</I><SUP>2</SUP></MATH>, by hypothesis, +or <MATH><I>y</I><SUP>2</SUP>:4<I>a.AN</I><SUB>1</SUB>=<I>N</I><SUB>1</SUB><I>H</I><SUP>2</SUP>:4<I>a</I><SUP>2</SUP></MATH>; +therefore <MATH><I>ay</I><SUP>2</SUP>=<I>AN</I><SUB>1</SUB>.<I>N</I><SUB>1</SUB><I>H</I><SUP>2</SUP>, +=4/27(<I>x</I>-2<I>a</I>)<SUP>3</SUP></MATH>, +or <MATH>27<I>ay</I><SUP>2</SUP>=4(<I>x</I>-2<I>a</I>)<SUP>3</SUP></MATH>. +<p>(2) In the case of the hyperbola or ellipse we naturally take +<I>CA, CB</I> as axes of <I>x</I> and <I>y.</I> The work is here rather more +complicated, but there is no difficulty in obtaining, as the +locus of <I>O</I>, the curve +<MATH>(<I>ax</I>)<SUP>2/3</SUP>∓(<I>by</I>)<SUP>2/3</SUP>=(<I>a</I><SUP>2</SUP>±<I>b</I><SUP>2</SUP>)<SUP>2/3</SUP></MATH>. +<pb n=166><head>APOLLONIUS OF PERGA</head> +<p>The propositions V. 53, 54 are particular cases of the pre- +ceding propositions. +<C><I>Construction of normals.</I></C> +<p>The next section of the Book (V. 55-63) relates to the con- +struction of normals through various points according to their +position within or without the conic and in relation to the +axes. It is proved that one normal can be drawn through any +internal point and through any external point which is not +on the axis through the vertex <I>A.</I> In particular, if <I>O</I> is any +point below the axis <I>AA</I>′ of an ellipse, and <I>OM</I> is perpen- +dicular to <I>AA</I>′, then, if <MATH><I>AM</I> > <I>AC</I></MATH>, one normal can always be +drawn through <I>O</I> cutting the axis between <I>A</I> and <I>C</I>, but never +more than one such normal (V. 55-7). The points on the +curve at which the straight lines through <I>O</I> are normals are +determined as the intersections of the conic with a certain +<FIG> +rectangular hyperbola. The procedure +of Apollonius is equivalent to the fol- +lowing analytical method. Let <I>AM</I> be +the axis of a conic, <I>PGO</I> one of the +normals which passes through the given +point <I>O, PN</I> the ordinate at <I>P</I>; and let +<I>OM</I> be drawn perpendicular to the axis. +Take as axes of coordinates the axes in the central conic and, +in the case of the parabola, the axis and the tangent at the +vertex. +<p>If then (<I>x, y</I>) be the coordinates of <I>P</I> and (<I>x</I><SUB>1</SUB>, <I>y</I><SUB>1</SUB>) those of <I>O</I> +we have <MATH><I>y</I>/(-<I>y</I><SUB>1</SUB>)=<I>NG</I>/(<I>x</I><SUB>1</SUB>-<I>x</I>-<I>NG</I>)</MATH>. +<p>Therefore (1) for the parabola +<MATH><I>y</I>/(-<I>y</I><SUB>1</SUB>)=1/2<I>p</I>/(<I>x</I><SUB>1</SUB>-<I>x</I>-1/2<I>p</I>)</MATH>, +or <MATH><I>xy</I>-(<I>x</I><SUB>1</SUB>-1/2<I>p</I>)<I>y</I>-<I>y</I><SUB>1</SUB>.1/2<I>p</I>=0</MATH>; (1) +<p>(2) in the ellipse or hyperbola +<MATH><I>xy</I>(1∓<I>b</I><SUP>2</SUP>/<I>a</I><SUP>2</SUP>)-<I>x</I><SUB>1</SUB><I>y</I>±<I>b</I><SUP>2</SUP>/<I>a</I><SUP>2</SUP>.<I>y</I><SUB>1</SUB><I>x</I>=0</MATH>. (2) +<p>The intersections of these rectangular hyperbolas respec- +<pb n=167><head>THE <I>CONICS</I>, BOOKS V, VI</head> +tively with the conics give the points at which the normals +passing through <I>O</I> are normals. +<p>Pappus criticizes the use of the rectangular hyperbola in +the case of the parabola as an unnecessary resort to a ‘<I>solid</I> +locus’; the meaning evidently is that the same points of +intersection can be got by means of a certain circle taking +the place of the rectangular hyperbola. We can, in fact, from +the equation (1) above combined with <MATH><I>y</I><SUP>2</SUP>=<I>px</I></MATH>, obtain the +circle +<MATH>(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>)-(<I>x</I><SUB>1</SUB>+1/2<I>p</I>)<I>x</I>-1/2<I>y</I><SUB>1</SUB><I>y</I>=0</MATH>. +<p>The Book concludes with other propositions about maxima +and minima. In particular V. 68-71 compare the lengths of +tangents <I>TQ, TQ</I>′, where <I>Q</I> is nearer to the axis than <I>Q</I>′. +V. 72, 74 compare the lengths of two normals from a point +<I>O</I> from which only two can be drawn and the lengths of other +straight lines from <I>O</I> to the curve; V. 75-7 compare the +lengths of three normals to an ellipse drawn from a point +<I>O</I> below the major axis, in relation to the lengths of other +straight lines from <I>O</I> to the curve. +<p>Book VI is of much less interest. The first part (VI. 1-27) +relates to equal (i.e. congruent) or similar conics and segments +of conics; it is naturally preceded by some definitions includ- +ing those of ‘equal’ and ‘similar’ as applied to conics and +segments of conics. Conics are said to be similar if, the same +number of ordinates being drawn to the axis at proportional +distances from the vertices, all the ordinates are respectively +proportional to the corresponding abscissae. The definition of +similar segments is the same with diameter substituted for +axis, and with the additional condition that the angles +between the base and diameter in each are equal. Two +parabolas are equal if the ordinates to a diameter in each are +inclined to the respective diameters at equal angles and the +corresponding parameters are equal; two ellipses or hyper- +bolas are equal if the ordinates to a diameter in each are +equally inclined to the respective diameters and the diameters +as well as the corresponding parameters are equal (VI. 1. 2). +Hyperbolas or ellipses are similar when the ‘figure’ on a +diameter of one is similar (instead of equal) to the ‘figure’ on +a diameter of the other, and the ordinates to the diameters in +<pb n=168><head>APOLLONIUS OF PERGA</head> +each make equal angles with them; all parabolas are similar +(VI. 11, 12, 13). No conic of one of the three kinds (para- +bolas, hyperbolas or ellipses) can be equal or similar to a conic +of either of the other two kinds (VI. 3, 14, 15). Let <I>QPQ</I>′, +<I>qpq</I>′ be two segments of similar conics in which <I>QQ</I>′, <I>qq</I>′ are +the bases and <I>PV, pv</I> are the diameters bisecting them; then, +if <I>PT, pt</I> be the tangents at <I>P, p</I> and meet the axes at <I>T, t</I> at +equal angles, and if <MATH><I>PV</I>:<I>PT</I>=<I>pv</I>:<I>pt</I></MATH>, the segments are similar +and similarly situated, and conversely (VI. 17, 18). If two +ordinates be drawn to the axes of two parabolas, or the major or +conjugate axes of two similar central conics, as <I>PN, P</I>′<I>N</I>′ and +<I>pn, p</I>′<I>n</I>′ respectively, such that the ratios <I>AN</I>:<I>an</I> and <I>AN</I>′:<I>an</I>′ +are each equal to the ratio of the respective <I>latera recta</I>, the +segments <I>PP</I>′, <I>pp</I>′ will be similar; also <I>PP</I>′ will not be similar +to any segment in the other conic cut off by two ordinates +other than <I>pn, p</I>′<I>n</I>′, and conversely (VI. 21, 22). If any cone +be cut by two parallel planes making hyperbolic or elliptic +sections, the sections will be similar but not equal (VI. 26, 27). +<p>The remainder of the Book consists of problems of con- +struction; we are shown how in a given right cone to find +a parabolic, hyperbolic or elliptic section equal to a given +parabola, hyperbola or ellipse, subject in the case of the +hyperbola to a certain <G>diorismo/s</G> or condition of possibility +(VI. 28-30); also how to find a right cone similar to a given +cone and containing a given parabola, hyperbola or ellipse as +a section of it, subject again in the case of the hyperbola to +a certain <G>diorismo/s</G> (VI. 31-3). These problems recall the +somewhat similar problems in I. 51-9. +<p>Book VII begins with three propositions giving expressions +for <MATH><I>AP</I><SUP>2</SUP>(=<I>AN</I><SUP>2</SUP>+<I>PN</I><SUP>2</SUP>)</MATH> in the same form as those for <I>PN</I><SUP>2</SUP> in +the statement of the ordinary property. In the parabola <I>AH</I> +is measured along the axis produced (i.e. in the opposite direc- +tion to <I>AN</I>) and of length equal to the <I>latus rectum</I>, and it is +proved that, for any point <MATH><I>P, AP</I><SUP>2</SUP>=<I>AN.NH</I></MATH> (VII. 1). In +the case of the central conics <I>AA</I>′ is divided at <I>H</I>, internally +for the hyperbola and externally for the ellipse (<I>AH</I> being the +segment adjacent to <I>A</I>) so that <MATH><I>AH</I>:<I>A</I>′<I>H</I>=<I>p</I>:<I>AA</I>′</MATH>, where <I>p</I> +is the parameter corresponding to <I>AA</I>′, or <MATH><I>p</I>=<I>BB</I>′<SUP>2</SUP>/<I>AA</I>′</MATH>, and +it is proved that +<MATH><I>AP</I><SUP>2</SUP>:<I>AN.NH</I>=<I>AA</I>′:<I>A</I>′<I>H</I></MATH>. +<pb n=169><head>THE <I>CONICS</I>, BOOKS VI, VII</head> +The same is true if <I>AA</I>′ is the minor axis of an ellipse and <I>p</I> +the corresponding parameter (VII. 2, 3). +<p>If <I>AA</I>′ be divided at <I>H</I>′ as well as <I>H</I> (internally for the +hyperbola and externally for the ellipse) so that <I>H</I> is adjacent +to <I>A</I> and <I>H</I>′ to <I>A</I>′, and if <MATH><I>A</I>′<I>H</I>:<I>AH</I>=<I>AH</I>′:<I>A</I>′<I>H</I>′=<I>AA</I>′:<I>p</I></MATH>, +the lines <I>AH, A</I>′<I>H</I>′ (corresponding to <I>p</I> in the proportion) are +called by Apollonius <I>homologues</I>, and he makes considerable +<FIG> +use of the auxiliary points <I>H, H</I>′ in later propositions from +VII. 6 onwards. Meantime he proves two more propositions, +which, like VII. 1-3, are by way of lemmas. First, if <I>CD</I> be +the semi-diameter parallel to the tangent at <I>P</I> to a central +conic, and if the tangent meet the axis <I>AA</I>′ in <I>T</I>, then +<MATH><I>PT</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=<I>NT</I>:<I>CN</I></MATH>. (VII. 4.) +Draw <I>AE, TF</I> at right angles to <I>CA</I> to meet <I>CP</I>, and let <I>AE</I> +meet <I>PT</I> in <I>O.</I> Then, if <I>p</I>′ be the parameter of the ordinates +to <I>CP</I>, we have +<MATH>1/2<I>p</I>′:<I>PT</I>=<I>OP</I>:<I>PE</I> (I. 49, 50.) +=<I>PT</I>:<I>PF</I></MATH>, +or <MATH>1/2<I>p</I>′.<I>PF</I>=<I>PT</I><SUP>2</SUP></MATH>. +<p>Therefore <MATH><I>PT</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=1/2<I>p</I>′.<I>PF</I>:1/2<I>p</I>′.<I>CP</I> +=<I>PF</I>:<I>CP</I> +=<I>NT</I>:<I>CN</I></MATH>. +<pb n=170><head>APOLLONIUS OF PERGA</head> +<p>Secondly, Apollonius proves that, if <I>PN</I> be a principal +ordinate in a parabola, <I>p</I> the principal parameter, <I>p</I>′ the +parameter of the ordinates to the diameter through <I>P</I>, then +<MATH><I>p</I>′=<I>p</I>+4<I>AN</I></MATH> (VII. 5); this is proved by means of the same +property as VII. 4, namely <MATH>1/2<I>p</I>′:<I>PT</I>=<I>OP</I>:<I>PE</I></MATH>. +<p>Much use is made in the remainder of the Book of two +points <I>Q</I> and <I>M</I>, where <I>AQ</I> is drawn parallel to the conjugate +diameter <I>CD</I> to meet the curve in <I>Q</I>, and <I>M</I> is the foot of +the principal ordinate at <I>Q</I>; since the diameter <I>CP</I> bisects +both <I>AA</I>′ and <I>QA</I>, it follows that <I>A</I>′<I>Q</I> is parallel to <I>CP.</I> +Many ratios between functions of <I>PP</I>′, <I>DD</I>′ are expressed in +terms of <I>AM, A</I>′<I>M, MH, MH</I>′, <I>AH, A</I>′<I>H</I>, &c. The first pro- +positions of the Book proper (VII. 6, 7) prove, for instance, +that <MATH><I>PP</I>′<SUP>2</SUP>:<I>DD</I>′<SUP>2</SUP>=<I>MH</I>′:<I>MH</I></MATH>. +<p>For <MATH><I>PT</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=<I>NT</I>:<I>CN</I>=<I>AM</I>:<I>A</I>′<I>M</I></MATH>, by similar triangles. +<p>Also <MATH><I>CP</I><SUP>2</SUP>:<I>PT</I><SUP>2</SUP>=<I>A</I>′<I>Q</I><SUP>2</SUP>:<I>AQ</I><SUP>2</SUP></MATH>. +<p>Therefore, <I>ex aequali</I>, +<MATH><I>CP</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=(<I>AM</I>:<I>A</I>′<I>M</I>)X(<I>A</I>′<I>Q</I><SUP>2</SUP>:<I>AQ</I><SUP>2</SUP>) +=(<I>AM</I>:<I>A</I>′<I>M</I>)X(<I>A</I>′<I>Q</I><SUP>2</SUP>:<I>A</I>′<I>M.MH</I>′) +X(<I>A</I>′<I>M.MH</I>′:<I>AM.MH</I>)X(<I>AM.MH</I>:<I>AQ</I><SUP>2</SUP>) +=(<I>AM</I>:<I>A</I>′<I>M</I>)X(<I>AA</I>′:<I>AH</I>′)X(<I>A</I>′<I>M</I>:<I>AM</I>) +X(<I>MH</I>′:<I>MH</I>)X(<I>A</I>′<I>H</I>:<I>AA</I>′)</MATH>, by aid of VII. 2, 3. +<p>Therefore <MATH><I>PP</I>′<SUP>2</SUP>:<I>DD</I>′<SUP>2</SUP>=<I>MH</I>′:<I>MH</I></MATH>. +<p>Next (VII. 8, 9, 10, 11) the following relations are proved, +namely +<p>(1) <MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′±<I>DD</I>′)<SUP>2</SUP>=<I>A</I>′<I>H.MH</I>′:{<I>MH</I>′±√(<I>MH.MH</I>′)}<SUP>2</SUP></MATH>, +<p>(2) <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′.<I>DD</I>′=<I>A</I>′<I>H</I>:√(<I>MH.MH</I>′)</MATH>, +<p>(3) <MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP>)=<I>A</I>′<I>H</I>:<I>MH</I>±<I>MH</I>′</MATH>. +<p>The steps by which these results are obtained are as follows. +<p>First, <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>=<I>A</I>′<I>H</I>:<I>MH</I>′ (<G>a</G>) +=<I>A</I>′<I>H.MH</I>′:<I>MH</I>′<SUP>2</SUP></MATH>. +<p>(This is proved thus: +<MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>=<I>CA</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP> +=<I>CN.CT</I>:<I>CP</I><SUP>2</SUP> +=<I>A</I>′<I>M.A</I>′<I>A</I>:<I>A</I>′<I>Q</I><SUP>2</SUP></MATH>. +<pb n=171><head>THE <I>CONICS</I>, BOOK VII</head> +<p>But <MATH><I>A</I>′<I>Q</I><SUP>2</SUP>:<I>A</I>′<I>M.MH</I>′=<I>AA</I>′:<I>AH</I>′ (VII. 2, 3) +=<I>AA</I>′:<I>A</I>′<I>H</I> +=<I>A</I>′<I>M.AA</I>′:<I>A</I>′<I>M.A</I>′<I>H</I></MATH>, +so that, alternately, +<MATH><I>A</I>′<I>M.AA</I>′:<I>A</I>′<I>Q</I><SUP>2</SUP>=<I>A</I>′<I>M.A</I>′<I>H</I>:<I>A</I>′<I>M.MH</I>′ +=<I>A</I>′<I>H</I>:<I>MH</I>′</MATH>.) +<p>Next, <MATH><I>PP</I>′<SUP>2</SUP>:<I>DD</I>′<SUP>2</SUP>=<I>MH</I>′:<I>MH</I>, as above, (<G>b</G>) +=<I>MH</I>′<SUP>2</SUP>:<I>MH.MH</I>′</MATH>, +whence <MATH><I>PP</I>′:<I>DD</I>′=<I>MH</I>′:√(<I>MH.MH</I>′)</MATH>, (<G>g</G>) +and <MATH><I>PP</I>′<SUP>2</SUP>:(<I>PP</I>′±<I>DD</I>′)<SUP>2</SUP>=<I>MH</I>′<SUP>2</SUP>:{<I>MH</I>′±√(<I>MH.MH</I>′)}<SUP>2</SUP></MATH>; +<p>(1) above follows from this relation and (<G>a</G>) <I>ex aequali</I>; +<p>(2) follows from (<G>a</G>) and (<G>g</G>) <I>ex aequali</I>, and (3) from (<G>a</G>) +and (<G>b</G>). +<p>We now obtain immediately the important proposition that +<MATH><I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP></MATH> is constant, whatever be the position of <I>P</I> on an +ellipse or hyperbola (the upper sign referring to the ellipse), +and is equal to <MATH><I>AA</I>′<SUP>2</SUP>±<I>BB</I>′<SUP>2</SUP></MATH> (VII. 12, 13, 29, 30). +<p>For <MATH><I>AA</I>′<SUP>2</SUP>:<I>BB</I>′<SUP>2</SUP>=<I>AA</I>′:<I>p</I>=<I>A</I>′<I>H</I>:<I>AH</I>=<I>A</I>′<I>H</I>:<I>A</I>′<I>H</I>′</MATH>, +by construction; +therefore <MATH><I>AA</I>′<SUP>2</SUP>:<I>AA</I>′<SUP>2</SUP>±<I>BB</I>′<SUP>2</SUP>=<I>A</I>′<I>H</I>:<I>HH</I>′</MATH>; +also, from (<G>a</G>) above, +<MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>=<I>A</I>′<I>H</I>:<I>MH</I>′</MATH>; +and, by means of (<G>b</G>), +<MATH><I>PP</I>′<SUP>2</SUP>:(<I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP>)=<I>MH</I>′:<I>MH</I>′±<I>MH</I> +=<I>MH</I>′:<I>HH</I>′</MATH>. +<p><I>Ex aequali</I>, from the last two relations, we have +<MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP>)=<I>A</I>′<I>H</I>:<I>HH</I>′ +=<I>AA</I>′<SUP>2</SUP>:<I>AA</I>′<SUP>2</SUP>±<I>BB</I>′<SUP>2</SUP></MATH>, from above, +whence <MATH><I>PP</I>′<SUP>2</SUP>±<I>DD</I>′<SUP>2</SUP>=<I>AA</I>′<SUP>2</SUP>±<I>BB</I>′<SUP>2</SUP></MATH>. +<pb n=172><head>APOLLONIUS OF PERGA</head> +<p>A number of other ratios are expressed in terms of the +straight lines terminating at <I>A, A</I>′, <I>H, H</I>′, <I>M, M</I>′ as follows +(VII. 14-20). +<p>In the ellipse <MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>-<I>DD</I>′<SUP>2</SUP>=<I>A</I>′<I>H</I>:2<I>CM</I></MATH>, +and in the hyperbola or ellipse (if <I>p</I> be the parameter of the +ordinates to <I>PP</I>′) +<MATH><I>AA</I>′<SUP>2</SUP>:<I>p</I><SUP>2</SUP>=<I>A</I>′<I>H.MH</I>′:<I>MH</I><SUP>2</SUP></MATH>, +<MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′±<I>p</I>)<SUP>2</SUP>=<I>A</I>′<I>H.MH</I>′:(<I>MH</I>±<I>MH</I>′)<SUP>2</SUP></MATH>, +<MATH><I>AA</I>′<SUP>2</SUP>:<I>PP</I>′.<I>p</I>=<I>A</I>′<I>H</I>:<I>MH</I></MATH>, +and <MATH><I>AA</I>′<SUP>2</SUP>:(<I>PP</I>′<SUP>2</SUP>±<I>p</I><SUP>2</SUP>)=<I>A</I>′<I>H.MH</I>′:(<I>MH</I>′<SUP>2</SUP>±<I>MH</I><SUP>2</SUP>)</MATH>. +<p>Apollonius is now in a position, by means of all these +relations, resting on the use of the auxiliary points <I>H, H</I>′, <I>M</I>, +to compare different functions of any conjugate diameters +with the same functions of the axes, and to show how the +former vary (by way of increase or diminution) as <I>P</I> moves +away from <I>A.</I> The following is a list of the functions com- +pared, where for brevity I shall use <I>a, b</I> to represent <I>AA</I>′, <I>BB</I>′; +<I>a</I>′, <I>b</I>′ to represent <I>PP</I>′, <I>DD</I>′; and <I>p, p</I>′ to represent the para- +meters of the ordinates to <I>AA</I>′, <I>PP</I>′ respectively. +<p>In a hyperbola, according as <I>a</I> > or < <I>b, a</I>′ > or < <I>b</I>′, and the +ratio <I>a</I>′:<I>b</I>′ decreases or increases as <I>P</I> moves from <I>A</I> on +either side; also, if <MATH><I>a</I>=<I>b</I></MATH>, <MATH><I>a</I>′=<I>b</I>′</MATH> (VII. 21-3); in an ellipse +<MATH><I>a</I>:<I>b</I> > <I>a</I>′:<I>b</I>′</MATH>, and the latter ratio diminishes as <I>P</I> moves from +<I>A</I> to <I>B</I> (VII. 24). +<p>In a hyperbola or ellipse <MATH><I>a</I>+<I>b</I> < <I>a</I>′+<I>b</I>′</MATH>, and <MATH><I>a</I>′+<I>b</I>′</MATH> in the +hyperbola increases continually as <I>P</I> moves farther from <I>A</I>, +but in the ellipse increases till <I>a</I>′, <I>b</I>′ take the position of the +equal conjugate diameters when it is a <I>maximum</I> (VII. +25, 26). +<p>In a hyperbola in which <I>a, b</I> are unequal, or in an ellipse, +<MATH><I>a</I>-<I>b</I> > <I>a</I>′-<I>b</I>′</MATH>, and <MATH><I>a</I>′-<I>b</I>′</MATH> diminishes as <I>P</I> moves away from <I>A</I>, +in the hyperbola continually, and in the ellipse till <I>a</I>′, <I>b</I>′ are +the equal conjugate diameters (VII. 27). +<p><MATH><I>ab</I> < <I>a</I>′<I>b</I>′</MATH>, and <I>a</I>′<I>b</I>′ increases as <I>P</I> moves away from <I>A</I>, in the +hyperbola continually, and in the ellipse till <I>a</I>′, <I>b</I>′ coincide with +the equal conjugate diameters (VII. 28). +<p>VII. 31 is the important proposition that, if <I>PP</I>′, <I>DD</I>′ are +<pb n=173><head>THE <I>CONICS</I>, BOOK VII</head> +conjugate diameters in an ellipse or conjugate hyperbolas, and +if the tangents at their extremities form the parallelogram +<I>LL</I>′<I>MM</I>′, then +the parallelogram <MATH><I>LL</I>′<I>MM</I>′=rect.<I>AA</I>′.<I>BB</I>′</MATH>. +<p>The proof is interesting. Let the tangents at <I>P, D</I> respec- +tively meet the major or transverse axis in <I>T, T</I>′. +<p>Now (by VII. 4) <MATH><I>PT</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>=<I>NT</I>:<I>CN</I></MATH>; +therefore <MATH>2▵<I>CPT</I>:2▵<I>T</I>′<I>DC</I>=<I>NT</I>:<I>CN</I></MATH>. +<FIG> +<p>But <MATH>2▵<I>CPT</I>:(<I>CL</I>)=<I>PT</I>:<I>CD</I>, +=<I>CP</I>:<I>DT</I>′, by similar triangles, +=(<I>CL</I>):2▵<I>T</I>′<I>DC</I></MATH>. +<p>That is, (<I>CL</I>) is a mean proportional between 2▵<I>CPT</I> and +2▵<I>T</I>′<I>DC.</I> +<p>Therefore, since <MATH>√(<I>NT.CN</I>)</MATH> is a mean proportional between +<I>NT</I> and <I>CN</I>, +<pb n=174><head>APOLLONIUS OF PERGA</head> +<MATH>2▵<I>CPT</I>:(<I>CL</I>)=√(<I>CN.NT</I>):<I>CN</I> +=<I>PN.CA</I>/<I>CB</I>:<I>CN</I> (I. 37, 39) +=<I>PN.CT</I>:<I>CT.CN.CB</I>/<I>CA</I> +=2▵<I>CPT</I>:<I>CA.CB</I></MATH>; +therefore <MATH>(<I>CL</I>)=<I>CA.CB</I></MATH>. +<p>The remaining propositions of the Book trace the variations +of different functions of the conjugate diameters, distinguishing +the maximum values, &c. The functions treated are the +following: +<p><I>p</I>′, the parameter of the ordinates to <I>PP</I>′ in the hyperbola, +according as <I>AA</I>′ is (1) not less than <I>p</I>, the parameter corre- +sponding to <I>AA</I>′, (2) less than <I>p</I> but not less than 1/2<I>p</I>, (3) less +than 1/2<I>p</I> (VII. 33-5). +<p><MATH><I>PP</I>′-<I>p</I>′</MATH>, as compared with <MATH><I>AA</I>′-<I>p</I></MATH> in the hyperbola (VII. 36) +or the ellipse (VII. 37). +<p><MATH><I>PP</I>′+<I>p</I>′</MATH> ”” <MATH><I>AA</I>′+<I>p</I></MATH> in the hyperbola (VII. +38-40) or the ellipse (VII. 41). +<p><MATH><I>PP</I>′.<I>p</I>′</MATH> ”” <MATH><I>AA</I>′.<I>p</I></MATH> in the hyperbola (VII. 42) +or the ellipse (VII. 43). +<p><MATH><I>PP</I>′<SUP>2</SUP>+<I>p</I>′<SUP>2</SUP></MATH> ”” <MATH><I>AA</I>′<SUP>2</SUP>+<I>p</I><SUP>2</SUP></MATH> in the hyperbola, accord- +ing as (1) <I>AA</I>′ is not less than +<I>p</I>, or (2) <MATH><I>AA</I>′ < <I>p</I></MATH>, but <I>AA</I>′<SUP>2</SUP> not +less than <MATH>1/2(<I>AA</I>′-<I>p</I>)<SUP>2</SUP></MATH>, or (3) +<MATH><I>AA</I>′<SUP>2</SUP> < 1/2(<I>AA</I>′-<I>p</I>)<SUP>2</SUP></MATH> (VII. 44-6). +<p><MATH><I>PP</I>′<SUP>2</SUP>+<I>p</I>′<SUP>2</SUP></MATH> ”” <MATH><I>AA</I>′<SUP>2</SUP>+<I>p</I><SUP>2</SUP></MATH> in the ellipse, according +as <I>AA</I>′<SUP>2</SUP> is not greater, or is +greater, than <MATH>(<I>AA</I>′+<I>p</I>)<SUP>2</SUP></MATH> (VII. +47, 48). +<p><MATH><I>PP</I>′<SUP>2</SUP>-<I>p</I>′<SUP>2</SUP></MATH> ”” <MATH><I>AA</I>′<SUP>2</SUP>-<I>p</I><SUP>2</SUP></MATH> in the hyperbola, accord- +ing as <I>AA</I>′ > or < <I>p</I> (VII. +49, 50). +<p><MATH><I>PP</I>′<SUP>2</SUP>-<I>p</I>′<SUP>2</SUP></MATH> ”” <MATH><I>AA</I>′<SUP>2</SUP>-<I>p</I><SUP>2</SUP></MATH> or <MATH><I>BB</I>′<SUP>2</SUP>-<I>p</I><SUB>b</SUB><SUP>2</SUP></MATH> in the ellipse, +according as <I>PP</I>′ > or < <I>p</I>′ +(VII. 51). +<pb n=175><head>THE <I>CONICS</I>, BOOK VII</head> +<p>As we have said, Book VIII is lost. The nature of its +contents can only be conjectured from Apollonius's own +remark that it contained determinate conic problems for +which Book VII was useful, particularly in determining +limits of possibility. Unfortunately, the lemmas of Pappus +do not enable us to form any clearer idea. But it is probable +enough that the Book contained a number of problems having +for their object the finding of conjugate diameters in a given +conic such that certain functions of their lengths have given +values. It was on this assumption that Halley attempted +a restoration of the Book. +<p>If it be thought that the above account of the <I>Conics</I> is +disproportionately long for a work of this kind, it must be +remembered that the treatise is a great classic which deserves +to be more known than it is. What militates against its +being read in its original form is the great extent of the +exposition (it contains 387 separate propositions), due partly +to the Greek habit of proving particular cases of a general +proposition separately from the proposition itself, but more to +the cumbrousness of the enunciations of complicated proposi- +tions in general terms (without the help of letters to denote +particular points) and to the elaborateness of the Euclidean +form, to which Apollonius adheres throughout. +<C>Other works by Apollonius.</C> +<p>Pappus mentions and gives a short indication of the con- +tents of six other works of Apollonius which formed part of the +<I>Treasury of Analysis.</I><note>Pappus, vii, pp. 640-8, 660-72.</note> Three of these should be mentioned +in close connexion with the <I>Conics.</I> +<C><I>(a) On the Cutting-off of a Ratio (<G>lo/gou a)potomh/</G>)</I>, +two Books.</C> +<p>This work alone of the six mentioned has survived, and +that only in the Arabic; it was published in a Latin trans- +lation by Edmund Halley in 1706. It deals with the general +problem, ‘<I>Given two straight lines, parallel to one another or +intersecting, and a fixed point on each line, to draw through +<pb n=176><head>APOLLONIUS OF PERGA</head> +a given point a straight line which shall cut off segments from +each line (measured from the fixed points) bearing a given +ratio to one another.</I>’ Thus, let <I>A, B</I> be fixed points on the +two given straight lines <I>AC, BK</I>, and let <I>O</I> be the given +point. It is required to draw through <I>O</I> a straight line +cutting the given straight lines in points <I>M, N</I> respectively +<FIG> +such that <I>AM</I> is to <I>BN</I> in a given ratio. The two Books of +the treatise discussed the various possible cases of this pro- +blem which arise according to the relative positions of the +given straight lines and points, and also the necessary condi- +tions and limits of possibility in cases where a solution is not +always possible. The first Book begins by supposing the +given lines to be parallel, and discusses the different cases +which arise; Apollonius then passes to the cases in which the +straight lines intersect, but one of the given points, <I>A</I> or <I>B</I>, is +at the intersection of the two lines. Book II proceeds to the +general case shown in the above figure, and first proves that +the general case can be reduced to the case in Book I where +one of the given points, <I>A</I> or <I>B</I>, is at the intersection of the +two lines. The reduction is easy. For join <I>OB</I> meeting <I>AC</I> +in <I>B</I>′, and draw <I>B</I>′<I>N</I>′ parallel to <I>BN</I> to meet <I>OM</I> in <I>N</I>′. Then +the ratio <MATH><I>B</I>′<I>N</I>′:<I>BN</I></MATH>, being equal to the ratio <MATH><I>OB</I>′:<I>OB</I></MATH>, is con- +stant. Since, therefore, <MATH><I>BN</I>:<I>AM</I></MATH> is a given ratio, the ratio +<MATH><I>B</I>′<I>N</I>′:<I>AM</I></MATH> is also given. +<p>Apollonius proceeds in all cases by the orthodox method of +analysis and synthesis. Suppose the problem solved and +<I>OMN</I> drawn through <I>O</I> in such a way that <MATH><I>B</I>′<I>N</I>′:<I>AM</I></MATH> is a +given ratio =<G>l</G>, say. +<pb n=177><head><I>ON THE CUTTING-OFF OF A RATIO</I></head> +<p>Draw <I>OC</I> parallel to <I>BN</I> or <I>B′N′</I> to meet <I>AM</I> in <I>C.</I> Take +<I>D</I> on <I>AM</I> such that <MATH><I>OC</I>:<I>AD</I> = <G>l</G> = <I>B′N′</I>:<I>AM</I></MATH>. +<p>Then <MATH><I>AM</I>:<I>AD</I> = <I>B′N′</I>:<I>OC</I></MATH> +<MATH>= <I>B′M</I>:<I>CM</I></MATH>; +therefore <MATH><I>MD</I>:<I>AD</I> = <I>B′C</I>:<I>CM</I></MATH>, +or <MATH><I>CM.MD</I> = <I>AD.B′C</I></MATH>, a given rectangle. +<p>Hence the problem is reduced to one of <I>applying to CD a +rectangle</I> (<I>CM.MD</I>) <I>equal to a given rectangle</I> (<I>AD.B′C</I>) <I>but +falling short by a square figure.</I> In the case as drawn, whatever +be the value of <G>l</G>, the solution is always possible because +the given rectangle <I>AD.CB′</I> is always less than <I>CA.AD,</I> and +therefore always less than 1/4<I>CD</I><SUP>2</SUP>; one of the positions of +<I>M</I> falls between <I>A</I> and <I>D</I> because <MATH><I>CM.MD</I><<I>CA.AD</I></MATH>. +<p>The proposition III. 41 of the <I>Conics</I> about the intercepts +made on two tangents to a parabola by a third tangent +(pp. 155-6 above) suggests an obvious application of our problem. +We had, with the notation of that proposition, +<MATH><I>Pr</I>:<I>rq</I> = <I>rQ</I>:<I>Qp</I> = <I>qp</I>:<I>pR</I></MATH>. +Suppose that the two tangents <I>qP, qR</I> are given as fixed +tangents with their points of contact <I>P, R.</I> Then we can +draw another tangent if we can draw a straight line +intersecting <I>qP, qR</I> in such a way that <MATH><I>Pr</I>:<I>rq</I> = <I>qp</I>:<I>pR</I></MATH> or +<MATH><I>Pq</I>:<I>qr</I> = <I>qR</I>:<I>pR</I></MATH>, i.e. <MATH><I>qr</I>:<I>pR</I> = <I>Pq</I>:<I>qR</I></MATH> (a constant ratio); +i.e. we have to draw a straight line such that the intercept by +it on <I>qP</I> measured from <I>q</I> has a given ratio to the intercept +by it on <I>qR</I> measured from <I>R.</I> This is a particular case of +our problem to which, as a matter of fact, Apollonius devotes +special attention. In the annexed figure the letters have the +<FIG> +same meaning as before, and <I>N′M</I> has to be drawn through <I>O</I> +such that <MATH><I>B′N′</I>:<I>AM</I> = <G>l</G></MATH>. In this case there are limits to +<pb n=178><head>APOLLONIUS OF PERGA</head> +the value of <G>l</G> in order that the solution may be possible. +Apollonius begins by stating the limiting case, saying that we +obtain a solution in a special manner in the case where <I>M</I> is +the middle point of <I>CD,</I> so that the rectangle <I>CM.MD</I> or +<I>CB′.AD</I> has its maximum value. +<p>The corresponding limiting value of <G>l</G> is determined by +finding the corresponding position of <I>D</I> or <I>M.</I> +<p>We have <MATH><I>B′C</I>:<I>MD</I> = <I>CM</I>:<I>AD</I></MATH>, as before, +<MATH>= <I>B′M</I>:<I>MA</I></MATH>; +whence, since <MATH><I>MD</I> = <I>CM</I></MATH>, +<MATH><I>B′C</I>:<I>B′M</I> = <I>CM</I>:<I>MA</I></MATH> +<MATH>= <I>B′M</I>:<I>B′A</I></MATH>, +so that <MATH><I>B′M</I><SUP>2</SUP> = <I>B′C.B′A</I></MATH>. +<p>Thus <I>M</I> is found and therefore <I>D</I> also. +<p>According, therefore, as <G>l</G> is less or greater than the particular +value of <I>OC</I>:<I>AD</I> thus determined, Apollonius finds no +solution or two solutions. +<p>Further, we have +<MATH><I>AD</I> = <I>B′A</I> + <I>B′C</I>-(<I>B′D</I> + <I>B′C</I>)</MATH> +<MATH>= <I>B′A</I> + <I>B′C</I>-2<I>B′M</I></MATH> +<MATH>= <I>B′A</I> + <I>B′C</I>-2√(<I>B′A. B′C</I>)</MATH>. +<p>If then we refer the various points to a system of coordinates +in which <I>B′A, B′N′</I> are the axes of <I>x</I> and <I>y,</I> and if +we denote <I>O</I> by (<I>x, y</I>) and the length <I>B′A</I> by <I>h</I>, +<MATH><G>l</G> = <I>OC</I>/<I>AD</I> = <I>y</I>/(<I>h</I> + <I>x</I> - 2√(<I>hx</I>)</MATH>. +<p>If we suppose Apollonius to have used these results for the +parabola, he cannot have failed to observe that the limiting +case described is that in which <I>O</I> is on the parabola, while +<I>N′OM</I> is the tangent at <I>O</I>; for, as above, +<MATH><I>B′M</I>:<I>B′A</I> = <I>B′C</I>:<I>B′M</I> = <I>N′O</I>:<I>N′M</I></MATH>, by parallels, +so that <I>B′A, N′M</I> are divided at <I>M, O</I> respectively in the same +proportion. +<pb n=179><head><I>ON THE CUTTING-OFF OF A RATIO</I></head> +<p>Further, if we put for <G>l</G> the ratio between the lengths of the +two fixed tangents, then if <I>h, k</I> be those lengths, +<MATH><I>k</I>/<I>h</I> = <I>y</I>/(<I>h</I>+<I>x</I>-2√(<I>hx</I>)</MATH>,........ +which can easily be reduced to +<MATH>(<I>x</I>/<I>h</I>)<SUP>1/2</SUP> + (<I>y</I>/<I>k</I>)<SUP>1/2</SUP>=1</MATH>, +the equation of the parabola referred to the two fixed tangents +as axes. +<C>(<G>b</G>) <I>On the cutting-off of an area</I> (<G>xwri/ou a)potomh/</G>), +two Books.</C> +<p>This work, also in two Books, dealt with a similar problem, +with the difference that the intercepts on the given straight +lines measured from the given points are required, not to +have a given ratio, but to contain a given rectangle. Halley +included an attempted restoration of this work in his edition +of the <I>De sectione rationis.</I> +<p>The general case can here again be reduced to the more +special one in which one of the fixed points is at the inter-section +of the two given straight lines. Using the same +figure as before, but with <I>D</I> taking the position shown by (<I>D</I>) +in the figure, we take that point such that +<MATH><I>OC.AD</I> = the given rectangle</MATH>. +<p>We have then to draw <I>ON′M</I> through <I>O</I> such that +<MATH><I>B′N′.AM</I> = <I>OC.AD</I></MATH>, +or <MATH><I>B′N′</I>:<I>OC</I> = <I>AD</I>:<I>AM</I></MATH>. +<p>But, by parallels, <MATH><I>B′N′</I>:<I>OC</I> = <I>B′M</I>:<I>CM</I></MATH>; +therefore <MATH><I>AM</I>:<I>CM</I> = <I>AD</I>:<I>B′M</I></MATH> +<MATH>= <I>MD</I>:<I>B′C</I></MATH>, +so that <MATH><I>B′M.MD</I> = <I>AD.B′C</I></MATH>. +<p>Hence, as before, the problem is reduced to an application +of a rectangle in the well-known manner. The complete +<pb n=180><head>APOLLONIUS OF PERGA</head> +treatment of this problem in all its particular cases with their +<G>diorismoi/</G> could present no difficulty to Apollonius. +<p>If the two straight lines are parallel, the solution of the +problem gives a means of drawing any number of tangents +to an ellipse when two parallel tangents, their points of contact, +and the length of the parallel semi-diameter are given +(see <I>Conics,</I> III. 42). In the case of the hyperbola (III. 43) +the intercepts made by any tangent on the asymptotes contain +a constant rectangle. Accordingly the drawing of tangents +depends upon the particular case of our problem in which both +fixed points are the intersection of the two fixed lines. +<C>(<G>g</G>) <I>On determinate section</I> (<G>diwrisme/nh tomh/</G>), two Books.</C> +<p>The general problem here is, Given four points <I>A, B, C, D</I> on +a straight line, to determine another point <I>P</I> on the same +straight line such that the ratio <I>AP.CP</I>:<I>BP.DP</I> has a +given value. It is clear from Pappus's account<note>Pappus, vii, pp. 642-4.</note> of the contents +of this work, and from his extensive collection of lemmas to +the different propositions in it, that the question was very +exhaustively discussed. To determine <I>P</I> by means of the +equation +<MATH><I>AP.CP</I> = <G>l</G>.<I>BP.DP</I></MATH>, +where <I>A, B, C, D</I>, <G>l</G> are given, is in itself an easy matter since +the problem can at once be put into the form of a quadratic +equation, and the Greeks would have no difficulty in reducing +it to the usual <I>application of areas.</I> If, however (as we may +fairly suppose), it was intended for application in further +investigations, the complete, discussion of it would naturally +include not only the finding of a solution, but also the determination +of the limits of possibility and the number of possible +solutions for different positions of the point-pairs <I>A, C</I> and +<I>B, D,</I> for the cases in which the points in either pair coincide, +or in which one of the points is infinitely distant, and so on. +This agrees with what we find in Pappus, who makes it clear +that, though we do not meet with any express mention of +<I>series</I> of point-pairs determined by the equation for different +values of <G>l</G>, yet the treatise contained what amounts to a com- +<pb n=181><head><I>ON DETERMINATE SECTION</I></head> +plete <I>Theory of Involution.</I> Pappus says that the separate +cases were dealt with in which the given ratio was that of +either (1) the square of one abscissa measured from the +required point or (2) the rectangle contained by two such +abscissae to any one of the following: (1) the square of one +abscissa, (2) the rectangle contained by one abscissa and +another separate line of given length independent of the +position of the required point, (3) the rectangle contained by +two abscissae. We learn also that maxima and minima were +investigated. From the lemmas, too, we may draw other +conclusions, e.g. +<p>(1) that, in the case where <G>l</G> = 1, or <MATH><I>AP.CP</I> = <I>BP.DP</I></MATH>, +Apollonius used the relation <MATH><I>BP</I>:<I>DP</I> = <I>AB.BC</I>:<I>AD.DC</I></MATH>, +<p>(2) that Apollonius probably obtained a double point <I>E</I> of the +involution determined by the point-pairs <I>A, C</I> and <I>B, D</I> by +means of the relation +<MATH><I>AB.BC</I>:<I>AD.DC</I> = <I>BE</I><SUP>2</SUP>:<I>DE</I><SUP>2</SUP></MATH>. +<p>A possible application of the problem was the determination +of the points of intersection of the given straight line with a +conic determined as a four-line locus, since <I>A, B, C, D</I> are in +fact the points of intersection of the given straight line with +the four lines to which the locus has reference. +<C>(<G>d</G>) <I>On Contacts</I> or <I>Tangencies</I> (<G>e)pafai/</G>), two Books.</C> +<p>Pappus again comprehends in one enunciation the varieties +of problems dealt with in the treatise, which we may repro- +duce as follows: <I>Given three things, each of which may be +either a point, a straight line or a circle, to draw a circle +which shall pass through each of the given points</I> (<I>so far as it +is points that are given</I>) <I>and touch the straight lines or +circles.</I><note>Pappus, vii, p. 644, 25-8.</note> The possibilities as regards the different data are +ten. We may have any one of the following: (1) three +points, (2) three straight lines, (3) two points and a straight +line, (4) two straight lines and a point, (5) two points and +a circle, (6) two circles and a point, (7) two straight lines and +<pb n=182><head>APOLLONIUS OF PERGA</head> +a circle, (8) two circles and a straight line, (9) a point, a circle +and a straight line, (10) three circles. Of these varieties the +first two are treated in Eucl. IV; Book I of Apollonius's +treatise treated of (3), (4), (5), (6), (8), (9), while (7), the case of +two straight lines and a circle, and (10), that of the three +circles, occupied the whole of Book II. +<p>The last problem (10), where the data are three circles, +has exercised the ingenuity of many distinguished geometers, +including Vieta and Newton. Vieta (1540-1603) set the pro- +blem to Adrianus Romanus (van Roomen, 1561-1615) who +solved it by means of a hyperbola. Vieta was not satisfied +with this, and rejoined with his <I>Apollonius Gallus</I> (1600) in +which he solved the problem by plane methods. A solution +of the same kind is given by Newton in his <I>Arithmetica +Universalis</I> (Prob. xlvii), while an equivalent problem is +solved by means of two hyperbolas in the <I>Principia</I>, Lemma +xvi. The problem is quite capable of a ‘plane’ solution, and, +as a matter of fact, it is not difficult to restore the actual +solution of Apollonius (which of course used the ‘plane’ method +depending on the straight line and circle only), by means of +the lemmas given by Pappus. Three things are necessary to +the solution. (1) A proposition, used by Pappus elsewhere<note>Pappus, iv, pp. 194-6.</note> +and easily proved, that, if two circles touch internally or +externally, any straight line through the point of contact +divides the circles into segments respectively similar. (2) The +proposition that, given three circles, their six centres of similitude +(external and internal) lie three by three on four straight +lines. This proposition, though not proved in Pappus, was +certainly known to the ancient geometers; it is even possible +that Pappus omitted to prove it because it was actually proved +by Apollonius in his treatise. (3) An auxiliary problem solved +by Pappus and enunciated by him as follows.<note><I>Ib.</I> vii, p. 848.</note> Given a circle +<I>ABC,</I> and given three points <I>D, E, F</I> in a straight line, to +inflect (the broken line) <I>DAE</I> (to the circle) so as to make <I>BC</I> +in a straight line with <I>CF</I>; in other words, to inscribe in the +circle a triangle the sides of which, when produced, pass +respectively through three given points lying in a straight +line. This problem is interesting as a typical example of the +ancient analysis followed by synthesis. Suppose the problem +<pb n=183><head><I>ON CONTACTS OR TANGENCIES</I></head> +solved, i.e. suppose <I>DA, EA</I> drawn to the circle cutting it in +points <I>B, C</I> such that <I>BC</I> produced passes through <I>F</I>. +<FIG> +<p>Draw <I>BG</I> parallel to <I>DF</I>; join <I>GC</I> +and produce it to meet <I>DE</I> in <I>H.</I> +<p>Then +<MATH>∠ <I>BAC</I> = ∠ <I>BGC</I> += ∠ <I>CHF</I> += supplement of ∠<I>CHD</I></MATH>; +therefore <I>A, D, H, C</I> lie on a circle, and +<MATH><I>DE.EH</I> = <I>AE.EC</I></MATH>. +<p>Now <I>AE.EC</I> is given, being equal to the square on the +tangent from <I>E</I> to the circle; and <I>DE</I> is given; therefore <I>HE</I> +is given, and therefore the point <I>H.</I> +<p>But <I>F</I> is also given; therefore the problem is reduced to +drawing <I>HC, FC</I> to meet the circle in such a way that, if +<I>HC, FC</I> produced meet the circle again in <I>G, B,</I> the straight +line <I>BG</I> is parallel to <I>HF</I>: a problem which Pappus has +previously solved.<note>Pappus, vii, pp. 830-2.</note> +<p>Suppose this done, and draw <I>BK</I> the tangent at <I>B</I> meeting +<I>HF</I> in <I>K</I>. Then +<MATH><I>∠ KBC</I> = <I>∠BGC</I>, in the alternate segment, += <I>∠CHF</I></MATH>. +<p>Also the angle <I>CFK</I> is common to the two triangles <I>KBF, +CHF</I>; therefore the triangles are similar, and +<MATH><I>CF</I>:<I>FH</I> = <I>KF</I>:<I>FB</I></MATH>, +or <MATH><I>HF.FK</I> = <I>BF.FC</I></MATH>. +<p>Now <I>BF.FC</I> is given, and so is <I>HF</I>; +therefore <I>FK</I> is given, and therefore <I>K</I> is given. +<p>The synthesis is as follows. Take a point <I>H</I> on <I>DE</I> such +that <I>DE.EH</I> is equal to the square on the tangent from <I>E</I> to +the circle. +<p>Next take <I>K</I> on <I>HF</I> such that <I>HF.FK</I> = the square on the +tangent from <I>F</I> to the circle. +<p>Draw the tangent to the circle from <I>K,</I> and let <I>B</I> be the +point of contact. Join <I>BF</I> meeting the circle in <I>C</I>, and join +<pb n=184><head>APOLLONIUS OF PERGA</head> +<I>HC</I> meeting the circle again in <I>G.</I> It is then easy to prove +that <I>BG</I> is parallel to <I>DF.</I> +<p>Now join <I>EC</I>, and produce it to meet the circle again at <I>A</I>; +join <I>AB.</I> +<p>We have only to prove that <I>AB, BD</I> are in one straight line. +<p>Since <I>DE.EH</I> = <I>AE.EC,</I> the points <I>A, D, H, C</I> are concyclic. +<p>Now the angle <I>CHF,</I> which is the supplement of the angle +<FIG> +<I>CHD,</I> is equal to the angle <I>BGC,</I> and therefore to the +angle <I>BAC.</I> +<p>Therefore the angle <I>BAC</I> is equal to the supplement of +angle <I>DHC,</I> so that the angle <I>BAC</I> is equal to the angle <I>DAC,</I> +and <I>AB, BD</I> are in a straight line. +<p>The problem of Apollonius is now easy. We will take the +case in which the required circle touches all the three given +circles externally as shown in the figure. Let the radii of the +<pb n=185><head><I>ON CONTACTS OR TANGENCIES</I></head> +given circles be <I>a, b, c</I> and their centres <I>A, B, C.</I> Let <I>D, E, F</I> +be the external centres of similitude so that <I>BD</I>:<I>DC</I>=<I>b</I>:<I>c</I>, &c. +<p>Suppose the problem solved, and let <I>P, Q, R</I> be the points +of contact. Let <I>PQ</I> produced meet the circles with centres +<I>A, B</I> again in <I>K, L.</I> Then, by the proposition (1) above, the +segments <I>KGP, QHL</I> are both similar to the segment <I>PYQ</I>; +therefore they are similar to one another. It follows that <I>PQ</I> +produced beyond <I>L</I> passes through <I>F.</I> Similarly <I>QR, PR</I> +produced pass respectively through <I>D, E.</I> +<p>Let <I>PE, QD</I> meet the circle with centre <I>C</I> again in <I>M, N.</I> +Then, the segments <I>PQR, RNM</I> being similar, the angles +<I>PQR, RNM</I> are equal, and therefore <I>MN</I> is parallel to <I>PQ.</I> +Produce <I>NM</I> to meet <I>EF</I> in <I>V.</I> +<p>Then <MATH><I>EV</I>:<I>EF</I> = <I>EM</I>:<I>EP</I> = <I>EC</I>:<I>EA</I> = <I>c</I>:<I>a</I></MATH>; +therefore the point <I>V</I> is given. +<p>Accordingly the problem reduces itself to this: Given three +points <I>V, E, D</I> in a straight line, it is required to draw <I>DR, ER</I> +to a point <I>R</I> on the circle with centre <I>C</I> so that, if <I>DR, ER</I> meet +the circle again in <I>N, M, NM</I> produced shall pass through <I>V.</I> +This is the problem of Pappus just solved. +<p>Thus <I>R</I> is found, and <I>DR, ER</I> produced meet the circles +with centres <I>B</I> and <I>A</I> in the other required points <I>Q, P</I> +respectively. +<C>(<G>e</G>) <I>Plane loci,</I> two Books.</C> +<p>Pappus gives a pretty full account of the contents of this +work, which has sufficed to enable restorations of it to +be made by three distinguished geometers, Fermat, van +Schooten, and (most completely) by Robert Simson. Pappus +prefaces his account by a classification of loci on two +different plans. Under the first classification loci are of three +kinds: (1) <G>e)fektikoi/</G>, <I>holding-in</I> or <I>fixed</I>; in this case the +locus of a point is a point, of a line a line, and of a solid +a solid, where presumably the line or solid can only move on +itself so that it does not change its position: (2) <G>diexo- +dikoi/</G>, <I>passing-along</I>: this is the ordinary sense of a locus, +where the locus of a point is a line, and of a line a solid: +(3) <G>a)nastrofikoi/</G>, <I>moving backwards and forwards,</I> as it were, +in which sense a plane may be the locus of a point and a solid +<pb n=186><head>APOLLONIUS OF PERGA</head> +of a line.<note>Pappus, vii, pp. 660. 18-662. 5.</note> The second classification is the familiar division into +<I>plane, solid,</I> and <I>linear</I> loci, <I>plane</I> loci being straight lines +and circles only, <I>solid</I> loci conic sections only, and <I>linear</I> loci +those which are not straight lines nor circles nor any of the +conic sections. The loci dealt with in our treatise are accordingly +all straight lines or circles. The proof of the propositions +is of course enormously facilitated by the use of +Cartesian coordinates, and many of the loci are really the +geometrical equivalent of fundamental theorems in analytical +or algebraical geometry. Pappus begins with a composite +enunciation, including a number of propositions, in these +terms, which, though apparently confused, are not difficult +to follow out: +<p>‘If two straight lines be drawn, from one given point or from +two, which are (<I>a</I>) in a straight line or (<I>b</I>) parallel or +(<I>c</I>) include a given angle, and either (<G>a</G>) bear a given ratio to +one another or (<G>b</G>) contain a given rectangle, then, if the locus +of the extremity of one of the lines is a plane locus given in +position, the locus of the extremity of the other will also be a +plane locus given in position, which will sometimes be of the +same kind as the former, sometimes of the other kind, and +will sometimes be similarly situated with reference to the +straight line, and sometimes contrarily, according to the +particular differences in the suppositions.’<note><I>Ib.</I> vii, pp. 662. 25-664. 7.</note> +<p>(The words ‘with reference to <I>the straight line</I>’ are obscure, but +the straight line is presumably some obvious straight line in +each figure, e.g., when there are two given points, the straight +line joining them.) After quoting three obvious loci ‘added +by Charmandrus’, Pappus gives three loci which, though containing +an unnecessary restriction in the third case, amount +to the statement that any equation of the first degree between +coordinates inclined at fixed angles to (<I>a</I>) two axes perpendicular +or oblique, (<I>b</I>) to any number of axes, represents a +straight line. The enunciations (5-7) are as follows.<note><I>Ib.,</I> pp. 664. 20-666. 6.</note> +<p>5. ‘If, when a straight line is given in magnitude and is +moved so as always to be parallel to a certain straight line +given in position, one of the extremities (of the moving +straight line), lies on a straight line given in position, the +<pb n=187><head><I>PLANE LOCI</I></head> +other extremity will also lie on a straight line given in +position.’ +<p>(That is, <MATH><I>x</I> = <I>a</I> or <I>y</I> = <I>b</I></MATH> in Cartesian coordinates represents a +straight line.) +<p>6. ‘If from any point straight lines be drawn to meet at given +angles two straight lines either parallel or intersecting, and if +the straight lines so drawn have a given ratio to one another +or if the sum of one of them and a line to which the other has +a given ratio be given (in length), then the point will lie on a +straight line given in position.’ +<p>(This includes the equivalent of saying that, if <I>x, y</I> be the +coordinates of the point, each of the equations <MATH><I>x</I> = <I>my, +x</I> + <I>my</I> = <I>c</I></MATH> represents a straight line.) +<p>7. ‘If any number of straight lines be given in position, and +straight lines be drawn from a point to meet them at given +angles, and if the straight lines so drawn be such that the +rectangle contained by one of them and a given straight line +added to the rectangle contained by another of them and +(another) given straight line is equal to the rectangle contained +by a third and a (third) given straight line, and similarly +with the others, the point will lie on a straight line given +in position.’ +<p>(Here we have trilinear or multilinear coordinates proportional +to the distances of the variable point from each of the +three or more fixed lines. When there are three fixed lines, +the statement is that <MATH><I>ax</I> + <I>by</I> = <I>cz</I></MATH> represents a straight line. +The precise meaning of the words ‘and similarly with the +the others’ or ‘of the others’—<G>kai\ tw=n loipw=n o(moi/ws</G>—is +uncertain; the words seem to imply that, when there were +more than three rectangles <I>ax, by, cz</I> ..., two of them were +taken to be equal to the sum of all the others; but it is quite +possible that Pappus meant that any linear equation between +these rectangles represented a straight line. Precisely how +far Apollonius went in generality we are not in a position to +judge.) +<p>The last enunciation (8) of Pappus referring to Book I +states that, +<p>‘If from any point (two) straight lines be drawn to meet (two) +parallel straight lines given in position at given angles, and +<pb n=188><head>APOLLONIUS OF PERGA</head> +cut off from the parallels straight lines measured from given +points on them such that (<I>a</I>) they have a given ratio or +(<I>b</I>) they contain a given rectangle or (<I>c</I>) the sum or difference +of figures of given species described on them respectively is +equal to a given area, the point will lie on a straight line +given in position.’<note>Pappus, vii, p. 666. 7-13.</note> +<p>The contents of Book II are equally interesting. Some of +the enunciations shall for brevity be given by means of letters +instead of in general terms. If from two given points <I>A, B</I> +two straight lines be ‘inflected’ (<G>klasqw=sin</G>) to a point <I>P,</I> then +(1), if <I>AP</I><SUP>2</SUP> <01> <I>BP</I><SUP>2</SUP> is given, the locus of <I>P</I> is a straight line; +(2) if <I>AP, BP</I> are in a given ratio, the locus is a straight line +or a circle [this is the proposition quoted by Eutocius in his +commentary on the <I>Conics,</I> but already known to Aristotle]; +(4) if <I>AP</I><SUP>2</SUP> is ‘greater by a given area than in a given ratio’ +to <I>BP</I><SUP>2</SUP>, i.e. if <I>AP</I><SUP>2</SUP> = <I>a</I><SUP>2</SUP> + <I>m.BP</I><SUP>2</SUP>, the locus is a circle given in +position. An interesting proposition is (5) that, ‘If from any +number of given points whatever straight lines be inflected to +one point, and the figures (given in species) described on all of +them be together equal to a given area, the point will lie on +a circumference (circle) given in position’; that is to say, if +<G>a</G>. <I>AP</I><SUP>2</SUP> + <G>b</G>.<I>BP</I><SUP>2</SUP> + <G>g</G>.<I>CP</I><SUP>2</SUP> + ... = a given area (where <G>a, b, g</G> ... +are constants), the locus of <I>P</I> is a circle. (3) states that, if +<I>AN</I> be a fixed straight line and <I>A</I> a fixed point on it, and if +<I>AP</I> be any straight line drawn to a point <I>P</I> such that, if <I>PN</I> +is perpendicular to <I>AN, AP</I><SUP>2</SUP> = <I>a.AN</I> or <I>a.BN,</I> where <I>a</I> is a +given length and <I>B</I> is another fixed point on <I>AN,</I> then the +locus of <I>P</I> is a circle given in position; this is equivalent +to the fact that, if <I>A</I> be the origin, <I>AN</I> the axis of <I>x,</I> and +<MATH><I>x</I> = <I>AN, y</I> = <I>PN</I></MATH> be the coordinates of <I>P,</I> the locus <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>ax</I></MATH> +or <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>a</I>(<I>x</I>-<I>b</I>)</MATH> is a circle. (6) is somewhat obscurely +enunciated: ‘If from two given points straight lines be inflected +(to a point), and from the point (of concourse) a straight +line be drawn parallel to a straight line given in position and +cutting off from another straight line given in position an +intercept measured from a given point on it, and if the sum of +figures (given in species) described on the two inflected lines +be equal to the rectangle contained by a given straight line +and the intercept, the point at which the straight lines are +<pb n=189><head><I>PLANE LOCI</I></head> +inflected lies on a circle given in position.’ The meaning +seems to be this: Given two fixed points <I>A, B,</I> a length <I>a,</I> +a straight line <I>OX</I> with a point <I>O</I> fixed upon it, and a direction +represented, say, by any straight line <I>OZ</I> through <I>O,</I> then, +if <I>AP, BP</I> be drawn to <I>P,</I> and <I>PM</I> parallel to <I>OZ</I> meets <I>OX</I> +in <I>M,</I> the locus of <I>P</I> will be a circle given in position if +<MATH><G>a</G>.<I>AP</I><SUP>2</SUP> + <G>b</G>.<I>BP</I><SUP>2</SUP> = <G>a</G>.<I>OM</I></MATH>, +where <G>a, b</G> are constants. The last two loci are again +obscurely expressed, but the sense is this: (7) If <I>PQ</I> be any +chord of a circle passing through a fixed internal point <I>O,</I> and +<I>R</I> be an external point on <I>PQ</I> produced such that either +<MATH>(<I>a</I>) <I>OR</I><SUP>2</SUP> = <I>PR.RQ</I> or (<I>b</I>) <I>OR</I><SUP>2</SUP> + <I>PO.OQ</I> = <I>PR.RQ</I></MATH>, the locus +of <I>R</I> is a straight line given in position. (8) is the reciprocal +of this: Given the fixed point <I>O,</I> the straight line which is +the locus of <I>R,</I> and also the relation (<I>a</I>) or (<I>b</I>), the locus of +<I>P, Q</I> is a circle. +<C>(<G>z</G>) <G>*neu/seis</G> (<I>Vergings</I> or <I>Inclinations</I>), two Books.</C> +<p>As we have seen, the problem in a <G>neu=sis</G> is to place +between two straight lines, a straight line and a curve, or +two curves, a straight line of given length in such a way +that it <I>verges</I> towards a fixed point, i.e. it will, if produced, +pass through a fixed point. Pappus observes that, +when we come to particular cases, the problem will be +‘plane’, ‘solid’ or ‘linear’, according to the nature of the +particular hypotheses; but a selection had been made from +the class which could be solved by plane methods, i.e. by +means of the straight line and circle, the object being to give +those which were more generally useful in geometry. The +following were the cases thus selected and proved.<note>Pappus, vii, pp. 670-2.</note> +<p>I. Given (<I>a</I>) a semicircle and a straight line at right angles +to the base, or (<I>b</I>) two semicircles with their bases in a straight +line, to insert a straight line of given length verging to an +angle of the semicircle [or of one of the semicircles]. +<p>II. Given a rhombus with one side produced, to insert +a straight line of given length in the external angle so that it +verges to the opposite angle. +<pb n=190><head>APOLLONIUS OF PERGA</head> +<p>III. Given a circle, to insert a chord of given length verging +to a given point. +<p>In Book I of Apollonius's work there were four cases of +I (<I>a</I>), two cases of III, and two of II; the second Book contained +ten cases of I (<I>b</I>). +<p>Restorations were attempted by Marino Ghetaldi (<I>Apollonius +redivivus,</I> Venice, 1607, and <I>Apollonius redivivus . . . Liber +secundus,</I> Venice, 1613), Alexander Anderson (in a <I>Supplementum +Apollonii redivivi,</I> 1612), and Samuel Horsley +(Oxford, 1770); the last is much the most complete. +<p>In the case of the rhombus (II) the construction of Apollonius +can be restored with certainty. It depends on a lemma given +by Pappus, which is as follows: Given a rhombus <I>AD</I> with +diagonal <I>BC</I> produced to <I>E,</I> if <I>F</I> be taken on <I>BC</I> such that <I>EF</I> +is a mean proportional between <I>BE</I> and <I>EC,</I> and if a circle be +<FIG> +described with <I>E</I> as centre and <I>EF</I> as radius cutting <I>CD</I> +in <I>K</I> and <I>AC</I> produced in <I>H,</I> then shall <I>B, K, H</I> be in one +straight line.<note>Pappus, vii, pp. 778-80.</note> +<p>Let the circle cut <I>AC</I> in <I>L,</I> join <I>LK</I> meeting <I>BC</I> in <I>M,</I> and +join <I>HE, LE, KE.</I> +<p>Since now <I>CL, CK</I> are equally inclined to the diameter of +the circle, <I>CL</I> = <I>CK.</I> Also <I>EL</I> = <I>EK,</I> and it follows that the +triangles <I>ECK, ECL</I> are equal in all respects, so that +<MATH>∠ <I>CKE</I> = ∠ <I>CLE</I> = ∠ <I>CHE</I></MATH>. +<p>By hypothesis, <MATH><I>EB</I>:<I>EF</I> = <I>EF</I>:<I>EC</I></MATH>, +or <MATH><I>EB</I>:<I>EK</I> = <I>EK</I>:<I>EC</I></MATH>. +<pb n=191><head><G>*n*e*g*s*e*i*s</G> (<I>VERGINGS OR INCLINATIONS</I>)</head> +<p>Therefore the triangles <I>BEK, KEC,</I> which have the angle +<I>BEK</I> common, are similar, and +<MATH>∠ <I>CBK</I> = ∠ <I>CKE</I> = ∠ <I>CHE</I> (from above)</MATH>. +<p>But <MATH>∠ <I>HCE</I> = ∠ <I>ACB</I> = ∠ <I>BCK</I></MATH>. +<p>Therefore in the triangles <I>CBK, CHE</I> two angles are +respectively equal, so that ∠ <I>CEH</I> = ∠ <I>CKB</I> also. +<p>But since ∠ <I>CKE</I> = ∠ <I>CHE</I> (from above), <I>K, C, E, H</I> are +concyclic. +<p>Hence <MATH>∠ <I>CEH</I> + ∠ <I>CKH</I> = (two right angles)</MATH>; +therefore, since <MATH>∠ <I>CEH</I> = ∠ <I>CKB</I></MATH>, +<MATH>∠ <I>CKB</I> + ∠ <I>CKH</I> = (two right angles)</MATH>, +and <I>BKH</I> is a straight line. +<p>It is certain, from the nature of this lemma, that Apollonius +made his construction by drawing the circle shown in the +figure. +<p>He would no doubt arrive at it by analysis somewhat as +follows. +<p>Suppose the problem solved, and <I>HK</I> inserted as required +(= <I>k</I>). +<p>Bisect <I>HK</I> in <I>N,</I> and draw <I>NE</I> at right angles to <I>KH</I> +meeting <I>BC</I> produced in <I>E.</I> Draw <I>KM</I> perpendicular to <I>BC,</I> +and produce it to meet <I>AC</I> in <I>L.</I> Then, by the property of +the rhombus, <I>LM</I> = <I>MK,</I> and, since <I>KN</I> = <I>NH</I> also, <I>MN</I> is +parallel to <I>LH.</I> +<p>Now, since the angles at <I>M, N</I> are right, <I>M, K, N, E</I> are +concyclic. +<p>Therefore ∠ <I>CEK</I> = ∠ <I>MNK</I> = ∠ <I>CHK,</I> so that <I>C, K, H, E</I> +are concyclic. +<p>Therefore ∠ <I>BCD</I> = supplement of <I>KCE</I> = ∠ <I>EHK</I> = ∠ <I>EKH,</I> +and the triangles <I>EKH, DCB</I> are similar. +<p>Lastly, +∠ <I>EBK</I> = ∠ <I>EKH</I> - ∠ <I>CEK</I> = ∠ <I>EHK</I> - ∠ <I>CHK</I> = ∠ <I>EHC</I> = ∠ <I>EKC</I>; +therefore the triangles <I>EBK, EKC</I> are similar, and +<MATH><I>BE</I>:<I>EK</I> = <I>EK</I>:<I>EC</I></MATH>, +or <MATH><I>BE.EC</I> = <I>EK</I><SUP>2</SUP></MATH>. +<pb n=192><head>APOLLONIUS OF PERGA</head> +<p>But, by similar triangles <I>EKH, DCB</I>, +<MATH><I>EK</I>:<I>KH</I> = <I>DC</I>:<I>CB</I></MATH>, +and, since the ratio <I>DC</I>:<I>CB,</I> as well as <I>KH,</I> is given, <I>EK</I> +is given. +<p>The construction then is as follows. +<p>If <I>k</I> be the given length, take a straight line <I>p</I> such that +<MATH><I>p</I>:<I>k</I>=<I>AB</I>:<I>BC</I></MATH>; +apply to <I>BC</I> a rectangle <I>BE.EC</I> equal to <I>p</I><SUP>2</SUP> and exceeding by +a square; then with <I>E</I> as centre and radius equal to <I>p</I> describe a +circle cutting <I>AC</I> produced in <I>H</I> and <I>CD</I> in <I>K.</I> <I>HK</I> is then +equal to <I>k</I> and, by Pappus's lemma, verges towards <I>B.</I> +<p>Pappus adds an interesting solution of the same problem +with reference to a square instead of a rhombus; the solution +is by one Heraclitus and depends on a lemma which Pappus +also gives.<note>Pappus, vii, pp. 780-4.</note> +<p>We hear of yet other lost works by Apollonius. +<p>(<G>h</G>) A <I>Comparison of the dodecahedron with the icosahedron.</I> +This is mentioned by Hypsicles in the preface to the so-called +Book XIV of Euclid. Like the <I>Conics,</I> it appeared in two +editions, the second of which contained the proposition that, +if there be a dodecahedron and an icosahedron inscribed in +one and the same sphere, the surfaces of the solids are in the +same ratio as their volumes; this was established by showing +that the perpendiculars from the centre of the sphere to +a pentagonal face of the dodecahedron and to a triangular +face of the icosahedron are equal. +<p>(<G>q</G>) Marinus on Euclid's <I>Data</I> speaks of a <I>General Treatise</I> +(<G>kaqo/lou pragmatei/a</G>) in which Apollonius used the word +<I>assigned</I> (<G>tetagme/non</G>) as a comprehensive term to describe the +<I>datum</I> in general. It would appear that this work must +have dealt with the fundamental principles of mathematics, +definitions, axioms, &c., and that to it must be referred the +various remarks on such subjects attributed to Apollonius by +Proclus, the elucidation of the notion of a line, the definition +<pb n=193><head>OTHER LOST WORKS</head> +of plane and solid angles, and his attempts to prove the axioms; +it must also have included the three definitions (13-15) in +Euclid's <I>Data</I> which, according to a scholium, were due to +Apollonius and must therefore have been interpolated (they +are definitions of <G>kathgme/nh, a)nhgme/nh</G>, and the elliptical +phrase <G>para\ qe/sei</G>, which means ‘parallel to a straight line +given in position’). Probably the same work also contained +Apollonius's alternative constructions for the problems of +Eucl. I. 10, 11 and 23 given by Proclus. Pappus speaks +of a mention by Apollonius ‘before his own elements’ of the +class of locus called <G>e)fektiko/s</G>, and it may be that the treatise +now in question is referred to rather than the <I>Plane Loci</I> +itself. +<p>(<G>i</G>) The work <I>On the Cochlias</I> was on the cylindrical helix. +It included the theoretical generation of the curve on the +surface of the cylinder, and the proof that the curve is +<I>homoeomeric</I> or uniform, i.e. such that any part will fit upon +or coincide with any other. +<p>(<G>k</G>) A work on <I>Unordered Irrationals</I> is mentioned by +Proclus, and a scholium on Eucl. X. 1 extracted from Pappus's +commentary remarks that ‘Euclid did not deal with all +rationals and irrationals, but only with the simplest kinds by +the combination of which an infinite number of irrationals +are formed, of which latter Apollonius also gave some’. +To a like effect is a passage of the fragment of Pappus's +commentary on Eucl. X discovered in an Arabic translation +by Woepcke: ‘it was Apollonius who, besides the <I>ordered</I> +irrational magnitudes, showed the existence of the <I>unordered,</I> +and by accurate methods set forth a great number of them’. +The hints given by the author of the commentary seem to imply +that Apollonius's extensions of the theory of irrationals took +two directions, (1) generalizing the <I>medial</I> straight line of +Euclid, on the basis that, between two lines commensurable in +square (only), we may take not only one sole medial line but +three or four, and so on <I>ad infinitum,</I> since we can take, +between any two given straight lines, as many lines as +we please in continued proportion, (2) forming compound +irrationals by the addition and subtraction of more than two +terms of the sort composing the <I>binomials, apotomes,</I> &c. +<pb n=194><head>APOLLONIUS OF PERGA</head> +<p>(<G>l</G>) <I>On the burning-mirror</I> (<G>peri\ tou= puri/ou</G>) is the title of +another work of Apollonius mentioned by the author of the +<I>Fragmentum mathematicum Bobiense,</I> which is attributed by +Heiberg to Anthemius but is more likely (judging by its survivals +of antiquated terminology) to belong to a much earlier +date. The fragment shows that Apollonius discussed the +spherical form of mirror among others. Moreover, the extant +fragment by Anthemius himself (on burning mirrors) proves the +property of mirrors of parabolic section, using the properties of +the parabola (<I>a</I>) that the tangent at any point makes equal +angles with the axis and with the focal distance of the point, +and (<I>b</I>) that the distance of any point on the curve from the +focus is equal to its distance from a certain straight line +(our ‘directrix’); and we can well believe that the parabolic +form of mirror was also considered in Apollonius's work, and +that he was fully aware of the focal properties of the parabola, +notwithstanding the omission from the <I>Conics</I> of all mention +of the focus of a parabola. +<p>(<G>m</G>) In a work called <G>w)kuto/kion</G> (‘quick-delivery’) Apollonius +is said to have found an approximation to the value of <G>p</G> ‘by +a different calculation (from that of Archimedes), bringing it +within closer limits’.<note>v. Eutocius on Archimedes, <I>Measurement of a Circle.</I></note> Whatever these closer limits may have +been, they were considered to be less suitable for practical use +than those of Archimedes. +<p>It is a moot question whether Apollonius's system of arithmetical +notation (by tetrads) for expressing large numbers +and performing the usual arithmetical operations with them, +as described by Pappus, was included in this same work. +Heiberg thinks it probable, but there does not seem to be any +necessary reason why the notation for large numbers, classifying +them into myriads, double myriads, triple myriads, &c., +i.e. according to powers of 10,000, need have been connected +with the calculation of the value of <G>p</G>, unless indeed the numbers +used in the calculation were so large as to require the +tetradic system for the handling of them. +<p>We have seen that Apollonius is credited with a solution +of the problem of the two mean proportionals (vol. i, +pp. 262-3). +<pb n=195><head>OTHER LOST WORKS</head> +<C><I>Astronomy.</I></C> +<p>We are told by Ptolemaeus Chennus<note><I>apud Photium,</I> Cod. cxc, p. 151 b 18, ed. Bekker.</note> that Apollonius was +famed for his astronomy, and was called <G>e</G> (Epsilon) because +the form of that letter is associated with that of the moon, to +which his accurate researches principally related. Hippolytus +says he made the distance of the moon's circle from the surface +of the earth to be 500 myriads of stades.<note>Hippol. <I>Refut.</I> iv. 8, p. 66, ed. Duncker.</note> This figure +can hardly be right, for, the diameter of the earth being, +according to Eratosthenes's evaluation, about eight myriads of +stades, this would make the distance of the moon from the +earth about 125 times the earth's radius. This is an unlikely +figure, seeing that Aristarchus had given limits for the ratios +between the distance of the moon and its diameter, and +between the diameters of the moon and the earth, which lead +to about 19 as the ratio of the moon's distance to the earth's +radius. Tannery suggests that perhaps Hippolytus made a +mistake in copying from his source and took the figure of +5,000,000 stades to be the length of the radius instead of the +<I>diameter</I> of the moon's orbit. +<p>But we have better evidence of the achievements of Apollonius +in astronomy. In Ptolemy's <I>Syntaxis</I><note>Ptolemy, <I>Syntaxis,</I> xii. 1.</note> he appears as +an authority upon the hypotheses of epicycles and eccentrics +designed to account for the apparent motions of the planets. +The propositions of Apollonius quoted by Ptolemy contain +exact statements of the alternative hypotheses, and from this +fact it was at one time concluded that Apollonius invented +the two hypotheses. This, however, is not the case. The +hypothesis of epicycles was already involved, though with +restricted application, in the theory of Heraclides of Pontus +that the two inferior planets, Mercury and Venus, revolve in +circles like satellites round the sun, while the sun itself +revolves in a circle round the earth; that is, the two planets +describe epicycles about the material sun as moving centre. +In order to explain the motions of the superior planets by +means of epicycles it was necessary to conceive of an epicycle +about a point as moving centre which is not a material but +a mathematical point. It was some time before this extension +of the theory of epicycles took place, and in the meantime +<pb n=196><head>APOLLONIUS OF PERGA</head> +another hypothesis, that of eccentrics, was invented to account +for the movements of the superior planets only. We are at this +stage when we come to Apollonius. His enunciations show +that he understood the theory of epicycles in all its generality, +but he states specifically that the theory of eccentrics can only +be applied to the three planets which can be at any distance +from the sun. The reason why he says that the eccentric +hypothesis will not serve for the inferior planets is that, in +order to make it serve, we should have to suppose the circle +described by the centre of the eccentric circle to be greater +than the eccentric circle itself. (Even this generalization was +made later, at or before the time of Hipparchus.) Apollonius +further says in his enunciation about the eccentric that ‘the +centre of the eccentric circle moves about the centre of the +zodiac in the direct order of the signs and <I>at a speed equal to +that of the sun,</I> while the star moves on the eccentric about +its centre in the inverse order of the signs and at a speed +equal to the anomaly’. It is clear from this that the theory +of eccentrics was invented for the specific purpose of explaining +the movements of Mars, Jupiter, and Saturn about the +sun and for that purpose alone. This explanation, combined +with the use of epicycles about the sun as centre to account +for the motions of Venus and Mercury, amounted to the +system of Tycho Brahe; that system was therefore anticipated +by some one intermediate in date between Heraclides and +Apollonius and probably nearer to the latter, or it may +have been Apollonius himself who took this important step. +If it was, then Apollonius, coming after Aristarchus of +Samos, would be exactly the Tycho Brahe of the Copernicus +of antiquity. The actual propositions quoted by Ptolemy as +proved by Apollonius among others show mathematically at +what points, under each of the two hypotheses, the apparent +forward motion changes into apparent retrogradation and +vice versa, or the planet appears to be <I>stationary.</I> +<pb> +<C>XV +THE SUCCESSORS OF THE GREAT GEOMETERS</C> +<p>WITH Archimedes and Apollonius Greek geometry reached +its culminating point. There remained details to be filled +in, and no doubt in a work such as, for instance, the <I>Conics</I> +geometers of the requisite calibre could have found proposi- +tions containing the germ of theories which were capable of +independent development. But, speaking generally, the fur- +ther progress of geometry on general lines was practically +barred by the restrictions of method and form which were +inseparable from the classical Greek geometry. True, it was +open to geometers to discover and investigate curves of a +higher order than conics, such as spirals, conchoids, and the +like. But the Greeks could not get very far even on these +lines in the absence of some system of coordinates and without +freer means of manipulation such as are afforded by modern +algebra, in contrast to the geometrical algebra, which could +only deal with equations connecting lines, areas, and volumes, +but involving no higher dimensions than three, except in so +far as the use of proportions allowed a very partial exemp- +tion from this limitation. The theoretical methods available +enabled quadratic, cubic and bi-quadratic equations or their +equivalents to be solved. But all the solutions were <I>geometri- +cal</I>; in other words, quantities could only be represented by +lines, areas and volumes, or ratios between them. There was +nothing corresponding to operations with general algebraical +quantities irrespective of what they represented. There were +no <I>symbols</I> for such quantities. In particular, the irrational +was discovered in the form of incommensurable <I>lines</I>; hence +irrationals came to be represented by straight lines as they +are in Euclid, Book X, and the Greeks had no other way of +representing them. It followed that a product of two irra- +tionals could only be represented by a <I>rectangle</I>, and so on. +Even when Diophantus came to use a symbol for an unknown +<pb n=198><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +quantity, it was only an abbreviation for the word <G>a)riqmo/s</G>, +with the meaning of ‘an undetermined multitude of units’, +not a general quantity. The restriction then of the algebra +employed by geometers to the geometrical form of algebra +operated as an insuperable obstacle to any really new depar- +ture in theoretical geometry. +<p>It might be thought that there was room for further exten- +sions in the region of solid geometry. But the fundamental +principles of solid geometry had also been laid down in Euclid, +Books XI-XIII; the theoretical geometry of the sphere had +been fully treated in the ancient <I>sphaeric</I>; and any further +application of solid geometry, or of loci in three dimensions, +was hampered by the same restrictions of method which +hindered the further progress of plane geometry. +<p>Theoretical geometry being thus practically at the end of +its resources, it was natural that mathematicians, seeking for +an opening, should turn to the <I>applications</I> of geometry. One +obvious branch remaining to be worked out was the geometry +of measurement, or <I>mensuration</I> in its widest sense, which of +course had to wait on pure theory and to be based on its +results. One species of mensuration was immediately required +for astronomy, namely the measurement of triangles, especially +spherical triangles; in other words, trigonometry plane and +spherical. Another species of mensuration was that in which +an example had already been set by Archimedes, namely the +measurement of areas and volumes of different shapes, and +arithmetical approximations to their true values in cases +where they involved surds or the ratio (<G>p</G>) between the +circumference of a circle and its diameter; the object of such +mensuration was largely practical. Of these two kinds of +mensuration, the first (trigonometry) is represented by Hip- +parchus, Menelaus and Ptolemy; the second by Heron of +Alexandria. These mathematicians will be dealt with in later +chapters; this chapter will be devoted to the successors of the +great geometers who worked on the same lines as the latter. +<p>Unfortunately we have only very meagre information as to +what these geometers actually accomplished in keeping up the +tradition. No geometrical works by them have come down +to us in their entirety, and we are dependent on isolated +extracts or scraps of information furnished by commen- +<pb n=199><head>NICOMEDES</head> +tators, and especially by Pappus and Eutocius. Some of +these are very interesting, and it is evident from the +extracts from the works of such writers as Diocles and +Dionysodorus that, for some time after Archimedes and +Apollonius, mathematicians had a thorough grasp of the +contents of the works of the great geometers, and were able +to use the principles and methods laid down therein with +ease and skill. +<p>Two geometers properly belonging to this chapter have +already been dealt with. The first is NICOMEDES, the inventor +of the conchoid, who was about intermediate in date between +Eratosthenes and Apollonius. The conchoid has already been +described above (vol. i, pp. 238-40). It gave a general method +of solving any <G>neu=sis</G> where one of the lines which cut off an +intercept of given length on the line verging to a given point +is a straight line; and it was used both for the finding of two +mean proportionals and for the trisection of any angle, these +problems being alike reducible to a <G>neu=sis</G> of this kind. How +far Nicomedes discussed the properties of the curve in itself +is uncertain; we only know from Pappus that he proved two +properties, (1) that the so-called ‘ruler’ in the instrument for +constructing the curve is an asymptote, (2) that any straight +line drawn in the space between the ‘ruler’ or asymptote and +the conchoid must, if produced, be cut by the conchoid.<note>Pappus, iv, p. 244. 21-8.</note> The +equation of the curve referred to polar coordinates is, as we +have seen, <MATH><I>r</I> = <I>a</I> + <I>b</I> sec <G>q</G></MATH>. According to Eutocius, Nicomedes +prided himself inordinately on his discovery of this curve, +contrasting it with Eratosthenes's mechanism for finding any +number of mean proportionals, to which he objected formally +and at length on the ground that it was impracticable and +entirely outside the spirit of geometry.<note>Eutoc. on Archimedes, <I>On the Sphere and Cylinder</I>, Archimedes, +vol. iii, p. 98.</note> +<p>Nicomedes is associated by Pappus with Dinostratus, the +brother of Menaechmus, and others as having applied to the +squaring of the circle the curve invented by Hippias and +known as the <I>quadratrix</I>,<note>Pappus, iv, pp. 250. 33-252. 4. Cf. vol. i, p. 225 sq.</note> which was originally intended for +the purpose of trisecting any angle. These facts are all that +we know of Nicomedes's achievements. +<pb n=200><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +<p>The second name is that of DIOCLES. We have already +(vol. i, pp. 264-6) seen him as the discoverer of the curve +known as the <I>cissoid</I>, which he used to solve the problem +of the two mean proportionals, and also (pp. 47-9 above) +as the author of a method of solving the equivalent of +a certain cubic equation by means of the intersection +of an ellipse and a hyperbola. We are indebted for our +information on both these subjects to Eutocius,<note>Eutocius, <I>loc. cit.</I>, p. 66. 8 sq., p. 160. 3 sq.</note> who tells +us that the fragments which he quotes came from Diocles's +work <G>peri\ purei/wn</G>, <I>On burning-mirrors.</I> The connexion of +the two things with the subject of this treatise is not obvious, +and we may perhaps infer that it was a work of considerable +scope. What exactly were the forms of the burning-mirrors +discussed in the treatise it is not possible to say, but it is +probably safe to assume that among them were concave +mirrors in the forms (1) of a sphere, (2) of a paraboloid, and +(3) of the surface described by the revolution of an ellipse +about its major axis. The author of the <I>Fragmentum mathe- +maticum Bobiense</I> says that Apollonius in his book <I>On the +burning-mirror</I> discussed the case of the concave spherical +mirror, showing about what point ignition would take place; +and it is certain that Apollonius was aware that an ellipse has +the property of reflecting all rays through one focus to the +other focus. Nor is it likely that the corresponding property +of a parabola with reference to rays parallel to the axis was +unknown to Apollonius. Diocles therefore, writing a century +or more later than Apollonius, could hardly have failed to +deal with all three cases. True, Anthemius (died about +A.D. 534) in his fragment on burning-mirrors says that the +ancients, while mentioning the usual burning-mirrors and +saying that such figures are conic sections, omitted to specify +which conic sections, and how produced, and gave no geo- +metrical proofs of their properties. But if the properties +were commonly known and quoted, it is obvious that they +must have been proved by the ancients, and the explanation +of Anthemius's remark is presumably that the original works +in which they were proved (e.g. those of Apollonius and +Diocles) were already lost when he wrote. There appears to +be no trace of Diocles's work left either in Greek or Arabic, +<pb n=201><head>DIOCLES</head> +unless we have a fragment from it in the <I>Fragmentum +mathematicum Bobiense.</I> But Moslem writers regarded Diocles +as the discoverer of the parabolic burning-mirror; ‘the ancients’, +says al Singārī (Sachāwī, An⋅ārī), ‘made mirrors of plane +surfaces. Some made them concave (i.e. spherical) until +Diocles (Diūklis) showed and proved that, if the surface of +these mirrors has its curvature in the form of a parabola, they +then have the greatest power and burn most strongly. There +is a work on this subject composed by Ibn al-Haitham.’ This +work survives in Arabic and in Latin translations, and is +reproduced by Heiberg and Wiedemann<note><I>Bibliotheca mathematica</I>, x<SUB>3</SUB>, 1910, pp. 201-37.</note>; it does not, how- +ever, mention the name of Diocles, but only those of Archi- +medes and Anthemius. Ibn al-Haitham says that famous +men like Archimedes and Anthemius had used mirrors made +up of a number of spherical rings; afterwards, he adds, they +considered the form of curves which would reflect rays to one +point, and found that the concave surface of a paraboloid of +revolution has this property. It is curious to find Ibn al- +Haitham saying that the ancients had not set out the proofs +sufficiently, nor the method by which they discovered them, +words which almost exactly recall those of Anthemius himself. +Nevertheless the whole course of Ibn al-Haitham's proofs is +on the Greek model, Apollonius being actually quoted by name +in the proof of the main property of the parabola required, +namely that the tangent at any point of the curve makes +equal angles with the focal distance of the point and the +straight line drawn through it parallel to the axis. A proof +of the property actually survives in the Greek <I>Fragmentum +mathematicum Bobiense</I>, which evidently came from some +treatise on the parabolic burning-mirror; but Ibn al-Haitham +does not seem to have had even this fragment at his disposal, +since his proof takes a different course, distinguishing three +different cases, reducing the property by analysis to the +known property <I>AN</I> = <I>AT</I>, and then working out the syn- +thesis. The proof in the <I>Fragmentum</I> is worth giving. It is +substantially as follows, beginning with the preliminary lemma +that, if <I>PT</I>, the tangent at any point <I>P</I>, meets the axis at <I>T</I>, +and if <I>AS</I> be measured along the axis from the vertex <I>A</I> +equal to 1/4<I>AL</I>, where <I>AL</I> is the parameter, then <I>SP</I> = <I>ST.</I> +<pb n=202><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +<p>Let <I>PN</I> be the ordinate from <I>P</I>; draw <I>AY</I> at right angles +to the axis meeting <I>PT</I> in <I>Y</I>, and join <I>SY.</I> +<p>Now <MATH><I>PN</I><SUP>2</SUP> = <I>AL.AN</I> += 4<I>AS.AN</I> += 4<I>AS.AT</I> (since <I>AN</I> = <I>AT</I>)</MATH>. +<p>But <MATH><I>PN</I> = 2<I>AY</I> (since <I>AN</I> = <I>AT</I>); +therefore <I>AY</I><SUP>2</SUP> = <I>TA.AS</I></MATH>, +and the angle <I>TYS</I> is right. +<p>The triangles <I>SYT, SYP</I> being right-angled, and <I>TY</I> being +equal to <I>YP</I>, it follows that <I>SP</I> = <I>ST.</I> +<FIG> +<p>With the same figure, let <I>BP</I> be a ray parallel to <I>AN</I> +impinging on the curve at <I>P.</I> It is required to prove that +the angles of incidence and reflection (to <I>S</I>) are equal. +<p>We have <I>SP</I> = <I>ST</I>, so that ‘the angles at the points <I>T, P</I> +are equal. So’, says the author, ‘are the angles <I>TPA, KPR</I> +[the angles between the tangent and the <I>curve</I> on each side of +the point of contact]. Let the difference between the angles +be taken; therefore the angles <I>SPA, RPB</I> which remain +[again ‘mixed’ angles] are equal. Similarly we shall show +that all the lines drawn parallel to <I>AS</I> will be reflected at +equal angles to the point <I>S.</I>’ +<p>The author then proceeds: ‘Thus burning-mirrors con- +structed with the surface of impact (in the form) of the +<I>section of a right-angled cone</I> may easily, in the manner +<pb n=203><head>DIOCLES. PERSEUS</head> +above shown, be proved to bring about ignition at the point +indicated.’ +<p>Heiberg held that the style of this fragment is Byzantine +and that it is probably by Anthemius. Cantor conjectured +that here we might, after all, have an extract from Diocles's +work. Heiberg's supposition seems to me untenable because +of the author's use (1) of the ancient terms ‘section of +a right-angled cone’ for parabola and ‘diameter’ for axis +(to say nothing of the use of the parameter, of which there is +no word in the genuine fragment of Anthemius), and (2) of +the mixed ‘angles of contact’. Nor does it seem likely that +even Diocles, living a century after Apollonius, would have +spoken of the ‘section of a right-angled cone’ instead of a +parabola, or used the ‘mixed’ angle of which there is only the +merest survival in Euclid. The assumption of the equality +of the two angles made by the curve with the tangent on +both sides of the point of contact reminds us of Aristotle's +assumption of the equality of the angles ‘<I>of</I> a segment’ of +a circle as prior to the truth proved in Eucl. I. 5. I am +inclined, therefore, to date the fragment much earlier even +than Diocles. Zeuthen suggested that the property of the +paraboloidal mirror may have been discovered by Archimedes, +who, according to a Greek tradition, wrote <I>Catoptrica.</I> This, +however, does not receive any confirmation in Ibn al-Haitham +or in Anthemius, and we can only say that the fragment at +least goes back to an original which was probably not later +than Apollonius. +<p>PERSEUS is only known, from allusions to him in Proclus,<note>Proclus on Eucl. I, pp. 111. 23-112. 8, 356. 12. Cf. vol. i, p. 226.</note> +as the discoverer and investigator of the <I>spiric sections.</I> They +are classed by Proclus among curves obtained by cutting +solids, and in this respect they are associated with the conic +sections. We may safely infer that they were discovered +after the conic sections, and only after the theory of conics +had been considerably developed. This was already the case +in Euclid's time, and it is probable, therefore, that Perseus was +not earlier than Euclid. On the other hand, by that time +the investigation of conics had brought the exponents of the +subject such fame that it would be natural for mathematicians +to see whether there was not an opportunity for winning a +<pb n=204><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +like renown as discoverers of other curves to be obtained by +cutting well-known solid figures other than the cone and +cylinder. A particular case of one such solid figure, the +<G>spei=ra</G>, had already been employed by Archytas, and the more +general form of it would not unnaturally be thought of as +likely to give sections worthy of investigation. Since Geminus +is Proclus's authority, Perseus may have lived at any date from +Euclid's time to (say) 75 B.C., but the most probable supposi- +tion seems to be that he came before Apollonius and near to +Euclid in date. +<p>The <I>spire</I> in one of its forms is what we call a <I>tore</I>, or an +anchor-ring. It is generated by the revolution of a circle +about a straight line in its plane in such a way that the plane +of the circle always passes through the axis of revolution. It +takes three forms according as the axis of revolution is +(<I>a</I>) altogether outside the circle, when the spire is <I>open</I> +(<G>diexh/s</G>), (<I>b</I>) a tangent to the circle, when the surface is <I>con- +tinuous</I> (<G>sunexh/s</G>), or (<I>c</I>) a chord of the circle, when it is <I>inter- +laced</I> (<G>e)mpeplegme/nh</G>), or <I>crossing-itself</I> (<G>e)palla/ttousa</G>); an +alternative name for the surface was <G>kri/kos</G>, a <I>ring.</I> Perseus +celebrated his discovery in an epigram to the effect that +‘Perseus on his discovery of three lines (curves) upon five +sections gave thanks to the gods therefor’.<note>Proclus on Eucl. I, p. 112. 2.</note> There is some +doubt about the meaning of ‘three lines <I>upon</I> five sections’ +(<G>trei=s gramma\s e)pi\ pe/nte tomai=s</G>). We gather from Proclus's +account of three sections distinguished by Perseus that the +plane of section was always parallel to the axis of revolution +or perpendicular to the plane which cuts the tore symmetri- +cally like the division in a split-ring. It is difficult to inter- +pret the phrase if it means three curves made by five different +sections. Proclus indeed implies that the three curves were +sections of the three kinds of tore respectively (the open, the +closed, and the interlaced), but this is evidently a slip. +Tannery interprets the phrase as meaning ‘three curves <I>in +addition to</I> five sections’.<note>See Tannery, <I>Mémoires scientifiques</I>, II, pp. 24-8.</note> Of these the five sections belong +to the open tore, in which the distance (<I>c</I>) of the centre of the +generating circle from the axis of revolution is greater than +the radius (<I>a</I>) of the generating circle. If <I>d</I> be the perpen- +<pb n=205><head>PERSEUS</head> +dicular distance of the plane of section from the axis of rota- +tion, we can distinguish the following cases: +<p>(1) <MATH><I>c</I> + <I>a</I> > <I>d</I> > <I>c</I></MATH>. Here the curve is an oval. +<p>(2) <MATH><I>d</I> = <I>c</I></MATH>: transition from case <B>(1)</B> to the next case. +<p>(3) <MATH><I>c</I> > <I>d</I> > <I>c</I> - <I>a</I></MATH>. The curve is now a closed curve narrowest +in the middle. +<p>(4) <MATH><I>d</I> = <I>c</I> - <I>a</I></MATH>. In this case the curve is the <I>hippopede</I> +(horse-fetter), a curve in the shape of the figure of 8. The +lemniscate of Bernoulli is a particular case of this curve, that +namely in which <I>c</I> = 2<I>a.</I> +<p>(5) <MATH><I>c</I> - <I>a</I> > <I>d</I> > 0</MATH>. In this case the section consists of two +ovals symmetrical with one another. +<p>The three curves specified by Proclus are those correspond- +ing to (1), (3) and (4). +<p>When the tore is ‘continuous’ or closed, <I>c</I> = <I>a</I>, and we have +sections corresponding to (1), (2) and (3) only; (4) reduces to +two circles touching one another. +<p>But Tannery finds in the third, the interlaced, form of tore +three new sections corresponding to (1) (2) (3), each with an +oval in the middle. This would make three curves in addi- +tion to the five sections, or eight curves in all. We cannot be +certain that this is the true explanation of the phrase in the +epigram; but it seems to be the best suggestion that has +been made. +<p>According to Proclus, Perseus worked out the property of +his curves, as Nicomedes did that of the conchoid, Hippias +that of the <I>quadratrix</I>, and Apollonius those of the three +conic sections. That is, Perseus must have given, in some +form, the equivalent of the Cartesian equation by which we +can represent the different curves in question. If we refer the +tore to three axes of coordinates at right angles to one another +with the centre of the tore as origin, the axis of <I>y</I> being taken +to be the axis of revolution, and those of <I>z, x</I> being perpen- +dicular to it in the plane bisecting the tore (making it a split- +ring), the equation of the tore is +<MATH>(<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP> - <I>a</I><SUP>2</SUP>)<SUP>2</SUP> = 4<I>c</I><SUP>2</SUP> (<I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP>)</MATH>, +<pb n=206><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +where <I>c, a</I> have the same meaning as above. The different +sections parallel to the axis of revolution are obtained by +giving (say) <I>z</I> any value between 0 and <I>c</I> + <I>a.</I> For the value +<I>z</I> = <I>a</I> the curve is the oval of Cassini which has the property +that, if <I>r, r</I>′ be the distances of any point on the curve from +two fixed points as poles, <I>rr</I>′ = const. For, if <I>z</I> = <I>a</I>, the equa- +tion becomes +<MATH>(<I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>)<SUP>2</SUP> = 4<I>c</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + 4<I>c</I><SUP>2</SUP><I>a</I><SUP>2</SUP>, +or {―(<I>c</I> - <I>x</I><SUP>2</SUP>) + <I>y</I><SUP>2</SUP>} {―(<I>c</I> + <I>x</I><SUP>2</SUP>) + <I>y</I><SUP>2</SUP>} = 4<I>c</I><SUP>2</SUP><I>a</I><SUP>2</SUP></MATH>; +and this is equivalent to <I>rr</I>′ = ± 2<I>ca</I> if <I>x, y</I> are the coordinates +of any point on the curve referred to <I>Ox, Oy</I> as axes, where <I>O</I> +is the middle point of the line (2<I>c</I> in length) joining the two +poles, and <I>Ox</I> lies along that line in either direction, while <I>Oy</I> +is perpendicular to it. Whether Perseus discussed this case +and arrived at the property in relation to the two poles is of +course quite uncertain. +<C>Isoperimetric figures.</C> +<p>The subject of isoperimetric figures, that is to say, the com- +parison of the areas of figures having different shapes but the +same perimeter, was one which would naturally appeal to the +early Greek mathematicians. We gather from Proclus's notes +on Eucl. I. 36, 37 that those theorems, proving that parallelo- +grams or triangles on the same or equal bases and between +the same parallels are equal in area, appeared to the ordinary +person paradoxical because they meant that, by moving the +side opposite to the base in the parallelogram, or the vertex +of the triangle, to the right or left as far as we please, we may +increase the perimeter of the figure to any extent while keep- +ing the same area. Thus the perimeter in parallelograms or +triangles is in itself no criterion as to their area. Misconcep- +tion on this subject was rife among non-mathematicians. +Proclus tells us of describers of countries who inferred +the size of cities from their perimeters; he mentions also +certain members of communistic societies in his own time who +cheated their fellow-members by giving them land of greater +perimeter but less area than the plots which they took +<pb n=207><head>ISOPERIMETRIC FIGURES. ZENODORUS</head> +themselves, so that, while they got a reputation for greater +honesty, they in fact took more than their share of the +produce.<note>Proclus on Eucl. I, p. 403. 5 sq.</note> Several remarks by ancient authors show the +prevalence of the same misconception. Thucydides estimates +the size of Sicily according to the time required for circum- +navigating it.<note>Thuc. vi. 1.</note> About 130 B.C. Polybius observed that there +were people who could not understand that camps of the same +periphery might have different capacities.<note>Polybius, ix. 21.</note> Quintilian has a +similar remark, and Cantor thinks he may have had in his +mind the calculations of Pliny, who compares the size of +different parts of the earth by adding their lengths to their +breadths.<note>Pliny, <I>Hist. nat</I>, vi. 208.</note> +<p>ZENODORUS wrote, at some date between (say) 200 B.C. and +A.D 90, a treatise <G>peri\ i)some/trwn sxhma/twn</G>, <I>On isometric +figures.</I> A number of propositions from it are preserved in +the commentary of Theon of Alexandria on Book I of +Ptolemy's <I>Syntaxis</I>; and they are reproduced in Latin in the +third volume of Hultsch's edition of Pappus, for the purpose +of comparison with Pappus's own exposition of the same +propositions at the beginning of his Book V, where he appears +to have followed Zenodorus pretty closely while making some +changes in detail.<note>Pappus, v, p. 308 sq.</note> From the closeness with which the style +of Zenodorus follows that of Euclid and Archimedes we may +judge that his date was not much later than that of Archi- +medes, whom he mentions as the author of the proposition +(<I>Measurement of a Circle</I>, Prop. 1) that the area of a circle is +half that of the rectangle contained by the perimeter of the +circle and its radius. The important propositions proved by +Zenodorus and Pappus include the following: (1) <I>Of all +regular polygons of equal perimeter, that is the greatest in +area which has the most angles.</I> (2) <I>A circle is greater than +any regular polygon of equal contour.</I> (3) <I>Of all polygons of +the same number of sides and equal perimeter the equilateral +and equiangular polygon is the greatest in area.</I> Pappus +added the further proposition that <I>Of all segments of a circle +having the same circumference the semicircle is the greatest in</I> +<pb n=208><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +<I>area.</I> Zenodorus's treatise was not confined to propositions +about plane figures, but gave also the theorem that <I>Of all +solid figures the surfaces of which are equal, the sphere is the +greatest in solid content.</I> +<p>We will briefly indicate Zenodorus's method of proof. To +begin with (1); let <I>ABC, DEF</I> be equilateral and equiangular +polygons of the same perimeter, <I>DEF</I> having more angles +than <I>ABC.</I> Let <I>G, H</I> be the centres of the circumscribing +circles, <I>GK, HL</I> the perpendiculars from <I>G, H</I> to the sides +<I>AB, DE</I>, so that <I>K, L</I> bisect those sides. +<FIG> +<p>Since the perimeters are equal, <I>AB</I> > <I>DE</I>, and <I>AK</I> > <I>DL.</I> +Make <I>KM</I> equal to <I>DL</I> and join <I>GM.</I> +<p>Since <I>AB</I> is the same fraction of the perimeter that the +angle <I>AGB</I> is of four right angles, and <I>DE</I> is the same fraction +of the same perimeter that the angle <I>DHE</I> is of four right +angles, it follows that +<MATH><I>AB</I> : <I>DE</I> = ∠ <I>AGB</I> : ∠ <I>DHE</I>, +that is, <I>AK</I> : <I>MK</I> = ∠ <I>AGK</I> : ∠ <I>DHL</I></MATH>. +<p>But <MATH><I>AK</I> : <I>MK</I> > ∠ <I>AGK</I> : ∠ <I>MGK</I></MATH> +(this is easily proved in a lemma following by the usual +method of drawing an arc of a circle with <I>G</I> as centre and <I>GM</I> +as radius cutting <I>GA</I> and <I>GK</I> produced. The proposition is of +course equivalent to tan <G>a</G> / tan <G>b</G> > <G>a</G> / <G>b</G>, where 1/2<G>p</G> > <G>a</G> > <G>b</G>). +<p>Therefore <MATH>∠ <I>MGK</I> > ∠ <I>DHL</I>, +and consequently ∠ <I>GMK</I> < ∠ <I>HDL.</I></MATH> +<p>Make the angle <I>NMK</I> equal to the angle <I>HDL</I>, so that <I>MN</I> +meets <I>KG</I> produced in <I>N.</I> +<pb n=209><head>ZENODORUS</head> +<p>The triangles <I>NMK, HDL</I> are now equal in all respects, and +<I>NK</I> is equal to <I>HL</I>, so that <I>GK</I> < <I>HL.</I> +<p>But the area of the polygon <I>ABC</I> is half the rectangle +contained by <I>GK</I> and the perimeter, while the area of the +polygon <I>DEF</I> is half the rectangle contained by <I>HL</I> and +the same perimeter. Therefore the area of the polygon <I>DEF</I> +is the greater. +<p>(2) The proof that a circle is greater than any regular +polygon with the same perimeter is deduced immediately from +Archimedes's proposition that the area of a circle is equal +to the right-angled triangle with perpendicular side equal to +the radius and base equal to the perimeter of the circle; +Zenodorus inserts a proof <I>in extenso</I> of Archimedes's pro- +position, with preliminary lemma. The perpendicular from +the centre of the circle circumscribing the polygon is easily +proved to be less than the radius of the given circle with +perimeter equal to that of the polygon; whence the proposition +follows. +<p>(3) The proof of this proposition depends on some pre- +liminary lemmas. The first proves that, if there be two +<FIG> +triangles on the same base and with the +same perimeter, one being isosceles and +the other scalene, the isosceles triangle +has the greater area. (Given the scalene +triangle <I>BDC</I> on the base <I>BC</I>, it is easy to +draw on <I>BC</I> as base the isosceles triangle +having the same perimeter. We have +only to take <I>BH</I> equal to 1/2(<I>BD</I> + <I>DC</I>), +bisect <I>BC</I> at <I>E</I>, and erect at <I>E</I> the per- +pendicular <I>AE</I> such that <MATH><I>AE</I><SUP>2</SUP> = <I>BH</I><SUP>2</SUP> - <I>BE</I><SUP>2</SUP></MATH>.) +<p>Produce <I>BA</I> to <I>F</I> so that <I>BA</I> = <I>AF</I>, and join <I>AD, DF.</I> +<p>Then <MATH><I>BD</I> + <I>DF</I> > <I>BF</I></MATH>, i.e. <I>BA</I> + <I>AC</I>, i.e. <I>BD</I> + <I>DC</I>, by hypo- +thesis; therefore <I>DF</I> > <I>DC</I>, whence in the triangles <I>FAD</I>, +<I>CAD</I> the angle <I>FAD</I> > the angle <I>CAD.</I> +<p>Therefore <MATH>∠ <I>FAD</I> > 1/2∠ <I>FAC</I> +> ∠ <I>BCA</I></MATH>. +<p>Make the angle <I>FAG</I> equal to the angle <I>BCA</I> or <I>ABC</I>, so +that <I>AG</I> is parallel to <I>BC</I>; let <I>BD</I> produced meet <I>AG</I> in <I>G</I>, +and join <I>GC.</I> +<pb n=210><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +<p>Then <MATH>▵ <I>ABC</I> = ▵ <I>GBC</I> +> ▵ <I>DBC</I></MATH>. +<p>The second lemma is to the effect that, given two isosceles +triangles not similar to one another, if we construct on the +same bases two triangles <I>similar to one another</I> such that the +sum of their perimeters is equal to the sum of the perimeters +of the first two triangles, then the sum of the areas of the +similar triangles is greater than the sum of the areas of +the non-similar triangles. (The easy construction of the +similar triangles is given in a separate lemma.) +<p>Let the bases of the isosceles triangles, <I>EB, BC</I>, be placed in +one straight line, <I>BC</I> being greater than <I>EB.</I> +<FIG> +<p>Let <I>ABC, DEB</I> be the similar isosceles triangles, and <I>FBC, +GEB</I> the non-similar, the triangles being such that +<MATH><I>BA</I> + <I>AC</I> + <I>ED</I> + <I>DB</I> = <I>BF</I> + <I>FC</I> + <I>EG</I> + <I>GB</I></MATH>. +<p>Produce <I>AF, GD</I> to meet the bases in <I>K, L.</I> Then clearly +<I>AK, GL</I> bisect <I>BC, EB</I> at right angles at <I>K, L.</I> +<p>Produce <I>GL</I> to <I>H</I>, making <I>LH</I> equal to <I>GL.</I> +<p>Join <I>HB</I> and produce it to <I>N</I>; join <I>HF.</I> +<p>Now, since the triangles <I>ABC, DEB</I> are similar, the angle +<I>ABC</I> is equal to the angle <I>DEB</I> or <I>DBE.</I> +<p>Therefore <MATH>∠ <I>NBC</I> (= ∠ <I>HBE</I> = ∠ <I>GBE</I>) > ∠ <I>DBE</I> or ∠ <I>ABC</I></MATH>; +therefore the angle <I>ABH</I>, and <I>a fortiori</I> the angle <I>FBH</I>, is +less than two right angles, and <I>HF</I> meets <I>BK</I> in some point <I>M.</I> +<pb n=211><head>ZENODORUS</head> +<p>Now, by hypothesis, <MATH><I>DB</I> + <I>BA</I> = <I>GB</I> + <I>BF</I>; +therefore <I>DB</I> + <I>BA</I> = <I>HB</I> + <I>BF</I> > <I>HF</I></MATH>. +<p>By an easy lemma, since the triangles <I>DEB, ABC</I> are similar, +<MATH>(<I>DB</I> + <I>BA</I>)<SUP>2</SUP> = (<I>DL</I> + <I>AK</I>)<SUP>2</SUP> + (<I>BL</I> + <I>BK</I>)<SUP>2</SUP> += (<I>DL</I> + <I>AK</I>)<SUP>2</SUP> + <I>LK</I><SUP>2</SUP></MATH>. +<p>Therefore <MATH>(<I>DL</I> + <I>AK</I>)<SUP>2</SUP> + <I>LK</I><SUP>2</SUP> > <I>HF</I><SUP>2</SUP> +> (<I>GL</I> + <I>FK</I>)<SUP>2</SUP> + <I>LK</I><SUP>2</SUP>, +whence <I>DL</I> + <I>AK</I> > <I>GL</I> + <I>FK</I>, +and it follows that <I>AF</I> > <I>GD</I></MATH>. +<p>But <I>BK</I> > <I>BL</I>; therefore <I>AF.BK</I> > <I>GD.BL.</I> +<p>Hence the ‘hollow-angled (figure)’ (<G>koilogw/nion</G>) <I>ABFC</I> is +greater than the hollow-angled (figure) <I>GEDB.</I> +<p>Adding ▵ <I>DEB</I> + ▵ <I>BFC</I> to each, we have +<MATH>▵ <I>DEB</I> + ▵ <I>ABC</I> > ▵ <I>GEB</I> + ▵ <I>FBC</I></MATH>. +<p>The above is the only case taken by Zenodorus. The proof +still holds if <I>EB</I> = <I>BC</I>, so that <I>BK</I> = <I>BL.</I> But it fails in the +case in which <I>EB</I> > <I>BC</I> and the vertex <I>G</I> of the triangle <I>EB</I> +belonging to the non-similar pair is still above <I>D</I> and not +below it (as <I>F</I> is below <I>A</I> in the preceding case). This was +no doubt the reason why Pappus gave a proof intended to +apply to all the cases without distinction. This proof is the +same as the above proof by Zenodorus up to the point where +it is proved that +<MATH><I>DL</I> + <I>AK</I> > <I>GL</I> + <I>FK</I></MATH>, +but there diverges. Unfortunately the text is bad, and gives +no sufficient indication of the course of the proof; but it would +seem that Pappus used the relations +<MATH><I>DL</I> : <I>GL</I> = ▵ <I>DEB</I> : ▵ <I>GEB</I>, +<I>AK</I> : <I>FK</I> = ▵ <I>ABC</I> : ▵ <I>FBC</I>, +and <I>AK</I><SUP>2</SUP> : <I>DL</I><SUP>2</SUP> = ▵ <I>ABC</I> : ▵ <I>DEB</I></MATH>, +combined of course with the fact that <MATH><I>GB</I> + <I>BF</I> = <I>DB</I> + <I>BA</I></MATH>, +in order to prove the proposition that, +according as <MATH><I>DL</I> + <I>AK</I> > or < <I>GL</I> + <I>FK</I>, +▵ <I>DEB</I> + ▵ <I>ABC</I> > or < ▵ <I>GEB</I> + ▵ <I>FBC</I></MATH>. +<pb n=212><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +<p>The proof of his proposition, whatever it was, Pappus +indicates that he will give later; but in the text as we have it +the promise is not fulfilled. +<p>Then follows the proof that the maximum polygon of given +perimeter is both equilateral and +equiangular. +<p>(1) It is equilateral. +<FIG> +<p>For, if not, let two sides of the +maximum polygon, as <I>AB, BC</I>, be +unequal. Join <I>AC</I>, and on <I>AC</I> as +base draw the isosceles triangle <I>AFC</I> +such that <I>AF</I> + <I>FC</I> = <I>AB</I> + <I>BC.</I> The +area of the triangle <I>AFC</I> is then +greater than the area of the triangle <I>ABC</I>, and the area of +the whole polygon has been increased by the construc- +tion: which is impossible, as by hypothesis the area is a +maximum. +<p>Similarly it can be proved that no other side is unequal +to any other. +<p>(2) It is also equiangular. +<p>For, if possible, let the maximum polygon <I>ABCDE</I> (which +<FIG> +we have proved to be equilateral) +have the angle at <I>B</I> greater than +the angle at <I>D.</I> Then <I>BAC, DEC</I> are +non-similar isosceles triangles. On +<I>AC, CE</I> as bases describe the two +isosceles triangles <I>FAC, GEC</I> similar +to one another which have the sum +of their perimeters equal to the +sum of the perimeters of <I>BAC, +DEC</I>. Then the sum of the areas of the two similar isosceles +triangles is greater than the sum of the areas of the triangles +<I>BAC, DEC</I>; the area of the polygon is therefore increased, +which is contrary to the hypothesis. +<p>Hence no two angles of the polygon can be unequal. +<p>The maximum polygon of given perimeter is therefore both +equilateral and equiangular. +<p>Dealing with the sphere in relation to other solids having +<pb n=213><head>ZENODORUS. HYPSICLES</head> +their surfaces equal to that of the sphere, Zenodorus confined +himself to proving (1) that the sphere is greater if the other +solid with surface equal to that of the sphere is a solid formed +by the revolution of a regular polygon about a diameter +bisecting it as in Archimedes, <I>On the Sphere and Cylinder</I>, +Book I, and (2) that the sphere is greater than any of +the regular solids having its surface equal to that of the +sphere. +<p>Pappus's treatment of the subject is more complete in that +he proves that the sphere is greater than the cone or cylinder +the surface of which is equal to that of the sphere, and further +that of the five regular solids which have the same surface +that which has more faces is the greater.<note>Pappus, v, Props. 19, 38-56.</note> +<p>HYPSICLES (second half of second century B.C.) has already +been mentioned (vol. i, pp. 419-20) as the author of the con- +tinuation of the <I>Elements</I> known as Book XIV. He is quoted +by Diophantus as having given a definition of a polygonal +number as follows: +<p>‘If there are as many numbers as we please beginning from +1 and increasing by the same common difference, then, when +the common difference is 1, the sum of all the numbers is +a triangular number; when 2, a square; when 3, a pentagonal +number [and so on]. And the number of angles is called +after the number which exceeds the common difference by 2, +and the side after the number of terms including 1.’ +<p>This definition amounts to saying that the <I>n</I>th <I>a</I>-gonal num- +ber (1 counting as the first) is <MATH>1/2<I>n</I> {2 + (<I>n</I>-1) (<I>a</I>-2)}</MATH>. If, as is +probable, Hypsicles wrote a treatise on polygonal numbers, it +has not survived. On the other hand, the <G>*)anaforiko/s</G> (<I>Ascen- +siones</I>) known by his name has survived in Greek as well as in +Arabic, and has been edited with translation.<note>Manitius, <I>Des Hypsikles Schrift Anaphorikos</I>, Dresden, Lehmannsche Buchdruckerei, 1888.</note> True, the +treatise (if it really be by Hypsicles, and not a clumsy effort +by a beginner working from an original by Hypsicles) +does no credit to its author; but it is in some respects +interesting, and in particular because it is the first Greek +<pb n=214><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +work in which we find the division of the ecliptic circle into +360 ‘parts’ or degrees. The author says, after the preliminary +propositions, +<p>‘The circle of the zodiac having been divided into 360 equal +circumferences (arcs), let each of the latter be called a <I>degree +in space</I> (<G>moi=ra topikh/</G>, ‘local’ or ‘spatial part’). And simi- +larly, supposing that the time in which the zodiac circle +returns to any position it has left is divided into 360 equal +times, let each of these be called a <I>degree in time</I> (<G>moi=ra +xronikh/</G>).’ +<p>From the word <G>kalei/sqw</G> (‘let it be called’) we may perhaps +infer that the terms were new in Greece. This brings us to +the question of the origin of the division (1) of the circle of +the zodiac, (2) of the circle in general, into 360 parts. On this +question innumerable suggestions have been made. With +reference to (1) it was suggested as long ago as 1788 (by For- +maleoni) that the division was meant to correspond to the +number of days in the year. Another suggestion is that it +would early be discovered that, in the case of any circle the +inscribed hexagon dividing the circumference into six parts +has each of its sides equal to the radius, and that this would +naturally lead to the circle being regularly divided into six +parts; after this, the very ancient sexagesimal system would +naturally come into operation and each of the parts would be +divided into 60 subdivisions, giving 360 of these for the whole +circle. Again, there is an explanation which is not even +geometrical, namely that in the Babylonian numeral system, +which combined the use of 6 and 10 as bases, the numbers 6, +60, 360, 3600 were fundamental round numbers, and these +numbers were transferred from arithmetic to the heavens. +The obvious objection to the first of these explanations +(referring the 360 to the number of days in the solar year) is +that the Babylonians were well acquainted, as far back as the +monuments go, with 365.2 as the number of days in the year. +A variant of the hexagon-theory is the suggestion that a +<I>natural</I> angle to be discovered, and to serve as a measure of +others, is the angle of an equilateral triangle, found by draw- +ing a star * like a six-spoked wheel without any circle. If +the base of a sundial was so divided into six angles, it would be +<pb n=215><head>HYPSICLES</head> +natural to divide each of the sixth parts into either 10 or 60 +parts; the former division would account for the attested +division of the day into 60 hours, while the latter division on +the sexagesimal system would give the 360 time-degrees (each +of 4 minutes) making up the day of 24 hours. The purely +arithmetical explanation is defective in that the series of +numbers for which the Babylonians had special names is not +60, 360, 3600 but 60 (Soss), <I>600</I> (Ner), and 3600 or 60<SUP>2</SUP> (Sar). +On the whole, after all that has been said, I know of no +better suggestion than that of Tannery.<note>Tannery, ‘La coudée astronomique et les anciennes divisions du cercle’ (<I>Mémoires scientifiques</I>, ii, pp. 256-68).</note> It is certain that +both the division of the ecliptic into 360 degrees and that of +the <G>nuxqh/meron</G> into 360 time-degrees were adopted by the +Greeks from Babylon. Now the earliest division of the +ecliptic was into 12 parts, the signs, and the question is, how +were the signs subdivided? Tannery observes that, accord- +ing to the cuneiform inscriptions, as well as the testimony of +Greek authors, the sign was divided into parts one of which +(<I>dargatu</I>) was double of the other (<I>murran</I>), the former being +1/30th, the other (called <I>stadium</I> by Manilius) 1/60th, of the +sign; the former division would give 360 parts, the latter 720 +parts for the whole circle. The latter division was more +natural, in view of the long-established system of sexagesimal +fractions; it also had the advantage of corresponding toler- +ably closely to the apparent diameter of the sun in comparison +with the circumference of the sun's apparent circle. But, on +the other hand, the double fraction, the 1/30th, was contained +in the circle of the zodiac approximately the same number of +times as there are days in the year, and consequently corre- +sponded nearly to the distance described by the sun along the +zodiac in one day. It would seem that this advantage was +sufficient to turn the scale in favour of diyiding each sign of +the zodiac into 30 parts, giving 360 parts for the whole +circle. While the Chaldaeans thus divided the ecliptic into +360 parts, it does not appear that they applied the same divi- +sion to the equator or any other circle. They measured angles +in general by <I>ells</I>, an ell representing 2°, so that the complete +circle contained 180, not 360, parts, which they called ells. +The explanation may perhaps be that the Chaldaeans divided +<pb n=216><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +the <I>diameter</I> of the circle into 60 ells in accordance with their +usual sexagesimal division, and then came to divide the cir- +cumference into 180 such ells on the ground that the circum- +ference is roughly three times the diameter. The measure- +ment in <I>ells</I> and <I>dactyli</I> (of which there were 24 to the ell) +survives in Hipparchus (<I>On the Phaenomena of Eudoxus and +Aratus</I>), and some measurements in terms of the same units +are given by Ptolemy. It was Hipparchus who first divided +the circle in general into 360 parts or degrees, and the +introduction of this division coincides with his invention of +trigonometry. +<p>The contents of Hypsicles's tract need not detain us long. +The problem is: If we know the ratio which the length of the +longest day bears to the length of the shortest day at any +given place, to find how many time-degrees it takes any given +sign to rise; and, after this has been found, the author finds +what length of time it takes any given degree in any sign to +rise, i.e. the interval between the rising of one degree-point on +the ecliptic and that of the next following. It is explained +that the longest day is the time during which one half of the +zodiac (Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius) rises, +and the shortest day the time during which the other half +(Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini) rises. +Now at Alexandria the longest day is to the shortest as 7 +to 5; the longest therefore contains 210 ‘time-degrees’, the +shortest 150. The two quadrants Cancer-Virgo and Libra- +Sagittarius take the same time to rise, namely 105 time- +degrees, and the two quadrants Capricornus-Pisces and Aries- +Gemini each take the same time, namely 75 time-degrees. +It is further assumed that the times taken by Virgo, Leo, +Cancer, Gemini, Taurus, Aries are in descending arithmetical +progression, while the times taken by Libra, Scorpio, Sagit- +tarius, Capricornus, Aquarius, Pisces continue the same de- +scending arithmetical series. The following lemmas are used +and proved: +<p>I. If <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB><I>n</I></SUB>, <I>a</I><SUB><I>n</I>+1</SUB>, <I>a</I><SUB><I>n</I>+2</SUB> ... <I>a</I><SUB>2<I>n</I></SUB> is a descending arithmeti- +cal progression of 2<I>n</I> terms with <G>d</G> (= <I>a</I><SUB>1</SUB> - <I>a</I><SUB>2</SUB> = <I>a</I><SUB>2</SUB> - <I>a</I><SUB>3</SUB> = ...) +as common difference, +<MATH><I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> + ... + <I>a</I><SUB><I>n</I></SUB> - (<I>a</I><SUB><I>n</I>+1</SUB> + <I>a</I><SUB><I>n</I>+2</SUB> + ... + <I>a</I><SUB>2<I>n</I></SUB>) = <I>n</I><SUP>2</SUP><G>d</G></MATH>. +<pb n=217><head>HYPSICLES</head> +<p>II. If <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB><I>n</I></SUB> ... <I>a</I><SUB>2<I>n</I>-1</SUB> is a descending arithmetical pro- +gression of 2<SUB><I>n</I>-1</SUB> terms with <G>d</G> as common difference and <I>a</I><SUB><I>n</I></SUB> +is the middle term, then +<MATH><I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> + ... + <I>a</I><SUB>2<I>n</I>-1</SUB> = (2<I>n</I> - 1)<I>a</I><SUB><I>n</I></SUB></MATH>. +<p>III. If <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB><I>n</I></SUB>, <I>a</I><SUB><I>n</I>+1</SUB> ... <I>a</I><SUB>2<I>n</I></SUB> is a descending arithmetical +progression of 2<I>n</I> terms, then +<MATH><I>a</I><SUB>1</SUB> + <I>a</I><SUB>2</SUB> + ... + <I>a</I><SUB>2<I>n</I></SUB> = <I>n</I>(<I>a</I><SUB>1</SUB> + <I>a</I><SUB>2<I>n</I></SUB>) = <I>n</I>(<I>a</I><SUB>2</SUB> + <I>a</I><SUB>2<I>n</I>-1</SUB>) = ... += <I>n</I>(<I>a</I><SUB><I>n</I></SUB> + <I>a</I><SUB><I>n</I>+1</SUB>)</MATH>. +<p>Now let <I>A, B, C</I> be the descending series the sum of which +is 105, and <I>D, E, F</I> the next three terms in the same series +the sum of which is 75, the common difference being <G>d</G>; we +then have, by (I), +<MATH><I>A</I> + <I>B</I> + <I>C</I> - (<I>D</I> + <I>E</I> + <I>F</I>) = 9<G>d</G>, or 30 = 9<G>d</G></MATH>, +and <G>d</G> = 3 1/3. +<p>Next, by (II), <MATH><I>A</I> + <I>B</I> + <I>C</I> = 3<I>B</I></MATH>, or 3<I>B</I> = 105, and <I>B</I> = 35; +therefore <I>A, B, C, D, E, F</I> are equal to 38 1/3, 35, 31 2/3, 28 1/3, 25, +21 2/3 time-degrees respectively, which the author of the tract +expresses in time-degrees and minutes as 38<SUP><I>t</I></SUP> 20′, 35<SUP><I>t</I></SUP>, 31<SUP><I>t</I></SUP> 40′, +28<SUP><I>t</I></SUP> 20′, 25<SUP><I>t</I></SUP>, 21<SUP><I>t</I></SUP> 40′. We have now to carry through the same +procedure for each degree in each sign. If the difference +between the times taken to rise by one sign and the next +is 3<SUP><I>t</I></SUP> 20′, what is the difference for each of the 30 degrees in +the sign? We have here 30 terms followed by 30 other terms +of the same descending arithmetical progression, and the +formula (I) gives <MATH>3<SUP><I>t</I></SUP> . 20′ = (30)<SUP>2</SUP><I>d</I></MATH>, where <I>d</I> is the common +difference; therefore <MATH><I>d</I> = 1/900 X 3<SUP><I>t</I></SUP> . 20′ = 0<SUP><I>t</I></SUP>0′13″20‴</MATH>. Lastly, +take the sign corresponding to 21<SUP><I>t</I></SUP> 40′. This is the sum of +a descending arithmetical progression of 30 terms <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB>30</SUB> +with common difference 0<SUP><I>t</I></SUP>0′13″20‴. Therefore, by (III), +<MATH>21<SUP><I>t</I></SUP> 40′ = 15 (<I>a</I><SUB>1</SUB> + <I>a</I><SUB>30</SUB>)</MATH>, whence <MATH><I>a</I><SUB>1</SUB> + <I>a</I><SUB>30</SUB> = 1<SUP><I>t</I></SUP> 26′ 40″</MATH>. Now, +since there are 30 terms <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a</I><SUB>30</SUB>, we have +<MATH><I>a</I><SUB>1</SUB> - <I>a</I><SUB>30</SUB> = 29<I>d</I> = 0<SUP><I>t</I></SUP> 6′ 26″ 40‴</MATH>. +It follows that <I>a</I><SUB>30</SUB> = 0<SUP><I>t</I></SUP> 40′ 6″ 40‴ and <I>a</I><SUB>1</SUB> = 0<SUP><I>t</I></SUP> 46′ 33″ 20‴, +<pb n=218><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +and from these and the common difference 0<SUP><I>t</I></SUP> 0′ 13″ 20‴ all +the times corresponding to all the degrees in the circle can be +found. +<p>The procedure was probably, as Tannery thinks, taken +direct from the Babylonians, who would no doubt use it for +the purpose of enabling the time to be determined at any +hour of the night. Another view is that the object was +astrological rather than astronomical (Manitius). In either +case the method was exceedingly rough, and the assumed +increases and decreases in the times of the risings of the signs +in arithmetical progression are not in accordance with the +facts. The book could only have been written before the in- +vention of trigonometry by Hipparchus, for the problem of +finding the times of rising of the signs is really one of +spherical trigonometry, and these times were actually cal- +culated by Hipparchus and Ptolemy by means of tables of +chords. +<p>DIONYSODORUS is known in the first place as the author of +a solution of the cubic equation subsidiary to the problem of +Archimedes, <I>On the Sphere and Cylinder</I>, II. 4, To cut a given +sphere by a plane so that the volumes of the segments have to +one another a given ratio (see above, p. 46). Up to recently +this Dionysodorus was supposed to be Dionysodorus of Amisene +in Pontus, whom Suidas describes as ‘a mathematician worthy +of mention in the field of education’. But we now learn from +a fragment of the Herculaneum Roll, No. 1044, that ‘Philonides +was a pupil, first of Eudemus, and afterwards of Dionysodorus, +the son of Dionysodorus the Caunian’. Now Eudemus is +evidently Eudemus of Pergamum to whom Apollonius dedi- +cated the first two Books of his <I>Conics</I>, and Apollonius actually +asks him to show Book II to Philonides. In another frag- +ment Philonides is said to have published some lectures by +Dionysodorus. Hence our Dionysodorus may be Dionysodorus +of Caunus and a contemporary of Apollonius, or very little +later.<note>W. Schmidt in <I>Bibliotheca mathematica</I>, iv<SUB>3</SUB>, pp. 321-5.</note> A Dionysodorus is also mentioned by Heron<note>Heron, <I>Metrica</I>, ii. 13, p. 128. 3.</note> as the +author of a tract <I>On the Spire</I> (or tore) in which he proved +that, if <I>d</I> be the diameter of the revolving circle which +<pb n=219><head>DIONYSODORUS</head> +generates the tore, and <I>c</I> the distance of its centre from the +axis of revolution, +<MATH>(volume of tore):<G>p</G><I>c</I><SUP>2</SUP> . <I>d</I> = 1/4<G>p</G><I>d</I><SUP>2</SUP>:1/2<I>cd</I></MATH>, +that is, <MATH>(volume of tore) = 1/2<G>p</G><SUP>2</SUP> . <I>cd</I><SUP>2</SUP></MATH>, +which is of course the product of the area of the generating +circle and the length of the path of its centre of gravity. The +form in which the result is stated, namely that the tore is to +the cylinder with height <I>d</I> and radius <I>c</I> as the generating +circle of the tore is to half the parallelogram <I>cd</I>, indicates +quite clearly that Dionysodorus proved his result by the same +procedure as that employed by Archimedes in the <I>Method</I> and +in the book <I>On Conoids and Spheroids</I>; and indeed the proof +on Archimedean lines is not difficult. +<p>Before passing to the mathematicians who are identified +with the discovery and development of trigonometry, it will +be convenient, I think, to dispose of two more mathematicians +belonging to the last century B.C., although this involves +a slight departure from chronological order; I mean Posidonius +and Geminus. +<p>POSIDONIUS, a Stoic, the teacher of Cicero, is known as +Posidonius of Apamea (where he was born) or of Rhodes +(where he taught); his date may be taken as approximately +135-51 B.C. In pure mathematics he is mainly quoted as the +author of certain definitions, or for views on technical terms, +e.g. ‘theorem’ and ‘problem’, and subjects belonging to ele- +mentary geometry. More important were his contributions +to mathematical geography and astronomy. He gave his +great work on geography the title <I>On the Ocean</I>, using the +word which had always had such a fascination for the Greeks; +its contents are known to us through the copious quotations +from it in Strabo; it dealt with physical as well as mathe- +matical geography, the zones, the tides and their connexion +with the moon, ethnography and all sorts of observations made +during extensive travels. His astronomical book bore the +title <I>Meteorologica</I> or <G>peri\ metew/rwn</G>, and, while Geminus +wrote a commentary on or exposition of this work, we may +assign to it a number of views quoted from Posidonius in +<pb n=220><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +Cleomedes's work <I>De motu circulari corporum caelestium.</I> +Posidonius also wrote a separate tract on the size of the sun. +<p>The two things which are sufficiently important to deserve +mention here are (1) Posidonius's measurement of the circum- +ference of the earth, (2) his hypothesis as to the distance and +size of the sun. +<p>(1) He estimated the circumference of the earth in this +way. He assumed (according to Cleomedes<note>Cleomedes, <I>De motu circulari</I>, i. 10, pp. 92-4.</note>) that, whereas +the star Canopus, invisible in Greece, was just seen to graze the +horizon at Rhodes, rising and setting again immediately, the +meridian altitude of the same star at Alexandria was ‘a fourth +part of a sign, that is, one forty-eighth part of the zodiac +circle’ (= 7 1/2°); and he observed that the distance between +the two places (supposed to lie on the same meridian) ‘was +considered to be 5,000 stades’. The circumference of the +earth was thus made out to be 240,000 stades. Unfortunately +the estimate of the difference of latitude, 7 1/2°, was very far +from correct, the true difference being 5 1/4° only; moreover +the estimate of 5,000 stades for the distance was incorrect, +being only the maximum estimate put upon it by mariners, +while some put it at 4,000 and Eratosthenes, by observations +of the shadows of gnomons, found it to be 3,750 stades only. +Strabo, on the other hand, says that Posidonius favoured ‘the +latest of the measurements which gave the smallest dimen- +sions to the earth, namely about 180,000 stades’.<note>Strabo, ii. c. 95.</note> This is +evidently 48 times 3,750, so that Posidonius combined Erato- +sthenes's figure of 3,750 stades with the incorrect estimate +of 7 1/2° for the difference of latitude, although Eratosthenes +presumably obtained the figure of 3,750 stades from his own +estimate (250,000 or 252,000) of the circumference of the earth +combined with an estimate of the difference of latitude which +was about 5 2/5° and therefore near the truth. +<p>(2) Cleomedes<note>Cleomedes, <I>op. cit.</I> ii. 1, pp. 144-6, p. 98. 1-5.</note> tells us that Posidonius supposed the circle +in which the sun apparently moves round the earth to be +10,000 times the size of a circular section of the earth through +its centre, and that with this assumption he combined the +<pb n=221><head>POSIDONIUS</head> +statement of Eratosthenes (based apparently upon hearsay) +that at Syene, which is under the summer tropic, and +throughout a circle round it of 300 stades in diameter, the +upright gnomon throws no shadow at noon. It follows from +this that the diameter of the sun occupies a portion of the +sun's circle 3,000,000 stades in length; in other words, the +diameter of the sun is 3,000,000 stades. The assumption that +the sun's circle is 10,000 times as large as a great circle of the +earth was presumably taken from Archimedes, who had proved +in the <I>Sand-reckoner</I> that the diameter of the sun's orbit is +<I>less</I> than 10,000 times that of the earth; Posidonius in fact +took the maximum value to be the true value; but his esti- +mate of the sun's size is far nearer the truth than the estimates +of Aristarchus, Hipparchus, and Ptolemy. Expressed in terms +of the mean diameter of the earth, the estimates of these +astronomers give for the diameter of the sun the figures 6 3/4, +12 1/3, and 5 1/2 respectively; Posidonius's estimate gives 39 1/4, the +true figure being 108.9. +<p>In elementary geometry Posidonius is credited by Proclus +with certain definitions. He defined ‘figure’ as ‘confining +limit’ (<G>pe/ras sugklei=on</G>)<note>Proclus on Eucl. I, p. 143. 8.</note> and ‘parallels’ as ‘those lines which, +being in one plane, neither converge nor diverge, but have all +the perpendiculars equal which are drawn from the points of +one line to the other’.<note><I>Ib.</I>, p. 176. 6-10.</note> (Both these definitions are included +in the <I>Definitions</I> of Heron.) He also distinguished seven +species of quadrilaterals, and had views on the distinction +between <I>theorem</I> and <I>problem.</I> Another indication of his +interest in the fundamentals of elementary geometry is the +fact<note><I>Ib.</I>, pp. 199. 14-200. 3.</note> that he wrote a separate work in refutation of the +Epicurean Zeno of Sidon, who had objected to the very begin- +nings of the <I>Elements</I> on the ground that they contained un- +proved assumptions. Thus, said Zeno, even Eucl.I. 1 requires it +to be admitted that ‘two straight lines cannot have a common +segment’; and, as regards the ‘proof’ of this fact deduced +from the bisection of a circle by its diameter, he would object +that it has to be assumed that two arcs of circles cannot have +a common part. Zeno argued generally that, even if we +admit the fundamental principles of geometry, the deductions +<pb n=222><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +from them cannot be proved without the admission of some- +thing else as well which has not been included in the said +principles, and he intended by means of these criticisms to +destroy the whole of geometry.<note>Proclus on Eucl. I, pp. 214. 18-215. 13, p. 216. 10-19, p. 217. 10-23.</note> We can understand, there- +fore, that the tract of Posidonius was a serious work. +<p>A definition of the centre of gravity by one ‘Posidonius a +Stoic’ is quoted in Heron's <I>Mechanics</I>, but, as the writer goes +on to say that Archimedes introduced a further distinction, we +may fairly assume that the Posidonius in question is not +Posidonius of Rhodes, but another, perhaps Posidonius of +Alexandria, a pupil of Zeno of Cittium in the third cen- +tury B.C. +<p>We now come to GEMINUS, a very important authority on +many questions belonging to the history of mathematics, as is +shown by the numerous quotations from him in Proclus's +<I>Commentary on Euclid, Book I.</I> His date and birthplace are +uncertain, and the discussions on the subject now form a whole +literature for which reference must be made to Manitius's +edition of the so-called <I>Gemini elementa astronomiae</I> (Teubner, +1898) and the article ‘Geminus’ in Pauly-Wissowa's <I>Real- +Encyclopädie.</I> The doubts begin with his name. Petau, who +included the treatise mentioned in his <I>Uranologion</I> (Paris, +1630), took it to be the Latin Gem&icaron;nus. Manitius, the latest +editor, satisfied himself that it was Gemīnus, a Greek name, +judging from the fact that it consistently appears with the +properispomenon accent in Greek (<G>*gemi=nos</G>), while it is also +found in inscriptions with the spelling <G>*gemei=nos</G>; Manitius +suggests the derivation from <G>gem</G>, as <G>*)ergi=vos</G> from <G>e)rg</G>, and +<G>*)alexi=nos</G> from <G>a)lex</G>; he compares also the unmistakably +Greek names <G>*)ikti=nos, *krati=nos</G>. Now, however, we are told +(by Tittel) that the name is, after all, the Latin Gém&ibreve;nus, +that <G>*gemi=nos</G> came to be so written through false analogy +with <G>*alexi=nos</G>, &c., and that <G>*ge[m]ei=nos</G>, if the reading is +correct, is also wrongly formed on the model of <G>*antwnei=nos, +*agrippei/na</G>. The occurrence of a Latin name in a centre +of Greek culture need not surprise us, since Romans settled in +such centres in large numbers during the last century B.C. +Geminus, however, in spite of his name, was thoroughly Greek. +<pb n=223><head>GEMINUS</head> +An upper limit for his date is furnished by the fact that he +wrote a commentary on or exposition of Posidonius's work +<G>peri\ metew/rwn</G>; on the other hand, Alexander Aphrodisiensis +(about A.D. 210) quotes an important passage from an ‘epitome’ +of this <G>e)xh/ghsis</G> by Geminus. The view most generally +accepted is that he was a Stoic philosopher, born probably +in the island of Rhodes, and a pupil of Posidonius, and that +he wrote about 73-67 B.C. +<p>Of Geminus's works that which has most interest for us +is a comprehensive work on mathematics. Proclus, though +he makes great use of it, does not mention its title, unless +indeed, in the passage where, after quoting from Geminus +a classification of lines which never meet, he says ‘these +remarks I have selected from the <G>filokali/a</G> of Geminus’,<note>Proclus on Eucl. I, p. 177. 24.</note> +the word <G>filokali/a</G> is a title or an alternative title. Pappus, +however, quotes a work of Geminus ‘on the classification of +the mathematics’ (<G>e)n tw=| peri\ th=s tw=n maqhma/twn ta/xews</G>), +while Eutocius quotes from ‘the sixth book of the doctrine of +the mathematics’ (<G>e)n tw=| e(/ktw| th=s tw=n maqhma/twn qewri/as</G>). +The former title corresponds well enough to the long extract +on the division of the mathematical sciences into arithmetic, +geometry, mechanics, astronomy, optics, geodesy, canonic +(musical harmony) and logistic which Proclus gives in his +first prologue, and also to the fragments contained in the +<I>Anonymi variae collectiones</I> published by Hultsch in his +edition of Heron; but it does not suit most of the other +passages borrowed by Proclus. The correct title was most +probably that given by Eutocius, <I>The Doctrine</I>, or <I>Theory, +of the Mathematics</I>; and Pappus probably refers to one +particular section of the work, say the first Book. If the +sixth Book treated of conics, as we may conclude from +Eutocius's reference, there must have been more Books to +follow; for Proclus has preserved us details about higher +curves, which must have come later. If again Geminus +finished his work and wrote with the same fullness about the +other branches of mathematics as he did about geometry, +there must have been a considerable number of Books +altogether. It seems to have been designed to give a com- +plete view of the whole science of mathematics, and in fact +<pb n=224><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +to have been a sort of encyclopaedia of the subject. The +quotations of Proclus from Geminus's work do not stand +alone; we have other collections of extracts, some more and +some less extensive, and showing varieties of tradition accord- +ing to the channel through which they came down. The +scholia to Euclid's <I>Elements</I>, Book I, contain a considerable +part of the commentary on the Definitions of Book I, and are +valuable in that they give Geminus pure and simple, whereas +Proclus includes extracts from other authors. Extracts from +Geminus of considerable length are included in the Arabic +commentary by an-Nairīzī (about A.D. 900) who got them +through the medium of Greek commentaries on Euclid, +especially that of Simplicius. It does not appear to be +doubted any longer that ‘Aganis’ in an-Nairīzī is really +Geminus; this is inferred from the close agreement between +an-Nairīzī's quotations from ‘Aganis’ and the correspond- +ing passages in Proclus; the difficulty caused by the fact +that Simplicius calls Agains ‘socius noster’ is met by the +suggestion that the particular word <I>socius</I> is either the +result of the double translation from the Greek or means +nothing more, in the mouth of Simplicius, than ‘colleague’ +in the sense of a worker in the same field, or ‘authority’. +A few extracts again are included in the <I>Anonymi variae +collectiones</I> in Hultsch's <I>Heron.</I> Nos. 5-14 give definitions of +geometry, logistic, geodesy and their subject-matter, remarks +on bodies as continuous magnitudes, the three dimensions as +‘principles’ of geometry, the purpose of geometry, and lastly +on optics, with its subdivisions, optics proper, <I>Catoptrica</I> and +<G>skhnografikh/</G>, scene-painting (a sort of perspective), with some +fundamental principles of optics, e.g. that all light travels +along straight lines (which are broken in the cases of reflection +and refraction), and the division between optics and natural +philosophy (the theory of light), it being the province of the +latter to investigate (what is a matter of indifference to optics) +whether (1) visual rays issue from the eye, (2) images proceed +from the object and impinge on the eye, or (3) the intervening +air is aligned or compacted with the beam-like breath or +emanation from the eye. +<p>Nos. 80-6 again in the same collection give the Peripatetic +explanation of the name mathematics, adding that the term +<pb n=225><head>GEMINUS</head> +was applied by the early Pythagoreans more particularly +to geometry and arithmetic, sciences which deal with the pure, +the eternal and the unchangeable, but was extended by later +writers to cover what we call ‘mixed’ or applied mathematics, +which, though theoretical, has to do with sensible objects, e.g. +astronomy and optics. Other extracts from Geminus are found +in extant manuscripts in connexion with Damianus's treatise +on optics (published by R. Schöne, Berlin, 1897). The defini- +tions of logistic and geometry also appear, but with decided +differences, in the scholia to Plato's <I>Charmides</I> 165 E. Lastly, +isolated extracts appear in Eutocius, (1) a remark reproduced +in the commentary on Archimedes's <I>Plane Equilibriums</I> to +the effect that Archimedes in that work gave the name of +postulates to what are really axioms, (2) the statement that +before Apollonius's time the conics were produced by cutting +different cones (right-angled, acute-angled, and obtuse-angled) +by sections perpendicular in each case to a generator.<note>Eutocius, <I>Comm. on Apollonius's Conics, ad init.</I></note> +<p>The object of Geminus's work was evidently the examina- +tion of the first principles, the logical building up of mathe- +matics on the basis of those admitted principles, and the +defence of the whole structure against the criticisms of +the enemies of the science, the Epicureans and Sceptics, some +of whom questioned the unproved principles, and others the +logical validity of the deductions from them. Thus in +geometry Geminus dealt first with the principles or hypotheses +(<G>a)rxai/, u(poqe/seis</G>) and then with the logical deductions, the +theorems and problems (<G>ta\ meta\ ta\s a)rxa/s</G>). The distinction +is between the things which must be taken for granted but +are incapable of proof and the things which must not be +assumed but are matter for demonstration. The principles +consisting of definitions, postulates, and axioms, Geminus +subjected them severally to a critical examination from this +point of view, distinguishing carefully between postulates and +axioms, and discussing the legitimacy or otherwise of those +formulated by Euclid in each class. In his notes on the defini- +tions Geminus treated them historically, giving the various +alternative definitions which had been suggested for each +fundamental concept such as ‘line’, ‘surface’, ‘figure’, ‘body’, +‘angle’, &c., and frequently adding instructive classifications +<pb n=226><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +of the different species of the thing defined. Thus in the +case of ‘lines’ (which include curves) he distinguishes, first, +the composite (e.g. a broken line forming an angle) and the +incomposite. The incomposite are subdivided into those +‘forming a figure’ (<G>sxhmatopoiou=sai</G>) or determinate (e.g. +circle, ellipse, cissoid) and those not forming a figure, inde- +terminate and extending without limit (e.g. straight line, +parabola, hyperbola, conchoid). In a second classification +incomposite lines are divided into (1) ‘simple’, namely the circle +and straight line, the one ‘making a figure’, the other extend- +ing without limit, and (2) ‘mixed’. ‘Mixed’ lines again are +divided into (<I>a</I>) ‘lines in planes’, one kind being a line meet- +ing itself (e.g. the cissoid) and another a line extending +without limit, and (<I>b</I>) ‘lines on solids’, subdivided into lines +formed by <I>sections</I> (e.g. conic sections, <I>spiric</I> curves) and +‘lines <I>round</I> solids’ (e.g. a helix round a cylinder, sphere, or +cone, the first of which is uniform, homoeomeric, alike in all +its parts, while the others are non-uniform). Geminus gave +a corresponding division of surfaces into simple and mixed, +the former being plane surfaces and spheres, while examples +of the latter are the tore or anchor-ring (though formed by +the revolution of a circle about an axis) and the conicoids of +revolution (the right-angled conoid, the obtuse-angled conoid, +and the two spheroids, formed by the revolution of a para- +bola, a hyperbola, and an ellipse respectively about their +axes). He observes that, while there are three <I>homoeomeric</I> +or uniform ‘lines’ (the straight line, the circle, and the +cylindrical helix), there are only two homoeomeric surfaces, +the plane and the sphere. Other classifications are those of +‘angles’ (according to the nature of the two lines or curves +which form them) and of figures and plane figures. +<p>When Proclus gives definitions, &c., by Posidonius, it is +evident that he obtained them from Geminus's work. Such +are Posidonius's definitions of ‘figure’ and ‘parallels’, and his +division of quadrilaterals into seven kinds. We may assume +further that, even where Geminus did not mention the name +of Posidonius, he was, at all events so far as the philosophy of +mathematics was concerned, expressing views which were +mainly those of his master. +<pb n=227><head>GEMINUS</head> +<C><I>Attempt to prove the Parallel-Postulate.</I></C> +<p>Geminus devoted much attention to the distinction between +postulates and axioms, giving the views of earlier philoso- +phers and mathematicians (Aristotle, Archimedes, Euclid, +Apollonius, the Stoics) on the subject as well as his own. It +was important in view of the attacks of the Epicureans and +Sceptics on mathematics, for (as Geminus says) it is as futile +to attempt to prove the indemonstrable (as Apollonius did +when he tried to prove the axioms) as it is incorrect to assume +what really requires proof, ‘as Euclid did in the fourth postu- +late [that all right angles are equal] and in the fifth postulate +[the parallel-postulate]’.<note>Proclus on Eucl. I, pp. 178-82. 4; 183. 14-184. 10.</note> +<p>The fifth postulate was the special stumbling-block. +Geminus observed that the converse is actually proved by +Euclid in I. 17; also that it is conclusively proved that an +angle equal to a right angle is not necessarily itself a right +angle (e.g. the ‘angle’ between the circumferences of two semi- +circles on two equal straight lines with a common extremity +and at right angles to one another); we cannot therefore admit +that the converses are incapable of demonstration.<note><I>Ib.</I>, pp. 183. 26-184. 5.</note> And +<p>‘we have learned from the very pioneers of this science not to +have regard to mere plausible imaginings when it is a ques- +tion of the reasonings to be included in our geometrical +doctrine. As Aristotle says, it is as justifiable to ask scien- +tific proofs from a rhetorician as to accept mere plausibilities +from a geometer ... So in this case (that of the parallel- +postulate) the fact that, when the right angles are lessened, the +straight lines converge is true and necessary; but the state- +ment that, since they converge more and more as they are +produced, they will sometime meet is plausible but not neces- +sary, in the absence of some argument showing that this is +true in the case of straight lines. For the fact that some lines +exist which approach indefinitely but yet remain non-secant +(<G>a)su/mptwtoi</G>), although it seems improbable and paradoxical, +is nevertheless true and fully ascertained with reference to +other species of lines [the hyperbola and its asymptote and +the conchoid and its asymptote, as Geminus says elsewhere]. +May not then the same thing be possible in the case of +<pb n=228><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +straight lines which happens in the case of the lines referred +to? Indeed, until the statement in the postulate is clinched +by proof, the facts shown in the case of the other lines may +direct our imagination the opposite way. And, though the +controversial arguments against the meeting of the straight +lines should contain much that is surprising, is there not all +the more reason why we should expel from our body of +doctrine this merely plausible and unreasoned (hypothesis)? +It is clear from this that we must seek a proof of the present +theorem, and that it is alien to the special character of +postulates.’<note>Proclus on Eucl. I, pp. 192. 5-193. 3.</note> +<p>Much of this might have been written by a modern +geometer. Geminus's attempted remedy was to substitute +a definition of parallels like that of Posidonius, based on the +notion of <I>equidistance.</I> An-Nairīzī gives the definition as +follows: ‘Parallel straight lines are straight lines situated in +the same plane and such that the distance between them, if +they are produced without limit in both directions at the same +time, is everywhere the same’, to which Geminus adds the +statement that the said distance is the shortest straight line +that can be drawn between them. Starting from this, +Geminus proved to his own satisfaction the propositions of +Euclid regarding parallels and finally the parallel-postulate. +He first gave the propositions (1) that the ‘distance’ between +the two lines as defined is perpendicular to both, and (2) that, +if a straight line is perpendicular to each of two straight lines +and meets both, the two straight lines are parallel, and the +‘distance’ is the intercept on the perpendicular (proved by +<I>reductio ad absurdum</I>). Next come (3) Euclid's propositions +I. 27, 28 that, if two lines are parallel, the alternate angles +made by any transversal are equal, &c. (easily proved by +drawing the two equal ‘distances’ through the points of +intersection with the transversal), and (4) Eucl. I. 29, the con- +verse of I. 28, which is proved by <I>reductio ad absurdum</I>, by +means of (2) and (3). Geminus still needs Eucl. I. 30, 31 +(about parallels) and I. 33, 34 (the first two propositions +relating to parallelograms) for his final proof of the postulate, +which is to the following effect. +<p>Let <I>AB, CD</I> be two straight lines met by the straight line +<pb n=229><head>GEMINUS</head> +<I>EF</I>, and let the interior angles <I>BEF, EFD</I> be together less +than two right angles. +<p>Take any point <I>H</I> on <I>FD</I> and draw <I>HK</I> parallel to <I>AB</I> +meeting <I>EF</I> in <I>K.</I> Then, if we bisect <I>EF</I> at <I>L, LF</I> at <I>M, MF</I> +at <I>N</I>, and so on, we shall at last have a length, as <I>FN</I>, less +<FIG> +than <I>FK.</I> Draw <I>FG, NOP</I> parallel to <I>AB.</I> Produce <I>FO</I> to <I>Q</I>, +and let <I>FQ</I> be the same multiple of <I>FO</I> that <I>FE</I> is of <I>FN</I>; +then shall <I>AB, CD</I> meet in <I>Q.</I> +<p>Let <I>S</I> be the middle point of <I>FQ</I> and <I>R</I> the middle point of +<I>FS.</I> Draw through <I>R, S, Q</I> respectively the straight lines +<I>RPG, STU, QV</I> parallel to <I>EF.</I> Join <I>MR, LS</I> and produce +them to <I>T, V.</I> Produce <I>FG</I> to <I>U.</I> +<p>Then, in the triangles <I>FON, ROP</I>, two angles are equal +respectively, the vertically opposite angles <I>FON, ROP</I> and +the alternate angles <I>NFO, PRO</I>; and <I>FO</I> = <I>OR</I>; therefore +<I>RP</I> = <I>FN.</I> +<p>And <I>FN, PG</I> in the parallelogram <I>FNPG</I> are equal; there- +fore <I>RG</I> = 2<I>FN</I> = <I>FM</I> (whence <I>MR</I> is parallel to <I>FG</I> or <I>AB</I>). +<p>Similarly we prove that <I>SU</I> = 2<I>FM</I> = <I>FL</I>, and <I>LS</I> is +parallel to <I>FG</I> or <I>AB.</I> +<p>Lastly, by the triangles <I>FLS, QVS</I>, in which the sides <I>FS</I>, +<I>SQ</I> are equal and two angles are respectively equal, <I>QV</I> = <I>FL.</I> +<p>Therefore <I>QV</I> = <I>LE.</I> +<p>Since then <I>EL, QV</I> are equal and parallel, so are <I>EQ, LV</I>, +and (says Geminus) it follows that <I>AB</I> passes through <I>Q.</I> +<pb n=230><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +<p>What follows is actually that both <I>EQ</I> and <I>AB</I> (or <I>EB</I>) +are parallel to <I>LV</I>, and Geminus assumes that <I>EQ, AB</I> +are coincident (in other words, that through a given point +only one parallel can be drawn to a given straight line, an +assumption known as Playfair's Axiom, though it is actually +stated in Proclus on Eucl. I. 31). +<p>The proof therefore, apparently ingenious as it is, breaks +down. Indeed the method is unsound from the beginning, +since (as Saccheri pointed out), before even the definition of +parallels by Geminus can be used, it has to be <I>proved</I> that +‘the geometrical locus of points equidistant from a straight +line is a straight line’, and this cannot be proved without a +postulate. But the attempt is interesting as the first which +has come down to us, although there must have been many +others by geometers earlier than Geminus. +<p>Coming now to the things which follow from the principles +(<G>ta\ meta\ ta\s a)rxa/s</G>), we gather from Proclus that Geminus +carefully discussed such generalities as the nature of <I>elements</I>, +the different views which had been held of the distinction +between theorems and problems, the nature and significance +of <G>diorismoi/</G> (conditions and limits of possibility), the meaning +of ‘porism’ in the sense in which Euclid used the word in his +<I>Porisms</I> as distinct from its other meaning of ‘corollary’, the +different sorts of problems and theorems, the two varieties of +converses (complete and partial), <I>topical</I> or <I>locus</I>-theorems, +with the classification of loci. He discussed also philosophical +questions, e.g. the question whether a line is made up of +indivisible parts (<G>e)x a)merw=n</G>), which came up in connexion +with Eucl. I. 10 (the bisection of a straight line). +<p>The book was evidently not less exhaustive as regards +higher geometry. Not only did Geminus mention the <I>spiric</I> +curves, conchoids and cissoids in his classification of curves; +he showed how they were obtained, and gave proofs, presum- +ably of their principal properties. Similarly he gave the +proof that there are three homoeomeric or uniform lines or +curves, the straight line, the circle and the cylindrical helix. +The proof of ‘uniformity’ (the property that any portion of +the line or curve will coincide with any other portion of the +same length) was preceded by a proof that, if two straight +lines be drawn from any point to meet a uniform line or curve +<pb n=231><head>GEMINUS</head> +and make equal angles with it, the straight lines are equal.<note>Proclus on Eucl. I, pp. 112. 22-113. 3, p. 251. 3-11.</note> +As Apollonius wrote on the cylindrical helix and proved the +fact of its uniformity, we may fairly assume that Geminus +was here drawing upon Apollonius. +<p>Enough has been said to show how invaluable a source of +information Geminus's work furnished to Proclus and all +writers on the history of mathematics who had access to it. +<p>In astronomy we know that Geminus wrote an <G>e)xh/ghsis</G> of +Posidonius's work, the <I>Meteorologica</I> or <G>peri\ metew/rwn</G>. This +is the source of the famous extract made from Geminus by +Alexander Aphrodisiensis, and reproduced by Simplicius in +his commentary on the <I>Physics</I> of Aristotle,<note>Simpl. <I>in Phys.</I>, pp. 291-2, ed. Diels.</note> on which Schia- +parelli relied in his attempt to show that it was Heraclides of +Pontus, not Aristarchus of Samos, who first put forward the +heliocentric hypothesis. The extract is on the distinction +between physical and astronomical inquiry as applied to the +heavens. It is the business of the physicist to consider the +substance of the heaven and stars, their force and quality, +their coming into being and decay, and he is in a position to +prove the facts about their size, shape, and arrangement; +astronomy, on the other hand, ignores the physical side, +proving the arrangement of the heavenly bodies by considera- +tions based on the view that the heaven is a real <G>ko/smos</G>, and, +when it tells us of the shapes, sizes and distances of the earth, +sun and moon, of eclipses and conjunctions, and of the quality +and extent of the movements of the heavenly bodies, it is +connected with the mathematical investigation of quantity, +size, form, or shape, and uses arithmetic and geometry to +prove its conclusions. Astronomy deals, not with causes, but +with facts; hence it often proceeds by hypotheses, stating +certain expedients by which the phenomena may be saved. +For example, why do the sun, the moon and the planets +appear to move irregularly? To explain the observed facts +we may assume, for instance, that the orbits are eccentric +circles or that the stars describe epicycles on a carrying +circle; and then we have to go farther and examine other +ways in which it is possible for the phenomena to be brought +about. ‘<I>Hence we actually find a certain person</I> [Heraclides +<pb n=232><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +of Pontus] <I>coming forward and saying that, even on the +assumption that the earth moves in a certain way, while +the sun is in a certain way at rest, the apparent irregularity +with reference to the sun may be saved.</I>’ Philological con- +siderations as well as the other notices which we possess +about Heraclides make it practically certain that ‘Heraclides +of Pontus’ is an interpolation and that Geminus said <G>tis</G> +simply, ‘a certain person’, without any name, though he +doubtless meant Aristarchus of Samos.<note>Cf. <I>Aristarchus of Samos</I>, pp. 275-83.</note> +<p>Simplicius says that Alexander quoted this extract from +the <I>epitome</I> of the <G>e)xh/ghsis</G> by Geminus. As the original +work was apparently made the subject of an abridgement, we +gather that it must have been of considerable scope. It is +a question whether <G>e)xh/ghsis</G> means ‘commentary’ or ‘ex- +position’; I am inclined to think that the latter interpretation +is the correct one, and that Geminus reproduced Posidonius's +work in its entirety with elucidations and comments; this +seems to me to be suggested by the words added by Simplicius +immediately after the extract ‘this is the account given by +Geminus, <I>or Posidonius in Geminus</I>, of the difference between +physics and astronomy’, which seems to imply that Geminus +in our passage reproduced Posidonius textually. +<p>‘<I>Introduction to the Phaenomena’ attributed to Geminus.</I> +<p>There remains the treatise, purporting to be by Geminus, +which has come down to us under the title <G>*gemi/nou ei)sagwgh\ +ei)s ta\ *faino/mena</G>.<note>Edited by Manitius (Teubner, 1898).</note> What, if any, is the relation of this work +to the exposition of Posidonius's <I>Meteorologica</I> or the epitome +of it just mentioned? One view is that the original <I>Isagoge</I> +of Geminus and the <G>e)xh/ghsis</G> of Posidonius were one and the +same work, though the <I>Isagoge</I> as we have it is not by +Geminus, but by an unknown compiler. The objections to +this are, first, that it does not contain the extract given by +Simplicius, which would have come in usefully at the begin- +ning of an Introduction to Astronomy, nor the other extract +given by Alexander from Geminus and relating to the rainbow +which seems likewise to have come from the <G>e)xh/ghsis</G><note>Alex. Aphr. on Aristotle's <I>Meteorologica</I>, iii. 4, 9 (Ideler, ii, p. 128; p. 152. 10, Hayduck).</note>; +<pb n=233><head>GEMINUS</head> +secondly, that it does not anywhere mention the name of +Posidonius (not, perhaps, an insuperable objection); and, +thirdly, that there are views expressed in it which are not +those held by Posidonius but contrary to them. Again, the +writer knows how to give a sound judgement as between +divergent views, writes in good style on the whole, and can +hardly have been the mere compiler of extracts from Posi- +donius which the view in question assumes him to be. It +seems in any case safer to assume that the <I>Isagoge</I> and the +<G>e)xh/ghsis</G> were separate works. At the same time, the <I>Isagoge</I>, +as we have it, contains errors which we cannot attribute to +Geminus. The choice, therefore, seems to lie between two +alternatives: either the book is by Geminus in the main, but +has in the course of centuries suffered deterioration by inter- +polations, mistakes of copyists, and so on, or it is a compilation +of extracts from an original <I>Isagoge</I> by Geminus with foreign +and inferior elements introduced either by the compiler him- +self or by other prentice hands. The result is a tolerable ele- +mentary treatise suitable for teaching purposes and containing +the most important doctrines of Greek astronomy represented +from the standpoint of Hipparchus. Chapter 1 treats of the +zodiac, the solar year, the irregularity of the sun's motion, +which is explained by the eccentric position of the sun's orbit +relatively to the zodiac, the order and the periods of revolution +of the planets and the moon. In ≻ 23 we are told that all +the fixed stars do not lie on one spherical surface, but some +are farther away than others—a doctrine due to the Stoics. +Chapter 2, again, treats of the twelve signs of the zodiac, +chapter 3 of the constellations, chapter 4 of the axis of +the universe and the poles, chapter 5 of the circles on the +sphere (the equator and the parallel circles, arctic, summer- +tropical, winter-tropical, antarctic, the colure-circles, the zodiac +or ecliptic, the horizon, the meridian, and the Milky Way), +chapter 6 of Day and Night, their relative lengths in different +latitudes, their lengthening and shortening, chapter 7 of +the times which the twelve signs take to rise. Chapter 8 +is a clear, interesting and valuable chapter on the calendar, +the length of months and years and the various cycles, the +octaëteris, the 16-years and 160-years cycles, the 19-years +cycle of Euctemon (and Meton), and the cycle of Callippus +<pb n=234><head>SUCCESSORS OF THE GREAT GEOMETERS</head> +(76 years). Chapter 9 deals with the moon's phases, chapters +10, 11 with eclipses of the sun and moon, chapter 12 with the +problem of accounting for the motions of the sun, moon and +planets, chapter 13 with Risings and Settings and the various +technical terms connected therewith, chapter 14 with the +circles described by the fixed stars, chapters 15 and 16 with +mathematical and physical geography, the zones, &c. (Geminus +follows Eratosthenes's evaluation of the circumference of the +earth, not that of Posidonius). Chapter 17, on weather indica- +tions, denies the popular theory that changes of atmospheric +conditions depend on the rising and setting of certain stars, +and states that the predictions of weather (<G>e)pishmasi/ai</G>) in +calendars (<G>paraph/gmata</G>) are only derived from experience +and observation, and have no scientific value. Chapter 18 is +on the <G>e)xeligmo/s</G>, the shortest period which contains an integral +number of synodic months, of days, and of anomalistic revolu- +tions of the moon; this period is three times the Chaldaean +period of 223 lunations used for predicting eclipses. The end +of the chapter deals with the maximum, mean, and minimum +daily motion of the moon. The chapter as a whole does not +correspond to the rest of the book; it deals with more difficult +matters, and is thought by Manitius to be a fragment only of +a discussion to which the compiler did not feel himself equal. +At the end of the work is a calendar (<I>Parapegma</I>) giving the +number of days taken by the sun to traverse each sign of +the zodiac, the risings and settings of various stars and the +weather indications noted by various astronomers, Democritus, +Eudoxus, Dositheus, Euctemon, Meton, Callippus; this calendar +is unconnected with the rest of the book and the contents +are in several respects inconsistent with it, especially the +division of the year into quarters which follows Callippus +rather than Hipparchus. Hence it has been, since Boeckh's +time, generally considered not to be the work of Geminus. +Tittel, however, suggests that it is not impossible that Geminus +may have reproduced an older <I>Parapegma</I> of Callippus. +<pb> +<C>XVI +SOME HANDBOOKS</C> +<p>THE description of the handbook on the elements of +astronomy entitled the <I>Introduction to the Phaenomena</I> and +attributed to Geminus might properly have been reserved +for this chapter. It was, however, convenient to deal with +Geminus in close connexion with Posidonius; for Geminus +wrote an exposition of Posidonius's <I>Meteorologica</I> related to the +original work in such a way that Simplicius, in quoting a long +passage from an epitome of this work, could attribute the +passage to either Geminus or ‘Posidonius in Geminus’; and it +is evident that, in other subjects too, Geminus drew from, and +was influenced by, Posidonius. +<p>The small work <I>De motu circulari corporum caelestium</I> by +CLEOMEDES (<G>*kleomh/dous kuklikh\ qewri/a</G>) in two Books is the +production of a much less competent person, but is much more +largely based on Posidonius. This is proved by several refer- +ences to Posidonius by name, but it is specially true of the +very long first chapter of Book II (nearly half of the Book) +which seems for the most part to be copied bodily from +Posidonius, in accordance with the author's remark at the +end of Book I that, in giving the refutation of the Epicurean +assertion that the sun is just as large as it looks, namely one +foot in diameter, he will give so much as suffices for such an +introduction of the particular arguments used by ‘certain +authors who have written whole treatises on this one topic +(i. e. the size of the sun), among whom is Posidonius’. The +interest of the book then lies mainly in what is quoted from +Posidonius; its mathematical interest is almost <I>nil.</I> +<p>The date of Cleomedes is not certainly ascertained, but, as +he mentions no author later than Posidonius, it is permissible +to suppose, with Hultsch, that he wrote about the middle of +<pb n=236><head>SOME HANDBOOKS</head> +the first century B. C. As he seems to know nothing of the +works of Ptolemy, he can hardly, in any case, have lived +later than the beginning of the second century A. D. +<p>Book I begins with a chapter the object of which is to +prove that the universe, which has the shape of a sphere, +is limited and surrounded by void extending without limit in +all directions, and to refute objections to this view. Then +follow chapters on the five parallel circles in the heaven and +the zones, habitable and uninhabitable (chap. 2); on the +motion of the fixed stars and the independent (<G>proairetikai/</G>) +movements of the planets including the sun and moon +(chap. 3); on the zodiac and the effect of the sun's motion in +it (chap. 4); on the inclination of the axis of the universe and +its effects on the lengths of days and nights at different places +(chap. 5); on the inequality in the rate of increase in the +lengths of the days and nights according to the time of year, +the different lengths of the seasons due to the motion of the +sun in an eccentric circle, the difference between a day-and- +night and an exact revolution of the universe owing to the +separate motion of the sun (chap. 6); on the habitable regions +of the globe including Britain and the ‘island of Thule’, said +to have been visited by Pytheas, where, when the sun is in +Cancer and visible, the day is a month long; and so on (chap. 7). +Chap. 8 purports to prove that the universe is a sphere by +proving first that the earth is a sphere, and then that the air +about it, and the ether about that, must necessarily make up +larger spheres. The earth is proved to be a sphere by the +method of exclusion; it is assumed that the only possibilities +are that it is (<I>a</I>) flat and plane, or (<I>b</I>) hollow and deep, or +(<I>c</I>) square, or (<I>d</I>) pyramidal, or (<I>e</I>) spherical, and, the first four +hypotheses being successively disposed of, only the fifth +remains. Chap. 9 maintains that the earth is in the centre of +the universe; chap. 10, on the size of the earth, contains the +interesting reproduction of the details of the measurements of +the earth by Posidonius and Eratosthenes respectively which +have been given above in their proper places (p.220, pp.106-7); +chap. 11 argues that the earth is in the relation of a point to, +i. e. is negligible in size in comparison with, the universe or +even the sun's circle, but not the moon's circle (cf. p. 3 above). +<p>Book II, chap. 1, is evidently the <I>pièce de résistance</I>, con- +<pb n=237><head>CLEOMEDES</head> +sisting of an elaborate refutation of Epicurus and his followers, +who held that the sun is just as large as it <I>looks</I>, and further +asserted (according to Cleomedes) that the stars are lit up as +they rise and extinguished as they set. The chapter seems to +be almost wholly taken from Posidonius; it ends with some +pages of merely vulgar abuse, comparing Epicurus with Ther- +sites, with more of the same sort. The value of the chapter +lies in certain historical traditions mentioned in it, and in the +account of Posidonius's speculation as to the size and distance +of the sun, which does, as a matter of fact, give results much +nearer the truth than those obtained by Aristarchus, Hippar- +chus, and Ptolemy. Cleomedes observes (1) that by means of +water-clocks it is found that the apparent diameter of the sun +is 1/750th of the sun's circle, and that this method of +measuring it is said to have been first invented by the +Egyptians; (2) that Hipparchus is said to have found that +the sun is 1,050 times the size of the earth, though, as regards +this, we have the better authority of Adrastus (in Theon of +Smyrna) and of Chalcidius, according to whom Hipparchus +made the sun nearly 1,880 times the size of the earth (both +figures refer of course to the solid content). We have already +described Posidonius's method of arriving at the size and +distance of the sun (pp. 220-1). After he has given this, Cleo- +medes, apparently deserting his guide, adds a calculation of +his own relating to the sizes and distances of the moon and +the sun which shows how little he was capable of any scien- +tific inquiry.<note>He says (pp. 146. 17-148. 27) that in an eclipse the breadth of the +earth's shadow is stated to be two moon-breadths; hence, he says, it +seems credible (<G>piqano/n</G>) that the earth is twice the size of the moon (this +practically assumes that the breadth of the earth's shadow is equal to +the diameter of the earth, or that the cone of the earth's shadow is +a cylinder!). Since then the circumference of the earth, according to +Eratosthenes, is 250,000 stades, and its diameter therefore ‘more than +80,000’ (he evidently takes <G>p</G>=3), the diameter of the moon will be +40,000 stades. Now, the moon's circle being 750 times the moon's +diameter, the radius of the moon's circle, i.e. the distance of the moon +from the earth, will be 1/6th of this (i.e. <G>p</G>=3) or 125 moon-diameters; +therefore the moon's distance is 5,000,000 stades (which is much too +great). Again, since the moon traverses its orbit 13 times to the sun's +once, he assumes that the sun's orbit is 13 times as large as the moon's, +and consequently that the diameter of the sun is 13 times that of the +moon, or 520,000 stades and its distance 13 times 5,000,000 or 65,000,000 +stades!</note> Chap. 2 purports to prove that the sun is +<pb n=238><head>SOME HANDBOOKS</head> +larger than the earth; and the remaining chapters deal with +the size of the moon and the stars (chap. 3), the illumination +of the moon by the sun (chap. 4), the phases of the moon and +its conjunctions with the sun (chap. 5), the eclipses of the +moon (chap. 6), the maximum deviation in latitude of the five +planets (given as 5° for Venus, 4° for Mercury, 2 1/2° for Mars +and Jupiter, 1° for Saturn), the maximum elongations of +Mercury and Venus from the sun (20° and 50° respectively), +and the synodic periods of the planets (Mercury 116 days, +Venus 584 days, Mars 780 days, Jupiter 398 days, Saturn +378 days) (chap. 7). +<p>There is only one other item of sufficient interest to be +mentioned here. In Book II, chap. 6, Cleomedes mentions +that there were stories of extraordinary eclipses which ‘the +more ancient of the mathematicians had vainly tried to ex- +plain’; the supposed ‘paradoxical’ case was that in which, +while the sun seems to be still above the horizon, the <I>eclipsed</I> +moon rises in the east. The passage has been cited above +(vol. i, pp. 6-7), where I have also shown that Cleomedes him- +self gives the true explanation of the phenomenon, namely +that it is due to atmospheric refraction. +<p>The first and second centuries of the Christian era saw +a continuation of the work of writing manuals or introduc- +tions to the different mathematical subjects. About A. D. 100 +came NICOMACHUS, who wrote an <I>Introduction to Arithmetic</I> +and an <I>Introduction to Harmony</I>; if we may judge by a +remark of his own,<note>Nicom. <I>Arith.</I> ii. 6. 1.</note> he would appear to have written an intro- +duction to geometry also. The <I>Arithmetical Introduction</I> has +been sufficiently described above (vol. i, pp. 97-112). +<p>There is yet another handbook which needs to be mentioned +separately, although we have had occasion to quote from it +several times already. This is the book by THEON OF SMYRNA +which goes by the title <I>Expositio rerum mathematicarum ad +legendum Platonem utilium.</I> There are two main divisions +of this work, contained in two Venice manuscripts respec- +tively. The first was edited by Bullialdus (Paris, 1644), the +second by T. H. Martin (Paris, 1849); the whole has been +<pb n=239><head>THEON OF SMYRNA</head> +edited by E. Hiller (Teubner, 1878) and finally, with a French +translation, by J. Dupuis (Paris, 1892). +<p>Theon's date is approximately fixed by two considerations. +He is clearly the person whom Theon of Alexandria called +‘the old Theon’, <G>to\n palaio\n *qe/wna</G>,<note>Theon of Alexandria, <I>Comm. on Ptolemy's Syntaxis</I>, Basel edition, +pp. 390, 395, 396.</note> and there is no reason +to doubt that he is the ‘Theon the mathematician’ (<G>o( maqh- +matiko/s</G>) who is credited by Ptolemy with four observations +of the planets Mercury and Venus made in A.D. 127, 129, 130 +and 132.<note>Ptolemy, <I>Syntaxis</I>, ix. 9, x. 1, 2.</note> The latest writers whom Theon himself mentions +are Thrasyllus, who lived in the reign of Tiberius, and +Adrastus the Peripatetic, who belongs to the middle of the +second century A.D. Theon's work itself is a curious medley, +valuable, not intrinsically, but for the numerous historical +notices which it contains. The title, which claims that the +book contains things useful for the study of Plato, must not +be taken too seriously. It was no doubt an elementary +<I>introduction</I> or vade-mecum for students of philosophy, but +there is little in it which has special reference to the mathe- +matical questions raised in Plato. The connexion consists +mostly in the long proem quoting the views of Plato on the +paramount importance of mathematics in the training of +the philosopher, and the mutual relation of the five different +branches, arithmetic, geometry, stereometry, astronomy and +music. The want of care shown by Theon in the quotations +from particular dialogues of Plato prepares us for the patch- +work character of the whole book. +<p>In the first chapter he promises to give the mathematical +theorems most necessary for the student of Plato to know, +in arithmetic, music, and geometry, with its application to +stereometry and astronomy.<note>Theon of Smyrna, ed. Hiller, p. 1. 10-17.</note> But the promise is by no means +kept as regards geometry and stereometry: indeed, in a +later passage Theon seems to excuse himself from including +theoretical geometry in his plan, on the ground that all those +who are likely to read his work or the writings of Plato may +be assumed to have gone through an elementary course of +theoretical geometry.<note><I>Ib.</I>, p. 16. 17-20.</note> But he writes at length on figured +<pb n=240><head>SOME HANDBOOKS</head> +numbers, plane and solid, which are of course analogous to +the corresponding geometrical figures, and he may have con- +sidered that he was in this way sufficiently fulfilling his +promise with regard to geometry and stereometry. Certain +geometrical definitions, of point, line, straight line, the three +dimensions, rectilinear plane and solid figures, especially +parallelograms and parallelepipedal figures including cubes, +<I>plinthides</I> (square bricks) and <G>doki/des</G> (beams), and <I>scalene</I> +figures with sides unequal every way (=<G>bwmi/skoi</G> in the +classification of solid numbers), are dragged in later (chaps. +53-5 of the section on music)<note>Theon of Smyrna, ed. Hiller, pp. 111-13.</note> in the middle of the discussion +of proportions and means; if this passage is not an inter- +polation, it confirms the supposition that Theon included in +his work only this limited amount of geometry and stereo- +metry. +<p>Section I is on Arithmetic in the same sense as Nicomachus's +<I>Introduction.</I> At the beginning Theon observes that arith- +metic will be followed by music. Of music in its three +aspects, music in instruments (<G>e)n o)rga/nois</G>), music in numbers, +i.e. musical intervals expressed in numbers or pure theoretical +music, and the music or harmony in the universe, the first +kind (instrumental music) is not exactly essential, but the other +two must be discussed immediately after arithmetic.<note><I>Ib.</I>, pp. 16. 24-17. 11.</note> The con- +tents of the arithmetical section have been sufficiently indicated +in the chapter on Pythagorean arithmetic (vol. i, pp. 112-13); +it deals with the classification of numbers, odd, even, and +their subdivisions, prime numbers, composite numbers with +equal or unequal factors, plane numbers subdivided into +square, oblong, triangular and polygonal numbers, with their +respective ‘gnomons’ and their properties as the sum of +successive terms of arithmetical progressions beginning with +1 as the first term, circular and spherical numbers, solid num- +bers with three factors, pyramidal numbers and truncated +pyramidal numbers, perfect numbers with their correlatives, +the over-perfect and the deficient; this is practically what +we find in Nicomachus. But the special value of Theon's +exposition lies in the fact that it contains an account of the +famous ‘side-’ and ‘diameter-’ numbers of the Pythagoreans.<note><I>Ib.</I>, pp. 42. 10-45. 9. Cf. vol. i, pp. 91-3.</note> +<pb n=241><head>THEON OF SMYRNA</head> +<p>In the Section on Music Theon says he will first speak of +the two kinds of music, the audible or instrumental, and the +intelligible or theoretical subsisting in numbers, after which +he promises to deal lastly with ratio as predicable of mathe- +matical entities in general and the ratio constituting the +harmony in the universe, ‘not scrupling to set out once again +the things discovered by our predecessors, just as we have +given the things handed down in former times by the Pytha- +goreans, with a view to making them better known, without +ourselves claiming to have discovered any of them’.<note>Theon of Smyrna, ed. Hiller, pp. 46. 20-47. 14.</note> Then +follows a discussion of audible music, the intervals which +give harmonies, &c., including substantial quotations from +Thrasyllus and Adrastus, and references to views of Aris- +toxenus, Hippasus, Archytas, Eudoxus and Plato. With +chap. 17 (p. 72) begins the account of the ‘harmony in +numbers’, which turns into a general discussion of ratios, +proportions and means, with more quotations from Plato, +Eratosthenes and Thrasyllus, followed by Thrasyllus's <I>divisio +canonis</I>, chaps. 35, 36 (pp. 87-93). After a promise to apply +the latter division to the sphere of the universe, Theon +purports to return to the subject of proportion and means. +This, however, does not occur till chap. 50 (p. 106), the +intervening chapters being taken up with a discussion of +the <G>deka/s</G> and <G>tetraktu/s</G> (with eleven applications of the +latter) and the mystic or curious properties of the numbers +from 2 to 10; here we have a part of the <I>theologumena</I> of +arithmetic. The discussion of proportions and the different +kinds of means after Eratosthenes and Adrastus is again +interrupted by the insertion of the geometrical definitions +already referred to (chaps. 53-5, pp. 111-13), after which +Theon resumes the question of means for ‘more precise’ +treatment. +<p>The Section on Astronomy begins on p. 120 of Hiller's +edition. Here again Theon is mainly dependent upon +Adrastus, from whom he makes long quotations. Thus, on +the sphericity of the earth, he says that for the neces- +sary conspectus of the arguments it will be sufficient to +refer to the grounds stated summarily by Adrastus. In +explaining (p. 124) that the unevennesses in the surface of +<pb n=242><head>SOME HANDBOOKS</head> +the earth, represented e.g. by mountains, are negligible in +comparison with the size of the whole, he quotes Eratosthenes +and Dicaearchus as claiming to have discovered that the +perpendicular height of the highest mountain above the normal +level of the land is no more than 10 stades; and to obtain the +diameter of the earth he uses Eratosthenes's figure of approxi- +mately 252,000 stades for the circumference of the earth, +which, with the Archimedean value of 22/7 for <G>p</G>, gives a +diameter of about 80,182 stades. The principal astronomical +circles in the heaven are next described (chaps. 5-12, pp. +129-35); then (chap. 12) the assumed maximum deviations in +latitude are given, that of the sun being put at 1°, that of the +moon and Venus at 12°, and those of the planets Mercury, +Mars, Jupiter and Saturn at 8°, 5°, 5° and 3° respectively; the +obliquity of the ecliptic is given as the side of a regular polygon +of 15 sides described in a circle, i.e. as 24° (chap. 23, p. 151). +Next the order of the orbits of the sun, moon and planets is ex- +plained (the system is of course geocentric); we are told (p.138) +that ‘some of the Pythagoreans’ made the order (reckoning +outwards from the earth) to be moon, Mercury, Venus, sun, +Mars, Jupiter, Saturn, whereas (p. 142) Eratosthenes put the +sun next to the moon, and the mathematicians, agreeing with +Eratosthenes in this, differed only in the order in which they +placed Venus and Mercury after the sun, some putting Mercury +next and some Venus (p. 143). The order adopted by ‘some +of the Pythagoreans’ is the Chaldaean order, which was not +followed by any Greek before Diogenes of Babylon (second +century B.C.); ‘some of the Pythagoreans’ are therefore the +later Pythagoreans (of whom Nicomachus was one); the other +order, moon, sun, Venus, Mercury, Mars, Jupiter, Saturn, was +that of Plato and the early Pythagoreans. In chap. 15 +(p. 138 sq.) Theon quotes verses of Alexander ‘the Aetolian’ +(not really the ‘Aetolian’, but Alexander of Ephesus, a con- +temporary of Cicero, or possibly Alexander of Miletus, as +Chalcidius calls him) assigning to each of the planets (includ- +ing the earth, though stationary) with the sun and moon and +the sphere of the fixed stars one note, the intervals between +the notes being so arranged as to bring the nine into an +octave, whereas with Eratosthenes and Plato the earth was +excluded, and the eight notes of the octachord were assigned +<pb n=243><head>THEON OF SMYRNA</head> +to the seven heavenly bodies and the sphere of the fixed stars. +The whole of this passage (chaps. 15 to 16, pp. 138-47) is no +doubt intended as the promised account of the ‘harmony in +the universe’, although at the very end of the work Theon +implies that this has still to be explained on the basis of +Thrasyllus's exposition combined with what he has already +given himself. +<p>The next chapters deal with the forward movements, the +stationary points, and the retrogradations, as they respectively +appear to us, of the five planets, and the ‘saving of the pheno- +mena’ by the alternative hypotheses of eccentric circles and +epicycles (chaps. 17-30, pp. 147-78). These hypotheses are +explained, and the identity of the motion produced by the +two is shown by Adrastus in the case of the sun (chaps. 26, 27, +pp. 166-72). The proof is introduced with the interesting +remark that ‘Hipparchus says it is worthy of investigation +by mathematicians why, on two hypotheses so different from +one another, that of eccentric circles and that of concentric +circles with epicycles, the same results appear to follow’. It +is not to be supposed that the proof of the identity could be +other than easy to a mathematician like Hipparchus; the +remark perhaps merely suggests that the two hypotheses were +discovered quite independently, and it was not till later that +the effect was discovered to be the same, when of course the +fact would seem to be curious and a mathematical proof would +immediately be sought. Another passage (p. 188) says that +Hipparchus preferred the hypothesis of the epicycle, as being +his own. If this means that Hipparchus claimed to have +discovered the epicycle-hypothesis, it must be a misapprehen- +sion; for Apollonius already understood the theory of epi- +cycles in all its generality. According to Theon, the epicycle- +hypothesis is more ‘according to nature’; but it was presum- +ably preferred because it was applicable to all the planets, +whereas the eccentric-hypothesis, when originally suggested, +applied only to the three superior planets; in order to make +it apply to the inferior planets it is necessary to suppose the +circle described by the centre of the eccentric to be greater +than the eccentric circle itself, which extension of the hypo- +thesis, though known to Hipparchus, does not seem to have +occurred to Apollonius. +<pb n=244><head>SOME HANDBOOKS</head> +<p>We next have (chap. 31, p. 178) an allusion to the systems +of Eudoxus, Callippus and Aristotle, and a description +(p. 180 sq.) of a system in which the ‘carrying’ spheres +(called ‘hollow’) have between them ‘solid spheres which by +their own motion will roll (<G>a)neli/xousi</G>) the carrying spheres in +the opposite direction, being in contact with them’. These +‘solid’ spheres (which carry the planet fixed at a point on +their surface) act in practically the same way as epicycles. +In connexion with this description Theon (i.e. Adrastus) +speaks (chap. 33, pp. 186-7) of two alternative hypotheses in +which, by comparison with Chalcidius,<note>Chalcidius, <I>Comm. on Timaeus</I>, c. 110. Cf. <I>Aristarchus of Samos</I>, pp. 256-8.</note> we recognize (after +eliminating epicycles erroneously imported into both systems) +the hypotheses of Plato and Heraclides respectively. It is +this passage which enables us to conclude for certain that +Heraclides made Venus and Mercury revolve in circles about +the sun, like satellites, while the sun in its turn revolves in +a circle about the earth as centre. Theon (p. 187) gives the +maximum arcs separating Mercury and Venus respectively +from the sun as 20° and 50°, these figures being the same as +those given by Cleomedes. +<p>The last chapters (chaps. 37-40), quoted from Adrastus, deal +with conjunctions, transits, occultations and eclipses. The +book concludes with a considerable extract from Dercyllides, +a Platonist with Pythagorean leanings, who wrote (before the +time of Tiberius and perhaps even before Varro) a book on +Plato's philosophy. It is here (p. 198. 14) that we have the +passage so often quoted from Eudemus: +<p>‘Eudemus relates in his Astronomy that it was Oenopides +who first discovered the girdling of the zodiac and the revolu- +tion (or cycle) of the Great Year, that Thales was the first to +discover the eclipse of the sun and the fact that the sun's +period with respect to the solstices is not always the same, +that Anaximander discovered that the earth is (suspended) on +high and lies (substituting <G>kei=tai</G> for the reading of the manu- +scripts, <G>kinei=tai</G>, moves) about the centre of the universe, and +that Anaximenes said that the moon has its light from the +sun and (explained) how its eclipses come about’ (Anaxi- +menes is here apparently a mistake for Anaxagoras). +<pb> +<C>XVII +TRIGONOMETRY: HIPPARCHUS, MENELAUS, +PTOLEMY</C> +<p>WE have seen that <I>Sphaeric</I>, the geometry of the sphere, +was very early studied, because it was required so soon as +astronomy became mathematical; with the Pythagoreans the +word <I>Sphaeric</I>, applied to one of the subjects of the quadrivium, +actually meant astronomy. The subject was so far advanced +before Euclid's time that there was in existence a regular +textbook containing the principal propositions about great +and small circles on the sphere, from which both Autolycus +and Euclid quoted the propositions as generally known. +These propositions, with others of purely astronomical in- +terest, were collected afterwards in a work entitled <I>Sphaerica</I>, +in three Books, by THEODOSIUS. +<p>Suidas has a notice, <I>s. v.</I> <G>*qeodo/sios</G>, which evidently con- +fuses the author of the <I>Sphaerica</I> with another Theodosius, +a Sceptic philosopher, since it calls him ‘Theodosius, a philoso- +pher’, and attributes to him, besides the mathematical works, +‘Sceptic chapters’ and a commentary on the chapters of +Theudas. Now the commentator on Theudas must have +belonged, at the earliest, to the second half of the second +century A.D., whereas our Theodosius was earlier than Mene- +laus (<I>fl.</I> about A. D. 100), who quotes him by name. The next +notice by Suidas is of yet another Theodosius, a poet, who +came from Tripolis. Hence it was at one time supposed that +our Theodosius was of Tripolis. But Vitruvius<note><I>De architectura</I> ix. 9.</note> mentions a +Theodosius who invented a sundial ‘for any climate’; and +Strabo, in speaking of certain Bithynians distinguished in +their particular sciences, refers to ‘Hipparchus, Theodosius +and his sons, mathematicians’<note>Strabo, xii. 4, 9, p. 566.</note>. We conclude that our Theo- +<pb n=246><head>TRIGONOMETRY</head> +dosius was of Bithynia and not later in date than Vitruvius +(say 20 B.C.); but the order in which Strabo gives the +names makes it not unlikely that he was contemporary with +Hipparchus, while the character of his <I>Sphaerica</I> suggests a +date even earlier rather than later. +<C>Works by Theodosius.</C> +<p>Two other works of Theodosius besides the <I>Sphaerica</I>, +namely <I>On habitations</I> and <I>On Days and Nights</I>, seem to +have been included in the ‘Little Astronomy’ (<G>mikro\s a)stro- +nomou/menos</G>, <I>sc.</I> <G>to/pos</G>). These two treatises need not detain us +long. They are extant in Greek (in the great MS. Vaticanus +Graecus 204 and others), but the Greek text has not appar- +ently yet been published. In the first, <I>On habitations</I>, in 12 +propositions, Theodosius explains the different phenomena due +to the daily rotation of the earth, and the particular portions +of the whole system which are visible to inhabitants of the +different zones. In the second, <I>On Days and Nights</I>, contain- +ing 13 and 19 propositions in the two Books respectively, +Theodosius considers the arc of the ecliptic described by the +sun each day, with a view to determining the conditions to be +satisfied in order that the solstice may occur in the meridian +at a given place, and in order that the day and the night may +really be equal at the equinoxes; he shows also that the +variations in the day and night must recur exactly after +a certain time, if the length of the solar year is commen- +surable with that of the day, while on the contrary assump- +tion they will not recur so exactly. +<p>In addition to the works bearing on astronomy, Theodosius +is said<note>Suidas, <I>loc. cit.</I></note> to have written a commentary, now lost, on the <G>e)fo/dion</G> +or <I>Method</I> of Archimedes (see above, pp. 27-34). +<C>Contents of the <I>Sphaerica.</I></C> +<p>We come now to the <I>Sphaerica</I>, which deserves a short +description from the point of view of this chapter. A text- +book on the geometry of the sphere was wanted as a supple- +ment to the <I>Elements</I> of Euclid. In the <I>Elements</I> themselves +<pb n=247><head>THEODOSIUS'S <I>SPHAERICA</I></head> +(Books XII and XIII) Euclid included no general properties +of the sphere except the theorem proved in XII. 16-18, that +the volumes of two spheres are in the triplicate ratio of their +diameters; apart from this, the sphere is only introduced in +the propositions about the regular solids, where it is proved +that they are severally inscribable in a sphere, and it was doubt- +less with a view to his proofs of this property in each case that +he gave a new definition of a sphere as the figure described by +the revolution of a semicircle about its diameter, instead of +the more usual definition (after the manner of the definition +of a circle) as the locus of all points (in space instead of in +a plane) which are equidistant from a fixed point (the centre). +No doubt the exclusion of the geometry of the sphere from +the <I>Elements</I> was due to the fact that it was regarded as +belonging to astronomy rather than pure geometry. +<p>Theodosius defines the sphere as ‘a solid figure contained +by one surface such that all the straight lines falling upon it +from one point among those lying within the figure are equal +to one another’, which is exactly Euclid's definition of a circle +with ‘solid’ inserted before ‘figure’ and ‘surface’ substituted +for ‘line’. The early part of the work is then generally +developed on the lines of Euclid's Book III on the circle. +Any plane section of a sphere is a circle (Prop. 1). The +straight line from the centre of the sphere to the centre of +a circular section is perpendicular to the plane of that section +(1, Por. 2; cf. 7, 23); thus a plane section serves for finding +the centre of the sphere just as a chord does for finding that +of a circle (Prop. 2). The propositions about tangent planes +(3-5) and the relation between the sizes of circular sections +and their distances from the centre (5, 6) correspond to +Euclid III. 16-19 and 15; as the small circle corresponds to +any chord, the great circle (‘greatest circle’ in Greek) corre- +sponds to the diameter. The poles of a circular section +correspond to the extremities of the diameter bisecting +a chord of a circle at right angles (Props. 8-10). Great +circles bisecting one another (Props. 11-12) correspond to +chords which bisect one another (diameters), and great circles +bisecting small circles at right angles and passing through +their poles (Props. 13-15) correspond to diameters bisecting +chords at right angles. The distance of any point of a great +<pb n=248><head>TRIGONOMETRY</head> +circle from its pole is equal to the side of a square inscribed +in the great circle and conversely (Props. 16, 17). Next come +certain problems: To find a straight line equal to the diameter +of any circular section or of the sphere itself (Props. 18, 19); +to draw the great circle through any two given points on +the surface (Prop. 20); to find the pole of any given circu- +lar section (Prop. 21). Prop. 22 applies Eucl. III. 3 to the +sphere. +<p>Book II begins with a definition of circles on a sphere +which touch one another; this happens ‘when the common +section of the planes (of the circles) touches both circles’. +Another series of propositions follows, corresponding again +to propositions in Eucl., Book III, for the circle. Parallel +circular sections have the same poles, and conversely (Props. +1, 2). Props. 3-5 relate to circles on the sphere touching +one another and therefore having their poles on a great +circle which also passes through the point of contact (cf. +Eucl. III. 11, [12] about circles touching one another). If +a great circle touches a small circle, it also touches another +small circle equal and parallel to it (Props. 6, 7), and if a +great circle be obliquely inclined to another circular section, +it touches each of two equal circles parallel to that section +(Prop. 8). If two circles on a sphere cut one another, the +great circle drawn through their poles bisects the intercepted +segments of the circles (Prop. 9). If there are any number of +parallel circles on a sphere, and any number of great circles +drawn through their poles, the arcs of the parallel circles +intercepted between any two of the great circles are similar, +and the arcs of the great circles intercepted between any two +of the parallel circles are equal (Prop. 10). +<p>The last proposition forms a sort of transition to the portion +of the treatise (II. 11-23 and Book III) which contains pro- +positions of purely astronomical interest, though expressed as +propositions in pure geometry without any specific reference +to the various circles in the heavenly sphere. The proposi- +tions are long and complicated, and it would neither be easy +nor worth while to attempt an enumeration. They deal with +circles or parts of circles (arcs intercepted on one circle by +series of other circles and the like). We have no difficulty in +recognizing particular circles which come into many proposi- +<pb n=249><head>THEODOSIUS'S <I>SPHAERICA</I></head> +tions. A particular small circle is the circle which is the +limit of the stars which do not set, as seen by an observer at +a particular place on the earth's surface; the pole of this +circle is the pole in the heaven. A great circle which touches +this circle and is obliquely inclined to the ‘parallel circles’ is the +circle of the horizon; the parallel circles of course represent +the apparent motion of the fixed stars in the diurnal rotation, +and have the pole of the heaven as pole. A second great +circle obliquely inclined to the parallel circles is of course the +circle of the zodiac or ecliptic. The greatest of the ‘parallel +circles’ is naturally the equator. All that need be said of the +various propositions (except two which will be mentioned +separately) is that the sort of result proved is like that of +Props. 12 and 13 of Euclid's <I>Phaenomena</I> to the effect that in +the half of the zodiac circle beginning with Cancer (or Capri- +cornus) equal arcs set (or rise) in unequal times; those which +are nearer the tropic circle take a longer time, those further +from it a shorter; those which take the shortest time are +those adjacent to the equinoctial points; those which are equi- +distant from the equator rise and set in equal times. In like +manner Theodosius (III. 8) in effect takes equal and con- +tiguous arcs of the ecliptic all on one side of the equator, +draws through their extremities great circles touching the +circumpolar ‘parallel’ circle, and proves that the correspond- +ing arcs of the equator intercepted between the latter great +circles are unequal and that, of the said arcs, that correspond- +ing to the arc of the ecliptic which is nearer the tropic circle +is the greater. The successive great circles touching the +circumpolar circle are of course successive positions of the +horizon as the earth revolves about its axis, that is to say, +the same length of arc on the ecliptic takes a longer or shorter +time to rise according as it is nearer to or farther from the +tropic, in other words, farther from or nearer to the equinoctial +points. +<p>It is, however, obvious that investigations of this kind, +which only prove that certain arcs are greater than others, +and do not give the actual numerical ratios between them, are +useless for any practical purpose such as that of telling the +hour of the night by the stars, which was one of the funda- +mental problems in Greek astronomy; and in order to find +<pb n=250><head>TRIGONOMETRY</head> +the required numerical ratios a new method had to be invented, +namely trigonometry. +<C><I>No actual trigonometry in Theodosius.</I></C> +<p>It is perhaps hardly correct to say that spherical triangles +are nowhere referred to in Theodosius, for in III. 3 the con- +gruence-theorem for spherical triangles corresponding to Eucl. +I. 4 is practically proved; but there is nothing in the book +that can be called trigonometrical. The nearest approach is +in III. 11, 12, where ratios between certain straight lines are +compared with ratios between arcs. <I>ACc</I> (Prop. 11) is a great +circle through the poles <I>A, A′</I>; <I>CDc, C′D</I> are two other great +circles, both of which are at right angles to the plane of <I>AC′c</I>, +but <I>CDc</I> is perpendicular to <I>AA′</I>, while <I>C′D</I> is inclined to it at +an acute angle. Let any other great circle <I>AB′BA′</I> through +<FIG> +<I>AA′</I> cut <I>CD</I> in any point <I>B</I> between <I>C</I> and <I>D</I>, and <I>C′D</I> in <I>B′.</I> +Let the ‘parallel’ circle <I>EB′e</I> be drawn through <I>B′</I>, and let +<I>C′c′</I> be the diameter of the ‘parallel’ circle touching the great +circle <I>C′D.</I> Let <I>L, K</I> be the centres of the ‘parallel’ circles, +and let <I>R</I>, <G>r</G> be the radii of the ‘parallel’ circles <I>CDc, C′c′</I> +respectively. It is required to prove that +<MATH>2<I>R</I>:2<G>r</G> > (arc <I>CB</I>):(arc <I>C′B′</I>)</MATH>. +<p>Let <I>C′O, Ee</I> meet in <I>N</I>, and join <I>NB′.</I> +<p>Then <I>B′N</I>, being the intersection of two planes perpendicu- +lar to the plane of <I>AC′CA′</I>, is perpendicular to that plane and +therefore to both <I>Ee</I> and <I>C′O.</I> +<pb n=251><head>THEODOSIUS'S <I>SPHAERICA</I></head> +<p>Now, the triangle <I>NLO</I> being right-angled at <I>L, NO</I> > <I>NL.</I> +<p>Measure <I>NT</I> along <I>NO</I> equal to <I>NL</I>, and join <I>TB′.</I> +<p>Then in the triangles <I>B′NT, B′NL</I> two sides <I>B′N, NT</I> are +equal to two sides <I>B′N, NL</I>, and the included angles (both +being right) are equal; therefore the triangles are equal in all +respects, and ∠ <I>NLB′</I> = ∠ <I>NTB′.</I> +<p>Now <MATH>2<I>R</I>:2<G>r</G> = <I>OC′</I>:<I>C′K</I> += <I>ON</I>:<I>NL</I> += <I>ON</I>:<I>NT</I> +[=tan <I>NTB′</I>:tan<I>NOB′</I>] +> ∠<I>NTB′</I>:∠<I>NOB′</I> +> ∠<I>NLB′</I>:∠<I>NOB′</I> +> ∠<I>COB</I>:∠<I>NOB′</I> +> (arc <I>BC</I>):(arc <I>B′C′</I>)</MATH>. +<p>If <I>a′, b′, c′</I> are the sides of the spherical triangle <I>AB′C′</I>, this +result is equivalent (since the angle <I>COB</I> subtended by the arc +<I>CB</I> is equal to <I>A</I>) to +<MATH>1:sin <I>b′</I> = tan <I>A</I>:tan <I>a′</I></MATH> +<MATH>> <I>a</I>:<I>a′</I></MATH>, +where <I>a</I> = <I>BC</I>, the side opposite <I>A</I> in the triangle <I>ABC.</I> +<p>The proof is based on the fact (proved in Euclid's <I>Optics</I> +and assumed as known by Aristarchus of Samos and Archi- +medes) that, if <G>a, b</G> are angles such that 1/2<G>p</G> > <G>a</G> > <G>b</G>, +tan <G>a</G>/tan <G>b</G> > <G>a</G>/<G>b</G>. +<p>While, therefore, Theodosius proves the equivalent of the +formula, applicable in the solution of a spherical triangle +right-angled at <I>C</I>, that tan <I>a</I> = sin <I>b</I> tan <I>A</I>, he is unable, for +want of trigonometry, to find the actual value of <I>a</I>/<I>a′</I>, and +can only find a limit for it. He is exactly in the same position +as Aristarchus, who can only approximate to the values of the +trigonometrical ratios which he needs, e.g. sin 1°, cos 1°, sin 3°, +by bringing them within upper and lower limits with the aid +of the inequalities +<MATH>tan<G>a</G>/tan<G>b</G> > <G>a</G>/<G>b</G> > sin<G>a</G>/sin<G>b</G></MATH>, +where 1/2 <G>p</G> > <G>a</G> > <G>b</G>. +<pb n=252><head>TRIGONOMETRY</head> +<p>We may contrast with this proposition of Theodosius the +corresponding proposition in Menelaus's <I>Sphaerica</I> (III. 15) +dealing with the more general case in which <I>C′</I>, instead of +being the tropical point on the ecliptic, is, like <I>B′</I>, any point +between the tropical point and <I>D.</I> If <I>R</I>, <G>r</G> have the same +meaning as above and <I>r</I><SUB>1</SUB>, <I>r</I><SUB>2</SUB> are the radii of the parallel circles +through <I>B′</I> and the new <I>C′</I>, Menelaus proves that +<MATH>sin<I>a</I>/sin<I>a′</I> = <I>R</I><G>r</G>/<I>r</I><SUB>1</SUB><I>r</I><SUB>2</SUB></MATH>, +which, of course, with the aid of Tables, gives the means +of finding the actual values of <I>a</I> or <I>a′</I> when the other elements +are given. +<p>The proposition III. 12 of Theodosius proves a result similar +to that of III. 11 for the case where the great circles <I>AB′B</I>, +<I>AC′C</I>, instead of being great circles through the poles, are +great circles touching ‘the circle of the always-visible stars’, +i.e. different positions of the horizon, and the points <I>C′, B′</I> are +any points on the arc of the oblique circle between the tropical +and the equinoctial points; in this case, with the same notation, +<MATH>4<I>R</I>:2<G>r</G> > (arc <I>BC</I>):(arc <I>B′C′</I>)</MATH>. +<p>It is evident that Theodosius was simply a laborious com- +piler, and that there was practically nothing original in his +work. It has been proved, by means of propositions quoted +<I>verbatim</I> or assumed as known by Autolycus in his <I>Moving +Sphere</I> and by Euclid in his <I>Phaenomena</I>, that the following +propositions in Theodosius are pre-Euclidean, I. 1, 6 a, 7, 8, 11, +12, 13, 15, 20; II. 1, 2, 3, 5, 8, 9, 10 a, 13, 15, 17, 18, 19, 20, 22; +III. 1 b, 2, 3, 7, 8, those shown in thick type being quoted +word for word. +<C>The beginnings of trigonometry.</C> +<p>But this is not all. In Menelaus's <I>Sphaerica</I>, III. 15, there +is a reference to the proposition (III. 11) of Theodosius proved +above, and in Gherard of Cremona's translation from the +Arabic, as well as in Halley's translation from the Hebrew +of Jacob b. Machir, there is an addition to the effect that this +proposition was used by Apollonius in a book the title of +which is given in the two translations in the alternative +<pb n=253><head>BEGINNINGS OF TRIGONOMETRY</head> +forms ‘<I>liber aggregativus</I>’ and ‘liber de principiis universa- +libus’. Each of these expressions may well mean the work +of Apollonius which Marinus refers to as the ‘General +Treatise’ (<G>h( kaqo/lou pragmatei/a</G>). There is no apparent +reason to doubt that the remark in question was really +contained in Menelaus's original work; and, even if it is an +Arabian interpolation, it is not likely to have been made +without some definite authority. If then Apollonius was the +discoverer of the proposition, the fact affords some ground for +thinking that the beginnings of trigonometry go as far back, +at least, as Apollonius. Tannery<note>Tannery, <I>Recherches sur l'hist. de l'astronomie ancienne</I>, p. 64.</note> indeed suggested that not +only Apollonius but Archimedes before him may have com- +piled a ‘table of chords’, or at least shown the way to such +a compilation, Archimedes in the work of which we possess +only a fragment in the <I>Measurement of a Circle</I>, and Apollonius +in the <G>w)kuto/kion</G>, where he gave an approximation to the value +of <G>p</G> closer than that obtained by Archimedes; Tannery +compares the Indian Table of Sines in the <I>Sūrya-Siddhānta</I>, +where the angles go by 24ths of a right angle (1/24th = 3°45′, +2/24ths = 7°30′, &c.), as possibly showing Greek influence. +This is, however, in the region of conjecture; the first person +to make systematic use of trigonometry is, so far as we know, +Hipparchus. +<p>HIPPARCHUS, the greatest astronomer of antiquity, was +born at Nicaea in Bithynia. The period of his activity is +indicated by references in Ptolemy to observations made by +him the limits of which are from 161 B.C. to 126 B.C. Ptolemy +further says that from Hipparchus's time to the beginning of +the reign of Antoninus Pius (A.D. 138) was 265 years.<note>Ptolemy, <I>Syntaxis</I>, vii. 2 (vol. ii, p. 15).</note> The +best and most important observations made by Hipparchus +were made at Rhodes, though an observation of the vernal +equinox at Alexandria on March 24, 146 B.C., recorded by him +may have been his own. His main contributions to theoretical +and practical astronomy can here only be indicated in the +briefest manner. +<pb n=254><head>TRIGONOMETRY</head> +<C>The work of Hipparchus.</C> +<C><I>Discovery of precession.</I></C> +<p>1. The greatest is perhaps his discovery of the precession +of the equinoxes. Hipparchus found that the bright star +Spica was, at the time of his observation of it, 6° distant +from the autumnal equinoctial point, whereas he deduced from +observations recorded by Timocharis that Timocharis had +made the distance 8°. Consequently the motion had amounted +to 2° in the period between Timocharis's observations, made in +283 or 295 B.C., and 129/8 B.C., a period, that is, of 154 or +166 years; this gives about 46.8″ or 43.4″ a year, as compared +with the true value of 50.3757″. +<C><I>Calculation of mean lunar month.</I></C> +<p>2. The same discovery is presupposed in his work <I>On the +length of the Year</I>, in which, by comparing an observation +of the summer solstice by Aristarchus in 281/0 B.C. with his +own in 136/5 B.C., he found that after 145 years (the interval +between the two dates) the summer solstice occurred half +a day-and-night earlier than it should on the assumption of +exactly 365 1/4 days to the year; hence he concluded that the +<I>tropical</I> year contained about 1/300th of a day-and-night less +than 365 1/4 days. This agrees very nearly with Censorinus's +statement that Hipparchus's cycle was 304 years, four times +the 76 years of Callippus, but with 111,035 days in it +instead of 111,036 (= 27,759 x 4). Counting in the 304 years +12 x 304 + 112 (intercalary) months, or 3,760 months in all, +Hipparchus made the mean lunar month 29 days 12 hrs. +44 min. 2 1/2 sec., which is less than a second out in comparison +with the present accepted figure of 29.53059 days! +<p>3. Hipparchus attempted a new determination of the sun's +motion by means of exact equinoctial and solstitial obser- +vations; he reckoned the eccentricity of the sun's course +and fixed the apogee at the point 5°30′ of <I>Gemini.</I> More +remarkable still was his investigation of the moon's +course. He determined the eccentricity and the inclination +of the orbit to the ecliptic, and by means of records of +observations of eclipses determined the moon's period with +extraordinary accuracy (as remarked above). We now learn +<pb n=255><head>HIPPARCHUS</head> +that the lengths of the mean synodic, the sidereal, the +anomalistic and the draconitic month obtained by Hipparchus +agree exactly with Babylonian cuneiform tables of date not +later than Hipparchus, and it is clear that Hipparchus was +in full possession of all the results established by Babylonian +astronomy. +<C><I>Improved estimates of sizes and distances of sun +and moon.</I></C> +<p>4. Hipparchus improved on Aristarchus's calculations of the +sizes and distances of the sun and moon, determining the +apparent diameters more exactly and noting the changes in +them; he made the mean distance of the sun 1,245<I>D</I>, the mean +distance of the moon 33 2/3<I>D</I>, the diameters of the sun and +moon 12 1/3<I>D</I> and 1/3<I>D</I> respectively, where <I>D</I> is the mean +diameter of the earth. +<C><I>Epicycles and eccentrics.</I></C> +<p>5. Hipparchus, in investigating the motions of the sun, moon +and planets, proceeded on the alternative hypotheses of epi- +cycles and eccentrics; he did not invent these hypotheses, +which were already fully understood and discussed by +Apollonius. While the motions of the sun and moon could +with difficulty be accounted for by the simple epicycle and +eccentric hypotheses, Hipparchus found that for the planets it +was necessary to combine the two, i.e. to superadd epicycles to +motion in eccentric circles. +<C><I>Catalogue of stars.</I></C> +<p>6. He compiled a catalogue of fixed stars including 850 or +more such stars; apparently he was the first to state their +positions in terms of coordinates in relation to the ecliptic +(latitude and longitude), and his table distinguished the +apparent sizes of the stars. His work was continued by +Ptolemy, who produced a catalogue of 1,022 stars which, +owing to an error in his solar tables affecting all his longi- +tudes, has by many erroneously been supposed to be a mere +reproduction of Hipparchus's catalogue. That Ptolemy took +many observations himself seems certain.<note>See two papers by Dr. J. L. E. Dreyer in the <I>Monthly Notices of the +Royal Astronomical Society</I>, 1917, pp. 528-39, and 1918, pp. 343-9.</note> +<pb n=256><head>TRIGONOMETRY</head> +<C><I>Improved Instruments.</I></C> +<p>7. He made great improvements in the instruments used for +observations. Among those which he used were an improved +dioptra, a ‘meridian-instrument’ designed for observations in +the meridian only, and a universal instrument (<G>a)strola/bon +o)/rganon</G>) for more general use. He also made a globe on +which he showed the positions of the fixed stars as determined +by him; it appears that he showed a larger number of stars +on his globe than in his catalogue. +<C><I>Geography.</I></C> +<p>In geography Hipparchus wrote a criticism of Eratosthenes, +in great part unfair. He checked Eratosthenes's data by +means of a sort of triangulation; he insisted on the necessity +of applying astronomy to geography, of fixing the position of +places by latitude and longitude, and of determining longitudes +by observations of lunar eclipses. +<p>Outside the domain of astronomy and geography, Hipparchus +wrote a book <I>On things borne down by their weight</I> from +which Simplicius (on Aristotle's <I>De caelo</I>, p. 264 sq.) quotes +two propositions. It is possible, however, that even in this +work Hipparchus may have applied his doctrine to the case of +the heavenly bodies. +<p>In pure mathematics he is said to have considered a problem +in permutations and combinations, the problem of finding the +number of different possible combinations of 10 axioms or +assumptions, which he made to be 103,049 (<I>v.l.</I> 101,049) +or 310,952 according as the axioms were affirmed or denied<note>Plutarch, <I>Quaest. Conviv.</I> viii. 9. 3, 732 F, <I>De Stoicorum repugn.</I> 29. +1047 D.</note>: +it seems impossible to make anything of these figures. When +the <I>Fihrist</I> attributes to him works ‘On the art of algebra, +known by the title of the Rules’ and ‘On the division of num- +bers’, we have no confirmation: Suter suspects some confusion, +in view of the fact that the article immediately following in +the <I>Fihrist</I> is on Diophantus, who also ‘wrote on the art of +algebra’. +<pb n=257><head>HIPPARCHUS</head> +<C>First systematic use of Trigonometry.</C> +<p>We come now to what is the most important from the +point of view of this work, Hipparchus's share in the develop- +ment of trigonometry. Even if he did not invent it, +Hipparchus is the first person of whose systematic use of +trigonometry we have documentary evidence. (1) Theon +of Alexandria says on the <I>Syntaxis</I> of Ptolemy, aà propos of +Ptolemy's Table of Chords in a circle (equivalent to sines), +that Hipparchus, too, wrote a treatise in twelve books on +straight lines (i.e. chords) in a circle, while another in six +books was written by Menelaus.<note>Theon, <I>Comm. on Syntaxis</I>, p. 110, ed. Halma.</note> In the <I>Syntaxis</I> I. 10 +Ptolemy gives the necessary explanations as to the notation +used in his Table. The circumference of the circle is divided +into 360 parts or degrees; the diameter is also divided into +120 parts, and one of such parts is the unit of length in terms +of which the length of each chord is expressed; each part, +whether of the circumference or diameter, is divided into 60 +parts, each of these again into 60, and so on, according to the +system of sexagesimal fractions. Ptolemy then sets out the +minimum number of propositions in plane geometry upon +which the calculation of the chords in the Table is based (<G>dia\ +th=s e)k tw=n grammw=n meqodikh=s au)tw=n susta/sews</G>). The pro- +positions are famous, and it cannot be doubted that Hippar- +chus used a set of propositions of the same kind, though his +exposition probably ran to much greater length. As Ptolemy +definitely set himself to give the necessary propositions in the +shortest form possible, it will be better to give them under +Ptolemy rather than here. (2) Pappus, in speaking of Euclid's +propositions about the inequality of the times which equal arcs +of the zodiac take to rise, observes that ‘Hipparchus in his book +<I>On the rising of the twelve signs of the zodiac</I> shows <I>by means +of numerical calculations</I> (<G>di' a)riqmw=n</G>) that equal arcs of the +semicircle beginning with Cancer which set in times having +a certain relation to one another do not everywhere show the +same relation between the times in which they rise’,<note>Pappus, vi, p. 600. 9-13.</note> and so +on. We have seen that Euclid, Autolycus, and even Theo- +dosius could only prove that the said times are greater or less +<pb n=258><head>TRIGONOMETRY</head> +in relation to one another; they could not calculate the actual +times. As Hipparchus proved corresponding propositions by +means of <I>numbers</I>, we can only conclude that he used proposi- +tions in spherical trigonometry, calculating arcs from others +which are given, by means of tables. (3) In the only work +of his which survives, the <I>Commentary on the Phaenomena +of Eudoxus and Aratus</I> (an early work anterior to the +discovery of the precession of the equinoxes), Hipparchus +states that (presumably in the latitude of Rhodes) a star which +lies 27 1/3° north of the equator describes above the horizon an +arc containing 3 minutes less than 15/24ths of the whole +circle<note>Ed. Manitius, pp. 148-50.</note>; then, after some more inferences, he says, ‘For each +of the aforesaid facts is proved <I>by means of lines</I> (<G>dia\ tw=n +grammw=n</G>) in the general treatises on these matters compiled +by me’. In other places<note><I>Ib.</I>, pp. 128. 5, 148. 20.</note> of the <I>Commentary</I> he alludes to +a work <I>On simultaneous risings</I> (<G>ta\ peri\ tw=n sunanatolw=n</G>), +and in II.4. 2 he says he will state summarily, about each of +the fixed stars, along with what sign of the zodiac it rises and +sets and from which degree to which degree of each sign it +rises or sets in the regions about Greece or wherever the +longest day is 14 1/2 equinoctial hours, adding that he has given +special proofs in another work designed so that it is possible +in practically every place in the inhabited earth to follow +the differences between the concurrent risings and settings.<note><I>Ib.</I>, pp. 182. 19-184. 5.</note> +Where Hipparchus speaks of proofs ‘by means of lines’, he +does not mean a merely graphical method, by construction +only, but theoretical determination by geometry, followed by +calculation, just as Ptolemy uses the expression <G>e)k tw=n gram- +mw=n</G> of his calculation of chords and the expressions <G>sfairikai\ +dei/xeis</G> and <G>grammikai\ dei/xeis</G> of the fundamental proposition +in spherical trigonometry (Menelaus's theorem applied to the +sphere) and its various applications to particular cases. It +is significant that in the <I>Syntaxis</I> VIII. 5, where Ptolemy +applies the proposition to the very problem of finding the +times of concurrent rising, culmination and setting of the +fixed stars, he says that the times can be obtained ‘by lines +only’ (<G>dia\ mo/nwn tw=n grammw=n</G>).<note><I>Syntaxis</I>, vol. ii, p. 193.</note> Hence we may be certain +that, in the other books of his own to which Hipparchus refers +<pb n=259><head>HIPPARCHUS</head> +in his <I>Commentary</I>, he used the formulae of spherical trigono- +metry to get his results. In the particular case where it is +required to find the time in which a star of 27 1/3° northern +declination describes, in the latitude of Rhodes, the portion of +its arc above the horizon, Hipparchus must have used the +equivalent of the formula in the solution of a right-angled +spherical triangle, tan <I>b</I> = cos <I>A</I> tan <I>c</I>, where <I>C</I> is the right +angle. Whether, like Ptolemy, Hipparchus obtained the +formulae, such as this one, which he used from different +applications of the one general theorem (Menelaus's theorem) +it is not possible to say. There was of course no difficulty +in calculating the tangent or other trigonometrical function +of an angle if only a table of sines was given; for Hippar- +chus and Ptolemy were both aware of the fact expressed by +sin<SUP>2</SUP> <G>a</G> + cos<SUP>2</SUP> <G>a</G> = 1 or, as they would have written it, +<MATH>(crd. 2<G>a</G>)<SUP>2</SUP> + {crd. (180° - 2<G>a</G>)}<SUP>2</SUP> = 4<I>r</I><SUP>2</SUP></MATH>, +where (crd. 2<G>a</G>) means the chord subtending an arc 2<G>a</G>, and <G>g</G> +is the radius, of the circle of reference. +<C>Table of Chords.</C> +<p>We have no details of Hipparchus's Table of Chords suffi- +cient to enable us to compare it with Ptolemy's, which goes +by half-degrees, beginning with angles of 1/2°, 1°, 1 1/2°, and so +on. But Heron<note>Heron, <I>Metrica</I>, i. 22, 24, pp. 58. 19 and 62. 17.</note> in his <I>Metrica</I> says that ‘it is proved in the +books about chords in a circle’ that, if <I>a</I><SUB>9</SUB> and <I>a</I><SUB>11</SUB> are the sides +of a regular enneagon (9-sided figure) and hendecagon (11-sided +figure) inscribed in a circle of diameter <I>d</I>, then (1) <I>a</I><SUB>9</SUB> = 1/3<I>d</I>, +(2) <I>a</I><SUB>11</SUB> = 7/25<I>d</I> very nearly, which means that sin 20° was +taken as equal to 0.3333 . . . (Ptolemy's table makes it +1/60(20 + 31/60 + 16 1/2/60<SUP>2</SUP>), so that the first approximation is 1/3), and +sin 1/11.180° or sin 16° 21′ 49″ was made equal to 0.28 (this cor- +responds to the chord subtending an angle of 32° 43′ 38″, nearly +half-way between 32 1/2° and 33°, and the mean between the two +chords subtending the latter angles gives 1/60(16 + 54/60 + 55/60<SUP>2</SUP>) as +the required sine, while 1/60(16 9/10) = 169/600, which only differs +<pb n=260><head>TRIGONOMETRY</head> +by 1/600 from 168/600 or 7/25, Heron's figure). There is little doubt +that it is to Hipparchus's work that Heron refers, though the +author is not mentioned. +<p>While for our knowledge of Hipparchus's trigonometry we +have to rely for the most part upon what we can infer from +Ptolemy, we fortunately possess an original source of infor- +mation about Greek trigonometry in its highest development +in the <I>Sphaerica</I> of Menelaus. +<p>The date of MENELAUS of Alexandria is roughly indi- +cated by the fact that Ptolemy quotes an observation of +his made in the first year of Trajan's reign (A.D. 98). He +was therefore a contemporary of Plutarch, who in fact +represents him as being present at the dialogue <I>De facie in +orbe lunae</I>, where (chap. 17) Lucius apologizes to Menelaus ‘the +mathematician’ for questioning the fundamental proposition +in optics that the angles of incidence and reflection are equal. +<p>He wrote a variety of treatises other than the <I>Sphaerica.</I> +We have seen that Theon mentions his work on <I>Chords in a +Circle</I> in six Books. Pappus says that he wrote a treatise +(<G>pragmatei/a</G>) on the setting (or perhaps only rising) of +different arcs of the zodiac.<note>Pappus, vi, pp. 600-2.</note> Proclus quotes an alternative +proof by him of Eucl. I. 25, which is direct instead of by +<I>reductio ad absurdum</I>,<note>Proclus on Eucl. I, pp. 345. 14-346. 11.</note> and he would seem to have avoided +the latter kind of proof throughout. Again, Pappus, speaking +of the many complicated curves ‘discovered by Demetrius of +Alexandria (in his “Linear considerations”) and by Philon +of Tyana as the result of interweaving plectoids and other +surfaces of all kinds’, says that one curve in particular was +investigated by Menelaus and called by him ‘paradoxical’ +(<G>para/doxos</G>)<note>Pappus, iv, p. 270. 25.</note>; the nature of this curve can only be conjectured +(see below). +<p>But Arabian tradition refers to other works by Menelaus, +(1) <I>Elements of Geometry</I>, edited by Thābit b. Qurra, in three +Books, (2) a Book on triangles, and (3) a work the title of +which is translated by Wenrich <I>de cognitione quantitatis +discretac corporum permixtorum.</I> Light is thrown on this +last title by one al-Chāzinī who (about A.D. 1121) wrote a +<pb n=261><head>MENELAUS OF ALEXANDRIA</head> +treatise about the hydrostatic balance, i.e. about the deter- +mination of the specific gravity of homogeneous or mixed +bodies, in the course of which he mentions Archimedes and +Menelaus (among others) as authorities on the subject; hence +the treatise (3) must have been a book on hydrostatics dis- +cussing such problems as that of the crown solved by Archi- +medes. The alternative proof of Eucl. I. 25 quoted by +Proclus might have come either from the <I>Elements of Geometry</I> +or the Book on triangles. With regard to the geometry, the +‘liber trium fratrum’ (written by three sons of Mūsā b. Shākir +in the ninth century) says that it contained a solution of the +duplication of the cube, which is none other than that of +Archytas. The solution of Archytas having employed the +intersection of a tore and a cylinder (with a cone as well), +there would, on the assumption that Menelaus reproduced the +solution, be a certain appropriateness in the suggestion of +Tannery<note>Tannery, <I>Mémoires scientifiques</I>, ii, p. 17.</note> that the curve which Menelaus called the <G>para/doxos +grammh/</G> was in reality the curve of double curvature, known +by the name of Viviani, which is the intersection of a sphere +with a cylinder touching it internally and having for its +diameter the radius of the sphere. This curve is a particular +case of Eudoxus's <I>hippopede</I>, and it has the property that the +portion left outside the curve of the surface of the hemisphere +on which it lies is equal to the square on the diameter of the +sphere; the fact of the said area being squareable would +justify the application of the word <G>para/doxos</G> to the curve, +and the quadrature itself would not probably be beyond the +powers of the Greek mathematicians, as witness Pappus's +determination of the area cut off between a complete turn of +a certain spiral on a sphere and the great circle touching it at +the origin.<note>Pappus, iv, pp. 264-8.</note> +<C>The <I>Sphaerica</I> of Menelaus.</C> +<p>This treatise in three Books is fortunately preserved in +the Arabic, and although the extant versions differ con- +siderably in form, the substance is beyond doubt genuine; +the original translator was apparently Ishāq b. Hunain +(died A. D. 910). There have been two editions, (1) a Latin +<pb n=262><head>TRIGONOMETRY</head> +translation by Maurolycus (Messina, 1558) and (2) Halley's +edition (Oxford, 1758). The former is unserviceable because +Maurolycus's manuscript was very imperfect, and, besides +trying to correct and restore the propositions, he added +several of his own. Halley seems to have made a free +translation of the Hebrew version of the work by Jacob b. +Machir (about 1273), although he consulted Arabic manuscripts +to some extent, following them, e.g., in dividing the work into +three Books instead of two. But an earlier version direct +from the Arabic is available in manuscripts of the thirteenth +to fifteenth centuries at Paris and elsewhere; this version is +without doubt that made by the famous translator Gherard +of Cremona (1114-87). With the help of Halley's edition, +Gherard's translation, and a Leyden manuscript (930) of +the redaction of the work by Abū-Nasr-Mansūr made in +A.D. 1007-8, Björnbo has succeeded in presenting an adequate +reproduction of the contents of the <I>Sphaerica.</I><note>Björnbo, <I>Studien über Menelaos' Sphärik</I> (Abhandlungen zur Gesch. d. +math. Wissenschaften, Heft xiv. 1902).</note> +<C>Book I.</C> +<p>In this Book for the first time we have the conception and +definition of a <I>spherical triangle.</I> Menelaus does not trouble +to give the usual definitions of points and circles related to +the sphere, e.g. pole, great circle, small circle, but begins with +that of a spherical triangle as ‘the area included by arcs of +great circles on the surface of a sphere’, subject to the restric- +tion (Def. 2) that each of the sides or legs of the triangle is an +arc less than a semicircle. The angles of the triangle are the +angles contained by the arcs of great circles on the sphere +(Def. 3), and one such angle is equal to or greater than another +according as the planes containing the arcs forming the first +angle are inclined at the same angle as, or a greater angle +than, the planes of the arcs forming the other (Defs. 4, 5). +The angle is a right angle if the planes of the arcs are at right +angles (Def. 6). Pappus tells us that Menelaus in his <I>Sphaerica</I> +calls the figure in question (the spherical triangle) a ‘three- +side’ (<G>tri/pleuron</G>)<note>Pappus, vi, p. 476. 16.</note>; the word <I>triangle</I> (<G>tri/gwnon</G>) was of course +<pb n=263><head>MENELAUS'S <I>SPHAERICA</I></head> +already appropriated for the plane triangle. We should gather +from this, as well as from the restriction of the definitions to +the spherical triangle and its parts, that the discussion of the +spherical triangle as such was probably new; and if the pre- +face in the Arabic version addressed to a prince and beginning +with the words, ‘O prince! I have discovered an excellent +method of proof . . .’ is genuine, we have confirmatory evidence +in the writer's own claim. +<p>Menelaus's object, so far as Book I is concerned, seems to +have been to give the main propositions about spherical +triangles corresponding to Euclid's propositions about plane +triangles. At the same time he does not restrict himself to +Euclid's methods of proof even where they could be adapted +to the case of the sphere; he avoids the form of proof by +<I>reductio ad absurdum</I>, but, subject to this, he prefers the +easiest proofs. In some respects his treatment is more com- +plete than Euclid's treatment of the analogous plane cases. +In the congruence-theorems, for example, we have I. 4a +corresponding to Eucl. I. 4, I. 4b to Eucl. I. 8, I. 14, 16 to +Eucl. I. 26a, b; but Menelaus includes (I. 13) what we know +as the ‘ambiguous case’, which is enunciated on the lines of +Eucl. VI. 7. I. 12 is a particular case of I. 16. Menelaus +includes also the further case which has no analogue in plane +triangles, that in which the three angles of one triangle are +severally equal to the three angles of the other (I. 17). He +makes, moreover, no distinction between the congruent and +the symmetrical, regarding both as covered by congruent. I. 1 +is a problem, to construct a spherical angle equal to a given +spherical angle, introduced only as a lemma because required +in later propositions. I. 2, 3 are the propositions about +isosceles triangles corresponding to Eucl. I. 5, 6; Eucl. I. 18, 19 +(greater side opposite greater angle and vice versa) have their +analogues in I. 7, 9, and Eucl. I. 24, 25 (two sides respectively +equal and included angle, or third side, in one triangle greater +than included angle, or third side, in the other) in I. 8. I. 5 +(two sides of a triangle together greater than the third) corre- +sponds to Eucl. I. 20. There is yet a further group of proposi- +tions comparing parts of spherical triangles, I. 6, 18, 19, where +I. 6 (corresponding to Eucl. I. 21) is deduced from I. 5, just as +the first part of Eucl. I. 21 is deduced from Eucl. I. 20. +<pb n=264><head>TRIGONOMETRY</head> +<p>Eucl. I. 16, 32 are not true of spherical triangles, and +Menelaus has therefore the corresponding but different pro- +positions. I. 10 proves that, with the usual notation <I>a, b, c, +A, B, C,</I> for the sides and opposite angles of a spherical +triangle, the exterior angle at <I>C,</I> or <MATH>180°-<I>C,</I> < = or > <I>A</I></MATH> +according as <MATH><I>c</I>+<I>a</I> > = or < 180°</MATH>, and vice versa. The proof +of this and the next proposition shall be given as specimens. +<p>In the triangle <I>ABC</I> suppose that <MATH><I>c</I>+<I>a</I> > = or < 180°</MATH>; let +<I>D</I> be the pole opposite to <I>A.</I> +<p>Then, according as <MATH><I>c</I>+<I>a</I> > = or < 180°, <I>BC</I> > = or < <I>BD</I> +(since <I>AD</I> = 180°)</MATH>, +and therefore <MATH>∠<I>D</I> > = or < ∠<I>BCD</I> (= 180°-<I>C</I>)</MATH>, [I. 9] +i.e. <MATH>(since ∠<I>D</I> = ∠<I>A</I>) 180°-<I>C</I> < = or > <I>A</I></MATH>. +<p>Menelaus takes the converse for granted. +<p>As a consequence of this, I. 11 proves that <MATH><I>A</I>+<I>B</I>+<I>C</I> > 180°</MATH>. +<p>Take the same triangle <I>ABC,</I> with the pole <I>D</I> opposite +<FIG> +to <I>A,</I> and from <I>B</I> draw the great circle <I>BE</I> such that +<MATH>∠<I>DBE</I> = ∠<I>BDE</I></MATH>. +<p>Then <MATH><I>CE</I>+<I>EB</I> = <I>CD</I> < 180°</MATH>, so that, by the preceding +proposition, the exterior angle <I>ACB</I> to the triangle <I>BCE</I> is +greater than ∠<I>CBE,</I> +i.e. <MATH><I>C</I> > ∠<I>CBE</I></MATH>. +<p>Add <I>A</I> or <I>D</I> (= ∠<I>EBD</I>) to the unequals; +therefore <MATH><I>C</I>+<I>A</I> > ∠<I>CBD</I></MATH>, +whence <MATH><I>A</I>+<I>B</I>+<I>C</I> > ∠<I>CBD</I>+<I>B</I> or 180°</MATH>. +<p>After two lemmas I. 21, 22 we have some propositions introducing +<I>M, N, P</I> the middle points of <I>a, b, c</I> respectively. I. 23 +proves, e.g., that the arc <I>MN</I> of a great circle > 1/2<I>c,</I> and I. 20 +that <MATH><I>AM</I> < = or > 1/2<I>a</I></MATH> according as <MATH><I>A</I> > = or < (<I>B</I>+<I>C</I>)</MATH>. The +last group of propositions, 26-35, relate to the figure formed +<pb n=265><head>MENELAUS'S <I>SPHAERICA</I></head> +by the triangle <I>ABC</I> with great circles drawn through <I>B</I> to +meet <I>AC</I> (between <I>A</I> and <I>C</I>) in <I>D, E</I> respectively, and the +case where <I>D</I> and <I>E</I> coincide, and they prove different results +arising from different relations between <I>a</I> and <I>c</I> (<I>a</I> > <I>c</I>), com- +bined with the equality of <I>AD</I> and <I>EC</I> (or <I>DC</I>), of the angles +<I>ABD</I> and <I>EBC</I> (or <I>DBC</I>), or of <I>a</I>+<I>c</I> and <I>BD</I>+<I>BE</I> (or 2 <I>BD</I>) +respectively, according as <MATH><I>a</I>+<I>c</I> < = or > 180°</MATH>. +<p>Book II has practically no interest for us. The object of it +is to establish certain propositions, of astronomical interest +only, which are nothing more than generalizations or exten- +sions of propositions in Theodosius's <I>Sphaerica,</I> Book III. +Thus Theodosius III. 5, 6, 9 are included in Menelaus II. 10, +Theodosius III. 7-8 in Menelaus II. 12, while Menelaus II. 11 +is an extension of Theodosius III. 13. The proofs are quite +different from those of Theodosius, which are generally very +long-winded. +<C>Book III. Trigonometry.</C> +<p>It will have been noticed that, while Book I of Menelaus +gives the geometry of the spherical triangle, neither Book I +nor Book II contains any trigonometry. This is reserved for +Book III. As I shall throughout express the various results +obtained in terms of the trigonometrical ratios, sine, cosine, +tangent, it is necessary to explain once for all that the Greeks +did not use this terminology, but, instead of sines, they used +<FIG> +the chords subtended by arcs of a +circle. In the accompanying figure +let the arc <I>AD</I> of a circle subtend an +angle <G>a</G> at the centre <I>O.</I> Draw <I>AM</I> +perpendicular to <I>OD,</I> and produce it +to meet the circle again in <I>A</I>′. Then +sin <MATH><G>a</G> = <I>AM/AO</I></MATH>, and <I>AM</I> is 1/2<I>AA</I>′ +or half the chord subtended by an +angle 2<G>a</G> at the centre, which may +shortly be denoted by 1/2(crd. 2<G>a</G>). +Since Ptolemy expresses the chords as so many 120th parts of +the diameter of the circle, while <MATH><I>AM/AO</I> = <I>AA</I>′/2<I>AO</I></MATH>, it +follows that sin <G>a</G> and 1/2(crd. 2<G>a</G>) are equivalent. Cos <G>a</G> is +of course sin (90°-<G>a</G>) and is therefore equivalent to 1/2 crd. +(180°-2<G>a</G>). +<pb n=266><head>TRIGONOMETRY</head> +<C>(<G>a</G>) <I>‘Menelaus's theorem’ for the sphere.</I></C> +<p>The first proposition of Book III is the famous ‘Menelaus's +theorem’ with reference to a spherical triangle and any trans- +versal (great circle) cutting the sides of a triangle, produced +if necessary. Menelaus does not, however, use a spherical +triangle in his enunciation, but enunciates the proposition in +terms of intersecting great circles. ‘Between two arcs <I>ADB, +AEC</I> of great circles are two other arcs of great circles <I>DFC</I> +and <I>BFE</I> which intersect them and also intersect each other +in <I>F.</I> All the arcs are less than a semicircle. It is required +to prove that +<MATH>sin <I>CE</I>/sin <I>EA</I> = sin <I>CF</I>/sin <I>FD</I>.sin <I>DB</I>/sin <I>BA</I></MATH>.’ +<p>It appears that Menelaus gave three or four cases, sufficient +to prove the theorem completely. The proof depends on two +simple propositions which Menelaus assumes without proof; +the proof of them is given by Ptolemy. +<p>(1) In the figure on the last page, if <I>OD</I> be a radius cutting +a chord <I>AB</I> in <I>C,</I> then +<MATH><I>AC</I>:<I>CB</I> = sin <I>AD</I>:sin <I>DB</I></MATH>. +<p>For draw <I>AM, BN</I> perpendicular to <I>OD.</I> Then +<MATH><I>AC</I>:<I>CB</I> = <I>AM</I>:<I>BN</I> += 1/2 (crd. 2<I>AD</I>):1/2 (crd. 2<I>DB</I>) += sin <I>AD</I>:sin <I>DB</I></MATH>. +<p>(2) If <I>AB</I> meet the radius <I>OC</I> produced in <I>T,</I> then +<MATH><I>AT</I>:<I>BT</I> = sin <I>AC</I>:sin <I>BC</I></MATH>. +<FIG> +<pb n=267><head>MENELAUS'S <I>SPHAERICA</I></head> +<p>For, if <I>AM, BN</I> are perpendicular to <I>OC,</I> we have, as before, +<MATH><I>AT</I>:<I>TB</I> = <I>AM</I>:<I>BN</I> += 1/2 (crd. 2 <I>AC</I>):1/2(crd. 2<I>BC</I>) += sin <I>AC</I>:sin <I>BC</I></MATH>. +<p>Now let the arcs of great circles <I>ADB, AEC</I> be cut by the +arcs of great circles <I>DFC, BFE</I> which themselves meet in <I>F.</I> +<p>Let <I>G</I> be the centre of the sphere and join <I>GB, GF, GE, AD.</I> +<p>Then the straight lines <I>AD, GB,</I> being in one plane, are +either parallel or not parallel. If they are not parallel, they +will meet either in the direction of <I>D, B</I> or of <I>A, G.</I> +<p>Let <I>AD, GB</I> meet in <I>T.</I> +<p>Draw the straight lines <I>AKC, DLC</I> meeting <I>GE, GF</I> in <I>K, L</I> +respectively. +<p>Then <I>K, L, T</I> must lie on a straight line, namely the straight +line which is the section of the planes determined by the arc +<I>EFB</I> and by the triangle <I>ACD.</I><note>So Ptolemy. In other words, since the straight lines <I>GB, GE, GF,</I> +which are in one plane, respectively intersect the straight lines <I>AD, AC, +CD</I> which are also in one plane, the points of intersection <I>T, K, L</I> are in +both planes, and therefore lie on the straight line in which the planes +intersect.</note> +<FIG> +<p>Thus we have two straight lines <I>AC, AT</I> cut by the two +straight lines <I>CD, TK</I> which themselves intersect in <I>L.</I> +<p>Therefore, by Menelaus's proposition in plane geometry, +<MATH><I>CK</I>/<I>KA</I> = <I>CL</I>/<I>LD</I>.<I>DT</I>/<I>TA</I></MATH>. +<pb n=268><head>TRIGONOMETRY</head> +<p>But, by the propositions proved above, +<MATH><I>CK</I>/<I>KA</I> = sin <I>CE</I>/sin <I>EA</I>, <I>CL</I>/<I>LD</I> = sin <I>CF</I>/sin <I>FD</I></MATH>, and <MATH><I>DT</I>/<I>TA</I> = sin <I>DB</I>/sin <I>BA</I></MATH>; +therefore, by substitution, we have +<MATH>sin <I>CE</I>/sin <I>EA</I> = sin <I>CF</I>/sin <I>FD</I>.sin <I>DB</I>/sin <I>BA</I></MATH>. +<p>Menelaus apparently also gave the proof for the cases in +which <I>AD, GB</I> meet towards <I>A, G,</I> and in which <I>AD, GB</I> are +parallel respectively, and also proved that in like manner, in +the above figure, +<MATH>sin <I>CA</I>/sin <I>AE</I> = sin <I>CD</I>/sin <I>DF</I>.sin <I>FB</I>/sin <I>BE</I></MATH> +(the triangle cut by the transversal being here <I>CFE</I> instead of +<I>ADC</I>). Ptolemy<note>Ptolemy, <I>Syntaxis,</I> i. 13, vol. i, p. 76.</note> gives the proof of the above case only, and +dismisses the last-mentioned result with a ‘similarly’. +<C>(<G>b</G>) <I>Deductions from Menelaus's Theorem.</I></C> +<p>III. 2 proves, by means of I. 14, 10 and III. 1, that, if <I>ABC, +A</I>′<I>B</I>′<I>C</I>′ be two spherical triangles in which <I>A</I> = <I>A</I>′, and <I>C, C</I>′ +are either equal or supplementary, <MATH>sin <I>c</I>/sin <I>a</I> = sin <I>c</I>′/sin <I>a</I>′</MATH> +and conversely. The particular case in which <I>C, C</I>′ are right +angles gives what was afterwards known as the ‘regula +quattuor quantitatum’ and was fundamental in Arabian +trigonometry.<note>See Braunmühl, <I>Gesch. der Trig.</I> i, pp. 17, 47, 58-60, 127-9.</note> A similar association attaches to the result of +III. 3, which is the so-called ‘tangent’ or ‘shadow-rule’ of the +Arabs. If <I>ABC, A</I>′<I>B</I>′<I>C</I>′ be triangles right-angled at <I>A, A</I>′, and +<I>C, C</I>′ are equal and both either > or < 90°, and if <I>P, P</I>′ be +the poles of <I>AC, A</I>′<I>C</I>′, then +<MATH>sin <I>AB</I>/sin <I>AC</I> = sin <I>A</I>′<I>B</I>′/sin <I>A</I>′<I>C</I>′.sin <I>BP</I>/sin <I>B</I>′<I>P</I>′</MATH>. +<p>Apply the triangles so that <I>C</I>′ falls on <I>C, C</I>′<I>B</I>′ on <I>CB</I> as <I>CE,</I> +and <I>C</I>′<I>A</I>′ on <I>CA</I> as <I>CD;</I> then the result follows directly from +III. 1. Since <MATH>sin <I>BP</I> = cos <I>AB</I></MATH>, and <MATH>sin <I>B</I>′<I>P</I>′ = cos <I>A</I>′<I>B</I>′</MATH>, the +result becomes +<MATH>sin <I>CA</I>/sin <I>C</I>′<I>A</I>′ = tan <I>AB</I>/tan <I>A</I>′<I>B</I>′</MATH>, +which is the ‘tangent-rule’ of the Arabs.<note>Cf. Braunmühl, <I>op. cit.</I> i, pp. 17-18, 58, 67-9, &c.</note> +<pb n=269><head>MENELAUS'S <I>SPHAERICA</I></head> +<p>It follows at once (Prop. 4) that, if <I>AM, A</I>′<I>M</I>′ are great +circles drawn perpendicular to the bases <I>BC, B</I>′<I>C</I>′ of two +spherical triangles <I>ABC, A</I>′<I>B</I>′<I>C</I>′ in which <I>B</I> = <I>B</I>′, <I>C</I> = <I>C</I>′, +<MATH>sin <I>BM</I>/sin <I>B</I>′<I>M</I>′ = sin <I>MC</I>/sin <I>M</I>′<I>C</I>′ (since both are equal to tan <I>AM</I>/tan <I>A</I>′<I>M</I>′</MATH>. +<p>III. 5 proves that, if there are two spherical triangles <I>ABC,</I> +<FIG> +<I>A</I>′<I>B</I>′<I>C</I>′ right-angled at <I>A, A</I>′ and such that <I>C</I> = <I>C</I>′, while <I>b</I> +and <I>b</I>′ are less than 90°, +<MATH>sin (<I>a</I>+<I>b</I>)/sin (<I>a</I>-<I>b</I>) = sin (<I>a</I>′+<I>b</I>′)/sin (<I>a</I>′-<I>b</I>′)</MATH>, +from which we may deduce<note>Braunmühl, <I>op. cit.</I> i, p. 18; Björnbo, p. 96.</note> the formula +<MATH>sin (<I>a</I>+<I>b</I>)/sin (<I>a</I>-<I>b</I>) = (1+cos <I>C</I>)/(1-cos <I>C</I>)</MATH>, +which is equivalent to tan <I>b</I> = tan <I>a</I> cos <I>C.</I> +<C>(<G>g</G>) <I>Anharmonic property of four great circles through +one point.</I></C> +<p>But more important than the above result is the fact that +<FIG> +the proof assumes as known the anhar- +monic property of four great circles +drawn from a point on a sphere in rela- +tion to any great circle intersecting them +all, viz. that, if <I>ABCD, A</I>′<I>B</I>′<I>C</I>′<I>D</I>′ be two +transversals, +<MATH>sin <I>AD</I>/sin <I>DC</I>.sin <I>BC</I>/sin <I>AB</I> = sin <I>A</I>′<I>D</I>′/sin <I>D</I>′<I>C</I>′.sin <I>B</I>′<I>C</I>′/sin <I>A</I>′<I>B</I>′</MATH>. +<pb n=270><head>TRIGONOMETRY</head> +<p>It follows that this proposition was known before Mene- +laus's time. It is most easily proved by means of ‘Menelaus's +Theorem’, III. 1, or alternatively it may be deduced for the +sphere from the corresponding proposition in plane geometry, +just as Menelaus's theorem is transferred by him from the +plane to the sphere in III. 1. We may therefore fairly con- +clude that both the anharmonic property and Menelaus's +theorem with reference to the sphere were already included +in some earlier text-book; and, as Ptolemy, who built so much +upon Hipparchus, deduces many of the trigonometrical +formulae which he uses from the one theorem (III. 1) of +Menelaus, it seems probable enough that both theorems were +known to Hipparchus. The corresponding plane theorems +appear in Pappus among his lemmas to Euclid's <I>Porisms,</I><note>Pappus, vii, pp. 870-2, 874.</note> and +there is therefore every probability that they were assumed +by Euclid as known. +<C>(<G>d</G>) <I>Propositions anulogous to Eucl. VI. 3.</I></C> +<p>Two theorems following, III. 6, 8, have their analogy in +Eucl. VI. 3. In III. 6 the vertical angle <I>A</I> of a spherical +triangle is bisected by an arc of a great circle meeting <I>BC</I> in +<I>D,</I> and it is proved that <MATH>sin <I>BD</I>/sin <I>DC</I> = sin <I>BA</I>/sin <I>AC</I></MATH>; +in III. 8 we have the vertical angle bisected both internally +and externally by arcs of great circles meeting <I>BC</I> in <I>D</I> and +<I>E,</I> and the proposition proves the harmonic property +<MATH>sin <I>BE</I>/sin <I>EC</I> = sin <I>BD</I>/sin <I>DC</I></MATH>. +<p>III. 7 is to the effect that, if arcs of great circles be drawn +through <I>B</I> to meet the opposite side <I>AC</I> of a spherical triangle +in <I>D, E</I> so that <MATH>∠<I>ABD</I> = ∠<I>EBC</I></MATH>, then +<MATH>(sin <I>EA</I>.sin <I>AD</I>)/(sin <I>DC</I>.sin <I>CE</I>) = sin<SUP>2</SUP> <I>AB</I>/sin<SUP>2</SUP> <I>BC</I></MATH>. +As this is analogous to plane propositions given by Pappus as +lemmas to different works included in the <I>Treasury of +Analysis,</I> it is clear that these works were familiar to +Menelaus. +<pb n=271><head>MENELAUS'S <I>SPHAERICA</I></head> +<p>III. 9 and III. 10 show, for a spherical triangle, that (1) the +great circles bisecting the three angles, (2) the great circles +through the angular points meeting the opposite sides at +right angles meet in a point. +<p>The remaining propositions, III. 11-15, return to the same +sort of astronomical problem as those dealt with in Euclid's +<I>Phaenomena,</I> Theodosius's <I>Sphaerica</I> and Book II of Mene- +laus's own work. Props. 11-14 amount to theorems in +spherical trigonometry such as the following. +<p>Given arcs <G>a</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB>, <G>a</G><SUB>3</SUB>, <G>a</G><SUB>4</SUB>, <G>b</G><SUB>1</SUB>, <G>b</G><SUB>2</SUB>, <G>b</G><SUB>3</SUB>, <G>b</G><SUB>4</SUB>, such that +<MATH>90°≥<G>a</G><SUB>1</SUB> > <G>a</G><SUB>2</SUB> > <G>a</G><SUB>3</SUB> > <G>a</G><SUB>4</SUB>, +90° > <G>b</G><SUB>1</SUB> > <G>b</G><SUB>2</SUB> > <G>b</G><SUB>3</SUB> > <G>b</G><SUB>4</SUB></MATH>, +and also <MATH><G>a</G><SUB>1</SUB> > <G>b</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB> > <G>b</G><SUB>2</SUB>, <G>a</G><SUB>3</SUB> > <G>b</G><SUB>3</SUB>, <G>a</G><SUB>4</SUB> > <G>b</G><SUB>4</SUB></MATH>, +<p>(1) If <MATH>sin <G>a</G><SUB>1</SUB>:sin <G>a</G><SUB>2</SUB>:sin <G>a</G><SUB>3</SUB>:sin <G>a</G><SUB>4</SUB> = sin <G>b</G><SUB>1</SUB>:sin <G>b</G><SUB>2</SUB>:sin <G>b</G><SUB>3</SUB>:sin <G>b</G><SUB>4</SUB></MATH>, +then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>3</SUB>-<G>a</G><SUB>4</SUB>)>(<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>3</SUB>-<G>b</G><SUB>4</SUB>)</MATH>. +<p>(2) If <MATH>sin (<G>a</G><SUB>1</SUB>+<G>b</G><SUB>1</SUB>)/sin (<G>a</G><SUB>1</SUB>-<G>b</G><SUB>1</SUB>) = sin (<G>a</G><SUB>2</SUB>+<G>b</G><SUB>2</SUB>)/sin (<G>a</G><SUB>2</SUB>-<G>b</G><SUB>2</SUB>) = sin +(<G>a</G><SUB>3</SUB>+<G>b</G><SUB>3</SUB>) += sin (<G>a</G><SUB>4</SUB>+<G>b</G><SUB>4</SUB>)/sin (<G>a</G><SUB>4</SUB>-<G>b</G><SUB>4</SUB>)</MATH>, +then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>3</SUB>-<G>a</G><SUB>4</SUB>) > (<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>3</SUB>-<G>b</G><SUB>4</SUB>)</MATH>. +<p>(3) If <MATH>sin (<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/sin (<G>a</G><SUB>3</SUB>-<G>a</G><SUB>4</SUB>) < sin (<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/sin (<G>b</G><SUB>3</SUB>-<G>b</G><SUB>4</SUB>)</MATH> +then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>3</SUB>-<G>a</G><SUB>4</SUB>) < (<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>3</SUB>-<G>b</G><SUB>4</SUB>)</MATH>. +<p>Again, given three series of three arcs such that +<MATH><G>a</G><SUB>1</SUB> > <G>a</G><SUB>2</SUB> > <G>a</G><SUB>3</SUB>, <G>b</G><SUB>1</SUB> > <G>b</G><SUB>2</SUB> > <G>b</G><SUB>3</SUB>, 90°><G>g</G><SUB>1</SUB> > <G>g</G><SUB>2</SUB> > <G>g</G><SUB>3</SUB></MATH>, +and <MATH>sin (<G>a</G><SUB>1</SUB>-<G>g</G><SUB>1</SUB>):sin (<G>a</G><SUB>2</SUB>-<G>g</G><SUB>2</SUB>):sin (<G>a</G><SUB>3</SUB>-<G>g</G><SUB>3</SUB>) += sin (<G>b</G><SUB>1</SUB>-<G>g</G><SUB>1</SUB>):sin (<G>b</G><SUB>2</SUB>-<G>g</G><SUB>2</SUB>):sin (<G>b</G><SUB>3</SUB>-<G>g</G><SUB>3</SUB>) += sin <G>g</G><SUB>1</SUB>:sin <G>g</G><SUB>2</SUB>:sin <G>g</G><SUB>3</SUB></MATH> +<pb n=272><head>TRIGONOMETRY</head> +<p>(1) If <MATH><G>a</G><SUB>1</SUB> > <G>b</G><SUB>1</SUB> > 2<G>g</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB> > <G>b</G><SUB>2</SUB> > 2<G>g</G><SUB>2</SUB>, <G>a</G><SUB>3</SUB> > <G>b</G><SUB>3</SUB> > 2<G>g</G><SUB>3</SUB></MATH>, +then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>2</SUB>-<G>a</G><SUB>3</SUB>)>(<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>2</SUB>-<G>b</G><SUB>3</SUB>)</MATH>; and +<p>(2) If <MATH><G>b</G><SUB>1</SUB> < <G>a</G><SUB>1</SUB> < <G>g</G><SUB>1</SUB>, <G>b</G><SUB>2</SUB> < <G>a</G><SUB>2</SUB> < <G>g</G><SUB>2</SUB>, <G>b</G><SUB>3</SUB> < <G>a</G><SUB>3</SUB> < <G>g</G><SUB>3</SUB></MATH>, +then <MATH>(<G>a</G><SUB>1</SUB>-<G>a</G><SUB>2</SUB>)/(<G>a</G><SUB>2</SUB>-<G>a</G><SUB>3</SUB>)<(<G>b</G><SUB>1</SUB>-<G>b</G><SUB>2</SUB>)/(<G>b</G><SUB>2</SUB>-<G>b</G><SUB>3</SUB>)</MATH>. +<p>III. 15, the last proposition, is in four parts. The first part +is the proposition corresponding to Theodosius III. 11 above +alluded to. Let <I>BA, BC</I> be two quadrants of great circles +(in which we easily recognize the equator and the ecliptic), +<I>P</I> the pole of the former, <I>PA</I><SUB>1</SUB>, <I>PA</I><SUB>3</SUB> quadrants of great circles +meeting the other quadrants in <I>A</I><SUB>1</SUB>, <I>A</I><SUB>3</SUB> and <I>C</I><SUB>1</SUB>, <I>C</I><SUB>3</SUB> respectively. +Let <I>R</I> be the radius of the sphere, <I>r, r</I><SUB>1</SUB>, <I>r</I><SUB>3</SUB> the radii of the +‘parallel circles’ (with pole <I>P</I>) through <I>C, C</I><SUB>1</SUB>, <I>C</I><SUB>3</SUB> respectively. +<p>Then shall <MATH>sin <I>A</I><SUB>1</SUB><I>A</I><SUB>3</SUB>/sin <I>C</I><SUB>1</SUB><I>C</I><SUB>3</SUB> = <I>Rr</I>/<I>r</I><SUB>1</SUB><I>r</I><SUB>3</SUB></MATH>. +<FIG> +<p>In the triangles <I>PCC</I><SUB>3</SUB>, <I>BA</I><SUB>3</SUB><I>C</I><SUB>3</SUB> the angles at <I>C, A</I><SUB>3</SUB> are right, +and the angles at <I>C</I><SUB>3</SUB> equal; therefore (III. 2) +<MATH>sin <I>PC</I>/sin <I>PC</I><SUB>3</SUB> = sin <I>BA</I><SUB>3</SUB>/sin <I>BC</I><SUB>3</SUB></MATH>. +<pb n=273><head>MENELAUS'S <I>SPHAERICA</I></head> +<p>But, by III. 1 applied to the triangle <I>BC</I><SUB>1</SUB><I>A</I><SUB>1</SUB> cut by the +transversal <I>PC</I><SUB>3</SUB><I>A</I><SUB>3</SUB>, +<MATH>sin <I>A</I><SUB>1</SUB><I>A</I><SUB>3</SUB>/sin <I>BA</I><SUB>3</SUB> = sin <I>C</I><SUB>1</SUB><I>C</I><SUB>3</SUB>/sin <I>BC</I><SUB>3</SUB>.sin <I>PA</I><SUB>1</SUB>/sin <I>PC</I><SUB>1</SUB></MATH>, +or <MATH>sin <I>A</I><SUB>1</SUB><I>A</I><SUB>3</SUB>/sin <I>C</I><SUB>1</SUB><I>C</I><SUB>3</SUB> = sin <I>PA</I><SUB>1</SUB>/sin <I>PC</I><SUB>1</SUB>.sin <I>BA</I><SUB>3</SUB>/sin <I>BC</I><SUB>3</SUB> = sin <I>PA</I><SUB>1</SUB>/sin +<I>PC</I><SUB>1</SUB>.sin <I>PC</I>/sin <I>PC</I><SUB>3</SUB>, +from above, += <I>Rr</I>/<I>r</I><SUB>1</SUB><I>r</I><SUB>3</SUB></MATH>. +<p>Part 2 of the proposition proves that, if <I>PC</I><SUB>2</SUB><I>A</I><SUB>2</SUB> be drawn +such that <MATH>sin<SUP>2</SUP> <I>PC</I><SUB>2</SUB> = sin <I>PA</I><SUB>2</SUB>.sin <I>PC,</I> or <I>r</I><SUB>2</SUB><SUP>2</SUP> = <I>Rr</I></MATH> (where <I>r</I><SUB>2</SUB> is +the radius of the parallel circle through <I>C</I><SUB>2</SUB>), <I>BC</I><SUB>2</SUB>-<I>BA</I><SUB>2</SUB> is a +maximum, while Parts 3, 4 discuss the limits to the value of +the ratio between the arcs <I>A</I><SUB>1</SUB><I>A</I><SUB>3</SUB> and <I>C</I><SUB>1</SUB><I>C</I><SUB>3</SUB>. +<p>Nothing is known of the life of CLAUDIUS PTOLEMY except +that he was of Alexandria, made observations between the +years A.D. 125 and 141 or perhaps 151, and therefore presum- +ably wrote his great work about the middle of the reign of +Antoninus Pius (A.D. 138-61). A tradition handed down by +the Byzantine scholar Theodorus Meliteniota (about 1361) +states that he was born, not at Alexandria, but at Ptolemais +<G>h( *(ermei/ou</G>. Arabian traditions, going back probably to +&Hdot;unain b. Is&hdot;āq, say that he lived to the age of 78, and give +a number of personal details to which too much weight must +not be attached. +<C>The <G>*maqhmatikh\ du/ntaxis</G> (Arab. <I>Almagest</I>).</C> +<p>Ptolemy's great work, the definitive achievement of Greek +astronomy, bore the title <G>*maqhmatikh=s *sunta/xews bibli/a ig</G>, +the <I>Mathematical Collection</I> in thirteen Books. By the time +of the commentators who distinguished the lesser treatises on +astronomy forming an introduction to Ptolemy's work as +<G>mikro\s a)stronomou/menos (to/pos)</G>, the ‘Little Astronomy’, the +book came to be called the ‘Great Collection’, <G>mega/lh su/n- +taxis</G>. Later still the Arabs, combining the article Al with +<pb n=274><head>TRIGONOMETRY</head> +the superlative <G>me/gistos</G>, made up a word Al-majisti, which +became <I>Almagest;</I> and it has been known by this name ever +since. The complicated character of the system expounded +by Ptolemy is no doubt responsible for the fact that it +speedily became the subject of elaborate commentaries. +<C>Commentaries on the <I>Syntaxis.</I></C> +<p>Pappus<note>Pappus, viii, p. 1106. 13.</note> cites a passage from his own commentary on +Book I of the <I>Mathematica,</I> which evidently means Ptolemy's +work. Part of Pappus's commentary on Book V, as well as +his commentary on Book VI, are actually extant in the +original. Theon of Alexandria, who wrote a commentary on +the <I>Syntaxis</I> in eleven Books, incorporated as much as was +available of Pappus's commentary on Book V with full +acknowledgement, though not in Pappus's exact words. In +his commentary on Book VI Theon made much more partial +quotations from Pappus; indeed the greater part of the com- +mentary on this Book is Theon's own or taken from other +sources. Pappus's commentaries are called <I>scholia,</I> Theon's +<G>u(pomnh/mata</G>. Passages in Pappus's commentary on Book V +allude to ‘the scholia preceding this one’ (in the plural), and +in particular to the scholium on Book IV. It is therefore all +but certain that he wrote on all the Books from I to VI at +least. The text of the eleven Books of Theon's commentary +was published at Basel by Joachim Camerarius in 1538, but +it is rare and, owing to the way in which it is printed, with +insufficient punctuation marks, gaps in places, and any number +of misprints, almost unusable; accordingly little attention has +so far been paid to it except as regards the first two Books, +which were included, in a more readable form and with a Latin +translation, by Halma in his edition of Ptolemy. +<C>Translations and editions.</C> +<p>The <I>Syntaxis</I> was translated into Arabic, first (we are told) +by translators unnamed at the instance of Ya&hdot;yā b. Khālid b. +Barmak, then by al-&Hdot;ajjāj, the translator of Euclid (about +786-835), and again by the famous translator Is&hdot;āq b. &Hdot;unain +(d. 910), whose translation, as improved by Thābit b. Quarra +<pb n=275><head>PTOLEMY'S <I>SYNTAXIS</I></head> +(died 901), is extant in part, as well as the version by Nasīrad- +dīn a⃛-&Tdot;ūsī (1201-74). +<p>The first edition to be published was the Latin translation +made by Gherard of Cremona from the Arabic, which was +finished in 1175 but was not published till 1515, when it was +brought out, without the author's name, by Peter Liechten- +stein at Venice. A translation from the Greek had been made +about 1160 by an unknown writer for a certain Henricus +Aristippus, Archdeacon of Catania, who, having been sent by +William I, King of Sicily, on a mission to the Byzantine +Emperor Manuel I. Comnenus in 1158, brought back with +him a Greek manuscript of the <I>Syntaxis</I> as a present; this +translation, however, exists only in manuscripts in the Vatican +and at Florence. The first Latin translation from the Greek +to be published was that made by Georgius ‘of Trebizond’ for +Pope Nicolas V in 1451; this was revised and published by +Lucas Gauricus at Venice in 1528. The <I>editio princeps</I> of the +Greek text was brought out by Grynaeus at Basel in 1538. +The next complete edition was that of Halma published +1813-16, which is now rare. All the more welcome, there- +fore, is the definitive Greek text of the astronomical works +of Ptolemy edited by Heiberg (1899-1907), to which is now +added, so far as the <I>Syntaxis</I> is concerned, a most valuable +supplement in the German translation (with notes) by Manitius +(Teubner, 1912-13). +<C>Summary of Contents.</C> +<p>The <I>Syntaxis</I> is most valuable for the reason that it con- +tains very full particulars of observations and investigations +by Hipparchus, as well as of the earlier observations recorded +by him, e.g. that of a lunar eclipse in 721 B.C. Ptolemy +based himself very largely upon Hipparchus, e.g. in the +preparation of a Table of Chords (equivalent to sines), the +theory of eccentrics and epicycles, &c.; and it is questionable +whether he himself contributed anything of great value except +a definite theory of the motion of the five planets, for which +Hipparchus had only collected material in the shape of obser- +vations made by his predecessors and himself. A very short +indication of the subjects of the different Books is all that can +<pb n=276><head>TRIGONOMETRY</head> +be given here. Book I: Indispensable preliminaries to the +study of the Ptolemaic system, general explanations of +the different motions of the heavenly bodies in relation to +the earth as centre, propositions required for the preparation +of Tables of Chords, the Table itself, some propositions in +spherical geometry leading to trigonometrical calculations of +the relations of arcs of the equator, ecliptic, horizon and +meridian, a ‘Table of Obliquity’, for calculating declinations +for each degree-point on the ecliptic, and finally a method of +finding the right ascensions for arcs of the ecliptic equal to +one-third of a sign or 10°. Book II: The same subject con- +tinued, i.e. problems on the sphere, with special reference to +the differences between various latitudes, the length of the +longest day at any degree of latitude, and the like. Book III: +On the length of the year and the motion of the sun on the +eccentric and epicycle hypotheses. Book IV: The length of the +months and the theory of the moon. Book V: The construc- +tion of the astrolabe, and the theory of the moon continued, +the diameters of the sun, the moon and the earth's shadow, +the distance of the sun and the dimensions of the sun, moon +and earth. Book VI: Conjunctions and oppositions of sun +and moon, solar and lunar eclipses and their periods. Books +VII and VIII are about the fixed stars and the precession of +the equinoxes, and Books IX-XIII are devoted to the move- +ments of the planets. +<C>Trigonometry in Ptolemy.</C> +<p>What interests the historian of mathematics is the trigono- +metry in Ptolemy. It is evident that no part of the trigono- +metry, or of the matter preliminary to it, in Ptolemy was new. +What he did was to abstract from earlier treatises, and to +condense into the smallest possible space, the minimum of +propositions necessary to establish the methods and formulae +used. Thus at the beginning of the preliminaries to the +Table of Chords in Book I he says: +<p>‘We will first show how we can establish a systematic and +speedy method of obtaining the lengths of the chords based on +the uniform use of the smallest possible number of proposi- +tions, so that we may not only have the lengths of the chords +<pb n=277><head>PTOLEMY'S <I>SYNTAXIS</I></head> +set out correctly, but may be in possession of a ready proof of +our method of obtaining them based on geometrical con- +siderations.’<note>Ptolemy, <I>Syntaxis,</I> i. 10, pp. 31 2.</note> +<p>He explains that he will use the division (1) of the circle into +360 equal parts or degrees and (2) of the diameter into 120 +equal parts, and will express fractions of these parts on the +sexagesimal system. Then come the geometrical propositions, +as follows. +<C>(<G>a</G>) <I>Lemma for finding</I> sin 18° <I>and</I> sin 36°.</C> +<p>To find the side of a pentagon and decagon inscribed in +a circle or, in other words, the chords subtending arcs of 72° +and 36° respectively. +<p>Let <I>AB</I> be the diameter of a circle, <I>O</I> the centre, <I>OC</I> the +radius perpendicular to <I>AB.</I> +<FIG> +<p>Bisect <I>OB</I> at <I>D,</I> join <I>DC,</I> and measure +<I>DE</I> along <I>DA</I> equal to <I>DC.</I> Join <I>EC.</I> +<p>Then shall <I>OE</I> be the side of the in- +scribed regular decagon, and <I>EC</I> the side +of the inscribed regular pentagon. +<p>For, since <I>OB</I> is bisected at <I>D,</I> +<MATH><I>BE.EO</I>+<I>OD</I><SUP>2</SUP> = <I>DE</I><SUP>2</SUP> += <I>DC</I><SUP>2</SUP> = <I>DO</I><SUP>2</SUP>+<I>OC</I><SUP>2</SUP></MATH>. +<p>Therefore <MATH><I>BE.EO</I> = <I>OC</I><SUP>2</SUP> = <I>OB</I><SUP>2</SUP></MATH>, +and <I>BE</I> is divided in extreme and mean ratio. +<p>But (Eucl. XIII. 9) the sides of the regular hexagon and the +regular decagon inscribed in a circle when placed in a straight +line with one another form a straight line divided in extreme +and mean ratio at the point of division. +<p>Therefore, <I>BO</I> being the side of the hexagon, <I>EO</I> is the side +of the decagon. +<p>Also (by Eucl. XIII. 10) +<MATH>(side of pentagon)<SUP>2</SUP> = (side of hexagon)<SUP>2</SUP>+(side of decagon)<SUP>2</SUP> += <I>CO</I><SUP>2</SUP>+<I>OE</I><SUP>2</SUP> = <I>EC</I><SUP>2</SUP></MATH>; +therefore <I>EC</I> is the side of the regular pentagon inscribed +in the circle. +<pb n=278><head>TRIGONOMETRY</head> +<p>The construction in fact easily leads to the results +<MATH><I>EO</I> = 1/2<I>a</I>(√5-1), <I>EC</I> = 1/2<I>a</I>√(10-2√5)</MATH>, +where <I>a</I> is the radius of the circle. +<p>Ptolemy does not however use these radicals, but calculates +the lengths in terms of ‘parts’ of the diameter thus. +<MATH><I>DO</I> = 30</MATH>, and <MATH><I>DO</I><SUP>2</SUP> = 900</MATH>; <MATH><I>OC</I> = 60</MATH> and <MATH><I>OC</I><SUP>2</SUP> = 3600</MATH>; +therefore <MATH><I>DE</I><SUP>2</SUP> = <I>DC</I><SUP>2</SUP> = 4500</MATH>, and <MATH><I>DE</I> = 67<SUP><I>p</I></SUP>4′55″</MATH> nearly; +therefore side of decagon or <MATH>(crd. 36°) = <I>DE</I>-<I>DO</I> = 37<SUP><I>p</I></SUP>4′55″</MATH>. +<p>Again <MATH><I>OE</I><SUP>2</SUP> = (37<SUP><I>p</I></SUP>4′55″)<SUP>2</SUP> = 1375.4′15″</MATH>, and <MATH><I>OC</I><SUP>2</SUP> = 3600</MATH>; +therefore <MATH><I>CE</I><SUP>2</SUP> = 4975.4′15″</MATH>, and <MATH><I>CE</I> = 70<SUP><I>p</I></SUP>32′3″</MATH> nearly, +i.e. side of pentagon or <MATH>(crd. 72°) = 70<SUP><I>p</I></SUP>32′3″</MATH>. +<p>The method of extracting the square root is explained by +Theon in connexion with the first of these cases, √4500 (see +above, vol. i, pp. 61-3). +<p>The chords which are the sides of other regular inscribed +figures, the hexagon, the square and the equilateral triangle, +are next given, namely, +<MATH>crd. 60° = 60<SUP><I>p</I></SUP>, +crd. 90° = √(2.60<SUP>2</SUP>) = √(7200) = 84<SUP><I>p</I></SUP>51′10″, +crd. 120° = √(3.60<SUP>2</SUP>) = √(10800) = 103<SUP><I>p</I></SUP>55′23″</MATH>. +<C>(<G>b</G>) <I>Equivalent of</I> <MATH>sin<SUP>2</SUP> <G>q</G>+cos<SUP>2</SUP> <G>q</G> = 1</MATH>.</C> +<p>It is next observed that, if <I>x</I> be any arc, +<MATH>(crd. <I>x</I>)<SUP>2</SUP>+{crd. (180°-<I>x</I>)}<SUP>2</SUP> = (diam.)<SUP>2</SUP> = 120<SUP>2</SUP></MATH>, +a formula which is of course equivalent to <MATH>sin<SUP>2</SUP> <G>q</G>+cos<SUP>2</SUP> <G>q</G> = 1</MATH>. +<p>We can therefore, from crd. 72°, derive crd. 108°, from +crd. 36°, crd. 144°, and so on. +<C>(<G>g</G>) ‘<I>Ptolemy's theorem</I>’, <I>giving the equivalent of</I> +<MATH>sin (<G>q</G>-<G>f</G>) = sin <G>q</G> cos <G>f</G>-cos <G>q</G> sin <G>f</G></MATH>.</C> +<p>The next step is to find a formula which will give us +crd. (<G>a</G>-<G>b</G>) when crd. <G>a</G> and crd. <G>b</G> are given. (This for +instance enables us to find crd. 12° from crd. 72° and crd. 60°.) +<pb n=279><head>PTOLEMY'S <I>SYNTAXIS</I></head> +<p>The proposition giving the required formula depends upon +a lemma, which is the famous ‘Ptolemy's theorem’. +<p>Given a quadrilateral <I>ABCD</I> inscribed in a circle, the +diagonals being <I>AC, BD,</I> to prove that +<MATH><I>AC.BD</I> = <I>AB.DC</I>+<I>AD.BC</I></MATH>. +<p>The proof is well known. Draw <I>BE</I> so that the angle <I>ABE</I> +is equal to the angle <I>DBC,</I> and let <I>BE</I> +meet <I>AC</I> in <I>E.</I> +<FIG> +<p>Then the triangles <I>ABE, DBC</I> are +equiangular, and therefore +<MATH><I>AB</I>:<I>AE</I> = <I>BD</I>:<I>DC</I></MATH>, +or <MATH><I>AB.DC</I> = <I>AE.BD</I></MATH>. (1) +<p>Again, to each of the equal angles +<I>ABE, DBC</I> add the angle <I>EBD;</I> +then the angle <I>ABD</I> is equal to the angle <I>EBC,</I> and the +triangles <I>ABD, EBC</I> are equiangular; +therefore <MATH><I>BC</I>:<I>CE</I> = <I>BD</I>:<I>DA</I></MATH>, +or <MATH><I>AD.BC</I> = <I>CE.BD</I></MATH>. (2) +<p>By adding (1) and (2), we obtain +<MATH><I>AB.DC</I>+<I>AD.BC</I> = <I>AC.BD</I></MATH>. +<p>Now let <I>AB, AC</I> be two arcs terminating at <I>A,</I> the extremity +<FIG> +of the diameter <I>AD</I> of a circle, and let +<I>AC</I> (= <G>a</G>) be greater than <I>AB</I> (= <G>b</G>). +Suppose that (crd. <I>AC</I>) and (crd. <I>AB</I>) +are given: it is required to find +(crd. <I>BC</I>). +<p>Join <I>BD, CD.</I> +<p>Then, by the above theorem, +<MATH><I>AC.BD</I> = <I>BC.AD</I>+<I>AB.CD</I></MATH>. +<p>Now <I>AB, AC</I> are given; therefore <MATH><I>BD</I> = crd. (180°-<I>AB</I>)</MATH> +and <MATH><I>CD</I> = crd. (180°-<I>AC</I>)</MATH> are known. And <I>AD</I> is known. +Hence the remaining chord <I>BC</I> (crd. <I>BC</I>) is known. +<pb n=280><head>TRIGONOMETRY</head> +<p>The equation in fact gives the formula, +<MATH>{crd. (<G>a</G>-<G>b</G>)}.(crd. 180°) = (crd. <G>a</G>).{crd. (180°-<G>b</G>)} +-(crd. <G>b</G>).{crd. (180°-<G>a</G>)}</MATH>, +which is, of course, equivalent to +<MATH>sin (<G>q</G>-<G>f</G>) = sin <G>q</G> cos <G>f</G>-cos <G>q</G> sin <G>f</G>, where <G>a</G> = 2<G>q</G>, <G>b</G> = 2<G>f</G></MATH>. +<p>By means of this formula Ptolemy obtained +<MATH>crd. 12° = crd. (72°-60°) = 12<SUP><I>p</I></SUP>32′36″</MATH>. +<C>(<G>d</G>) <I>Equivalent of</I> <MATH>sin<SUP>2</SUP> 1/2<G>q</G> = 1/2 (1-cos <G>q</G>)</MATH>.</C> +<p>But, in order to get the chords of smaller angles still, we +want a formula for finding the chord of half an arc when the +chord of the arc is given. This is the subject of Ptolemy's +next proposition. +<p>Let <I>BC</I> be an arc of a circle with diameter <I>AC,</I> and let the +arc <I>BC</I> be bisected at <I>D.</I> Given (crd. <I>BC</I>), it is required to +find (crd. <I>DC</I>). +<FIG> +<p>Draw <I>DF</I> perpendicular to <I>AC,</I> +and join <I>AB, AD, BD, DC.</I> Measure +<I>AE</I> along <I>AC</I> equal to <I>AB,</I> and join +<I>DE.</I> +<p>Then shall <I>FC</I> be equal to <I>EF,</I> or +<I>FC</I> shall be half the difference be- +tween <I>AC</I> and <I>AB.</I> +<p>For the triangles <I>ABD, AED</I> are +equal in all respects, since two sides +of the one are equal to two sides of the other and the included +angles <I>BAD, EAD,</I> standing on equal arcs, are equal. +<p>Therefore <MATH><I>ED</I> = <I>BD</I> = <I>DC</I></MATH>, +and the right-angled triangles <I>DEF, DCF</I> are equal in all +respects, whence <MATH><I>EF</I> = <I>FC</I></MATH>, or <MATH><I>CF</I> = 1/2(<I>AC</I>-<I>AB</I>)</MATH>. +<p>Now <MATH><I>AC.CF</I> = <I>CD</I><SUP>2</SUP></MATH>, +whence <MATH>(crd. <I>CD</I>)<SUP>2</SUP> = 1/2 <I>AC</I> (<I>AC</I>-<I>AB</I>) += 1/2 (crd. 180°).{(crd. 180°)-(crd.―(180°-<I>BC</I>))}</MATH>. +<p>This is, of course, equivalent to the formula +<MATH>sin<SUP>2</SUP> 1/2<G>q</G> = 1/2(1-cos <G>q</G>)</MATH>. +<pb n=281><head>PTOLEMY'S <I>SYNTAXIS</I></head> +<p>By successively applying this formula, Ptolemy obtained +(crd. 6°), (crd. 3°) and finally <MATH>(crd. 1 1/2°) = 1<SUP><I>p</I></SUP>34′15″</MATH> and +<MATH>(crd. 3/4°) = 0<SUP><I>p</I></SUP>47′8″</MATH>. But we want a table going by half- +degrees, and hence two more things are necessary; we have to +get a value for (crd. 1°) lying between (crd. 1 1/2°) and (crd. 3/4°), +and we have to obtain an <I>addition</I> formula enabling us when +(crd. <G>a</G>) is given to find {crd. (<G>a</G>+1/2°)}, and so on. +<C>(<G>e</G>) <I>Equivalent of</I> <MATH>cos (<G>q</G>+<G>f</G>) = cos <G>q</G> cos <G>f</G>-sin <G>q</G> sin <G>f</G></MATH>.</C> +<p>To find the addition formula. Suppose <I>AD</I> is the diameter +of a circle, and <I>AB, BC</I> two arcs. Given (crd. <I>AB</I>) and +(crd. <I>BC</I>), to find (crd. <I>AC</I>). Draw the diameter <I>BOE,</I> and +join <I>CE, CD, DE, BD.</I> +<FIG> +<p>Now, (crd. <I>AB</I>) being known, +(crd. <I>BD</I>) is known, and therefore +also (crd. <I>DE</I>), which is equal to +(crd. <I>AB</I>); and, (crd. <I>BC</I>) being +known, (crd. <I>CE</I>) is known. +<p>And, by Ptolemy's theorem, +<MATH><I>BD.CE</I> = <I>BC.DE</I>+<I>BE.CD</I></MATH>. +<p>The diameter <I>BE</I> and all the chords in this equation except +<I>CD</I> being given, we can find <I>CD</I> or crd. (180°-<I>AC</I>). We have +in fact +<MATH>(crd. 180°).{crd. (180°-<I>AC</I>)} += {crd. (180°-<I>AB</I>)}.{crd. (180°-<I>BC</I>)}-(crd. <I>AB</I>).(crd. <I>BC</I>)</MATH>; +thus crd. (180°-<I>AC</I>) and therefore (crd. <I>AC</I>) is known. +<p>If <MATH><I>AB</I> = 2<G>q</G>, <I>BC</I> = 2<G>f</G></MATH>, the result is equivalent to +<MATH>cos (<G>q</G>+<G>f</G>) = cos <G>q</G> cos <G>f</G>-sin <G>q</G> sin <G>f</G></MATH>. +<C>(<G>z</G>) <I>Method of interpolation based on formula</I> +<MATH>sin <G>a</G>/sin <G>b</G> < <G>a</G>/<G>b</G> (<I>where</I> 1/2<G>p</G> > <G>a</G> > <G>b</G>)</MATH>.</C> +<p>Lastly we have to find (crd. 1°), having given (crd. 1 1/2°) and +(crd. 3/4°). +<p>Ptolemy uses an ingenious method of <I>interpolation</I> based on +a proposition already assumed as known by Aristarchus. +<p>If <I>AB, BC</I> be unequal chords in a circle, <I>BC</I> being the +<pb n=282><head>TRIGONOMETRY</head> +greater, then shall the ratio of <I>CB</I> to <I>BA</I> be less than the +ratio of the arc <I>CB</I> to the arc <I>BA.</I> +<p>Let <I>BD</I> bisect the angle <I>ABC,</I> meeting <I>AC</I> in <I>E</I> and +<FIG> +the circumference in <I>D.</I> The arcs +<I>AD, DC</I> are then equal, and so are +the chords <I>AD, DC.</I> Also <MATH><I>CE</I> > <I>EA</I> +(since <I>CB</I>:<I>BA</I> = <I>CE</I>:<I>EA</I>)</MATH>. +<p>Draw <I>DF</I> perpendicular to <I>AC;</I> +then <I>AD</I> > <I>DE</I> > <I>DF,</I> so that the +circle with centre <I>D</I> and radius <I>DE</I> +will meet <I>DA</I> in <I>G</I> and <I>DF</I> produced +in <I>H.</I> +<p>Now <MATH><I>FE</I>:<I>EA</I> = ▵<I>FED</I>:▵<I>AED</I> +<(sector <I>HED</I>):(sector <I>GED</I>) +<∠<I>FDE</I>:∠<I>EDA</I></MATH>. +<I>Componendo,</I> <MATH><I>FA</I>:<I>AE</I><∠<I>FDA</I>:∠<I>ADE</I></MATH>. +<p>Doubling the antecedents, we have +<MATH><I>CA</I>:<I>AE</I><∠<I>CDA</I>:∠<I>ADE</I></MATH>, +and, <I>separando,</I> <MATH><I>CE</I>:<I>EA</I><∠<I>CDE</I>:∠<I>EDA</I></MATH>; +therefore <MATH>(since <I>CB</I>:<I>BA</I> = <I>CE</I>:<I>EA</I>) +<I>CB</I>:<I>BA</I><∠<I>CDB</I>:∠<I>BDA</I> +<(arc <I>CB</I>):(arc <I>BA</I>)</MATH>, +i.e. <MATH>(crd. <I>CB</I>):(crd. <I>BA</I>)<(arc <I>CB</I>):(arc <I>BA</I>)</MATH>. +[This is of course equivalent to <MATH>sin <G>a</G>:sin <G>b</G> < <G>a</G>:<G>b</G></MATH>, where +<MATH>1/2<G>p</G> > <G>a</G> > <G>b</G></MATH>.] +<p>It follows (1) that <MATH>(crd. 1°):(crd. 3/4°)<1:3/4</MATH>, +and (2) that <MATH>(crd. 1 1/2°):(crd. 1°)<1 1/2:1</MATH>. +<p>That is, <MATH>4/3.(crd. 3/4°)>(crd. 1°)>2/3.(crd. 1 1/2°)</MATH>. +<p>But <MATH>(crd. 3/4°) = 0<SUP><I>p</I></SUP>47′8″</MATH>, so that <MATH>4/3(crd. 3/4°) = 1<SUP><I>p</I></SUP>2′50″</MATH> +nearly (actually 1<SUP><I>p</I></SUP>2′50 2/3″); +and <MATH>(crd. 1 1/2°) = 1<SUP><I>p</I></SUP>34′15″</MATH>, so that <MATH>2/3(crd. 1 1/2°) = 1<SUP><I>p</I></SUP>2′50″</MATH>. +<p>Since, then, (crd. 1°) is both less and greater than a length +which only differs inappreciably from 1<SUP><I>p</I></SUP>2′50″, we may say +that <MATH>(crd. 1°) = 1<SUP><I>p</I></SUP>2′50″</MATH> as nearly as possible. +<pb n=283><head>PTOLEMY'S <I>SYNTAXIS</I></head> +<C>(<G>h</G>) <I>Table of Chords.</I></C> +<p>From this Ptolemy deduces that (crd. 1/2°) is very nearly +0<SUP>p</SUP> 31′ 25″, and by the aid of the above propositions he is in +a position to complete his Table of Chords for arcs subtending +angles increasing from 1/2° to 180° by steps of 1/2°; in other +words, a Table of Sines for angles from 1/4° to 90° by steps +of 1/4°. +<C>(<G>q</G>) <I>Further use of proportional increase.</I></C> +<p>Ptolemy carries further the principle of proportional in- +crease as a method of finding approximately the chords of +arcs containing an odd number of minutes between 0′ and 30′. +Opposite each chord in the Table he enters in a third column +1/30th of the excess of that chord over the one before, i.e. the +chord of the arc containing 30′ less than the chord in question. +For example (crd. 2 1/2°) is stated in the second column of the +Table as 2<SUP>p</SUP> 37′ 4″. The excess of (crd. 2 1/2°) over (crd. 2°) in the +Table is 0<SUP>p</SUP> 31′ 24″; 1/30th of this is 0<SUP>p</SUP> 1′ 2″ 48‴, which is +therefore the amount entered in the third column opposite +(crd. 2 1/2°). Accordingly, if we want (crd. 2° 25′), we take +(crd. 2°) or 2<SUP>p</SUP> 5′ 40″ and add 25 times 0<SUP>p</SUP> 1′ 2″ 48‴; or we +take (crd. 2 1/2°) or 2<SUP>p</SUP> 37′ 4″ and subtract 5 times 0<SUP>p</SUP> 1′ 2″ 48‴. +Ptolemy adds that if, by using the approximation for 1° and +1/2°, we gradually accumulate an error, we can check the calcu- +lation by comparing the chord with that of other related arcs, +e.g. the double, or the supplement (the difference between the +arc and the semicircle). +<p>Some particular results obtained from the Table may be +mentioned. Since <MATH>(crd. 1°) = 1<SUP>p</SUP> 2′ 50″</MATH>, the whole circumference +<MATH>= 360 (1<SUP>p</SUP> 2′ 50″)</MATH>, nearly, and, the length of the diameter +being 120<SUP>p</SUP>, the value of <G>p</G> is <MATH>3 (1 + 2/60 + 50/3600) = 3 + 8/60 + 30/3600</MATH>, +which is the value used later by Ptolemy and is equivalent to +3.14166... Again, <MATH>√3 = 2 sin 60°</MATH> and, 2 (crd. 120°) being +equal to 2 (103<SUP>p</SUP> 55′ 23″), we have <MATH>√3 = 2/120 (103 + 55/60 + 23/3600) += 1 + 43/60 + 55/60<SUP>2</SUP> + 23/60<SUP>3</SUP> = 1.7320509</MATH>, +which is correct to 6 places of decimals. Speaking generally, +<pb n=284><head>TRIGONOMETRY</head> +the sines obtained from Ptolemy's Table are correct to 5 +places. +<C>(<G>i</G>) <I>Plane trigonometry in effect used.</I></C> +<p>There are other cases in Ptolemy in which plane trigono- +metry is in effect used, e.g. in the determination of the +eccentricity of the sun's orbit.<note>Ptolemy, <I>Syntaxis</I>, iii. 4, vol. i, pp. 234-7.</note> Suppose that <I>ACBD</I> is +<FIG> +the eccentric circle with centre <I>O</I>, +and <I>AB, CD</I> are chords at right +angles through <I>E</I>, the centre of the +earth. To find <I>OE.</I> The are <I>BC</I> +is known (= <G>a</G>, say) as also the arc +<I>CA</I> (= <G>b</G>). If <I>BF</I> be the chord +parallel to <I>CD</I>, and <I>CG</I> the chord +parallel to <I>AB</I>, and if <I>N, P</I> be the +middle points of the arcs <I>BF, GC</I>, +Ptolemy finds (1) the arc <I>BF</I> +<MATH>(= <G>a</G> + <G>b</G> - 180°)</MATH>, then the chord <I>BF</I>, +crd. <MATH>(<G>a</G> + <G>b</G> - 180°)</MATH>, then the half of it, (2) the arc <MATH><I>GC</I> += arc (<G>a</G> + <G>b</G> - 2<G>b</G>)</MATH> or arc (<G>a</G> - <G>b</G>), then the chord <I>GC</I>, and +lastly half of it. He then adds the squares on the half- +chords, i.e. he obtains +<MATH><I>OE</I><SUP>2</SUP> = 1/4{crd. (<G>a</G> + <G>b</G> - 180)}<SUP>2</SUP> + 1/4 {crd. (<G>a</G> - <G>b</G>)}<SUP>2</SUP></MATH>, +that is, <MATH><I>OE</I><SUP>2</SUP>/r<SUP>2</SUP> = cos<SUP>2</SUP> 1/2 (<G>a</G> + <G>b</G>) + sin<SUP>2</SUP>1/2(<G>a</G> - <G>b</G>).</MATH> +He proceeds to obtain the angle <I>OEC</I> from its sine <I>OR / OE</I>, +which he expresses as a chord of double the angle in the +circle on <I>OE</I> as diameter in relation to that diameter. +<C>Spherical trigonometry: formulae in solution of +spherical triangles.</C> +<p>In spherical trigonometry, as already stated, Ptolemy +obtains everything that he wants by using the one funda- +mental proposition known as ‘Menelaus's theorem’ applied +to the sphere (Menelaus III. 1), of which he gives a proof +following that given by Menelaus of the first case taken in +his proposition. Where Ptolemy has occasion for other pro- +positions of Menelaus's <I>Sphaerica</I>, e.g. III. 2 and 3, he does +<pb n=285><head>PTOLEMY'S <I>SYNTAXIS</I></head> +not quote those propositions, as he might have done, but proves +them afresh by means of Menelaus's theorem.<note><I>Syntaxis</I>, vol. i, p. 169 and pp. 126-7 respectively.</note> The appli- +cation of the theorem in other cases gives in effect the +following different formulae belonging to the solution of +a spherical triangle <I>ABC</I> right-angled at <I>C</I>, viz. +<MATH>sin <I>a</I> = sin <I>c</I> sin <I>A</I></MATH>, +<MATH>tan <I>a</I> = sin <I>b</I> tan <I>A</I></MATH>, +<MATH>cos <I>c</I> = cos <I>a</I> cos <I>b</I></MATH>, +<MATH>tan <I>b</I> = tan <I>c</I> cos <I>A.</I></MATH> +<p>One illustration of Ptolemy's procedure will be sufficient.<note><I>Ib.</I>, vol. i, pp. 121-2.</note> +Let <I>HAH′</I> be the horizon, <I>PEZH</I> the meridian circle, <I>EE′</I> +<FIG> +the equator, <I>ZZ′</I> the ecliptic, <I>F</I> an +equinoctial point. Let <I>EE′, ZZ′</I> +cut the horizon in <I>A, B</I>. Let <I>P</I> be +the pole, and let the great circle +through <I>P, B</I> cut the equator at <I>C.</I> +Now let it be required to find the +time which the arc <I>FB</I> of the ecliptic +takes to rise; this time will be +measured by the arc <I>FA</I> of the +equator. (Ptolemy has previously found the length of the +arcs <I>BC</I>, the declination, and <I>FC</I>, the right ascension, of <I>B</I>, +I. 14, 16.) +<p>By Menelaus's theorem applied to the arcs <I>AE′, E′P</I> cut by +the arcs <I>AH′, PC</I> which also intersect one another in <I>B</I>, +<MATH>(crd. 2 <I>PH′</I>)/(crd. 2 <I>H′E′</I>) = (crd. 2 <I>PB</I>/crd. 2 <I>BC</I>) . (crd. 2 <I>CA</I>/crd. 2 <I>AE′</I>)</MATH>; +that is, <MATH>sin <I>PH′</I>/sin <I>H′E′</I> = (sin <I>PB</I>/sin <I>BC</I>) . (sin <I>CA</I>/sin <I>AE′</I>)</MATH>. +<p>Now <MATH>sin <I>PH′</I> = cos <I>H′E′</I>, sin <I>PB</I> = cos <I>BC</I></MATH>, and <MATH>sin <I>AE′</I> = 1</MATH>; +therefore <MATH>cot <I>H′E′</I> = cot <I>BC</I> . sin <I>CA</I></MATH>, +in other words, in the triangle <I>ABC</I> right-angled at <I>C</I>, +<MATH>cot <I>A</I> = cot <I>a</I> sin <I>b</I></MATH>, +or <MATH>tan <I>a</I> = sin <I>b</I> tan <I>A</I></MATH>. +<pb n=286><head>TRIGONOMETRY</head> +<p>Thus <I>AC</I> is found, and therefore <I>FC-AC</I> or <I>FA.</I> +<p>The lengths of <I>BC, FC</I> are found in I. 14, 16 by the same +method, the four intersecting great circles used in the figure +being in that case the equator <I>EE′</I>, the ecliptic <I>ZZ′</I>, the great +circle <I>PBCP′</I> through the poles, and the great circle <I>PKLP′</I> +passing through the poles of both the ecliptic and the equator. +In this case the two arcs <I>PL, AE′</I> are cut by the intersecting +great circles <I>PC, FK</I>, and Menelaus's theorem gives (1) +<MATH>sin <I>PL</I>/sin <I>KL</I> = (sin <I>CP</I>/sin <I>BC</I>) . (sin <I>BF</I>/sin <I>FK</I>)</MATH>. +<p>But <MATH>sin <I>PL</I> = 1, sin <I>KL</I> = sin <I>BFC</I>, sin <I>CP</I> = 1, sin <I>FK</I> = 1</MATH>, +and it follows that +<MATH>sin <I>BC</I> = sin <I>BF</I> sin <I>BFC</I></MATH>, +corresponding to the formula for a triangle right-angled at <I>C</I>, +<MATH>sin <I>a</I> = sin <I>c</I> sin <I>A</I></MATH>. +<p>(2) We have +<MATH>sin <I>PK</I>/sin <I>KL</I> = (sin <I>PB</I>/sin <I>BC</I>) . (sin <I>CF</I>/sin <I>FL</I>)</MATH>, +and <MATH>sin <I>PK</I> = cos <I>KL</I> = cos <I>BFC</I>, sin <I>PB</I> = cos <I>BC</I>, sin <I>FL</I> = 1</MATH>, +so that <MATH>tan <I>BC</I> = sin <I>CF</I> tan <I>BFC</I></MATH>, +corresponding to the formula +<MATH>tan <I>a</I> = sin <I>b</I> tan <I>A</I></MATH>. +<p>While, therefore, Ptolemy's method implicitly gives the +formulae for the solution of right-angled triangles above +quoted, he does not speak of right-angled triangles at all, but +only of arcs of intersecting great circles. The advantage +from his point of view is that he works in sines and cosines +only, avoiding tangents as such, and therefore he requires +tables of only one trigonometrical ratio, namely the sine (or, +as he has it, the chord of the double arc). +<C>The <I>Analemma</I>.</C> +<p>Two other works of Ptolemy should be mentioned here. +The first is the <I>Analemma.</I> The object of this is to explain +a method of representing on one plane the different points +<pb n=287><head>THE <I>ANALEMMA</I> OF PTOLEMY</head> +and arcs of the heavenly sphere by means of <I>orthogonal +projection</I> upon three planes mutually at right angles, the +meridian, the horizon, and the ‘prime vertical’. The definite +problem attacked is that of showing the position of the sun at +any given time of the day, and the use of the method and +of the instruments described in the book by Ptolemy was +connected with the construction of sundials, as we learn from +Vitruvius.<note>Vitruvius, <I>De architect.</I> ix. 4.</note> There was another <G>a)na/lhmma</G> besides that of +Ptolemy; the author of it was Diodorus of Alexandria, a con- +temporary of Caesar and Cicero (‘Diodorus, famed among the +makers of gnomons, tell me the time!’ says the Anthology<note><I>Anth. Palat.</I> xiv. 139.</note>), +and Pappus wrote a commentary upon it in which, as he tells +us,<note>Pappus, iv, p. 246. 1.</note> he used the conchoid in order to trisect an angle, a problem +evidently required in the <I>Analemma</I> in order to divide any +arc of a circle into six equal parts (hours). The word +<G>a)na/lhmma</G> evidently means ‘taking up’ (‘Aufnahme’) in the +sense of ‘making a graphic representation’ of something, in +this case the representation on a plane of parts of the heavenly +sphere. Only a few fragments remain of the Greek text of +the <I>Analemma</I> of Ptolemy; these are contained in a palimpsest +(Ambros. Gr. L. 99 sup., now 491) attributed to the seventh +century but probably earlier. Besides this, we have a trans- +lation by William of Moerbeke from an Arabic version. +This Latin translation was edited with a valuable commentary +by the indefatigable Commandinus (Rome, 1562); but it is +now available in William of Moerbeke's own words, Heiberg +having edited it from Cod. Vaticanus Ottobon. lat. 1850 of the +thirteenth century (written in William's own hand), and in- +cluded it with the Greek fragments (so far as they exist) in +parallel columns in vol. ii of Ptolemy's works (Teubner, 1907). +<p>The figure is referred to three fixed planes (1) the meridian, +(2) the horizon, (3) the prime vertical; these planes are the +planes of the three circles <I>APZB, ACB, ZQC</I> respectively +shown in the diagram below. Three other great circles are +used, one of which, the equator with pole <I>P</I>, is fixed; the +other two are movable and were called by special names; +the first is the circle represented by any position of the circle +of the horizon as it revolves round <I>COC′</I> as diameter (<I>CSM</I> in +<pb n=288><head>TRIGONOMETRY</head> +the diagram is one position of it, coinciding with the equator), +and it was called <G>e(kth/moros ku/klos</G> (‘the circle in six parts’) +because the highest point of it above the horizon corresponds +to the lapse of six hours; the second, called the <I>hour-circle</I>, is +the circle represented by any position, as <I>BSQA</I>, of the circle +of the horizon as it revolves round <I>BA</I> as axis. +<p>The problem is, as above stated, to find the position of the +sun at a given hour of the day. In order to illustrate +the method, it is sufficient, with A. v. Braunmühl,<note>Braunmühl, <I>Gesch. der Trigonometrie</I>, i, pp. 12, 13.</note> to take the +simplest case where the sun is on the equator, i.e. at one of +the equinoctial points, so that the <I>hectemoron</I> circle coincides +with the equator. +<p>Let <I>S</I> be the position of the sun, lying on the equator <I>MSC, +P</I> the pole, <I>MZA</I> the meridian, <I>BCA</I> the horizon, <I>BSQA</I> the +<I>hour-circle</I>, and let the vertical great circle <I>ZSV</I> be drawn +through <I>S</I>, and the vertical great circle <I>ZQC</I> through <I>Z</I> the +zenith and <I>C</I> the east-point. +<p>We are given the arc <MATH><I>SC</I> = 90° - <I>t</I></MATH>, where <I>t</I> is the hour- +angle, and the arc <MATH><I>MB</I> = 90° - <G>f</G></MATH>, where <G>f</G> is the elevation of +the pole; and we have to find the arcs <I>SV</I> (the sun's altitude), +<FIG> +<I>VC</I>, the ‘ascensional difference’, <I>SQ</I> and <I>QC.</I> Ptolemy, in +fact, practically determines the position of <I>S</I> in terms of +certain spherical coordinates. +<p>Draw the perpendiculars, <I>SF</I> to the plane of the meridian, +<I>SH</I> to that of the horizon, and <I>SE</I> to the plane of the prime +<pb n=289><head>THE <I>ANALEMMA</I> OF PTOLEMY</head> +vertical; and draw <I>FG</I> perpendicular to <I>BA</I>, and <I>ET</I> to <I>OZ.</I> +Join <I>HG</I>, and we have <MATH><I>FG</I> = <I>SH</I>, <I>GH</I> = <I>FS</I> = <I>ET</I></MATH>. +<p>We now represent <I>SF</I> in a separate figure (for clearness' +sake, as Ptolemy uses only one figure), where <I>B′Z′A′</I> corre- +sponds to <I>BZA, P′</I> to <I>P</I> and <I>O′M′</I> to <I>OM.</I> Set off the arc +<I>P′S′</I> equal to <I>CS</I> (= 90° - <I>t</I>), and draw <I>S′F′</I> perpendicular +to <I>O′M′.</I> Then <MATH><I>S′M′</I> = <I>SM</I></MATH>, and <MATH><I>S′F′</I> = <I>SF</I></MATH>; it is as if in the +original figure we had turned the quadrant <I>MSC</I> round <I>MO</I> +till it coincided with the meridian circle. +<p>In the two figures draw <I>IFK, I′F′K′</I> parallel to <I>BA, B′A′</I>, +and <I>LFG, L′F′G′</I> parallel to <I>OZ, O′Z′.</I> +<p>Then (1) arc <MATH><I>ZI</I> = arc <I>ZS</I> = arc (90° - <I>SV</I>)</MATH>, because if we +turn the quadrant <I>ZSV</I> about <I>ZO</I> till it coincides with the +<FIG> +meridian, <I>S</I> falls on <I>I</I>, and <I>V</I> on <I>B.</I> It follows that the +required arc <I>SV</I> = arc <I>B′I′</I> in the second figure. +<p>(2) To find the arc <I>VC</I>, set off <I>G′X</I> (in the second figure) +along <I>G′F′</I> equal to <I>FS</I> or <I>F′S′</I>, and draw <I>O′X</I> through to +meet the circle in <I>X′.</I> Then arc <MATH><I>Z′X′</I> = arc <I>VC</I></MATH>; for it is as if +we had turned the quadrant <I>BVC</I> about <I>BO</I> till it coincided +with the meridian, when (since <MATH><I>G′X</I> = <I>FS</I> = <I>GH</I></MATH>) <I>H</I> would +coincide with <I>X</I> and <I>V</I> with <I>X′.</I> Therefore <I>BV</I> is also equal +to <I>B′X′.</I> +<p>(3) To find <I>QC</I> or <I>ZQ</I>, set off along <I>T′F′</I> in the second figure +<I>T′Y</I> equal to <I>F′S′</I>, and draw <I>O′Y</I> through to <I>Y′</I> on the circle. +<p>Then arc <I>B′Y′</I> = arc <I>QC</I>; for it is as if we turned the prime +vertical <I>ZQC</I> about <I>ZO</I> till it coincided with the meridian, +when (since <I>T′Y</I> = <I>S′F′</I> = <I>TE</I>) <I>E</I> would fall on <I>Y</I>, the radius +<I>OEQ</I> on <I>O′YY′</I> and <I>Q</I> on <I>Y′.</I> +<p>(4) Lastly, arc <MATH><I>BS</I> = arc <I>BL</I> = arc <I>B′L′</I></MATH>, because <I>S, L</I> are +<pb n=290><head>TRIGONOMETRY</head> +both in the plane <I>LSHG</I> at right angles to the meridian; +therefore arc <I>SQ</I> = arc <I>L′Z′.</I> +<p>Hence all four arcs <I>SV, VC, QC, QS</I> are represented in the +auxiliary figure in one plane. +<p>So far the procedure amounts to a method of <I>graphically</I> +constructing the arcs required as parts of an auxiliary circle +in one plane. But Ptolemy makes it clear that practical +calculation followed on the basis of the figure.<note>See Zeuthen in <I>Bibliotheca mathematica</I>, i<SUB>3</SUB>, 1900, pp. 23-7.</note> The lines +used in the construction are <MATH><I>SF</I> = sin <I>t</I></MATH> (where the radius = 1), +<MATH><I>FT</I> = <I>OF</I> sin <G>f</G>, <I>FG</I> = <I>OF</I> sin (90° - <G>f</G>)</MATH>, and this was fully +realized by Ptolemy. Thus he shows how to calculate the +arc <I>SZ</I>, the zenith distance (= <I>d</I>, say) or its complement <I>SV</I>, +the height of the sun (= <I>h</I>, say), in the following way. He +says in effect: Since <I>G</I> is known, and <MATH>∠<I>F′O′G′</I> = 90° - <G>f</G></MATH>, the +ratios <I>O′F′</I>:<I>F′T′</I> and <I>O′F′</I>:<I>O′T′</I> are known. +<p>[In fact <MATH><I>O′F′</I>/<I>O′T′</I> = <I>D</I>/crd. (180° - 2<G>f</G>)</MATH>, where <I>D</I> is the diameter +of the sphere.] +<p>Next, since the arc <I>MS</I> or <I>M′S′</I> is known [= <I>t</I>], and there- +fore the arc <I>P′S′</I> [= 90° - <I>t</I>], the ratio of <I>O′F′</I> to <I>D</I> is known +[in fact <MATH><I>O′F′</I> / <I>D</I> = {crd. (180 - 2<I>t</I>)} / 2 <I>D.</I></MATH> +<p>It follows from these two results that +<MATH><I>O</I>′<I>T</I>′ = crd. (180° - 2<I>t</I>)/2<I>D</I>.crd. (180° - 2<G>f</G>)].</MATH> +<p>Lastly, the arc <I>SV</I> (= <I>h</I>) being equal to <I>B′I′</I>, the angle <I>h</I> is +equal to the angle <I>O′I′T′</I> in the triangle <I>I′O′T′.</I> And in this +triangle <I>O′I′</I>, the radius, is known, while <I>O′T′</I> has been found; +and we have therefore +<MATH><I>O′T′</I>/<I>O′I′</I> = crd. (2<I>h</I>)/<I>D</I> = (crd. (180° - 2<I>t</I>)/<I>D</I>).(crd. (180° - 2<G>f</G>)/<I>D</I>)</MATH>, from above. +<p>[In other words, <MATH>sin <I>h</I> = cos<I>t</I>cos <G>f</G></MATH>; or, if <MATH><I>u</I> = <I>SC</I> = 90° - <I>t</I>, +sin <I>h</I> = sin <I>u</I> cos <G>f</G></MATH>, the formula for finding sin <I>h</I> in the right- +angled spherical triangle <I>SVC.</I>] +<p>For the azimuth <MATH><G>w</G> (arc <I>BV</I> = arc <I>B′X′</I>)</MATH>, the figure gives +<MATH>tan <G>w</G> = <I>XG′</I>/<I>G′O′</I> = <I>S′F′</I>/<I>F′T′</I> = (<I>S′F′</I>/<I>O′F′</I>) . (<I>O′F′</I>/<I>F′T′</I>) = tan <I>t</I> . (1/sin +<G>f</G>)</MATH>, +<pb n=291><head>THE <I>ANALEMMA</I> OF PTOLEMY</head> +or tan <I>VC</I> = tan <I>SC</I> cos <I>SCV</I> in the right-angled spherical +triangle <I>SVC.</I> +<p>Thirdly, +<MATH>tan <I>QZ</I> = tan <I>Z′Y′</I> = <I>S′F′</I>/<I>O′T′</I> = (<I>S′F′</I>/<I>O′F′</I>) . (<I>O′F′</I>/<I>O′T′</I>) = tan<I>t</I> . (1/cos<G>f</G>);</MATH> +that is, <MATH>tan <I>QZ</I>/tan <I>SM</I> = sin <I>BZ</I>/sin <I>BM</I></MATH>, which is Menelaus, <I>Sphaerica</I>, +III. 3, applied to the right-angled spherical triangles <I>ZBQ</I>, +<I>MBS</I> with the angle <I>B</I> common. +<p>Zeuthen points out that later in the same treatise Ptolemy +finds the arc 2<G>a</G> described above the horizon by a star of +given declination <G>d</G>′, by a procedure equivalent to the formula +<MATH>cos <G>a</G> = tan <G>d</G>′ tan <G>f</G></MATH>, +and this is the same formula which, as we have seen, +Hipparchus must in effect have used in his <I>Commentary on +the Phaenomena of Eudoxus and Aratus.</I> +<p>Lastly, with regard to the calculations of the height <I>h</I> and +the azimuth <G>w</G> in the general case where the sun's declination +is <G>d</G>′, Zeuthen has shown that they may be expressed by the +formulae +<MATH>sin <I>h</I> = (cos <G>d</G>′ cos <I>t</I> - sin <G>d</G>′ tan <G>f</G>) cos <G>f</G></MATH>, +and <MATH>tan <G>w</G> = cos <G>d</G>′ sin <I>t</I>/(sin <G>d</G>′/cos <G>f</G> + (cos <G>d</G>′ cos <I>t</I> - sin <G>d</G>′ tan <G>f</G>) sin <G>f</G>)</MATH>, +or <MATH>cos <G>d</G>′ sin <I>t</I>/(sin <G>d</G>′ cos <G>f</G> + cos <G>d</G>′ cos <I>t</I> sin <G>f</G>).</MATH> +<p>The statement therefore of A. v. Braunmühl<note>Braunmühl, i, pp. 13, 14, 38-41.</note> that the +Indians were the first to utilize the method of projection +contained in the <I>Analemma</I> for actual trigonometrical calcu- +lations with the help of the Table of Chords or Sines requires +modification in so far as the Greeks at all events showed the +way to such use of the figure. Whether the practical applica- +tion of the method of the <I>Analemma</I> for what is equivalent +to the solution of spherical triangles goes back as far as +Hipparchus is not certain; but it is quite likely that it does, +<pb n=292><head>TRIGONOMETRY</head> +seeing that Diodorus wrote his <I>Analemma</I> in the next cen- +tury. The other alternative source for Hipparchus's spherical +trigonometry is the Menelaus-theorem applied to the sphere, +on which alone Ptolemy, as we have seen, relies in his +<I>Syntaxis.</I> In any case the Table of Chords or Sines was in +full use in Hipparchus's works, for it is presupposed by either +method. +<C>The <I>Planisphaerium</I>.</C> +<p>With the <I>Analemma</I> of Ptolemy is associated another +work of somewhat similar content, the <I>Planisphaerium.</I> +This again has only survived in a Latin translation from an +Arabic version made by one Maslama b. Ahmad al-Majriti, of +Cordova (born probably at Madrid, died 1007/8); the transla- +tion is now found to be, not by Rudolph of Bruges, but by +‘Hermannus Secundus’, whose pupil Rudolph was; it was +first published at Basel in 1536, and again edited, with com- +mentary, by Commandinus (Venice, 1558). It has been +re-edited from the manuscripts by Heiberg in vol. ii. of his +text of Ptolemy. The book is an explanation of the system +of projection known as <I>stereographic</I>, by which points on the +heavenly sphere are represented on the plane of the equator +by projection from one point, a pole; Ptolemy naturally takes +the south pole as centre of projection, as it is the northern +hemisphere which he is concerned to represent on a plane. +Ptolemy is aware that the projections of all circles on the +sphere (great circles—other than those through the poles +which project into straight lines—and small circles either +parallel or not parallel to the equator) are likewise circles. +It is curious, however, that he does not give any general +proof of the fact, but is content to prove it of particular +circles, such as the ecliptic, the horizon, &c. This is remark- +able, because it is easy to show that, if a cone be described +with the pole as vertex and passing through any circle on the +sphere, i.e. a circular cone, in general oblique, with that circle +as base, the section of the cone by the plane of the equator +satisfies the criterion found for the ‘subcontrary sections’ by +Apollonius at the beginning of his <I>Conics</I>, and is therefore a +circle. The fact that the method of stereographic projection is +so easily connected with the property of subcontrary sections +<pb n=293><head>THE <I>PLANISPHAERIUM</I> OF PTOLEMY</head> +of oblique circular cones has led to the conjecture that Apollo- +nius was the discoverer of the method. But Ptolemy makes no +mention of Apollonius, and all that we know is that Synesius +of Cyrene (a pupil of Hypatia, and born about A.D. 365-370) +attributes the discovery of the method and its application to +Hipparchus; it is curious that he does not mention Ptolemy's +treatise on the subject, but speaks of himself alone as having +perfected the theory. While Ptolemy is fully aware that +circles on the sphere become circles in the projection, he says +nothing about the other characteristic of this method of pro- +jection, namely that the angles on the sphere are represented +by equal angles on the projection. +<p>We must content ourselves with the shortest allusion to +other works of Ptolemy. There are, in the first place, other +minor astronomical works as follows: +<p>(1) <G>*fa/seis a)planw=n a)ste/rwn</G> of which only Book II sur- +vives, (2) <G>*(gpoqe/seis tw=n planwme/nwn</G> in two Books, the first +of which is extant in Greek, the second in Arabic only, (3) the +inscription in Canobus, (4) <G>*proxei/rwn kano/nwn dia/tasis kai\ +yhfofori/a</G>. All these are included in Heiberg's edition, +vol. ii. +<C>The <I>Optics.</I></C> +<p>Ptolemy wrote an <I>Optics</I> in five Books, which was trans- +lated from an Arabic version into Latin in the twelfth +century by a certain Admiral Eugenius Siculus<note>See G. Govi, <I>L'ottica di Claudio Tolomeo di Eugenio Ammiraglio di +Sicilia</I>, . . . Torino, 1884; and particulars in G. Loria. <I>Le scienze esatte +nell' antica Grecia</I>, pp. 570, 571.</note>; Book I, +however, and the end of Book V are wanting. Books I, II +were physical, and dealt with generalities; in Book III +Ptolemy takes up the theory of mirrors, Book IV deals with +concave and composite mirrors, and Book V with refraction. +The theoretical portion would suggest that the author was +not very proficient in geometry. Many questions are solved +incorrectly owing to the assumption of a principle which is +clearly false, namely that ‘the image of a point on a mirror is +at the point of concurrence of two lines, one of which is drawn +from the luminous point to the centre of curvature of the +mirror, while the other is the line from the eye to the point +<pb n=294><head>TRIGONOMETRY</head> +on the mirror where the reflection takes place’; Ptolemy uses +the principle to solve various special cases of the following +problem (depending in general on a biquadratic equation and +now known as the problem of Alhazen), ‘Given a reflecting +surface, the position of a luminous point, and the position +of a point through which the reflected ray is required to pass, +to find the point on the mirror where the reflection will take +place.’ Book V is the most interesting, because it seems to +be the first attempt at a theory of refraction. It contains +many details of experiments with different media, air, glass, +and water, and gives tables of angles of refraction (<I>r</I>) corre- +sponding to different angles of incidence (<I>i</I>); these are calcu- +lated on the supposition that <I>r</I> and <I>i</I> are connected by an +equation of the following form, +<MATH><I>r</I> = <I>ai</I> - <I>bi</I><SUP>2</SUP></MATH>, +where <I>a, b</I> are constants, which is worth noting as the first +recorded attempt to state a law of refraction. +<p>The discovery of Ptolemy's <I>Optics</I> in the Arabic at once +made it clear that the work <I>De speculis</I> formerly attributed +to Ptolemy is not his, and it is now practically certain that it +is, at least in substance, by Heron. This is established partly +by internal evidence, e.g. the style and certain expressions +recalling others which are found in the same author's <I>Auto- +mata</I> and <I>Dioptra</I>, and partly by a quotation by Damianus +(<I>On hypotheses in Optics</I>, chap. 14) of a proposition proved by +‘the mechanician Heron in his own <I>Catoptrica</I>’, which appears +in the work in question, but is not found in Ptolemy's <I>Optics</I>, +or in Euclid's. The proposition in question is to the effect +that of all broken straight lines from the eye to the mirror +and from that again to the object, that particular broken line +is shortest in which the two parts make equal angles with the +surface of the mirror; the inference is that, as nature does +nothing in vain, we must assume that, in reflection from a +mirror, the ray takes the shortest course, i.e. the angles of +incidence and reflection are equal. Except for the notice in +Damianus and a fragment in Olympiodorus<note>Olympiodorus on Aristotle, <I>Meteor.</I> iii. 2, ed. Ideler, ii, p. 96, ed. +Stüve, pp. 212. 5-213. 20.</note> containing the +proof of the proposition, nothing remains of the Greek text; +<pb n=295><head>THE <I>OPTICS</I> OF PTOLEMY</head> +but the translation into Latin (now included in the Teubner +edition of Heron, ii, 1900, pp. 316-64), which was made by +William of Moerbeke in 1269, was evidently made from the +Greek and not from the Arabic, as is shown by Graecisms in +the translation. +<C>A mechanical work, <G>*peri\ r(opw=n</G>.</C> +<p>There are allusions in Simplicius<note>Simplicius on Arist. <I>De caelo</I>, p. 710. 14, Heib. (Ptolemy, ed. Heib., +vol. ii, p. 263).</note> and elsewhere to a book +by Ptolemy of mechanical content, <G>peri\ r(opw=n</G>, on balancings +or turnings of the scale, in which Ptolemy maintained as +against Aristotle that air or water (e.g.) in their own ‘place’ +have no weight, and, when they are in their own ‘place’, either +remain at rest or rotate simply, the tendency to go up or to +fall down being due to the desire of things which are not in +their own places to move to them. Ptolemy went so far as to +maintain that a bottle full of air was not only not heavier +than the same bottle empty (as Aristotle held), but actually +lighter when inflated than when empty. The same work is +apparently meant by the ‘book on the elements’ mentioned +by Simplicius.<note><I>Ib.</I>, p. 20. 10 sq.</note> Suidas attributes to Ptolemy three Books of +<I>Mechanica.</I> +<p>Simplicius<note><I>Ib.</I>, p. 9. 21 sq., (Ptolemy, ed. Heib., vol. ii, p. 265).</note> also mentions a single book, <G>peri\ diasta/sews</G>, +‘<I>On dimension</I>’, i.e. dimensions, in which Ptolemy tried to +show that the possible number of dimensions is limited to +three. +<C>Attempt to prove the Parallel-Postulate.</C> +<p>Nor should we omit to notice Ptolemy's attempt to prove +the Parallel-Postulate. Ptolemy devoted a tract to this +subject, and Proclus<note>Proclus on Eucl. I, pp. 362. 14 sq., 365. 7-367. 27 (Ptolemy, ed. Heib., +vol. ii, pp. 266-70).</note> has given us the essentials of the argu- +ment used. Ptolemy gives, first, a proof of Eucl. I. 28, and +then an attempted proof of I. 29, from which he deduces +Postulate 5. +<pb n=296><head>TRIGONOMETRY</head> +<p>I. To prove I. 28, Ptolemy takes two straight lines <I>AB, CD</I>, +and a transversal <I>EFGH.</I> We have to prove that, if the sum +<FIG> +of the angles <I>BFG, FGD</I> is equal to two right angles, the +straight lines <I>AB, CD</I> are parallel, i.e. non-secant. +<p>Since <I>AFG</I> is the supplement of <I>BFG</I>, and <I>FGC</I> of <I>FGD</I>, it +follows that the sum of the angles <I>AFG, FGC</I> is also equal to +two right angles. +<p>Now suppose, if possible, that <I>FB, GD</I>, making the sum of +the angles <I>BFG, FGD</I> equal to two right angles, meet at <I>K</I>; +then similarly <I>FA, GC</I> making the sum of the angles <I>AFG, +FGC</I> equal to two right angles must also meet, say at <I>L.</I> +<p>[Ptolemy would have done better to point out that not +only are the two sums equal but the angles themselves are +equal in pairs, i.e. <I>AFG</I> to <I>FGD</I> and <I>FGC</I> to <I>BFG</I>, and we can +therefore take the triangle <I>KFG</I> and apply it to <I>FG</I> on the other +side so that the sides <I>FK, GK</I> may lie along <I>GC, FA</I> respec- +tively, in which case <I>GC, FA</I> will meet at the point where +<I>K</I> falls.] +<p>Consequently the straight lines <I>LABK, LCDK</I> enclose a +space: which is impossible. +<p>It follows that <I>AB, CD</I> cannot meet in either direction; +they are therefore parallel. +<p>II. To prove I. 29, Ptolemy takes two parallel lines <I>AB, +CD</I> and the transversal <I>FG</I>, and argues thus. It is required +to prove that <MATH>∠<I>AFG</I> + ∠<I>CGF</I> =</MATH> two right angles. +<p>For, if the sum is not equal to two right angles, it must be +either (1) greater or (2) less. +<p>(1) If it is greater, the sum of the angles on the other side, +<I>BFG, FGD</I>, which are the supplements of the first pair of +angles, must be <I>less</I> than two right angles. +<p>But <I>AF, CG</I> are no more parallel than <I>FB, GD, so that, if +FG makes one pair of angles AFG, FGC together greater than</I> +<pb n=297><head>PTOLEMY ON THE PARALLEL-POSTULATE</head> +<I>two right angles, it must also make the other pair BFG, FGD +together greater than two right angles.</I> +<p>But the latter pair of angles were proved less than two +right angles: which is impossible. +<p>Therefore the sum of the angles <I>AFG, FGC</I> cannot be +<I>greater</I> than two right angles. +<p>(2) Similarly we can show that the sum of the two angles +<I>AFG, FGC</I> cannot be <I>less</I> than two right angles. +<p>Therefore <MATH>∠<I>AFG</I> + ∠<I>CGF</I> =</MATH> two right angles. +<p>[The fallacy here lies in the inference which I have marked +by italics. When Ptolemy says that <I>AF, CG</I> are no more +parallel than <I>FB, GD</I>, he is in effect assuming that <I>through +any one point only one parallel can be drawn to a given straight +line</I>, which is an equivalent for the very Postulate he is +endeavouring to prove. The alternative Postulate is known +as ‘Playfair's axiom’, but it is of ancient origin, since it is +distinctly enunciated in Proclus's note on Eucl. I. 31.] +<p>III. Post. 5 is now deduced, thus. +<p>Suppose that the straight lines making with a transversal +angles the sum of which is less than two right angles do not +meet on the side on which those angles are. +<p>Then, <I>a fortiori</I>, they will not meet on the other side on +which are the angles the sum of which is <I>greater</I> than two +right angles. [This is enforced by a supplementary proposi- +tion showing that, if the lines met on that side, Eucl. I. 16 +would be contradicted.] +<p>Hence the straight lines cannot meet in either direction: +they are therefore <I>parallel.</I> +<p>But in that case the angles made with the transversal are +<I>equal</I> to two right angles: which contradicts the assumption. +<p>Therefore the straight lines will meet. +<pb> +<C>XVIII +MENSURATION: HERON OF ALEXANDRIA +Controversies as to Heron's date.</C> +<p>THE vexed question of Heron's date has perhaps called +forth as much discussion as any doubtful point in the history +of mathematics. In the early stages of the controversy much +was made of the supposed relation of Heron to Ctesibius. +The <I>Belopoeïca</I> of Heron has, in the best manuscript, the +heading <G>*(/hrwnos *kthsibi/ou *belopoii+ka/</G>, and from this, coupled +with an expression used by an anonymous Byzantine writer +of the tenth century, <G>o( *)askrhno\s *kthsi/bios o( tou= *)alexandre/ws +*(/hrwnos kaqhghth/s</G>, ‘Ctesibius of Ascra, the teacher of Heron +of Alexandria’, it was inferred that Heron was a pupil of +Ctesibius. The question then was, when did Ctesibius live? +Martin took him to be a certain barber of that name who +lived in the time of Ptolemy Euergetes II, that is, Ptolemy VII, +called Physcon (died 117 B.C.), and who is said to have made +an improved water-organ<note>Athenaeus, <I>Deipno-Soph.</I> iv. c. 75, p. 174 b-e: cf. Vitruvius, x. 9, 13.</note>; Martin therefore placed Heron at +the beginning of the first century (say 126-50) B.C. But +Philon of Byzantium, who repeatedly mentions Ctesibius by +name, says that the first mechanicians (<G>texni=tai</G>) had the +great advantage of being under kings who loved fame and +supported the arts.<note>Philon, <I>Mechan. Synt.,</I> p. 50. 38, ed. Schöne.</note> This description applies much better +to Ptolemy II Philadelphus (285-247) and Ptolemy III Euer- +getes I (247-222). It is more probable, therefore, that Ctesibius +was the mechanician Ctesibius who is mentioned by Athenaeus +as having made an elegant drinking-horn in the time of +Ptolemy Philadelphus<note>Athenaeus, xi. c. 97, p. 497 b-e.</note>; a pupil then of Ctesibius would +probably belong to the end of the third and the beginning of +the second century B.C. But in truth we cannot safely con- +clude that Heron was an immediate pupil of Ctesibius. The +Byzantine writer probably only inferred this from the title +<pb n=299><head>CONTROVERSIES AS TO HERON'S DATE</head> +above quoted; the title, however, in itself need not imply +more than that Heron's work was a new edition of a similar +work by Ctesibius, and the <G>*kthsibi/ou</G> may even have been added +by some well-read editor who knew both works and desired to +indicate that the greater part of the contents of Heron's work +was due to Ctesibius. One manuscript has <G>*(/hrwnos *)alexan- +dre/ws *belopoii+ka/</G>, which corresponds to the titles of the other +works of Heron and is therefore more likely to be genuine. +<p>The discovery of the Greek text of the <I>Metrica</I> by R. Schöne +in 1896 made it possible to fix with certainty an upper limit. +In that work there are a number of allusions to Archimedes, +three references to the <G>xwri/ou a)potomh/</G> of Apollonius, and +two to ‘the (books) about straight lines (chords) in a circle’ +(<G>de/deiktai de\ e)n toi=s peri\ tw=n e)n ku/klw| eu)qeiw=n</G>). Now, although +the first beginnings of trigonometry may go back as far as +Apollonius, we know of no work giving an actual Table of +Chords earlier than that of Hipparchus. We get, therefore, +at once the date 150 B.C. or thereabouts as the <I>terminus post +quem.</I> A <I>terminus ante quem</I> is furnished by the date of the +composition of Pappus's <I>Collection;</I> for Pappus alludes to, and +draws upon, the works of Heron. As Pappus was writing in +the reign of Diocletian (A.D. 284-305), it follows that Heron +could not be much later than, say, A.D. 250. In speaking of +the solutions by ‘the old geometers’ (<G>oi( palaioi\ gewme/trai</G>) of +the problem of finding the two mean proportionals, Pappus may +seem at first sight to include Heron along with Eratosthenes, +Nicomedes and Philon in that designation, and it has been +argued, on this basis, that Heron lived long before Pappus. +But a close examination of the passage<note>Pappus, iii, pp. 54-6.</note> shows that this is +by no means necessary. The relevant words are as follows: +<p>‘The ancient geometers were not able to solve the problem +of the two straight lines [the problem of finding two mean +proportionals to them] by ordinary geometrical methods, since +the problem is by nature “solid” . . . but by attacking it with +mechanical means they managed, in a wonderful way, to +reduce the question to a practical and convenient construction, +as may be seen in the <I>Mesolabon</I> of Eratosthenes and in the +mechanics of Philon and Heron . . . Nicomedes also solved it +by means of the cochloid curve, with which he also trisected +an angle.’ +<pb n=300><head>HERON OF ALEXANDRIA</head> +<p>Pappus goes on to say that he will give four solutions, one +of which is his own; the first, second, and third he describes +as those of Eratosthenes, Nicomedes and Heron. But in the +earlier sentence he mentions Philon along with Heron, and we +know from Eutocius that Heron's solution is practically the +same as Philon's. Hence we may conclude that by the third +solution Pappus really meant Philon's, and that he only men- +tioned Heron's <I>Mechanics</I> because it was a convenient place in +which to find the same solution. +<p>Another argument has been based on the fact that the +extracts from Heron's <I>Mechanics</I> given at the end of Pappus's +Book VIII, as we have it, are introduced by the author with +a complaint that the copies of Heron's works in which he +found them were in many respects corrupt, having lost both +beginning and end.<note>Pappus, viii, p. 1116. 4-7.</note> But the extracts appear to have been +added, not by Pappus, but by some later writer, and the +argument accordingly falls to the ground. +<p>The limits of date being then, say, 150 B.C. to A.D. 250, our +only course is to try to define, as well as possible, the relation +in time between Heron and the other mathematicians who +come, roughly, within the same limits. This method has led +one of the most recent writers on the subject (Tittel<note>Art. ‘Heron von Alexandreia’ in Pauly-Wissowa's <I>Real-Encyclopädie +der class. Altertumswissenschaft,</I> vol. 8. 1, 1912.</note>) to +place Heron not much later than 100 B.C., while another,<note>I. Hammer-Jensen in <I>Hermes,</I> vol. 48, 1913, pp. 224-35.</note> +relying almost entirely on a comparison between passages in +Ptolemy and Heron, arrives at the very different conclusion +that Heron was later than Ptolemy and belonged in fact to +the second century A.D. +<p>In view of the difference between these results, it will be +convenient to summarize the evidence relied on to establish +the earlier date, and to consider how far it is or is not con- +clusive against the later. We begin with the relation of +Heron to Philon. Philon is supposed to come not more than +a generation later than Ctesibius, because it would appear that +machines for throwing projectiles constructed by Ctesibius +and Philon respectively were both available at one time for +inspection by experts on the subject<note>Philon, <I>Mech. Synt.</I> iv, pp. 68. 1, 72. 36.</note>; it is inferred that +<pb n=301><head>CONTROVERSIES AS TO HERON'S DATE</head> +Philon's date cannot be later than the end of the second +century B.C. (If Ctesibius flourished before 247 B.C. the argu- +ment would apparently suggest rather the beginning than the +end of the second century.) Next, Heron is supposed to have +been a younger contemporary of Philon, the grounds being +the following. (1) Heron mentions a ‘stationary-automaton’ +representation by Philon of the Nauplius-story,<note>Heron, <I>Autom.,</I> pp. 404. 11-408. 9.</note> and this is +identified by Tittel with a representation of the same story by +some contemporary of Heron's (<G>oi( kaq' h(ma=s</G><note><I>Ib.,</I> p. 412. 13.</note>). But a careful +perusal of the whole passage seems to me rather to suggest +that the latter representation was not Philon's, and that +Philon was included by Heron among the ‘ancient’ auto- +maton-makers, and not among his contemporaries.<note>The relevant remarks of Heron are as follows. (1) He says that he +has found no arrangements of ‘stationary automata’ better or more +instructive than those described by Philon of Byzantium (p. 404. 11). +As an instance he mentions Philon's setting of the Nauplius-story, in +which he found everything good except two things (<I>a</I>) the mechanism +for the appearance of Athene, which was too difficult (<G>e)rgwde/steron</G>), and +(<I>b</I>) the absence of an incident promised by Philon in his description, +namely the falling of a thunderbolt on Ajax with a sound of thunder +accompanying it (pp. 404. 15-408. 9). This latter incident Heron could +not find anywhere in Philon, though he had consulted a great number +of copies of his work. He continues (p. 408. 9-13) that we are not to +suppose that he is running down Philon or charging him with not being +capable of carrying out what he promised. On the contrary, the omission +was probably due to a slip of memory, for it is easy enough to make +stage-thunder (he proceeds to show how to do it). But the rest of +Philon's arrangements seemed to him satisfactory, and this, he says, is +why he has not ignored Philon's work: ‘for I think that my readers will +get the most benefit if they are shown, first what has been well said by +the ancients and then, separately from this, what the ancients overlooked +or what in their work needed improvement’ (pp. 408.22-410.6). (2) The +next chapter (pp. 410. 7-412. 2) explains generally the sort of thing the +automaton-picture has to show, and Heron says he will give one example +which he regards as the best. Then (3), after drawing a contrast between +the simpler pictures made by ‘the ancients’, which involved three different +movements only, and the contemporary (<G>oi( kaq' h(ma=s</G>) representations of +interesting stories by means of more numerous and varied movements +(p. 412. 3-15), he proceeds to describe a setting of the Nauplius-story. +This is the representation which Tittel identifies with Philon's. But it +is to be observed that the description includes that of the episode of the +thunderbolt striking Ajax (c. 30, pp. 448. 1-452. 7) which Heron expressly +says that Philon omitted. Further, the mechanism for the appearance +of Athene described in c. 29 is clearly not. Philon's ‘more difficult’ +arrangement, but the simpler device described (pp. 404. 18-408. 5) as +possible and preferable to Philon's (cf. Heron, vol. i, ed. Schmidt, pp. +lxviii-lxix).</note> (2) Another +argument adduced to show that Philon was contemporary +<pb n=302><head>HERON OF ALEXANDRIA</head> +with Heron is the fact that Philon has some criticisms of +details of construction of projectile-throwers which are found +in Heron, whence it is inferred that Philon had Heron's work +specifically in view. But if Heron's <G>*belopoii+ka/</G> was based on +the work of Ctesibius, it is equally possible that Philon may +be referring to Ctesibius. +<p>A difficulty in the way of the earlier date is the relation in +which Heron stands to Posidonius. In Heron's <I>Mechanics,</I> +i. 24, there is a definition of ‘centre of gravity’ which is +attributed by Heron to ‘Posidonius a Stoic’. But this can +hardly be Posidonius of Apamea, Cicero's teacher, because the +next sentence in Heron, stating a distinction drawn by Archi- +medes in connexion with this definition, seems to imply that +the Posidonius referred to lived before Archimedes. But the +<I>Definitions</I> of Heron do contain definitions of geometrical +notions which are put down by Proclus to Posidonius of +Apamea or Rhodes, and, in particular, definitions of ‘figure’ +and of ‘parallels’. Now Posidonius lived from 135 to 51 B.C., +and the supporters of the earlier date for Heron can only +suggest that either Posidonius was not the first to give these +definitions, or alternatively, if he was, and if they were +included in Heron's <I>Definitions</I> by Heron himself and not by +some later editor, all that this obliges us to admit is that +Heron cannot have lived before the first century B.C. +<p>Again, if Heron lived at the beginning of the first cen- +tury B.C., it is remarkable that he is nowhere mentioned by +Vitruvius. The <I>De architectura</I> was apparently brought out +in 14 B.C. and in the preface to Book VII Vitruvius gives +a list of authorities on <I>machinationes</I> from whom he made +extracts. The list contains twelve names and has every +appearance of being scrupulously complete; but, while it +includes Archytas (second), Archimedes (third), Ctesibius +(fourth), and Philon of Byzantium (sixth), it does not men- +tion Heron. Nor is it possible to establish interdependence +between Heron and Vitruvius; the differences seem, on the +whole, to be more numerous than the resemblances. A few of +the differences may be mentioned. Vitruvius uses 3 as the +value of <G>p</G>, whereas Heron always uses the Archimedean value +3 1/7. Both writers make extracts from the Aristotelian +<G>*mhxanika\ problh/mata</G>, but their selections are different. The +<pb n=303><head>CONTROVERSIES AS TO HERON'S DATE</head> +machines used by the two for the same purpose frequently +differ in details; e.g. in Vitruvius's hodometer a pebble drops +into a box at the end of each Roman mile,<note>Vitruvius, x. 14.</note> while in Heron's +the distance completed is marked by a pointer.<note>Heron, <I>Dioptra,</I> c. 34.</note> It is indeed +pointed out that the water-organ of Heron is in many respects +more primitive than that of Vitruvius; but, as the instru- +ments are altogether different, this can scarcely be said to +prove anything. +<p>On the other hand, there are points of contact between +certain propositions of Heron and of the Roman <I>agrimen- +sores.</I> Columella, about A.D. 62, gave certain measurements of +plane figures which agree with the formulae used by Heron, +notably those for the equilateral triangle, the regular hexagon +(in this case not only the formula but the actual figures agree +with Heron's) and the segment of a circle which is less than +a semicircle, the formula in the last case being +<MATH>1/2(<I>s</I>+<I>h</I>)<I>h</I>+1/14(1/2<I>s</I>)<SUP>2</SUP>,</MATH> +where <I>s</I> is the chord and <I>h</I> the height of the segment. Here +there might seem to be dependence, one way or the other; +but the possibility is not excluded that the two writers may +merely have drawn from a common source; for Heron, in +giving the formula for the area of the segment of a circle, +states that it was the formula used by ‘the more accurate +investigators’ (<G>oi( a)kribe/steron e)zhthko/tes</G>).<note>Heron, <I>Metrica,</I> i. 31, p. 74. 21.</note> +<p>We have, lastly, to consider the relation between Ptolemy +and Heron. If Heron lived about 100 B.C., he was 200 years +earlier than Ptolemy (A.D. 100-178). The argument used to +prove that Ptolemy came some time after Heron is based on +a passage of Proclus where Ptolemy is said to have remarked +on the untrustworthiness of the method in vogue among the +‘more ancient’ writers of measuring the apparent diameter of +the sun by means of water-clocks.<note>Proclus, <I>Hypotyposis,</I> pp. 120. 9-15, 124. 7-26.</note> Hipparchus, says Pro- +clus, used his dioptra for the purpose, and Ptolemy followed +him. Proclus proceeds: +<p>‘Let us then set out here not only the observations of +the ancients but also the construction of the dioptra of +<pb n=304><head>HERON OF ALEXANDRIA</head> +Hipparchus. And first we will show how we can measure an +interval of time by means of the regular efflux of water, +a procedure which was explained by Heron the mechanician +in his treatise on water-clocks.’ +<p>Theon of Alexandria has a passage to a similar effect.<note>Theon, <I>Comm. on the Syntaxis,</I> Basel, 1538, pp. 261 sq. (quoted in +Proclus, <I>Hypotyposis,</I> ed. Manitius, pp. 309-11).</note> He +first says that the most ancient mathematicians contrived +a vessel which would let water flow out uniformly through a +small aperture at the bottom, and then adds at the end, almost +in the same words as Proclus uses, that Heron showed how +this is managed in the first book of his work on water- +clocks. Theon's account is from Pappus's Commentary on +the <I>Syntaxis,</I> and this is also Proclus's source, as is shown by +the fact that Proclus gives a drawing of the water-clock +which appears to have been lost in Theon's transcription from +Pappus, but which Pappus must have reproduced from the +work of Heron. Tittel infers that Heron must have ranked +as one of the ‘more ancient’ writers as compared with +Ptolemy. But this again does not seem to be a necessary +inference. No doubt Heron's work was a convenient place to +refer to for a description of a water-clock, but it does not +necessarily follow that Ptolemy was referring to Heron's +clock rather than some earlier form of the same instrument. +<p>An entirely different conclusion from that of Tittel is +reached in the article ‘Ptolemaios and Heron’ already alluded +to.<note>Hammer-Jensen, <I>op. cit.</I></note> The arguments are shortly these. (1) Ptolemy says in +his <I>Geography</I> (c. 3) that his predecessors had only been able +to measure the distance between two places (as an are of a +great circle on the earth's circumference) in the case where +the two places are on the same meridian. He claims that he +himself invented a way of doing this even in the case where +the two places are neither on the same meridian nor on the +same parallel circle, provided that the heights of the pole at +the two places respectively, and the angle between the great +circle passing through both and the meridian circle through +one of the places, are known. Now Heron in his <I>Dioptra</I> +deals with the problem of measuring the distance between +two places by means of the dioptra, and takes as an example +<pb n=305><head>CONTROVERSIES AS TO HERON'S DATE</head> +the distance between Rome and Alexandria.<note>Heron, <I>Dioptra,</I> c. 35 (vol. iii, pp. 302-6).</note> Unfortunately +the text is in places corrupt and deficient, so that the method +cannot be reconstructed in detail. But it involved the obser- +vation of the same lunar eclipse at Rome and Alexandria +respectively and the drawing of the <I>analemma</I> for Rome. +That is to say, the mathematical method which Ptolemy +claims to have invented is spoken of by Heron as a thing +generally known to experts and not more remarkable than +other technical matters dealt with in the same book. Conse- +quently Heron must have been later than Ptolemy. (It is +right to add that some hold that the chapter of the <I>Dioptra</I> +in question is not germane to the subject of the treatise, and +was probably not written by Heron but interpolated by some +later editor; if this is so, the argument based upon it falls to +the ground.) (2) The dioptra described in Heron's work is a +fine and accurate instrument, very much better than anything +Ptolemy had at his disposal. If Ptolemy had been aware of +its existence, it is highly unlikely that he would have taken +the trouble to make his separate and imperfect ‘parallactic’ +instrument, since it could easily have been grafted on to +Heron's dioptra. Not only, therefore, must Heron have been +later than Ptolemy but, seeing that the technique of instru- +ment-making had made such strides in the interval, he must +have been considerably later. (3) In his work <G>peri\ r(opw=n</G><note>Simplicius on <I>De caelo,</I> p. 710. 14, Heib. (Ptolemy, vol. ii, p. 263).</note> +Ptolemy, as we have seen, disputed the view of Aristotle that +air has weight even when surrounded by air. Aristotle +satisfied himself experimentally that a vessel full of air is +heavier than the same vessel empty; Ptolemy, also by ex- +periment, convinced himself that the former is actually the +lighter. Ptolemy then extended his argument to water, and +held that water with water round it has no weight, and that +the diver, however deep he dives, does not feel the weight of +the water above him. Heron<note>Heron, <I>Pneumatica,</I> i. Pref. (vol. i, p. 22. 14 sq.).</note> asserts that water has no +appreciable weight and has no appreciable power of com- +pressing the air in a vessel inverted and forced down into +the water. In confirmation of this he cites the case of the +diver, who is not prevented from breathing when far below +<pb n=306><head>HERON OF ALEXANDRIA</head> +the surface. He then inquires what is the reason why the +diver is not oppressed though he has an unlimited weight of +water on his back. He accepts, therefore, the view of Ptolemy +as to the fact, however strange this may seem. But he is not +satisfied with the explanation given: ‘Some say’, he goes on, +‘it is because water in itself is uniformly heavy (<G>i)sobare\s au)to\ +kaq' au(to/</G>)’—this seems to be equivalent to Ptolemy's dictum +that water in water has no weight—‘but they give no ex- +planation whatever why divers...’ He himself attempts an +explanation based on Archimedes. It is suggested, therefore, +that Heron's criticism is directed specifically against Ptolemy +and no one else. (4) It is suggested that the Dionysius to whom +Heron dedicated his <I>Definitions</I> is a certain Dionysius who +was <I>praefectus urbi</I> at Rome in A.D. 301. The grounds are +these (<I>a</I>) Heron addresses Dionysius as <G>*dionu/sie lampro/tate</G>, +where <G>lampro/tatos</G> obviously corresponds to the Latin <I>clarissi- +mus,</I> a title which in the third century and under Diocletian +was not yet in common use. Further, this Dionysius was +<I>curator aquarum</I> and <I>curator operum publicorum,</I> so that he +was the sort of person who would have to do with the +engineers, architects and craftsmen for whom Heron wrote. +Lastly, he is mentioned in an inscription commemorating an +improvement of water supply and dedicated ‘to Tiberinus, +father of all waters, and to the ancient inventors of marvel- +lous constructions’ (<I>repertoribus admirabilium fabricarum +priscis viris</I>), an expression which is not found in any other +inscription, but which recalls the sort of tribute that Heron +frequently pays to his predecessors. This identification of the +two persons named Dionysius is an ingenious conjecture, but +the evidence is not such as to make it anything more.<note>Dionysius was of course a very common name. Diophantus dedicated +his <I>Arithmetica</I> to a person of this name (<G>timiw/tate/ moi *dionu/sie</G>), whom he +praised for his ambition to learn the solutions of arithmetical problems. +This Dionysius must have lived in the second half of the third century +A.D., and if Heron also belonged to this time, is it not possible that +Heron's Dionysius was the same person?</note> +<p>The result of the whole investigation just summarized is to +place Heron in the third century A.D., and perhaps little, if +anything, earlier than Pappus. Heiberg accepts this conclu- +sion,<note>Heron, vol. v, p. ix.</note> which may therefore, I suppose, be said to hold the field +for the present. +<pb n=307><head>CONTROVERSIES AS TO HERON'S DATE</head> +<p>Heron was known as <G>o( *)alexandreu/s</G> (e.g. by Pappus) or +<G>o( mhxaniko/s</G> (<I>mechanicus</I>), to distinguish him from other +persons of the same name; Proclus and Damianus use the +latter title, while Pappus also speaks of <G>oi( peri\ to\n *(/hrwna +mhxanikoi/</G>. +<C>Character of works.</C> +<p>Heron was an almost encyclopaedic writer on mathematical +and physical subjects. Practical utility rather than theoreti- +cal completeness was the object aimed at; his environment in +Egypt no doubt accounts largely for this. His <I>Metrica</I> begins +with the old legend of the traditional origin of geometry in +Egypt, and in the <I>Dioptra</I> we find one of the very problems +which geometry was intended to solve, namely that of re- +establishing boundaries of lands when the flooding of the +Nile had destroyed the land-marks: ‘When the boundaries +of an area have become obliterated to such an extent that +only two or three marks remain, in addition to a plan of the +area, to supply afresh the remaining marks.’<note>Heron, <I>Dioptra,</I> c. 25, p. 268. 17-19.</note> Heron makes +little or no claim to originality; he often quotes authorities, +but, in accordance with Greek practice, he more frequently +omits to do so, evidently without any idea of misleading any +one; only when he has made what is in his opinion any +slight improvement on the methods of his predecessors does +he trouble to mention the fact, a habit which clearly indi- +cates that, except in these cases, he is simply giving the best +traditional methods in the form which seemed to him easiest +of comprehension and application. The <I>Metrica</I> seems to be +richest in definite references to the discoveries of prede- +cessors; the names mentioned are Archimedes; Dionysodorus, +Eudoxus, Plato; in the <I>Dioptra</I> Eratosthenes is quoted, and +in the introduction to the <I>Catoptrica</I> Plato and Aristotle are +mentioned. +<p>The practical utility of Heron's manuals being so great, it +was natural that they should have great vogue, and equally +natural that the most popular of them at any rate should be +re-edited, altered and added to by later writers; this was +inevitable with books which, like the <I>Elements</I> of Euclid, +were in regular use in Greek, Byzantine, Roman, and Arabian +<pb n=308><head>HERON OF ALEXANDRIA</head> +education for centuries. The geometrical or mensurational +books in particular gave scope for expansion by multiplication +of examples, so that it is difficult to disentangle the genuine +Heron from the rest of the collections which have come down +to us under his name. Hultsch's considered criterion is as +follows: ‘The Heron texts which have come down to our +time are authentic in so far as they bear the author's name +and have kept the original design and form of Heron's works, +but are unauthentic in so far as, being constantly in use for +practical purposes, they were repeatedly re-edited and, in the +course of re-editing, were rewritten with a view to the +particular needs of the time.’ +<C>List of Treatises.</C> +<p>Such of the works of Heron as have survived have reached +us in very different ways. Those which have come down in +the Greek are: +<p>I. The <I>Metrica,</I> first discovered in 1896 in a manuscript +of the eleventh (or twelfth) century at Constantinople by +R. Schöne and edited by his son, H. Schöne (<I>Heronis Opera,</I> iii, +Teubner, 1903). +<p>II. <I>On the Dioptra,</I> edited in an Italian version by Venturi +in 1814; the Greek text was first brought out by A. J. H. +Vincent<note><I>Notices et extraits des manuscrits de la Bibliothèque impériale,</I> xix, pt. 2, +pp. 157-337.</note> in 1858, and the critical edition of it by H. Schöne is +included in the Teubner vol. iii just mentioned. +<p>III. The <I>Pneumatica,</I> in two Books, which appeared first in +a Latin translation by Commandinus, published after his +death in 1575; the Greek text was first edited by Thévenot +in <I>Veterum mathematicorum opera Graece et Latine edita</I> +(Paris, 1693), and is now available in <I>Heronis Opera,</I> i (Teub- +ner, 1899), by W. Schmidt. +<p>IV. <I>On the art of constructing automata</I> (<G>peri\ au)tomato- +poihtikh=s</G>), or <I>The automaton-theatre,</I> first edited in an Italian +translation by B. Baldi in 1589; the Greek text was included +in Thévenot's <I>Vet. math.,</I> and now forms part of <I>Heronis +Opera,</I> vol. i, by W. Schmidt. +<p>V. <I>Belopoeïca</I> (on the construction of engines of war), edited +<pb n=309><head>LIST OF TREATISES</head> +by B. Baldi (Augsburg, 1616), Thévenot (<I>Vet. math.</I>), Köchly +and Rüstow (1853) and by Wescher (<I>Poliorcétique des Grecs,</I> +1867, the first critical edition). +<p>VI. The <I>Cheirobalistra</I> (<G>*(/hrwnos xeiroballi/stras kataskeuh\ +kai\ summetri/a</G> (?)), edited by V. Prou, <I>Notices et extraits,</I> xxvi. 2 +(Paris, 1877). +<p>VII. The geometrical works, <I>Definitiones, Geometria, Geo- +daesia, Stereometrica</I> I and II, <I>Mensurae, Liber Geeponicus,</I> +edited by Hultsch with <I>Variae collectiones</I> (<I>Heronis Alexan- +drini geometrioorum et stereometricorum reliquiae,</I> 1864). +This edition will now be replaced by that of Heiberg in the +Teubner collection (vols. iv, v), which contains much addi- +tional matter from the Constantinople manuscript referred to, +but omits the <I>Liber Geeponicus</I> (except a few extracts) and the +<I>Geodaesia</I> (which contains only a few extracts from the +<I>Geometry</I> of Heron). +<p>Only fragments survive of the Greek text of the <I>Mechanics</I> +in three Books, which, however, is extant in the Arabic (now +edited, with German translation, in <I>Heronis Opera,</I> vol. ii, +by L. Nix and W. Schmidt, Teubner, 1901). +<p>A smaller separate mechanical treatise, the <G>*baroulko/s</G>, is +quoted by Pappus.<note>Pappus, viii, p. 1060. 5.</note> The object of it was ‘to move a given +weight by means of a given force’, and the machine consisted +of an arrangement of interacting toothed wheels with different +diameters. +<p>At the end of the <I>Dioptra</I> is a description of a <I>hodometer</I> for +measuring distances traversed by a wheeled vehicle, a kind of +taxameter, likewise made of a combination of toothed wheels. +<p>A work on <I>Water-clocks</I> (<G>peri\ u(dri/wn w(roskopei/wn</G>) is men- +tioned in the <I>Pneumatica</I> as having contained four Books, +and is also alluded to by Pappus.<note><I>Ib.,</I> p. 1026. 1.</note> Fragments are preserved +in Proclus (<I>Hypotyposis,</I> chap. 4) and in Pappus's commentary +on Book V of Ptolemy's <I>Syntaxis</I> reproduced by Theon. +<p>Of Heron's <I>Commentary on Euclid's Elements</I> only very +meagre fragments survive in Greek (Proclus), but a large +number of extracts are fortunately preserved in the Arabic +commentary of an-Nairīzī, edited (1) in the Latin version of +Gherard of Cremona by Curtze (Teubner, 1899); and (2) by +<pb n=310><head>HERON OF ALEXANDRIA</head> +Besthorn and Heiberg (<I>Codex Leidensis</I> 399. 1, five parts of +which had appeared up to 1910). The commentary extended +as far as <I>Elem.</I> VIII. 27 at least. +<p>The <I>Catoptrica,</I> as above remarked under Ptolemy, exists in +a Latin translation from the Greek, presumed to be by William +of Moerbeke, and is included in vol. ii of <I>Heronis Opera,</I> +edited, with introduction, by W. Schmidt. +<p>Nothing is known of the <I>Camarica</I> (‘on vaultings’) men- +tioned by Eutocius (on Archimedes, <I>Sphere and Cylinder</I>), the +<I>Zygia</I> (balancings) associated by Pappus with the <I>Automata,</I><note>Pappus, viii, p. 1024. 28.</note> +or of a work on the use of the astrolabe mentioned in the +<I>Fihrist.</I> +<p>We are in this work concerned with the treatises of mathe- +matical content, and therefore can leave out of account such +works as the <I>Pneumatica,</I> the <I>Automata,</I> and the <I>Belopoeïca.</I> +The <I>Pneumatica</I> and <I>Automata</I> have, however, an interest to +the historian of physics in so far as they employ the force of +compressed air, water, or steam. In the <I>Pneumatica</I> the +reader will find such things as siphons, ‘Heron's fountain’, +‘penny-in-the-slot’ machines, a fire-engine, a water-organ, and +many arrangements employing the force of steam. +<C>Geometry.</C> +<C>(<I>a</I>) <I>Commentary on Euclid's Elements.</I></C> +<p>In giving an account of the geometry and mensuration +(or geodesy) of Heron it will be well, I think, to begin +with what relates to the <I>elements,</I> and first the Commen- +tary on Euclid's <I>Elements,</I> of which we possess a number +of extracts in an-Nairīzī and Proclus, enabling us to form +a general idea of the character of the work. Speaking +generally, Heron's comments do not appear to have contained +much that can be called important. They may be classified +as follows: +<p>(1) A few general notes, e.g. that Heron would not admit +more than three axioms. +<p>(2) Distinctions of a number of particular <I>cases</I> of Euclid's +propositions according as the figure is drawn in one way +or another. +<pb n=311><head>GEOMETRY</head> +<p>Of this class are the different cases of I. 35, 36, III. 7, 8 +(where the chords to be compared are drawn on different sides +of the diameter instead of on the same side), III. 12 (which is +not Euclid's at all but Heron's own, adding the case of +external to that of internal contact in III. 11, VI. 19 (where +the triangle in which an additional line is drawn is taken to +be the <I>smaller</I> of the two), VII. 19 (where the particular case +is given of <I>three</I> numbers in continued proportion instead of +four proportionals). +<p>(3) Alternative proofs. +<p>It appears to be Heron who first introduced the easy but +uninstructive semi-algebraical method of proving the proposi- +tions II. 2-10 which is now so popular. On this method the +propositions are proved ‘without figures’ as consequences of +II. 1 corresponding to the algebraical formula +<MATH><I>a</I>(<I>b</I>+<I>c</I>+<I>d</I>+...)=<I>ab</I>+<I>ac</I>+<I>ad</I>+...</MATH> +<p>Heron explains that it is not possible to prove II. 1 without +drawing a number of lines (i.e. without actually drawing the +rectangles), but that the following propositions up to II. 10 +can be proved by merely drawing one line. He distinguishes +two varieties of the method, one by <I>dissolutio,</I> the other by +<I>compositio,</I> by which he seems to mean <I>splitting-up</I> of rect- +angles and squares and <I>combination</I> of them into others. +But in his proofs he sometimes combines the two varieties. +<p>Alternative proofs are given (<I>a</I>) of some propositions of +Book III, namely III. 25 (placed after III. 30 and starting +from the <I>are</I> instead of the chord), III. 10 (proved by means +of III. 9), III. 13 (a proof preceded by a lemma to the effect +that a straight line cannot meet a circle in more than two +points). +<p>A class of alternative proof is (<I>b</I>) that which is intended to +meet a particular objection (<G>e)/nstasis</G>) which had been or might +be raised to Euclid's constructions. Thus in certain cases +Heron avoids <I>producing</I> a certain straight line, where Euclid +produces it, the object being to meet the objection of one who +should deny our right to assume that there is <I>any space +available.</I> Of this class are his proofs of I. 11, 20 and his +note on I. 16. Similarly in I. 48 he supposes the right-angled +<pb n=312><head>HERON OF ALEXANDRIA</head> +triangle which is constructed to be constructed on the same +side of the common side as the given triangle is. +<p>A third class (<I>c</I>) is that which avoids <I>reductio ad absurdum,</I> +e.g. a direct proof of I. 19 (for which he requires and gives +a preliminary lemma) and of I. 25. +<p>(4) Heron supplies certain <I>converses</I> of Euclid's propositions +e.g. of II. 12, 13 and VIII. 27. +<p>(5) A few additions to, and extensions of, Euclid's propositions +are also found. Some are unimportant, e.g. the construction +of isosceles and scalene triangles in a note on I. 1 and the +construction of <I>two</I> tangents in III. 17. The most important +extension is that of III. 20 to the case where the angle at the +circumference is greater than a right angle, which gives an +easy way of proving the theorem of III. 22. Interesting also +are the notes on I. 37 (on I. 24 in Proclus), where Heron +proves that two triangles with two sides of the one equal +to two sides of the other and with the included angles <I>supple- +mentary</I> are equal in area, and compares the areas where the +sum of the included angles (one being supposed greater than +the other) is less or greater than two right angles, and on I. 47, +where there is a proof (depending on preliminary lemmas) of +the fact that, in the figure of Euclid's proposition (see next +page), the straight lines <I>AL, BG, CE</I> meet in a point. This +last proof is worth giving. First come the lemmas. +<p>(1) If in a triangle <I>ABC</I> a straight line <I>DE</I> be drawn +parallel to the base <I>BC</I> cutting the sides <I>AB, AC</I> or those +sides produced in <I>D, E,</I> and if <I>F</I> be the +<FIG> +middle point of <I>BC,</I> then the straight line +<I>AF</I> (produced if necessary) will also bisect +<I>DE.</I> (<I>HK</I> is drawn through <I>A</I> parallel to +<I>DE,</I> and <I>HDL, KEM</I> through <I>D, E</I> parallel +to <I>AF</I> meeting the base in <I>L, M</I> respec- +tively. Then the triangles <I>ABF, AFC</I> +between the same parallels are equal. So are the triangles +<I>DBF, EFC.</I> Therefore the differences, the triangles <I>ADF, +AEF,</I> are equal and so therefore are the parallelograms <I>HF, +KF.</I> Therefore <MATH><I>LF</I>=<I>FM,</I></MATH> or <MATH><I>DG</I>=<I>GE.</I></MATH>) +<p>(2) is the converse of Eucl. I. 43. If a parallelogram is +<pb n=313><head>GEOMETRY</head> +cut into four others <I>ADGE, DF, FGCB, CE</I>, so that <I>DF, CE</I> +are equal, the common vertex <I>G</I> will lie on the diagonal <I>AB.</I> +<p>Heron produces <I>AG</I> to meet <I>CF</I> in <I>H,</I> and then proves that +<I>AHB</I> is a straight line. +<FIG> +<p>Since <I>DF, CE</I> are equal, so are +the triangles <I>DGF, ECG.</I> Adding +the triangle <I>GCF,</I> we have the +triangles <I>ECF, DCF</I> equal, and +<I>DE, CF</I> are parallel. +<p>But (by I. 34, 29, 26) the tri- +angles <I>AKE, GKD</I> are congruent, +so that <MATH><I>EK</I>=<I>KD</I></MATH>; and by lemma (1) it follows that <MATH><I>CH</I>=<I>HF.</I></MATH> +<p>Now, in the triangles <I>FHB, CHG,</I> two sides (<I>BF, FH</I> and +<I>GC, CH</I>) and the included angles are equal; therefore the +triangles are congruent, and the angles <I>BHF, GHC</I> are equal. +<p>Add to each the angle <I>GHF,</I> and +<MATH>∠<I>BHF</I>+∠<I>FHG</I>=∠<I>CHG</I>+∠<I>GHF</I>=two right angles.</MATH> +<p>To prove his substantive proposition Heron draws <I>AKL</I> +perpendicular to <I>BC,</I> and joins <I>EC</I> meeting <I>AK</I> in <I>M.</I> Then +we have only to prove that <I>BMG</I> is a straight line. +<FIG> +<p>Complete the parallelogram <I>FAHO,</I> and draw the diagonals +<I>OA, FH</I> meeting in <I>Y.</I> Through <I>M</I> draw <I>PQ, SR</I> parallel +respectively to <I>BA, AC.</I> +<pb n=314><head>HERON OF ALEXANDRIA</head> +<p>Now the triangles <I>FAH</I>, <I>BAC</I> are equal in all respects; +therefore <MATH>∠<I>HFA</I>=∠<I>ABC</I> +=∠<I>CAK</I></MATH> (since <I>AK</I> is at right angles to <I>BC</I>). +<p>But, the diagonals of the rectangle <I>FH</I> cutting one another +in <I>Y</I>, we have <I>FY</I>=<I>YA</I> and ∠<I>HFA</I>=∠<I>OAF</I>; +therefore ∠<I>OAF</I>=∠<I>CAK</I>, and <I>OA</I> is in a straight line +with <I>AKL.</I> +<p>Therefore, <I>OM</I> being the diagonal of <I>SQ</I>, <I>SA</I>=<I>AQ</I>, and, if +we add <I>AM</I> to each, <I>FM</I>=<I>MH.</I> +<p>Also, since <I>EC</I> is the diagonal of <I>FN</I>, <I>FM</I>=<I>MN.</I> +<p>Therefore the parallelograms <I>MH</I>, <I>MN</I> are equal; and +hence, by the preceding lemma, <I>BMG</I> is a straight line. Q.E.D. +<C>(<G>b</G>) The <I>Definitions.</I></C> +<p>The elaborate collection of <I>Definitions</I> is dedicated to one +Dionysius in a preface to the following effect: +<p>‘In setting out for you a sketch, in the shortest possible +form, of the technical terms premised in the elements of +geometry, I shall take as my point of departure, and shall +base my whole arrangement upon, the teaching of Euclid, the +author of the elements of theoretical geometry; for by this +means I think that I shall give you a good general under- +standing not only of Euclid's doctrine but of many other +works in the domain of geometry. I shall begin then with +the <I>point.</I>’ +<p>He then proceeds to the definitions of the point, the line, +the different sorts of lines, straight, circular, ‘curved’ and +‘spiral-shaped’ (the Archimedean spiral and the cylindrical +helix), Defs. 1-7; surfaces, plane and not plane, solid body, +Defs. 8-11; angles and their different kinds, plane, solid, +rectilinear and not rectilinear, right, acute and obtuse angles, +Defs. 12-22; figure, boundaries of figure, varieties of figure, +plane, solid, composite (of homogeneous or non-homogeneous +parts) and incomposite, Defs. 23-6. The incomposite plane +figure is the circle, and definitions follow of its parts, segments +(which are composite of non-homogeneous parts), the semi- +circle, the <G>a(yi/s</G> (less than a semicircle), and the segment +greater than a semicircle, angles in segments, the sector, +<pb n=315><head>THE <I>DEFINITIONS</I></head> +‘concave’ and ‘convex’, lune, garland (these last two are +composite of homogeneous parts) and <I>axe</I> (<G>pe/lekus</G>), bounded by +four circular arcs, two concave and two convex, Defs. 27-38. +Rectilineal figures follow, the various kinds of triangles and +of quadrilaterals, the gnomon in a parallelogram, and the +gnomon in the more general sense of the figure which added +to a given figure makes the whole into a similar figure, +polygons, the parts of figures (side, diagonal, height of a +triangle), perpendicular, parallels, the three figures which will +fill up the space round a point, Defs. 39-73. Solid figures are +next classified according to the surfaces bounding them, and +lines on surfaces are divided into (1) simple and circular, +(2) mixed, like the conic and spiric curves, Defs. 74, 75. The +sphere is then defined, with its parts, and stated to be +the figure which, of all figures having the same surface, is the +greatest in content, Defs. 76-82. Next the cone, its different +species and its parts are taken up, with the distinction +between the three conics, the section of the acute-angled cone +(‘by some also called <I>ellipse</I>’) and the sections of the right- +angled and obtuse-angled cones (also called <I>parabola</I> and +<I>hyperbola</I>), Defs. 83-94; the cylinder, a section in general, +the <I>spire</I> or <I>tore</I> in its three varieties, open, continuous (or +just closed) and ‘crossing-itself’, which respectively have +sections possessing special properties, ‘square rings’ which +are cut out of cylinders (i.e. presumably rings the cross-section +of which through the centre is two squares), and various other +figures cut out of spheres or mixed surfaces, Defs. 95-7; +rectilineal solid figures, pyramids, the five regular solids, the +semi-regular solids of Archimedes two of which (each with +fourteen faces) were known to Plato, Defs. 98-104; prisms +of different kinds, parallelepipeds, with the special varieties, +the cube, the <I>beam</I>, <G>doko/s</G> (length longer than breadth and +depth, which may be equal), the <I>brick</I>, <G>plinqi/s</G> (length less +than breadth and depth), the <G>sfhni/skos</G> or <G>bwmi/skos</G> with +length, breadth and depth unequal, Defs. 105-14. +<p>Lastly come definitions of relations, equality of lines, sur- +faces, and solids respectively, similarity of figures, ‘reciprocal +figures’, Defs. 115-18; indefinite increase in magnitude, +parts (which must be homogeneous with the wholes, so that +e.g. the horn-like angle is not a part or submultiple of a right +<pb n=316><head>HERON OF ALEXANDRIA</head> +or any angle), multiples, Defs. 119-21; proportion in magni- +tudes, what magnitudes can have a ratio to one another, +magnitudes in the same ratio or magnitudes in proportion, +definition of greater ratio, Defs. 122-5; transformation of +ratios (<I>componendo, separando, convertendo, alternando, in- +vertendo</I> and <I>ex aequali</I>), Defs. 126-7; commensurable and +incommensurable magnitudes and straight lines, Defs. 128, +129. There follow two tables of measures, Defs. 130-2. +<p>The <I>Definitions</I> are very valuable from the point of view of +the historian of mathematics, for they give the different alter- +native definitions of the fundamental conceptions; thus we +find the Archimedean ‘definition’ of a straight line, other +definitions which we know from Proclus to be due to Apol- +lonius, others from Posidonius, and so on. No doubt the +collection may have been recast by some editor or editors +after Heron's time, but it seems, at least in substance, to go +back to Heron or earlier still. So far as it contains original +definitions of Posidonius, it cannot have been compiled earlier +than the first century B.C.; but its content seems to belong in +the main to the period before the Christian era. Heiberg +adds to his edition of the <I>Definitions</I> extracts from Heron's +Geometry, postulates and axioms from Euclid, extracts from +Geminus on the classification of mathematics, the principles +of geometry, &c., extracts from Proclus or some early collec- +tion of scholia on Euclid, and extracts from Anatolius and +Theon of Smyrna, which followed the actual definitions in the +manuscripts. These various additions were apparently collected +by some Byzantine editor, perhaps of the eleventh century. +<C>Mensuration.</C> +<C>The <I>Metrica, Geometrica, Stereometrica, Geodaesia, +Mensurae.</I></C> +<p>We now come to the mensuration of Heron. Of the +different works under this head the <I>Metrica</I> is the most +important from our point of view because it seems, more than +any of the others, to have preserved its original form. It is +also more fundamental in that it gives the theoretical basis of +the formulae used, and is not a mere application of rules to +particular examples. It is also more akin to theory in that it +<pb n=317><head>MENSURATION</head> +does not use concrete measures, but simple numbers or units +which may then in particular cases be taken to be feet, cubits, +or any other unit of measurement. Up to 1896, when a +manuscript of it was discovered by R. Schöne at Constanti- +nople, it was only known by an allusion to it in Eutocius +(on Archimedes's <I>Measurement of a Circle</I>), who states that +the way to obtain an approximation to the square root of +a non-square number is shown by Heron in his <I>Metrica</I>, as +well as by Pappus, Theon, and others who had commented on +the <I>Syntaxis</I> of Ptolemy.<note>Archimedes, vol. iii, p. 232. 13-17.</note> Tannery<note>Tannery, <I>Mémoires scientifiques</I>, ii, 1912, pp. 447-54.</note> had already in 1894 +discovered a fragment of Heron's <I>Metrica</I> giving the particular +rule in a Paris manuscript of the thirteenth century contain- +ing Prolegomena to the <I>Syntaxis</I> compiled presumably from +the commentaries of Pappus and Theon. Another interesting +difference between the <I>Metrica</I> and the other works is that in +the former the Greek way of writing fractions (which is our +method) largely preponderates, the Egyptian form (which +expresses a fraction as the sum of diminishing submultiples) +being used comparatively rarely, whereas the reverse is the +case in the other works. +<p>In view of the greater authority of the <I>Metrica</I>, we shall +take it as the basis of our account of the mensuration, while +keeping the other works in view. It is desirable at the +outset to compare broadly the contents of the various collec- +tions. Book I of the <I>Metrica</I> contains the mensuration of +squares, rectangles and triangles (chaps. 1-9), parallel-trapezia, +rhombi, rhomboids and quadrilaterals with one angle right +(10-16), regular polygons from the equilateral triangle to the +regular dodecagon (17-25), a ring between two concentric +circles (26), segments of circles (27-33), an ellipse (34), a para- +bolic segment (35), the surfaces of a cylinder (36), an isosceles +cone (37), a sphere (38) and a segment of a sphere (39). +Book II gives the mensuration of certain solids, the solid +content of a cone (chap. 1), a cylinder (2), rectilinear solid +figures, a parallelepiped, a prism, a pyramid and a frustum, +&c. (3-8), a frustum of a cone (9, 10), a sphere and a segment +of a sphere (11, 12), a <I>spire</I> or <I>tore</I> (13), the section of a +cylinder measured in Archimedes's <I>Method</I> (14), and the solid +<pb n=318><head>HERON OF ALEXANDRIA</head> +formed by the intersection of two cylinders with axes at right +angles inscribed in a cube, also measured in the <I>Method</I> (15), +the five regular solids (16-19). Book III deals with the divi- +sion of figures into parts having given ratios to one another, +first plane figures (1-19), then solids, a pyramid, a cone and a +frustum, a sphere (20-3). +<p>The <I>Geometria</I> or <I>Geometrumena</I> is a collection based upon +Heron, but not his work in its present form. The addition of +a theorem due to Patricius<note><I>Geometrica</I>, 21 26 (vol. iv, p. 386. 23).</note> and a reference to him in the +<I>Stereometrica</I> (I. 22) suggest that Patricius edited both works, +but the date of Patricius is uncertain. Tannery identifies +him with a mathematical professor of the tenth century, +Nicephorus Patricius; if this is correct, he would be contem- +porary with the Byzantine writer (erroneously called Heron) +who is known to have edited genuine works of Heron, and +indeed Patricius and the anonymous Byzantine might be one +and the same person. The mensuration in the <I>Geometry</I> has +reference almost entirely to the same figures as those +measured in Book I of the <I>Metrica</I>, the difference being that +in the <I>Geometry</I> (1) the rules are not explained but merely +applied to examples, (2) a large number of numerical illustra- +tions are given for each figure, (3) the Egyptian way of +writing fractions as the sum of submultiples is followed, +(4) lengths and areas are given in terms of particular +measures, and the calculations are lengthened by a consider- +able amount of conversion from one measure into another. +The first chapters (1-4) are of the nature of a general intro- +duction, including certain definitions and ending with a table +of measures. Chaps. 5-99, Hultsch (=5-20, 14, Heib.), though +for the most part corresponding in content to <I>Metrica</I> I, +seem to have been based on a different collection, because +chaps. 100-3 and 105 (=21, 1-25, 22, 3-24, Heib.) are clearly +modelled on the <I>Metrica</I>, and 101 is headed ‘A definition +(really ‘measurement’) of a circle in another book of Heron’. +Heiberg transfers to the <I>Geometrica</I> a considerable amount of +the content of the so-called <I>Liber Geeponicus</I>, a badly ordered +collection consisting to a large extent of extracts from the +other works. Thus it begins with 41 definitions identical +with the same number of the <I>Definitiones.</I> Some sections +<pb n=319><head>MENSURATION</head> +Heiberg puts side by side with corresponding sections of the +<I>Geometrica</I> in parallel columns; others he inserts in suitable +places; sections 78. 79 contain two important problems in +indeterminate analysis (=<I>Geom.</I> 24, 1-2, Heib.). Heiberg +adds, from the Constantinople manuscript containing the +<I>Metrica</I>, eleven more sections (chap. 24, 3-13) containing +indeterminate problems, and other sections (chap. 24, 14-30 and +37-51) giving the mensuration, mainly, of figures inscribed in or +circumscribed to others, e.g. squares or circles in triangles, +circles in squares, circles about triangles, and lastly of circles +and segments of circles. +<p>The <I>Stereometrica</I> I has at the beginning the title <G>*ei)sa- +gwgai\ tw=n stereometroume/nwn *(/hrwnos</G> but, like the <I>Geometrica</I>, +seems to have been edited by Patricius. Chaps. 1-40 give the +mensuration of the geometrical solid figures, the sphere, the +cone, the frustum of a cone, the obelisk with circular base, +the cylinder, the ‘pillar’, the cube, the <G>sfhni/skos</G> (also called +<G>o)/nux</G>), the <G>mei/ouron proeskarifeume/non</G>, pyramids, and frusta. +Some portions of this section of the book go back to Heron; +thus in the measurement of the sphere chap. 1=<I>Metrica</I> +II. 11, and both here and elsewhere the ordinary form of +fractions appears. Chaps. 41-54 measure the contents of cer- +tain buildings or other constructions, e.g. a theatre, an amphi- +theatre, a swimming-bath, a well, a ship, a wine-butt, and +the like. +<p>The second collection, <I>Stereometrica</I> II, appears to be of +Byzantine origin and contains similar matter to <I>Stereometrica</I> I, +parts of which are here repeated. Chap. 31 (27, Heib.) gives +the problem of Thales, to find the height of a pillar or a tree +by the measurement of shadows; the last sections measure +various pyramids, a prism, a <G>bwmi/skos</G> (little altar). +<p>The <I>Geodaesia</I> is not an independent work, but only con- +tains extracts from the <I>Geometry</I>; thus chaps. 1-16=<I>Geom.</I> +5-31, Hultsch (=5, 2-12, 32, Heib.); chaps. 17-19 give the +methods of finding, in any scalene triangle the sides of which +are given, the segments of the base made by the perpendicular +from the vertex, and of finding the area direct by the well- +known ‘formula of Heron’; i.e. we have here the equivalent of +<I>Metrica</I> I. 5-8. +<p>Lastly, the <G>metrh/seis</G>, or <I>Mensurae</I>, was attributed to Heron +<pb n=320><head>HERON OF ALEXANDRIA</head> +in an Archimedes manuscript of the ninth century, but can- +not in its present form be due to Heron, although portions of +it have points of contact with the genuine works. Sects. 2-27 +measure all sorts of objects, e.g. stones of different shapes, +a pillar, a tower, a theatre, a ship, a vault, a hippodrome; but +sects. 28-35 measure geometrical figures, a circle and segments +of a circle (cf. <I>Metrica</I> I), and sects. 36-48 on spheres, segments +of spheres, pyramids, cones and frusta are closely connected +with <I>Stereom.</I> I and <I>Metrica</I> II; sects. 49-59, giving the men- +suration of receptacles and plane figures of various shapes, +seem to have a different origin. We can now take up the +<C>Contents of the <I>Metrica.</I></C> +<C>Book I. Measurement of Areas.</C> +<p>The preface records the tradition that the first geometry +arose out of the practical necessity of measuring and dis- +tributing land (whence the name ‘geometry’), after which +extension to three dimensions became necessary in order to +measure solid bodies. Heron then mentions Eudoxus and +Archimedes as pioneers in the discovery of difficult measure- +ments, Eudoxus having been the first to prove that a cylinder +is three times the cone on the same base and of equal height, +and that circles are to one another as the squares on their +diameters, while Archimedes first proved that the surface of +a sphere is equal to four times the area of a great circle in it, +and the volume two-thirds of the cylinder circumscribing it. +<C>(<I>a</I>) <I>Area of scalene triangle.</I></C> +<p>After the easy cases of the rectangle, the right-angled +triangle and the isosceles triangle, Heron gives two methods +of finding the area of a scalene triangle (acute-angled or +obtuse-angled) when the lengths of the three sides are given. +<p>The first method is based on Eucl. II. 12 and 13. If <I>a</I>, <I>b</I>, <I>c</I> +be the sides of the triangle opposite to the angles <I>A</I>, <I>B</I>, <I>C</I> +respectively, Heron observes (chap. 4) that any angle, e.g. <I>C</I>, is +acute, right or obtuse according as <MATH><I>c</I><SUP>2</SUP><=or><I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP></MATH>, and this +is the criterion determining which of the two propositions is +applicable. The method is directed to determining, first the +segments into which any side is divided by the perpendicular +<pb n=321><head>AREA OF SCALENE TRIANGLE</head> +from the opposite vertex, and thence the length of the per- +pendicular itself. We have, in the cases of the triangle acute- +angled at <I>C</I> and the triangle obtuse-angled at <I>C</I> respectively, +<MATH><I>c</I><SUP>2</SUP>=<I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>∓2<I>a</I>.<I>CD</I></MATH>, +or <MATH><I>CD</I>={(<I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>)-<I>c</I><SUP>2</SUP>}/2<I>a</I></MATH>, +<FIG> +whence <MATH><I>AD</I><SUP>2</SUP>(=<I>b</I><SUP>2</SUP>-<I>CD</I><SUP>2</SUP>)</MATH> is found, so that we know the area +<MATH>(=1/2<I>a</I>.<I>AD</I>)</MATH>. +<p>In the cases given in <I>Metrica</I> I. 5, 6 the sides are (14, 15, 13) +and (11, 13, 20) respectively, and <I>AD</I> is found to be rational +(=12). But of course both <I>CD</I> (or <I>BD</I>) and <I>AD</I> may be surds, +in which case Heron gives approximate values. Cf. <I>Geom.</I> +53, 54, Hultsch (15, 1-4, Heib.), where we have a triangle +in which <MATH><I>a</I>=8, <I>b</I>=4, <I>c</I>=6</MATH>, so that <MATH><I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>=44</MATH> and +<MATH><I>CD</I>=44/16=2 1/2 1/4</MATH>. Thus <MATH><I>AD</I><SUP>2</SUP>=16-(2 1/2 1/4)<SUP>2</SUP>=16-7 1/2 1/(16) +=8 1/4 1/8 1/(16)</MATH>, and <MATH><I>AD</I>=√(8 1/4 1/8 1/(16))=2 2/3 1/4</MATH> approximately, whence +the area <MATH>=4x2 2/3 1/4=11 2/3</MATH>. Heron then observes that we get +a nearer result still if we multiply <I>AD</I><SUP>2</SUP> by (1/2<I>a</I>)<SUP>2</SUP> before +extracting the square root, for the area is then <MATH>√(16x8 1/4 1/8 1/(16))</MATH> +or √(135), which is very nearly 11 1/2 1/(14) 1/(21) or 11 (13)/(21). +<p>So in <I>Metrica</I> I. 9, where the triangle is 10, 8, 12 (10 being +the base), Heron finds the perpendicular to be √(63), but he +obtains the area as <MATH>√(1/4<I>AD</I><SUP>2</SUP>.<I>BC</I><SUP>2</SUP>)</MATH>, or √(1575), while observing +that we <I>can</I>, of course, take the approximation to √(63), or +7 1/2 1/4 1/8 1/(16), and multiply it by half 10, obtaining 39 1/2 1/8 1/(16) as +the area. +<p><I>Proof of the formula</I> <MATH>▵=√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)}</MATH>. +<p>The second method is that known as the ‘formula of +Heron’, namely, in our notation, <MATH>▵=√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)}</MATH>. +The proof of the formula is given in <I>Metrica</I> I. 8 and also in +<pb n=322><head>HERON OF ALEXANDRIA</head> +chap. 30 of the <I>Dioptra</I>; but it is now known (from Arabian +sources) that the proposition is due to Archimedes. +<p>Let the sides of the triangle <I>ABC</I> be given in length. +<p>Inscribe the circle <I>DEF</I>, and let <I>O</I> be the centre. +<p>Join <I>AO</I>, <I>BO</I>, <I>CO</I>, <I>DO</I>, <I>EO</I>, <I>FO.</I> +<p>Then <MATH><I>BC</I>.<I>OD</I>=2▵<I>BOC</I>, +<I>CA</I>.<I>OE</I>=2▵<I>COA</I>, +<I>AB</I>.<I>OF</I>=2▵<I>AOB</I></MATH>; +<FIG> +whence, by addition, +<MATH><I>p</I>.<I>OD</I>=2▵<I>ABC</I></MATH>, +where <I>p</I> is the perimeter. +<p>Produce <I>CB</I> to <I>H</I>, so that <I>BH</I>=<I>AF.</I> +<p>Then, since <MATH><I>AE</I>=<I>AF</I>, <I>BF</I>=<I>BD</I></MATH>, and <I>CE</I>=<I>CD</I>, we have +<MATH><I>CH</I>=1/2<I>p</I>=<I>s.</I></MATH> +<p>Therefore <MATH><I>CH</I>.<I>OD</I>=▵<I>ABC.</I></MATH> +<p>But <I>CH</I>.<I>OD</I> is the ‘side’ of the product <I>CH</I><SUP>2</SUP>.<I>OD</I><SUP>2</SUP>, i.e. +<MATH>√(<I>CH</I><SUP>2</SUP>.<I>OD</I><SUP>2</SUP>)</MATH>, +so that <MATH>(▵<I>ABC</I>)<SUP>2</SUP>=<I>CH</I><SUP>2</SUP>.<I>OD</I><SUP>2</SUP></MATH>. +<pb n=323><head>PROOF OF THE ‘FORMULA OF HERON’</head> +<p>Draw <I>OL</I> at right angles to <I>OC</I> cutting <I>BC</I> in <I>K</I>, and <I>BL</I> at +right angles to <I>BC</I> meeting <I>OL</I> in <I>L.</I> Join <I>CL.</I> +<p>Then, since each of the angles <I>COL</I>, <I>CBL</I> is right, <I>COBL</I> is +a quadrilateral in a circle. +<p>Therefore <MATH>∠<I>COB</I>+∠<I>CLB</I>=2<I>R.</I></MATH> +<p>But <MATH>∠<I>COB</I>+∠<I>AOF</I>=2<I>R</I></MATH>, because <I>AO</I>, <I>BO</I>, <I>CO</I> bisect the +angles round <I>O</I>, and the angles <I>COB</I>, <I>AOF</I> are together equal +to the angles <I>AOC</I>, <I>BOF</I>, while the sum of all four angles +is equal to 4<I>R.</I> +<p>Consequently <MATH>∠<I>AOF</I>=∠<I>CLB.</I></MATH> +<p>Therefore the right-angled triangles <I>AOF</I>, <I>CLB</I> are similar; +therefore <MATH><I>BC</I>:<I>BL</I>=<I>AF</I>:<I>FO</I> +=<I>BH</I>:<I>OD</I></MATH>, +and, alternately, <MATH><I>CB</I>:<I>BH</I>=<I>BL</I>:<I>OD</I> +=<I>BK</I>:<I>KD</I></MATH>; +whence, <I>componendo</I>, <MATH><I>CH</I>:<I>HB</I>=<I>BD</I>:<I>DK.</I></MATH> +<p>It follows that +<MATH><I>CH</I><SUP>2</SUP>:<I>CH</I>.<I>HB</I>=<I>BD</I>.<I>DC</I>:<I>CD</I>.<I>DK</I> +=<I>BD</I>.<I>DC</I>:<I>OD</I><SUP>2</SUP></MATH>, since the angle <I>COK</I> is right. +<p>Therefore <MATH>(▵<I>ABC</I>)<SUP>2</SUP>=<I>CH</I><SUP>2</SUP>.<I>OD</I><SUP>2</SUP></MATH> (from above) +<MATH>=<I>CH</I>.<I>HB</I>.<I>BD</I>.<I>DC</I> +=<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)</MATH>. +<C>(<G>b</G>) <I>Method of approximating to the square root of +a non-square number.</I></C> +<p>It is à propos of the triangle 7, 8, 9 that Heron gives the +important statement of his method of approximating to the +value of a surd, which before the discovery of the passage +of the <I>Metrica</I> had been a subject of unlimited conjecture +as bearing on the question how Archimedes obtained his +approximations to √(3). +<p>In this case <MATH><I>s</I>=12, <I>s</I>-<I>a</I>=5, <I>s</I>-<I>b</I>=4, <I>s</I>-<I>c</I>=3</MATH>, so that +<MATH>▵=√(12.5.4.3)=√(720)</MATH>. +<pb n=324><head>HERON OF ALEXANDRIA</head> +<p>‘Since’, says Heron,<note><I>Metrica</I>, i. 8, pp. 18, 22-20. 5.</note> ‘720 has not its side rational, we can +obtain its side within a very small difference as follows. Since +the next succeeding square number is 729, which has 27 for +its side, divide 720 by 27. This gives 26 2/3. Add 27 to this, +making 53 2/3, and take half of this or 26 1/2 1/3. The side of 720 +will therefore be very nearly 26 1/2 1/3. In fact, if we multiply +26 1/2 1/3 by itself, the product is 720 1/36, so that the difference (in +the square) is 1/36. +<p>‘If we desire to make the difference still smaller than 1/36, we +shall take 720 1/36 instead of 729 [or rather we should take +26 1/2 1/3 instead of 27], and by proceeding in the same way we +shall find that the resulting difference is much less than 1/36.’ +<p>In other words, if we have a non-square number <I>A</I>, and <I>a</I><SUP>2</SUP> +is the nearest square number to it, so that <MATH><I>A</I>=<I>a</I><SUP>2</SUP>±<I>b</I></MATH>, then we +have, as the first approximation to √(<I>A</I>), +<MATH><G>a</G><SUB>1</SUB>=1/2(<I>a</I>+<I>A</I>/<I>a</I>)</MATH>; (1) +for a second approximation we take +<MATH><G>a</G><SUB>2</SUB>=1/2(<G>a</G><SUB>1</SUB>+<I>A</I>/(<G>a</G><SUB>1</SUB>))</MATH>, (2) +and so on.<note>The method indicated by Heron was known to Barlaam and Nicolas +Rhabdas in the fourteenth century. The equivalent of it was used by +Luca Paciuolo (fifteenth-sixteenth century), and it was known to the other +Italian algebraists of the sixteenth century. Thus Luca Paciuolo gave +2 1/2, 2 9/(20) and 2 (881)/(1960) as successive approximations to √(6). He obtained +the first as <MATH>2+2/(2.2)</MATH>, the second as <MATH>2 1/2-((2 1/2)<SUP>2</SUP>-6)/(2.2 1/2)</MATH>, and the third as +<MATH>(49)/(20)-(((49)/(20))<SUP>2</SUP>-6)/(2.(49)/(20))</MATH>. The above rule gives <MATH>1/2(2+6/2)=2 1/2, 1/2 (5/2+2/5)=2 9/(20), +1/2((49)/(20)+(120)/(49))=2 (881)/(1960)</MATH>. +<p>The formula of Heron was again put forward, in modern times, by +Buzengeiger as a means of accounting for the Archimedean approxima- +tion to √(3), apparently without knowing its previous history. Bertrand +also stated it in a treatise on arithmetic (1851). The method, too, by +which Oppermann and Alexeieff sought to account for Archimedes's +approximations is in reality the same. The latter method depends on +the formula +<MATH>1/2 (<G>a</G>+<G>b</G>):√(<G>ab</G>)=√(<G>ab</G>):(2<G>ab</G>)/(<G>a</G>+<G>b</G>)</MATH>. +Alexeieff separated <I>A</I> into two factors <I>a</I><SUB>0</SUB>, <I>b</I><SUB>0</SUB>, and pointed out that if, say, +<MATH><I>a</I><SUB>0</SUB>>√(<I>A</I>)><I>b</I><SUB>0</SUB></MATH>, +then, <MATH>1/2 (<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)>√(<I>A</I>)>(2<I>A</I>)/(<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)</MATH> or <MATH>(2<I>a</I><SUB>0</SUB><I>b</I><SUB>0</SUB>)/(<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)</MATH>, +and again, if <MATH>1/2(<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)=<I>a</I><SUB>1</SUB>, 2<I>A</I>/(<I>a</I><SUB>0</SUB>+<I>b</I><SUB>0</SUB>)=<I>b</I><SUB>1</SUB>, +1/2(<I>a</I><SUB>1</SUB>+<I>b</I><SUB>1</SUB>)>√(<I>A</I>)>(2<I>A</I>)/(<I>a</I><SUB>1</SUB>+<I>b</I><SUB>1</SUB>)</MATH>, +and so on. +<p>Now suppose that, in Heron's formulae, we put <MATH><I>a</I>=<I>X</I><SUB>0</SUB>, <I>A</I>/<I>a</I>=<I>x</I><SUB>0</SUB>, +<G>a</G><SUB>1</SUB>=<I>X</I><SUB>1</SUB>, <I>A</I>/<G>a</G><SUB>1</SUB>=<I>x</I><SUB>1</SUB></MATH>, and so on. We then have +<MATH><I>X</I><SUB>1</SUB>=1/2(<I>a</I>+<I>A</I>/<I>a</I>)=1/2(<I>X</I><SUB>0</SUB>+<I>x</I><SUB>0</SUB>), <I>x</I><SUB>1</SUB>=<I>A</I>/<I>X</I><SUB>1</SUB>=<I>A</I>/(1/2(<I>X</I><SUB>0</SUB>+<I>x</I><SUB>0</SUB>))</MATH> or +<MATH>(2<I>X</I><SUB>0</SUB><I>x</I><SUB>0</SUB>)/(<I>X</I><SUB>0</SUB>+<I>x</I><SUB>0</SUB>)</MATH>; +that is, <I>X</I><SUB>1</SUB>, <I>x</I><SUB>1</SUB> are, respectively, the arithmetic and harmonic means +between <I>X</I><SUB>0</SUB>, <I>x</I><SUB>0</SUB>; <I>X</I><SUB>2</SUB>, <I>x</I><SUB>2</SUB> are the arithmetic and harmonic means between +<I>X</I><SUB>1</SUB>, <I>x</I><SUB>1</SUB>, and so on, exactly as in Alexeieff's formulae. +<p>Let us now try to apply the method to Archimedes's case, √(3), and we +shall see to what extent it serves to give what we want. Suppose +we begin with 3>√(3)>1. We then have +<MATH>1/2 (3+1)>√(3)>3/1/2(3+1)</MATH>, or <MATH>2>√(3)>3/2</MATH>, +and from this we derive successively +<MATH>7/4>√(3)>(12)/7, (97)/(56)>√(3)>(168)/(97), (18817)/(10864)>√(3)>(32592)/(18817)</MATH>. +But, if we start from 5/3, obtained by the formula <MATH><I>a</I>+<I>b</I>/(2<I>a</I>+1)<√(<I>a</I><SUP>2</SUP>+<I>b</I>)</MATH>, +we obtain the following approximations by excess, +<MATH>1/2(5/3+9/5)=(26)/(15), 1/2((26)/(15)+(45)/(26))=(1351)/(780)</MATH>. +The second process then gives one of Archimedes's results, (1351)/(780), but +neither of the two processes gives the other, (265)/(153), directly. The latter +can, however, be obtained by using the formula that, if <MATH><I>a</I>/<I>b</I><<I>c</I>/<I>d</I></MATH>, then +<MATH><I>a</I>/<I>b</I><(<I>ma</I>+<I>nc</I>)/(<I>mb</I>+<I>nd</I>)<<I>c</I>/<I>d</I></MATH>. +<p>For we can obtain (265)/(153) from (97)/(56) and (168)/(97) thus: <MATH>(97+168)/(56+97)=(265)/(153)</MATH>, or from +(97)/(56) and 7/4 thus: <MATH>(11.97-7)/(11.56-4)=(1060)/(612)=(265)/(153)</MATH>; and so on. Or again (1351)/(780) can +be obtained from (18817)/(10864) and (97)/(56) thus: <MATH>(18817+97)/(10864+56)=(18914)/(10920)=(1351)/(780)</MATH>. +<p>The advantage of the method is that, as compared with that of con- +tinued fractions, it is a very rapid way of arriving at a close approxi- +mation. Günther has shown that the (<I>m</I>+1)th approximation obtained +by Heron's formula is the 2<SUP><I>m</I></SUP>th obtained by continued fractions. (‘Die +quadratischen Irrationalitäten der Alten und deren Entwickelungs- +methoden in <I>Abhandlungen zur Gesch. d. Math.</I> iv. 1882, pp. 83-6.)</note> +<pb n=325><head>APPROXIMATIONS TO SURDS</head> +<p>Substituting in (1) the value <I>a</I><SUP>2</SUP>±<I>b</I> for <I>A</I>, we obtain +<MATH><G>a</G><SUB>1</SUB>=<I>a</I>±<I>b</I>/(2<I>a</I>)</MATH>. +<p>Heron does not seem to have used this formula with a nega- +tive sign, unless in <I>Stereom.</I> I. 33 (34, Hultsch), where √(63) +<pb n=326><head>HERON OF ALEXANDRIA</head> +is given as approximately 8-1/(16). In <I>Metrica</I> I. 9, as we +have seen, √(63) is given as 7 1/2 1/4 1/8 1/(16), which was doubtless +obtained from the formula (1) as +<MATH>1/2 (8+(63)/8)=1/2 (8+7 7/8)=7 1/2 1/4 1/8 1/(16)</MATH>. +<p>The above seems to be the only <I>classical</I> rule which has +been handed down for finding second and further approxi- +mations to the value of a surd. But, although Heron thus +shows how to obtain a second approximation, namely by +formula (2), he does not seem to make any direct use of +this method himself, and consequently the question how the +approximations closer than the first which are to be found in +his works were obtained still remains an open one. +<C>(<G>g</G>) <I>Quadrilaterals.</I></C> +<p>It is unnecessary to give in detail the methods of measuring +the areas of quadrilaterals (chaps. 11-16). Heron deals with +the following kinds, the parallel-trapezium (isosceles or non- +isosceles), the rhombus and rhomboid, and the quadrilateral +which has one angle right and in which the four sides have +given lengths. Heron points out that in the rhombus or +rhomboid, and in the general case of the quadrilateral, it is +necessary to know a diagonal as well as the four sides. The +mensuration in all the cases reduces to that of the rectangle +and triangle. +<C>(<G>d</G>) <I>The regular polygons with 3, 4, 5, 6, 7, 8, 9, 10, 11, +or 12 sides.</I></C> +<p>Beginning with the <I>equilateral triangle</I> (chap. 17), Heron +proves that, if <I>a</I> be the side and <I>p</I> the perpendicular from +a vertex on the opposite side, <MATH><I>a</I><SUP>2</SUP>:<I>p</I><SUP>2</SUP>=4:3</MATH>, whence +<MATH><I>a</I><SUP>4</SUP>:<I>p</I><SUP>2</SUP><I>a</I><SUP>2</SUP>=4:3=16:12</MATH>, +so that <MATH><I>a</I><SUP>4</SUP>:(▵<I>ABC</I>)<SUP>2</SUP>=16:3</MATH>, +and <MATH>(▵<I>ABC</I>)<SUP>2</SUP>=3/(16)<I>a</I><SUP>4</SUP></MATH>. In the particular case taken <I>a</I>=10 +and <MATH>▵<SUP>2</SUP>=1875</MATH>, whence <MATH>▵=43 1/3</MATH> nearly. +<p>Another method is to use an approximate value for √(3) in +the formula √(3).<I>a</I><SUP>2</SUP>/4. This is what is done in the <I>Geometrica</I> +14 (10, Heib.), where we are told that the area is <MATH>(1/3+1/(10))<I>a</I><SUP>2</SUP></MATH>; +<pb n=327><head>THE REGULAR POLYGONS</head> +now <MATH>1/3+1/(10)=(13)/(30)=1/4((26)/(15))</MATH>, so that the approximation used by +Heron for √(3) is here (26)/(15). For the side 10, the method gives +the same result as above, for <MATH>(13)/(30).100=43 1/3</MATH>. +<p>The regular <I>pentagon</I> is next taken (chap. 18). Heron +premises the following lemma. +<p>Let <I>ABC</I> be a right-angled triangle, with the angle <I>A</I> equal +<FIG> +to 2/5<I>R.</I> Produce <I>AC</I> to <I>O</I> so that <I>CO</I>=<I>AC</I> +If now <I>AO</I> is divided in extreme and +mean ratio, <I>AB</I> is equal to the greater +segment. (For produce <I>AB</I> to <I>D</I> so that +<I>AD</I>=<I>AO</I>, and join <I>BO</I>, <I>DO.</I> Then, since +<I>ADO</I> is isosceles and the angle at <MATH><I>A</I>=2/5<I>R</I>, +∠<I>ADO</I>=∠<I>AOD</I>=4/5<I>R</I></MATH>, and, from the +equality of the triangles <MATH><I>ABC</I>, <I>OBC</I>, +∠<I>AOB</I>=∠<I>BAO</I>=2/5<I>R</I></MATH>. It follows that +the triangle <I>ADO</I> is the isosceles triangle of Eucl. IV. 10, and +<I>AD</I> is divided in extreme and mean ratio in <I>B.</I>) Therefore, +says Heron, <MATH>(<I>BA</I>+<I>AC</I>)<SUP>2</SUP>=5<I>AC</I><SUP>2</SUP></MATH>. [This is Eucl. XIII. 1.] +<p>Now, since <MATH>∠<I>BOC</I>=2/5<I>R</I></MATH>, if <I>BC</I> be produced to <I>E</I> so that +<I>CE</I>=<I>BC</I>, <I>BE</I> subtends at <I>O</I> an angle equal to 4/5<I>R</I>, and there- +fore <I>BE</I> is the side of a regular pentagon inscribed in the +circle with <I>O</I> as centre and <I>OB</I> as radius. (This circle also +passes through <I>D</I>, and <I>BD</I> is the side of a regular decagon in +the same circle). If now <MATH><I>BO</I>=<I>AB</I>=<I>r</I>, <I>OC</I>=<I>p</I>, <I>BE</I>=<I>a</I></MATH>, +we have from above, <MATH>(<I>r</I>+<I>p</I>)<SUP>2</SUP>=5<I>p</I><SUP>2</SUP></MATH>, whence, since √(5) is +approximately 9/4, we obtain approximately <MATH><I>r</I>=5/4<I>p</I></MATH>, and +<MATH>1/2<I>a</I>=3/4<I>p</I></MATH>, so that <MATH><I>p</I>=2/3<I>a</I></MATH> Hence <MATH>1/2<I>pa</I>=1/3<I>a</I><SUP>2</SUP></MATH>, and the area +of the pentagon=5/3<I>a</I><SUP>2</SUP>. Heron adds that, if we take a closer +approximation to √(5) than 9/4, we shall obtain the area still +more exactly. In the <I>Geometry</I><note><I>Geom.</I> 102 (21, 14, Heib.).</note> the formula is given as (12)/7<I>a</I><SUP>2</SUP>. +<p>The regular <I>hexagon</I> (chap. 19) is simply 6 times the +equilateral triangle with the same side. If ▵ be the area +of the equilateral triangle with side <I>a</I>, Heron has proved +that <MATH>▵<SUP>2</SUP>=3/(16)<I>a</I><SUP>4</SUP></MATH> (<I>Metrica</I> I. 17), hence <MATH>(hexagon)<SUP>2</SUP>=(27)/4<I>a</I><SUP>4</SUP></MATH>. If, +e.g. <MATH><I>a</I>=10, (hexagon)<SUP>2</SUP>=67500</MATH>, and (hexagon)=259 nearly. +In the <I>Geometry</I><note><I>Ib.</I> 102 (21, 16, 17, Heib.).</note> the formula is given as (13)/5<I>a</I><SUP>2</SUP>, while ‘another +book’ is quoted as giving <MATH>6(1/3+1/(10))<I>a</I><SUP>2</SUP></MATH>; it is added that the +latter formula, obtained from the area of the triangle, <MATH>(1/3+1/(10))<I>a</I><SUP>2</SUP></MATH>, +represents the more accurate procedure, and is fully set out by +<pb n=328><head>HERON OF ALEXANDRIA</head> +Heron. As a matter of fact, however, <MATH>6(1/3+1/(10))=(13)/5</MATH> exactly, +and only the <I>Metrica</I> gives the more accurate calculation. +<p>The regular <I>heptagon.</I> +<p>Heron assumes (chap. 20) that, if <I>a</I> be the side and <I>r</I> the +radius of the circumscribing circle, <I>a</I>=7/8<I>r</I>, being approxi- +mately equal to the perpendicular from the centre of the +circle to the side of the regular hexagon inscribed in it (for 7/8 +is the approximate value of 1/2√(3)). This theorem is quoted by +Jordanus Nemorarius (d. 1237) as an ‘Indian rule’; he pro- +bably obtained it from Abŭ'l Wafā (940-98). The <I>Metrica</I> +shows that it is of Greek origin, and, if Archimedes really +wrote a book on the heptagon in a circle, it may be due to +him. If then <I>p</I> is the perpendicular from the centre of the +circle on the side (<I>a</I>) of the inscribed heptagon, <MATH><I>r</I>/(1/2<I>a</I>)=8/3 1/2</MATH> +or 16/7, whence <MATH><I>p</I><SUP>2</SUP>/(1/2<I>a</I>)<SUP>2</SUP>=(207)/(49)</MATH>, and <I>p</I>/1/2<I>a</I>=(approxi- +mately) 14 1/3/7 or 43/21. Consequently the area of the +heptagon <MATH>=7.1/2<I>pa</I>=7.(43)/(84)<I>a</I><SUP>2</SUP>=(43)/(12)<I>a</I><SUP>2</SUP></MATH>. +<p>The regular <I>octagon, decagon</I> and <I>dodecagon.</I> +<p>In these cases (chaps. 21, 23, 25) Heron finds <I>p</I> by drawing +<FIG> +the perpendicular <I>OC</I> from <I>O</I>, the centre of the +circumscribed circle, on a side <I>AB</I>, and then making +the angle <I>OAD</I> equal to the angle <I>AOD.</I> +<p>For the octagon, +<MATH>∠<I>ADC</I>=1/2<I>R</I></MATH>, and <MATH><I>p</I>=1/2<I>a</I>(1+√(2))=1/2<I>a</I>(1+(17)/(12))</MATH> +or 1/2<I>a</I>.(29)/(12) approximately. +<p>For the decagon, +<MATH>∠<I>ADC</I>=2/5<I>R</I></MATH>, and <MATH><I>AD</I>:<I>DC</I>=5:4</MATH> nearly (see preceding page); +hence <MATH><I>AD</I>:<I>AC</I>=5:3</MATH>, and <MATH><I>p</I>=1/2<I>a</I>(5/3+4/3)=3/2<I>a.</I></MATH> +<p>For the dodecagon, +<MATH>∠<I>ADC</I>=1/3<I>R</I></MATH>, and <MATH><I>p</I>=1/2<I>a</I>(2+√(3))=1/2<I>a</I>(2+7/4)=(15)/8<I>a</I></MATH> +approximately. +<p>Accordingly <MATH><I>A</I><SUB>8</SUB>=(29)/6<I>a</I><SUP>2</SUP>, <I>A</I><SUB>10</SUB>=(15)/2<I>a</I><SUP>2</SUP>, <I>A</I><SUB>12</SUB>=(45)/4<I>a</I><SUP>2</SUP></MATH>, where <I>a</I> is +the side in each case. +<p>The regular <I>enneagon</I> and <I>hendecagon.</I> +<p>In these cases (chaps. 22, 24) the Table of Chords (i.e. +<pb n=329><head>THE REGULAR POLYGONS</head> +presumably Hipparchus's Table) is appealed to. If <I>AB</I> be the +side (<I>a</I>) of an enneagon or hendecagon inscribed in a circle, <I>AC</I> +the diameter through <I>A</I>, we are told that the Table of Chords +gives 1/3 and 7/(25) as the respective approximate values of the +ratio <I>AB</I>/<I>AC.</I> The angles subtended at the centre <I>O</I> by the +<FIG> +side <I>AB</I> are 40° and 32 8/(11)° respec- +tively, and Ptolemy's Table gives, +as the chords subtended by angles of +40° and 33° respectively, 41<SUP><I>p</I></SUP>2′33″ +and 34<SUP><I>p</I></SUP>4′55″ (expressed in 120th +parts of the diameter); Heron's +figures correspond to 40<SUP><I>p</I></SUP> and 33<SUP><I>p</I></SUP> +36′ respectively. For the <I>enneagon +AC</I><SUP>2</SUP>=9<I>AB</I><SUP>2</SUP>, whence <I>BC</I><SUP>2</SUP>=8<I>AB</I><SUP>2</SUP> +or approximately (289)/(36)<I>AB</I><SUP>2</SUP>, and +<I>BC</I>=(17)/6<I>a</I>; therefore <MATH>(area of +enneagon)=9/2.▵<I>ABC</I>=(51)/8<I>a</I><SUP>2</SUP></MATH>. For +the <MATH><I>hendecagon AC</I><SUP>2</SUP>=(625)/(49)<I>AB</I><SUP>2</SUP></MATH> and <MATH><I>BC</I><SUP>2</SUP>=(576)/(49)<I>AB</I><SUP>2</SUP></MATH>, so that +<I>BC</I>=(24)/7<I>a</I>, and area of hendecagon <MATH>=(11)/2.▵<I>ABC</I>=(66)/7<I>a</I><SUP>2</SUP></MATH>. +<p>An ancient formula for the ratio between the side of any +regular polygon and the diameter of the circumscribing circle +is preserved in <I>Geëpon.</I> 147 sq. (= Pseudo-Dioph. 23-41), +namely <MATH><I>d</I><SUB><I>n</I></SUB>=<I>n</I>(<I>a</I><SUB><I>n</I></SUB>)/3</MATH>. Now the ratio <I>na</I><SUB><I>n</I></SUB>/<I>d</I><SUB><I>n</I></SUB> tends to <G>p</G> as the +number (<I>n</I>) of sides increases, and the formula indicates a time +when <G>p</G> was generally taken as = 3. +<C>(<G>e</G>) <I>The Circle.</I></C> +<p>Coming to the circle (<I>Metrica</I> I. 26) Heron uses Archi- +medes's value for <G>p</G>, namely (22)/7, making the circumference of +a circle (44)/7<I>r</I> and the area (11)/(14)<I>d</I><SUP>2</SUP>, where <I>r</I> is the radius and <I>d</I> the +diameter. It is here that he gives the more exact limits +for <G>p</G> which he says that Archimedes found in his work <I>On +Plinthides and Cylinders</I>, but which are not convenient for +calculations. The limits. as we have seen, are given in the +text as <MATH>(211875)/(67441)<<G>p</G><(197888)/(62351)</MATH>, and with Tannery's alteration to +<MATH>(211872)/(67441)<<G>p</G><(195882)/(62351)</MATH> are quite satisfactory.<note>See vol. i, pp. 232-3.</note> +<pb n=330><head>HERON OF ALEXANDRIA</head> +<C>(<G>z</G>) <I>Segment of a circle.</I></C> +<p>According to Heron (<I>Metrica</I> I. 30) the ancients measured +the area of a segment rather inaccurately, taking the area +to be <MATH>1/2(<I>b</I>+<I>h</I>)<I>h</I></MATH>, where <I>b</I> is the base and <I>h</I> the height. He +conjectures that it arose from taking <G>p</G>=3, because, if we +apply the formula to the semicircle, the area becomes 1/2.3<I>r</I><SUP>2</SUP>, +where <I>r</I> is the radius. Those, he says (chap. 31), who have +investigated the area more accurately have added 1/(14) (1/2<I>b</I>)<SUP>2</SUP> +to the above formula, making it <MATH>1/2(<I>b</I>+<I>h</I>)<I>h</I>+1/(14)(1/2<I>b</I>)<SUP>2</SUP></MATH>, and this +seems to correspond to the value 3 1/7 for <G>p</G>, since, when applied +to the semicircle, the formula gives <MATH>1/2(3<I>r</I><SUP>2</SUP>+1/7<I>r</I><SUP>2</SUP>)</MATH>. He adds +that this formula should only be applied to segments of +a circle less than a semicircle, and not even to all of these, but +only in cases where <I>b</I> is not greater than 3<I>h.</I> Suppose e.g. +that <I>b</I>=60, <I>h</I>=1; in that case even <MATH>1/(14)(1/2<I>b</I>)<SUP>2</SUP>=1/(14).900=64 2/7</MATH>, +which is greater even than the parallelogram with 60, 1 as +sides, which again is greater than the segment. Where there- +fore <I>b</I>>3<I>h</I>, he adopts another procedure. +<p>This is exactly modelled on Archimedes's quadrature of +a segment of a parabola. Heron proves (<I>Metrica</I> I. 27-29, 32) +that, if <I>ADB</I> be a segment of a circle, and <I>D</I> the middle point +<FIG> +of the arc, and if the arcs <I>AD</I>, <I>DB</I> be +similarly bisected at <I>E</I>, <I>F</I>, +<MATH>▵<I>ADB</I><4(▵<I>AED</I>+▵<I>DFB</I>)</MATH>. +<p>Similarly, if the same construction be +made for the segments <I>AED</I>, <I>BFD</I>, each +of them is less than 4 times the sum of the two small triangles +in the segments left over. It follows that +(area of segmt. <MATH><I>ADB</I>)>▵<I>ADB</I>{1+1/4+(1/4)<SUP>2</SUP>+...} +>4/3▵<I>ADB.</I></MATH> +‘If therefore we measure the triangle, and add one-third of +it, we shall obtain the area of the segment as nearly as +possible.’ That is, for segments in which <I>b</I>>3<I>h</I>, Heron +takes the area to be equal to that of the parabolic segment +with the same base and height, or 2/3<I>bh.</I> +<p>In addition to these three formulae for <I>S</I>, the area of +a segment, there are yet others, namely +<MATH><I>S</I>=1/2(<I>b</I>+<I>h</I>)<I>h</I>(1+1/(21))</MATH>, <I>Mensurae</I> 29, +<MATH><I>S</I>=1/2(<I>b</I>+<I>h</I>)<I>h</I>(1+1/(16))</MATH>, ” 31. +<pb n=331><head>SEGMENT OF A CIRCLE</head> +The first of these formulae is applied to a segment greater +than a semicircle, the second to a segment less than a semi- +circle. +<p>In the <I>Metrica</I> the area of a segment greater than a semi- +circle is obtained by subtracting the area of the complementary +segment from the area of the circle. +<p>From the <I>Geometrica</I><note>Cf. <I>Geom.</I>, 94, 95 (19. 2, 4, Heib.), 97. 4 (20. 7, Heib.).</note> we find that the circumference of the +segment less than a semicircle was taken to be <MATH>√(<I>b</I><SUP>2</SUP>+4<I>h</I><SUP>2</SUP>)+1/4<I>h</I></MATH> +or alternatively <MATH>√(<I>b</I><SUP>2</SUP>+4<I>h</I><SUP>2</SUP>)+{√(<I>b</I><SUP>2</SUP>+4<I>h</I><SUP>2</SUP>)-<I>b</I>}<I>h</I>/<I>b.</I></MATH> +<C>(<G>h</G>) <I>Ellipse, parabolic segment, surface of cylinder, right +cone, sphere and segment of sphere.</I></C> +<p>After the area of an ellipse (<I>Metrica</I> I. 34) and of a parabolic +segment (chap. 35), Heron gives the surface of a cylinder +(chap. 36) and a right cone (chap. 37); in both cases he unrolls +the surface on a plane so that the surface becomes that of a +parallelogram in the one case and a sector of a circle in the +other. For the surface of a sphere (chap. 38) and a segment of +it (chap. 39) he simply uses Archimedes's results. +<p>Book I ends with a hint how to measure irregular figures, +plane or not. If the figure is plane and bounded by an +irregular curve, neighbouring points are taken on the curve +such that, if they are joined in order, the contour of the +polygon so formed is not much different from the curve +itself, and the polygon is then measured by dividing it into +triangles. If the surface of an irregular solid figure is to be +found, you wrap round it pieces of very thin paper or cloth, +enough to cover it, and you then spread out the paper or +cloth and measure that. +<C>Book II. Measurement of volumes.</C> +<p>The preface to Book II is interesting as showing how +vague the traditions about Archimedes had already become. +<p>‘After the measurement of surfaces, rectilinear or not, it is +proper to proceed to the solid bodies, the surfaces of which we +have already measured in the preceding book, surfaces plane +and spherical, conical and cylindrical, and irregular surfaces +as well. The methods of dealing with these solids are, in +<pb n=332><head>HERON OF ALEXANDRIA</head> +view of their surprising character, referred to Archimedes by +certain writers who give the traditional account of their +origin. But whether they belong to Archimedes or another, +it is necessary to give a sketch of these methods as well.’ +<p>The Book begins with generalities about figures all the +sections of which parallel to the base are equal to the base +and similarly situated, while the centres of the sections are on +a straight line through the centre of the base, which may be +either obliquely inclined or perpendicular to the base; whether +the said straight line (‘the axis’) is or is not perpendicular to +the base, the volume is equal to the product of the area of the +base and the <I>perpendicular</I> height of the top of the figure +from the base. The term ‘height’ is thenceforward restricted +to the length of the perpendicular from the top of the figure +on the base. +<C>(<I>a</I>) <I>Cone, cylinder, parallelepiped</I> (<I>prism</I>), <I>pyramid, and +frustum.</I></C> +<p>II. 1-7 deal with a cone, a cylinder, a ‘parallelepiped’ (the +base of which is not restricted to the parallelogram but is in +the illustration given a regular hexagon, so that the figure is +more properly a prism with polygonal bases), a triangular +prism, a pyramid with base of any form, a frustum of a +triangular pyramid; the figures are in general <I>oblique.</I> +<C>(<G>b</G>) <I>Wedge-shaped solid</I> (<G>bwmi/skos</G> <I>or</I> <G>sfhni/skos</G>).</C> +<p>II. 8 is a case which is perhaps worth giving. It is that of +a rectilineal solid, the base of which is a rectangle <I>ABCD</I> and +has opposite to it another rectangle <I>EFGH</I>, the sides of which +are respectively parallel but not necessarily proportional to +those of <I>ABCD.</I> Take <I>AK</I> equal to <I>EF</I>, and <I>BL</I> equal to <I>FG.</I> +Bisect <I>BK</I>, <I>CL</I> in <I>V</I>, <I>W</I>, and draw <I>KRPU</I>, <I>VQOM</I> parallel to +<I>AD</I>, and <I>LQRN</I>, <I>WOPT</I> parallel to <I>AB.</I> Join <I>FK</I>, <I>GR</I>, <I>LG</I>, +<I>GU</I>, <I>HN.</I> +<p>Then the solid is divided into (1) the parallelepiped with +<I>AR</I>, <I>EG</I> as opposite faces, (2) the prism with <I>KL</I> as base and +<I>FG</I> as the opposite edge, (3) the prism with <I>NU</I> as base and +<I>GH</I> as opposite edge, and (4) the pyramid with <I>RLCU</I> as base +and <I>G</I> as vertex. Let <I>h</I> be the ‘height’ of the figure. Now +<pb n=333><head>MEASUREMENT OF SOLIDS</head> +the parallelepiped (1) is on <I>AR</I> as base and has height <I>h</I>; the +prism (2) is equal to a parallelepiped on <I>KQ</I> as base and with +height <I>h</I>; the prism (3) is equal to a parallelepiped with <I>NP</I> +as base and height <I>h</I>; and finally the pyramid (4) is equal to +a parallelepiped of height <I>h</I> and one-third of <I>RC</I> as base. +<FIG> +<p>Therefore the whole solid is equal to one parallelepiped +with height <I>h</I> and base equal to <MATH>(<I>AR</I>+<I>KQ</I>+<I>NP</I>+<I>RO</I>+1/3<I>RO</I>)</MATH> +or <I>AO</I>+1/3<I>RO.</I> +<p>Now, if <MATH><I>AB</I>=<I>a</I>, <I>BC</I>=<I>b</I>, <I>EF</I>=<I>c</I>, <I>FG</I>=<I>d</I>, +<I>AV</I>=1/2(<I>a</I>+<I>c</I>), <I>AT</I>=1/2(<I>b</I>+<I>d</I>), <I>RQ</I>=1/2(<I>a</I>-<I>c</I>), <I>RP</I>=1/2(<I>b</I>-<I>d</I>)</MATH>. +<p>Therefore volume of solid +<MATH>={1/4(<I>a</I>+<I>c</I>)(<I>b</I>+<I>d</I>)+1/(12)(<I>a</I>-<I>c</I>)(<I>b</I>-<I>d</I>)}<I>h.</I></MATH> +<p>The solid in question is evidently the true <G>bwmi/skos</G> (‘little +altar’), for the formula is used to calculate the content of +a <G>bwmi/skos</G> in <I>Stereom.</I> II. 40 (68, Heib.) It is also, I think, +the <G>sfhni/skos</G> (‘little wedge’), a measurement of which is +given in <I>Stereom.</I> I. 26 (25, Heib.) It is true that the second +term of the first factor <MATH>1/(12)(<I>a</I>-<I>c</I>)(<I>b</I>-<I>d</I>)</MATH> is there neglected, +perhaps because in the case taken <MATH>(<I>a</I>=7, <I>b</I>=6, <I>c</I>=5, <I>d</I>=4)</MATH> +this term (=1/3) is small compared with the other (=30). A +particular <G>sfhni/skos</G>, in which either <I>c</I>=<I>a</I> or <I>d</I>=<I>b</I>, was +called <G>o)/nux</G>; the second term in the factor of the content +vanishes in this case, and, if e.g. <I>c</I>=<I>a</I>, the content is <MATH>1/2(<I>b</I>+<I>d</I>)<I>ah.</I></MATH> +Another <G>bwmi/skos</G> is measured in <I>Stereom.</I> I. 35 (34, Heib.), +where the solid is inaccurately called ‘a pyramid oblong +(<G>e(teromh/khs</G>) and truncated (<G>ko/louros</G>) or half-perfect’. +<pb n=334><head>HERON OF ALEXANDRIA</head> +<p>The method is the same <I>mutatis mutandis</I> as that used in +II. 6 for the frustum of a pyramid with any triangle for base, +and it is applied in II. 9 to the case of a frustum of a pyramid +with a square base, the formula for which is +<MATH>[{1/2(<I>a</I>+<I>a</I>′)}<SUP>2</SUP>+1/3{1/2(<I>a</I>-<I>a</I>′)}<SUP>2</SUP>]<I>h</I></MATH>, +where <I>a, a</I>′ are the sides of the larger and smaller bases +respectively, and <I>h</I> the height; the expression is of course +easily reduced to <MATH>1/3<I>h</I>(<I>a</I><SUP>2</SUP>+<I>aa</I>′+<I>a</I>′<SUP>2</SUP>)</MATH>. +<C>(<G>g</G>) <I>Frustum of cone, sphere, and segment thereof</I>.</C> +<p>A <I>frustum of a cone</I> is next measured in two ways, (1) by +comparison with the corresponding frustum of the circum- +scribing pyramid with square base, (2) directly as the +difference between two cones (chaps. 9, 10). The volume of +the frustum of the cone is to that of the frustum of the +circumscribing pyramid as the area of the base of the cone to +that of the base of the pyramid; i.e. the volume of the frus- +tum of the cone is 1/4<G>p</G>, or 11/14, times the above expression for +the frustum of the pyramid with <I>a</I><SUP>2</SUP>, <I>a</I>′<SUP>2</SUP> as bases, and it +reduces to <MATH>1/12<G>p</G><I>h</I> (<I>a</I><SUP>2</SUP>+<I>aa</I>′+<I>a</I>′<SUP>2</SUP>)</MATH>, where <I>a, a</I>′ are the <I>diameters</I> +of the two bases. For the <I>sphere</I> (chap. 11) Heron uses +Archimedes's proposition that the circumscribing cylinder is +1 1/2 times the sphere, whence the volume of the sphere +=2/3.<I>d</I>.11/14<I>d</I><SUP>2</SUP> or 11/21<I>d</I><SUP>3</SUP>; for a <I>segment of a sphere</I> (chap. 12) he +likewise uses Archimedes's result (<I>On the Sphere and Cylinder</I>, +II. 4). +<C>(<G>d</G>) <I>Anchor-ring or tore</I>.</C> +<p>The anchor-ring or <I>tore</I> is next measured (chap. 13) by +means of a proposition which Heron quotes from Dionyso- +dorus, and which is to the effect that, if <I>a</I> be the radius of either +circular section of the <I>tore</I> through the axis of revolution, and +<I>c</I> the distance of its centre from that axis, +<MATH><G>p</G><I>a</I><SUP>2</SUP>:<I>ac</I>=(volume of tore):<G>p</G><I>c</I><SUP>2</SUP>.2<I>a</I></MATH> +<MATH>[whence volume of tore=2<G>p</G><SUP>2</SUP><I>ca</I><SUP>2</SUP>]</MATH>. In the particular case +taken <MATH><I>a</I>=6, <I>c</I>=14</MATH>, and Heron obtains, from the proportion +113 1/7:84=<I>V</I>:7392, <I>V</I>=9956 4/7. But he shows that he is +aware that the volume is the product of the area of the +<pb n=335><head>MEASUREMENT OF SOLIDS</head> +describing circle and the length of the path of its centre. +For, he says, since 14 is a radius (of the path of the centre), +28 is its diameter and 88 its circumference. ‘If then the tore +be straightened out and made into a cylinder, it will have 88 +for its length, and the diameter of the base of the cylinder is +12; so that the solid content of the cylinder is, as we have +seen, <MATH>9956 4/7’(=88.11/14.144)</MATH>. +<C>(<G>e</G>) <I>The two special solids of Archimedes's ‘Method’</I>.</C> +<p>Chaps. 14, 15 give the measurement of the two remarkable +solids of Archimedes's <I>Method</I>, following Archimedes's results. +<C>(<G>z</G>) <I>The five regular solids</I>.</C> +<p>In chaps. 16-18 Heron measures the content of the five +regular solids after the cube. He has of course in each case +to find the perpendicular from the centre of the circumscrib- +ing sphere on any face. Let <I>p</I> be this perpendicular, <I>a</I> the +edge of the solid, <I>r</I> the radius of the circle circumscribing any +face. Then (1) for the <I>tetrahedron</I> +<MATH><I>a</I><SUP>2</SUP>=3<I>r</I><SUP>2</SUP>,<I>p</I><SUP>2</SUP>=<I>a</I><SUP>2</SUP>-1/3<I>a</I><SUP>2</SUP>=2/3<I>a</I><SUP>2</SUP></MATH>. +(2) In the case of the <I>octahedron</I>, which is the sum of two +equal pyramids on a square base, the content is one-third +of that base multiplied by the diagonal of the figure, +i.e. 1/3.<I>a</I><SUP>2</SUP>.√2<I>a</I> or 1/3√2.<I>a</I><SUP>3</SUP>; in the case taken <MATH><I>a</I>=7</MATH>, and +Heron takes 10 as an approximation to √(2.7<SUP>2</SUP>) or √98, the +result being 1/3.10.49 or 163 1/3. (3) In the case of the <I>icosa- +hedron</I> Heron merely says that +<MATH><I>p</I>:<I>a</I>=93:127 (the real value of the ratio is 1/2√(7+3√5)/6)</MATH>. +(4) In the case of the <I>dodecahedron</I>, Heron says that +<MATH><I>p</I>:<I>a</I>=9:8 (the true value is 1/2√(25+11√5)/10</MATH>, and, if √5 is +put equal to 9/4, Heron's ratio is readily obtained). +<p>Book II ends with an allusion to the method attributed to +Archimedes for measuring the contents of irregular bodies by +immersing them in water and measuring the amount of fluid +displaced. +<pb n=336><head>HERON OF ALEXANDRIA</head> +<C>Book III. Divisions of figures.</C> +<p>This book has much in common with Euclid's book <I>On divi- +sions</I> (<I>of figures</I>), the problem being to divide various figures, +plane or solid, by a straight line or plane into parts having +a given ratio. In III. 1-3 a triangle is divided into two parts +in a given ratio by a straight line (1) passing through a vertex, +(2) parallel to a side, (3) through any point on a side. +III. 4 is worth description: ‘Given a triangle <I>ABC</I>, to cut +out of it a triangle <I>DEF</I> (where <I>D, E, F</I> are points on the +sides respectively) given in magnitude and such that the +triangles <I>AEF, BFD, CED</I> may be equal in area.’ Heron +<I>assumes</I> that, if <I>D, E, F</I> divide the sides so that +<MATH><I>AF</I>:<I>FB</I>=<I>BD</I>:<I>DC</I>=<I>CE</I>:<I>EA</I></MATH>, +the latter three triangles are equal in area. +<FIG> +<p>He then has to find the value of +each of the three ratios which will +result in the triangle <I>DEF</I> having a +given area. +<p>Join <I>AD.</I> +<p>Since <MATH><I>BD</I>:<I>CD</I>=<I>CE</I>:<I>EA</I>, +<I>BC</I>:<I>CD</I>=<I>CA</I>:<I>AE</I></MATH>, +and <MATH>▵<I>ABC</I>:▵<I>ADC</I>=▵<I>ADC</I>:▵<I>ADE</I></MATH>. +<p>Also <MATH>▵<I>ABC</I>:▵<I>ABD</I>=▵<I>ADC</I>:▵<I>EDC</I></MATH>. +<p>But (since the area of the triangle <I>DEF</I> is given) ▵<I>EDC</I> is +given, as well as ▵<I>ABC.</I> Therefore ▵<I>ABD</I>x▵<I>ADC</I> is given. +<p>Therefore, if <I>AH</I> be perpendicular to <I>BC</I>, +<MATH><I>AH</I><SUP>2</SUP>.<I>BD.DC</I></MATH> is given; +therefore <I>BD.DC</I> is given, and, since <I>BC</I> is given, <I>D</I> is given +in position (we have to apply to <I>BC</I> a rectangle equal to +<I>BD.DC</I> and falling short by a square). +<p>As an example Heron takes <MATH><I>AB</I>=13, <I>BC</I>=14, <I>CA</I>=15, +▵<I>DEF</I>=24. ▵<I>ABC</I></MATH> is then 84, and <MATH><I>AH</I>=12</MATH>. +<p>Thus <MATH>▵<I>EDC</I>=20</MATH>, and <MATH><I>AH</I><SUP>2</SUP>.<I>BD.DC</I>=4.84.20=6720</MATH>; +therefore <MATH><I>BD.DC</I>=6720/144 or 46 2/3</MATH> (the text omits the 2/3). +<p>Therefore, says Heron, <MATH><I>BD</I>=8</MATH> approximately. For 8 we +<pb n=337><head>DIVISIONS OF FIGURES</head> +should apparently have 8 1/2, since <I>DC</I> is immediately stated to +be 5 1/2 (not 6). That is, in solving the equation +<MATH><I>x</I><SUP>2</SUP>-14<I>x</I>+46 2/3=0</MATH>, +which gives <MATH><I>x</I>=7±√(2 1/3)</MATH>, Heron apparently substituted 2 1/4 or +9/4 for 2 1/3, thereby obtaining 1 1/2 as an approximation to the +surd. +<p>(The lemma assumed in this proposition is easily proved. +Let <I>m</I>:<I>n</I> be the ratio <MATH><I>AF</I>:<I>FB</I>=<I>BD</I>:<I>DC</I>=<I>CE</I>:<I>EA</I></MATH>. +<p>Then <MATH><I>AF</I>=<I>mc</I>/(<I>m</I>+<I>n</I>), <I>FB</I>=<I>nc</I>/(<I>m</I>+<I>n</I>), <I>CE</I>=<I>mb</I>/(<I>m</I>+<I>n</I>), +<I>EA</I>=<I>nb</I>/(<I>m</I>+<I>n</I>), &c</MATH>. +Hence +<MATH>▵<I>AFE</I>/▵<I>ABC</I>=(<I>mn</I>)/((<I>m</I>+<I>n</I>)<SUP>2</SUP>)=▵<I>BDF</I>/▵<I>ABC</I>=▵<I>CDE</I>/▵<I>ABC</I></MATH>, +and the triangles <I>AFE, BDF, CDE</I> are equal. +<p>Pappus<note>Pappus, viii, pp. 1034-8. Cf. pp. 430-2 <I>post.</I></note> has the proposition that the triangles <I>ABC, DEF</I> +have the same centre of gravity.) +<p>Heron next shows how to divide a parallel-trapezium into +two parts in a given ratio by a straight line (1) through the +point of intersection of the non-parallel sides, (2) through a +given point on one of the parallel sides, (3) parallel to the +parallel sides, (4) through a point on one of the non-parallel +sides (III. 5-8). III. 9 shows how to divide the area of a +circle into parts which have a given ratio by means of an +inner circle with the same centre. For the problems begin- +ning with III. 10 Heron says that numerical calculation alone +no longer suffices, but geometrical methods must be applied. +Three problems are reduced to problems solved by Apollonius +in his treatise <I>On cutting off an area.</I> The first of these is +III. 10, to cut off from the angle of a triangle a given +proportion of the triangle by a straight line through a point +on the opposite side produced. III. 11, 12, 13 show how +to cut any quadrilateral into parts in a given ratio by a +straight line through a point (1) on a side (<I>a</I>) dividing the +side in the given ratio, (<I>b</I>) not so dividing it, (2) not on any +side, (<I>a</I>) in the case where the quadrilateral is a trapezium, +i.e. has two sides parallel, (<I>b</I>) in the case where it is not; the +last case (<I>b</I>) is reduced (like III. 10) to the ‘cutting-off of an +<pb n=338><head>HERON OF ALEXANDRIA</head> +area’. These propositions are ingenious and interesting. +III. 11 shall be given as a specimen. +<p>Given any quadrilateral <I>ABCD</I> and a point <I>E</I> on the side +<I>AD</I>, to draw through <I>E</I> a straight line <I>EF</I> which shall cut +<FIG> +the quadrilateral into two parts in +the ratio of <I>AE</I> to <I>ED.</I> (We omit +the analysis.) Draw <I>CG</I> parallel +to <I>DA</I> to meet <I>AB</I> produced in <I>G.</I> +<p>Join <I>BE</I>, and draw <I>GH</I> parallel +to <I>BE</I> meeting <I>BC</I> in <I>H.</I> +<p>Join <I>CE, EH, EG.</I> +<p>Then <MATH>▵<I>GBE</I>=▵<I>HBE</I></MATH> and, adding ▵<I>ABE</I> to each, we have +<MATH>▵<I>AGE</I>=(quadrilateral <I>ABHE</I>)</MATH>. +<p>Therefore <MATH>(quadr. <I>ABHE</I>):▵<I>CED</I>=▵<I>GAE</I>:▵<I>CED</I> +=<I>AE</I>:<I>ED</I></MATH>. +<p>But (quadr. <I>ABHE</I>) and ▵<I>CED</I> are parts of the quadri- +lateral, and they leave over only the triangle <I>EHC.</I> We have +therefore only to divide ▵<I>EHC</I> in the same ratio <I>AE:ED</I> by +the straight line <I>EF.</I> This is done by dividing <I>HC</I> at <I>F</I> in +the ratio <I>AE</I>:<I>ED</I> and joining <I>EF.</I> +<p>The next proposition (III. 12) is easily reduced to this. +<p>If <I>AE</I>:<I>ED</I> is not equal to the given ratio, let <I>F</I> divide <I>AD</I> +<FIG> +in the given ratio, and through <I>F</I> +draw <I>FG</I> dividing the quadri- +lateral in the given ratio (III. 11). +<p>Join <I>EG</I>, and draw <I>FH</I> parallel +to <I>EG.</I> Let <I>FH</I> meet <I>BC</I> in <I>H</I>, +and join <I>EH.</I> +<p>Then is <I>EH</I> the required straight +line through <I>E</I> dividing the quad- +rilateral in the given ratio. +<p>For <MATH>▵<I>FGE</I>=▵<I>HGE</I></MATH>. Add to each (quadr. <I>GEDC</I>). +<p>Therefore <MATH>(quadr. <I>CGFD</I>)=(quadr. <I>CHED</I>)</MATH>. +<p>Therefore <I>EH</I> divides the quadrilateral in the given ratio, +just as <I>FG</I> does. +<p>The case (III. 13) where <I>E</I> is not on a side of the quadri- +lateral [(2) above] takes two different forms according as the +<pb n=339><head>DIVISIONS OF FIGURES</head> +two opposite sides which the required straight line cuts are +(<I>a</I>) parallel or (<I>b</I>) not parallel. In the first case (<I>a</I>) the +problem reduces to drawing a straight line through <I>E</I> inter- +secting the parallel sides in points <I>F, G</I> such that <I>BF</I>+<I>AG</I> +<FIG> +is equal to a given length. In the second case (<I>b</I>) where +<I>BC, AD</I> are not parallel Heron supposes them to meet in <I>H.</I> +The angle at <I>H</I> is then given, and the area <I>ABH.</I> It is then +a question of cutting off from a triangle with vertex <I>H</I> a +triangle <I>HFG</I> of given area by a straight line drawn from <I>E</I>, +which is again a problem in Apollonius's <I>Cutting-off of an</I> +<FIG> +<I>area.</I> The auxiliary problem in case (<I>a</I>) is easily solved in +III. 16. Measure <I>AH</I> equal to the given length. Join <I>BH</I> +and bisect it at <I>M.</I> Then <I>EM</I> meets <I>BC, AD</I> in points such +that <MATH><I>BF</I>+<I>AG</I>=the given length</MATH>. For, by congruent triangles, +<MATH><I>BF</I>=<I>GH</I></MATH>. +<p>The same problems are solved for the case of any polygon +in III. 14, 15. A sphere is then divided (III. 17) into segments +such that their surfaces are in a given ratio, by means of +Archimedes, <I>On the Sphere and Cylinder</I>, II. 3, just as, in +III. 23, Prop. 4 of the same Book is used to divide a sphere +into segments having their volumes in a given ratio. +<p>III. 18 is interesting because it recalls an ingenious pro- +position in Euclid's book <I>On Divisions.</I> Heron's problem is +‘To divide a given circle into three equal parts by two straight +<pb n=340><head>HERON OF ALEXANDRIA</head> +lines’, and he observes that, ‘as the problem is clearly not +rational, we shall, for practical convenience, make the division, +<FIG> +as exactly as possible, in the follow- +ing way.’ <I>AB</I> is the side of an +equilateral triangle inscribed in the +circle. Let <I>CD</I> be the parallel +diameter, <I>O</I> the centre of the circle, +and join <I>AO, BO, AD, DB.</I> Then +shall the segment <I>ABD</I> be very +nearly one-third of the circle. For, +since <I>AB</I> is the side of an equi- +lateral triangle in the circle, the +sector <I>OAEB</I> is one-third of the +circle. And the triangle <I>AOB</I> forming part of the sector +is equal to the triangle <I>ADB</I>; therefore the segment <I>AEB +plus</I> the triangle <I>ABD</I> is equal to one-third of the circle, +and the segment <I>ABD</I> only differs from this by the small +segment on <I>BD</I> as base, which may be neglected. Euclid's +proposition is to cut off one-third (or any fraction) of a circle +between two parallel chords (see vol. i, pp. 429-30). +<p>III. 19 finds a point <I>D</I> within any triangle <I>ABC</I> such that +the triangles <I>DBC, DCA, DAB</I> are all equal; and then Heron +passes to the division of solid figures. +<p>The solid figures divided in a given ratio (besides the +sphere) are the pyramid with base of any form (III. 20), +the cone (III. 21) and the frustum of a cone (III. 22), the +cutting planes being parallel to the base in each case. These +problems involve the extraction of the cube root of a number +which is in general not an exact cube, and the point of +interest is Heron's method of approximating to the cube root +in such a case. Take the case of the cone, and suppose that +the portion to be cut off at the top is to the rest of the cone as +<I>m</I> to <I>n.</I> We have to find the ratio in which the height or the +edge is cut by the plane parallel to the base which cuts +the cone in the given ratio. The volume of a cone being +1/3<G>p</G><I>c</I><SUP>2</SUP><I>h</I>, where <I>c</I> is the radius of the base and <I>h</I> the height, +we have to find the height of the cone the volume of which +is <I>m</I>/(<I>m</I>+<I>n</I>).1/3<G>p</G><I>c</I><SUP>2</SUP><I>h</I>, and, as the height <I>h</I>′ is to the radius <I>c</I>′ of +its base as <I>h</I> is to <I>c</I>, we have simply to find <I>h</I>′ where +<pb n=341><head>DIVISIONS OF FIGURES</head> +<MATH><I>h</I>′<SUP>3</SUP>/<I>h</I><SUP>3</SUP>=<I>m</I>/(<I>m</I>+<I>n</I>)</MATH>. Or, if we take the edges <I>e, e</I>′ instead +of the heights, <MATH><I>e</I>′<SUP>3</SUP>/<I>e</I><SUP>3</SUP>=<I>m</I>/(<I>m</I>+<I>n</I>)</MATH>. In the case taken by +Heron <MATH><I>m</I>:<I>n</I>=4:1</MATH>, and <MATH><I>e</I>=5</MATH>. Consequently <MATH><I>e</I>′<SUP>3</SUP>=4/5.5<SUP>3</SUP>=100</MATH>. +Therefore, says Heron, <MATH><I>e</I>′=4 9/14</MATH> approximately, and in III. 20 +he shows how this is arrived at. +<C><I>Approximation to the cube root of a non-cube number.</I></C> +<p>‘Take the nearest cube numbers to 100 both above and +below; these are 125 and 64. +<p>Then 125-100=25, +and 100-64=36. +<p>Multiply 5 into 36; this gives 180. Add 100, making 280. +<Divide 180 by 280>; this gives 9/14. Add this to the side of +the smaller cube: this gives 4 9/14. This is as nearly as possible +the cube root (“cubic side”) of 100 units.’ +<p>We have to conjecture Heron's formula from this example. +Generally, if <MATH><I>a</I><SUP>3</SUP><<I>A</I><(<I>a</I>+1)<SUP>3</SUP></MATH>, suppose that <MATH><I>A</I>-<I>a</I><SUP>3</SUP>=<I>d</I><SUB>1</SUB></MATH>, and +<MATH>(<I>a</I>+1)<SUP>3</SUP>-<I>A</I>=<I>d</I><SUB>2</SUB></MATH>. The best suggestion that has been made +is Wertheim's,<note><I>Zeitschr. f. Math. u. Physik</I>, xliv, 1899, hist.-litt. Abt., pp. 1-3.</note> namely that Heron's formula for the approxi- +mate cube root was <I>a</I>+((<I>a</I>+1)<I>d</I><SUB>1</SUB>)/((<I>a</I>+1)<I>d</I><SUB>1</SUB>+<I>ad</I><SUB>2</SUB>). The 5 multiplied +into the 36 might indeed have been the square root of 25 or +√<I>d</I><SUB>2</SUB>, and the 100 added to the 180 in the denominator of the +fraction might have been the original number 100 (<I>A</I>) and not +4.25 or <I>ad</I><SUB>2</SUB>, but Wertheim's conjecture is the more satisfactory +because it can be evolved out of quite elementary considera- +tions. This is shown by G. Enestrőm as follows.<note><I>Bibliotheca Mathematica</I>, viii<SUB>3</SUB>, 1907-8, pp. 412-13.</note> Using the +same notation, Enestrőm further supposes that <I>x</I> is the exact +value of √<SUP>3</SUP><I>A</I> and that <MATH>(<I>x</I>-<I>a</I>)<SUP>3</SUP>=<G>d</G><SUB>1</SUB>, (<I>a</I>+1-<I>x</I>)<SUP>3</SUP>=<G>d</G><SUB>2</SUB></MATH>. +<p>Thus +<MATH><G>d</G><SUB>1</SUB>=<I>x</I><SUP>3</SUP>-3<I>x</I><SUP>2</SUP><I>a</I>+3<I>xa</I><SUP>2</SUP>-<I>a</I><SUP>3</SUP></MATH>, and +<MATH>3<I>ax</I>(<I>x</I>-<I>a</I>)=<I>x</I><SUP>3</SUP>-<I>a</I><SUP>3</SUP>-<G>d</G><SUB>1</SUB>=<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB></MATH>. +<p>Similarly from <MATH><G>d</G><SUB>2</SUB>=(<I>a</I>+1-<I>x</I>)<SUP>3</SUP></MATH> we derive +<MATH>3(<I>a</I>+1)<I>x</I>(<I>a</I>+1-<I>x</I>)=(<I>a</I>+1)<SUP>3</SUP>-<I>x</I><SUP>3</SUP>-<G>d</G><SUB>2</SUB>=<I>d</I><SUB>2</SUB>-<G>d</G><SUB>2</SUB></MATH>. +<p>Therefore +<MATH>(<I>d</I><SUB>2</SUB>-<G>d</G><SUB>2</SUB>)/(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>)=(3(<I>a</I>+1)<I>x</I>(<I>a</I>+1-<I>x</I>))/(3<I>ax</I>(<I>x</I>-<I>a</I>))=((<I>a</I>+1){1-(<I>x</I>-<I>a</I>)})/(<I>a</I>(<I>x</I>-<I>a</I>)) +=(<I>a</I>+1)/<I>a</I>(<I>x</I>-<I>a</I>)-(<I>a</I>+1)/<I>a</I></MATH>; +<pb n=342><head>HERON OF ALEXANDRIA</head> +and, solving for <I>x</I>-<I>a</I>, we obtain +<MATH><I>x</I>-<I>a</I>=((<I>a</I>+1)(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>))/((<I>a</I>+1)(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>)+<I>a</I>(<I>d</I><SUB>2</SUB>-<G>d</G><SUB>2</SUB>))</MATH>, +or <MATH>√<SUP>3</SUP><I>A</I>=<I>a</I>+((<I>a</I>+1)(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>))/((<I>a</I>+1)(<I>d</I><SUB>1</SUB>-<G>d</G><SUB>1</SUB>)+<I>a</I>(<I>d</I><SUB>2</SUB>-<G>d</G><SUB>2</SUB>))</MATH>. +<p>Since <G>d</G><SUB>1</SUB>, <G>d</G><SUB>2</SUB> are in any case the cubes of fractions, we may +neglect them for a first approximation, and we have +<MATH>√<SUP>3</SUP><I>A</I>=<I>a</I>+((<I>a</I>+1)<I>d</I><SUB>1</SUB>)/((<I>a</I>+1)<I>d</I><SUB>1</SUB>+<I>ad</I><SUB>2</SUB>)</MATH>. +<FIG> +<p>III. 22, which shows how to cut a frustum of a cone in a given +ratio by a section parallel to the bases, shall end our account +of the <I>Metrica.</I> I shall give the general formulae on the left +and Heron's case on the right. Let <I>ABED</I> be the frustum, +let the <I>diameters</I> of the bases be <I>a, a</I>′, and the height <I>h.</I> +Complete the cone, and let the height of <I>CDE</I> be <I>x.</I> +<p>Suppose that the frustum has to be cut by a plane <I>FG</I> in +such a way that +<MATH>(frustum <I>DG</I>):(frustum <I>FB</I>)=<I>m</I>:<I>n</I></MATH>. +<p>In the case taken by Heron +<MATH><I>a</I>=28, <I>a</I>′=21, <I>h</I>=12, <I>m</I>=4, <I>n</I>=1</MATH>. +<p>Draw <I>DH</I> perpendicular to <I>AB.</I> +<pb n=343><head>DIVISIONS OF FIGURES</head> +<p>Since <MATH>(<I>DG</I>):(<I>FB</I>)=<I>m</I>:<I>n</I>, +(<I>DB</I>):(<I>DG</I>)=(<I>m</I>+<I>n</I>):<I>m</I></MATH>. +<p>Now +<MATH>(<I>DB</I>)=1/12<G>p</G><I>h</I>(<I>a</I><SUP>2</SUP>+<I>aa</I>′+<I>a</I>′<SUP>2</SUP>)</MATH>, +and <MATH>(<I>DG</I>)=<I>m</I>/(<I>m</I>+<I>n</I>)(<I>DB</I>)</MATH>. +<p>Let <I>y</I> be the height (<I>CM</I>) of the +cone <I>CFG.</I> +<p>Then <MATH><I>DH</I>:<I>AH</I>=<I>CK</I>:<I>KA</I></MATH>, +or <MATH><I>h</I>:1/2(<I>a</I>-<I>a</I>′)=(<I>x</I>+<I>h</I>):1/2<I>a</I></MATH>, +whence <I>x</I> is known. +<p><MATH>Cone <I>CDE</I>=1/12<G>p</G><I>a</I>′<SUP>2</SUP><I>x</I>, +cone <I>CFG</I>=(<I>CDE</I>)+<I>m</I>/(<I>m</I>+<I>n</I>)(<I>DB</I>), +cone <I>CAB</I>=(<I>CDE</I>)+(<I>DB</I>)</MATH>. +<p>Now, says Heron, +<MATH>((<I>CAB</I>)+(<I>CDE</I>))/(<I>CFG</I>)=((<I>x</I>+<I>h</I>)<SUP>3</SUP>+<I>x</I><SUP>3</SUP>)/<I>y</I><SUP>3</SUP></MATH>. +<p>[He might have said simply +<MATH>(<I>CDE</I>):(<I>CFG</I>)=<I>x</I><SUP>3</SUP>:<I>y</I><SUP>3</SUP></MATH>.] +<p>This gives <I>y</I> or <I>CM</I>, +whence <I>LM</I> is known. +<p>Now <MATH><I>AD</I><SUP>2</SUP>=<I>AH</I><SUP>2</SUP>+<I>DH</I><SUP>2</SUP> +={1/2(<I>a</I>-<I>a</I>′)}<SUP>2</SUP>+<I>h</I><SUP>2</SUP></MATH>, +so that <I>AD</I> is known. +<p>Therefore <MATH><I>DF</I>=(<I>y</I>-<I>x</I>)/<I>h</I>.<I>AD</I></MATH> is +known. +<MATH>(<I>DG</I>):(<I>FB</I>)=4:1, +(<I>DB</I>):(<I>DG</I>)=5:4. +(<I>DB</I>)=5698, +(<I>DG</I>)=4558 2/5. +<I>x</I>+<I>h</I>=(14.12)/3 1/2=48</MATH>, +and <MATH><I>x</I>=48-12=36. +(cone <I>CDE</I>)=4158, +(cone <I>CFG</I>)=4158+4558 2/5=8716 2/5, +(cone <I>CAB</I>)=4158+5698=9856. +<I>y</I><SUP>3</SUP>=8716 2/5/(9856+4158).(48<SUP>3</SUP>+36<SUP>3</SUP>) +=8716 2/5.157248/14014=97805</MATH>, +whence <MATH><I>y</I>=46</MATH> approximately. +<p>Therefore <MATH><I>LM</I>=<I>y</I>-<I>x</I>=10. +<I>AD</I><SUP>2</SUP>=(3 1/2)<SUP>2</SUP>+12<SUP>2</SUP> +=156 1/4</MATH>, +and <MATH><I>AD</I>=12 1/2</MATH>. +<p>Therefore <MATH><I>DF</I>=10/12.12 1/2 +=10 5/12</MATH>. +<pb n=344><head>HERON OF ALEXANDRIA</head> +<C><I>Quadratic equations solved in Heron.</I></C> +<p>We have already met with one such equation (in <I>Metrica</I> +III. 4), namely <MATH><I>x</I><SUP>2</SUP>-14<I>x</I>+46 2/3=0</MATH>, the result only <MATH>(<I>x</I>=8 1/2)</MATH> +being given. There are others in the <I>Geometrica</I> where the +process of solution is shown. +<p>(1) <I>Geometrica</I> 24, 3 (Heib.). ‘Given a square such that the +sum of its area and perimeter is 896 feet: to separate the area +from the perimenter’: i.e. <MATH><I>x</I><SUP>2</SUP>+4<I>x</I>=896</MATH>. Heron takes half of +4 and adds its square, completing the square on the left side. +<p>(2) <I>Geometrica</I> 21, 9 and 24, 46 (Heib.) give one and the same +equation, <I>Geom.</I> 24, 47 another like it. ‘Given the sum of +the diameter, perimeter and area of a circle, to find each +of them.’ +<p>The two equations are +<MATH>11/14<I>d</I><SUP>2</SUP>+29/7<I>d</I>=212</MATH>, +and <MATH>11/14<I>d</I><SUP>2</SUP>+29/7<I>d</I>=67 1/2</MATH>. +<p>Our usual method is to begin by dividing by 11/14 throughout, +so as to leave <I>d</I><SUP>2</SUP> as the first term. Heron's is to <I>multiply</I> by +such a number as will leave a square as the first term. In this +case he multiplies by 154, giving <MATH>11<SUP>2</SUP><I>d</I><SUP>2</SUP>+58.11<I>d</I>=212.154 +or 67 1/2.154</MATH> as the case may be. Completing the square, +he obtains <MATH>(11<I>d</I>+29)<SUP>2</SUP>=32648+841 or 10395+841</MATH>. Thus +<MATH>11<I>d</I>+29=√(33489) or √(11236)</MATH>, that is, 183 or 106. +Thus <MATH>11<I>d</I>=154 or 77</MATH>, and <MATH><I>d</I>=14 or 7</MATH>, as the case may be. +<C>Indeterminate problems in the <I>Geometrica.</I></C> +<p>Some very interesting indeterminate problems are now +included by Heiberg in the <I>Geometrica.</I><note><I>Heronis Alexandrini opera</I>, vol. iv, p. 414. 28 sq.</note> Two of them (chap. +24, 1-2) were included in the <I>Ge&edblac;ponicus</I> in Hultsch's edition +(sections 78, 79); the rest are new, having been found in the +Constantinople manuscript from which Schőne edited the +<I>Metrica.</I> As, however, these problems, to whatever period +they belong, are more akin to algebra than to mensuration, +they will be more properly described in a later chapter on +Algebra. +<pb n=345><head>THE <I>DIOPTRA</I></head> +<C>The <I>Dioptra</I> (<G>peri\ dio/ptras</G>).</C> +<p>This treatise begins with a careful description of the +<I>dioptra</I>, an instrument which served with the ancients for +the same purpose as a theodolite with us (chaps. 1-5). The +problems with which the treatise goes on to deal are +(<I>a</I>) problems of ‘heights and distances’, (<I>b</I>) engineering pro- +blems, (<I>c</I>) problems of mensuration, to which is added +(chap. 34) a description of a ‘hodometer’, or taxameter, con- +sisting of an arrangement of toothed wheels and endless +screws on the same axes working on the teeth of the next +wheels respectively. The book ends with the problem +(chap. 37), ‘With a given force to move a given weight by +means of interacting toothed wheels’, which really belongs +to mechanics, and was apparently added, like some other +problems (e.g. 31, ‘to measure the outflow of, i.e. the volume +of water issuing from, a spring’), in order to make the book +more comprehensive. The essential problems dealt with are +such as the following. To determine the difference of level +between two given points (6), to draw a straight line connect- +ing two points the one of which is not visible from the other +(7), to measure the least breadth of a river (9), the distance of +two inaccessible points (10), the height of an inaccessible point +(12), to determine the difference between the heights of two +inaccessible points and the position of the straight line joining +them (13), the depth of a ditch (14); to bore a tunnel through +a mountain going straight from one mouth to the other (15), to +sink a shaft through a mountain perpendicularly to a canal +flowing underneath (16); given a subterranean canal of any +form, to find on the ground above a point from which a +vertical shaft must be sunk in order to reach a given point +on the canal (for the purpose e.g. of removing an obstruction) +(20); to construct a harbour on the model of a given segment +of a circle, given the ends (17), to construct a vault so that it +may have a spherical surface modelled on a given segment +(18). The mensuration problems include the following: to +measure an irregular area, which is done by inscribing a +rectilineal figure and then drawing perpendiculars to the +sides at intervals to meet the contour (23), or by drawing one +straight line across the area and erecting perpendiculars from +<pb n=346><head>HERON OF ALEXANDRIA</head> +that to meet the contour on both sides (24); given that all +the boundary stones of a certain area have disappeared except +two or three, but that the plan of the area is forthcoming, +to determine the position of the lost boundary stones (25). +Chaps. 26-8 remind us of the <I>Metrica</I>: to divide a given +area into given parts by straight lines drawn from one point +(26); to measure a given area without entering it, whether +because it is thickly covered with trees, obstructed by houses, +or entry is forbidden! (27); chaps. <MATH>28-30=<I>Metrica</I> III. 7, +III. 1, and I. 7</MATH>, the last of these three propositions being the +proof of the ‘formula of Heron’ for the area of a triangle in +terms of the sides. Chap. 35 shows how to find the distance +between Rome and Alexandria along a great circle of the +earth by means of the observation of the same eclipse at +the two places, the <I>analemma</I> for Rome, and a concave hemi- +sphere constructed for Alexandria to show the position of the +sun at the time of the said eclipse. It is here mentioned that +the estimate by Eratosthenes of the earth's circumference in +his book <I>On the Measurement of the Earth</I> was the most +accurate that had been made up to date.<note>Heron, vol. iii, p. 302. 13-17.</note> Some hold that +the chapter, like some others which have no particular con- +nexion with the real subject of the <I>Dioptra</I> (e.g. chaps. 31, 34, +37-8) were probably inserted by a later editor, ‘in order to +make the treatise as complete as possible’.<note><I>Ib.</I>, p. 302. 9.</note> +<C>The <I>Mechanics.</I></C> +<p>It is evident that the <I>Mechanics</I>, as preserved in the Arabic, +is far from having kept its original form, especially in +Book I. It begins with an account of the arrangement of +toothed wheels designed to solve the problem of moving a +given weight by a given force; this account is the same as +that given at the end of the Greek text of the <I>Dioptra</I>, and it +is clearly the same description as that which Pappus<note>Pappus, viii, p. 1060 sq.</note> found in +the work of Heron entitled <G>*baroulko/s</G> (‘weight-lifter’) and +himself reproduced with a ratio of force to weight altered +from 5:1000 to 4:160 and with a ratio of 2:1 substituted for +5:1 in the diameters of successive wheels. It would appear +that the chapter from the <G>*baroulko/s</G> was inserted in place of +<pb n=347><head>THE <I>MECHANICS</I></head> +the first chapter or chapters of the real <I>Mechanics</I> which had +been lost. The treatise would doubtless begin with generalities +introductory to mechanics such as we find in the (much +interpolated) beginning of Pappus, Book VIII. It must then +apparently have dealt with the properties of circles, cylinders, +and spheres with reference to their importance in mechanics; +for in Book II. 21 Heron says that the circle is of all figures +the most movable and most easily moved, the same thing +applying also to the cylinder and sphere, and he adds in +support of this a reference to a proof ‘in the preceding Book’. +This reference may be to I. 21, but at the end of that chapter +he says that ‘cylinders, even when heavy, if placed on the +ground so that they touch it in one line only, are easily +moved, and the same is true of spheres also, a matter <I>which +we have already discussed’</I>; the discussion may have come +earlier in the Book, in a chapter now lost. +<p>The treatise, beginning with chap. 2 after the passage +interpolated from the <G>*baroulko/s</G>, is curiously disconnected. +Chaps. 2-7 discuss the motion of circles or wheels, equal or +unequal, moving on different axes (e.g. interacting toothed +wheels), or fixed on the same axis, much after the fashion of +the Aristotelian <I>Mechanical problems.</I> +<C><I>Aristotle's Wheel</I>.</C> +<p>In particular (chap. 7) Heron attempts to explain the puzzle +of the ‘Wheel of Aristotle’, which remained a puzzle up to quite +modern times, and gave rise to the proverb, ‘rotam Aristotelis +magis torquere, quo magis torqueretur’.<note>See Van Capelle, <I>Aristotelis quaestiones mechanicae</I>, 1812, p. 263 sq.</note> ‘The question is’, says +the Aristotelian problem 24, ‘why does the greater circle roll an +equal distance with the lesser circle when they are placed about +the same centre, whereas, when they roll separately, as the +size of one is to the size of the other, so are the straight lines +traversed by them to one another?’<note>Arist. <I>Mechanica</I>, 855 a 28.</note> Let <I>AC, BD</I> be quadrants +of circles with centre <I>O</I> bounded by the same radii, and draw +tangents <I>AE, BF</I> at <I>A</I> and <I>B.</I> In the first case suppose the +circle <I>BD</I> to roll along <I>BF</I> till <I>D</I> takes the position <I>H</I>; then +the radius <I>ODC</I> will be at right angles to <I>AE</I>, and <I>C</I> will be +at <I>G</I>, a point such that <I>AG</I> is equal to <I>BH.</I> In the second +<pb n=348><head>HERON OF ALEXANDRIA</head> +case suppose the circle <I>AC</I> to roll along <I>AE</I> till <I>ODC</I> takes +the position <I>O</I>′<I>FE</I>; then <I>D</I> will be at <I>F</I> where <I>AE</I>=<I>BF</I>. +And similarly if a whole revolution is performed and <I>OBA</I> is +again perpendicular to <I>AE.</I> Contrary, therefore, to the prin- +ciple that the greater circle moves quicker than the smaller on +the same axis, it would appear that the movement of the +<FIG> +smaller in this case is as quick as that of the greater, since +<MATH><I>BH</I>=<I>AG</I></MATH>, and <MATH><I>BF</I>=<I>AE</I></MATH>. Heron's explanation is that, e.g. +in the case where the larger circle rolls on <I>AE</I>, the lesser +circle maintains the same speed as the greater because it has +<I>two</I> motions; for if we regard the smaller circle as merely +fastened to the larger, and not rolling at all, its centre <I>O</I> will +move to <I>O</I>′ traversing a distance <I>OO</I>′ equal to <I>AE</I> and <I>BF</I>; +hence the greater circle will take the lesser with it over an +equal distance, the rolling of the lesser circle having no effect +upon this. +<C><I>The parallelogram of velocities</I>.</C> +<p>Heron next proves the parallelogram of velocities (chap. 8); +he takes the case of a rectangle, but the proof is applicable +generally. +<FIG> +<p>The way it is put is this. A +point moves with uniform velocity +along a straight line <I>AB</I>, from <I>A</I> +to <I>B</I>, while at the same time <I>AB</I> +moves with uniform velocity always +parallel to itself with its extremity +<I>A</I> describing the straight line <I>AC.</I> +Suppose that, when the point arrives at <I>B</I>, the straight line +<pb n=349><head>THE PARALLELOGRAM OF VELOCITIES</head> +reaches the position <I>CD.</I> Let <I>EF</I> be any intermediate +position of <I>AB</I>, and <I>G</I> the position at the same instant +of the moving point on it. Then clearly <MATH><I>AE</I>:<I>AC</I>=<I>EG</I>:<I>EF</I></MATH>; +therefore <MATH><I>AE</I>:<I>EG</I>=<I>AC</I>:<I>EF</I>=<I>AC</I>:<I>CD</I></MATH>, and it follows that +<I>G</I> lies on the diagonal <I>AD</I>, which is therefore the actual path +of the moving point. +<p>Chaps. 9-19 contain a digression on the construction of +plane and solid figures similar to given figures but greater or +less in a given ratio. Heron observes that the case of plane +figures involves the finding of a mean proportional between +two straight lines, and the case of solid figures the finding of +<I>two</I> mean proportionals; in chap. 11 he gives his solution of +the latter problem, which is preserved in Pappus and Eutocius +as well, and has already been given above (vol. i, pp. 262-3). +<p>The end of chap. 19 contains, quite inconsequently, the con- +struction of a toothed wheel to move on an endless screw, +after which chap. 20 makes a fresh start with some observa- +tions on weights in equilibrium on a horizontal plane but +tending to fall when the plane is inclined, and on the ready +mobility of objects of cylindrical form which touch the plane +in one line only. +<C><I>Motion on an inclined plane</I>.</C> +<p>When a weight is hanging freely by a rope over a pulley, +no force applied to the other end of the rope less than the +weight itself will keep it up, but, if the weight is placed on an +inclined plane, and both the plane and the portion of the +weight in contact with it are smooth, the case is different. +Suppose, e.g., that a weight in the form of a cylinder is placed +on an inclined plane so that the line in which they touch is +horizontal; then the force required to be applied to a rope +parallel to the line of greatest slope in the plane in order to +keep the weight in equilibrium is less than the weight. For +the vertical plane passing through the line of contact between +the cylinder and the plane divides the cylinder into two +unequal parts, that on the downward side of the plane being +the greater, so that the cylinder will tend to roll down; but +the force required to support the cylinder is the ‘equivalent’, +not of the weight of the whole cylinder, but of the difference +<pb n=350><head>HERON OF ALEXANDRIA</head> +between the two portions into which the vertical plane cuts it +(chap. 23). +<C><I>On the centre of gravity</I>.</C> +<p>This brings Heron to the centre of gravity (chap. 24). Here +a definition by Posidonius, a Stoic, of the ‘centre of gravity’ +or ‘centre of inclination’ is given, namely ‘a point such that, +if the body is hung up at it, the body is divided into two +equal parts’ (he should obviously have said ‘divided <I>by any +vertical plane through the point of suspension</I> into two equal +parts’). But, Heron says, Archimedes distinguished between +the ‘centre of gravity’ and the ‘point of suspension’, defining +the latter as a point on the body such that, if the body is +hung up at it, all the parts of the body remain in equilibrium +and do not oscillate or incline in any direction. ‘“Bodies”, said +Archimedes, “may rest (without inclining one way or another) +with either a line, or only one point, in the body fixed”.’ The +‘centre of inclination’, says Heron, ‘is one single point in any +particular body to which all the vertical lines through the +points of suspension converge.’ Comparing Simplicius's quo- +tation of a definition by Archimedes in his <G>*kentrobarika/</G>, to +the effect that the centre of gravity is a certain point in the +body such that, if the body is hung up by a string attached to +that point, it will remain in its position without inclining in +any direction,<note>Simplicius on <I>De caelo</I>, p. 543. 31-4, Heib.</note> we see that Heron directly used a certain +treatise of Archimedes. So evidently did Pappus, who has +a similar definition. Pappus also speaks of a body supported +at a point by a vertical stick: if, he says, the body is in +equilibrium, the line of the stick produced upwards must pass +through the centre of gravity.<note>Pappus, viii, p. 1032. 5-24.</note> Similarly Heron says that +the same principles apply when the body is supported as when +it is suspended. Taking up next (chaps. 25-31) the question +of ‘supports’, he considers cases of a heavy beam or a wall +supported on a number of pillars, equidistant or not, even +or not even in number, and projecting or not projecting +beyond one or both of the extreme pillars, and finds how +much of the weight is supported on each pillar. He says +that Archimedes laid down the principles in his ‘Book on +<pb n=351><head>ON THE CENTRE OF GRAVITY</head> +Supports’. As, however, the principles are the same whether +the body is supported or hung up, it does not follow that +this was a different work from that known as <G>peri\ zugw=n</G>. +Chaps. 32-3, which are on the principles of the lever or of +weighing, end with an explanation amounting to the fact +that ‘greater circles overpower smaller when their movement +is about the same centre’, a proposition which Pappus says +that Archimedes proved in his work <G>peri\ zugw=n</G>.<note>Pappus, viii, p. 1068. 20-3.</note> In chap. 32, +too, Heron gives as his authority a proof given by Archimedes +in the same work. With I. 33 may be compared II. 7, +where Heron returns to the same subject of the greater and +lesser circles moving about the same centre and states the +fact that weights reciprocally proportional to their radii are +in equilibrium when suspended from opposite ends of the +horizontal diameters, observing that Archimedes proved the +proposition in his work ‘On the equalization of inclination’ +(presumably <G>i)sorropi/ai</G>). +<C>Book II. The five mechanical powers.</C> +<p>Heron deals with the wheel and axle, the lever, the pulley, +the wedge and the screw, and with combinations of these +powers. The description of the powers comes first, chaps. 1-6, +and then, after II. 7, the proposition above referred to, and the +theory of the several powers based upon it (chaps. 8-20). +Applications to specific cases follow. Thus it is shown how +to move a weight of 1000 talents by means of a force of +5 talents, first by the system of wheels described in the +<G>*baroulko/s</G>, next by a system of pulleys, and thirdly by a +combination of levers (chaps. 21-5). It is possible to combine +the different powers (other than the wedge) to produce the +same result (chap. 29). The wedge and screw are discussed +with reference to their angles (chaps. 30-1), and chap. 32 refers +to the effect of friction. +<C><I>Mechanics in daily life; queries and answers</I>.</C> +<p>After a prefatory chapter (33), a number of queries resem- +bling the Aristotelian problems are stated and answered +(chap. 34), e.g. ‘Why do waggons with two wheels carry +a weight more easily than those with four wheels?’, ‘Why +<pb n=352><head>HERON OF ALEXANDRIA</head> +do great weights fall to the ground in a shorter time than +lighter ones?’, ‘Why does a stick break sooner when one +puts one's knee against it in the middle?’, ‘Why do people +use pincers rather than the hand to draw a tooth?’, ‘Why +is it easy to move weights which are suspended?’, and +‘Why is it the more difficult to move such weights the farther +the hand is away from them, right up to the point of suspension +or a point near it?’, ‘Why are great ships turned by a rudder +although it is so small?’, ‘Why do arrows penetrate armour +or metal plates but fail to penetrate cloth spread out?’ +<C><I>Problems on the centre of gravity, &c.</I></C> +<p>II. 35, 36, 37 show how to find the centre of gravity of +a triangle, a quadrilateral and a pentagon respectively. Then, +assuming that a triangle of uniform thickness is supported by +a prop at each angle, Heron finds what weight is supported +by each prop, (<I>a</I>) when the props support the triangle only, +(<I>b</I>) when they support the triangle plus a given weight placed +at any point on it (chaps. 38, 39). Lastly, if known weights +are put on the triangle at each angle, he finds the centre of +gravity of the system (chap. 40); the problem is then extended +to the case of any polygon (chap. 41). +<p>Book III deals with the practical construction of engines +for all sorts of purposes, machines employing pulleys with +one, two, or more supports for lifting weights, oil-presses, &c. +<C>The <I>Catoptrica.</I></C> +<p>This work need not detain us long. Several of the theoretical +propositions which it contains are the same as propositions +in the so-called <I>Catoptrica</I> of Euclid, which, as we have +seen, was in all probability the work of Theon of Alexandria +and therefore much later in date. In addition to theoretical +propositions, it contains problems the purpose of which is to +construct mirrors or combinations of mirrors of such shape +as will reflect objects in a particular way, e.g. to make the +right side appear as the right in the picture (instead of the +reverse), to enable a person to see his back or to appear in +the mirror head downwards, with face distorted, with three +eyes or two noses, and so forth. Concave and convex +<pb n=353><head>THE <I>CATOPTRICA</I></head> +cylindrical mirrors play a part in these arrangements. The +whole theory of course ultimately depends on the main pro- +positions 4 and 5 that the angles of incidence and reflection +are equal whether the mirror is plane or circular. +<C><I>Heron's proof of equality of angles of incidence and reflection</I>.</C> +<p>Let <I>AB</I> be a plane mirror, <I>C</I> the eye, <I>D</I> the object seen. +The argument rests on the fact that nature ‘does nothing in +vain’. Thus light travels in a straight line, that is, by the +<FIG> +quickest road. Therefore, even +when the ray is a line broken +at a point by reflection, it must +mark the shortest broken line +of the kind connecting the eye +and the object. Now, says +Heron, I maintain that the +shortest of the broken lines +(broken at the mirror) which +connect <I>C</I> and <I>D</I> is the line, as +<I>CAD</I>, the parts of which make equal angles with the mirror. +Join <I>DA</I> and produce it to meet in <I>F</I> the perpendicular from +<I>C</I> to <I>AB.</I> Let <I>B</I> be any point on the mirror other than <I>A</I>, +and join <I>FB, BD.</I> +<p>Now <MATH>∠<I>EAF</I>=∠<I>BAD</I> +=∠<I>CAE</I></MATH>, by hypothesis. +<p>Therefore the triangles <I>AEF, AEC</I>, having two angles equal +and <I>AE</I> common, are equal in all respects. +<p>Therefore <MATH><I>CA</I>=<I>AF</I></MATH>, and <MATH><I>CA</I>+<I>AD</I>=<I>DF</I></MATH>. +<p>Since <MATH><I>FE</I>=<I>EC</I></MATH>, and <I>BE</I> is perpendicular to <MATH><I>FC, BF</I>=<I>BC</I></MATH>. +<p>Therefore <MATH><I>CB</I>+<I>BD</I>=<I>FB</I>+<I>BD</I> +> <I>FD</I>, +i.e. > <I>CA</I>+<I>AD</I></MATH>. +<p>The proposition was of course known to Archimedes. We +gather from a scholium to the Pseudo-Euclidean <I>Catoptrica</I> +that he proved it in a different way, namely by <I>reductio ad +absurdum</I>, thus: Denote the angles <I>CAE, DAB</I> by <G>a, b</G> re- +spectively. Then, <G>a</G> is >= or <<G>b</G>. Suppose <G>a>b</G>. Then, +<pb n=354><head>HERON OF ALEXANDRIA</head> +reversing the ray so that the eye is at <I>D</I> instead of <I>C</I>, and the +object at <I>C</I> instead of <I>D</I>, we must have <G>b>a</G>. But <G>b</G> was +less than <G>a</G>, which is impossible. (Similarly it can be proved +that <G>a</G> is not less than <G>b</G>.) Therefore <MATH><G>a=b</G></MATH>. +<p>In the Pseudo-Euclidean <I>Catoptrica</I> the proposition is +practically assumed; for the third assumption or postulate +at the beginning states in effect that, in the above figure, if <I>A</I> +be the point of incidence, <MATH><I>CE</I>:<I>EA</I>=<I>DH</I>:<I>HA</I></MATH> (where <I>DH</I> is +perpendicular to <I>AB</I>). It follows instantaneously (Prop. 1) +that ∠<I>CAE</I>=∠<I>DAH</I>. +<p>If the mirror is the convex side of a circle, the same result +<FIG> +follows <I>a fortiori.</I> Let <I>CA, AD</I> meet +the arc at equal angles, and <I>CB, BD</I> at +unequal angles. Let <I>AE</I> be the tan- +gent at <I>A</I>, and complete the figure. +Then, says Heron, (the angles <I>GAC, +BAD</I> being by hypothesis equal), if we +subtract the equal angles <I>GAE, BAF</I> +from the equal angles <I>GAC, BAD</I> (both +pairs of angles being ‘mixed’, be it +observed), we have ∠<I>EAC</I>=∠<I>FAD.</I> Therefore <MATH><I>CA</I>+<I>AD</I> +<<I>CF</I>+<I>FD</I></MATH> and <I>a fortiori</I> <<I>CB</I>+<I>BD</I>. +<p>The problems solved (though the text is so corrupt in places +that little can be made of it) were such as the following: +11, To construct a right-handed mirror (i.e. a mirror which +makes the right side right and the left side left instead of +the opposite); 12, to construct the mirror called <I>polytheoron</I> +(‘with many images’); 16, to construct a mirror inside the +window of a house, so that you can see in it (while inside +the room) everything that passes in the street; 18, to arrange +mirrors in a given place so that a person who approaches +cannot actually see either himself or any one else but can see +any image desired (a ‘ghost-seer’). +<pb> +<C>XIX +PAPPUS OF ALEXANDRIA</C> +<p>WE have seen that the Golden Age of Greek geometry +ended with the time of Apollonius of Perga. But the influence +of Euclid, Archimedes and Apollonius continued, and for some +time there was a succession of quite competent mathematicians +who, although not originating anything of capital importance, +kept up the tradition. Besides those who were known for +particular investigations, e.g. of new curves or surfaces, there +were such men as Geminus who, it cannot be doubted, were +thoroughly familiar with the great classics. Geminus, as we +have seen, wrote a comprehensive work of almost encyclopaedic +character on the classification and content of mathematics, +including the history of the development of each subject. +But the beginning of the Christian era sees quite a different +state of things. Except in sphaeric and astronomy (Menelaus +and Ptolemy), production was limited to elementary text- +books of decidedly feeble quality. In the meantime it would +seem that the study of higher geometry languished or was +completely in abeyance, until Pappus arose to revive interest +in the subject. From the way in which he thinks it necessary +to describe the contents of the classical works belonging to +the <I>Treasury of Analysis</I>, for example, one would suppose +that by his time many of them were, if not lost, completely +forgotten, and that the great task which he set himself was +the re-establishment of geometry on its former high plane of +achievement. Presumably such interest as he was able to +arouse soon flickered out, but for us his work has an in- +estimable value as constituting, after the works of the great +mathematicians which have actually survived, the most im- +portant of all our sources. +<pb n=356><head>PAPPUS OF ALEXANDRIA</head> +<C>Date of Pappus.</C> +<p>Pappus lived at the end of the third century A.D. The +authority for this date is a marginal note in a Leyden manu- +script of chronological tables by Theon of Alexandria, where, +opposite to the name of Diocletian, a scholium says, ‘In his +time Pappus wrote’. Diocletian reigned from 284 to 305, +and this must therefore be the period of Pappus's literary +activity. It is true that Suidas makes him a contemporary +of Theon of Alexandria, adding that they both lived under +Theodosius I (379-395). But Suidas was evidently not well +acquainted with the works of Pappus; though he mentions +a description of the earth by him and a commentary on four +Books of Ptolemy's <I>Syntaxis</I>, he has no word about his greatest +work, the <I>Synagoge.</I> As Theon also wrote a commentary on +Ptolemy and incorporated a great deal of the commentary of +Pappus, it is probable that Suidas had Theon's commentary +before him and from the association of the two names wrongly +inferred that they were contemporaries. +<C>Works (commentaries) other than the <I>Collection.</I></C> +<p>Besides the <I>Synagoge</I>, which is the main subject of this +chapter, Pappus wrote several commentaries, now lost except for +fragments which have survived in Greek or Arabic. One was +a commentary on the <I>Elements</I> of Euclid. This must presum- +ably have been pretty complete, for, while Proclus (on Eucl. I) +quotes certain things from Pappus which may be assumed to +have come in the notes on Book I, fragments of his commen- +tary on Book X actually survive in the Arabic (see above, +vol. i, pp. 154-5, 209), and again Eutocius in his note on Archi- +medes, <I>On the Sphere and Cylinder</I>, I. 13, says that Pappus +explained in his commentary on the <I>Elements</I> how to inscribe +in a circle a polygon similar to a polygon inscribed in another +circle, which problem would no doubt be solved by Pappus, as +it is by a scholiast, in a note on XII. 1. Some of the references +by Proclus deserve passing mention. (1) Pappus said that +the converse of Post. 4 (equality of all right angles) is not +true, i.e. it is not true that all angles equal to a right angle are +themselves right, since the ‘angle’ between the conterminous +arcs of two semicircles which are equal and have their +<pb n=357><head>WORKS OTHER THAN THE <I>COLLECTION</I></head> +diameters at right angles and terminating at one point is +equal to, but is not, a right angle.<note>Proclus on Eucl. I, pp. 189-90.</note> (2) Pappus said that, +in addition to the genuine axioms of Euclid, there were others +<FIG> +on record about unequals added to +equals and equals added to unequals. +Others given by Pappus are (says +Proclus) involved by the definitions, +e.g. that ‘all parts of the plane and of +the straight line coincide with one +another’, that ‘a point divides a line, +a line a surface, and a surface a solid’, and that ‘the infinite +is (obtained) in magnitudes both by addition and diminution’.<note><I>Ib.</I>, pp. 197. 6-198. 15.</note> +(3) Pappus gave a pretty proof of Eucl. I. 5, which modern +editors have spoiled when introducing it into text-books. If +<I>AB, AC</I> are the equal sides in an isosceles triangle, Pappus +compares the triangles <I>ABC</I> and <I>ACB</I> (i.e. as if he were com- +paring the triangle <I>ABC</I> seen from the front with the same +triangle seen from the back), and shows that they satisfy the +conditions of I. 4, so that they are equal in all respects, whence +the result follows.<note><I>Ib.</I>, pp. 249. 20-250. 12.</note> +<p>Marinus at the end of his commentary on Euclid's <I>Data</I> +refers to a commentary by Pappus on that book. +<p>Pappus's commentary on Ptolemy's <I>Syntaxis</I> has already +been mentioned (p. 274); it seems to have extended to six +Books, if not to the whole of Ptolemy's work. The <I>Fihrist</I> +says that he also wrote a commentary on Ptolemy's <I>Plani- +sphaerium</I>, which was translated into Arabic by Thābit b. +Qurra. Pappus himself alludes to his own commentary on +the <I>Analemma</I> of Diodorus, in the course of which he used the +conchoid of Nicomedes for the purpose of trisecting an angle. +<p>We come now to Pappus's great work. +<C>The <I>Synagoge</I> or <I>Collection.</I></C> +<C>(<G>a</G>) <I>Character of the work; wide range.</I></C> +<p>Obviously written with the object of reviving the classical +Greek geometry, it covers practically the whole field. It is, +<pb n=358><head>PAPPUS OF ALEXANDRIA</head> +however, a handbook or guide to Greek geometry rather than +an encyclopaedia; it was intended, that is, to be read with the +original works (where still extant) rather than to enable them +to be dispensed with. Thus in the case of the treatises +included in the <I>Treasury of Analysis</I> there is a general intro- +duction, followed by a general account of the contents, with +lemmas, &c., designed to facilitate the reading of the treatises +themselves. On the other hand, where the history of a subject +is given, e.g. that of the problem of the duplication of the +cube or the finding of the two mean proportionals, the various +solutions themselves are reproduced, presumably because they +were not easily accessible, but had to be collected from various +sources. Even when it is some accessible classic which is +being described, the opportunity is taken to give alternative +methods, or to make improvements in proofs, extensions, and +so on. Without pretending to great originality, the whole +work shows, on the part of the author, a thorough grasp of +all the subjects treated, independence of judgment, mastery +of technique; the style is terse and clear; in short, Pappus +stands out as an accomplished and versatile mathematician, +a worthy representative of the classical Greek geometry. +<C>(<G>b</G>) <I>List of authors mentioned.</I></C> +<p>The immense range of the <I>Collection</I> can be gathered from +a mere enumeration of the names of the various mathematicians +quoted or referred to in the course of it. The greatest of +them, Euclid, Archimedes and Apollonius, are of course con- +tinually cited, others are mentioned for some particular +achievement, and in a few cases the mention of a name by +Pappus is the whole of the information we possess about the +person mentioned. In giving the list of the names occurring +in the book, it will, I think, be convenient and may economize +future references if I note in brackets the particular occasion +of the reference to the writers who are mentioned for one +achievement or as the authors of a particular book or investi- +gation. The list in alphabetical order is: Apollonius of Perga, +Archimedes, Aristaeus the elder (author of a treatise in five +Books on the Elements of Conics or of ‘five Books on Solid +Loci connected with the conics’), Aristarchus of Samos (<I>On the</I> +<pb n=359><head>THE <I>COLLECTION</I></head> +<I>sizes and distances of the sun and moon</I>), Autolycus (<I>On the +moving sphere</I>), Carpus of Antioch (who is quoted as having +said that Archimedes wrote only one mechanical book, that +on sphere-making, since he held the mechanical appliances +which made him famous to be nevertheless unworthy of +written description: Carpus himself, who was known as +<I>mechanicus</I>, applied geometry to other arts of this practical +kind), Charmandrus (who added three simple and obvious loci +to those which formed the beginning of the <I>Plane Loci</I> of +Apollonius), Conon of Samos, the friend of Archimedes (cited +as the propounder of a theorem about the spiral in a plane +which Archimedes proved: this would, however, seem to be +a mistake, as Archimedes says at the beginning of his treatise +that he sent certain theorems, without proofs, to Conon, who +would certainly have proved them had he lived), Demetrius of +Alexandria (mentioned as the author of a work called ‘Linear +considerations’, <G>grammikai\ e)pista/seis</G>, i.e. considerations on +curves, as to which nothing more is known), Dinostratus, +the brother of Menaechmus (cited, with Nicomedes, as having +used the curve of Hippias, to which they gave the name of +<I>quadratrix</I>, <G>tetragwni/zousa</G>, for the squaring of the circle), +Diodorus (mentioned as the author of an <I>Analemma</I>), Erato- +sthenes (whose <I>mean-finder</I>, an appliance for finding two or +any number of geometric means, is described, and who is +further mentioned as the author of two Books ‘On means’ +and of a work entitled ‘Loci with reference to means’), +Erycinus (from whose <I>Paradoxa</I> are quoted various problems +seeming at first sight to be inconsistent with Eucl. I. 21, it +being shown that straight lines can be drawn from two points +on the base of a triangle to a point within the triangle which +are together greater than the other two sides, provided that the +points in the base may be points other than the extremities), +Euclid, Geminus the mathematician (from whom is cited a +remark on Archimedes contained in his book ‘On the classifica- +tion of the mathematical sciences’, see above, p. 223), Heraclitus +(from whom Pappus quotes an elegant solution of a <G>neu=sis</G> +with reference to a square), Hermodorus (Pappus's son, to +whom he dedicated Books VII, VIII of his <I>Collection</I>), Heron +of Alexandria (whose mechanical works are extensively quoted +from), Hierius the philosopher (a contemporary of Pappus, +<pb n=360><head>PAPPUS OF ALEXANDRIA</head> +who is mentioned as having asked Pappus's opinion on the +attempted solution by ‘plane’ methods of the problem of the two +means, which actually gives a method of approximating to +a solution<note>See vol. i, pp. 268-70.</note>), Hipparchus (quoted as practically adopting three +of the hypotheses of Aristarchus of Samos), Megethion (to +whom Pappus dedicated Book V of his <I>Collection</I>), Menelaus +of Alexandria (quoted as the author of <I>Sphaerica</I> and as having +applied the name <G>para/doxos</G> to a certain curve), Nicomachus +(on three means additional to the first three), Nicomedes, +Pandrosion (to whom Book III of the <I>Collection</I> is dedicated), +Pericles (editor of Euclid's <I>Data</I>), Philon of Byzantium (men- +tioned along with Heron), Philon of Tyana (mentioned as the +discoverer of certain complicated curves derived from the inter- +weaving of plectoid and other surfaces), Plato (with reference +to the five regular solids), Ptolemy, Theodosius (author of the +<I>Sphaerica</I> and <I>On Days and Nights</I>). +<C>(<G>g</G>) <I>Translations and editions.</I></C> +<p>The first published edition of the <I>Collection</I> was the Latin +translation by Commandinus (Venice 1589, but dated at the +end ‘Pisauri apud Hieronymum Concordiam 1588’; reissued +with only the title-page changed ‘Pisauri . . . 1602’). Up to +1876 portions only of the Greek text had appeared, namely +Books VII, VIII in Greek and German, by C. J. Gerhardt, 1871, +chaps. 33-105 of Book V, by Eisenmann, Paris 1824, chaps. +45-52 of Book IV in <I>Iosephi Torelli Veronensis Geometrica</I>, +1769, the remains of Book II, by John Wallis (in <I>Opera +mathematica</I>, III, Oxford 1699); in addition, the restorers +of works of Euclid and Apollonius from the indications +furnished by Pappus give extracts from the Greek text +relating to the particular works, Bretonde Champ on Euclid's +<I>Porisms</I>, Halley in his edition of the <I>Conics</I> of Apollonius +(1710) and in his translation from the Arabic and restoration +respectively of the <I>De sectione rationis</I> and <I>De sectione spatii</I> +of Apollonius (1706), Camerer on Apollonius's <I>Tactiones</I> (1795), +Simson and Horsley in their restorations of Apollonius's <I>Plane +Loci</I> and <I>Inclinationes</I> published in the years 1749 and 1770 +respectively. In the years 1876-8 appeared the only com- +<pb n=361><head>THE <I>COLLECTION.</I> BOOKS I, II, III</head> +plete Greek text, with apparatus, Latin translation, com- +mentary, appendices and indices, by Friedrich Hultsch; this +great edition is one of the first monuments of the revived +study of the history of Greek mathematics in the last half +of the nineteenth century, and has properly formed the model +for other definitive editions of the Greek text of the other +classical Greek mathematicians, e.g. the editions of Euclid, +Archimedes, Apollonius, &c., by Heiberg and others. The +Greek index in this edition of Pappus deserves special mention +because it largely serves as a dictionary of mathematical +terms used not only in Pappus but by the Greek mathe- +maticians generally. +<C>(<G>d</G>) <I>Summary of contents.</I></C> +<p>At the beginning of the work, Book I and the first 13 pro- +positions (out of 26) of Book II are missing. The first 13 +propositions of Book II evidently, like the rest of the Book, +dealt with Apollonius's method of working with very large +numbers expressed in successive powers of the myriad, 10000. +This system has already been described (vol. i, pp. 40, 54-7). +The work of Apollonius seems to have contained 26 proposi- +tions (25 leading up to, and the 26th containing, the final +continued multiplication). +<p>Book III consists of four sections. Section (1) is a sort of +history of the problem of <I>finding two mean proportionals, in +continued proportion, between two given struight lines.</I> +<p>It begins with some general remarks about the distinction +between theorems and problems. Pappus observes that, +whereas the ancients called them all alike by one name, some +regarding them all as problems and others as theorems, a clear +distinction was drawn by those who favoured more exact +terminology. According to the latter a problem is that in +which it is proposed to <I>do</I> or <I>construct</I> something, a theorem +that in which, given certain hypotheses, we investigate that +which follows from and is necessarily implied by them. +Therefore he who propounds a theorem, no matter how he has +become aware of the fact which is a necessary consequence of +the premisses, must state, as the object of inquiry, the right +result and no other. On the other hand, he who propounds +<pb n=362><head>PAPPUS OF ALEXANDRIA</head> +a problem may bid us do something which is in fact im- +possible, and that without necessarily laying himself open +to blame or criticism. For it is part of the solver's duty +to determine the conditions under which the problem is +possible or impossible, and, ‘if possible, when, how, and in +how many ways it is possible’. When, however, a man pro- +fesses to know mathematics and yet commits some elementary +blunder, he cannot escape censure. Pappus gives, as an +example, the case of an unnamed person ‘who was thought to +be a great geometer’ but who showed ignorance in that he +claimed to know how to solve the problem of the two mean +proportionals by ‘plane’ methods (i.e. by using the straight +line and circle only). He then reproduces the argument of +the anonymous person, for the purpose of showing that it +does not solve the problem as its author claims. We have +seen (vol. i, pp. 269-70) how the method, though not actually +solving the problem, does furnish a series of successive approxi- +mations to the real solution. Pappus adds a few simple +lemmas assumed in the exposition. +<p>Next comes the passage<note>Pappus, iii, p. 54. 7-22.</note>, already referred to, on the dis- +tinction drawn by the ancients between (1) <I>plane</I> problems or +problems which can be solved by means of the straight line +and circle, (2) <I>solid</I> problems, or those which require for their +solution one or more conic sections, (3) <I>linear</I> problems, or +those which necessitate recourse to higher curves still, curves +with a more complicated and indeed a forced or unnatural +origin (<G>bebiasme/nhn</G>) such as spirals, quadratrices, cochloids +and cissoids, which have many surprising properties of their +own. The problem of the two mean proportionals, being +a <I>solid</I> problem, required for its solution either conics or some +equivalent, and, as conics could not be constructed by purely +geometrical means, various mechanical devices were invented +such as that of Eratosthenes (the <I>mean-finder</I>), those described +in the <I>Mechanics</I> of Philon and Heron, and that of Nicomedes +(who used the ‘cochloidal’ curve). Pappus proceeds to give the +solutions of Eratosthenes, Nicomedes and Heron, and then adds +a fourth which he claims as his own, but which is practically +the same as that attributed by Eutocius to Sporus. All these +solutions have been given above (vol. i, pp. 258-64, 266-8). +<pb n=363><head>THE <I>COLLECTION.</I> BOOK III</head> +<C>Section (2). <I>The theory of means.</I></C> +<p>Next follows a section (pp. 69-105) on the theory of the +different kinds of <I>means.</I> The discussion takes its origin +from the statement of the ‘second problem’, which was that +of ‘exhibiting the three means’ (i.e. the arithmetic, geometric +and harmonic) ‘in a semicircle’. Pappus first gives a con- +struction by which another geometer (<G>a)/llos tis</G>) claimed to +have solved this problem, but he does not seem to have under- +stood it, and returns to the same problem later (pp. 80-2). +<p>In the meantime he begins with the definitions of the +three means and then shows how, given any two of three +terms <I>a, b, c</I> in arithmetical, geometrical or harmonical pro- +gression, the third can be found. The definition of the mean +(<I>b</I>) of three terms <I>a, b, c</I> in harmonic progression being that it +satisfies the relation <MATH><I>a:c</I>=<I>a--b:b--c</I>,</MATH> Pappus gives alternative +definitions for the arithmetic and geometric means in corre- +sponding form, namely for the arithmetic mean <MATH><I>a:a</I>=<I>a--b:b--c</I></MATH> +and for the geometric <MATH><I>a:b</I>=<I>a--b:b--c.</I></MATH> +<p>The construction for the harmonic mean is perhaps worth +giving. Let <I>AB, BG</I> be two given straight lines. At <I>A</I> draw +<I>DAE</I> perpendicular to <I>AB</I>, and make <I>DA, AE</I> equal. Join +<I>DB, BE.</I> From <I>G</I> draw <I>GF</I> at right +<FIG> +angles to <I>AB</I> meeting <I>DB</I> in <I>F.</I> +Join <I>EF</I> meeting <I>AB</I> in <I>C.</I> Then +<I>BC</I> is the required harmonic mean. +<p>For +<MATH><I>AB:BG</I>=<I>DA:FG</I> +=<I>EA:FG</I> +=<I>AC:CG</I> +=<I>(AB--BC):(BC--BG).</I></MATH> +<p>Similarly, by means of a like figure, we can find <I>BG</I> when +<I>AB, BC</I> are given, and <I>AB</I> when <I>BC, BG</I> are given (in +the latter case the perpendicular <I>DE</I> is drawn through <I>G</I> +instead of <I>A</I>). +<p>Then follows a proposition that, if the three means and the +several extremes are represented in one set of lines, there must +be five of them at least, and, after a set of five such lines have +been found in the smallest possible integers, Pappus passes to +<pb n=364><head>PAPPUS OF ALEXANDRIA</head> +the problem of representing the three means with the respective +extremes by <I>six</I> lines drawn in a semicircle. +<p>Given a semicircle on the diameter <I>AC</I>, and <I>B</I> any point on +the diameter, draw <I>BD</I> at right angles to <I>AC.</I> Let the tangent +<FIG> +at <I>D</I> meet <I>AC</I> produced in <I>G</I>, and measure <I>DH</I> along the +tangent equal to <I>DG.</I> Join <I>HB</I> meeting the radius <I>OD</I> in <I>K.</I> +Let <I>BF</I> be perpendicular to <I>OD.</I> +<p>Then, exactly as above, it is shown that <I>OK</I> is a harmonic +mean between <I>OF</I> and <I>OD.</I> Also <I>BD</I> is the geometric mean +between <I>AB, BC</I>, while <I>OC</I> (=<I>OD</I>) is the arithmetic mean +between <I>AB, BC.</I> +<p>Therefore the <I>six</I> lines <I>DO</I> (=<I>OC</I>), <I>OK, OF, AB, BC, BD</I> +supply the three means with the respective extremes. +<p>But Pappus seems to have failed to observe that the ‘certain +other geometer’, who has the same figure excluding the dotted +lines, supplied the same in <I>five</I> lines. For he said that <I>DF</I> +is ‘a harmonic mean’. It is in fact the harmonic mean +between <I>AB, BC</I>, as is easily seen thus. +<p>Since <I>ODB</I> is a right-angled triangle, and <I>BF</I> perpendicular +to <I>OD</I>, +<MATH><I>DF:BD</I>=<I>BD:DO</I>,</MATH> +or <MATH><I>DF.DO</I>=<I>BD</I><SUP>2</SUP>=<I>AB.BC.</I></MATH> +<p>But <MATH><I>DO</I>=1/2(<I>AB</I>+<I>BC</I>);</MATH> +therefore <MATH><I>DF.</I>(<I>AB</I>+<I>BC</I>)=2<I>AB.BC.</I></MATH> +<p>Therefore <MATH><I>AB.(DF--BC)</I>=<I>BC.(AB--DF)</I>,</MATH> +that is, <MATH><I>AB:BC</I>=<I>(AB--DF):(DF--BC)</I>,</MATH> +and <I>DF</I> is the harmonic mean between <I>AB, BC.</I> +<p>Consequently the <I>five</I> lines <I>DO</I> (=<I>OC</I>), <I>DF, AB, BC, BD</I> +exhibit all the three means with the extremes. +<pb n=365><head>THE <I>COLLECTION.</I> BOOK III</head> +<p>Pappus does not seem to have seen this, for he observes +that the geometer in question, though saying that <I>DF</I> is +a harmonic mean, does not say how it is a harmonic mean +or between what straight liues. +<p>In the next chapters (pp. 84-104) Pappus, following Nico- +machus and others, defines seven more means, three of which +were ancient and the last four more modern, and shows how +we can form all ten means as linear functions of <G>a, b, g</G>, where +<G>a, b, g</G> are in geometrical progression. The exposition has +already been described (vol. i, pp. 86-9). +<C>Section (3). <I>The ‘Paradoxes’ of Erycinus.</I></C> +<p>The third section of Book III (pp. 104-30) contains a series +of propositions, all of the same sort, which are curious rather +than geometrically important. They appear to have been +taken direct from a collection of <I>Paradoxes</I> by one Erycinus.<note>Pappus, iii, p. 106. 5-9.</note> +The first set of these propositions (Props. 28-34) are connected +with Eucl. I. 21, which says that, if from the extremities +of the base of any triangle two straight lines be drawn meeting +at any point within the triangle, the straight lines are together +less than the two sides of the triangle other than the base, +but contain a greater angle. It is pointed out that, if the +straight lines are allowed to be drawn from points in the base +other than the extremities, their sum may be greater than the +other two sides of the triangle. +<p>The first case taken is that of a right-angled triangle <I>ABC</I> +right-angled at <I>B.</I> Draw <I>AD</I> to any point <I>D</I> on <I>BC.</I> Measure +on it <I>DE</I> equal to <I>AB</I>, bisect <I>AE</I> +in <I>F</I>, and join <I>FC.</I> Then shall +<FIG> +<MATH><I>DF</I>+<I>FC</I> be><I>BA</I>+<I>AC.</I></MATH> +<p>For <MATH><I>EF</I>+<I>FC</I>=<I>AF</I>+<I>FC</I>><I>AC.</I></MATH> +<p>Add <I>DE</I> and <I>AB</I> respectively, +and we have +<MATH><I>DF</I>+<I>FC</I>><I>BA</I>+<I>AC.</I></MATH> +<p>More elaborate propositions are next proved, such as the +following. +<p>1. In any triangle, except an equilateral triangle or an isosceles +<pb n=366><head>PAPPUS OF ALEXANDRIA</head> +triangle with base less than one of the other sides, it is possible +to construct on the base and within the triangle two straight +lines meeting at a point, the sum of which is <I>equal</I> to the sum +of the other two sides of the triangle (Props. 29, 30). +<p>2. In any triangle in which it is possible to construct two +straight lines from the base to one internal point the sum +of which is equal to the sum of the two sides of the triangle, +it is also possible to construct two other such straight lines the +sum of which is <I>greater</I> than that sum (Prop. 31). +<p>3. Under the same conditions, if the base is greater than either +of the other two sides, two straight lines can be so constructed +from the base to an internal point which are <I>respectively</I> +greater than the other two sides of the triangle; and the lines +may be constructed so as to be respectively <I>equal</I> to the two +sides, if one of those two sides is less than the other and each +of them is less than the base (Props. 32, 33). +<p>4. The lines may be so constructed that their sum will bear to +the sum of the two sides of the triangle any ratio less than +2:1 (Prop. 34). +<p>As examples of the proofs, we will take the case of the +scalene triangle, and prove the first and Part 1 of the third of +the above propositions for such a triangle. +<p>In the triangle <I>ABC</I> with base <I>BC</I> let <I>AB</I> be greater +than <I>AC.</I> +<p>Take <I>D</I> on <I>BA</I> such that <MATH><I>BD</I>=1/2 (<I>BA</I>+<I>AC</I>).</MATH> +<FIG> +<p>On <I>DA</I> between <I>D</I> and <I>A</I> take any point <I>E</I>, and draw <I>EF</I> +parallel to <I>BC.</I> Let <I>G</I> be any point on <I>EF</I>; draw <I>GH</I> parallel +to <I>AB</I> and join <I>GC.</I> +<pb n=367><head>THE <I>COLLECTION.</I> BOOK III</head> +<p>Now <MATH><I>EA</I>+<I>AC</I>><I>EF</I>+<I>FC</I> +><I>EG</I>+<I>GC</I> and ><I>GC</MATH>, a fortiori.</I> +<p>Produce <I>GC</I> to <I>K</I> so that <MATH><I>GK</I>=<I>EA</I>+<I>AC</I>,</MATH> and with <I>G</I> as +centre and <I>GK</I> as radius describe a circle. This circle will +meet <I>HC</I> and <I>HG</I>, because <MATH><I>GH</I>=<I>EB</I>><I>BD</I> or <I>DA</I>+<I>AC</I> and +><I>GK</MATH>, a fortiori.</I> +<p>Then <MATH><I>HG</I>+<I>GL</I>=<I>BE</I>+<I>EA</I>+<I>AC</I>=<I>BA</I>+<I>AC.</I></MATH> +<p>To obtain two straight lines <I>HG</I>′, <I>G</I>′<I>L</I> such that <MATH><I>HG</I>′+<I>G</I>′<I>L</I> +><I>BA</I>+<I>AC</I>,</MATH> we have only to choose <I>G</I>′ so that <I>HG</I>′, <I>G</I>′<I>L</I> +enclose the straight lines <I>HG, GL</I> completely. +<p>Next suppose that, given a triangle <I>ABC</I> in which <I>BC</I>><I>BA</I> +<FIG> +><I>AC</I>, we are required to draw from two points on <I>BC</I> to +an internal point two straight lines greater <I>respectively</I> than +<I>BA, AC.</I> +<p>With <I>B</I> as centre and <I>BA</I> as radius describe the arc <I>AEF.</I> +Take any point <I>E</I> on it, and any point <I>D</I> on <I>BE</I> produced +but within the triangle. Join <I>DC</I>, and produce it to <I>G</I> so +that <MATH><I>DG</I>=<I>AC.</I></MATH> Then with <I>D</I> as centre and <I>DG</I> as radius +describe a circle. This will meet both <I>BC</I> and <I>BD</I> because +<MATH><I>BA</I>><I>AC</I>,</MATH> and <I>a fortiori</I> <MATH><I>DB</I>><I>DG.</I></MATH> +<p>Then, if <I>L</I> be any point on <I>BH</I>, it is clear that <I>BD, DL</I> +are two straight lines satisfying the conditions. +<p>A point <I>L</I>′ on <I>BH</I> can be found such that <I>DL</I>′ is <I>equal</I> +to <I>AB</I> by marking off <I>DN</I> on <I>DB</I> equal to <I>AB</I> and drawing +with <I>D</I> as centre and <I>DN</I> as radius a circle meeting <I>BH</I> +in <I>L</I>′. Also, if <I>DH</I> be joined, <MATH><I>DH</I>=<I>AC.</I></MATH> +<p>Propositions follow (35-9) having a similar relation to the +Postulate in Archimedes, <I>On the Sphere and Cylinder</I>, I, +about conterminous broken lines one of which wholly encloses +<pb n=368><head>PAPPUS OF ALEXANDRIA</head> +the other, i.e. it is shown that broken lines, consisting of +several straight lines, can be drawn with two points on the +base of a triangle or parallelogram as extremities, and of +greater total length than the remaining two sides of the +triangle or three sides of the parallelogram. +<p>Props. 40-2 show that triangles or parallelograms can be +constructed with sides respectively greater than those of a given +triangle or parallelogram but having a less area. +<C>Section (4). <I>The inscribing of the five regular solids +in a sphere.</I></C> +<p>The fourth section of Book III (pp. 132-62) solves the +problems of inscribing each of the five regular solids in a +given sphere. After some preliminary lemmas (Props. 43-53), +Pappus attacks the substantive problems (Props. 54-8), using +the method of analysis followed by synthesis in the case of +each solid. +<p>(<I>a</I>) In order to inscribe a regular pyramid or tetrahedron in +the sphere, he finds two circular sections equal and parallel +to one another, each of which contains one of two opposite +edges as its diameter. If <I>d</I> be the diameter of the sphere, the +parallel circular sections have <I>d</I>′ as diameter, where <MATH><I>d</I><SUP>2</SUP>=3/2<I>d</I>′<SUP>2</SUP>.</MATH> +<p>(<I>b</I>) In the case of the cube Pappus again finds two parallel +circular sections with diameter <I>d</I>′ such that <MATH><I>d</I><SUP>2</SUP>=3/2<I>d</I>′<SUP>2</SUP>;</MATH> a square +inscribed in one of these circles is one face of the cube and +the square with sides parallel to those of the first square +inscribed in the second circle is the opposite face. +<p>(<I>c</I>) In the case of the octahedron the same two parallel circular +sections with diameter <I>d</I>′ such that <MATH><I>d</I><SUP>2</SUP>=3/2<I>d</I>′<SUP>2</SUP></MATH> are used; an +equilateral triangle inscribed in one circle is one face, and the +opposite face is an equilateral triangle inscribed in the other +circle but placed in exactly the opposite way. +<p>(<I>d</I>) In the case of the icosahedron Pappus finds four parallel +circular sections each passing through three of the vertices of +the icosahedron; two of these are small circles circumscribing +two opposite triangular faces respectively, and the other two +circles are between these two circles, parallel to them, and +equal to one another. The pairs of circles are determined in +<pb n=369><head>THE <I>COLLECTION.</I> BOOKS III, IV</head> +this way. If <I>d</I> be the diameter of the sphere, set out two +straight lines <I>x, y</I> such that <I>d, x, y</I> are in the ratio of the sides +of the regular pentagon, hexagon and decagon respectively +described in one and the same circle. The smaller pair of +circles have <I>r</I> as radius where <MATH><I>r</I><SUP>2</SUP>=1/3<I>y</I><SUP>2</SUP>,</MATH> and the larger pair +have <I>r</I>′ as radius where <MATH><I>r</I>′<SUP>2</SUP>=1/3<I>x</I><SUP>2</SUP>.</MATH> +<p>(<I>e</I>) In the case of the dodecahedron the <I>same</I> four parallel +circular sections are drawn as in the case of the icosahedron. +Inscribed pentagons set the opposite way are inscribed in the +two smaller circles; these pentagons form opposite faces. +Regular pentagons inscribed in the larger circles with vertices +at the proper points (and again set the opposite way) determine +ten more vertices of the inscribed dodecahedron. +<p>The constructions are quite different from those in Euclid +XIII. 13, 15, 14, 16, 17 respectively, where the problem is first +to construct the particular regular solid and then to ‘com- +prehend it in a sphere’, i.e. to determine the circumscribing +sphere in each case. I have set out Pappus's propositions in +detail elsewhere.<note><I>Vide</I> notes to Euclid's propositions in <I>The Thirteen Books of Euclid's +Elements</I>, pp. 473, 480, 477, 489-91, 501-3.</note> +<C>Book IV.</C> +<p>At the beginning of Book IV the title and preface are +missing, and the first section of the Book begins immediately +with an enunciation. The first section (pp. 176-208) contains +Propositions 1-12 which, with the exception of Props. 8-10, +seem to be isolated propositions given for their own sakes and +not connected by any general plan. +<C>Section (1). <I>Extension of the theorem of Pythagoras.</I></C> +<p>The first proposition is of great interest, being the generaliza- +tion of Eucl. I. 47, as Pappus himself calls it, which is by this +time pretty widely known to mathematicians. The enunciation +is as follows. +<p>‘If <I>ABC</I> be a triangle and on <I>AB, AC</I> any parallelograms +whatever be described, as <I>ABDE, ACFG</I>, and if <I>DE, FG</I> +produced meet in <I>H</I> and <I>HA</I> be joined, then the parallelo- +grams <I>ABDE, ACFG</I> are together equal to the parallelogram +<pb n=370><head>PAPPUS OF ALEXANDRIA</head> +contained by <I>BC, HA</I> in an angle which is equal to the sum of +the angles <I>ABC, DHA.</I>’ +<p>Produce <I>HA</I> to meet <I>BC</I> in <I>K</I>, draw <I>BL, CM</I> parallel to <I>KH</I> +meeting <I>DE</I> in <I>L</I> and <I>FG</I> in <I>M</I>, and join <I>LNM.</I> +<p>Then <I>BLHA</I> is a parallelogram, and <I>HA</I> is equal and +parallel to <I>BL.</I> +<FIG> +<p>Similarly <I>HA, CM</I> are equal and parallel; therefore <I>BL, CM</I> +are equal and parallel. +<p>Therefore <I>BLMC</I> is a parallelogram; and its angle <I>LBK</I> is +equal to the sum of the angles <I>ABC, DHA.</I> +<p>Now <MATH><I>▭ ABDE</I>=<I>▭ BLHA</I>,</MATH> in the same parallels, +<MATH>=<I>▭ BLNK</I>,</MATH> for the same reason. +<p>Similarly <MATH><I>▭ ACFG</I>=<I>▭ ACMH</I>=<I>▭ NKCM.</I></MATH> +<p>Therefore, by addition, <MATH><I>▭ ABDE</I>+<I>▭ ACFG</I>=<I>▭ BLMC.</I></MATH> +<p>It has been observed (by Professor Cook Wilson<note><I>Mathematical Gazette</I>, vii, p. 107 (May 1913).</note>) that the +parallelograms on <I>AB, AC</I> need not necessarily be erected +<I>outwards</I> from <I>AB, AC.</I> If one of them, e.g. that on <I>AC</I>, be +drawn inwards, as in the second figure above, and Pappus's +construction be made, we have a similar result with a negative +sign, namely, +<MATH><I>▭ BLMC</I>=<I>▭ BLNK -- ▭ CMNK</I> +=<I>▭ ABDE -- ▭ ACFG.</I></MATH> +<p>Again, if both <I>ABDE</I> and <I>ACFG</I> were drawn inwards, their +sum would be equal to <I>BLMC</I> drawn <I>outwards.</I> Generally, if +the areas of the parallelograms described outwards are regarded +as of opposite sign to those of parallelograms drawn inwards, +<pb n=371><head>THE <I>COLLECTION.</I> BOOK IV</head> +we may say that the algebraic sum of the three parallelograms +is equal to zero. +<p>Though Pappus only takes one case, as was the Greek habit, +I see no reason to doubt that he was aware of the results +in the other possible cases. +<p>Props. 2, 3 are noteworthy in that they use the method and +phraseology of Eucl. X, proving that a certain line in one +figure is the irrational called <I>minor</I> (see Eucl. X. 76), and +a certain line in another figure is ‘the excess by which the +<I>binomial</I> exceeds the <I>straight line which produces with a +rational area a medial whole’</I> (Eucl. X. 77). The propositions +4-7 and 11-12 are quite interesting as geometrical exercises, +but their bearing is not obvious: Props. 4 and 12 are remark- +able in that they are cases of analysis followed by synthesis +applied to the proof of <I>theorems.</I> Props. 8-10 belong to the +subject of <I>tangencies</I>, being the sort of propositions that would +come as particular cases in a book such as that of Apollonius +<I>On Contacts</I>; Prop. 8 shows that, if there are two equal +circles and a given point outside both, the diameter of the +circle passing through the point and touching both circles +is ‘given’; the proof is in many places obscure and assumes +lemmas of the same kind as those given later à propos of +Apollonius's treatise; Prop. 10 purports to show how, given +three unequal circles touching one another two and two, to +find the diameter of the circle including them and touching +all three. +<C>Section (2). <I>On circles inscribed in the</I> <G>a)/bhlos</G> +(<I>‘shoemaker's knife’</I>).</C> +<p>The next section (pp. 208-32), directed towards the demon- +stration of a theorem about the relative sizes of successive +circles inscribed in the <G>a)/rbhlos</G> (shoemaker's knife), is ex- +tremely interesting and clever, and I wish that I had space +to reproduce it completely. The <G>a)/rbhlos</G>, which we have +already met with in Archimedes's ‘Book of Lemmas’, is +formed thus. <I>BC</I> is the diameter of a semicircle <I>BGC</I> and +<I>BC</I> is divided into two parts (in general unequal) at <I>D</I>; +semicircles are described on <I>BD, DC</I> as diameters on the same +side of <I>BC</I> as <I>BGC</I> is; the figure included between the three +semicircles is the <G>a)/rbhlos</G>. +<pb n=372><head>PAPPUS OF ALEXANDRIA</head> +<p>There is, says Pappus, on record an ancient proposition to +the following effect. Let successive circles be inscribed in the +<G>a)/rbhlos</G> touching the semicircles and one another as shown +in the figure on p. 376, their centres being <I>A, P, O</I> .... Then, if +<I>p</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB> ... be the perpendiculars from the centres <I>A, P, O</I> ... +on <I>BC</I> and <I>d</I><SUB>1</SUB>, <I>d</I><SUB>2</SUB>, <I>d</I><SUB>3</SUB> ... the diameters of the corresponding +circles, +<MATH><I>p</I><SUB>1</SUB>=<I>d</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>=2<I>d</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB>=3<I>d</I><SUB>3</SUB>....</MATH> +<p>He begins by some lemmas, the course of which I shall +reproduce as shortly as I can. +<p>I. If (Fig. 1) two circles with centres <I>A, C</I> of which the +former is the greater touch externally at <I>B,</I> and another circle +with centre <I>G</I> touches the two circles at <I>K, L</I> respectively, +then <I>KL</I> produced cuts the circle <I>BL</I> again in <I>D</I> and meets +<I>AC</I> produced in a point <I>E</I> such that <MATH><I>AB</I>:<I>BC</I>=<I>AE</I>:<I>EC.</I></MATH> +This is easily proved, because the circular segments <I>DL, LK</I> +are similar, and <I>CD</I> is parallel to <I>AG.</I> Therefore +<MATH><I>AB</I>:<I>BC</I>=<I>AK</I>:<I>CD</I>=<I>AE</I>:<I>EC.</I></MATH> +<p>Also <MATH><I>KE.EL</I>=<I>EB</I><SUP>2</SUP></MATH>. +<p>For <MATH><I>AE</I>:<I>EC</I>=<I>AB</I>:<I>BC</I>=<I>AB</I>:<I>CF</I>=(<I>AE</I>-<I>AB</I>):(<I>EC</I>-<I>CF</I>) +=<I>BE</I>:<I>EF.</I></MATH> +<FIG> +<CAP>FIG 1.</CAP> +<p>But <MATH><I>AE</I>:<I>EC</I>=<I>KE</I>:<I>ED</I></MATH>; therefore <MATH><I>KE</I>:<I>ED</I>=<I>BE</I>:<I>EF.</I></MATH> +<p>Therefore <MATH><I>KE.EL</I>:<I>EL.ED</I>=<I>BE</I><SUP>2</SUP>:<I>BE.EF.</I></MATH> +<p>And <MATH><I>EL.ED</I>=<I>BE.EF</I></MATH>; therefore <MATH><I>KE.EL</I>=<I>EB</I><SUP>2</SUP></MATH>. +<pb n=373><head>THE <I>COLLECTION.</I> BOOK IV</head> +<p>II. Let (Fig. 2) <I>BC, BD,</I> being in one straight line, be the +diameters of two semicircles <I>BGC, BED,</I> and let any circle as +<I>FGH</I> touch both semicircles, <I>A</I> being the centre of the circle. +Let <I>M</I> be the foot of the perpendicular from <I>A</I> on <I>BC, r</I> the +radius of the circle <I>FGH.</I> There are two cases according +as <I>BD</I> lies along <I>BC</I> or <I>B</I> lies between <I>D</I> and <I>C;</I> i.e. in the +first case the two semicircles are the outer and one of the inner +semicircles of the <G>a)/rbhlos</G>, while in the second case they are +the two inner semicircles; in the latter case the circle <I>FGH</I> +may either include the two semicircles or be entirely external +to them. Now, says Pappus, it is to be proved that +in case (1) <MATH><I>BM</I>:<I>r</I>=(<I>BC</I>+<I>BD</I>):(<I>BC</I>-<I>BD</I>)</MATH>, +and in case (2) <MATH><I>BM</I>:<I>r</I>=(<I>BC</I>-<I>BD</I>):(<I>BC</I>+<I>BD</I>).</MATH> +<FIG> +<CAP>FIG 2.</CAP> +<p>We will confine ourselves to the first case, represented in +the figure (Fig. 2). +<p>Draw through <I>A</I> the diameter <I>HF</I> parallel to <I>BC.</I> Then, +since the circles <I>BGC, HGF</I> touch at <I>G,</I> and <I>BC, HF</I> are +parallel diameters, <I>GHB, GFC</I> are both straight lines. +<p>Let <I>E</I> be the point of contact of the circles <I>FGH</I> and <I>BED;</I> +then, similarly, <I>BEF, HED</I> are straight lines. +<p>Let <I>HK, FL</I> be drawn perpendicular to <I>BC.</I> +<p>By the similar triangles <I>BGC, BKH</I> we have +<MATH><I>BC</I>:<I>BG</I>=<I>BH</I>:<I>BK,</I></MATH> or <MATH><I>CB.BK</I>=<I>GB.BH</I></MATH>; +and by the similar triangles <I>BLF, BED</I> +<MATH><I>BF</I>:<I>BL</I>=<I>BD</I>:<I>BE,</I></MATH> or <MATH><I>DB.BL</I>=<I>FB.BE.</I></MATH> +<pb n=374><head>PAPPUS OF ALEXANDRIA</head> +<p>But <MATH><I>GB.BH</I>=<I>FB.BE</I></MATH>; +therefore <MATH><I>CB.BK</I>=<I>DB.BL</I></MATH>, +or <MATH><I>BC</I>:<I>BD</I>=<I>BL</I>:<I>BK.</I></MATH> +<p>Therefore <MATH>(<I>BC</I>+<I>BD</I>):(<I>BC</I>-<I>BD</I>)=(<I>BL</I>+<I>BK</I>):(<I>BL</I>-<I>BK</I>) +=2<I>BM</I>:<I>KL.</I></MATH> +<p>And <MATH><I>KL</I>=<I>HF</I>=2<I>r</I></MATH>; +therefore <MATH><I>BM</I>:<I>r</I>=(<I>BC</I>+<I>BD</I>):(<I>BC</I>-<I>BD</I>).</MATH> (<I>a</I>) +<p>It is next proved that <MATH><I>BK.LC</I>=<I>AM</I><SUP>2</SUP></MATH>. +<p>For, by similar triangles <I>BKH, FLC,</I> +<MATH><I>BK</I>:<I>KH</I>=<I>FL</I>:<I>LC,</I></MATH> or <MATH><I>BK.LC</I>=<I>KH.FL</I> +=<I>AM</I><SUP>2</SUP></MATH>. (<I>b</I>) +<p>Lastly, since <MATH><I>BC</I>:<I>BD</I>=<I>BL</I>:<I>BK,</I></MATH> from above, +<MATH><I>BC</I>:<I>CD</I>=<I>BL</I>:<I>KL,</I></MATH> or <MATH><I>BL.CD</I>=<I>BC.KL</I> +=<I>BC.</I>2<I>r.</I></MATH> (<I>c</I>) +<p>Also <MATH><I>BD</I>:<I>CD</I>=<I>BK</I>:<I>KL,</I></MATH> or <MATH><I>BK.CD</I>=<I>BD.KL</I> +=<I>BD.</I>2<I>r.</I></MATH> (<I>d</I>) +<p>III. We now (Fig. 3) take any two circles touching the +semicircles <I>BGC, BED</I> and one another. Let their centres be +<I>A</I> and <I>P, H</I> their point of contact, <I>d, d</I>′ their diameters respec- +tively. Then, if <I>AM, PN</I> are drawn perpendicular to <I>BC,</I> +Pappus proves that +<MATH>(<I>AM</I>+<I>d</I>):<I>d</I>=<I>PN</I>:<I>d</I>′</MATH>. +<p>Draw <I>BF</I> perpendicular to <I>BC</I> and therefore touching the +semicircles <I>BGC, BED</I> at <I>B.</I> Join <I>AP,</I> and produce it to +meet <I>BF</I> in <I>F.</I> +<p>Now, by II. (<I>a</I>) above, +<MATH>(<I>BC</I>+<I>BD</I>):(<I>BC</I>-<I>BD</I>)=<I>BM</I>:<I>AH,</I></MATH> +and for the same reason =<I>BN</I>:<I>PH;</I> +it follows that <MATH><I>AH</I>:<I>PH</I>=<I>BM</I>:<I>BN</I> +=<I>FA</I>:<I>FP.</I></MATH> +<pb n=375><head>THE <I>COLLECTION,</I> BOOK IV</head> +<p>Therefore (Lemma I), if the two circles touch the semi- +circle <I>BED</I> in <I>R, E</I> respectively, <I>FRE</I> is a straight line and +<MATH><I>EF.FR</I>=<I>FH</I><SUP>2</SUP></MATH>. +<p>But <MATH><I>EF.FR</I>=<I>FB</I><SUP>2</SUP></MATH>; therefore <MATH><I>FH</I>=<I>FB.</I></MATH> +<p>If now <I>BH</I> meets <I>PN</I> in <I>O</I> and <I>MA</I> produced in <I>S,</I> we have, +by similar triangles, <MATH><I>FH</I>:<I>FB</I>=<I>PH</I>:<I>PO</I>=<I>AH</I>:<I>AS,</I></MATH> whence +<MATH><I>PH</I>=<I>PO</I></MATH> and <MATH><I>SA</I>=<I>AH,</I></MATH> so that <I>O, S</I> are the intersections +of <I>PN, AM</I> with the respective circles. +<FIG> +<CAP>FIG 3.</CAP> +<p>Join <I>BP,</I> and produce it to meet <I>MA</I> in <I>K.</I> +<p>Now <MATH><I>BM</I>:<I>BN</I>=<I>FA</I>:<I>FP</I> +=<I>AH</I>:<I>PH,</I> from above, +=<I>AS</I>:<I>PO.</I></MATH> +<p>And <MATH><I>BM</I>:<I>BN</I>=<I>BK</I>:<I>BP</I> +=<I>KS</I>:<I>PO.</I></MATH> +<p>Therefore <MATH><I>KS</I>=<I>AS,</I></MATH> and <MATH><I>KA</I>=<I>d,</I></MATH> the diameter of the +circle <I>EHG.</I> +<p>Lastly, <MATH><I>MK</I>:<I>KS</I>=<I>PN</I>:<I>PO,</I></MATH> +that is, <MATH>(<I>AM</I>+<I>d</I>):1/2<I>d</I>=<I>PN</I>:1/2<I>d</I>′</MATH>, +or <MATH>(<I>AM</I>+<I>d</I>):<I>d</I>=<I>PN</I>:<I>d</I>′</MATH>. +<pb n=376><head>PAPPUS OF ALEXANDRIA</head> +<p>IV. We now come to the substantive theorem. +<p>Let <I>FGH</I> be the circle touching all three semicircles (Fig. 4). +We have then, as in Lemma II, +<MATH><I>BC.BK</I>=<I>BD.BL,</I></MATH> +and for the same reason (regarding <I>FGH</I> as touching the +semicircles <I>BGC, DUC</I>) +<MATH><I>BC.CL</I>=<I>CD.CK.</I></MATH> +<p>From the first relation we have +<MATH><I>BC</I>:<I>BD</I>=<I>BL</I>:<I>BK,</I></MATH> +<FIG> +<CAP>FIG 4.</CAP> +whence <MATH><I>DC</I>:<I>BD</I>=<I>KL</I>:<I>BK,</I></MATH> and inversely <MATH><I>BD</I>:<I>DC</I>=<I>BK</I>:<I>KL,</I></MATH> +while, from the second relation, <MATH><I>BC</I>:<I>CD</I>=<I>CK</I>:<I>CL,</I></MATH> +whence <MATH><I>BD</I>:<I>DC</I>=<I>KL</I>:<I>CL.</I></MATH> +<p>Consequently <MATH><I>BK</I>:<I>KL</I>=<I>KL</I>:<I>CL,</I></MATH> +or <MATH><I>BK.LC</I>=<I>KL</I><SUP>2</SUP></MATH>. +<p>But we saw in Lemma II (<I>b</I>) that <MATH><I>BK.LC</I>=<I>AM</I><SUP>2</SUP></MATH>. +<p>Therefore <MATH><I>KL</I>=<I>AM,</I></MATH> or <MATH><I>p</I><SUB>1</SUB>=<I>d</I><SUB>1</SUB></MATH>. +<p>For the second circle Lemma III gives us +<MATH>(<I>p</I><SUB>1</SUB>+<I>d</I><SUB>1</SUB>):<I>d</I><SUB>1</SUB>=<I>p</I><SUB>2</SUB>:<I>d</I><SUB>2</SUB></MATH>, +whence, since <MATH><I>p</I><SUB>1</SUB>=<I>d</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>=2<I>d</I><SUB>2</SUB></MATH>. +<p>For the third circle +<MATH>(<I>p</I><SUB>2</SUB>+<I>d</I><SUB>2</SUB>):<I>d</I><SUB>2</SUB>=<I>p</I><SUB>3</SUB>:<I>d</I><SUB>3</SUB></MATH>, +whence <MATH><I>p</I><SUB>3</SUB>=3<I>d</I><SUB>3</SUB></MATH>. +<p>And so on <I>ad infinitum.</I> +<pb n=377><head>THE <I>COLLECTION.</I> BOOK IV</head> +<p>The same proposition holds when the successive circles, +instead of being placed between the large and one of the small +semicircles, come down between the two small semicircles. +<p>Pappus next deals with special cases (1) where the two +smaller semicircles become straight lines perpendicular to the +diameter of the other semicircle at its extremities, (2) where +we replace one of the smaller semicircles by a straight line +through <I>D</I> at right angles to <I>BC,</I> and lastly (3) where instead +of the semicircle <I>DUC</I> we simply have the straight line <I>DC</I> +and make the first circle touch it and the two other semi- +circles. +<p>Pappus's propositions of course include as particular cases +the partial propositions of the same kind included in the ‘Book +of Lemmas’ attributed to Archimedes (Props. 5, 6); cf. p. 102. +<C>Sections (3) and (4). <I>Methods of squaring the circle, and of +trisecting (or dividing in any ratio) any given angle.</I></C> +<p>The last sections of Book IV (pp. 234-302) are mainly +devoted to the solutions of the problems (1) of squaring or +rectifying the circle and (2) of trisecting any given angle +or dividing it into two parts in any ratio. To this end Pappus +gives a short account of certain curves which were used for +the purpose. +<C>(<G>a</G>) <I>The Archimedean spiral.</I></C> +<p>He begins with the spiral of Archimedes, proving some +of the fundamental properties. His method of finding the +area included (1) between the first turn and the initial line, +(2) between any radius vector on the first turn and the curve, +is worth giving because it differs from the method of Archi- +medes. It is the area of the whole first turn which Pappus +works out in detail. We will take the area up to the radius +vector <I>OB,</I> say. +<p>With centre <I>O</I> and radius <I>OB</I> draw the circle <I>A</I>′<I>BCD.</I> +<p>Let <I>BC</I> be a certain fraction, say 1/<I>n</I>th, of the arc <I>BCDA</I>′, +and <I>CD</I> the same fraction, <I>OC, OD</I> meeting the spiral in <I>F, E</I> +respectively. Let <I>KS, SV</I> be the same fraction of a straight +line <I>KR,</I> the side of a square <I>KNLR.</I> Draw <I>ST, VW</I> parallel +to <I>KN</I> meeting the diagonal <I>KL</I> of the square in <I>U, Q</I> respec- +tively, and draw <I>MU, PQ</I> parallel to <I>KR.</I> +<pb n=378><head>PAPPUS OF ALEXANDRIA</head> +<p>With <I>O</I> as centre and <I>OE, OF</I> as radii draw arcs of circles +meeting <I>OF, OB</I> in <I>H, G</I> respectively. +<p>For brevity we will now denote a cylinder in which <I>r</I> is the +radius of the base and <I>h</I> the height by (cyl. <I>r, h</I>) and the cone +with the same base and height by (cone <I>r, h</I>). +<FIG> +<p>By the property of the spiral, +<MATH><I>OB</I>:<I>BG</I>=(arc <I>A</I>′<I>DCB</I>):(arc <I>CB</I>) +=<I>RK</I>:<I>KS</I> +=<I>NK</I>:<I>KM</I></MATH>, +whence <MATH><I>OB</I>:<I>OG</I>=<I>NK</I>:<I>NM.</I></MATH> +<p>Now +<MATH>(sector <I>OBC</I>):(sector <I>OGF</I>)=<I>OB</I><SUP>2</SUP>:<I>OG</I><SUP>2</SUP>=<I>NK</I><SUP>2</SUP>:<I>MN</I><SUP>2</SUP> +=(cyl. <I>KN, NT</I>):(cyl. <I>MN, NT</I>)</MATH>. +<p>Similarly +<MATH>(sector <I>OCD</I>):(sector <I>OEH</I>)=(cyl. <I>ST, TW</I>):(cyl. <I>PT, TW</I>)</MATH>, +and so on. +<p>The sectors <I>OBC, OCD</I> ... form the sector <I>OA</I>′<I>DB,</I> and the +sectors <I>OFG, OEH</I> ... form a figure inscribed to the spiral. +In like manner the cylinders (<I>KN, TN</I>), (<I>ST, TW</I>) ... form the +cylinder (<I>KN, NL</I>), while the cylinders (<I>MN, NT</I>), (<I>PT, TW</I>) ... +form a figure inscribed to the cone (<I>KN, NL</I>). +<p>Consequently +<MATH>(sector <I>OA</I>′<I>DB</I>):(fig. inscr. in spiral) +=(cyl. <I>KN, NL</I>):(fig. inscr. in cone <I>KN, NL</I>)</MATH>. +<pb n=379><head>THE <I>COLLECTION.</I> BOOK IV</head> +<p>We have a similar proportion connecting a figure circum- +scribed to the spiral and a figure circumscribed to the cone. +<p>By increasing <I>n</I> the inscribed and circumscribed figures can +be compressed together, and by the usual method of exhaustion +we have ultimately +<MATH>(sector <I>OA</I>′<I>DB</I>):(area of spiral)=(cyl. <I>KN, NL</I>):(cone <I>KN, NL</I>) +=3:1</MATH>, +or <MATH>(area of spiral cut off by <I>OB</I>)=1/3(sector <I>OA</I>′<I>DB</I>)</MATH>. +<p>The ratio of the sector <I>OA</I>′<I>DB</I> to the complete circle is that +of the angle which the radius vector describes in passing from +the position <I>OA</I> to the position <I>OB</I> to four right angles, that +is, by the property of the spiral, <I>r</I>:<I>a,</I> where <MATH><I>r</I>=<I>OB,</I> <I>a</I>=<I>OA.</I></MATH> +<p>Therefore <MATH>(area of spiral cut off by <I>OB</I>)=1/3<I>r/a</I>.<G>p</G><I>r</I><SUP>2</SUP></MATH>. +<p>Similarly the area of the spiral cut off by any other radius +vector <MATH><I>r</I>′=1/3<I>r</I>′/<I>a</I>.<G>p</G><I>r</I>′<SUP>2</SUP></MATH>. +<p>Therefore (as Pappus proves in his next proposition) the +first area is to the second as <I>r</I><SUP>3</SUP> to <I>r</I>′<SUP>3</SUP>. +<p>Considering the areas cut off by the radii vectores at the +points where the revolving line has passed through angles +of 1/2<G>p</G>, <G>p</G>, 3/2<G>p</G> and 2<G>p</G> respectively, we see that the areas are in +the ratio of (1/4)<SUP>3</SUP>, (1/2)<SUP>3</SUP>, (3/4)<SUP>3</SUP>, 1 or 1, 8, 27, 64, so that the areas of +the spiral included in the four quadrants are in the ratio +of 1, 7, 19, 37 (Prop. 22). +<C>(<G>b</G>) <I>The conchoid of Nicomedes.</I></C> +<p>The conchoid of Nicomedes is next described (chaps. 26-7), +and it is shown (chaps. 28, 29) how it can be used to find two +geometric means between two straight lines, and consequently +to find a cube having a given ratio to a given cube (see vol. i, +pp. 260-2 and pp. 238-40, where I have also mentioned +Pappus's remark that the conchoid which he describes is the +<I>first</I> conchoid, while there also exist a <I>second,</I> a <I>third</I> and a +<I>fourth</I> which are of use for other theorems). +<C>(<G>g</G>) <I>The quadratrix.</I></C> +<p>The <I>quadratrix</I> is taken next (chaps. 30-2), with Sporus's +criticism questioning the construction as involving a <I>petitio</I> +<pb n=380><head>PAPPUS OF ALEXANDRIA</head> +<I>principii.</I> Its use for squaring the circle is attributed to +Dinostratus and Nicomedes. The whole substance of this +subsection is given above (vol. i, pp. 226-30). +<C><I>Two constructions for the quadratrix by means of +‘surface-loci’.</I></C> +<p>In the next chapters (chaps. 33, 34, Props. 28, 29) Pappus +gives two alternative ways of producing the <I>quadratrix</I> ‘by +means of surface-loci’, for which he claims the merit that +they are geometrical rather than ‘too mechanical’ as the +traditional method (of Hippias) was. +<p>(1) The first method uses a cylindrical helix thus. +<p>Let <I>ABC</I> be a quadrant of a circle with centre <I>B,</I> and +let <I>BD</I> be any radius. Suppose +that <I>EF,</I> drawn from a point <I>E</I> +on the radius <I>BD</I> perpendicular +to <I>BC,</I> is (for all such radii) in +a given ratio to the arc <I>DC.</I> +<FIG> +<p>‘I say’, says Pappus, ‘that the +locus of <I>E</I> is a certain curve.’ +<p>Suppose a right cylinder +erected from the quadrant and +a cylindrical helix <I>CGH</I> drawn +upon its surface. Let <I>DH</I> be +the generator of this cylinder through <I>D,</I> meeting the helix +in <I>H.</I> Draw <I>BL, EI</I> at right angles to the plane of the +quadrant, and draw <I>HIL</I> parallel to <I>BD.</I> +<p>Now, by the property of the helix, <MATH><I>EI</I>(=<I>DH</I>)</MATH> is to the +arc <I>CD</I> in a given ratio. Also <MATH><I>EF</I>:(arc <I>CD</I>)=a given ratio.</MATH> +<p>Therefore the ratio <I>EF</I>:<I>EI</I> is given. <I>A</I>nd since <I>EF, EI</I> are +given in position, <I>FI</I> is given in position. But <I>FI</I> is perpen- +dicular to <I>BC.</I> Therefore <I>FI</I> is in a plane given in position, +and so therefore is <I>I.</I> +<p>But <I>I</I> is also on a certain surface described by the line <I>LH,</I> +which moves always parallel to the plane <I>ABC,</I> with one +extremity <I>L</I> on <I>BL</I> and the other extremity <I>H</I> on the helix. +Therefore <I>I</I> lies on the intersection of this surface with the +plane through <I>FI.</I> +<pb n=381><head>THE <I>COLLECTION.</I> BOOK IV</head> +<p>Hence <I>I</I> lies on a certain curve. Therefore <I>E,</I> its projection +on the plane <I>ABC,</I> also lies on a curve. +<p>In the particular case where the given ratio of <I>EF</I> to the +arc <I>CD</I> is equal to the ratio of <I>BA</I> to the arc <I>CA,</I> the locus of +<I>E</I> is a <I>quadratrix.</I> +<p>[The surface described by the straight line <I>LH</I> is a <I>plectoid.</I> +The shape of it is perhaps best realized as a <I>continuous</I> spiral +staircase, i.e. a spiral staircase with infinitely small steps. +The <I>quadratrix</I> is thus produced as the orthogonal projection +of the curve in which the plectoid is intersected by a plane +through <I>BC</I> inclined at a given angle to the plane <I>ABC.</I> It is +not difficult to verify the result analytically.] +<p>(2) The second method uses a right cylinder the base of which +is an Archimedean spiral. +<p>Let <I>ABC</I> be a quadrant of a circle, as before, and <I>EF,</I> per- +pendicular at <I>F</I> to <I>BC,</I> a straight +line of such length that <I>EF</I> is +to the arc <I>DC</I> as <I>AB</I> is to the +arc <I>ADC.</I> +<FIG> +<p>Let a point on <I>AB</I> move uni- +formly from <I>A</I> to <I>B</I> while, in the +same time, <I>AB</I> itself revolves +uniformly about <I>B</I> from the position <I>BA</I> to the position <I>BC.</I> +The point thus describes the spiral <I>AGB.</I> If the spiral cuts +<I>BD</I> in <I>G,</I> +<MATH><I>BA</I>:<I>BG</I>=(arc <I>ADC</I>):(arc <I>DC</I>)</MATH>, +or <MATH><I>BG</I>:(arc <I>DC</I>)=<I>BA</I>:(arc <I>ADC</I>)</MATH>. +<p>Therefore <MATH><I>BG</I>=<I>EF.</I></MATH> +<p>Draw <I>GK</I> at right angles to the plane <I>ABC</I> and equal to <I>BG.</I> +Then <I>GK,</I> and therefore <I>K,</I> lies on a right cylinder with the +spiral as base. +<p>But <I>BK</I> also lies on a conical surface with vertex <I>B</I> such that +its generators all make an angle of 1/4<G>p</G> with the plane <I>ABC.</I> +<p>Consequently <I>K</I> lies on the intersection of two surfaces, +and therefore on a curve. +<p>Through <I>K</I> draw <I>LKI</I> parallel to <I>BD,</I> and let <I>BL, EI</I> be at +right angles to the plane <I>ABC.</I> +<p>Then <I>LKI,</I> moving always parallel to the plane <I>ABC,</I> with +one extremity on <I>BL</I> and passing through <I>K</I> on a certain +<pb n=382><head>PAPPUS OF ALEXANDRIA</head> +curve, describes a certain plectoid, which therefore contains the +point <I>I.</I> +<p>Also <MATH><I>IE</I>=<I>EF</I></MATH>, <I>IF</I> is perpendicular to <I>BC,</I> and hence <I>IF,</I> and +therefore <I>I,</I> lies on a fixed plane through <I>BC</I> inclined to <I>ABC</I> +at an angle of 1/4<G>p</G>. +<p>Therefore <I>I,</I> lying on the intersection of the plectoid and the +said plane, lies on a certain curve. So therefore does the +projection of <I>I</I> on <I>ABC,</I> i.e. the point <I>E.</I> +<p>The locus of <I>E</I> is clearly the <I>quadratrix.</I> +<p>[This result can also be verified analytically.] +<C>(<G>d</G>) <I>Digression: a spiral on a sphere.</I></C> +<p>Prop. 30 (chap. 35) is a digression on the subject of a certain +spiral described on a sphere, suggested by the discussion of +a spiral in a plane. +<p>Take a hemisphere bounded by the great circle <I>KLM,</I> +with <I>H</I> as pole. Suppose that the quadrant of a great circle +<I>HNK</I> revolves uniformly about the radius <I>HO</I> so that <I>K</I> +describes the circle <I>KLM</I> and returns to its original position +at <I>K,</I> and suppose that a point moves uniformly at the same +time from <I>H</I> to <I>K</I> at such speed that the point arrives at <I>K</I> +at the same time that <I>HK</I> resumes its original position. The +point will thus describe a spiral on the surface of the sphere +between the points <I>H</I> and <I>K</I> as shown in the figure. +<FIG> +<p>Pappus then sets himself to prove that the portion of the +surface of the sphere cut off towards the pole between the +spiral and the arc <I>HNK</I> is to the surface of the hemisphere in +<pb n=383><head>THE <I>COLLECTION.</I> BOOK IV</head> +a certain ratio shown in the second figure where <I>ABC</I> is +a quadrant of a circle equal to a great circle in the sphere, +namely the ratio of the segment <I>ABC</I> to the sector <I>DABC.</I> +<FIG> +<p>Draw the tangent <I>CF</I> to the quadrant at <I>C.</I> With <I>C</I> as +centre and radius <I>CA</I> draw the circle <I>AEF</I> meeting <I>CF</I> in <I>F.</I> +<p>Then the sector <I>CAF</I> is equal to the sector <I>ADC</I> (since +<MATH><I>CA</I><SUP>2</SUP>=2<I>AD</I><SUP>2</SUP></MATH>, while <MATH>∠<I>ACF</I>=1/2∠<I>ADC</I>)</MATH>. +<p>It is required, therefore, to prove that, if <I>S</I> be the area cut +off by the spiral as above described, +<MATH><I>S</I>:(surface of hemisphere)=(segmt. <I>ABC</I>):(sector <I>CAF</I>)</MATH>. +<p>Let <I>KL</I> be a (small) fraction, say 1/<I>n</I>th, of the circum- +ference of the circle <I>KLM,</I> and let <I>HPL</I> be the quadrant of the +great circle through <I>H, L</I> meeting the spiral in <I>P.</I> Then, by +the property of the spiral, +<MATH>(arc <I>HP</I>):(arc <I>HL</I>)=(arc <I>KL</I>):(circumf. of <I>KLM</I>) +=1:<I>n.</I></MATH> +<p>Let the small circle <I>NPQ</I> passing through <I>P</I> be described +about the pole <I>H.</I> +<p>Next let <I>FE</I> be the same fraction, 1/<I>n</I>th, of the arc <I>FA</I> +that <I>KL</I> is of the circumference of the circle <I>KLM,</I> and join <I>EC</I> +meeting the arc <I>ABC</I> in <I>B.</I> With <I>C</I> as centre and <I>CB</I> as +radius describe the arc <I>BG</I> meeting <I>CF</I> in <I>G.</I> +<p>Then the arc <I>CB</I> is the same fraction, 1/<I>n</I>th, of the arc +<I>CBA</I> that the arc <I>FE</I> is of <I>FA</I> (for it is easily seen that +<MATH>∠<I>FCE</I>=1/2∠<I>BDC,</I></MATH> while <MATH>∠<I>FCA</I>=1/2∠<I>CDA</I>)</MATH>. Therefore, since +<MATH>(arc <I>CBA</I>)=(arc <I>HPL</I>), (arc <I>CB</I>)=(arc <I>HP</I>)</MATH>, and chord <MATH><I>CB</I> +=chord <I>HP.</I></MATH> +<pb n=384><head>PAPPUS OF ALEXANDRIA</head> +<p>Now <MATH>(sector <I>HPN</I> on sphere):(sector <I>HKL</I> on sphere) +=(chord <I>HP</I>)<SUP>2</SUP>:(chord <I>HL</I>)<SUP>2</SUP></MATH> +(a consequence of Archimedes, <I>On Sphere and Cylinder,</I> I. 42). +<p>And <MATH><I>HP</I><SUP>2</SUP>:<I>HL</I><SUP>2</SUP>=<I>CB</I><SUP>2</SUP>:<I>CA</I><SUP>2</SUP> +=<I>CB</I><SUP>2</SUP>:<I>CE</I><SUP>2</SUP></MATH>. +<p>Therefore +<MATH>(sector <I>HPN</I>):(sector <I>HKL</I>)=(sector <I>CBG</I>):(sector <I>CEF</I>)</MATH>. +<p>Similarly, if the arc <I>LL</I>′ be taken equal to the arc <I>KL</I> and +the great circle through <I>H, L</I>′ cuts the spiral in <I>P</I>′, and a small +circle described about <I>H</I> and through <I>P</I>′ meets the arc <I>HPL</I> +in <I>p;</I> and if likewise the arc <I>BB</I>′ is made equal to the arc <I>BC,</I> +and <I>CB</I>′ is produced to meet <I>AF</I> in <I>E</I>′, while again a circular +arc with <I>C</I> as centre and <I>CB</I>′ as radius meets <I>CE</I> in <I>b,</I> +<MATH>(sector <I>HP</I>′<I>p</I> on sphere):(sector <I>HLL</I>′ on sphere) +=(sector <I>CB</I>′<I>b</I>):(sector <I>CE</I>′<I>E</I>)</MATH>. +<p>And so on. +<p>Ultimately then we shall get a figure consisting of sectors +on the sphere circumscribed about the area <I>S</I> of the spiral and +a figure consisting of sectors of circles circumscribed about the +segment <I>CBA;</I> and in like manner we shall have inscribed +figures in each case similarly made up. +<p>The method of exhaustion will then give +<MATH><I>S</I>:(surface of hemisphere)=(segmt. <I>ABC</I>):(sector <I>CAF</I>) +=(segmt. <I>ABC</I>):(sector <I>DAC</I>)</MATH>. +<p>[We may, as an illustration, give the analytical equivalent +of this proposition. If <G>r, w</G> be the spherical coordinates of <I>P</I> +with reference to <I>H</I> as pole and the arc <I>HNK</I> as polar axis, +the equation of Pappus's curve is obviously <MATH><G>w</G>=4<G>r</G>.</MATH> +<p>If now the radius of the sphere is taken as unity, we have as +the element of area +<MATH><I>dA</I>=<I>d</I><G>w</G>(1-cos<G>r</G>)=4<I>d</I><G>r</G>(1-cos<G>r</G>)</MATH>. +<p>Therefore <MATH><I>A</I>=∫<SUP>1/2<G>p</G></SUP><SUB>0</SUB>4<I>d</I><G>r</G>(1-cos<G>r</G>)=2<G>p</G>-4</MATH>. +<pb n=385><head>THE <I>COLLECTION.</I> BOOK IV</head> +<p>Therefore +<MATH><I>A</I>/(surface of hemisphere)=(2<G>p</G>-4)/(2<G>p</G>)=(1/4<G>p</G>-1/2)/(1/4<G>p</G>) +=(segment <I>ABC</I>)/(sector <I>DABC</I>).]</MATH> +<p>The second part of the last section of Book IV (chaps. 36-41, +pp. 270-302) is mainly concerned with the problem of tri- +secting any given angle or dividing it into parts in any given +ratio. Pappus begins with another account of the distinction +between <I>plane, solid</I> and <I>linear</I> problems (cf. Book III, chaps. +20-2) according as they require for their solution (1) the +straight line and circle only, (2) conics or their equivalent, +(3) higher curves still, ‘which have a more complicated and +forced (or unnatural) origin, being produced from more +irregular surfaces and involved motions. Such are the curves +which are discovered in the so-called <I>loci on surfaces,</I> as +well as others more complicated still and many in number +discovered by Demetrius of Alexandria in his <I>Linear con- +siderations</I> and by Philon of Tyana by means of the inter- +lacing of plectoids and other surfaces of all sorts, all of which +curves possess many remarkable properties peculiar to them. +Some of these curves have been thought by the more recent +writers to be worthy of considerable discussion; one of them is +that which also received from Menelaus the name of the +<I>paradoxical</I> curve. Others of the same class are spirals, +quadratrices, cochloids and cissoids.’ He adds the often-quoted +reflection on the error committed by geometers when they +solve a problem by means of an ‘inappropriate class’ (of +curve or its equivalent), illustrating this by the use in +Apollonius, Book V, of a rectangular hyperbola for finding the +feet of normals to a <I>parabola</I> passing through one point +(where a circle would serve the purpose), and by the assump- +tion by Archimedes of a <I>solid</I> <G>neu=sis</G> in his book <I>On Spirals</I> +(see above, pp. 65-8). +<C><I>Trisection (or division in any ratio) of any angle.</I></C> +<p>The method of trisecting any angle based on a certain <G>neu=sis</G> +is next described, with the solution of the <G>neu=sis</G> itself by +<pb n=386><head>PAPPUS OF ALEXANDRIA</head> +means of a hyperbola which has to be constructed from certain +data, namely the asymptotes and a certain point through +which the curve must pass (this easy construction is given in +Prop. 33, chap. 41-2). Then the problem is directly solved +(chaps. 43, 44) by means of a hyperbola in two ways prac- +tically equivalent, the hyperbola being determined in the one +case by the ordinary Apollonian property, but in the other by +means of the <I>focus-directrix</I> property. Solutions follow of +the problem of dividing any angle in a given ratio by means +(1) of the <I>quadratrix,</I> (2) of the spiral of Archimedes (chaps. +45, 46). All these solutions have been sufficiently described +above (vol. i, pp. 235-7, 241-3, 225-7). +<p>Some problems follow (chaps. 47-51) depending on these +results, namely those of constructing an isosceles triangle in +which either of the base angles has a given ratio to the vertical +angle (Prop. 37), inscribing in a circle a regular polygon of +any number of sides (Prop. 38), drawing a circle the circum- +ference of which shall be equal to a given straight line (Prop. +39), constructing on a given straight line <I>AB</I> a segment of +a circle such that the arc of the segment may have a given +ratio to the base (Prop. 40), and constructing an angle incom- +mensurable with a given angle (Prop. 41). +<C>Section (5). <I>Solution of the</I> <G>neu=sis</G> <I>of Archimedes, ‘On Spirals’, +Prop. 8, by means of conics.</I></C> +<p>Book IV concludes with the solution of the <G>neu=sis</G> which, +according to Pappus, Archimedes unnecessarily assumed in +<I>On Spirals,</I> Prop. 8. Archimedes's assumption is this. Given +a circle, a chord (<I>BC</I>) in it less than the diameter, and a point +<I>A</I> on the circle the perpendicular from which to <I>BC</I> cuts <I>BC</I> +in a point <I>D</I> such that <I>BD</I>><I>DC</I> and meets the circle again +in <I>E,</I> it is possible to draw through <I>A</I> a straight line <I>ARP</I> +cutting <I>BC</I> in <I>R</I> and the circle in <I>P</I> in such a way that <I>RP</I> +shall be equal to <I>DE</I> (or, in the phraseology of <G>neu/seis</G>, to +place between the straight line <I>BC</I> and the circumference +of the circle a straight line equal to <I>DE</I> and <I>verging</I> +towards <I>A</I>). +<p>Pappus makes the problem rather more general by not +requiring <I>PR</I> to be equal to <I>DE,</I> but making it of any given +<pb n=387><head>THE <I>COLLECTION.</I> BOOK IV</head> +length (consistent with a real solution). The problem is best +exhibited by means of analytical geometry. +<p>If <MATH><I>BD</I>=<I>a,</I> <I>DC</I>=<I>b,</I> <I>AD</I>=<I>c</I></MATH> (so that <MATH><I>DE</I>=<I>ab/c</I></MATH>), we have +<FIG> +to find the point <I>R</I> on <I>BC</I> such that <I>AR</I> produced solves the +problem by making <I>PR</I> equal to <I>k,</I> say. +<p>Let <MATH><I>DR</I>=<I>x.</I></MATH> Then, since <MATH><I>BR.RC</I>=<I>PR.RA,</I></MATH> we have +<MATH>(<I>a</I>-<I>x</I>)(<I>b</I>+<I>x</I>)=<I>k</I>√(<I>c</I><SUP>2</SUP>+<I>x</I><SUP>2</SUP>).</MATH> +<p>An obvious expedient is to put <I>y</I> for √(<I>c</I><SUP>2</SUP>+<I>x</I><SUP>2</SUP>, when +we have +<MATH>(<I>a</I>-<I>x</I>)(<I>b</I>+<I>x</I>)=<I>ky,</I></MATH> (1) +and <MATH><I>y</I><SUP>2</SUP>=<I>c</I><SUP>2</SUP>+<I>x</I><SUP>2</SUP></MATH>. (2) +<p>These equations represent a parabola and a hyperbola +respectively, and Pappus does in fact solve the problem by +means of the intersection of a parabola and a hyperbola; one +of his preliminary lemmas is, however, again a little more +general. In the above figure <I>y</I> is represented by <I>RQ.</I> +<p>The first lemma of Pappus (Prop. 42, p. 298) states that, if +from a given point <I>A</I> any straight line be drawn meeting +a straight line <I>BC</I> given in position in <I>R,</I> and if <I>RQ</I> be drawn +at right angles to <I>BC</I> and of length bearing a given ratio +to <I>AR,</I> the locus of <I>Q</I> is a <I>hyperbola.</I> +<p>For draw <I>AD</I> perpendicular to <I>BC</I> and produce it to <I>A</I>′ +so that +<MATH><I>QR</I>:<I>RA</I>=<I>A</I>′<I>D</I>:<I>DA</I>=the given ratio</MATH>. +<pb n=388><head>PAPPUS OF ALEXANDRIA</head> +<p>Measure <I>DA</I>″ along <I>DA</I> equal to <I>DA</I>′. +<p>Then, if <I>QN</I> be perpendicular to <I>AD,</I> +<MATH>(<I>AR</I><SUP>2</SUP>-<I>AD</I><SUP>2</SUP>):(<I>QR</I><SUP>2</SUP>-<I>A</I>′<I>D</I><SUP>2</SUP>)=(const.)</MATH>, +that is, <MATH><I>QN</I><SUP>2</SUP>:<I>A</I>′<I>N.A</I>″<I>N</I>=(const.)</MATH>, +and the locus of <I>Q</I> is a hyperbola. +<p>The equation of the hyperbola is clearly +<MATH><I>x</I><SUP>2</SUP>=<G>m</G>(<I>y</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>)</MATH>, +where <G>m</G> is a constant. In the particular case taken by +Archimedes <MATH><I>QR</I>=<I>RA,</I></MATH> or <MATH><G>m</G>=1</MATH>, and the hyperbola becomes +the rectangular hyperbola (2) above. +<p>The second lemma (Prop. 43, p. 300) proves that, if <I>BC</I> is +given in length, and <I>Q</I> is such a point that, when <I>QR</I> is drawn +perpendicular to <I>BC</I>, <MATH><I>BR.RC</I>=<I>k.QR,</I></MATH> where <I>k</I> is a given +length, the locus of <I>Q</I> is a <I>parabola.</I> +<p>Let <I>O</I> be the middle point of <I>BC,</I> and let <I>OK</I> be drawn at +right angles to <I>BC</I> and of length such that +<MATH><I>OC</I><SUP>2</SUP>=<I>k.KO.</I></MATH> +<p>Let <I>QN</I>′ be drawn perpendicular to <I>OK.</I> +<p>Then <MATH><I>QN</I>′<SUP>2</SUP>=<I>OR</I><SUP>2</SUP> +=<I>OC</I><SUP>2</SUP>-<I>BR.RC</I> +=<I>k.</I>(<I>KO</I>-<I>QR</I>), by hypothesis, +=<I>k.KN</I>′.</MATH> +<p>Therefore the locus of <I>Q</I> is a parabola. +<p>The equation of the parabola referred to <I>DB, DE</I> as axes of +<I>x</I> and <I>y</I> is obviously +<MATH>{1/2(<I>a</I>-<I>b</I>)-<I>x</I>}<SUP>2</SUP>=<I>k</I>{(<I>a</I>+<I>b</I>)<SUP>2</SUP>/4<I>k</I>-<I>y</I>}</MATH>, +which easily reduces to +<MATH>(<I>a</I>-<I>x</I>)(<I>b</I>+<I>x</I>)=<I>ky,</I></MATH> as above (1). +<p>In Archimedes's particular case <MATH><I>k</I>=<I>ab/c.</I></MATH> +<p>To solve the problem then we have only to draw the para- +bola and hyperbola in question, and their intersection then +gives <I>Q,</I> whence <I>R,</I> and therefore <I>ARP,</I> is determined. +<pb n=389> +<head>THE <I>COLLECTION.</I> BOOKS IV, V</head> +<C>Book V. Preface on the Sagacity of Bees.</C> +<p>It is characteristic of the great Greek mathematicians that, +whenever they were free from the restraint of the technical +language of mathematics, as when for instance they had occa- +sion to write a preface, they were able to write in language of +the highest literary quality, comparable with that of the +philosophers, historians, and poets. We have only to recall +the introductions to Archimedes's treatises and the prefaces +to the different Books of Apollonius's <I>Conics.</I> Heron, though +severely practical, is no exception when he has any general +explanation, historical or other, to give. We have now to +note a like case in Pappus, namely the preface to Book V of +the <I>Collection.</I> The editor, Hultsch, draws attention to the +elegance and purity of the language and the careful writing; +the latter is illustrated by the studied avoidance of hiatus.<note>Pappus, vol. iii, p. 1233.</note> +The subject is one which a writer of taste and imagination +would naturally find attractive, namely the practical intelli- +gence shown by bees in selecting the hexagonal form for the +cells in the honeycomb. Pappus does not disappoint us; the +passage is as attractive as the subject, and deserves to be +reproduced. +<p>‘It is of course to men that God has given the best and +most perfect notion of wisdom in general and of mathematical +science in particular, but a partial share in these things he +allotted to some of the unreasoning animals as well. To men, +as being endowed with reason, he vouchsafed that they should +do everything in the light of reason and demonstration, but to +the other animals, while denying them reason, he granted +that each of them should, by virtue of a certain natural +instinct, obtain just so much as is needful to support life. +This instinct may be observed to exist in very many other +species of living creatures, but most of all in bees. In the first +place their orderliness and their submission to the queens who +rule in their state are truly admirable, but much more admirable +still is their emulation, the cleanliness they observe in the +gathering of honey, and the forethought and housewifely care +they devote to its custody. Presumably because they know +themselves to be entrusted with the task of bringing from +the gods to the accomplished portion of mankind a share of +<pb n=390> +<head>PAPPUS OF ALEXANDRIA</head> +ambrosia in this form, they do not think it proper to pour it +carelessly on ground or wood or any other ugly and irregular +material; but, first collecting the sweets of the most beautiful +flowers which grow on the earth, they make from them, for +the reception of the honey, the vessels which we call honey- +combs, (with cells) all equal, similar and contiguous to one +another, and hexagonal in form. And that they have con- +trived this by virtue of a certain geometrical forethought we +may infer in this way. They would necessarily think that +the figures must be such as to be contiguous to one another, +that is to say, to have their sides common, in order that no +foreign matter could enter the interstices between them and +so defile the purity of their produce. Now only three recti- +lineal figures would satisfy the condition, I mean regular +figures which are equilateral and equiangular; for the bees +would have none of the figures which are not uniform. . . . +There being then three figures capable by themselves of +exactly filling up the space about the same point, the bees by +reason of their instinctive wisdom chose for the construction +of the honeycomb the figure which has the most angles, +because they conceived that it would contain more honey than +either of the two others. +<p>‘Bees, then, know just this fact which is of service to them- +selves, that the hexagon is greater than the square and the +triangle and will hold more honey for the same expenditure of +material used in constructing the different figures. We, how- +ever, claiming as we do a greater share in wisdom than bees, +will investigate a problem of still wider extent, namely that, +of all equilateral and equiangular plane figures having an +equal perimeter, that which has the greater number of angles +is always greater, and the greatest plane figure of all those +which have a perimeter equal to that of the polygons is the +circle.’ +<p>Book V then is devoted to what we may call <I>isoperimetry</I>, +including in the term not only the comparison of the areas of +different plane figures with the same perimeter, but that of the +contents of different solid figures with equal surfaces. +<C>Section (1). <I>Isoperimetry after Zenodorus.</I></C> +<p>The first section of the Book relating to plane figures +(chaps. 1-10, pp. 308-34) evidently followed very closely +the exposition of Zenodorus <G>peri\ i)some/trwn sxhma/twn</G> (see +pp. 207-13, above); but before passing to solid figures Pappus +inserts the proposition that <I>of all circular segments having</I> +<pb n=391> +<head>THE <I>COLLECTION.</I> BOOK V</head> +<I>the same circumference the semicircle is the greatest</I>, with some +preliminary lemmas which deserve notice (chaps. 15, 16). +<p>(1) <I>ABC</I> is a triangle right-angled at <I>B.</I> With <I>C</I> as centre +and radius <I>CA</I> describe the arc +<I>AD</I> cutting <I>CB</I> produced in <I>D.</I> +To prove that (<I>R</I> denoting a right +angle) +<FIG> +<MATH>(sector <I>CAD</I>):(area <I>ABD</I>) +><I>R</I>:∠<I>BCA</I></MATH>. +<p>Draw <I>AF</I> at right angles to <I>CA</I> meeting <I>CD</I> produced in <I>F</I>, +and draw <I>BH</I> perpendicular to <I>AF.</I> With <I>A</I> as centre and +<I>AB</I> as radius describe the arc <I>GBE.</I> +<p>Now <MATH>(area <I>EBF</I>):(area <I>EBH</I>)>(area <I>EBF</I>):(sector <I>ABE</I>)</MATH>, +and, <I>componendo</I>, <MATH>▵<I>FBH</I>:(<I>EBH</I>)>▵<I>ABF</I>:(<I>ABE</I>)</MATH>. +<p>But (by an easy lemma which has just preceded) +<MATH>▵<I>FBH</I>:(<I>EBH</I>)=▵<I>ABF</I>:(<I>ABD</I>)</MATH>, +whence <MATH>▵<I>ABF</I>:(<I>ABD</I>)>▵<I>ABF</I>:(<I>ABE</I>)</MATH>, +and <MATH>(<I>ABE</I>)>(<I>ABD</I>)</MATH>. +<p>Therefore <MATH>(<I>ABE</I>):(<I>ABG</I>)>(<I>ABD</I>):(<I>ABG</I>) +>(<I>ABD</I>):▵<I>ABC</I></MATH>, <I>a fortiori.</I> +<p>Therefore <MATH>∠<I>BAF</I>:∠<I>BAC</I>>(<I>ABD</I>):▵<I>ABC</I></MATH>, +whence, inversely, <MATH>▵<I>ABC</I>:(<I>ABD</I>)>∠<I>BAC</I>:∠<I>BAF</I></MATH>. +and, <I>componendo</I>, <MATH>(sector <I>ACD</I>):(<I>ABD</I>)><I>R</I>:∠<I>BCA</I></MATH>. +<p>[If <G>a</G> be the circular measure of ∠<I>BCA</I>, this gives (if <MATH><I>AC</I>=<I>b</I></MATH>) +<MATH>1/2 <G>a</G><I>b</I><SUP>2</SUP>:(1/2 <G>a</G><I>b</I><SUP>2</SUP>-1/2 sin<G>a</G>cos<G>a</G>.<I>b</I><SUP>2</SUP>)>1/2 <G>p</G>:<G>a</G></MATH>, +or <MATH>2<G>a</G>:(2<G>a</G>-sin2<G>a</G>)><G>p</G>:2<G>a</G></MATH>; +that is, <MATH><G>q</G>/(<G>q</G>-sin<G>q</G>)><G>p</G>/<G>q</G></MATH>, where <MATH>0<<G>q</G><<G>p</G></MATH>.] +<p>(2) <I>ABC</I> is again a triangle right-angled at <I>B.</I> With <I>C</I> as +centre and <I>CA</I> as radius draw a circle <I>AD</I> meeting <I>BC</I> pro- +duced in <I>D.</I> To prove that +<MATH>(sector <I>CAD</I>):(area <I>ABD</I>)><I>R</I>:∠<I>ACD</I></MATH>. +<pb n=392> +<head>PAPPUS OF ALEXANDRIA</head> +<p>Draw <I>AE</I> at right angles to <I>AC.</I> With <I>A</I> as centre and +<I>AC</I> as radius describe the circle <I>FCE</I> meeting <I>AB</I> produced +in <I>F</I> and <I>AE</I> in <I>E.</I> +<p>Then, since <MATH>∠<I>ACD</I>>∠<I>CAE</I>, (sector <I>ACD</I>)>(sector <I>ACE</I>)</MATH>. +<p>Therefore <MATH>(<I>ACD</I>):▵<I>ABC</I>>(<I>ACE</I>):▵<I>ABC</I> +>(<I>ACE</I>):(<I>ACF</I>)</MATH>, <I>a fortiori</I>, +<MATH>>∠<I>EAC</I>:∠<I>CAB</I></MATH>. +<FIG> +Inversely, +<MATH>▵<I>ABC</I>:(<I>ACD</I>)<∠<I>CAB</I>:∠<I>EAC</I></MATH>, +and, <I>componendo</I>, +<MATH>(<I>ABD</I>):(<I>ACD</I>)<∠<I>EAB</I>:∠<I>EAC</I></MATH>. +<p>Inversely, <MATH>(<I>ACD</I>):(<I>ABD</I>)>∠<I>EAC</I>:∠<I>EAB</I> +><I>R</I>:∠<I>ACD</I></MATH>. +<p>We come now to the application of these lemmas to the +proposition comparing the area of a semicircle with that of +other segments of equal circumference (chaps. 17, 18). +<C><I>A semicircle is the greatest of all segments of circles which +have the same circumference.</I></C> +<p>Let <I>ABC</I> be a semicircle with centre <I>G</I>, and <I>DEF</I> another +segment of a circle such that the circumference <I>DEF</I> is equal +<FIG> +to the circumference <I>ABC.</I> I say that the area of <I>ABC</I> is +greater than the area of <I>DEF.</I> +<p>Let <I>H</I> be the centre of the circle <I>DEF.</I> Draw <I>EHK, BG</I> at +right angles to <I>DF, AC</I> respectively. Join <I>DH</I>, and draw +<I>LHM</I> parallel to <I>DF.</I> +<pb n=393> +<head>THE <I>COLLECTION.</I> BOOK V</head> +<p>Then <MATH><I>LH</I>:<I>AG</I>=(arc <I>LE</I>):(arc <I>AB</I>) +=(arc <I>LE</I>):(arc <I>DE</I>) +=(sector <I>LHE</I>):(sector <I>DHE</I>)</MATH>. +<p>Also <MATH><I>LH</I><SUP>2</SUP>:<I>AG</I><SUP>2</SUP>=(sector <I>LHE</I>):(sector <I>AGB</I>)</MATH>. +<p>Therefore the sector <I>LHE</I> is to the sector <I>AGB</I> in the +ratio duplicate of that which the sector <I>LHE</I> has to the +sector <I>DHE.</I> +<p>Therefore +<MATH>(sector <I>LHE</I>):(sector <I>DHE</I>)=(sector <I>DHE</I>):(sector <I>AGB</I>)</MATH>. +<p>Now (1) in the case of the segment less than a semicircle +and (2) in the case of the segment greater than a semicircle +<MATH>(sector <I>EDH</I>):(<I>EDK</I>)><I>R</I>:∠<I>DHE</I></MATH>, +by the lemmas (1) and (2) respectively. +<p>That is, +<MATH>(sector <I>EDH</I>):(<I>EDK</I>)>∠<I>LHE</I>:∠<I>DHE</I> +>(sector <I>LHE</I>):(sector <I>DHE</I>) +>(sector <I>EDH</I>):(sector <I>AGB</I>)</MATH>, +from above. +<p>Therefore the half segment <I>EDK</I> is less than the half +semicircle <I>AGB</I>, whence the semicircle <I>ABC</I> is greater than +the segment <I>DEF.</I> +<p>We have already described the content of Zenodorus's +treatise (pp. 207-13, above) to which, so far as plane figures +are concerned, Pappus added nothing except the above pro- +position relating to segments of circles. +<C>Section (2). <I>Comparison of volumes of solids having their +surfaces equal. Case of the sphere.</I></C> +<p>The portion of Book V dealing with solid figures begins +(p. 350. 20) with the statement that the philosophers who +considered that the creator gave the universe the form of a +sphere because that was the most beautiful of all shapes also +asserted that the sphere is the greatest of all solid figures +<pb n=394> +<head>PAPPUS OF ALEXANDRIA</head> +which have their surfaces equal; this, however, they had not +proved, nor could it be proved without a long investigation. +Pappus himself does not attempt to prove that the sphere is +greater than <I>all</I> solids with the same surface, but only that +the sphere is greater than any of the five regular solids having +the same surface (chap. 19) and also greater than either a cone +or a cylinder of equal surface (chap. 20). +<C>Section (3). <I>Digression on the semi-regular solids +of Archimedes.</I></C> +<p>He begins (chap. 19) with an account of the thirteen <I>semi- +regular</I> solids discovered by Archimedes, which are contained +by polygons all equilateral and all equiangular but not all +similar (see pp. 98-101, above), and he shows how to determine +the number of solid angles and the number of edges which +they have respectively; he then gives them the go-by for his +present purpose because they are not completely regular; still +less does he compare the sphere with any irregular solid +having an equal surface. +<C><I>The sphere is greater than any of the regular solids which +has its surface equal to that of the sphere.</I></C> +<p>The proof that the sphere is greater than any of the regular +solids with surface equal to that of the sphere is the same as +that given by Zenodorus. Let <I>P</I> be any one of the regular solids, +<I>S</I> the sphere with surface equal to that of <I>P.</I> To prove that +<MATH><I>S</I>><I>P</I></MATH>. Inscribe in the solid a sphere <I>s</I>, and suppose that <I>r</I> is its +radius. Then the surface of <I>P</I> is greater than the surface of <I>s</I>, +and accordingly, if <I>R</I> is the radius of <I>S</I>, <MATH><I>R</I>><I>r</I></MATH>. But the +volume of <I>S</I> is equal to the cone with base equal to the surface +of <I>S</I>, and therefore of <I>P</I>, and height equal to <I>R</I>; and the volume +of <I>P</I> is equal to the cone with base equal to the surface of <I>P</I> +and height equal to <I>r.</I> Therefore, since <MATH><I>R</I>><I>r</I></MATH>, volume of <I>S</I>> +volume of <I>P.</I> +<C>Section (4). <I>Propositions on the lines of Archimedes, +‘On the Sphere and Cylinder’.</I></C> +<p>For the fact that the volume of a sphere is equal to the cone +with base equal to the surface, and height equal to the radius, +<pb n=395> +<head>THE <I>COLLECTION.</I> BOOK V</head> +of the sphere, Pappus quotes Archimedes, <I>On the Sphere and +Cylinder</I>, but thinks proper to add a series of propositions +(chaps. 20-43, pp. 362-410) on much the same lines as those of +Archimedes and leading to the same results as Archimedes +obtains for the surface of a segment of a sphere and of the whole +sphere (Prop. 28), and for the volume of a sphere (Prop. 35). +Prop. 36 (chap. 42) shows how to divide a sphere into two +segments such that their surfaces are in a given ratio and +Prop. 37 (chap. 43) proves that the volume as well as the +surface of the cylinder circumscribing a sphere is 1 1/2 times +that of the sphere itself. +<p>Among the lemmatic propositions in this section of the +Book Props. 21, 22 may be mentioned. Prop. 21 proves that, +if <I>C, E</I> be two points on the tangent at <I>H</I> to a semicircle such +that <MATH><I>CH</I>=<I>HE</I></MATH>, and if <I>CD, EF</I> be drawn perpendicular to the +diameter <I>AB</I>, then <MATH>(<I>CD</I>+<I>EF</I>)<I>CE</I>=<I>AB.DF</I></MATH>; Prop. 22 proves +a like result where <I>C, E</I> are points on the semicircle, <I>CD, EF</I> +are as before perpendicular to <I>AB</I>, and <I>EH</I> is the chord of +the circle subtending the arc which with <I>CE</I> makes up a semi- +circle; in this case <MATH>(<I>CD</I>+<I>EF</I>)<I>CE</I>=<I>EH.DF</I></MATH>. Both results +are easily seen to be the equivalent of the trigonometrical +formula +<MATH>sin(<I>x</I>+<I>y</I>)+sin(<I>x</I>-<I>y</I>)=2sin<I>x</I>cos<I>y</I></MATH>, +or, if certain different angles be taken as <I>x, y</I>, +<MATH>(sin<I>x</I>+sin<I>y</I>)/(cos<I>y</I>-cos<I>x</I>)=cot1/2(<I>x</I>-<I>y</I>)</MATH>. +<C>Section (5). <I>Of regular solids with surfaces equal, that is +greater which has more faces.</I></C> +<p>Returning to the main problem of the Book, Pappus shows +that, of the five regular solid figures assumed to have their +surfaces equal, that is greater which has the more faces, so +that the pyramid, the cube, the octahedron, the dodecahedron +and the icosahedron of equal surface are, as regards solid +content, in ascending order of magnitude (Props. 38-56). +Pappus indicates (p. 410. 27) that ‘some of the ancients’ had +worked out the proofs of these propositions by the analytical +method; for himself, he will give a method of his own by +<pb n=396> +<head>PAPPUS OF ALEXANDRIA</head> +synthetical deduction, for which he claims that it is clearer +and shorter. We have first propositions (with auxiliary +lemmas) about the perpendiculars from the centre of the +circumscribing sphere to a face of (<I>a</I>) the octahedron, (<I>b</I>) the +icosahedron (Props. 39, 43), then the proposition that, if a +dodecahedron and an icosahedron be inscribed in the same +sphere, the same small circle in the sphere circumscribes both +the pentagon of the dodecahedron and the triangle of the +icosahedron (Prop. 48); this last is the proposition proved by +Hypsicles in the so-called ‘Book XIV of Euclid’, Prop. 2, and +Pappus gives two methods of proof, the second of which (chap. +56) corresponds to that of Hypsicles. Prop. 49 proves that +twelve of the regular pentagons inscribed in a circle are together +greater than twenty of the equilateral triangles inscribed in +the same circle. The final propositions proving that the cube +is greater than the pyramid with the same surface, the octa- +hedron greater than the cube, and so on, are Props. 52-6 +(chaps. 60-4). Of Pappus's auxiliary propositions, Prop. 41 +is practically contained in Hypsicles's Prop. 1, and Prop. 44 +in Hypsicles's last lemma; but otherwise the exposition is +different. +<C>Book VI.</C> +<p>On the contents of Book VI we can be brief. It is mainly +astronomical, dealing with the treatises included in the so- +called <I>Little Astronomy</I>, that is, the smaller astronomical +treatises which were studied as an introduction to the great +<I>Syntaxis</I> of Ptolemy. The preface says that many of those +who taught the <I>Treasury of Astronomy</I>, through a careless +understanding of the propositions, added some things as being +necessary and omitted others as unnecessary. Pappus mentions +at this point an incorrect addition to Theodosius, <I>Sphaerica</I>, +III. 6, an omission from Euclid's <I>Phaenomena</I>, Prop. 2, an +inaccurate representation of Theodosius, <I>On Days and Nights</I>, +Prop. 4, and the omission later of certain other things as +being unnecessary. His object is to put these mistakes +right. Allusions are also found in the Book to Menelaus's +<I>Sphaerica</I>, e.g. the statement (p. 476. 16) that Menelaus in +his <I>Sphaerica</I> called a spherical triangle <G>tri/pleurov</G>, <I>three-side.</I> +<pb n=397> +<head>THE <I>COLLECTION.</I> BOOKS V, VI</head> +The <I>Sphaerica</I> of Theodosius is dealt with at some length +(chaps. 1-26, Props. 1-27), and so are the theorems of +Autolycus <I>On the moving Sphere</I> (chaps. 27-9), Theodosius +<I>On Days and Nights</I> (chaps. 30-6, Props. 29-38), Aristarchus +<I>On the sizes and distances of the Sun and Moon</I> (chaps. 37-40, +including a proposition, Prop. 39 with two lemmas, which is +corrupt at the end and is not really proved), Euclid's <I>Optics</I> +(chaps. 41-52, Props. 42-54), and Euclid's <I>Phaenomena</I> (chaps. +53-60, Props. 55-61). +<C><I>Problem arising out of Euclid's ‘Optics’.</I></C> +<p>There is little in the Book of general mathematical interest +except the following propositions which occur in the section on +Euclid's <I>Optics.</I> +<p>Two propositions are fundamental in solid geometry, +namely: +<p>(<I>a</I>) If from a point <I>A</I> above a plane <I>AB</I> be drawn perpen- +dicular to the plane, and if from <I>B</I> a straight line <I>BD</I> be +drawn perpendicular to any straight line <I>EF</I> in the plane, +then will <I>AD</I> also be perpendicular to <I>EF</I> (Prop. 43). +<p>(<I>b</I>) If from a point <I>A</I> above a plane <I>AB</I> be drawn to the plane +but not at right angles to it, and <I>AM</I> be drawn perpendicular +to the plane (i.e. if <I>BM</I> be the orthogonal projection of <I>BA</I> on +the plane), the angle <I>ABM</I> is the least of all the angles which +<I>AB</I> makes with any straight lines through <I>B</I>, as <I>BP</I>, in the +plane; the angle <I>ABP</I> increases as <I>BP</I> moves away from <I>BM</I> +on either side; and, given any straight line <I>BP</I> making +a certain angle with <I>BA</I>, only one other straight line in the +plane will make the same angle with <I>BA</I>, namely a straight +line <I>BP</I>′ on the other side of <I>BM</I> making the same angle with +it that <I>BP</I> does (Prop. 44). +<p>These are the first of a series of lemmas leading up to the +main problem, the investigation of the apparent form of +a circle as seen from a point outside its plane. In Prop. 50 +(=Euclid, <I>Optics</I>, 34) Pappus proves the fact that all the +diameters of the circle will appear equal if the straight line +drawn from the point representing the eye to the centre of +the circle is either (<I>a</I>) at right angles to the plane of the circle +or (<I>b</I>), if not at right angles to the plane of the circle, is equal +<pb n=398> +<head>PAPPUS OF ALEXANDRIA</head> +in length to the radius of the circle. In all other cases +(Prop. 51=Eucl. <I>Optics</I>, 35) the diameters will appear unequal. +Pappus's other propositions carry farther Euclid's remark +that the circle seen under these conditions will appear +deformed or distorted (<G>parespasme/nos</G>), proving (Prop. 53, +pp. 588-92) that the apparent form will be an ellipse with its +centre not, ‘as some think’, at the centre of the circle but +at another point in it, determined in this way. Given a circle +<I>ABDE</I> with centre <I>O</I>, let the eye be at a point <I>F</I> above the +plane of the circle such that <I>FO</I> is neither perpendicular +to that plane nor equal to the radius of the circle. Draw <I>FG</I> +perpendicular to the plane of the circle and let <I>ADG</I> be the +diameter through <I>G</I>. Join <I>AF, DF,</I> and bisect the angle <I>AFD</I> +by the straight line <I>FC</I> meeting <I>AD</I> in <I>C</I>. Through <I>C</I> draw +<I>BE</I> perpendicular to <I>AD</I>, and let the tangents at <I>B, E</I> meet +<I>AG</I> produced in <I>K</I>. Then Pappus proves that <I>C</I> (not <I>O</I>) is the +centre of the apparent ellipse, that <I>AD, BE</I> are its major and +minor axes respectively, that the ordinates to <I>AD</I> are parallel +to <I>BE</I> both really and apparently, and that the ordinates to +<I>BE</I> will pass through <I>K</I> but will appear to be parallel to <I>AD.</I> +Thus in the figure, <I>C</I> being the centre of the apparent ellipse, +<FIG> +it is proved that, if <I>LCM</I> is any straight line through <I>C, LC</I> is +apparently equal to <I>CM</I> (it is practically assumed--a proposi- +tion proved later in Book VII, Prop. 156--that, if <I>LK</I> meet +the circle again in <I>P</I>, and if <I>PM</I> be drawn perpendicular to +<I>AD</I> to meet the circle again in <I>M, LM</I> passes through <I>C</I>). +<pb n=399> +<head>THE <I>COLLECTION.</I> BOOKS VI, VII</head> +The test of apparent equality is of course that the two straight +lines should subtend equal angles at <I>F.</I> +<p>The main points in the proof are these. The plane through +<I>CF, CK</I> is perpendicular to the planes <I>BFE, PFM</I> and <I>LFR</I>; +hence <I>CF</I> is perpendicular to <I>BE, QF</I> to <I>PM</I> and <I>HF</I> to <I>LR</I>, +whence <I>BC</I> and <I>CE</I> subtend equal angles at <I>F</I>: so do <I>LH, HR</I>, +and <I>PQ, QM</I>. +<p>Since <I>FC</I> bisects the angle <I>AFD</I>, and <MATH><I>AC</I>:<I>CD</I>=<I>AK</I>:<I>KD</I></MATH> +(by the polar property), ∠<I>CFK</I> is a right angle. And <I>CF</I> is +the intersection of two planes at right angles, namely <I>AFK</I> +and <I>BFE</I>, in the former of which <I>FK</I> lies; therefore <I>KF</I> is +perpendicular to the plane <I>BFE</I>, and therefore to <I>FN</I>. Since +therefore (by the polar property) <MATH><I>LN</I>:<I>NP</I>=<I>LK</I>:<I>KP</I></MATH>, it +follows that the angle <I>LFP</I> is bisected by <I>FN</I>; hence <I>LN, NP</I> +are apparently equal. +<p>Again <MATH><I>LC</I>:<I>CM</I>=<I>LN</I>:<I>NP</I>=<I>LF</I>:<I>FP</I>=<I>LF</I>:<I>FM</I></MATH>. +<p>Therefore the angles <I>LFC, CFM</I> are equal, and <I>LC, CM</I> +are apparently equal. +<p>Lastly <MATH><I>LR</I>:<I>PM</I>=<I>LK</I>:<I>KP</I>=<I>LN</I>:<I>NP</I>=<I>LF</I>:<I>FP</I></MATH>; therefore +the isosceles triangles <I>FLR, FPM</I> are equiangular; there- +fore the angles <I>PFM, LFR</I>, and consequently <I>PFQ, LFH</I>, are +equal. Hence <I>LP, RM</I> will appear to be parallel to <I>AD</I>. +<p>We have, based on this proposition, an easy method of +solving Pappus's final problem (Prop. 54). ‘Given a circle +<I>ABDE</I> and any point within it, to find outside the plane of +the circle a point from which the circle will have the appear- +ance of an ellipse with centre <I>C.</I>’ +<p>We have only to produce the diameter <I>AD</I> through <I>C</I> to the +pole <I>K</I> of the chord <I>BE</I> perpendicular to <I>AD</I> and then, in +the plane through <I>AK</I> perpendicular to the plane of the circle, +to describe a semicircle on <I>CK</I> as diameter. Any point <I>F</I> on +this semicircle satisfies the condition. +<C>Book VII. <I>On the ‘Treasury of Analysis’</I>.</C> +<p>Book VII is of much greater importance, since it gives an +account of the books forming what was called the <I>Treasury of +Analysis</I> (<G>a)naluo/menos to/pos</G>) and, as regards those of the books +which are now lost, Pappus's account, with the hints derivable +from the large collection of lemmas supplied by him to each +<pb n=400> +<head>PAPPUS OF ALEXANDRIA</head> +book, practically constitutes our only source of information. +The Book begins (p. 634) with a definition of <I>analysis</I> and +<I>synthesis</I> which, as being the most elaborate Greek utterance +on the subject, deserves to be quoted in full. +<p>‘The so-called <G>*)analuo/menos</G> is, to put it shortly, a special +body of doctrine provided for the use of those who, after +finishing the ordinary Elements, are desirous of acquiring the +power of solving problems which may be set them involving +(the construction of) lines, and it is useful for this alone. It is +the work of three men, Euclid the author of the Elements, +Apollonius of Perga and Aristaeus the elder, and proceeds by +way of analysis and synthesis.’ +<C><I>Definition of Analysis and Synthesis.</I></C> +<p>‘<I>Analysis</I>, then, takes that which is sought as if it were +admitted and passes from it through its successive conse- +quences to something which is admitted as the result of +synthesis: for in analysis we assume that which is sought +as if it were already done (<G>lelono/s</G>), and we inquire what it is +from which this results, and again what is the antecedent +cause of the latter, and so on, until by so retracing our steps +we come upon something already known or belonging to the +class of first principles, and such a method we call analysis +as being solution backwards (<G>a)na/palin lu/sin</G>). +<p>‘But in <I>synthesis</I>, reversing the process, we take as already +done that which was last arrived at in the analysis and, by +arranging in their natural order as consequences what before +were antecedents, and successively connecting them one with +another, we arrive finally at the construction of what was +sought; and this we call synthesis. +<p>‘Now analysis is of two kinds, the one directed to searching +for the truth and called <I>theoretical</I>, the other directed to +finding what we are told to find and called <I>problematical.</I> +(1) In the <I>theoretical</I> kind we assume what is sought as if +it were existent and true, after which we pass through its +successive consequences, as if they too were true and established +by virtue of our hypothesis, to something admitted: then +(<I>a</I>), if that something admitted is true, that which is sought +will also be true and the proof will correspond in the reverse +order to the analysis, but (<I>b</I>), if we come upon something +admittedly false, that which is sought will also be false. +(2) In the <I>problematical</I> kind we assume that which is pro- +pounded as if it were known, after which we pass through its +<pb n=401> +<head>THE <I>COLLECTION.</I> BOOK VII</head> +successive consequences, taking them as true, up to something +admitted: if then (<I>a</I>) what is admitted is possible and obtain- +able, that is, what mathematicians call <I>given</I>, what was +originally proposed will also be possible, and the proof will +again correspond in the reverse order to the analysis, but if (<I>b</I>) +we come upon something admittedly impossible, the problem +will also be impossible.’ +<p>This statement could hardly be improved upon except that +it ought to be added that each step in the chain of inference +in the analysis must be <I>unconditionally convertible</I>; that is, +when in the analysis we say that, if <I>A</I> is true, <I>B</I> is true, +we must be sure that each statement is a necessary conse- +quence of the other, so that the truth of <I>A</I> equally follows +from the truth of <I>B.</I> This, however, is almost implied by +Pappus when he says that we inquire, not what it is (namely +<I>B</I>) which follows from <I>A</I>, but what it is (<I>B</I>) from which <I>A</I> +follows, and so on. +<C><I>List of works in the ‘Treasury of Analysis’.</I></C> +<p>Pappus adds a list, in order, of the books forming the +<G>*)analuo/menos</G>, namely: +<p>‘Euclid's <I>Data</I>, one Book, Apollonius's <I>Cutting-off of a ratio</I>, +two Books, <I>Cutting-off of an area</I>, two Books, <I>Determinate +Section</I>, two Books, <I>Contacts</I>, two Books, Euclid's <I>Porisms</I>, +three Books, Apollonius's <I>Inclinations</I> or <I>Vergings</I> (<G>neu/seis</G>), +two Books, the same author's <I>Plane Loci</I>, two Books, and +<I>Conics</I>, eight Books, Aristaeus's <I>Solid Loci</I>, five Books, Euclid's +<I>Surface-Loci</I>, two Books, Eratosthenes's <I>On means</I>, two Books. +There are in all thirty-three Books, the contents of which up +to the <I>Conics</I> of Apollonius I have set out for your considera- +tion, including not only the number of the propositions, the +<I>diorismi</I> and the cases dealt with in each Book, but also the +lemmas which are required; indeed I have not, to the best +of my belief, omitted any question arising in the study of the +Books in question.’ +<C><I>Description of the treatises.</I></C> +<p>Then follows the short description of the contents of the +various Books down to Apollonius's <I>Conics</I>; no account is +given of Aristaeus's <I>Solid Loci</I>, Euclid's <I>Surface-Loci</I> and +<pb n=402> +<head>PAPPUS OF ALEXANDRIA</head> +Eratosthenes's <I>On means</I>, nor are there any lemmas to these +works except two on the <I>Surface-Loci</I> at the end of the Book. +<p>The contents of the various works, including those of the +lost treatises so far as they can be gathered from Pappus, +have been described in the chapters devoted to their authors, +and need not be further referred to here, except for an +<I>addendum</I> to the account of Apollonius's <I>Conics</I> which is +remarkable. Pappus has been speaking of the ‘locus with +respect to three or four lines’ (which is a conic), and proceeds +to say (p. 678. 26) that we may in like manner have loci with +reference to five or six or even more lines; these had not up +to his time become generally known, though the synthesis +of one of them, not by any means the most obvious, had been +worked out and its utility shown. Suppose that there are +five or six lines, and that <I>p</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB>, <I>p</I><SUB>4</SUB>, <I>p</I><SUB>5</SUB> or <I>p</I><SUB>1</SUB>, <I>p</I><SUB>2</SUB>, <I>p</I><SUB>3</SUB>, <I>p</I><SUB>4</SUB>, <I>p</I><SUB>5</SUB>, +<I>p</I><SUB>6</SUB> +are the lengths of straight lines drawn from a point to meet +the five or six at given angles, then, if in the first case +<MATH><I>p</I><SUB>1</SUB><I>p</I><SUB>2</SUB><I>p</I><SUB>3</SUB>=<G>l</G><I>p</I><SUB>4</SUB><I>p</I><SUB>5</SUB><I>a</I></MATH> (where <G>l</G> is a constant ratio and <I>a</I> a given +length), and in the second case <MATH><I>p</I><SUB>1</SUB><I>p</I><SUB>2</SUB><I>p</I><SUB>3</SUB>=<G>l</G><I>p</I><SUB>4</SUB><I>p</I><SUB>5</SUB><I>p</I><SUB>6</SUB></MATH>, the locus +of the point is in each case a certain curve given in position. +The relation could not be expressed in the same form if +there were more lines than six, because there are only three +dimensions in geometry, although certain recent writers had +allowed themselves to speak of a rectangle multiplied by +a square or a rectangle without giving any intelligible idea of +what they meant by such a thing (is Pappus here alluding to +Heron's proof of the formula for the area of a triangle in +terms of its sides given on pp. 322-3, above?). But the system +of compounded ratios enables it to be expressed for any +number of lines thus, <MATH>(<I>p</I><SUB>1</SUB>/<I>p</I><SUB>2</SUB>).(<I>p</I><SUB>3</SUB>/<I>p</I><SUB>4</SUB>)....(<I>p<SUB>n</SUB>/a</I>)</MATH> (or <MATH><I>p</I><SUB><I>n</I>-1</SUB>/<I>p<SUB>n</SUB></I>))=<G>l</G></MATH>. Pappus +proceeds in language not very clear (p. 680. 30); but the gist +seems to be that the investigation of these curves had not +attracted men of light and leading, as, for instance, the old +geometers and the best writers. Yet there were other impor- +tant discoveries still remaining to be made. For himself, he +noticed that every one in his day was occupied with the elements, +the first principles and the natural origin of the subject- +matter of investigation; ashamed to pursue such topics, he had +himself proved propositions of much more importance and +<pb n=403> +<head>THE <I>COLLECTION.</I> BOOK VII</head> +utility. In justification of this statement and ‘in order that +he may not appear empty-handed when leaving the subject’, +he will present his readers with the following. +<C>(<I>Anticipation of Guldin's Theorem.</I>)</C> +<p>The enunciations are not very clearly worded, but there +is no doubt as to the sense. +<p>‘<I>Figures generated by a complete revolution of a plane figure +about an axis are in a ratio compounded (1) of the ratio +of the areas of the figures, and (2) of the ratio of the straight +lines similarly drawn to (i.e. drawn to meet at the same angles) +the axes of rotation from the respective centres of gravity. +Figures generated by incomplete revolutions are in the ratio +compounded (1) of the ratio of the areas of the figures and +(2) of the ratio of the arcs described by the centres of gravity +of the respective figures, the latter ratio being itself compounded +(a) of the ratio of the straight lines similarly drawn (from +the respective centres of gravity to the axes of rotation) and +(b) of the ratio of the angles contained (i.e. described) about +the axes of revolution by the extremities of the said straight +lines (i.e. the centres of gravity).</I>’ +<p>Here, obviously, we have the essence of the celebrated +theorem commonly attributed to P. Guldin (1577-1643), +‘quantitas rotunda in viam rotationis ducta producit Pote- +statem Rotundam uno grado altiorem Potestate sive Quantitate +Rotata’.<note><I>Centrobaryca</I>, Lib. ii, chap. viii, Prop. 3. Viennae 1641.</note> +<p>Pappus adds that +<p>‘these propositions, which are practically one, include any +number of theorems of all sorts about curves, surfaces, and +solids, all of which are proved at once by one demonstration, +and include propositions both old and new, and in particular +those proved in the twelfth Book of these Elements.’ +<p>Hultsch attributes the whole passage (pp. 680. 30-682. 20) +to an interpolator, I do not know for what reason; but it +seems to me that the propositions are quite beyond what +could be expected from an interpolator, indeed I know of +no Greek mathematician from Pappus's day onward except +Pappus himself who was capable of discovering such a pro- +position. +<pb n=404><head>PAPPUS OF ALEXANDRIA</head> +<p>If the passage is genuine, it seems to indicate, what is not +elsewhere confirmed, that the <I>Collection</I> originally contained, +or was intended to contain, twelve Books. +<C><I>Lemmas to the different treatises.</I></C> +<p>After the description of the treatises forming the <I>Treasury +of Analysis</I> come the collections of lemmas given by Pappus +to assist the student of each of the books (except Euclid's +<I>Data</I>) down to Apollonius's <I>Conics</I>, with two isolated lemmas +to the <I>Surface-Loci</I> of Euclid. It is difficult to give any +summary or any general idea of these lemmas, because they +are very numerous, extremely various, and often quite diffi- +cult, requiring first-rate ability and full command of all the +resources of pure geometry. Their number is also greatly +increased by the addition of alternative proofs, often requiring +lemmas of their own, and by the separate formulation of +particular cases where by the use of algebra and conventions +with regard to sign we can make one proposition cover all the +cases. The style is admirably terse, often so condensed as to +make the argument difficult to follow without some little +filling-out; the hand is that of a master throughout. The +only misfortune is that, the books elucidated being lost (except +the <I>Conics</I> and the <I>Cutting-off of a ratio</I> of Apollonius), it is +difficult, often impossible, to see the connexion of the lemmas +with one another and the problems of the book to which they +relate. In the circumstances, all that I can hope to do is to +indicate the types of propositions included in the lemmas and, +by way of illustration, now and then to give a proof where it +is sufficiently out of the common. +<p>(<I>a</I>) Pappus begins with Lemmas to the <I>Sectio rationis</I> and +<I>Sectio spatii</I> of Apollonius (Props. 1-21, pp. 684-704). The +first two show how to divide a straight line in a given ratio, +and how, given the first, second and fourth terms of a pro- +portion between straight lines, to find the third term. The +next section (Props. 3-12 and 16) shows how to manipulate +relations between greater and less ratios by transforming +them, e.g. <I>componendo, convertendo</I>, &c., in the same way +as Euclid transforms <I>equal</I> ratios in Book V; Prop. 16 proves +that, according as <I>a</I>:<I>b</I> > or < <I>c</I>:<I>d, ad</I> > or < <I>bc.</I> Props. +<pb n=405><head>THE <I>COLLECTION.</I> BOOK VII</head> +17-20 deal with three straight lines <I>a, b, c</I> in geometrical +progression, showing how to mark on a straight line containing +<I>a, b, c</I> as segments (including the whole among ‘segments’), +lengths equal to <MATH><I>a</I>+<I>c</I>±2√(<I>ac</I>)</MATH>; the lengths are of course equal +to <MATH><I>a</I>+<I>c</I>±2<I>b</I></MATH> respectively. These lemmas are preliminary to +the problem (Prop. 21), Given two straight lines <I>AB, BC</I> +(<I>C</I>lying between <I>A</I> and <I>B</I>), to find a point <I>D</I> on <I>BA</I> produced +such that <MATH><I>BD</I>:<I>DA</I>=<I>CD</I>:(<I>AB</I>+<I>BC</I>-2√(<I>AB.BC</I>))</MATH>. This is, +of course, equivalent to the quadratic equation <MATH>(<I>a</I>+<I>x</I>):<I>x</I> +=(<I>a</I>-<I>c</I>+<I>x</I>):(<I>a</I>+<I>c</I>-2√(<I>ac</I>))</MATH>, and, after marking off <I>AE</I> along +<I>AD</I> equal to the fourth term of this proportion, Pappus solves +the equation in the usual way by application of areas. +<C>(<G>b</G>) <I>Lemmas to the</I> ‘<I>Determinate Section</I>’ <I>of Apollonius.</I></C> +<p>The next set of Lemmas (Props. 22-64, pp. 704-70) belongs +to the <I>Determinate Section</I> of Apollonius. As we have seen +(pp. 180-1, above), this work seems to have amounted to +a <I>Theory of Involution.</I> Whether the application of certain +of Pappus's lemmas corresponded to the conjecture of Zeuthen +or not, we have at all events in this set of lemmas some +remarkable applications of ‘geometrical algebra’. They may +be divided into groups as follows +<p>I. Props. 22, 25, 29. +<p>If in the figure <MATH><I>AD.DC</I>=<I>BD.DE</I></MATH>, then +<MATH><I>BD</I>:<I>DE</I>=<I>AB.BC</I>:<I>AE.EC</I></MATH>. +<FIG> +<p>The proofs by proportions are not difficult. Prop. 29 is an +alternative proof by means of Prop. 26 (see below). The +algebraic equivalent may be expressed thus: if <MATH><I>ax</I>=<I>by</I></MATH>, then +<MATH><I>b</I>/<I>y</I>=((<I>a</I>+<I>b</I>)(<I>b</I>+<I>x</I>))/((<I>a</I>+<I>y</I>)(<I>x</I>+<I>y</I>))</MATH>. +<p>II. Props. 30, 32, 34. +<p>If in the same figure <MATH><I>AD.DE</I>=<I>BD.DC</I></MATH>, then +<MATH><I>BD</I>:<I>DC</I>=(<I>AB.BE</I>):(<I>EC.CA</I>)</MATH>. +<pb n=406><head>PAPPUS OF ALEXANDRIA</head> +<p>Props. 32, 34 are alternative proofs based on other lemmas +(Props. 31, 33 respectively). The algebraic equivalent may be +stated thus: if <MATH><I>ax</I>=<I>by</I>, then <I>b</I>/<I>y</I>=((<I>a</I>+<I>b</I>)(<I>b</I>-<I>x</I>))/((<I>x</I>+<I>y</I>)(<I>a</I>-<I>y</I>))</MATH>. +<p>III. Props. 35, 36. +<p>If <MATH><I>AB.BE</I>=<I>CB.BD</I>, then <I>AB</I>:<I>BE</I>=(<I>DA.AC</I>):(<I>CE.ED</I>)</MATH>, +and <MATH><I>CB</I>:<I>BD</I>=(<I>AC.CE</I>):(<I>AD.DE</I>)</MATH>, results equivalent to the +following: if <MATH><I>ax</I>=<I>by</I></MATH>, then +<MATH><I>a</I>/<I>x</I>=((<I>a</I>-<I>y</I>)(<I>a</I>-<I>b</I>))/((<I>b</I>-<I>x</I>)(<I>y</I>-<I>x</I>)) and <I>b</I>/<I>y</I>=((<I>a</I>-<I>b</I>)(<I>b</I>-<I>x</I>))/((<I>a</I>-<I>y</I>)(<I>y</I>-<I>x</I>))</MATH>. +<p>IV. Props. 23, 24, 31, 57, 58. +<FIG> +<p>If <MATH><I>AB</I>=<I>CD</I></MATH>, and <I>E</I> is any point in <I>CD</I>, +<MATH><I>AB.CD</I>=<I>AE.ED</I>+<I>BE.EC</I></MATH>, +and similar formulae hold for other positions of <I>E.</I> If <I>E</I> is +between <I>B</I> and <I>C</I>, <MATH><I>AC.CD</I>=<I>AE.ED</I>-<I>BE.EC</I></MATH>; and if <I>E</I> +is on <I>AD</I> produced, <MATH><I>BE.EC</I>=<I>AE.ED</I>+<I>BD.DC</I></MATH>. +<p>V. A small group of propositions relate to a triangle <I>ABC</I> +with two straight lines <I>AD, AE</I> drawn from the vertex <I>A</I> to +points on the base <I>BC</I> in accordance with one or other of the +conditions (<I>a</I>) that the angles <I>BAC, DAE</I> are supplementary, +(<I>b</I>) that the angles <I>BAE, DAC</I> are both right angles or, as we +<FIG> +may add from Book VI, Prop. 12, (<I>c</I>) that the angles <I>BAD, +EAC</I> are equal. The theorems are: +In case <MATH>(<I>a</I>) <I>BC.CD</I>:<I>BE.ED</I>=<I>CA</I><SUP>2</SUP>:<I>AE</I><SUP>2</SUP></MATH>, +” <MATH>(<I>b</I>) <I>BC.CE</I>:<I>BD.DE</I>=<I>CA</I><SUP>2</SUP>:<I>AD</I><SUP>2</SUP></MATH>, +” <MATH>(<I>c</I>) <I>DC.CE</I>:<I>EB.BD</I>=<I>AC</I><SUP>2</SUP>:<I>AB</I><SUP>2</SUP></MATH>. +<pb n=407><head>THE <I>COLLECTION.</I> BOOK VII</head> +Two proofs are given of the first theorem. We will give the +first (Prop. 26) because it is a case of <I>theoretical analysis</I> +followed by <I>synthesis</I>. Describe a circle about <I>ABD</I>: produce +<I>EA, CA</I> to meet the circle again in <I>F, G</I>, and join <I>BF, FG</I>. +<p>Substituting <I>GC.CA</I> for <I>BC.CD</I> and <I>FE.EA</I> for <I>BE.ED</I>, +we have to inquire whether <MATH><I>GC.CA</I>:<I>CA</I><SUP>2</SUP>=<I>FE.EA</I>:<I>AE</I><SUP>2</SUP></MATH>, +i.e. whether <MATH><I>GC</I>:<I>CA</I>=<I>FE</I>:<I>EA</I></MATH>, +i.e. whether <MATH><I>GA</I>:<I>AC</I>=<I>FA</I>:<I>AE</I></MATH>, +i.e. whether the triangles <I>GAF, CAE</I> are similar or, in other +words, whether <I>GF</I> is parallel to <I>BC</I>. +<p>But <I>GF is</I> parallel to <I>BC</I>, because, the angles <I>BAC, DAE</I> +being supplementary, <MATH>∠<I>DAE</I>=∠<I>GAB</I>=∠<I>GFB</I></MATH>, while at the +same time <MATH>∠<I>DAE</I>=suppt. of ∠<I>FAD</I>=∠<I>FBD</I></MATH>. +<p>The synthesis is obvious. +<p>An alternative proof (Prop. 27) dispenses with the circle, +and only requires <I>EKH</I> to be drawn parallel to <I>CA</I> to meet +<I>AB, AD</I> in <I>H, K</I>. +<p>Similarly (Prop. 28) for case (<I>b</I>) it is only necessary to draw +<I>FG</I> through <I>D</I> parallel to <I>AC</I> meeting <I>BA</I> in <I>F</I> and <I>AE</I> +produced in <I>G</I>. +<FIG> +<p>Then, <MATH>∠<I>FAG</I>, ∠<I>ADF</I> (=∠<I>DAC</I>)</MATH> being both right angles, +<MATH><I>FD.DG</I>=<I>DA</I><SUP>2</SUP></MATH>. +<p>Therefore <MATH><I>CA</I><SUP>2</SUP>:<I>AD</I><SUP>2</SUP>=<I>CA</I><SUP>2</SUP>:<I>FD.DG</I>=(<I>CA</I>:<I>FD</I>).(<I>CA</I>:<I>DG</I>) +=(<I>BC</I>:<I>BD</I>).(<I>CE</I>:<I>DE</I>) +=<I>BC.CE</I>:<I>BD.DE</I></MATH>. +<p>In case (<I>c</I>) a circle is circumscribed to <I>ADE</I> cutting <I>AB</I> in <I>F</I> +and <I>AC</I> in <I>G</I>. Then, since <MATH>∠<I>FAD</I>=∠<I>GAE</I></MATH>, the arcs <I>DF, EG</I> +are equal and therefore <I>FG</I> is parallel to <I>DE</I>. The proof is +like that of case (<I>a</I>). +<pb n=408><head>PAPPUS OF ALEXANDRIA</head> +VI. Props. 37, 38. +<p>If <MATH><I>AB</I>:<I>BC</I>=<I>AD</I><SUP>2</SUP>:<I>DC</I><SUP>2</SUP></MATH>, whether <I>AB</I> be greater or less +than <I>AD</I>, then +<MATH><I>AB.BC</I>=<I>BD</I><SUP>2</SUP></MATH>. +<p>[<I>E</I> in the figure is a point such that <MATH><I>ED</I>=<I>CD</I></MATH>.] +<FIG> +<p>The algebraical equivalent is: If <MATH><I>a</I>/<I>c</I>=(<I>a</I>±<I>b</I>)<SUP>2</SUP>/(<I>b</I>±<I>c</I>)<SUP>2</SUP>, then <I>ac</I>=<I>b</I><SUP>2</SUP></MATH>. +<p>These lemmas are subsidiary to the next (Props. 39, 40), +being used in the first proofs of them. +<p>Props. 39, 40 prove the following: +<p>If <I>ACDEB</I> be a straight line, and if +<MATH><I>BA.AE</I>:<I>BD.DE</I>=<I>AC</I><SUP>2</SUP>:<I>CD</I><SUP>2</SUP>, +then <I>AB.BD</I>:<I>AE.ED</I>=<I>BC</I><SUP>2</SUP>:<I>CE</I><SUP>2</SUP>; +if, again, <I>AC.CB</I>:<I>AE.EB</I>=<I>CD</I><SUP>2</SUP>:<I>DE</I><SUP>2</SUP>, +then <I>EA.AC</I>:<I>CB.BE</I>=<I>AD</I><SUP>2</SUP>:<I>DB</I><SUP>2</SUP></MATH>. +<p>If <MATH><I>AB</I>=<I>a</I>, <I>BC</I>=<I>b</I>, <I>BD</I>=<I>c</I>, <I>BE</I>=<I>d</I></MATH>, the algebraic equiva- +lents are the following. +<p>If <MATH>(<I>a</I>(<I>a</I>-<I>d</I>))/(<I>c</I>(<I>c</I>-<I>d</I>))=(<I>a</I>-<I>b</I>)<SUP>2</SUP>/(<I>b</I>-<I>c</I>)<SUP>2</SUP>, + then <I>ac</I>/((<I>a</I>-<I>d</I>)(<I>c</I>-<I>d</I>))=<I>b</I><SUP>2</SUP>/(<I>b</I>-<I>d</I>)<SUP>2</SUP></MATH>; +and if <MATH>((<I>a</I>-<I>b</I>)<I>b</I>)/((<I>a</I>-<I>d</I>)<I>d</I>)=(<I>b</I>-<I>c</I>)<SUP>2</SUP>/(<I>c</I>-<I>d</I>)<SUP>2</SUP>, + then ((<I>a</I>-<I>d</I>)(<I>a</I>-<I>b</I>))/<I>bd</I>=(<I>a</I>-<I>c</I>)<SUP>2</SUP>/<I>e</I><SUP>2</SUP></MATH>. +<p>VII. Props. 41, 42, 43. +<p>If <MATH><I>AD.DC</I>=<I>BD.DE</I></MATH>, suppose that in Figures (1) and (2) +<FIG> +<MATH><I>k</I>=<I>AE</I>+<I>CB</I></MATH>, and in Figure (3) <MATH><I>k</I>=<I>AE</I>-<I>BC</I></MATH>, then +<MATH><I>k.AD</I>=<I>BA.AE</I>, <I>k.CD</I>=<I>BC.CE</I>, <I>k.BD</I>=<I>AB.BC</I>, +<I>k.DE</I>=<I>AE.EC</I></MATH>. +<pb n=409><head>THE <I>COLLECTION.</I> BOOK VII</head> +<p>The algebraical equivalents for Figures (1) and (2) re- +spectively may be written (if <MATH><I>a</I>=<I>AD</I>, <I>b</I>=<I>DC</I>, <I>c</I>=<I>BD</I>, +<I>d</I>=<I>DE</I></MATH>): +<p>If <MATH><I>ab</I>=<I>cd</I>, then (<I>a</I>±<I>d</I>+<I>c</I>±<I>b</I>) <I>a</I>=(<I>a</I>+<I>c</I>)(<I>a</I>±<I>d</I>), +(<I>a</I>±<I>d</I>+<I>c</I>±<I>b</I>) <I>b</I>=(<I>c</I>±<I>b</I>)(<I>b</I>+<I>d</I>), +(<I>a</I>±<I>d</I>+<I>c</I>±<I>b</I>) <I>c</I>=(<I>c</I>+<I>a</I>)(<I>c</I>±<I>b</I>), +(<I>a</I>±<I>d</I>+<I>c</I>±<I>b</I>) <I>d</I>=(<I>a</I>±<I>d</I>)(<I>d</I>+<I>b</I>)</MATH>. +<p>Figure (3) gives other varieties of sign. Troubles about +sign can be avoided by measuring all lengths in one direction +from an origin <I>O</I> outside the line. Thus, if <I>OA</I>=<I>a</I>, <I>OB</I>=<I>b</I>, +&c., the proposition may be as follows: +<p>If <MATH>(<I>d</I>-<I>a</I>)(<I>d</I>-<I>c</I>)=(<I>b</I>-<I>d</I>)(<I>e</I>-<I>d</I>) and <I>k</I>=<I>e</I>-<I>a</I>+<I>b</I>-<I>c</I></MATH>, +then <MATH><I>k</I>(<I>d</I>-<I>a</I>)=(<I>b</I>-<I>a</I>)(<I>e</I>-<I>a</I>), <I>k</I>(<I>d</I>-<I>c</I>)=(<I>b</I>-<I>c</I>)(<I>e</I>-<I>c</I>), +<I>k</I>(<I>b</I>-<I>d</I>)=(<I>b</I>-<I>a</I>)(<I>b</I>-<I>c</I>) and <I>k</I>(<I>e</I>-<I>d</I>)=(<I>e</I>-<I>a</I>)(<I>e</I>-<I>c</I>)</MATH>. +<p>VIII. Props. 45-56. +<p>More generally, if <MATH><I>AD.DC</I>=<I>BD.DE</I></MATH> and <MATH><I>k</I>=<I>AE</I>±<I>BC</I></MATH>, +then, if <I>F</I> be any point on the line, we have, according to the +position of <I>F</I> in relation to <I>A, B, C, D, E</I>, +<MATH>±<I>AF.FC</I>±<I>EF.FB</I>=<I>k.DF</I></MATH>. +<p>Algebraically, if <MATH><I>OA</I>=<I>a</I>, <I>OB</I>=<I>b...OF</I>=<I>x</I></MATH>, the equivalent +is: If <MATH>(<I>d</I>-<I>a</I>)(<I>d</I>-<I>c</I>)=(<I>b</I>-<I>d</I>)(<I>e</I>-<I>d</I>), and <I>k</I>=(<I>e</I>-<I>a</I>)+(<I>b</I>-<I>c</I>)</MATH>, +then <MATH>(<I>x</I>-<I>a</I>)(<I>x</I>-<I>c</I>)+(<I>x</I>-<I>e</I>)(<I>b</I>-<I>x</I>)=<I>k</I>(<I>x</I>-<I>d</I>)</MATH>. +<p>By making <MATH><I>x</I>=<I>a, b, c, e</I></MATH> successively in this equation, we +obtain the results of Props. 41-3 above. +<p>IX. Props. 59-64. +<p>In this group Props. 59, 60, 63 are lemmas required for the +remarkable propositions (61, 62, 64) in which Pappus investi- +gates ‘singular and minimum’ values of the ratio +<MATH><I>AP.PD</I>:<I>BP.PC</I></MATH>, +where (<I>A, D</I>), (<I>B, C</I>) are point-pairs on a straight line and <I>P</I> +is another point on the straight line. He finds, not only when +the ratio has the ‘singular and minimum (or maximum)’ value, +<pb n=410><head>PAPPUS OF ALEXANDRIA</head> +but also what the value is, for three different positions of <I>P</I> in +relation to the four given points. +<p>I will give, as an illustration, the first case, on account of its +elegance. It depends on the following <I>Lemma. AEB</I> being +a semicircle on <I>AB</I> as diameter, <I>C, D</I> any two points on <I>AB</I>, +and <I>CE, DF</I> being perpendicular to <I>AB</I>, let <I>EF</I> be joined and +<FIG> +produced, and let <I>BG</I> be drawn perpendicular to <I>EG</I>. To +prove that +<MATH><I>CB.BD</I>=<I>BG</I><SUP>2</SUP></MATH>, (1) +<MATH><I>AC.DB</I>=<I>FG</I><SUP>2</SUP></MATH>, (2) +<MATH><I>AD.BC</I>=<I>EG</I><SUP>2</SUP></MATH>. (3) +<p>Join <I>GC, GD, FB, EB, AF</I>. +<p>(1) Since the angles at <I>G, D</I> are right, <I>F, G, B, D</I> are concyclic. +Similarly <I>E, G, B, C</I> are concyclic. +<p>Therefore +<MATH>∠<I>BGD</I>=∠<I>BFD</I> +=∠<I>FAB</I> +=∠<I>FEB</I>, in the same segment of the semicircle, +=∠<I>GCB</I></MATH>, in the same segment of the circle <I>EGBC</I>. +<p>And the triangles <I>GCB, DGB</I> also have the angle <I>CBG</I> +common; therefore they are similar, and <MATH><I>CB</I>:<I>BG</I>=<I>BG</I>:<I>BD</I></MATH>, +or <MATH><I>CB.BD</I>=<I>BG</I><SUP>2</SUP></MATH>. +<p>(2) We have <MATH><I>AB.BD</I>=<I>BF</I><SUP>2</SUP></MATH>; +therefore, by subtraction, <MATH><I>AC.DB</I>=<I>BF</I><SUP>2</SUP>-<I>BG</I><SUP>2</SUP>=<I>FG</I><SUP>2</SUP></MATH>. +<p>(3) Similarly <MATH><I>AB.BC</I>=<I>BE</I><SUP>2</SUP></MATH>; +therefore, by subtraction, from the same result (1), +<MATH><I>AD.BC</I>=<I>BE</I><SUP>2</SUP>-<I>BG</I><SUP>2</SUP>=<I>EG</I><SUP>2</SUP></MATH>. +<p>Thus the lemma gives an extremely elegant construction for +squares equal to each of the three rectangles. +<pb n=411><head>THE <I>COLLECTION.</I> BOOK VII</head> +<p>Now suppose (<I>A, D</I>), (<I>B, C</I>) to be two point-pairs on a +straight line, and let <I>P</I>, another point on it, be determined by +the relation +<MATH><I>AB.BD</I>:<I>AC.CD</I>=<I>BP</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>; +then, says Pappus, the ratio <I>AP.PD</I>:<I>BP.PC</I> is singular and +a minimum, and is equal to +<MATH><I>AD</I><SUP>2</SUP>:(√(<I>AC.BD</I>)-√(<I>AB.CD</I>))<SUP>2</SUP></MATH>. +<p>On <I>AD</I> as diameter draw a circle, and draw <I>BF, CG</I> perpen- +dicular to <I>AD</I> on opposite sides. +<FIG> +<p>Then, by hypothesis, <MATH><I>AB.BD</I>:<I>AC.CD</I>=<I>BP</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>; +therefore <MATH><I>BF</I><SUP>2</SUP>:<I>CG</I><SUP>2</SUP>=<I>BP</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>, +or <MATH><I>BF</I>:<I>CG</I>=<I>BP</I>:<I>CP</I></MATH>, +whence the triangles <I>FBP, GCP</I> are similar and therefore +equiangular, so that <I>FPG</I> is a straight line. +<p>Produce <I>GC</I> to meet the circle in <I>H</I>, join <I>FH</I>, and draw <I>DK</I> +perpendicular to <I>FH</I> produced. Draw the diameter <I>FL</I> and +join <I>LH</I>. +<p>Now, by the lemma, <MATH><I>FK</I><SUP>2</SUP>=<I>AC.BD</I>, and <I>HK</I><SUP>2</SUP>=<I>AB.CD</I></MATH>; +therefore <MATH><I>FH</I>=<I>FK</I>-<I>HK</I>=√(<I>AC.BD</I>)-√(<I>AB.CD</I>)</MATH>. +<p>Since, in the triangles <I>FHL, PCG</I>, the angles at <I>H, C</I> are +right and <MATH>∠<I>FLH</I>=∠<I>PGC</I></MATH>, the triangles are similar, and +<MATH><I>GP</I>:<I>PC</I>=<I>FL</I>:<I>FH</I>=<I>AD</I>:<I>FH</I> +=<I>AD</I>:{√(<I>AC.BD</I>)-√(<I>AB.CD</I>)}</MATH>. +But <MATH><I>GP</I>:<I>PC</I>=<I>FP</I>:<I>PB</I></MATH>; +therefore <MATH><I>GP</I><SUP>2</SUP>:<I>PC</I><SUP>2</SUP>=<I>FP.PG</I>:<I>BP.PC</I> +=<I>AP.PD</I>:<I>BP.PC</I></MATH>. +<pb n=412><head>PAPPUS OF ALEXANDRIA</head> +<p>Therefore +<MATH><I>AP.PD</I>:<I>BP.PC</I>=<I>AD</I><SUP>2</SUP>:{√(<I>AC.BD</I>)-√(<I>AB.CD</I>)}<SUP>2</SUP></MATH>. +<p>The proofs of Props. 62 and 64 are different, the former +being long and involved. The results are: +<p>Prop. 62. If <I>P</I> is between <I>C</I> and <I>D</I>, and +<MATH><I>AD.DB</I>:<I>AC.CB</I>=<I>DP</I><SUP>2</SUP>:<I>PC</I><SUP>2</SUP></MATH>, +then the ratio <MATH><I>AP.PB</I>:<I>CP.PD</I></MATH> is singular and a minimum +and is equal to <MATH>{√(<I>AC.BD</I>)+√(<I>AD.BC</I>)}<SUP>2</SUP>:<I>DC</I><SUP>2</SUP></MATH>. +<p>Prop. 64. If <I>P</I> is on <I>AD</I> produced, and +<MATH><I>AB.BD</I>:<I>AC.CD</I>=<I>BP</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>, +then the ratio <MATH><I>AP.PD</I>:<I>BP.PC</I></MATH> is singular and a maximum, +and is equal to <MATH><I>AD</I><SUP>2</SUP>:{√(<I>AC.BD</I>)+√(<I>AB.CD</I>)}<SUP>2</SUP></MATH>. +<C>(<G>g</G>) <I>Lemmas on the</I> <G>*neu/seis</G> <I>of Apollonius</I>.</C> +<p>After a few easy propositions (e.g. the equivalent of the +proposition that, if <MATH><I>ax</I>+<I>x</I><SUP>2</SUP>=<I>by</I>+<I>y</I><SUP>2</SUP></MATH>, then, according as <I>a</I> > +or < <I>b</I>, <I>a</I>+<I>x</I> > or < <I>b</I>+<I>y</I>), Pappus gives (Prop. 70) the +lemma leading to the solution of the <G>neu=sis</G> with regard to +the rhombus (see pp. 190-2, above), and after that the solu- +tion by one Heraclitus of the same problem with respect to +a square (Props. 71, 72, pp. 780-4). The problem is, <I>Given a +square ABCD, to draw through B a straight line, meeting CD +in H and AD produced in E, such that HE is equal to a given +length</I>. +<p>The solution depends on a lemma to the effect that, if any +straight line <I>BHE</I> through <I>B</I> meets <I>CD</I> in <I>H</I> and <I>AD</I> pro- +<FIG> +duced in <I>E</I>, and if <I>EF</I> be drawn perpendicular to <I>BE</I> meeting +<I>BC</I> produced in <I>F</I>, then +<MATH><I>CF</I><SUP>2</SUP>=<I>BC</I><SUP>2</SUP>+<I>HE</I><SUP>2</SUP></MATH>. +<pb n=413><head>THE <I>COLLECTION.</I> BOOK VII</head> +<p>Draw <I>EG</I> perpendicular to <I>BF.</I> +<p>Then the triangles <I>BCH, EGF</I> are similar and since +<MATH><I>BG</I>=<I>EG</I></MATH>) equal in all respects; therefore <MATH><I>EF</I>=<I>BH</I></MATH>. +<p>Now <MATH><I>BF</I><SUP>2</SUP>=<I>BE</I><SUP>2</SUP>+<I>EF</I><SUP>2</SUP></MATH>, +or <MATH><I>BC.BF</I>+<I>BF.FC</I>=<I>BH.BE</I>+<I>BE.EH</I>+<I>EF</I><SUP>2</SUP></MATH>. +<p>But, the angles <I>HCF, HEF</I> being right, <I>H, C, F, E</I> are +concyclic, and <MATH><I>BC.BF</I>=<I>BH.BE</I></MATH>. +<p>Therefore, by subtraction, +<MATH><I>BF.FC</I>=<I>BE.EH</I>+<I>EF</I><SUP>2</SUP> +=<I>BE.EH</I>+<I>BH</I><SUP>2</SUP> +=<I>BH.HE</I>+<I>EH</I><SUP>2</SUP>+<I>BH</I><SUP>2</SUP> +=<I>EB.BH</I>+<I>EH</I><SUP>2</SUP> +=<I>FB.BC</I>+<I>EH</I><SUP>2</SUP></MATH>. +<p>Taking away the common part, <I>BC.CF</I>, we have +<MATH><I>CF</I><SUP>2</SUP>=<I>BC</I><SUP>2</SUP>+<I>EH</I><SUP>2</SUP></MATH>. +<p>Now suppose that we have to draw <I>BHE</I> through <I>B</I> in +such a way that <MATH><I>HE</I>=<I>k</I></MATH>. Since <I>BC, EH</I> are both given, we +have only to determine a length <I>x</I> such that <MATH><I>x</I><SUP>2</SUP>=<I>BC</I><SUP>2</SUP>+<I>k</I><SUP>2</SUP></MATH>, +produce <I>BC</I> to <I>F</I> so that <MATH><I>CF</I>=<I>x</I></MATH>, draw a semicircle on <I>BF</I> as +diameter, produce <I>AD</I> to meet the semicircle in <I>E</I>, and join +<I>BE. BE</I> is thus the straight line required. +<p>Prop. 73 (pp. 784-6) proves that, if <I>D</I> be the middle point +of <I>BC</I>, the base of an isosceles triangle <I>ABC</I>, then <I>BC</I> is the +shortest of all the straight lines through <I>D</I> terminated by +the straight lines <I>AB, AC</I>, and the nearer to <I>BC</I> is shorter than +the more remote. +<p>There follows a considerable collection of lemmas mostly +showing the equality of certain intercepts made on straight +lines through one extremity of the diameter of one of two +semicircles having their diameters in a straight line, either +one including or partly including the other, or wholly ex- +ternal to one another, on the same or opposite sides of the +diameter. +<pb n=414><head>PAPPUS OF ALEXANDRIA</head> +<p>I need only draw two figures by way of illustration. +<p>In the first figure (Prop. 83), <I>ABC, DEF</I> being the semi- +circles, <I>BEKC</I> is any straight line through <I>C</I> cutting both; +<I>FG</I> is made equal to <I>AD; AB</I> is joined; <I>GH</I> is drawn per- +pendicular to <I>BK</I> produced. It is required to prove that +<FIG> +<CAP>FIG. 1.</CAP> +<MATH><I>BE</I>=<I>KH</I></MATH>. (This is obvious when from <I>L</I>, the centre of the +semicircle <I>DEF, LM</I> is drawn perpendicular to <I>BK</I>.) If <I>E, K</I> +coincide in the point <I>M</I>′ of the semicircle so that <I>B</I>′<I>CH</I>′ is +a tangent, then <MATH><I>B</I>′<I>M</I>′=<I>M</I>′<I>H</I>′</MATH> (Props. 83, 84). +<p>In the second figure (Prop. 91) <I>D</I> is the centre of the +semicircle <I>ABC</I> and is also the extremity of the diameter +of the semicircle <I>DEF</I>. If <I>BEGF</I> be any straight line through +<FIG> +<CAP>FIG. 2.</CAP> +<I>F</I> cutting both semicircles, <MATH><I>BE=EG</I></MATH>. This is clear, since <I>DE</I> +is perpendicular to <I>BG</I>. +<p>The only problem of any difficulty in this section is Prop. +85 (p. 796). Given a semicircle <I>ABC</I> on the diameter <I>AC</I> +and a point <I>D</I> on the diameter, to draw a semicircle passing +through <I>D</I> and having its diameter along <I>DC</I> such that, if +<I>CEB</I> be drawn touching it at <I>E</I> and meeting the semicircle +<I>ABC</I> in <I>B, BE</I> shall be equal to <I>AD</I>. +<pb n=415><head>THE <I>COLLECTION.</I> BOOK VII</head> +<p>The problem is reduced to a problem contained in Apollo- +nius's <I>Determinate Section</I> thus. +<p>Suppose the problem solved by the semicircle <I>DEF, BE</I> +being equal to <I>AD</I>. Join <I>E</I> to the centre <I>G</I> of the semicircle +<FIG> +<I>DEF</I>. Produce <I>DA</I> to <I>H</I>, making <I>HA</I> equal to <I>AD</I>. Let <I>K</I> +be the middle point of <I>DC</I>. +<p>Since the triangles <I>ABC, GEC</I> are similar, +<MATH><I>AG</I><SUP>2</SUP>:<I>GC</I><SUP>2</SUP>=<I>BE</I><SUP>2</SUP>:<I>EC</I><SUP>2</SUP> +=<I>AD</I><SUP>2</SUP>:<I>EC</I><SUP>2</SUP>, by hypothesis, +=<I>AD</I><SUP>2</SUP>:<I>GC</I><SUP>2</SUP>-<I>DG</I><SUP>2</SUP> (since <I>DG</I>=<I>GE</I>) +=<I>AG</I><SUP>2</SUP>-<I>AD</I><SUP>2</SUP>:<I>DG</I><SUP>2</SUP> +=<I>HG.DG</I>:<I>DG</I><SUP>2</SUP> +=<I>HG</I>:<I>DG</I></MATH>. +<p>Therefore +<MATH><I>HG</I>:<I>DG</I>=<I>AD</I><SUP>2</SUP>:<I>GC</I><SUP>2</SUP>-<I>DG</I><SUP>2</SUP> +=<I>AD</I><SUP>2</SUP>:2<I>DC.GK</I></MATH>. +<p>Take a straight line <I>L</I> such that <MATH><I>AD</I><SUP>2</SUP>=<I>L</I>.2<I>DC</I></MATH>; +therefore <MATH><I>HG</I>:<I>DG</I>=<I>L</I>:<I>GK</I></MATH>, +or <MATH><I>HG.GK</I>=<I>L.DG</I></MATH>. +<p>Therefore, given the two straight lines <I>HD, DK</I> (or the +three points <I>H, D, K</I> on a straight line), we have to find +a point <I>G</I> between <I>D</I> and <I>K</I> such that +<MATH><I>HG.GK</I>=<I>L.DG</I></MATH>, +which is the second <I>epitagma</I> of the third Problem in the +<I>Determinate Section</I> of Apollonius, and therefore may be +taken as solved. (The problem is the equivalent of the +<pb n=416><head>PAPPUS OF ALEXANDRIA</head> +solution of a certain quadratic equation.) Pappus observes +that the problem is always possible (requires no <G>diorismo/s</G>), +and proves that it has only one solution. +<C>(<G>d</G>) <I>Lemmas on the treatise</I> ‘<I>On contacts</I>’ <I>by Apollonius</I>.</C> +<p>These lemmas are all pretty obvious except two, which are +important, one belonging to Book I of the treatise, and the other +to Book II. The two lemmas in question have already been set +out à propos of the treatise of Apollonius (see pp. 182-5, above). +As, however, there are several cases of the first (Props. 105, +107, 108, 109), one case (Prop. 108, pp. 836-8), different from +that before given, may be put down here: <I>Given a circle and +two points D, E within it, to draw straight lines through D, E +to a point A on the circumference in such a way that, if they +meet the circle again in B, C, BC shall be parallel to DE</I>. +<p>We proceed by analysis. Suppose the problem solved and +<I>DA, EA</I> drawn (‘inflected’) to <I>A</I> in such a way that, if <I>AD</I>, +<FIG> +<I>AE</I> meet the circle again in <I>B, C, +BC</I> is parallel to <I>DE</I>. +<p>Draw the tangent at <I>B</I> meeting +<I>ED</I> produced in <I>F</I>. +<p>Then <MATH>∠<I>FBD</I>=∠<I>ACB</I>=∠<I>AED</I></MATH>; +therefore <I>A, E, B, F</I> are concyclic, +and consequently +<MATH><I>FD.DE</I>=<I>AD.DB</I></MATH>. +<p>But the rectangle <I>AD.DB</I> is given, since it depends only +on the position of <I>D</I> in relation to the circle, and the circle +is given. +<p>Therefore the rectangle <I>FD.DE</I> is given. +<p>And <I>DE</I> is given; therefore <I>FD</I> is given, and therefore <I>F</I>. +<p>If follows that the tangent <I>FB</I> is given in position, and +therefore <I>B</I> is given. Therefore <I>BDA</I> is given and conse- +quently <I>AE</I> also. +<p>To solve the problem, therefore, we merely take <I>F</I> on <I>ED</I> +produced such that <MATH><I>FD.DE</I>=</MATH> the given rectangle made by +the segments of any chord through <I>D</I>, draw the tangent <I>FB</I>, +join <I>BD</I> and produce it to <I>A</I>, and lastly draw <I>AE</I> through to +<I>C</I>; <I>BC</I> is then parallel to <I>DE</I>. +<pb n=417><head>THE <I>COLLECTION.</I> BOOK VII</head> +<p>The other problem (Prop. 117, pp. 848-50) is, as we have +seen, equivalent to the following: <I>Given a circle and three +points D, E, F in a straight line external to it, to inscribe in +the circle a triangle ABC such that its sides pass severally +through the three points D, E, F</I>. For the solution, see +pp. 182-4, above. +<p>(<G>e</G>) The Lemmas to the <I>Plane Loci</I> of Apollonius (Props. +119-26, pp. 852-64) are mostly propositions in geometrical +algebra worked out by the methods of Eucl., Books II and VI. +We may mention the following: +<p>Prop. 122 is the well-known proposition that, if <I>D</I> be the +middle point of the side <I>BC</I> in a triangle <I>ABC</I>, +<MATH><I>BA</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>=2(<I>AD</I><SUP>2</SUP>+<I>DC</I><SUP>2</SUP>)</MATH>. +<p>Props. 123 and 124 are two cases of the same proposition, +the enunciation being marked by an expression which is also +found in Euclid's <I>Data</I>. Let <I>AB:BC</I> be a given ratio, and +<FIG> +let the rectangle <I>CA.AD</I> be given; then, if <I>BE</I> is a mean +proportional between <I>DB, BC</I>, ‘the square on <I>AE</I> is greater +by the rectangle <I>CA.AD</I> than in the ratio of <I>AB</I> to <I>BC</I> to the +square on <I>EC</I>’, by which is meant that +<MATH><I>AE</I><SUP>2</SUP>=<I>CA.AD</I>+<I>AB</I>/<I>BC</I> <I>EC</I><SUP>2</SUP></MATH>, +or <MATH>(<I>AE</I><SUP>2</SUP>-<I>CA.AD</I>):<I>EC</I><SUP>2</SUP>=<I>AB</I>:<I>BC</I></MATH>. +<p>The algebraical equivalent may be expressed thus (if <MATH><I>AB</I>=<I>a</I>, +<I>BC</I>=<I>b</I>, <I>AD</I>=<I>c</I>, <I>BE</I>=<I>x</I></MATH>): +<p>If <MATH><I>x</I>=√(<I>a</I>-<I>c</I>)<I>b</I></MATH>, then <MATH>((<I>a</I>∓<I>x</I>)<SUP>2</SUP>-(<I>a</I>-<I>b</I>)<I>c</I>)/(<I>x</I>∓<I>b</I>)<SUP>2</SUP>=<I>a</I>/<I>b</I></MATH>. +<p>Prop. 125 is remarkable: If <I>C, D</I> be two points on a straight +line <I>AB</I>, +<MATH><I>AD</I><SUP>2</SUP>+<I>AC</I>/<I>BC</I>.<I>DB</I><SUP>2</SUP>=<I>AC</I><SUP>2</SUP>+<I>AC.CB</I>+<I>AB</I>/<I>BC</I>.<I>CD</I><SUP>2</SUP></MATH>. +<pb n=418><head>PAPPUS OF ALEXANDRIA</head> +<p>This is equivalent to the general relation between four +points on a straight line discovered by Simson and therefore +wrongly known as Stewart's theorem: +<MATH><I>AD</I><SUP>2</SUP>.<I>BC</I>+<I>BD</I><SUP>2</SUP>.<I>CA</I>+<I>CD</I><SUP>2</SUP>.<I>AB</I>+<I>BC.CA.AB</I>=0</MATH>. +<p>(Simson discovered this theorem for the more general case +where <I>D</I> is a point outside the line <I>ABC</I>.) +<p>An algebraical equivalent is the identity +<MATH>(<I>d</I>-<I>a</I>)<SUP>2</SUP>(<I>b</I>-<I>c</I>)+(<I>d</I>-<I>b</I>)<SUP>2</SUP>(<I>c</I>-<I>a</I>)+(<I>d</I>-<I>c</I>)<SUP>2</SUP>(<I>a</I>-<I>b</I>) ++(<I>b</I>-<I>c</I>)(<I>c</I>-<I>a</I>)(<I>a</I>-<I>b</I>)=0</MATH>. +<p>Pappus's proof of the last-mentioned lemma is perhaps +worth giving. +<FIG> +<p><I>C, D</I> being two points on the straight line <I>AB</I>, take the +point <I>F</I> on it such that +<MATH><I>FD</I>:<I>DB</I>=<I>AC</I>:<I>CB</I></MATH>. (1) +<p>Then <MATH><I>FB</I>:<I>BD</I>=<I>AB</I>:<I>BC</I></MATH>, +and <MATH>(<I>AB</I>-<I>FB</I>):(<I>BC</I>-<I>BD</I>)=<I>AB</I>:<I>BC</I></MATH>, +or <MATH><I>AF</I>:<I>CD</I>=<I>AB</I>:<I>BC</I></MATH>, +and therefore +<MATH><I>AF.CD</I>:<I>CD</I><SUP>2</SUP>=<I>AB</I>:<I>BC</I></MATH>. (2) +<p>From (1) we derive +<MATH><I>AC</I>/<I>CB</I>.<I>DB</I><SUP>2</SUP>=<I>FD.DB</I></MATH>, +and from (2) +<MATH><I>AB</I>/<I>BC</I>.<I>CD</I><SUP>2</SUP>=<I>AF.CD</I></MATH>. +<p>We have now to prove that +<MATH><I>AD</I><SUP>2</SUP>+<I>BD.DF</I>=<I>AC</I><SUP>2</SUP>+<I>AC.CB</I>+<I>AF.CD</I></MATH>, +or <MATH><I>AD</I><SUP>2</SUP>+<I>BD.DF</I>=<I>CA.AB</I>+<I>AF.CD</I></MATH>, +<pb n=419><head>THE <I>COLLECTION.</I> BOOK VII</head> +i.e. (if <I>DA.AC</I> be subtracted from each side) +that <MATH><I>AD.DC</I>+<I>FD.DB</I>=<I>AC.DB</I>+<I>AF.CD</I></MATH>, +i.e. (if <I>AF.CD</I> be subtracted from each side) +that <MATH><I>FD.DC</I>+<I>FD.DB</I>=<I>AC.DB</I></MATH>, +or <MATH><I>FD.CB</I>=<I>AC.DB</I></MATH>: +which is true, since, by (1) above, <MATH><I>FD</I>:<I>DB</I>=<I>AC</I>:<I>CB</I></MATH>. +<C>(<G>z</G>) <I>Lemmas to the ‘Porisms’ of Euclid</I>.</C> +<p>The 38 Lemmas to the <I>Porisms</I> of Euclid form an important +collection which, of course, has been included in one form or +other in the ‘restorations’ of the original treatise. Chasles<note>Chasles, <I>Les trois livres de Porismes d'Euclide</I>, Paris, 1860, pp. 74 sq.</note> +in particular gives a classification of them, and we cannot +do better than use it in this place: ‘23 of the Lemmas relate +to rectilineal figures, 7 refer to the harmonic ratio of four +points, and 8 have reference to the circle. +<p>‘Of the 23 relating to rectilineal figures, 6 deal with the +quadrilateral cut by a transversal; 6 with the equality of +the anharmonic ratios of two systems of four points arising +from the intersections of four straight lines issuing from +one point with two other straight lines; 4 may be regarded as +expressing a property of the hexagon inscribed in two straight +lines; 2 give the relation between the areas of two triangles +which have two angles equal or supplementary; 4 others refer +to certain systems of straight lines; and the last is a case +of the problem of the <I>Cutting-off of an area</I>.’ +<p>The lemmas relating to the quadrilateral and the transversal +are 1, 2, 4, 5, 6 and 7 (Props. 127, 128, 130, 131, 132, 133). +Prop. 130 is a general proposition about any transversal +<FIG> +whatever, and is equivalent to one of the equations by which +we express the involution of six points. If <I>A</I>, <I>A</I>′; <I>B</I>, <I>B</I>′; +<I>C, C</I>′ be the points in which the transversal meets the pairs of +<pb n=420><head>PAPPUS OF ALEXANDRIA</head> +opposite sides and the two diagonals respectively, Pappus's +result is equivalent to +<MATH>(<I>AB</I>.<I>B</I>′<I>C</I>)/(<I>A</I>′<I>B</I>′.<I>BC</I>′)=<I>CA</I>/<I>C</I>′<I>A</I>′</MATH>. +Props. 127, 128 are particular cases in which the transversal +is parallel to a side; in Prop. 131 the transversal passes +through the points of concourse of opposite sides, and the +result is equivalent to the fact that the two diagonals divide +into proportional parts the straight line joining the points of +concourse of opposite sides; Prop. 132 is the particular case +of Prop. 131 in which the line joining the points of concourse +of opposite sides is parallel to a diagonal; in Prop. 133 the +transversal passes through one only of the points of concourse +of opposite sides and is parallel to a diagonal, the result being +<MATH><I>CA</I><SUP>2</SUP>=<I>CB.CB</I>′</MATH>. +<p>Props. 129, 136, 137, 140, 142, 145 (Lemmas 3, 10, 11, 14, 16, +19) establish the equality of the anharmonic ratios which +four straight lines issuing from a point determine on two +transversals; but both transversals are supposed to be drawn +from the same point on one of the four straight lines. Let +<FIG> +<I>AB, AC, AD</I> be cut by transversals <I>HBCD, HEFG</I>. It is +required to prove that +<MATH>(<I>HE.FG</I>)/(<I>HG.EF</I>)=(<I>HB.CD</I>)/(<I>HD.BC</I>)</MATH>. +Pappus gives (Prop. 129) two methods of proof which are +practically equivalent. The following is the proof ‘by com- +pound ratios’. +<p>Draw <I>HK</I> parallel to <I>AF</I> meeting <I>DA</I> and <I>AE</I> produced +<pb n=421><head>THE <I>COLLECTION.</I> BOOK VII</head> +in <I>K, L</I>; and draw <I>LM</I> parallel to <I>AD</I> meeting <I>GH</I> pro- +duced in <I>M</I>. +<p>Then <MATH>(<I>HE.FG</I>)/(<I>HG.EF</I>)=(<I>HE</I>/<I>EF</I>).(<I>FG</I>/<I>HG</I>)=(<I>LH</I>/<I>AF</I>).(<I>AF</I>/<I>HK</I>)=<I>LH</I>/<I>HK</I></MATH>. +<p>In exactly the same way, if <I>DH</I> produced meets <I>LM</I> in <I>M</I>′ +we prove that +<MATH>(<I>HB.CD</I>)/(<I>HD.BC</I>)=<I>LH</I>/<I>HK</I></MATH>. +Therefore <MATH>(<I>HE.FG</I>)/(<I>HG.EF</I>)=(<I>HB.CD</I>)/(<I>HD.BC</I>)</MATH>. +<p>(The proposition is proved for <I>HBCD</I> and any other trans- +versal not passing through <I>H</I> by applying our proposition +twice, as usual.) +<p>Props. 136, 142 are the reciprocal; Prop. 137 is a particular +case in which one of the transversals is parallel to one of the +straight lines, Prop. 140 a reciprocal of Prop. 137, Prop. 145 +another case of Prop. 129. +<p>The Lemmas 12, 13, 15, 17 (Props. 138, 139, 141, 143) are +equivalent to the property of the hexagon inscribed in two +straight lines, viz. that, if the vertices of a hexagon are +situate, three and three, on two straight lines, the points of +concourse of opposite sides are in a straight line; in Props. +138, 141 the straight lines are parallel, in Props. 139, 143 not +parallel. +<p>Lemmas 20, 21 (Props. 146, 147) prove that, when one angle +of one triangle is equal or supplementary to one angle of +another triangle, the areas of the triangles are in the ratios +of the rectangles contained by the sides containing the equal +or supplementary angles. +<p>The seven Lemmas 22, 23, 24, 25, 26, 27, 34 (Props. 148-53 +and 160) are propositions relating to the segments of a straight +line on which two intermediate points are marked. Thus: +<p>Props. 148, 150. +<p>If <I>C, D</I> be two points on <I>AB</I>, then +<p>(<I>a</I>) if <MATH>2<I>AB.CD</I>=<I>CB</I><SUP>2</SUP>, <I>AD</I><SUP>2</SUP>=<I>AC</I><SUP>2</SUP>+<I>DB</I><SUP>2</SUP></MATH>; +<FIG> +<p>(<I>b</I>) if <MATH>2<I>AC.BD</I>=<I>CD</I><SUP>2</SUP>, <I>AB</I><SUP>2</SUP>=<I>AD</I><SUP>2</SUP>+<I>CB</I><SUP>2</SUP></MATH>. +<pb n=422><head>PAPPUS OF ALEXANDRIA</head> +<p>Props. 149, 151. +<p>If <MATH><I>AB.BC</I>=<I>BD</I><SUP>2</SUP></MATH>, +then <MATH>(<I>AD</I>±<I>DC</I>)<I>BD</I>=<I>AD.DC</I>, +(<I>AD</I>±<I>DC</I>)<I>BC</I>=<I>DC</I><SUP>2</SUP></MATH>, +<FIG> +and <MATH>(<I>AD</I>±<I>DC</I>)<I>BA</I>=<I>AD</I><SUP>2</SUP></MATH>. +<p>Props. 152, 153. +<p>If <MATH><I>AB</I>:<I>BC</I>=<I>AD</I><SUP>2</SUP>:<I>DC</I><SUP>2</SUP></MATH>, then <MATH><I>AB.BC</I>=<I>BD</I><SUP>2</SUP></MATH>. +<FIG> +<p>Prop. 160. +<p>If <MATH><I>AB</I>:<I>BC</I>=<I>AD</I>:<I>DC</I></MATH>, then, if <I>E</I> be the middle point of <I>AC</I>, +<MATH><I>BE.ED</I>=<I>EC</I><SUP>2</SUP>, +<I>BD.DE</I>=<I>AD.DC</I>, +<I>EB.BD</I>=<I>AB.BC</I></MATH>. +<FIG> +<p>The Lemmas about the circle include the harmonic proper- +ties of the pole and polar, whether the pole is external to the +circle (Prop. 154) or internal (Prop. 161). Prop. 155 is a +problem, Given a segment of a circle on <I>AB</I> as base, to inflect +straight lines <I>AC</I>, <I>BC</I> to the segment in a given ratio to one +another. +<p>Prop. 156 is one which Pappus has already used earlier +in the <I>Collection</I>. It proves that the straight lines drawn +from the extremities of a chord (<I>DE</I>) to any point (<I>F</I>) of the +circumference divide harmonically the diameter (<I>AB</I>) perpen- +dicular to the chord. Or, if <I>ED</I>, <I>FK</I> be parallel chords, and +<I>EF</I>, <I>DK</I> meet in <I>G</I>, and <I>EK</I>, <I>DF</I> in <I>H</I>, then +<MATH><I>AH</I>:<I>HB</I>=<I>AG</I>:<I>GB</I></MATH>. +<pb n=423><head>THE <I>COLLECTION</I>. BOOK VII</head> +<p>Since <I>AB</I> bisects <I>DE</I> perpendicularly, <MATH>(arc <I>AE</I>)=(arc <I>AD</I>)</MATH> +and <MATH>∠<I>EFA</I>=∠<I>AFD</I></MATH>, or <I>AF</I> bisects the angle <I>EFD</I>. +<FIG> +<p>Since the angle <I>AFB</I> is right, <I>FB</I> bisects ∠<I>HFG</I>, the supple- +ment of ∠<I>EFD</I>. +<p>Therefore (Eucl. VI. 3) <MATH><I>GB</I>:<I>BH</I>=<I>GF</I>:<I>FH</I>=<I>GA</I>:<I>AH</I></MATH>, +and, alternately and inversely, <MATH><I>AH</I>:<I>HB</I>=<I>AG</I>:<I>GB</I></MATH>. +<p>Prop. 157 is remarkable in that (without any mention of +a conic) it is practically identical with Apollonius's <I>Conics</I> +III. 45 about the foci of a central conic. Pappus's theorem +is as follows. Let <I>AB</I> be the diameter of a semicircle, and +<FIG> +from <I>A</I>, <I>B</I> let two straight lines <I>AE</I>, <I>BD</I> be drawn at right +angles to <I>AB</I>. Let any straight line <I>DE</I> meet the two perpen- +diculars in <I>D</I>, <I>E</I> and the semicircle in <I>F</I>. Further, let <I>FG</I> be +drawn at right angles to <I>DE</I>, meeting <I>AB</I> produced in <I>G</I>. +<p>It is to be proved that +<MATH><I>AG.GB</I>=<I>AE.BD</I></MATH> +<p>Since <I>F</I>, <I>D</I>, <I>G</I>, <I>B</I> are concyclic, <MATH>∠<I>BDG</I>=∠<I>BFG</I></MATH>. +<pb n=424><head>PAPPUS OF ALEXANDRIA</head> +<p>And, since <I>AFB</I>, <I>EFG</I> are both right angles, <MATH>∠<I>BFG</I>=∠<I>AFE</I></MATH>. +<p>But, since <I>A</I>, <I>E</I>, <I>G</I>, <I>F</I> are concyclic, <MATH>∠<I>AFE</I>=∠<I>AGE</I></MATH>. +<p>Therefore <MATH>∠<I>BDG</I>=∠<I>AGE</I></MATH>; +and the right-angled triangles <I>DBG</I>, <I>GAE</I> are similar. +<p>Therefore <MATH><I>AG</I>:<I>AE</I>=<I>BD</I>:<I>GB</I></MATH>, +or <MATH><I>AG.GB</I>=<I>AE.DB</I></MATH>. +<p>In Apollonius <I>G</I> and the corresponding point <I>G</I>′ on <I>BA</I> +produced which is obtained by drawing <I>F</I>′<I>G</I>′ perpendicular to +<I>ED</I> (where <I>DE</I> meets the circle again in <I>F</I>′) are the foci +of a central conic (in this case a hyperbola), and <I>DE</I> is any +tangent to the conic; the rectangle <I>AE.BD</I> is of course equal +to the square on half the conjugate axis. +<p>(<G>h</G>) The Lemmas to the <I>Conics</I> of Apollonius (pp. 918-1004) +do not call for any extended notice. There are a large number +of propositions in geometrical algebra of the usual kind, +relating to the segments of a straight line marked by a number +of points on it; propositions about lines divided into proportional +segments and about similar figures; two propositions +relating to the construction of a hyperbola (Props. 204, 205) +and a proposition (208) proving that two hyperbolas with the +same asymptotes do not meet one another. There are also +two propositions (221, 222) equivalent to an obvious trigono- +<FIG> +metrical formula. Let <I>ABCD</I> be a rectangle, and let any +straight line through <I>A</I> meet <I>DC</I> produced in <I>E</I> and <I>BC</I> +(produced if necessary) in <I>F</I>. +<p>Then <MATH><I>EA.AF</I>=<I>ED.DC</I>+<I>CB.BF</I></MATH>. +<pb n=425><head>THE <I>COLLECTION</I>. BOOK VII</head> +<p>For <MATH><I>EA</I><SUP>2</SUP>+<I>AF</I><SUP>2</SUP>=<I>ED</I><SUP>2</SUP>+<I>DA</I><SUP>2</SUP>+<I>AB</I><SUP>2</SUP>+<I>BF</I><SUP>2</SUP> +=<I>ED</I><SUP>2</SUP>+<I>BC</I><SUP>2</SUP>+<I>CD</I><SUP>2</SUP>+<I>BF</I><SUP>2</SUP></MATH>. +<p>Also <MATH><I>EA</I><SUP>2</SUP>+<I>AF</I><SUP>2</SUP>=<I>EF</I><SUP>2</SUP>+2<I>EA.AF</I></MATH>. +<p>Therefore +<MATH>2<I>EA.AF</I>=<I>EA</I><SUP>2</SUP>+<I>AF</I><SUP>2</SUP>-<I>EF</I><SUP>2</SUP> +=<I>ED</I><SUP>2</SUP>+<I>BC</I><SUP>2</SUP>+<I>CD</I><SUP>2</SUP>+<I>BF</I><SUP>2</SUP>-<I>EF</I><SUP>2</SUP> +=(<I>ED</I><SUP>2</SUP>+<I>CD</I><SUP>2</SUP>)+(<I>BC</I><SUP>2</SUP>+<I>BF</I><SUP>2</SUP>)-<I>EF</I><SUP>2</SUP> +=<I>EC</I><SUP>2</SUP>+2<I>ED.DC</I>+<I>CF</I><SUP>2</SUP>+2<I>CB.BF</I>-<I>EF</I><SUP>2</SUP> +=2<I>ED.DC</I>+2<I>CB.BF</I></MATH>; +i.e. <MATH><I>EA.AF</I>=<I>ED.DC</I>+<I>CB.BF</I></MATH>. +<p>This is equivalent to <MATH>sec <G>q</G> cosec <G>q</G>=tan <G>q</G>+cot <G>q</G></MATH>. +<p>The algebraical equivalents of some of the results obtained +by the usual geometrical algebra may be added. +<p>Props. 178, 179, 192-4. +<MATH>(<I>a</I>+2<I>b</I>)<I>a</I>+(<I>b</I>+<I>x</I>)(<I>b</I>-<I>x</I>)=(<I>a</I>+<I>b</I>+<I>x</I>)(<I>a</I>+<I>b</I>-<I>x</I>)</MATH>. +<p>Prop. 195. <MATH>4<I>a</I><SUP>2</SUP>=2{(<I>a</I>-<I>x</I>)(<I>a</I>+<I>x</I>)+(<I>a</I>-<I>y</I>)(<I>a</I>+<I>y</I>)+<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>}</MATH>. +<p>Prop. 196. +<MATH>(<I>a</I>+<I>b</I>-<I>x</I>)<SUP>2</SUP>+(<I>a</I>+<I>b</I>+<I>x</I>)<SUP>2</SUP>=(<I>x</I>-<I>b</I>)<SUP>2</SUP>+(<I>x</I>+<I>b</I>)<SUP>2</SUP>+2(<I>a</I>+2<I>b</I>)<I>a</I></MATH>. +<p>Props. 197, 199, 198. +<MATH><BRACE><note>then <I>x</I>=<I>y</I>.</note> +<p>If (<I>x</I>+<I>y</I>+<I>a</I>)<I>a</I>+<I>x</I><SUP>2</SUP>=(<I>a</I>+<I>x</I>)<SUP>2</SUP>, +or if (<I>x</I>+<I>y</I>+<I>a</I>)<I>a</I>+<I>x</I><SUP>2</SUP>=(<I>a</I>+<I>y</I>)<SUP>2</SUP>, +or if (<I>x</I>+<I>y</I>-<I>a</I>)<I>a</I>+(<I>x</I>-<I>a</I>)<SUP>2</SUP>=<I>y</I><SUP>2</SUP>,</BRACE></MATH> +<p>Props. 200, 201. If <MATH>(<I>a</I>+<I>b</I>)<I>x</I>=<I>b</I><SUP>2</SUP></MATH>, then <MATH>(2<I>b</I>+<I>a</I>)/<I>a</I>=(<I>b</I>+<I>x</I>)/(<I>b</I>-<I>x</I>)</MATH> and +<MATH>(2<I>b</I>+<I>a</I>)<I>a</I>=(<I>a</I>+<I>b</I>)(<I>a</I>+<I>b</I>-<I>x</I>)</MATH>. +<p>Prop. 207. If <MATH>(<I>a</I>+<I>b</I>)<I>b</I>=2<I>a</I><SUP>2</SUP></MATH>, then <MATH><I>a</I>=<I>b</I></MATH>. +<p>(<G>q</G>) The two Lemmas to the <I>Surface-Loci</I> of Euclid have +already been mentioned as significant. The first has the +appearance of being a general enunciation, such as Pappus +<pb n=426><head>PAPPUS OF ALEXANDRIA</head> +is fond of giving, to cover a class of propositions. The +enunciation may be translated as follows: ‘If <I>AB</I> be a straight +line, and <I>CD</I> a straight line parallel to a straight line given in +position, and if the ratio <I>AD.DB</I>:<I>DC</I><SUP>2</SUP> be given, the point <I>C</I> +lies on a conic section. If now <I>AB</I> be no longer given in +position, and the points <I>A</I>, <I>B</I> are no longer given but lie +(respectively) on straight lines <I>AE</I>, <I>EB</I> given in position, the +point <I>C</I> raised above (the plane containing <I>AE</I>, <I>EB</I>) lies on +a surface given in position. And this was proved.’ Tannery +<FIG> +was the first to explain this intelligibly; +and his interpretation only requires the +very slight change in the text of sub- +stituting <G>eu)qei/ais</G> for <G>eu)qei=a</G> in the phrase +<G>ge/nhtai de\ pro\s qe/sei eu)qei=a tai=s</G> <I>AE</I>, <I>EB</I>. +It is not clear whether, when <I>AB</I> ceases +to be given in <I>position</I>, it is still given +in <I>length</I>. If it is given in <I>length</I> and <I>A</I>, <I>B</I> move on the lines +<I>AE</I>, <I>EB</I> respectively, the surface which is the locus of <I>C</I> is +a complicated one such as Euclid would hardly have been +in a position to investigate. But two possible cases are +indicated which he may have discussed, (1) that in which <I>AB</I> +moves always parallel to itself and varies in length accord- +ingly, (2) that in which the two lines on which <I>A</I>, <I>B</I> move are +parallel instead of meeting at a point. The loci in these two +cases would of course be a cone and a cylinder respectively. +<p>The second Lemma is still more important, since it is the +first statement on record of the focus-directrix property of +the three conic sections. The proof, after Pappus, has been +set out above (pp. 119-21). +<C>(<G>i</G>) <I>An unallocated Lemma</I></C>. +<p>Book VII ends (pp. 1016-18) with a lemma which is not +given under any particular treatise belonging to the <I>Treasury +of Analysis</I>, but is simply called ‘Lemma to the <G>*)analuo/menos</G>’. +If <I>ABC</I> be a triangle right-angled at <I>B</I>, and <I>AB</I>, <I>BC</I> be +divided at <I>F</I>, <I>G</I> so that <MATH><I>AF</I>:<I>FB</I>=<I>BG</I>:<I>GC</I>=<I>AB</I>:<I>BC</I></MATH>, and +if <I>AEG</I>, <I>CEF</I> be joined and <I>BE</I> joined and produced to <I>D</I>, +then shall <I>BD</I> be perpendicular to <I>AC</I>. +<p>The text is unsatisfactory, for there is a long interpolation +containing an attempt at a proof by <I>reductio ad absurdum</I>; +<pb n=427><head>THE <I>COLLECTION</I>. BOOKS VII, VIII</head> +but the genuine proof is indicated, although it breaks off +before it is quite complete. +<p>Since <MATH><I>AF</I>:<I>FB</I>=<I>BG</I>:<I>GC</I>, +<I>AB</I>:<I>FB</I>=<I>BC</I>:<I>GC</I></MATH>, +or <MATH><I>AB</I>:<I>BC</I>=<I>FB</I>:<I>GC</I></MATH>. +But, by hypothesis, <MATH><I>AB</I>:<I>BC</I>=<I>BG</I>:<I>GC</I></MATH>; +therefore <MATH><I>BF</I>=<I>BG</I></MATH>. +<p>From this point the proof apparently proceeded by analysis. +‘Suppose it done’ (<G>gegone/tw</G>), i.e. suppose the proposition true, +and <I>BED</I> perpendicular to <I>AC</I>. +<FIG> +<p>Then, by similarity of triangles, <MATH><I>AD</I>:<I>DB</I>=<I>AB</I>:<I>BC</I></MATH>; +therefore <MATH><I>AF</I>:<I>FB</I>=<I>AD</I>:<I>DB</I></MATH>, and consequently the angle +<I>ADB</I> is bisected by <I>DF</I>. +<p>Similarly the angle <I>BDC</I> is bisected by <I>DG</I>. +<p>Therefore each of the angles <I>BDF</I>, <I>BDG</I> is half a right +angle, and consequently the angle <I>FDG</I> is a right angle. +<p>Therefore <I>B</I>, <I>G</I>, <I>D</I>, <I>F</I> are concyclic; and, since the angles +<I>FDB</I>, <I>BDG</I> are equal, <MATH><I>FB</I>=<I>BG</I></MATH>. +<p>This is of course the result above proved. +<p>Evidently the interpolator tried to clinch the argument by +proving that the angle <I>BDA</I> could not be anything but a right +angle. +<C>Book VIII.</C> +<p>Book VIII of the <I>Collection</I> is mainly on mechanics, although +it contains, in addition, some propositions of purely geometrical +interest. +<pb n=428><head>PAPPUS OF ALEXANDRIA</head> +<C><I>Historical preface</I></C>. +<p>It begins with an interesting preface on the claim of +theoretical mechanics, as distinct from the merely practical +or industrial, to be regarded as a mathematical subject. +Archimedes, Philon, Heron of Alexandria are referred to as +the principal exponents of the science, while Carpus of Antioch +is also mentioned as having applied geometry to ‘certain +(practical) arts’. +<p>The date of Carpus is uncertain, though it is probable that +he came after Geminus; the most likely date seems to be the +first or second century A.D. Simplicius gives the authority of +Iamblichus for the statement that Carpus squared the circle +by means of a certain curve, which he simply called a curve +generated by a double motion.<note>Simplicius on Arist. <I>Categ.</I>, p. 192, Kalbfleisch.</note> Proclus calls him ‘Carpus the +writer on mechanics (<G>o( mhxaniko/s</G>)’, and quotes from a work of +his on Astronomy some remarks about the relation between +problems and theorems and the ‘priority in order’ of the +former.<note>Proclus on Eucl. I, pp. 241-3.</note> Proclus also mentions him as having held that an +angle belongs to the category of <I>quantity</I> (<G>poso/n</G>), since it +represents a sort of ‘distance’ between the two lines forming +it, this distance being ‘extended one way’ (<G>e)f) e(\n diestw/s</G>) +though in a different sense from that in which a line represents +extension one way, so that Carpus's view appeared to be ‘the +greatest possible paradox’<note><I>Ib.</I>, pp. 125. 25-126. 6.</note>; Carpus seems in reality to have +been anticipating the modern view of an angle as representing +<I>divergence</I> rather than distance, and to have meant by <G>e)f) e(\n</G> +<I>in one sense</I> (rotationally), as distinct from one way or in one +dimension (linearly). +<p>Pappus tells us that Heron distinguished the logical, i.e. +theoretical, part of mechanics from the practical or manual +(<G>xeirourgiko/n</G>), the former being made up of geometry, arith- +metic, astronomy and physics, the latter of work in metal, +architecture, carpentering and painting; the man who had +been trained from his youth up in the <I>sciences</I> aforesaid as well +as practised in the said <I>arts</I> would naturally prove the best +architect and inventor of mechanical devices, but, as it is diffi- +cult or impossible for the same person to do both the necessary +<pb n=429><head>THE <I>COLLECTION</I>. BOOK VIII</head> +mathematics and the practical work, he who has not the former +must perforce use the resources which practical experience in +his particular art or craft gives him. Other varieties of +mechanical work included by the ancients under the general +term mechanics were (1) the use of the mechanical powers, +or devices for moving or lifting great weights by means of +a small force, (2) the construction of engines of war for +throwing projectiles a long distance, (3) the pumping of water +from great depths, (4) the devices of ‘wonder-workers’ +(<G>qaumasiourgoi/</G>), some depending on pneumatics (like Heron +in the <I>Pneumatica</I>), some using strings, &c., to produce move- +ments like those of living things (like Heron in ‘Automata and +Balancings’), some employing floating bodies (like Archimedes +in ‘Floating Bodies’), others using water to measure time +(like Heron in his ‘Water-clocks’), and lastly ‘sphere-making’, +or the construction of mechanical imitations of the movements +of the heavenly bodies with the uniform circular motion of +water as the motive power. Archimedes, says Pappus, was +held to be the one person who had understood the cause and +the reason of all these various devices, and had applied his +extraordinarily versatile genius and inventiveness to all the +purposes of daily life, and yet, although this brought him +unexampled fame the world over, so that his name was on +every one's lips, he disdained (according to Carpus) to write +any mechanical work save a tract on sphere-making, but +diligently wrote all that he could in a small compass of the +most advanced parts of geometry and of subjects connected +with arithmetic. Carpus himself, says Pappus, as well as +others applied geometry to practical arts, and with reason: +‘for geometry is in no wise injured, nay it is by nature +capable of giving substance to many arts by being associated +with them, and, so far from being injured, it may be said, +while itself advancing those arts, to be honoured and adorned +by them in return.’ +<C><I>The object of the Book</I></C>. +<p>Pappus then describes the object of the Book, namely +to set out the propositions which the ancients established by +geometrical methods, besides certain useful theorems dis- +covered by himself, but in a shorter and clearer form and +<pb n=430><head>PAPPUS OF ALEXANDRIA</head> +in better logical sequence than his predecessors had attained. +The sort of questions to be dealt with are (1) a comparison +between the force required to move a given weight along +a horizontal plane and that required to move the same weight +upwards on an inclined plane, (2) the finding of two mean +proportionals between two unequal straight lines, (3) given +a toothed wheel with a certain number of teeth, to find the +diameter of, and to construct, another wheel with a given num- +ber of teeth to work on the former. Each of these things, he says, +will be clearly understood in its proper place if the principles +on which the ‘centrobaric doctrine’ is built up are first set out. +It is not necessary, he adds, to define what is meant by ‘heavy’ +and ‘light’ or upward and downward motion, since these +matters are discussed by Ptolemy in his <I>Mathematica</I>; but +the notion of the centre of gravity is so fundamental in the +whole theory of mechanics that it is essential in the first +place to explain what is meant by the ‘centre of gravity’ +of any body. +<C><I>On the centre of gravity</I></C>. +<p>Pappus then defines the centre of gravity as ‘the point +within a body which is such that, if the weight be conceived +to be suspended from the point, it will remain at rest in any +position in which it is put’.<note>Pappus, viii, p. 1030. 11-13.</note> The method of determining the +point by means of the intersection, first of planes, and then of +straight lines, is next explained (chaps. 1, 2), and Pappus then +proves (Prop. 2) a proposition of some difficulty, namely that, +if <I>D</I>, <I>E</I>, <I>F</I> be points on the sides <I>BC</I>, <I>CA</I>, <I>AB</I> of a triangle <I>ABC</I> +such that +<MATH><I>BD</I>:<I>DC</I>=<I>CE</I>:<I>EA</I>=<I>AF</I>:<I>FB</I></MATH>, +then the centre of gravity of the triangle <I>ABC</I> is also the +centre of gravity of the triangle <I>DEF</I>. +<p>Let <I>H</I>, <I>K</I> be the middle points of <I>BC</I>, <I>CA</I> respectively; +join <I>AH</I>, <I>BK</I>. Join <I>HK</I> meeting <I>DE</I> in <I>L</I>. +<p>Then <I>AH</I>, <I>BK</I> meet in <I>G</I>, the centre of gravity of the +triangle <I>ABC</I>, and <MATH><I>AG</I>=2<I>GH</I>, <I>BG</I>=2<I>GK</I></MATH>, so that +<MATH><I>CA</I>:<I>AK</I>=<I>AB</I>:<I>HK</I>=<I>BG</I>:<I>GK</I>=<I>AG</I>:<I>GH</I></MATH>. +<pb n=431><head>THE <I>COLLECTION</I>. BOOK VIII</head> +<p>Now, by hypothesis, +<MATH><I>CE</I>:<I>EA</I>=<I>BD</I>:<I>DC</I></MATH>, +whence <MATH><I>CA</I>:<I>AE</I>=<I>BC</I>:<I>CD</I></MATH>, +and, if we halve the antecedents, +<MATH><I>AK</I>:<I>AE</I>=<I>HC</I>:<I>CD</I></MATH>; +therefore <MATH><I>AK</I>:<I>EK</I>=<I>HC</I>:<I>HD</I></MATH> or <MATH><I>BH</I>:<I>HD</I></MATH>, +<FIG> +whence, <I>componendo</I>, <MATH><I>CE</I>:<I>EK</I>=<I>BD</I>:<I>DH</I></MATH>. (1) +<p>But <MATH><I>AF</I>:<I>FB</I>=<I>BD</I>:<I>DC</I>=(<I>BD</I>:<I>DH</I>).(<I>DH</I>:<I>DC</I>) +=(<I>CE</I>:<I>EK</I>).(<I>DH</I>:<I>DC</I>)</MATH>. (2) +<p>Now, <I>ELD</I> being a transversal cutting the sides of the +triangle <I>KHC</I>, we have +<MATH><I>HL</I>:<I>KL</I>=(<I>CE</I>:<I>EK</I>).(<I>DH</I>:<I>DC</I>)</MATH>. (3) +[This is ‘Menelaus's theorem’; Pappus does not, however, +quote it, but proves the relation <I>ad hoc</I> in an added lemma by +drawing <I>CM</I> parallel to <I>DE</I> to meet <I>HK</I> produced in <I>M</I>. The +proof is easy, for <MATH><I>HL</I>:<I>LK</I>=(<I>HL</I>:<I>LM</I>).(<I>LM</I>:<I>LK</I>) +=(<I>HD</I>:<I>DC</I>).(<I>CE</I>:<I>EK</I>)</MATH>.] +<p>It follows from (2) and (3) that +<MATH><I>AF</I>:<I>FB</I>=<I>HL</I>:<I>LK</I></MATH>, +and, since <I>AB</I> is parallel to <I>HK</I>, and <I>AH</I>, <I>BK</I> are straight +lines meeting in <I>G</I>, <I>FGL</I> is a straight line. +<p>[This is proved in another easy lemma by <I>reductio ad +absurdum</I>.] +<pb n=432><head>PAPPUS OF ALEXANDRIA</head> +<p>We have next to prove that <MATH><I>EL</I>=<I>LD</I></MATH>. +<p>Now [again by ‘Menelaus's theorem’, proved <I>ad hoc</I> by +drawing <I>CN</I> parallel to <I>HK</I> to meet <I>ED</I> produced in <I>N</I>] +<MATH><I>EL</I>:<I>LD</I>=(<I>EK</I>:<I>KC</I>).(<I>CH</I>:<I>HD</I>)</MATH>. (4) +<p>But, by (1) above, <MATH><I>CE</I>:<I>EK</I>=<I>BD</I>:<I>DH</I></MATH>; +therefore <MATH><I>CK</I>:<I>KE</I>=<I>BH</I>:<I>HD</I>=<I>CH</I>:<I>HD</I></MATH>, +so that <MATH>(<I>EK</I>:<I>KC</I>).(<I>CH</I>:<I>HD</I>)=1</MATH>, and therefore, from (4), +<MATH><I>EL</I>=<I>LD</I></MATH>. +<p>It remains to prove that <MATH><I>FG</I>=2<I>GL</I></MATH>, which is obvious by +parallels, since <MATH><I>FG</I>:<I>GL</I>=<I>AG</I>:<I>GH</I>=2:1</MATH>. +<p>Two more propositions follow with reference to the centre +of gravity. The first is, Given a rectangle with <I>AB</I>, <I>BC</I> as +adjacent sides, to draw from <I>C</I> a straight line meeting the side +opposite <I>BC</I> in a point <I>D</I> such that, if the trapezium <I>ADCB</I> is +hung from the point <I>D</I>, it will rest with <I>AD</I>, <I>BC</I> horizontal. +<FIG> +In other words, the centre of gravity must be in <I>DL</I> drawn +perpendicular to <I>BC</I>. Pappus proves by analysis that +<MATH><I>CL</I><SUP>2</SUP>=3<I>BL</I><SUP>2</SUP></MATH>, so that the problem is reduced to that of +dividing <I>BC</I> into parts <I>BL</I>, <I>LC</I> such that this relation holds. +The latter problem is solved (Prop. 6) by taking a point, +say <I>X</I>, in <I>CB</I> such that <MATH><I>CX</I>=3<I>XB</I></MATH>, describing a semicircle on +<I>BC</I> as diameter and drawing <I>XY</I> at right angles to <I>BC</I> to +meet the semicircle in <I>Y</I>, so that <MATH><I>XY</I><SUP>2</SUP>=(3/16)<I>BC</I><SUP>2</SUP></MATH>, and then +dividing <I>CB</I> at <I>L</I> so that +<MATH><I>CL</I>:<I>LB</I>=<I>CX</I>:<I>XY</I>(=3/4:1/4√3=√3:1)</MATH>. +<p>The second proposition is this (Prop. 7). Given two straight +lines <I>AB</I>, <I>AC</I>, and <I>B</I> a fixed point on <I>AB</I>, if <I>CD</I> be drawn +<pb n=433><head>THE <I>COLLECTION</I>. BOOK VIII</head> +with its extremities on <I>AC</I>, <I>AB</I> and so that <I>AC</I>:<I>BD</I> is a given +ratio, then the centre of gravity of the triangle <I>ADC</I> will lie +on a straight line. +<p>Take <I>E</I>, the middle point of <I>AC</I>, and <I>F</I> a point on <I>DE</I> such +that <MATH><I>DF</I>=2<I>FE</I></MATH>. Also let <I>H</I> be a point on <I>BA</I> such that +<MATH><I>BH</I>=2<I>HA</I></MATH>. Draw <I>FG</I> parallel to <I>AC</I>. +Then <MATH><I>AG</I>=(1/3)<I>AD</I></MATH>, and <MATH><I>AH</I>=(1/3)<I>AB</I></MATH>; +therefore <MATH><I>HG</I>=(1/3)<I>BD</I></MATH>. +<p>Also <MATH><I>FG</I>=(2/3)<I>AE</I>=(1/3)<I>AC</I></MATH>. Therefore, +since the ratio <I>AC</I>:<I>BD</I> is given, the +ratio <I>GH</I>:<I>GF</I> is given. +<FIG> +<p>And the angle <I>FGH</I> (=<I>A</I>) is given; +therefore the triangle <I>FGH</I> is given in +species, and consequently the angle <I>GHF</I> +is given. And <I>H</I> is a given point. +Therefore <I>HF</I> is a given straight line, and it contains the +centre of gravity of the triangle <I>ADC</I>. +<C><I>The inclined plane</I></C>. +<p>Prop. 8 is on the construction of a plane at a given inclination +to another plane parallel to the horizon, and with this +Pappus leaves theory and proceeds to the practical part. +Prop. 9 (p. 1054. 4 sq.) investigates the problem ‘Given +a weight which can be drawn along a plane parallel to the +horizon by a given force, and a plane inclined to the horizon +at a given angle, to find the force required to draw the weight +upwards on the inclined plane’. This seems to be the first +or only attempt in ancient times to investigate motion on +an inclined plane, and as such it is curious, though of no +value. +<p>Let <I>A</I> be the weight which can be moved by a force <I>C</I> along +a horizontal plane. Conceive a sphere with weight equal to <I>A</I> +placed in contact at <I>L</I> with the given inclined plane; the circle +<I>OGL</I> represents a section of the sphere by a vertical plane +passing through <I>E</I> its centre and <I>LK</I> the line of greatest slope +drawn through the point <I>L</I>. Draw <I>EGH</I> horizontal and therefore +parallel to <I>MN</I> in the plane of section, and draw <I>LF</I> +perpendicular to <I>EH</I>. Pappus seems to regard the plane +as rough, since he proceeds to make a system in equilibrium +<pb n=434><head>PAPPUS OF ALEXANDRIA</head> +about <I>FL</I> as if <I>L</I> were the fulcrum of a lever. Now the +weight <I>A</I> acts vertically downwards along a straight line +through <I>E</I>. To balance it, Pappus supposes a weight <I>B</I> +attached with its centre of gravity at <I>G</I>. +<FIG> +<p>Then <MATH><I>A</I>:<I>B</I>=<I>GF</I>:<I>EF</I> +=(<I>EL</I>-<I>EF</I>):<I>EF</I> +[=(1-sin <G>q</G>):sin <G>q</G></MATH>, +where <MATH>∠<I>KMN</I>=<G>q</G></MATH>]; +and, since ∠<I>KMN</I> is given, the ratio <I>EF</I>:<I>EL</I>, +and therefore the ratio (<I>EL</I>-<I>EF</I>):<I>EF</I>, is +given; thus <I>B</I> is found. +<p>Now, says Pappus, if <I>D</I> is the force which will move <I>B</I> +along a horizontal plane, as <I>C</I> is the force which will move +<I>A</I> along a horizontal plane, the sum of <I>C</I> and <I>D</I> will be the +force required to move the sphere upwards on the inclined +plane. He takes the particular case where <MATH><G>q</G>=60°</MATH>. Then +sin <G>q</G> is approximately 104/120 (he evidently uses 1/2.26/15 for 1/2√3), +and <MATH><I>A</I>:<I>B</I>=16:104</MATH>. +Suppose, for example, that <MATH><I>A</I>=200</MATH> talents; then <I>B</I> is 1300 +talents. Suppose further that <I>C</I> is 40 man-power; then, since +<MATH><I>D</I>:<I>C</I>=<I>B</I>:<I>A</I>, <I>D</I>=260</MATH> man-power; and it will take <I>D</I>+<I>C</I>, or +300 man-power, to move the weight up the plane! +<p>Prop. 10 gives, from Heron's <I>Barulcus</I>, the machine con- +sisting of a pulley, interacting toothed wheels, and a spiral +screw working on the last wheel and turned by a handle; +Pappus merely alters the proportions of the weight to the +force, and of the diameter of the wheels. At the end of +the chapter (pp. 1070-2) he repeats his construction for the +finding of two mean proportionals. +<C><I>Construction of a conic through five points</I></C>. +<p>Chaps. 13-17 are more interesting, for they contain the +solution of the problem of <I>constructing a conic through five +given points</I>. The problem arises in this way. Suppose we +are given a broken piece of the surface of a cylindrical column +such that no portion of the circumference of either of its base +<pb n=435><head>THE <I>COLLECTION</I>. BOOK VIII</head> +is left intact, and let it be required to find the diameter of +a circular section of the cylinder. We take any two points +<I>A</I>, <I>B</I> on the surface of the fragment and by means of these we +find five points on the surface all lying in one plane section, +in general oblique. This is done by taking five different radii +and drawing pairs of circles with <I>A</I>, <I>B</I> as centres and with +each of the five radii successively. These pairs of circles with +equal radii, intersecting at points on the surface, determine +five points on the plane bisecting <I>AB</I> at right angles. The five +points are then represented on any plane by triangulation. +<p>Suppose the points are <I>A</I>, <I>B</I>, <I>C</I>, <I>D</I>, <I>E</I> and are such that +no two of the lines connecting the different pairs are parallel. +<FIG> +This case can be reduced to the construction of a conic through +the five points <I>A</I>, <I>B</I>, <I>D</I>, <I>E</I>, <I>F</I> where <I>EF</I> is parallel to <I>AB</I>. +This is shown in a subsequent lemma (chap. 16). +<p>For, if <I>EF</I> be drawn through <I>E</I> parallel to <I>AB</I>, and if <I>CD</I> +meet <I>AB</I> in <I>O</I> and <I>EF</I> in <I>O</I>′, we have, by the well-known +proposition about intersecting chords, +<MATH><I>CO.OD</I>:<I>AO.OB</I>=<I>CO</I>′.<I>O</I>′<I>D</I>:<I>EO</I>′.<I>O</I>′<I>F</I></MATH>, +whence <I>O</I>′<I>F</I> is known, and <I>F</I> is determined. +<p>We have then (Prop. 13) to construct a conic through <I>A</I>, <I>B</I>, +<I>D</I>, <I>E</I>, <I>F</I>, where <I>EF</I> is parallel to <I>AB</I>. +<p>Bisect <I>AB</I>, <I>EF</I> at <I>V</I>, <I>W</I>; then <I>VW</I> produced both ways +is a diameter. Draw <I>DR</I>, the chord through <I>D</I> parallel +<pb n=436><head>PAPPUS OF ALEXANDRIA</head> +to this diameter. Then <I>R</I> is determined by means of the +relation +<MATH><I>RG.GD</I>:<I>BG.GA</I>=<I>RH.HD</I>:<I>FH.HE</I></MATH> (1) +in this way. +<p>Join <I>DB</I>, <I>RA</I>, meeting <I>EF</I> in <I>K</I>, <I>L</I> respectively. +<p>Then, by similar triangles, +<MATH><I>RG.GD</I>:<I>BG.GA</I>=(<I>RH</I>:<I>HL</I>).(<I>DH</I>:<I>HK</I>) +=<I>RH.HD</I>:<I>KH.HL</I></MATH>. +<p>Therefore, by (1), <MATH><I>FH.HE</I>=<I>KH.HL</I></MATH>, +whence <I>HL</I> is determined, and therefore <I>L</I>. The intersection +of <I>AL</I>, <I>DH</I> determines <I>R</I>. +<p>Next, in order to find the extremities <I>P</I>, <I>P</I>′ of the diameter +through <I>V</I>, <I>W</I>, we draw <I>ED</I>, <I>RF</I> meeting <I>PP</I>′ in <I>M</I>, <I>N</I> respectively. +<p>Then, as before, +<MATH><I>FW.WE</I>:<I>P</I>′<I>W.WP</I>=<I>FH.HE</I>:<I>RH.HD</I></MATH>, by the ellipse, +<MATH>=<I>FW.WE</I>:<I>NW.WM</I></MATH>, by similar triangles. +<p>Therefore <MATH><I>P</I>′<I>W.WP</I>=<I>NW.WM</I></MATH>; +and similarly we can find the value of <I>P</I>′<I>V.VP</I>. +<p>Now, says Pappus, since <I>P</I>′<I>W.WP</I> and <I>P</I>′<I>V.VP</I> are given +areas and the points <I>V</I>, <I>W</I> are given, <I>P</I>, <I>P</I>′ are given. His +determination of <I>P</I>, <I>P</I>′ amounts (Prop. 14 following) to an +elimination of one of the points and the finding of the other +by means of an equation of the second degree. +<p>Take two points <I>Q</I>, <I>Q</I>′ on the diameter such that +<MATH><I>P</I>′<I>V.VP</I>=<I>WV.VQ</I></MATH>, (<G>a</G>) +<MATH><I>P</I>′<I>W.WP</I>=<I>VW.WQ</I>′</MATH>; (<G>b</G>) +<I>Q</I>, <I>Q</I>′ are thus known, while <I>P</I>, <I>P</I>′ remain to be found. +<p>By (<G>a</G>) <MATH><I>P</I>′<I>V</I>:<I>VW</I>=<I>QV</I>:<I>VP</I></MATH>, +whence <MATH><I>P</I>′<I>W</I>:<I>VW</I>=<I>PQ</I>:<I>PV</I></MATH>. +<p>Therefore, by means of (<G>b</G>), +<MATH><I>PQ</I>:<I>PV</I>=<I>Q</I>′<I>W</I>:<I>WP</I></MATH>, +<pb n=437><head>THE <I>COLLECTION</I>. BOOK VIII</head> +so that <MATH><I>PQ</I>:<I>QV</I>=<I>Q</I>′<I>W</I>:<I>PQ</I>′</MATH>, +or <MATH><I>PQ.PQ</I>′=<I>QV.Q</I>′<I>W</I></MATH>. +<p>Thus <I>P</I> can be found, and similarly <I>P</I>′. +<p>The conjugate diameter is found by virtue of the relation +<MATH>(conjugate diam.)<SUP>2</SUP>:<I>PP</I>′<SUP>2</SUP>=<I>p</I>:<I>PP</I>′</MATH>. +where <I>p</I> is the latus rectum to <I>PP</I>′ determined by the property +of the curve +<MATH><I>p</I>:<I>PP</I>′=<I>AV</I><SUP>2</SUP>:<I>PV.VP</I>′</MATH>. +<C><I>Problem, Given two conjugate diameters of an ellipse, +to find the axes</I></C>. +<p>Lastly, Pappus shows (Prop. 14, chap. 17) how, when we are +given two conjugate diameters, we can find the axes. The +construction is as follows. Let <I>AB</I>, <I>CD</I> be conjugate diameters +(<I>CD</I> being the greater), <I>E</I> the centre. +<p>Produce <I>EA</I> to <I>H</I> so that +<MATH><I>EA.AH</I>=<I>DE</I><SUP>2</SUP></MATH>. +<p>Through <I>A</I> draw <I>FG</I> parallel to <I>CD</I>. Bisect <I>EH</I> in <I>K</I>, and +draw <I>KL</I> at right angles to <I>EH</I> meeting <I>FG</I> in <I>L</I>. +<FIG> +<p>With <I>L</I> as centre, and <I>LE</I> as radius, describe a circle cutting +<I>GF</I> in <I>G</I>, <I>F</I>. +<p>Join <I>EF</I>, <I>EG</I>, and from <I>A</I> draw <I>AM</I>, <I>AN</I> parallel to <I>EF</I>, <I>EG</I> +respectively. +<pb n=438><head>PAPPUS OF ALEXANDRIA</head> +<p>Take points <I>P</I>, <I>R</I> on <I>EG</I>, <I>EF</I> such that +<MATH><I>EP</I><SUP>2</SUP>=<I>GE.EM</I></MATH>, and <MATH><I>ER</I><SUP>2</SUP>=<I>FE.EN</I></MATH>. +<p>Then <I>EP</I> is half the major axis, and <I>ER</I> half the minor axis. +<p>Pappus omits the proof. +<C><I>Problem of seven hexagons in a circle</I></C>. +<p>Prop. 19 (chap. 23) is a curious problem. To inscribe seven +equal regular hexagons in a circle in such a way that one +<FIG> +is about the centre of the circle, while six others stand on its +sides and have the opposite sides in each case placed as chords +in the circle. +<p>Suppose <I>GHKLNM</I> to be the hexagon so described on <I>HK</I>, +a side of the inner hexagon; <I>OKL</I> will then be a straight line. +Produce <I>OL</I> to meet the circle in <I>P</I>. +<p>Then <MATH><I>OK</I>=<I>KL</I>=<I>LN</I></MATH>. Therefore, in the triangle <I>OLN</I>, +<MATH><I>OL</I>=2<I>LN</I></MATH>, while the included angle <I>OLN</I> (=120°) is also +given. Therefore the triangle is given in species; therefore +the ratio <I>ON</I>:<I>NL</I> is given, and, since <I>ON</I> is given, the side <I>NL</I> +of each of the hexagons is given. +<p>Pappus gives the auxiliary construction thus. Let <I>AF</I> be +taken equal to the radius <I>OP</I>. Let <MATH><I>AC</I>=(1/3)<I>AF</I></MATH>, and on <I>AC</I> as +base describe a segment of a circle containing an angle of 60°. +Take <I>CE</I> equal to 4/5<I>AC</I>, and draw <I>EB</I> to touch the circle at <I>B</I>. +<pb n=439><head>THE <I>COLLECTION</I>. BOOK VIII</head> +Then he proves that, if we join <I>AB</I>, <I>AB</I> is equal to the length +of the side of the hexagon required. +<p>Produce <I>BC</I> to <I>D</I> so that <MATH><I>BD</I>=<I>BA</I></MATH>, and join <I>DA</I>. <I>ABD</I> +is then equilateral. +<p>Since <I>EB</I> is a tangent to the segment, <MATH><I>AE.EC</I>=<I>EB</I><SUP>2</SUP></MATH> or +<MATH><I>AE</I>:<I>EB</I>=<I>EB</I>:<I>EC</I></MATH>, and the triangles <I>EAB</I>, <I>EBC</I> are similar. +<p>Therefore <MATH><I>BA</I><SUP>2</SUP>:<I>BC</I><SUP>2</SUP>=<I>AE</I><SUP>2</SUP>:<I>EB</I><SUP>2</SUP>=<I>AE</I>:<I>EC</I>=9:4</MATH>; +and <MATH><I>BC</I>=(2/3)<I>BA</I>=(2/3)<I>BD</I></MATH>, so that <MATH><I>BC</I>=2<I>CD</I></MATH>. +<p>But <MATH><I>CF</I>=2<I>CA</I></MATH>; therefore <MATH><I>AC</I>:<I>CF</I>=<I>DC</I>:<I>CB</I></MATH>, and <I>AD</I>, <I>BF</I> +are parallel. +<p>Therefore <MATH><I>BF</I>:<I>AD</I>=<I>BC</I>:<I>CD</I>=2:1</MATH>, so that +<MATH><I>BF</I>=2<I>AD</I>=2<I>AB</I></MATH>. +<p>Also <MATH>∠<I>FBC</I>=∠<I>BDA</I>=60°</MATH>, so that <MATH>∠<I>ABF</I>=120°</MATH>, and +the triangle <I>ABF</I> is therefore equal and similar to the required +triangle <I>NLO</I>. +<C><I>Construction of toothed wheels and indented screws</I></C>. +<p>The rest of the Book is devoted to the construction (1) of +toothed wheels with a given number of teeth equal to those of +a given wheel, (2) of a cylindrical helix, the <I>cochlias</I>, indented +so as to work on a toothed wheel. The text is evidently +defective, and at the end an interpolator has inserted extracts +about the mechanical powers from Heron's <I>Mechanics</I>. +<pb><C>XX +ALGEBRA: DIOPHANTUS OF ALEXANDRIA</C> +<C>Beginnings learnt from Egypt.</C> +<p>IN algebra, as in geometry, the Greeks learnt the beginnings +from the Egyptians. Familiarity on the part of the Greeks +with Egyptian methods of calculation is well attested. (1) +These methods are found in operation in the Heronian writings +and collections. (2) Psellus in the letter published by Tannery +in his edition of Diophantus speaks of ‘the method of arith- +metical calculations used by the Egyptians, by which problems +in analysis are handled’; he adds details, doubtless taken +from Anatolius, of the technical terms used for different kinds +of numbers, including the powers of the unknown quantity. +(3) The scholiast to Plato's <I>Charmides</I> 165 E says that ‘parts +of <G>logistikh/</G>, the science of calculation, are the so-called Greek +and Egyptian methods in multiplications and divisions, and +the additions and subtractions of fractions’. (4) Plato himself +in the <I>Laws</I> 819 A-C says that free-born boys should, as is the +practice in Egypt, learn, side by side with reading, simple +mathematical calculations adapted to their age, which should +be put into a form such as to combine amusement with +instruction: problems about the distribution of, say, apples or +garlands, the calculation of mixtures, and other questions +arising in military or civil life. +<C>‘Hau’-calculations.</C> +<p>The Egyptian calculations here in point (apart from their +method of writing and calculating in fractions, which, with +the exception of 2/3, were always decomposed and written +as the sum of a diminishing series of aliquot parts or sub- +multiples) are the <I>hau</I>-calculations. <I>Hau</I>, meaning a <I>heap</I>, is +the term denoting the unknown quantity, and the calculations +<pb n=441><head>‘HAU’-CALCULATIONS</head> +in terms of it are equivalent to the solutions of simple equations +with one unknown quantity. Examples from the Papyrus +Rhind correspond to the following equations: +<MATH>1/7<I>x</I>+<I>x</I>=19, +2/3<I>x</I>+1/2<I>x</I>+1/7<I>x</I>+<I>x</I>=33, +(<I>x</I>+2/3<I>x</I>)-1/3(<I>x</I>+2/3<I>x</I>)=10</MATH>. +<p>The Egyptians anticipated, though only in an elementary +form, a favourite method of Diophantus, that of the ‘false +supposition’ or ‘regula falsi’. An arbitrary assumption is +made as to the value of the unknown, and the true value +is afterwards found by a comparison of the result of sub- +stituting the wrong value in the original expression with the +actual data. Two examples may be given. The first, from +the Papyrus Rhind, is the problem of dividing 100 loaves +among five persons in such a way that the shares are in +arithmetical progression, and one-seventh of the sum of the +first three shares is equal to the sum of the other two. If +<MATH><I>a</I>+4<I>d</I>, <I>a</I>+3<I>d</I>, <I>a</I>+2<I>d</I>, <I>a</I>+<I>d</I></MATH>, <I>a</I> be the shares, then +<MATH>3<I>a</I>+9<I>d</I>=7(2<I>a</I>+<I>d</I>)</MATH>, +or <MATH><I>d</I>=5 1/2<I>a</I></MATH>. +Ahmes says, without any explanation, ‘make the difference, +as it is, 5 1/2’, and then, assuming <MATH><I>a</I>=1</MATH>, writes the series +23, 17 1/2, 12, 6 1/2, 1. The addition of these gives 60, and 100 is +1 2/3 times 60. Ahmes says simply ‘multiply 1 2/3 times’ and +thus gets the correct values 38 1/3, 29 1/6, 20, 10 2/3 1/6, 1 2/3. +<p>The second example (taken from the Berlin Papyrus 6619) +is the solution of the equations +<MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>=100, +<I>x</I>:<I>y</I>=1:(3/4), or <I>y</I>=3/4<I>x</I></MATH>. +<I>x</I> is first assumed to be 1, and <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP></MATH> is thus found to be 25/16. +In order to make 100, 25/16 has to be multiplied by 64 or 8<SUP>2</SUP>. +The true value of <I>x</I> is therefore 8 times 1, or 8. +<C>Arithmetical epigrams in the Greek Anthology.</C> +<p>The simple equations solved in the Papyrus Rhind are just +the kind of equations of which we find many examples in the +<pb n=442><head>ALGEBRA: DIOPHANTUS OF ALEXANDRIA</head> +arithmetical epigrams contained in the Greek Anthology. Most +of these appear under the name of Metrodorus, a grammarian, +probably of the time of the Emperors Anastasius I (A.D. 491- +518) and Justin I (A.D. 518-27). They were obviously only +collected by Metrodorus, from ancient as well as more recent +sources. Many of the epigrams (46 in number) lead to simple +equations, and several of them are problems of dividing a num- +ber of apples or nuts among a certain number of persons, that +is to say, the very type of problem mentioned by Plato. For +example, a number of apples has to be determined such that, +if four persons out of six receive one-third, one-eighth, one- +fourth and one-fifth respectively of the whole number, while +the fifth person receives 10 apples, there is one apple left over +for the sixth person, i.e. +<MATH>1/3<I>x</I>+1/8<I>x</I>+1/4<I>x</I>+1/5<I>x</I>+10+1=<I>x</I></MATH>. +Just as Plato alludes to bowls (<G>fia/lai</G>) of different metals, +there are problems in which the weights of bowls have to +be found. We are thus enabled to understand the allusions of +Proclus and the scholiast on <I>Charmides</I> 165 E to <G>mhli=tai</G> +and <G>fiali=tai a)riqmoi/</G>, ‘numbers of apples or of bowls’. +It is evident from Plato's allusions that the origin of such +simple algebraical problems dates back, at least, to the fifth +century B.C. +<p>The following is a classification of the problems in the +<I>Anthology</I>. (1) Twenty-three are simple equations in one +unknown and of the type shown above; one of these is an +epigram on the age of Diophantus and certain incidents of +his life (xiv. 126). (2) Twelve are easy simultaneous equations +with two unknowns, like Dioph. I. 6; they can of course be +reduced to a simple equation with one unknown by means of +an easy elimination. One other (xiv. 51) gives simultaneous +equations in three unknowns +<MATH><I>x</I>=<I>y</I>+1/3<I>z</I>, <I>y</I>=<I>z</I>+1/3<I>x</I>, <I>z</I>=10+1/3<I>y</I></MATH>, +and one (xiv. 49) gives four equations in four unknowns, +<MATH><I>x</I>+<I>y</I>=40, <I>x</I>+<I>z</I>=45, <I>x</I>+<I>u</I>=36, <I>x</I>+<I>y</I>+<I>z</I>+<I>u</I>=60</MATH>. +<p>With these may be compared Dioph. I. 16-21, as well as the +general solution of any number of simultaneous linear equa- +<pb n=443><head>EPIGRAMS IN THE GREEK ANTHOLOGY</head> +tions of this type with the same number of unknown quantities +which was given by Thymaridas, an early Pythagorean, and +was called the <G>e)pa/nqhma</G>, ‘flower’ or ‘bloom’ of Thymaridas +(see vol. i, pp. 94-6). (3) Six more are problems of the usual +type about the filling and emptying of vessels by pipes; e.g. +(xiv. 130) one pipe fills the vessel in one day, a second in two +and a third in three; how long will all three running together +take to fill it? Another about brickmakers (xiv. 136) is of +the same sort. +<C>Indeterminate equations of the first degree.</C> +<p>The Anthology contains (4) two <I>indeterminate</I> equations of +the first degree which can be solved in positive integers in an +infinite number of ways (xiv. 48, 144); the first is a distribu- +tion of apples, 3<I>x</I> in number, into parts satisfying the equation +<MATH><I>x</I>-3<I>y</I>=<I>y</I></MATH>, where <I>y</I> is not less than 2; the second. leads to +three equations connecting four unknown quantities: +<MATH><I>x</I>+<I>y</I>=<I>x</I><SUB>1</SUB>+<I>y</I><SUB>1</SUB>, +<I>x</I>=2<I>y</I><SUB>1</SUB>, +<I>x</I><SUB>1</SUB>=3<I>y</I></MATH>, +the general solution of which is <MATH><I>x</I>=4<I>k</I>, <I>y</I>=<I>k</I>, <I>x</I><SUB>1</SUB>=3<I>k</I>, +<I>y</I><SUB>1</SUB>=2<I>k</I></MATH>. These very equations, which, however, are made +determinate by assuming that <MATH><I>x</I>+<I>y</I>=<I>x</I><SUB>1</SUB>+<I>y</I><SUB>1</SUB>=100</MATH>, are solved +in Dioph. I. 12. +<p>Enough has been said to show that Diophantus was not +the inventor of Algebra. Nor was he the first to solve inde- +terminate problems of the second degree. +<C>Indeterminate equations of second degree before +Diophantus.</C> +<p>Take first the problem (Dioph. II. 8) of dividing a square +number into two squares, or of finding a right-angled triangle +with sides in rational numbers. We have already seen that +Pythagoras is credited with the discovery of a general formula +for finding such triangles, namely, +<MATH><I>n</I><SUP>2</SUP>+{1/2(<I>n</I><SUP>2</SUP>-1)}<SUP>2</SUP>={1/2(<I>n</I><SUP>2</SUP>+1)}<SUP>2</SUP></MATH>, +<pb n=444><head>ALGEBRA: DIOPHANTUS OF ALEXANDRIA</head> +where <I>n</I> is any odd number, and Plato with another formula +of the same sort, namely <MATH>(2<I>n</I>)<SUP>2</SUP>+(<I>n</I><SUP>2</SUP>-1)<SUP>2</SUP>=(<I>n</I><SUP>2</SUP>+1)<SUP>2</SUP></MATH>. Euclid +(Lemma following X. 28) finds the following more general +formula +<MATH><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>p</I><SUP>2</SUP><I>q</I><SUP>2</SUP>={1/2(<I>mnp</I><SUP>2</SUP>+<I>mnq</I><SUP>2</SUP>)}<SUP>2</SUP>-{1/2(<I>mnp</I><SUP>2</SUP>-<I>mnq</I><SUP>2</SUP>)}<SUP>2</SUP></MATH>. +<p>The Pythagoreans too, as we have seen (vol. i, pp. 91-3), +solved another indeterminate problem, discovering, by means +of the series of ‘side-’ and ‘diameter-numbers’, any number +of successive integral solutions of the equations +<MATH>2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>=±1</MATH>. +<p>Diophantus does not particularly mention this equation, +but from the Lemma to VI. 15 it is clear that he knew how +to find any number of solutions when one is known. Thus, +seeing that <MATH>2<I>x</I><SUP>2</SUP>-1=<I>y</I><SUP>2</SUP></MATH> is satisfied by <MATH><I>x</I>=1, <I>y</I>=1</MATH>, he would +put +<MATH>2(1+<I>x</I>)<SUP>2</SUP>-1=a square +=(<I>px</I>-1)<SUP>2</SUP></MATH>, say; +whence <MATH><I>x</I>=(4+2<I>p</I>)/(<I>p</I><SUP>2</SUP>-2)</MATH>. +Take the value <MATH><I>p</I>=2</MATH>, and we have <MATH><I>x</I>=4</MATH>, and <MATH><I>x</I>+1=5</MATH>; +in this case <MATH>2.5<SUP>2</SUP>-1=49=7<SUP>2</SUP></MATH>. Putting <I>x</I>+5 in place of <I>x</I>, +we can find a still higher value, and so on. +<C>Indeterminate equations in the Heronian collections.</C> +<p>Some further Greek examples of indeterminate analysis are +now available. They come from the Constantinople manuscript +(probably of the twelfth century) from which Schöne edited +the <I>Metrica</I> of Heron; they have been published and translated +by Heiberg, with comments by Zeuthen.<note><I>Bibliotheca mathematica</I>, viii<SUB>s</SUB>, 1907-8, pp. 118-34. See now <I>Geom</I>. +24. 1-13 in Heron, vol. iv (ed. Heiberg), pp. 414-26.</note> Two of the problems +(thirteen in number) had been published in a less complete +form in Hultsch's Heron (<I>Geëponicus</I>, 78, 79); the others +are new. +<p>I. The first problem is to find two rectangles such that the +perimeter of the second is three times that of the first, and +the area of the first is three times that of the second. The +<pb n=445><head>HERONIAN INDETERMINATE EQUATIONS</head> +number 3 is of course only an illustration, and the problem is +equivalent to the solution of the equations +<MATH><BRACE> +(1) <I>u</I>+<I>v</I>=<I>n</I>(<I>x</I>+<I>y</I>) +(2) <I>xy</I>=<I>n.uv</I></BRACE></MATH>. +<p>The solution given in the text is equivalent to +<MATH><BRACE> +<I>x</I>=2<I>n</I><SUP>3</SUP>-1, <I>y</I>=2<I>n</I><SUP>3</SUP> +<I>u</I>=<I>n</I>(4<I>n</I><SUP>3</SUP>-2), <I>v</I>=<I>n</I></BRACE></MATH>. +<p>Z<*>uthen suggests that the solution may have been obtained +thus. As the problem is indeterminate, it would be natural +to start with some hypothesis, e.g. to put <I>v</I>=<I>n</I>. It would +follow from equation (1) that <I>u</I> is a multiple of <I>n</I>, say <I>nz</I>. +We have then +<MATH><I>x</I>+<I>y</I>=1+<I>z</I></MATH>, +while, by (2), <MATH><I>xy</I>=<I>n</I><SUP>3</SUP><I>z</I></MATH>, +whence <MATH><I>xy</I>=<I>n</I><SUP>3</SUP>(<I>x</I>+<I>y</I>)-<I>n</I><SUP>3</SUP></MATH>, +or <MATH>(<I>x</I>-<I>n</I><SUP>3</SUP>)(<I>y</I>-<I>n</I><SUP>3</SUP>)=<I>n</I><SUP>3</SUP>(<I>n</I><SUP>3</SUP>-1)</MATH>. +<p>An obvious solution is +<MATH><I>x</I>-<I>n</I><SUP>3</SUP>=<I>n</I><SUP>3</SUP>-1, <I>y</I>-<I>n</I><SUP>3</SUP>=<I>n</I><SUP>3</SUP></MATH>, +which gives <MATH><I>z</I>=2<I>n</I><SUP>3</SUP>-1+2<I>n</I><SUP>3</SUP>-1=4<I>n</I><SUP>3</SUP>-2</MATH>, so that +<MATH><I>u</I>=<I>nz</I>=<I>n</I>(4<I>n</I><SUP>3</SUP>-2)</MATH>. +<p>II. The second is a similar problem about two rectangles, +equivalent to the solution of the equations +<MATH><BRACE> +(1) <I>x</I>+<I>y</I>=<I>u</I>+<I>v</I> +(2) <I>xy</I>=<I>n.uv</I></BRACE></MATH>, +and the solution given in the text is +<MATH><I>x</I>+<I>y</I>=<I>u</I>+<I>v</I>=<I>n</I><SUP>3</SUP>-1, (3)</MATH> +<MATH><BRACE> +<I>u</I>=<I>n</I>-1, <I>v</I>=<I>n</I>(<I>n</I><SUP>2</SUP>-1) +<I>x</I>=<I>n</I><SUP>2</SUP>-1, <I>y</I>=<I>n</I><SUP>2</SUP>(<I>n</I>-1)</BRACE></MATH>. (4) +<p>In this case trial may have been made of the assumptions +<MATH><I>v</I>=<I>nx</I>, <I>y</I>=<I>n</I><SUP>2</SUP><I>u</I></MATH>, +<pb n=446><head>ALGEBRA: DIOPHANTUS OF ALEXANDRIA</head> +when equation (1) would give +<MATH>(<I>n</I>-1)<I>x</I>=(<I>n</I><SUP>2</SUP>-1)<I>u</I></MATH>, +a solution of which is <MATH><I>x</I>=<I>n</I><SUP>2</SUP>-1, <I>u</I>=<I>n</I>-1</MATH>. +<p>III. The fifth problem is interesting in one respect. We are +asked to find a right-angled triangle (in rational numbers) +with area of 5 feet. We are told to multiply 5 by some +square <I>containing 6 as a factor</I>, e.g. 36. This makes 180, +and this is the area of the triangle (9, 40, 41). Dividing each +side by 6, we have the triangle required. The author, then, +is aware that the area of a right-angled triangle with sides in +whole numbers is divisible by 6. If we take the Euclidean +formula for a right-angled triangle, making the sides <I>a.mn</I>, +<MATH><I>a</I>.1/2(<I>m</I><SUP>2</SUP>-<I>n</I><SUP>2</SUP>), <I>a</I>.1/2(<I>m</I><SUP>2</SUP>+<I>n</I><SUP>2</SUP>)</MATH>, where <I>a</I> is any number, and <I>m, n</I> +are numbers which are both odd or both even, the area is +<MATH>1/4<I>mn</I>(<I>m</I>-<I>n</I>) (<I>m</I>+<I>n</I>)<I>a</I><SUP>2</SUP></MATH>, +and, as a matter of fact, the number <MATH><I>mn</I>(<I>m</I>-<I>n</I>) (<I>m</I>+<I>n</I>)</MATH> is +divisible by 24, as was proved later (for another purpose) by +Leonardo of Pisa. +<p>IV. The last four problems (10 to 13) are of great interest. +They are different particular cases of one problem, that of +finding a rational right-angled triangle such that the numerical +sum of its area and its perimeter is a given number. The +author's solution depends on the following formulae, where +<I>a, b</I> are the perpendiculars, and <I>c</I> the hypotenuse, of a right- +angled triangle, <I>S</I> its area, <I>r</I> the radius of the inscribed circle, +and <MATH><I>s</I>=1/2(<I>a</I>+<I>b</I>+<I>c</I>)</MATH>; +<MATH><I>S</I>=<I>rs</I>=1/2<I>ab</I>, <I>r</I>+<I>s</I>=<I>a</I>+<I>b</I>, <I>c</I>=<I>s</I>-<I>r</I></MATH>. +(The proof of these formulae by means of the usual figure, +namely that used by Heron to prove the formula +<MATH><I>S</I>=√{<I>s</I>(<I>s</I>-<I>a</I>)(<I>s</I>-<I>b</I>)(<I>s</I>-<I>c</I>)}</MATH>, +is easy.) +<p>Solving the first two equations, in order to find <I>a</I> and <I>b</I>, +we have +<MATH><BRACE><note>=1/2[<I>r</I>+<I>s</I>∓√{(<I>r</I>+<I>s</I>)<SUP>2</SUP>-8<I>rs</I>}],</note> +<I>a</I> +<I>b</I></BRACE></MATH> +which formula is actually used by the author for finding <I>a</I> +<pb n=447><head>HERONIAN INDETERMINATE EQUATIONS</head> +and <I>b</I>. The method employed is to take the sum of the area +and the perimeter <I>S</I>+2<I>s</I>, separated into its two obvious +factors <MATH><I>s</I>(<I>r</I>+2)</MATH>, to put <MATH><I>s</I>(<I>r</I>+2)=<I>A</I></MATH> (the given number), and +then to separate <I>A</I> into suitable factors to which <I>s</I> and <I>r</I>+2 +may be equated. They must obviously be such that <I>sr</I>, the +area, is divisible by 6. To take the first example where +<I>A</I>=280: the possible factors are 2X140, 4X70, 5X56, 7X40, +8X35, 10X28, 14X20. The suitable factors in this case are +<MATH><I>r</I>+2=8, <I>s</I>=35</MATH>, because <I>r</I> is then equal to 6, and <I>rs</I> is +a multiple of 6. +<p>The author then says that +<MATH><I>a</I>=1/2[6+35-√{(6+35)<SUP>2</SUP>-8.6.35}]=1/2(41-1)=20, +<I>b</I>=1/2(41+1)=21, +<I>c</I>=35-6=29</MATH>. +<p>The triangle is therefore (20, 21, 29) in this case. The +triangles found in the other three cases, by the same method, +are (9, 40, 41), (8, 15, 17) and (9, 12, 15). +<p>Unfortunately there is no guide to the date of the problems +just given. The probability is that the original formulation +of the most important of the problems belongs to the period +between Euclid and Diophantus. This supposition best agrees +with the fact that the problems include nothing taken from +the great collection in the <I>Arithmetica</I>. On the other hand, +it is strange that none of the seven problems above mentioned +is found in Diophantus. The five relating to rational right- +angled triangles might well have been included by him; thus +he finds rational right-angled triangles such that the area <I>plus</I> +or <I>minus</I> one of the perpendiculars is a given number, but not +the rational triangle which has a given area; and he finds +rational right-angled triangles such that the area <I>plus</I> or <I>minus</I> +the sum of <I>two</I> sides is a given number, but not the rational +triangle such that the sum of the area and the <I>three</I> sides is +a given number. The omitted problems might, it is true, have +come in the lost Books; but, on the other hand, Book VI would +have been the appropriate place for them. +<p>The crowning example of a difficult indeterminate problem +propounded before Diophantus's time is the Cattle-Problem +attributed to Archimedes, described above (pp. 97-8). +<pb n=448><head>ALGEBRA: DIOPHANTUS OF ALEXANDRIA</head> +<C>Numerical solution of quadratic equations.</C> +<p>The <I>geometrical</I> algebra of the Greeks has been in evidence +all through our history from the Pythagoreans downwards, +and no more need be said of it here except that its arithmetical +application was no new thing in Diophantus. It is probable, +for example, that the solution of the quadratic equation, +discovered first by geometry, was applied for the purpose of +finding <I>numerical</I> values for the unknown as early as Euclid, +if not earlier still. In Heron the numerical solution of +equations is well established, so that Diophantus was not the +first to treat equations algebraically. What he did was to +take a step forward towards an algebraic <I>notation</I>. +<p>The date of DIOPHANTUS can now be fixed with fair certainty. +He was later than Hypsicles, from whom he quotes a definition +of a polygonal number, and earlier than Theon of Alexandria, +who has a quotation from Diophantus's definitions. The +possible limits of date are therefore, say, 150 B.C. to A.D. 350. +But the letter of Psellus already mentioned says that Anatolius +(Bishop of Laodicea about A.D. 280) dedicated to Diophantus +a concise treatise on the Egyptian method of reckoning; +hence Diophantus must have been a contemporary, so that he +probably flourished A.D. 250 or not much later. +<p>An epigram in the Anthology gives some personal particulars: +his boyhood lasted 1/6th of his life; his beard grew after 1/12th +more; he married after 1/7th more, and his son was born 5 years +later; the son lived to half his father's age, and the father +died 4 years after his son. Thus, if <I>x</I> was his age when +he died, +<MATH>1/6<I>x</I>+1/12<I>x</I>+1/7<I>x</I>+5+1/2<I>x</I>+4=<I>x</I></MATH>, +which gives <MATH><I>x</I>=84</MATH>. +<C>Works of Diophantus.</C> +<p>The works on which the fame of Diophantus rests are: +<p>(1) the <I>Arithmetica</I> (originally in thirteen Books), +<p>(2) a tract <I>On Polygonal Numbers</I>. +<pb n=449><head>WORKS</head> +<p>Six Books only of the former and a fragment of the latter +survive. +<p>Allusions in the <I>Arithmetica</I> imply the existence of +<p>(3) A collection of propositions under the title of <I>Porisms</I>; +in three propositions (3, 5, 16) of Book V, Diophantus quotes +as known certain propositions in the Theory of Numbers, +prefixing to the statement of them the words ‘We have it in +the <I>Porisms</I> that ...’ +<p>A scholium on a passage of Iamblichus, where Iamblichus +cites a dictum of certain Pythagoreans about the unit being +the dividing line (<G>meqo/rion</G>) between number and aliquot parts, +says ‘thus Diophantus in the <I>Moriastica</I> .... for he describes +as “parts” the progression without limit in the direction of +less than the unit’. The <I>Moriastica</I> may be a separate work +by Diophantus giving rules for reckoning with fractions; but +I do not feel sure that the reference may not simply be to the +definitions at the beginning of the <I>Arithmetica</I>. +<C>The <I>Arithmetica</I>.</C> +<C><I>The seven lost Books and their place</I>.</C> +<p>None of the manuscripts which we possess contain more +than six Books of the <I>Arithmetica</I>, the only variations being +that some few divide the six Books into seven, while one or +two give the fragment on Polygonal Numbers as VIII. The +missing Books were evidently lost at a very early date. +Tannery suggests that Hypatia's commentary extended only +to the first six Books, and that she left untouched the remain- +ing seven, which, partly as a consequence, were first forgotten +and then lost (cf. the case of Apollonius's <I>Conics</I>, where the +only Books which have survived in Greek, I-IV, are those +on which Eutocius commented). There is no sign that even +the Arabians ever possessed the missing Books. The <I>Fakhrī</I>, +an algebraical treatise by Abū Bekr Muh. b. al-Hasan al- +Karkhī (d. about 1029), contains a collection of problems in +determinate and indeterminate analysis which not only show +that their author had deeply studied Diophantus but in many +cases are taken direct from the <I>Arithmetica</I>, sometimes with +a change in constants; in the fourth section of the work, +<pb n=450><head>DIOPHANTUS OF ALEXANDRIA</head> +between problems corresponding to problems in Dioph. II +and III, are 25 problems not found in Diophantus, but +internal evidence, and especially the admission of irrational +results (which are always avoided by Diophantus), exclude +the hypothesis that we have here one of the lost Books. +Nor is there any sign that more of the work than we possess +was known to Abū'l Wafā al-Būzjānī (A.D. 940-98) who wrote +a ‘commentary on the algebra of Diophantus’, as well as +a ‘Book of proofs of propositions used by Diophantus in his +work’. These facts again point to the conclusion that the +lost Books were lost before the tenth century. +<p>The old view of the place originally occupied by the lost +seven Books is that of Nesselmann, who argued it with great +ability.<note>Nesselmann, <I>Algebra der Griechen</I>, pp. 264-73.</note> According to him (1) much less of Diophantus is +wanting than would naturally be supposed on the basis of +the numerical proportion of 7 lost to 6 extant Books, (2) the +missing portion came, not at the end, but in the middle of +the work, and indeed mostly between the first and second +Books. Nesselmann's general argument is that, if we care- +fully read the last four Books, from the third to the sixth, +we shall find that Diophantus moves in a rigidly defined and +limited circle of methods and artifices, and seems in fact to be +at the end of his resources. As regards the possible contents +of the lost portion on this hypothesis, Nesselmann can only +point to (1) topics which we should expect to find treated, +either because foreshadowed by the author himself or as +necessary for the elucidation or completion of the whole +subject, (2) the <I>Porisms</I>; under head (1) come, (<I>a</I>) deter- +minate equations of the second degree, and (<I>b</I>) indeterminate +equations of the first degree. Diophantus does indeed promise +to show how to solve the general quadratic <MATH><I>ax</I><SUP>2</SUP>±<I>bx</I>±<I>c</I>=0</MATH> so +far as it has rational and positive solutions; the suitable place +for this would have been between Books I and II. But there +is nothing whatever to show that indeterminate equations +of the first degree formed part of the writer's plan. Hence +Nesselmann is far from accounting for the contents of seven +whole Books; and he is forced to the conjecture that the six +Books may originally have been divided into even more than +seven Books; there is, however, no evidence to support this. +<pb n=451><head>RELATION OF WORKS</head> +<C><I>Relation of the ‘Porisms’ to the Arithmetica</I>.</C> +<p>Did the <I>Porisms</I> form part of the <I>Arithmetica</I> in its original +form? The phrase in which they are alluded to, and which +occurs three times, ‘We have it in the <I>Porisms</I> that ...’ + suggests that they were a distinct collection of propositions concerning +the properties of certain numbers, their divisibility into a +certain number of squares, and so on; and it is possible that +it was from the same collection that Diophantus took the +numerous other propositions which he assumes, explicitly or +implicitly. If the collection was part of the <I>Arithmetica</I>, it +would be strange to quote the propositions under a separate +title ‘The Porisms’ when it would have been more natural +to refer to particular propositions of particular Books, and +more natural still to say <G>tou=to ga\r prode/deiktai</G>, or some such +phrase, ‘for this has been proved’, without any reference to +the particular place where the proof occurred. The expression +‘We have it in the <I>Porisms</I>’ (in the plural) would be still +more inappropriate if the <I>Porisms</I> had been, as Tannery +supposed, not collected together as one or more Books of the +<I>Arithmetica</I>, but scattered about in the work as <I>corollaries</I> to +particular propositions. Hence I agree with the view of +Hultsch that the <I>Porisms</I> were not included in the <I>Arith- +metica</I> at all, but formed a separate work. +<p>If this is right, we cannot any longer hold to the view of +Nesselmann that the lost Books were in the middle and not at +the end of the treatise; indeed Tannery produces strong +arguments in favour of the contrary view, that it is the last +and most difficult Books which are lost. He replies first to +the assumption that Diophantus could not have proceeded +to problems more difficult than those of Book V. ‘If the +fifth or the sixth Book of the <I>Arithmetica</I> had been lost, who, +pray, among us would have believed that such problems had +ever been attempted by the Greeks? It would be the greatest +error, in any case in which a thing cannot clearly be proved +to have been unknown to all the ancients, to maintain that +it could not have been known to some Greek mathematician. +If we do not know to what lengths Archimedes brought the +theory of numbers (to say nothing of other things), let us +admit our ignorance. But, between the famous problem of the +<pb n=452><head>DIOPHANTUS OF ALEXANDRIA</head> +cattle and the most difficult of Diophantus's problems, is there +not a sufficient gap to require seven Books to fill it? And, +without attributing to the ancients what modern mathe- +maticians have discovered, may not a number of the things +attributed to the Indians and Arabs have been drawn from +Greek sources? May not the same be said of a problem +solved by Leonardo of Pisa, which is very similar to those of +Diophantus but is not now to be found in the <I>Arithmetica</I>? +In fact, it may fairly be said that, when Chasles made his +reasonably probable restitution of the <I>Porisms</I> of Euclid, he, +notwithstanding that he had Pappus's lemmas to help him, +undertook a more difficult task than he would have undertaken +if he had attempted to fill up seven Diophantine Books with +numerical problems which the Greeks may reasonably be +supposed to have solved.’<note>Diophantus, ed. Tannery, vol. ii, p. xx.</note> +<p>It is not so easy to agree with Tannery's view of the relation +of the treatise <I>On Polygonal Numbers</I> to the <I>Arithmetica</I>. +According to him, just as Serenus's treatise on the sections +of cones and cylinders was added to the mutilated <I>Conics</I> of +Apollonius consisting of four Books only, in order to make up +a convenient volume, so the tract on Polygonal Numbers was +added to the remains of the <I>Arithmetica</I>, though forming no +part of the larger work.<note><I>Ib</I>., p. xviii.</note> Thus Tannery would seem to deny +the genuineness of the whole tract on Polygonal Numbers, +though in his text he only signalizes the portion beginning +with the enunciation of the problem ‘Given a number, to find +in how many ways it can be a polygonal number’ as ‘a vain +attempt by a commentator’ to solve this problem. Hultsch, +on the other hand, thinks that we may conclude that Dio- +phantus really solved the problem. The tract begins, like +Book I of the <I>Arithmetica</I>, with definitions and preliminary +propositions; then comes the difficult problem quoted, the +discussion of which breaks off in our text after a few pages, +and to these it would be easy to tack on a great variety of +other problems. +<p>The name of Diophantus was used, as were the names of +Euclid, Archimedes and Heron in their turn, for the pur- +pose of palming off the compilations of much later authors. +<pb n=453><head>RELATION OF WORKS</head> +Tannery includes in his edition three fragments under the +heading ‘Diophantus Pseudepigraphus’. The first, which is +not ‘from the Arithmetic of Diophantus’ as its heading states, +is worth notice as containing some particulars of one of ‘two +methods of finding the square root of any square number’; +we are told to begin by writing the number ‘according to +the arrangement of the Indian method’, i.e. in the Indian +numerical notation which reached us through the Arabs. The +second fragment is the work edited by C. Henry in 1879 as +<I>Opusculum de multiplicatione et divisione sexagesimalibus +Diophanto vel Pappo attribuendum</I>. The third, beginning +with <G>*diofa/ntou e)pipedometrika/</G> is a Byzantine compilation +from later reproductions of the <G>gewmetrou/mena</G> and <G>stereo- +metrou/mena</G> of Heron. Not one of the three fragments has +anything to do with Diophantus. +<C><I>Commentators from Hypatia downwards</I>.</C> +<p>The first commentator on Diophantus of whom we hear +is Hypatia, the daughter of Theon of Alexandria; she +was murdered by Christian fanatics in A.D. 415. I have +already mentioned the attractive hypothesis of Tannery that +Hypatia's commentary extended only to our six Books, and +that this accounts for their survival when the rest were lost. +It is possible that the remarks of Psellus (eleventh century) at +the beginning of his letter about Diophantus, Anatolius and +the Egyptian method of arithmetical reckoning were taken +from Hypatia's commentary. +<p>Georgius Pachymeres (1240 to about 1310) wrote in Greek +a paraphrase of at least a portion of Diophantus. Sections +25-44 of this commentary relating to Book I, Def. 1 to Prop. +11, survive. Maximus Planudes (about 1260-1310) also wrote +a systematic commentary on Books I, II. Arabian commen- +tators were Abū'l Wafā al-Būzjānī (940-98), Qustā b. Lūqā +al-Ba'labakkī (d. about 912) and probably Ibn al-Haitham +(about 965-1039). +<C><I>Translations and editions</I>.</C> +<p>To Regiomontanus belongs the credit of being the first to +call attention to the work of Diophantus as being extant in +<pb n=454><head>DIOPHANTUS OF ALEXANDRIA</head> +Greek. In an <I>Oratio</I> delivered at the end of 1463 as an +introduction to a course of lectures on astronomy which he +gave at Padua in 1463-4 he observed: ‘No one has yet +translated from the Greek into Latin the fine thirteen Books +of Diophantus, in which the very flower of the whole of +arithmetic lies hid, the <I>ars rei et census</I> which to-day they +call by the Arabic name of Algebra.’ Again, in a letter dated +February 5, 1464, to Bianchini, he writes that he has found at +Venice ‘Diofantus, a Greek arithmetician who has not yet +been translated into Latin’. Rafael Bombelli was the first to +find a manuscript in the Vatican and to conceive the idea of +publishing the work; this was towards 1570, and, with +Antonio Maria Pazzi, he translated five Books out of the +seven into which the manuscript was divided. The translation +was not published, but Bombelli took all the problems of the +first four Books and some of those of the fifth and embodied +them in his <I>Algebra</I> (1572), interspersing them with his own +problems. +<p>The next writer on Diophantus was Wilhelm Holzmann, +who called himself Xylander, and who with extraordinary +industry and care produced a very meritorious Latin trans- +lation with commentary (1575). Xylander was an enthusiast +for Diophantus, and his preface and notes are often delightful +reading. Unfortunately the book is now very rare. The +standard edition of Diophantus till recent years was that of +Bachet, who in 1621 published for the first time the Greek +text with Latin translation and notes. A second edition +(1670) was carelessly printed and is untrustworthy as regards +the text; on the other hand it contained the epoch-making +notes of Fermat; the editor was S. Fermat, his son. The +great blot on the work of Bachet is his attitude to Xylander, +to whose translation he owed more than he was willing to +avow. Unfortunately neither Bachet nor Xylander was able +to use the best manuscripts; that used by Bachet was Parisinus +2379 (of the middle of the sixteenth century), with the help +of a transcription of part of a Vatican MS. (Vat. gr. 304 of +the sixteenth century), while Xylander's manuscript was the +Wolfenbüttel MS. Guelferbytanus Gudianus 1 (fifteenth cen- +tury). The best and most ancient manuscript is that of +Madrid (Matritensis 48 of the thirteenth century) which was +<pb n=455><head>TRANSLATIONS AND EDITIONS</head> +unfortunately spoiled by corrections made, especially in Books +I, II, from some manuscript of the ‘Planudean’ class; where +this is the case recourse must be had to Vat. gr. 191 which +was copied from it before it had suffered the general alteration +referred to: these are the first two of the manuscripts used by +Tannery in his definitive edition of the Greek text (Teubner, +1893, 1895). +<p>Other editors can only be shortly enumerated. In 1585 +Simon Stevin published a French version of the first four +Books, based on Xylander. Albert Girard added the fifth and +sixth Books, the complete edition appearing in 1625. German +translations were brought out by Otto Schulz in 1822 and by +G. Wertheim in 1890. Poselger translated the fragment on +Polygonal Numbers in 1810. All these translations depended +on the text of Bachet. +<p>A reproduction of Diophantus in modern notation with +introduction and notes by the present writer (second edition +1910) is based on the text of Tannery and may claim to be the +most complete and up-to-date edition. +<p>My account of the <I>Arithmetica</I> of Diophantus will be most +conveniently arranged under three main headings (1) the +notation and definitions, (2) the principal methods employed, +so far as they can be generally stated, (3) the nature of the +contents, including the assumed Porisms, with indications of +the devices by which the problems are solved. +<C><B>Notation and definitions</B>.</C> +<p>In his work <I>Die Algebra der Griechen</I> Nesselmann distin- +guishes three stages in the evolution of algebra. (1) The +first stage he calls ‘Rhetorical Algebra’ or reckoning by +means of complete words. The characteristic of this stage +is the absolute want of all symbols, the whole of the calcula- +tion being carried on by means of complete words and forming +in fact continuous prose. This first stage is represented by +such writers as Iamblichus, all Arabian and Persian algebraists, +and the oldest Italian algebraists and their followers, including +Regiomontanus. (2) The second stage Nesselmann calls the +‘Syncopated Algebra’, essentially like the first as regards +<pb n=456><head>DIOPHANTUS OF ALEXANDRIA</head> +literary style, but marked by the use of certain abbreviational +symbols for constantly recurring quantities and operations. +To this stage belong Diophantus and, after him, all the later +Europeans until about the middle of the seventeenth century +(with the exception of Vieta, who was the first to establish, +under the name of <I>Logistica speciosa</I>, as distinct from <I>Logistica +numerosa</I>, a regular system of reckoning with letters denoting +magnitudes as well as numbers). (3) To the third stage +Nesselmann gives the name of ‘Symbolic Algebra’, which +uses a complete system of notation by signs having no visible +connexion with the words or things which they represent, +a complete language of symbols, which entirely supplants the +‘rhetorical’ system, it being possible to work out a solution +without using a single word of ordinary language with the +exception of a connecting word or two here and there used for +clearness' sake. +<C><I>Sign for the unknown</I> (=<I>x</I>), <I>and its origin</I>.</C> +<p>Diophantus's system of notation then is merely abbrevia- +tional. We will consider first the representation of the +unknown quantity (our <I>x</I>). Diophantus defines the unknown +quantity as ‘<I>containing an indeterminate or undefined multi- +tude of units</I>’ (<G>plh=qos mona/dwn a)o/riston</G>), adding that it is +called <G>a)riqmo/s</G>, i.e. <I>number</I> simply, and is denoted by a certain +sign. This sign is then used all through the book. In the +earliest (the Madrid) MS. the sign takes the form <FIG>, in +Marcianus 308 it appears as <B>S</B>. In the printed editions of +Diophantus before Tannery's it was represented by the final +sigma with an accent, <G>s</G>′, which is sufficiently like the second +of the two forms. Where the symbol takes the place of +inflected forms <G>a)riqmo/n</G>, <G>a)riqmou=</G>, &c., the termination was put +above and to the right of the sign like an exponent, e.g. <G>s</G>″ for +<G>a)riqmo/n</G> as <G>t</G>″ for <G>to\n</G>, <G>s<SUP>ou=</SUP></G> for <G>a)riqmou=</G>; the symbol was, in +addition, doubled in the plural cases, thus <G>ss<SUP>oi/</SUP>, ss<SUP>ou/s</SUP></G>, &c. The +coefficient is expressed by putting the required Greek numeral +immediately after it; thus <G>s<SUP>oi\</SUP> ia</G>=<B>11</B> <G>a)riqmoi/</G>, equivalent +to 11<I>x</I>, <G>s</G>′<G>a</G>=<I>x</I>, and so on. Tannery gives reasons for think- +ing that in the archetype the case-endings did not appear, and +<pb n=457><head>NOTATION AND DEFINITIONS</head> +that the sign was not duplicated for the plural, although such +duplication was the practice of the Byzantines. That the +sign was merely an abbreviation for the word <G>a)riqmo/s</G> and no +algebraical symbol is shown by the fact that it occurs in the +manuscripts for <G>a)riqmo/s</G> in the ordinary sense as well as for +<G>a)riqmo/s</G> in the technical sense of the unknown quantity. Nor +is it confined to Diophantus. It appears in more or less +similar forms in the manuscripts of other Greek mathe- +maticians, e.g. in the Bodleian MS. of Euclid (D'Orville 301) +of the ninth century (in the forms <FIG>, or as a curved line +similar to the abbreviation for <G>kai/</G>), in the manuscripts of +the <I>Sand-reckoner</I> of Archimedes (in a form approximat- +ing to <G>s</G>), where again there is confusion caused by the +similarity of the signs for <G>a)riqmo/s</G> and <G>kai/</G>, in a manuscript +of the <I>Geodaesia</I> included in the Heronian collections edited +by Hultsch (where it appears in various forms resembling +sometimes <G>z</G>, sometimes <G>r</G>, sometimes <G>o</G>, and once <G>x</G>, with +case-endings superposed) and in a manuscript of Theon of +Smyrna. +<p>What is the origin of the sign? It is certainly not the +final sigma, as is proved by several of the forms which it +takes. I found that in the Bodleian manuscript of Diophantus +it is written in the form <FIG>, larger than and quite unlike the +final sigma. This form, combined with the fact that in one +place Xylander's manuscript read <G>ar</G> for the full word, suggested +to me that the sign might be a simple contraction of the first +two letters of <G>a)riqmo/s</G>. This seemed to be confirmed by +Gardthausen's mention of a contraction for <G>ar</G>, in the form <FIG> +occurring in a papyrus of A.D. 154, since the transition to the +form found in the manuscripts of Diophantus might easily +have been made through an intermediate form <FIG>. The loss of +the downward stroke, or of the loop, would give a close +approximation to the forms which we know. This hypothesis +as to the origin of the sign has not, so far as I know, been +improved upon. It has the immense advantage that it makes +the sign for <G>a)riqmo/s</G> similar to the signs for the powers of +the unknown, e.g. <G>*d<SUP>g</SUP></G> for <G>du/namis</G>, <G>*k<SUP>g</SUP></G> for <G>ku/bos</G>, and to the +sign <FIG> for the unit, the sole difference being that the two +letters coalesce into one instead of being separate. +<pb n=458><head>DIOPHANTUS OF ALEXANDRIA</head> +<C><I>Signs for the powers of the unknown and their reciprocals</I>.</C> +<p>The powers of the unknown, corresponding to our <MATH><I>x</I><SUP>2</SUP>, <I>x</I><SUP>3</SUP> ... <I>x</I><SUP>6</SUP></MATH>, +are defined and denoted as follows: +<C><I>x</I><SUP>2</SUP> is <G>du/namis</G> and is denoted by <G>*d<SUP>g</SUP></G></C>, +<C><I>x</I><SUP>3</SUP> ” <G>ku/bos</G> ” ” ” <G>*k<SUP>g</SUP></G></C>, +<C><I>x</I><SUP>4</SUP> ” <G>dunamodu/namis</G> ” ” <G>*d<SUP>g</SUP>*d</G></C>, +<C><I>x</I><SUP>5</SUP> ” <G>dunamo/kubos</G> ” ” <G>*d*k<SUP>g</SUP></G></C>, +<C><I>x</I><SUP>6</SUP> ” <G>kubo/kubos</G> ” ” ” <G>*k<SUP>g</SUP>*k</G></C>. +Beyond the sixth power Diophantus does not go. It should +be noted that, while the terms from <G>ku/bos</G> onwards may be +used for the powers of any ordinary known number as well as +for the powers of the unknown, <G>du/namis</G> is restricted to the +square of the unknown; wherever a particular square number +is spoken of, the term is <G>tetra/gwnos a)riqmo/s</G>. The term +<G>dunamodu/namis</G> occurs once in another author, namely in the +<I>Metrica</I> of Heron,<note>Heron, <I>Metrica</I>, p. 48. 11, 19, Schöne.</note> where it is used for the fourth power of +the side of a triangle. +<p>Diophantus has also terms and signs for the reciprocals of +the various powers of the unknown, i.e. for 1/<I>x</I>, 1/<I>x</I><SUP>2</SUP> .... +As an aliquot part was ordinarily denoted by the corresponding +numeral sign with an accent, e.g. <MATH><G>g</G>′=1/3, <G>ia</G>′=1/11</MATH>, Diophantus +has a mark appended to the symbols for <I>x</I>, <I>x</I><SUP>2</SUP> ... to denote the +reciprocals; this, which is used for aliquot parts as well, is +printed by Tannery thus, χ. With Diophantus then +<C><G>a)riqmosto/n</G>, denoted by <G>s</G><SUP>χ</SUP>, is equivalent to 1/<I>x</I>,</C> +<C><G>dunamosto/n</G>, ” <G>*d</G><SUP><G>g</G>χ</SUP> ” ” 1/<I>x</I><SUP>2</SUP>,</C> +and so on. +<p>The coefficient of the term in <I>x</I>, <I>x</I><SUP>2</SUP> ... or 1/<I>x</I>, 1/<I>x</I><SUP>2</SUP> ... is +expressed by the ordinary numeral immediately following, +e.g. <MATH><G>*d*k<SUP>g</SUP> ks</G>=26<I>x</I><SUP>5</SUP>, <G>*d</G><SUP><G>g</G>χ</SUP> <G>sn</G>=250/<I>x</I><SUP>2</SUP></MATH>. +<p>Diophantus does not need any signs for the operations of +multiplication and division. Addition is indicated by mere +juxtaposition; thus <G>*k<SUP>g</SUP> a *d<SUP></SUP> igse</G> corresponds to <MATH><I>x</I><SUP>3</SUP>+13<I>x</I><SUP>2</SUP>+5<I>x</I></MATH>. +<pb n=459><head>NOTATION AND DEFINITIONS</head> +When there are units in addition, the units are indicated by +the abbreviation <FIG>; thus <G>*k<SUP>g</SUP> a *d<SUP>g</SUP> ig s e</G> <FIG> <G>b</G> corresponds to +<MATH><I>x</I><SUP>3</SUP>+13<I>x</I><SUP>2</SUP>+5<I>x</I>+2</MATH>. +<C><I>The sign (<FIG>) for minus and its meaning</I>.</C> +<p>For subtraction alone is a sign used. The full term for +<I>wanting</I> is <G>lei=yis</G>, as opposed to <G>u(/parxis</G>, a <I>forthcoming</I>, +which denotes a <I>positive</I> term. The symbol used to indicate +a <I>wanting</I>, corresponding to our sign for <I>minus</I>, is <FIG>, which +is described in the text as a ‘<G>y</G> turned downwards and +truncated’ (<G>*y e)llipe\s ka/tw neu=on</G>). The description is evidently +interpolated, and it is now certain that the sign has nothing +to do with <G>y</G>. Nor is it confined to Diophantus, for it appears +in practically the same form in Heron's <I>Metrica</I>,<note>Heron, <I>Metrica</I>, p. 156. 8, 10.</note> where in one +place the reading of the manuscript is <G>mona/dwn od *t i′d′</G>, +74-1/14. In the manuscripts of Diophantus, when the sign +is resolved by writing the full word instead of it, it is +generally resolved into <G>lei/yei</G>, the dative of <G>lei=yis</G>, but in +other places the symbol is used instead of parts of the verb +<G>lei/pein</G>, namely <G>lipw/n</G> or <G>lei/yas</G> and once even <G>li/pwsi</G>; +sometimes <G>lei/yei</G> in the manuscripts is followed by the +<I>accusative</I>, which shows that in these cases the sign was +wrongly resolved. It is therefore a question whether Dio- +phantus himself ever used the dative <G>lei/yei</G> for <I>minus</I> at all. +The use is certainly foreign to classical Greek. Ptolemy has +in two places <G>lei=yan</G> and <G>lei/pousan</G> respectively followed, +properly, by the accusative, and in one case he has <G>to\ a)po\ +th=s *g*l leifqe\n u(po\ tou= a)po\ th=s *z*g</G> (where the meaning is +<G>*z*g</G><SUP>2</SUP>-<G>*g*l</G><SUP>2</SUP>). Hence Heron would probably have written a +participle where the <G>*t</G> occurs in the expression quoted above, +say <G>mona/dwn od leiyasw=n tessarakaide/katon</G>. On the whole, +therefore, it is probable that in Diophantus, and wherever else +it occurred, <FIG> is a compendium for the root of the verb <G>lei/pein</G>, +in fact a <G>*l</G> with <G>*i</G> placed in the middle (cf. <FIG>, an abbreviation +for <G>ta/lanton</G>). This is the hypothesis which I put forward +in 1885, and it seems to be confirmed by the fresh evidence +now available as shown above. +<pb n=460><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>Attached to the definition of <I>minus</I> is the statement that +‘a <I>wanting</I> (i.e. a <I>minus</I>) multiplied by a <I>wanting</I> makes +a <I>forthcoming</I> (i.e. a <I>plus</I>); and a <I>wanting</I> (a <I>minus</I>) multi- +plied by a <I>forthcoming</I> (a <I>plus</I>) makes a <I>wanting</I> (a <I>minus</I>)’. +<p>Since Diophantus uses no sign for <I>plus</I>, he has to put all +the positive terms in an expression together and write all the +negative terms together after the sign for <I>minus</I>; e.g. for +<MATH><I>x</I><SUP>3</SUP>-5<I>x</I><SUP>2</SUP>+8<I>x</I>-1</MATH> he necessarily writes <G>*k<SUP>g</SUP a s h <*> *d<SUP>g</SUP> e <FIG> a</G>. +<p>The Diophantine notation for fractions as well as for large +numbers has been fully explained with many illustrations +in Chapter II above. It is only necessary to add here that, +when the numerator and denominator consist of composite +expressions in terms of the unknown and its powers, he puts +the numerator first followed by <G>e)n mori/w|</G> or <G>mori/ou</G> and the +denominator. +Thus <MATH><G>*d<SUP>g</SUP> x <FIG><*>bfk e)n mori/w| *d<SUP>g</SUP>*d a <FIG> <*> <*> *d<SUP>g</SUP> x</G> +=(60<I>x</I><SUP>2</SUP>+2520)/(<I>x</I><SUP>4</SUP>+900-60<I>x</I><SUP>2</SUP>)</MATH>, [VI. 12] +and <MATH><G>*d<SUP>g</SUP> ie <*> <FIG> ls e)n mori/w| *d<SUP>g</SUP> *d a <FIG> ls <*> *d<SUP>g</SUP> ib</G> +=(15<I>x</I><SUP>2</SUP>-36)/(<I>x</I><SUP>4</SUP>+36-12<I>x</I><SUP>2</SUP>)</MATH> [VI. 14]. +<p>For a <I>term</I> in an algebraical expression, i.e. a power of <I>x</I> +with a certain coefficient, and the term containing a certain +number of units, Diophantus uses the word <G>ei)=dos</G>, ‘species’, +which primarily means the particular power of the variable +without the coefficient. At the end of the definitions he gives +directions for simplifying equations until each side contains +positive terms only, by the addition or subtraction of coeffi- +cients, and by getting rid of the negative terms (which is done +by adding the necessary quantities to both sides); the object, +he says, is to reduce the equation until one term only is left +on each side; ‘but’, he adds, ‘I will show you later how, in +the case also where two terms are left equal to one term, +such a problem is solved’. We find in fact that, when he has +to solve a quadratic equation, he endeavours by means of +suitable assumptions to reduce it either to a simple equation +or a <I>pure</I> quadratic. The solution of the mixed quadratic +<pb n=461><head>NOTATION AND DEFINITIONS</head> +in three terms is clearly assumed in several places of the +<I>Arithmetica</I>, but Diophantus never gives the necessary ex- +planation of this case as promised in the preface. +<p>Before leaving the notation of Diophantus, we may observe +that the form of it limits him to the use of one unknown at +a time. The disadvantage is obvious. For example, where +we can begin with any number of unknown quantities and +gradually eliminate all but one, Diophantus has practically to +perform his eliminations beforehand so as to express every +quantity occurring in the problem in terms of only one +unknown. When he handles problems which are by nature +indeterminate and would lead in our notation to an inde- +terminate equation containing two or three unknowns, he has +to assume for one or other of these some particular number +arbitrarily chosen, the effect being to make the problem +determinate. However, in doing so, Diophantus is careful +to say that we may for such and such a quantity put any +number whatever, say such and such a number; there is +therefore (as a rule) no real loss of generality. The particular +devices by which he contrives to express all his unknowns +in terms of one unknown are extraordinarily various and +clever. He can, of course, use the same variable <G>s</G> in the +same problem with different significations <I>successively</I>, as +when it is necessary in the course of the problem to solve +a subsidiary problem in order to enable him to make the +coefficients of the different terms of expressions in <I>x</I> such +as will answer his purpose and enable the original problem +to be solved. There are, however, two cases, II. 28, 29, where +for the proper working-out of the problem two unknowns are +imperatively necessary. We should of course use <I>x</I> and <I>y</I>; +Diophantus calls the first <G>s</G> as usual; the second, for want +of a term, he agrees to call in the first instance ‘<I>one unit</I>’, +i.e. 1. Then later, having completed the part of the solution +necessary to find <I>x</I>, he substitutes its value and uses <G>s</G> over +again for what he had originally called 1. That is, he has to +put his finger on the place to which the 1 has passed, so as +to substitute <G>s</G> for it. This is a <I>tour de force</I> in the particular +cases, and would be difficult or impossible in more complicated +problems. +<pb n=462><head>DIOPHANTUS OF ALEXANDRIA</head> +<C>The methods of Diophantus.</C> +<p>It should be premised that Diophantus will have in his +solutions no numbers whatever except ‘rational’ numbers; +he admits fractional solutions as well as integral, but he +excludes not only surds and imaginary quantities but also +negative quantities. Of a negative quantity <I>per se</I>, i.e. with- +out some greater positive quantity to subtract it from, he +had apparently no conception. Such equations then as lead +to imaginary or negative roots he regards as useless for his +purpose; the solution is in these cases <G>a)du/natos</G>, impossible. +So we find him (V. 2) describing the equation <MATH>4=4<I>x</I>+20</MATH> as +<G>a)/topos</G>, absurd, because it would give <MATH><I>x</I>=-4</MATH>. He does, it is +true, make occasional use of a quadratic which would give +a root which is positive but a surd, but only for the purpose +of obtaining limits to the root which are integers or numerical +fractions; he never uses or tries to express the actual root of +such an equation. When therefore he arrives in the course +of solution at an equation which would give an ‘irrational’ +result, he retraces his steps, finds out how his equation has +arisen, and how he may, by altering the previous work, +substitute for it another which shall give a rational result. +This gives rise in general to a subsidiary problem the solution +of which ensures a rational result for the problem itself. +<p>It is difficult to give a complete account of Diophantus's +methods without setting out the whole book, so great is the +variety of devices and artifices employed in the different +problems. There are, however, a few general methods which +do admit of differentiation and description, and these we pro- +ceed to set out under subjects. +<C>I. Diophantus's treatment of equations.</C> +<C>(A) <I>Determinate equations</I>.</C> +<p>Diophantus solved without difficulty determinate equations +of the first and second degrees; of a cubic we find only one +example in the <I>Arithmetica</I>, and that is a very special case. +<p>(1) <I>Pure determinate equations</I>. +<p>Diophantus gives a general rule for this case without regard +to degree. We have to take like from like on both sides of an +<pb n=463><head>DETERMINATE EQUATIONS</head> +equation and neutralize negative terms by adding to both +sides, then take like from like again, until we have one term +left equal to one term. After these operations have been +performed, the equation (after dividing out, if both sides +contain a power of <I>x</I>, by the lesser power) reduces to <MATH><I>Ax<SUP>m</SUP></I>=<I>B</I></MATH>, +and is considered solved. Diophantus regards this as giving +one root only, excluding any negative value as ‘impossible’. +No equation of the kind is admitted which does not give +a ‘rational’ value, integral or fractional. The value <I>x</I>=0 is +ignored in the case where the degree of the equation is reduced +by dividing out by any power of <I>x</I>. +<p>(2) <I>Mixed quadratic equations</I>. +<p>Diophantus never gives the explanation of the method of +solution which he promises in the preface. That he had +a definite method like that used in the Geometry of Heron +is proved by clear verbal explanations in different propositions. +As he requires the equation to be in the form of two positive +terms being equal to one positive term, the possible forms for +Diophantus are +<MATH>(<I>a</I>) <I>mx</I><SUP>2</SUP>+<I>px</I>=<I>q</I>, (<I>b</I>) <I>mx</I><SUP>2</SUP>=<I>px</I>+<I>q</I>, (<I>c</I>) <I>mx</I><SUP>2</SUP>+<I>q</I>=<I>px</I></MATH>. +It does not appear that Diophantus divided by <I>m</I> in order to +make the first term a square; rather he multiplied by <I>m</I> for +this purpose. It is clear that he stated the roots in the above +cases in a form equivalent to +<MATH>(<I>a</I>) (-1/2<I>p</I>+√(1/4<I>p</I><SUP>2</SUP>+<I>mq</I>))/<I>m</I>, (<I>b</I>) (1/2<I>p</I>+√(1/4<I>p</I><SUP>2</SUP>+<I>mq</I>))/<I>m</I>, +(<I>c</I>) (1/2<I>p</I>+√(1/4<I>p</I><SUP>2</SUP>-<I>mq</I>))/<I>m</I></MATH>. +The explanations which show this are to be found in VI. 6, +in IV. 39 and 31, and in V. 10 and VI. 22 respectively. For +example in V. 10 he has the equation <MATH>17<I>x</I><SUP>2</SUP>+17<72<I>x</I></MATH>, and he +says ‘Multiply half the coefficient of <I>x</I> into itself and we have +1296; subtract the product of the coefficient of <I>x</I><SUP>2</SUP> and the +term in units, or 289. The remainder is 1007, the square root +of which is not greater than 31. Add half the coefficient of <I>x</I> +and the result is not greater than 67. Divide by the coefficient +of <I>x</I><SUP>2</SUP>, and <I>x</I> is not greater than (67)/(17).’ In IV. 39 he has the +<pb n=464><head>DIOPHANTUS OF ALEXANDRIA</head> +equation <MATH>2<I>x</I><SUP>2</SUP>>6<I>x</I>+18</MATH> and says, ‘To solve this, take the square +of half the coefficient of <I>x</I>, i.e. 9, and the product of the unit- +term and the coefficient of <I>x</I><SUP>2</SUP>, i.e. 36. Adding, we have 45, +the square root of which is not less than 7. Add half the +coefficient of <I>x</I> [and divide by the coefficient of <I>x</I><SUP>2</SUP>]; whence <I>x</I> +is not less than 5.’ In these cases it will be observed that 31 +and 7 are not accurate limits, but are the nearest integral +limits which will serve his purpose. +<p>Diophantus always uses the positive sign with the radical, +and there has been much discussion as to whether he knew +that a quadratic equation has <I>two</I> roots. The evidence of the +text is inconclusive because his only object, in every case, is to +get one solution; in some cases the other root would be +negative, and would therefore naturally be ignored as ‘absurd’ +or ‘impossible’. In yet other cases where the second root is +possible it can be shown to be useless from Diophantus's point +of view. For my part, I find it difficult or impossible to +believe that Diophantus was unaware of the existence of two +real roots in such cases. It is so obvious from the geometrical +form of solution based on Eucl. II. 5, 6 and that contained in +Eucl. VI. 27-9; the construction of VI. 28, too, corresponds +in fact to the <I>negative</I> sign before the radical in the case of the +particular equation there solved, while a quite obvious and +slight variation of the construction would give the solution +corresponding to the <I>positive</I> sign. +<p>The following particular cases of quadratics occurring in +the <I>Arithmetica</I> may be quoted, with the results stated by +Diophantus. +<MATH><I>x</I><SUP>2</SUP>=4<I>x</I>-4; therefore <I>x</I>=2. (IV. 22) +325<I>x</I><SUP>2</SUP>=3<I>x</I>+18; <I>x</I>=(78)/(325) or 6/(25). (IV. 31) +84<I>x</I><SUP>2</SUP>+7<I>x</I>=7; <I>x</I>=1/4. (VI. 6) +84<I>x</I><SUP>2</SUP>-7<I>x</I>=7; <I>x</I>=1/3. (VI. 7) +630<I>x</I><SUP>2</SUP>-73<I>x</I>=6; <I>x</I>=6/(35). (VI. 9) +630<I>x</I><SUP>2</SUP>+73<I>x</I>=6; <I>x</I> is rational. (VI. 8) +5<I>x<x</I><SUP>2</SUP>-60<8<I>x</I>; <I>x</I> not<11 and not>12. (V. 30) +17<I>x</I><SUP>2</SUP>+17<72<I>x</I><19<I>x</I><SUP>2</SUP>+19; <I>x</I> not>(67)/(17) and not<(66)/(19). (V. 10) +22<I>x<x</I><SUP>2</SUP>+60<24<I>x</I>; <I>x</I> not<19 but<21. (V. 30)</MATH> +<pb n=465><head>DETERMINATE EQUATIONS</head> +In the first and third of the last three cases the limits are not +accurate, but are <I>integral</I> limits which are <I>a fortiori</I> safe. +In the second (66)/(19) should have been (67)/(19), and it would have been +more correct to say that, if <I>x</I> is not greater than (67)/(17) and not +less than (67)/(19), the given conditions are <I>a fortiori</I> satisfied. +<p>For comparison with Diophantus's solutions of quadratic +equations we may refer to a few of his solutions of +<p>(3) <I>Simultaneous equations involving quadratics</I>. +<p>In I. 27, 28, and 30 we have the following pairs of equations. +<MATH><BRACE>(<G>a</G>) <G>x</G>+<G>h</G>=2<I>a</I> <G>xh</G>=<I>B</I></BRACE>, <BRACE>(<G>b</G>) <G>x</G>+<G>h</G>=2<I>a</I> <G>x</G><SUP>2</SUP>+<G>h</G><SUP>2</SUP>=<I>B</I></BRACE>, <BRACE>(<G>g</G>) <G>x</G>-<G>h</G>=2<I>a</I> <G>xh</G>=<I>B</I></BRA +E></MATH>. +<p>I use the Greek letters for the numbers required to be found +as distinct from the one unknown which Diophantus uses, and +which I shall call <I>x</I>. +<p>In (<G>a</G>), he says, let <MATH><G>x</G>-<G>h</G>=2<I>x</I> (<G>x</G>><G>h</G>)</MATH>. +<p>It follows, by addition and subtraction, that <MATH><G>x</G>=<I>a</I>+<I>x</I>, <G>h</G>=<I>a</I>-<I>x</I></MATH>; +therefore <MATH><G>xh</G>=(<I>a</I>+<I>x</I>) (<I>a</I>-<I>x</I>)=<I>a</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>=<I>B</I></MATH>, +and <I>x</I> is found from the pure quadratic equation. +<p>In (<G>b</G>) similarly he assumes <MATH><G>x</G>-<G>h</G>=2<I>x</I></MATH>, and the resulting +equation is <MATH><G>x</G><SUP>2</SUP>+<G>h</G><SUP>2</SUP>=(<I>a</I>+<I>x</I>)<SUP>2</SUP>+(<I>a</I>-<I>x</I>)<SUP>2</SUP>=2(<I>a</I><SUP>2</SUP>+<I>x</I><SUP>2</SUP>)=<I>B</I></MATH>. +<p>In (<G>g</G>) he puts <MATH><G>x</G>+<G>h</G>=2<I>x</I></MATH> and solves as in the case of (<G>a</G>). +<p>(4) <I>Cubic equation</I>. +<p>Only one very particular case occurs. In VI. 17 the problem +leads to the equation +<MATH><I>x</I><SUP>2</SUP>+2<I>x</I>+3=<I>x</I><SUP>3</SUP>+3<I>x</I>-3<I>x</I><SUP>2</SUP>-1</MATH>. +Diophantus says simply ‘whence <I>x</I> is found to be 4’. In fact +the equation reduces to +<MATH><I>x</I><SUP>3</SUP>+<I>x</I>=4<I>x</I><SUP>2</SUP>+4</MATH>. +Diophantus no doubt detected, and divided out by, the common +factor <MATH><I>x</I><SUP>2</SUP>+1</MATH>, leaving <MATH><I>x</I>=4</MATH>. +<pb n=466><head>DIOPHANTUS OF ALEXANDRIA</head> +<C>(B) <I>Indeterminate equations</I>.</C> +<p>Diophantus says nothing of indeterminate equations of the +first degree. The reason is perhaps that it is a principle with +him to admit rational <I>fractional</I> as well as integral solutions, +whereas the whole point of indeterminate equations of the +first degree is to obtain a solution in <I>integral</I> numbers. +Without this limitation (foreign to Diophantus) such equa- +tions have no significance. +<C>(<G>a</G>) <I>Indeterminate equations of the second degree</I>.</C> +<p>The form in which these equations occur is invariably this: +one or two (but never more) functions of <B><I>x</I></B> of the form +<MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I></MATH> or simpler forms are to be made rational square +numbers by finding a suitable value for <I>x</I>. That is, we have +to solve, in the most general case, one or two equations of the +form <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>. +<p>(1) <I>Single equation</I>. +<p>The solutions take different forms according to the particular +values of the coefficients. Special cases arise when one or +more of them vanish or they satisfy certain conditions. +<p>1. When <I>A</I> or <I>C</I> or both vanish, the equation can always +be solved rationally. +<p>Form <MATH><I>Bx</I>=<I>y</I><SUP>2</SUP></MATH>. +<p>Form <MATH><I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>. +<p>Diophantus puts for <I>y</I><SUP>2</SUP> any determinate square <I>m</I><SUP>2</SUP>, and <I>x</I> is +immediately found. +<p>Form <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>=<I>y</I><SUP>2</SUP></MATH>. +<p>Diophantus puts for <I>y</I> any multiple of <I>x</I>, as <I>m</I>/<I>n</I><I>x</I>. +<p>2. The equation <MATH><I>Ax</I><SUP>2</SUP>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH> can be rationally solved accord- +ing to Diophantus: +<p>(<I>a</I>) when <I>A</I> is positive and a square, say <I>a</I><SUP>2</SUP>; +in this case we put <MATH><I>a</I><SUP>2</SUP><I>x</I><SUP>2</SUP>+<I>C</I>=(<I>ax</I>±<I>m</I>)<SUP>2</SUP></MATH>, whence +<MATH><I>x</I>=±(<I>C</I>-<I>m</I><SUP>2</SUP>)/(2<I>ma</I>)</MATH> +(<I>m</I> and the sign being so chosen as to give <I>x</I> a positive value); +<pb n=467><head>INDETERMINATE EQUATIONS</head> +<p>(<G>b</G>) when <I>C</I> is positive and a square, say <I>c</I><SUP>2</SUP>; +in this case Diophantus puts <MATH><I>Ax</I><SUP>2</SUP>+<I>c</I><SUP>2</SUP>=(<I>mx</I>±<I>c</I>)<SUP>2</SUP></MATH>, and obtains +<MATH><I>x</I>=±(2<I>mc</I>)/(<I>A</I>-<I>m</I><SUP>2</SUP>)</MATH>. +<p>(<G>g</G>) When one solution is known, any number of other +solutions can be found. This is stated in the Lemma to +VI. 15. It would be true not only of the cases <MATH>±<I>Ax</I><SUP>2</SUP>∓<I>C</I>=<I>y</I><SUP>2</SUP></MATH>, +but of the general case <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>. Diophantus, how- +ever, only states it of the case <MATH><I>Ax</I><SUP>2</SUP>-<I>C</I>=<I>y</I><SUP>2</SUP></MATH>. +<p>His method of finding other (greater) values of <I>x</I> satisfy- +ing the equation when one (<I>x</I><SUB>0</SUB>) is known is as follows. If +<MATH><I>Ax</I><SUB>0</SUB><SUP>2</SUP>-<I>C</I>=<I>q</I><SUP>2</SUP></MATH>, he substitutes in the original equation (<I>x</I><SUB>0</SUB>+<I>x</I>) +for <I>x</I> and (<I>q</I>-<I>kx</I>) for <I>y</I>, where <I>k</I> is some integer. +<p>Then, since <MATH><I>A</I>(<I>x</I><SUB>0</SUB>+<I>x</I>)<SUP>2</SUP>-<I>C</I>=(<I>q</I>-<I>kx</I>)<SUP>2</SUP></MATH>, while <MATH><I>Ax</I><SUB>0</SUB><SUP>2</SUP>-<I>C</I>=<I>q</I><SUP>2</SUP></MATH>, +it follows by subtraction that +<MATH>2<I>x</I>(<I>Ax</I><SUB>0</SUB>+<I>kq</I>)=<I>x</I><SUP>2</SUP>(<I>k</I><SUP>2</SUP>-<I>A</I>)</MATH>, +whence <MATH><I>x</I>=2(<I>Ax</I><SUB>0</SUB>+<I>kq</I>)/(<I>k</I><SUP>2</SUP>-<I>A</I>)</MATH>, +and the new value of <I>x</I> is <MATH><I>x</I><SUB>0</SUB>+(2(<I>Ax</I><SUB>0</SUB>+<I>kq</I>))/(<I>k</I><SUP>2</SUP>-<I>A</I>)</MATH>. +<p>Form <MATH><I>Ax</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>=<I>y</I><SUP>2</SUP></MATH>. +<p>Diophantus says (VI. 14) that a rational solution of this +case is only possible when <I>A</I> is the sum of two squares. +<p>[In fact, if <I>x</I>=<I>p/q</I> satisfies the equation, and <MATH><I>Ax</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>=<I>k</I><SUP>2</SUP></MATH>, +we have <MATH><I>Ap</I><SUP>2</SUP>=<I>c</I><SUP>2</SUP><I>q</I><SUP>2</SUP>+<I>k</I><SUP>2</SUP><I>q</I><SUP>2</SUP></MATH>, +or <MATH><I>A</I>=((<I>cq</I>)/<I>p</I>)<SUP>2</SUP>+((<I>kq</I>)/<I>p</I>)<SUP>2</SUP>.]</MATH> +<p>Form <MATH><I>Ax</I><SUP>2</SUP>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>. +<p>Diophantus proves in the Lemma to VI. 12 that this equa- +tion has an infinite number of solutions when <I>A</I>+<I>C</I> is a square, +i.e. in the particular case where <I>x</I>=1 is a solution. (He does +not, however, always bear this in mind, for in III. 10 he +regards the equation <MATH>52<I>x</I><SUP>2</SUP>+12=<I>y</I><SUP>2</SUP></MATH> as impossible, though +<MATH>52+12=64</MATH> is a square, just as, in III. 11, <MATH>266<I>x</I><SUP>2</SUP>-10=<I>y</I><SUP>2</SUP></MATH> +is regarded as impossible.) +<p>Suppose that <MATH><I>A</I>+<I>C</I>=<I>q</I><SUP>2</SUP></MATH>; the equation is then solved by +<pb n=468><head>DIOPHANTUS OF ALEXANDRIA</head> +substituting in the original equation 1+<I>x</I> for <I>x</I> and (<I>q</I>-<I>kx</I>) +for <I>y</I>, where <I>k</I> is some integer. +<p>3. Form <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH>. +<p>This can be reduced to the form in which the second term is +wanting by replacing <I>x</I> by <I>z</I>- <I>B</I>/(2<I>A</I>). +<p>Diophantus, however, treats this case separately and less +fully. According to him, a rational solution of the equation +<MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>2</SUP></MATH> is only possible +<p>(<G>a</G>) when <I>A</I> is positive and a square, say <I>a</I><SUP>2</SUP>; +<p>(<G>b</G>) when <I>C</I> is positive and a square, say <I>c</I><SUP>2</SUP>; +<p>(<G>g</G>) when 1/4<I>B</I><SUP>2</SUP>-<I>AC</I> is positive and a square. +<p>In case (<G>a</G>) <I>y</I> is put equal to (<I>ax</I>-<I>m</I>), and in case (<G>b</G>) <I>y</I> is put +equal to (<I>mx</I>-<I>c</I>). +<p>Case (<G>g</G>) is not expressly enunciated, but occurs, as it +were, accidentally (IV. 31). The equation to be solved is +<MATH>3<I>x</I>+18-<I>x</I><SUP>2</SUP>=<I>y</I><SUP>2</SUP></MATH>. Diophantus first assumes <MATH>3<I>x</I>+18-<I>x</I><SUP>2</SUP>=4<I>x</I><SUP>2</SUP></MATH>, +which gives the quadratic <MATH>3<I>x</I>+18=5<I>x</I><SUP>2</SUP></MATH>; but this ‘is not +rational’. Therefore the assumption of 4<I>x</I><SUP>2</SUP> for <I>y</I><SUP>2</SUP> will not do, +‘and we must find a square [to replace 4] such that 18 times +(this square+1)+(3/2)<SUP>2</SUP> may be a square’. The auxiliary +equation is therefore <MATH>18(<I>m</I><SUP>2</SUP>+1)+9/4=<I>y</I><SUP>2</SUP></MATH>, or <MATH>72<I>m</I><SUP>2</SUP>+81=a</MATH> +square, and Diophantus assumes <MATH>72<I>m</I><SUP>2</SUP>+81=(8<I>m</I>+9)<SUP>2</SUP></MATH>, whence +<I>m</I>=18. Then, assuming <MATH>3<I>x</I>+18-<I>x</I><SUP>2</SUP>=(18)<SUP>2</SUP><I>x</I><SUP>2</SUP></MATH>, he obtains the +equation <MATH>325<I>x</I><SUP>2</SUP>-3<I>x</I>-18=0</MATH>, whence <MATH><I>x</I>=(78)/(325)</MATH>, that is, 6/25. +<p>(2) <I>Double equation</I>. +<p>The Greek term is <G>diploi+so/ths, diplh= i)so/ths</G> or <G>diplh= i)/swsis</G>. +Two different functions of the unknown have to be made +simultaneously squares. The general case is to solve in +rational numbers the equations +<MATH><BRACE><I>mx</I><SUP>2</SUP>+<G>a</G><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP><I>nx</I><SUP>2</SUP>+<G>b</G><I>x</I>+<I>b</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>. +The necessary preliminary condition is that each of the two +expressions can be made a square. This is always possible +when the first term (in <I>x</I><SUP>2</SUP>) is wanting. We take this simplest +case first. +<pb n=469><head>INDETERMINATE EQUATIONS</head> +<p>1. <I>Double equation of the first degree</I>. +<p>The equations are +<MATH><G>a</G><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP>, +<G>b</G><I>x</I>+<I>b</I>=<I>w</I><SUP>2</SUP></MATH>. +<p>Diophantus has one general method taking slightly different +forms according to the nature of the coefficients. +<p>(<I>a</I>) First method of solution. +<p>This depends upon the identity +<MATH>{1/2(<I>p</I>+<I>q</I>)}<SUP>2</SUP>-{1/2(<I>p</I>-<I>q</I>)}<SUP>2</SUP>=<I>pq</I></MATH>. +<p>If the difference between the two expressions in <I>x</I> can be +separated into two factors <I>p, q</I>, the expressions themselves +are equated to <MATH>{1/2(<I>p</I>+<I>q</I>)}<SUP>2</SUP></MATH> and <MATH>{1/2(<I>p</I>-<I>q</I>)}<SUP>2</SUP></MATH> respectively. As +Diophantus himself says in II. 11, we ‘equate either the square +of half the difference of the two factors to the lesser of the +expressions, or the square of half the sum to the greater’. +<p>We will consider the general case and investigate to what +particular classes of cases the method is applicable from +Diophantus's point of view, remembering that the final quad- +ratic in <I>x</I> must always reduce to a single equation. +<p>Subtracting, we have <MATH>(<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>)=<I>u</I><SUP>2</SUP>-<I>w</I><SUP>2</SUP></MATH>. +<p>Separate <MATH>(<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>)</MATH> into the factors +<MATH><I>p</I>, {(<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>)}/<I>p</I></MATH>. +<p>We write accordingly +<MATH><I>u</I>±<I>w</I>=((<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>))/<I>p</I>, +<I>u</I>∓<I>w</I>=<I>p</I></MATH>. +<p>Thus <MATH><BRACE><I>u</I><SUP>2</SUP>=<G>a</G><I>x</I>+<I>a</I>=1/4((<G>a</G>-<G>b</G>)<I>x</I>+(<I>a</I>-<I>b</I>))/<I>p</I>+<I>p</I><SUP>2</SUP></BRACE></MATH>; +therefore <MATH>{(<G>a</G>-<G>b</G>)<I>x</I>+<I>a</I>-<I>b</I>+<I>p</I><SUP>2</SUP>}<SUP>2</SUP>=4<I>p</I><SUP>2</SUP>(<G>a</G><I>x</I>+<I>a</I>)</MATH>. +<p>This reduces to +<MATH>(<G>a</G>-<G>b</G>)<SUP>2</SUP><I>x</I><SUP>2</SUP>+2<I>x</I>{(<G>a</G>-<G>b</G>)(<I>a</I>-<I>b</I>)-<I>p</I><SUP>2</SUP>(<G>a</G>+<G>b</G>)} ++(<I>a</I>-<I>b</I>)<SUP>2</SUP>-2<I>p</I><SUP>2</SUP>(<I>a</I>+<I>b</I>)+<I>p</I><SUP>4</SUP>=0</MATH>. +<pb n=470><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>In order that this equation may reduce to a simple equation, +either +<p>(1) the coefficient of <I>x</I><SUP>2</SUP> must vanish, or <MATH><G>a</G>-<G>b</G>=0</MATH>, +or (2) the absolute term must vanish, that is, +<MATH><I>p</I><SUP>4</SUP>-2<I>p</I><SUP>2</SUP>(<I>a</I>+<I>b</I>)+(<I>a</I>-<I>b</I>)<SUP>2</SUP>=0</MATH>, +or <MATH>{<I>p</I><SUP>2</SUP>-(<I>a</I>+<I>b</I>}<SUP>2</SUP>=4<I>ab</I></MATH>, +so that <I>ab</I> must be a square number. +<p>As regards condition (1) we observe that it is really sufficient +if <G>a</G><I>n</I><SUP>2</SUP>=<G>b</G><I>m</I><SUP>2</SUP>, since, if <G>a</G><I>x</I>+<I>a</I> is a square, (<G>a</G><I>x</I>+<I>a</I>)<I>n</I><SUP>2</SUP> is equally +a square, and, if <G>b</G><I>x</I>+<I>b</I> is a square, so is (<G>b</G><I>x</I>+<I>b</I>)<I>m</I><SUP>2</SUP>, and +vice versa. +<p>That is, (1) we can solve any pair of equations of the form +<MATH><BRACE><G>a</G><I>m</I><SUP>2</SUP><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP> +<G>a</G><I>n</I><SUP>2</SUP><I>x</I>+<I>b</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>. +<p>Multiply by <I>n</I><SUP>2</SUP>, <I>m</I><SUP>2</SUP> respectively, and we have to solve the +equations +<MATH><BRACE><G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>x</I>+<I>an</I><SUP>2</SUP>=<I>u</I>′<SUP>2</SUP> +<G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>x</I>+<I>bm</I><SUP>2</SUP>=<I>w</I>′<SUP>2</SUP></BRACE></MATH>. +<p>Separate the difference, <I>an</I><SUP>2</SUP>-<I>bm</I><SUP>2</SUP>, into two factors <I>p</I>, <I>q</I> and +put <MATH><I>u</I>′±<I>w</I>′=<I>p</I>, +<I>u</I>′∓<I>w</I>′=<I>q</I></MATH>; +therefore <MATH><I>u</I>′<SUP>2</SUP>=1/4(<I>p</I>+<I>q</I>)<SUP>2</SUP>, <I>w</I>′<SUP>2</SUP>=1/4(<I>p</I>-<I>q</I>)<SUP>2</SUP></MATH>, +and <MATH><G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>x</I>+<I>an</I><SUP>2</SUP>=1/4(<I>p</I>+<I>q</I>)<SUP>2</SUP>, +<G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><I>x</I>+<I>bm</I><SUP>2</SUP>=1/4(<I>p</I>-<I>q</I>)<SUP>2</SUP></MATH>; +and from either of these equations we get +<MATH><I>x</I>=(1/4(<I>p</I><SUP>2</SUP>+<I>q</I><SUP>2</SUP>)-1/2(<I>an</I><SUP>2</SUP>+<I>bm</I><SUP>2</SUP>))/(<G>a</G><I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP>)</MATH>, +since <MATH><I>pq</I>=<I>an</I><SUP>2</SUP>-<I>bm</I><SUP>2</SUP></MATH>. +<p>Any factors <I>p, q</I> can be chosen provided that the resulting +value of <I>x</I> is <I>positive</I>. +<pb n=471><head>INDETERMINATE EQUATIONS</head> +<p>Ex. from Diophantus: +<MATH><BRACE>65-6<I>x</I>=<I>u</I><SUP>2</SUP> +65-24<I>x</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>; (IV. 32) +therefore <MATH><BRACE>260-24<I>x</I>=<I>u</I>′<SUP>2</SUP> +65-24<I>x</I>=<I>w</I>′<SUP>2</SUP></BRACE></MATH>. +<p>The difference=195=15.13, say; +therefore <MATH>1/4(15-13)<SUP>2</SUP>=65-24<I>x</I></MATH>; that is, <MATH>24<I>x</I>=64</MATH>, and <MATH><I>x</I>=8/3</MATH>. +<p>Taking now the condition (2) that <I>ab</I> is a square, we see +that the equations can be solved in the cases where either +<I>a</I> and <I>b</I> are both squares, or the ratio of <I>a</I> to <I>b</I> is the ratio of +a square to a square. If the equations are +<MATH><G>a</G><I>x</I>+<I>c</I><SUP>2</SUP>=<I>u</I><SUP>2</SUP>, +<G>b</G><I>x</I>+<I>d</I><SUP>2</SUP>=<I>w</I><SUP>2</SUP></MATH>, +and factors are taken of the difference between the expressions +as they stand, then, since one factor <I>p</I>, as we saw, satisfies the +equation <MATH>{<I>p</I><SUP>2</SUP>-(<I>c</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>}<SUP>2</SUP>=4<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP></MATH>, +we must have <MATH><I>p</I>=<I>c</I>±<I>d</I></MATH>. +<p>Ex. from Diophantus: +<MATH><BRACE>10<I>x</I>+9=<I>u</I><SUP>2</SUP> +5<I>x</I>+4=<I>w</I><SUP>2</SUP></BRACE></MATH>. (III. 15) +The difference is <MATH>5<I>x</I>+5=5(<I>x</I>+1)</MATH>; the solution is given by +<MATH>(1/2<I>x</I>+3)<SUP>2</SUP>=10<I>x</I>+9, and <I>x</I>=28</MATH>. +<p>Another method is to multiply the equations by squares +such that, when the expressions are subtracted, the absolute +term vanishes. The case can be worked out generally, thus. +<p>Multiply by <I>d</I><SUP>2</SUP> and <I>c</I><SUP>2</SUP> respectively, and we have to solve +<MATH><BRACE><G>a</G><I>d</I><SUP>2</SUP><I>x</I>+<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=<I>u</I><SUP>2</SUP> +<G>b</G><I>c</I><SUP>2</SUP><I>x</I>+<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=<I>w</I><SUP>2</SUP></BRACE></MATH>. +<p><MATH>Difference =(<G>a</G><I>d</I><SUP>2</SUP>-<G>b</G><I>c</I><SUP>2</SUP>)<I>x</I>=<I>px.q</I></MATH> say. +<p>Then <I>x</I> is found from the equation +<MATH><G>a</G><I>d</I><SUP>2</SUP><I>x</I>+<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=1/4(<I>px</I>+<I>q</I>)<SUP>2</SUP></MATH>, +which gives <MATH><I>p</I><SUP>2</SUP><I>x</I><SUP>2</SUP>+2<I>x</I>(<I>pq</I>-2<G>a</G><I>d</I><SUP>2</SUP>)+<I>q</I><SUP>2</SUP>-4<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=0</MATH>, +<pb n=472><head>DIOPHANTUS OF ALEXANDRIA</head> +or, since <MATH><I>pq</I>=<G>a</G><I>d</I><SUP>2</SUP>-<G>b</G><I>c</I><SUP>2</SUP>, +<I>p</I><SUP>2</SUP><I>x</I><SUP>2</SUP>-2<I>x</I>(<G>a</G><I>d</I><SUP>2</SUP>+<G>b</G><I>c</I><SUP>2</SUP>)+<I>q</I><SUP>2</SUP>-4<I>c</I><SUP>2</SUP><I>d</I><SUP>2</SUP>=0</MATH>. +<p>In order that this may reduce to a simple equation, as +Diophantus requires, the absolute term must vanish, so that +<MATH><I>q</I>=2<I>cd</I></MATH>. The method therefore only gives one solution, since +<I>q</I> is restricted to the value 2<I>cd</I>. +<p>Ex. from Diophantus: +<MATH><BRACE>8<I>x</I>+4=<I>u</I><SUP>2</SUP> +6<I>x</I>+4=<I>w</I><SUP>2</SUP></BRACE></MATH>. (IV. 39) +Difference 2<I>x</I>; <I>q</I> necessarily taken to be 2√4 or 4; factors +therefore 1/2<I>x</I>, 4. Therefore <MATH>8<I>x</I>+4=1/4(1/2<I>x</I>+4)<SUP>2</SUP></MATH>, and <MATH><I>x</I>=112</MATH>. +<p>(<G>b</G>) Second method of solution of a double equation of the +first degree. +<p>There is only one case of this in Diophantus, the equations +being of the form +<MATH><BRACE><I>hx</I>+<I>n</I><SUP>2</SUP>=<I>u</I><SUP>2</SUP> +(<I>h</I>+<I>f</I>)<I>x</I>+<I>n</I><SUP>2</SUP>=<I>w</I><SUP>2</SUP></BRACE></MATH>. +<p>Suppose <MATH><I>hx</I>+<I>n</I><SUP>2</SUP>=(<I>y</I>+<I>n</I>)<SUP>2</SUP></MATH>; therefore <MATH><I>hx</I>=<I>y</I><SUP>2</SUP>+2<I>ny</I></MATH>, +and <MATH>(<I>h</I>+<I>f</I>)<I>x</I>+<I>n</I><SUP>2</SUP>=(<I>y</I>+<I>n</I>)<SUP>2</SUP>+<I>f/h</I>(<I>y</I><SUP>2</SUP>+2<I>ny</I>)</MATH>. +<p>It only remains to make the latter expression a square, +which is done by equating it to (<I>py</I>-<I>n</I>)<SUP>2</SUP>. +<p>The case in Diophantus is the same as that last mentioned +(IV. 39). Where I have used <I>y</I>, Diophantus as usual contrives +to use his one unknown a second time. +<p>2. <I>Double equations of the second degree</I>. +<p>The general form is +<MATH><BRACE><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>u</I><SUP>2</SUP> +<I>A</I>′<I>x</I><SUP>2</SUP>+<I>B</I>′<I>x</I>+<I>C</I>′=<I>w</I><SUP>2</SUP></BRACE></MATH>; +but only three types appear in Diophantus, namely +(1) <MATH><BRACE><G>r</G><SUP>2</SUP><I>x</I><SUP>2</SUP>+<G>a</G><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP> +<G>r</G><SUP>2</SUP><I>x</I><SUP>2</SUP>+<G>b</G><I>x</I>+<I>b</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>, where, except in one case, <I>a</I>=<I>b</I>. +<pb n=473><head>INDETERMINATE EQUATIONS</head> +(2) <MATH><BRACE><I>x</I><SUP>2</SUP>+<G>a</G><I>x</I>+<I>a</I>=<I>u</I><SUP>2</SUP> +<G>b</G><I>x</I><SUP>2</SUP>+<I>a</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>. +(The case where the absolute terms are in the ratio of a square +to a square reduces to this.) +<p>In all examples of these cases the usual method of solution +applies. +(3) <MATH><BRACE><G>a</G><I>x</I><SUP>2</SUP>+<I>ax</I>=<I>u</I><SUP>2</SUP> +<G>b</G><I>x</I><SUP>2</SUP>+<I>bx</I>=<I>w</I><SUP>2</SUP></BRACE></MATH>. +<p>The usual method does not here serve, and a special artifice +is required. +<p>Diophantus assumes <MATH><I>u</I><SUP>2</SUP>=<I>m</I><SUP>2</SUP><I>x</I><SUP>2</SUP></MATH>. +<p>Then <MATH><I>x</I>=<I>a</I>/(<I>m</I><SUP>2</SUP>-<G>a</G>)</MATH> and, by substitution in the second +equation, we have +<MATH><G>b</G>(<I>a</I>/(<I>m</I><SUP>2</SUP>-<G>a</G>))<SUP>2</SUP>+(<I>ba</I>)/(<I>m</I><SUP>2</SUP>-<G>a</G>)</MATH>, which must be made a square, +or <MATH><I>a</I><SUP>2</SUP><G>b</G>+<I>ba</I>(<I>m</I><SUP>2</SUP>-<G>a</G>)</MATH> must be a square. +<p>We have therefore to solve the equation +<MATH><I>abm</I><SUP>2</SUP>+<I>a</I>(<I>a</I><G>b</G>-<G>a</G><I>b</I>)=<I>y</I><SUP>2</SUP></MATH>, +which can or cannot be solved by Diophantus's methods +according to the nature of the coefficients. Thus it can be +solved if <MATH>(<I>a</I><G>b</G>-<G>a</G><I>b</I>)/<I>a</I></MATH> is a square, or if <I>a/b</I> is a square. +Examples in VI. 12, 14. +<C>(<I>b</I>) <I>Indeterminate equations of a degree higher than the +second</I>.</C> +<p>(1) <I>Single equations</I>. +<p>There are two classes, namely those in which expressions +in <I>x</I> have to be made squares or cubes respectively. The +general form is therefore +<MATH><I>Ax</I><SUP><I>n</I></SUP>+<I>Bx</I><SUP><I>n</I>-1</SUP>+...+<I>Kx</I>-<I>L</I>=<I>y</I><SUP>2</SUP> or <I>y</I><SUP>3</SUP></MATH>. +<p>In Diophantus <I>n</I> does not exceed 6, and in the second class +of cases, where the expression has to be made a cube, <I>n</I> does +not generally exceed 3. +<pb n=474><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>The species of the first class found in the <I>Arithmetica</I> are +as follows. +<p>1. Equation <MATH><I>Ax</I><SUP>3</SUP>+<I>Bx</I><SUP>2</SUP>+<I>Cx</I>+<I>d</I><SUP>2</SUP>=<I>y</I><SUP>2</SUP></MATH>. +<p>As the absolute term is a square, we can assume +<MATH><I>y</I>=<I>C</I>/(2<I>d</I>)<I>x</I>+<I>d</I></MATH>, +or we might assume <MATH><I>y</I>=<I>m</I><SUP>2</SUP><I>x</I><SUP>2</SUP>+<I>nx</I>+<I>d</I></MATH> and determine <I>m</I>, <I>n</I> so +that the coefficients of <I>x</I>, <I>x</I><SUP>2</SUP> in the resulting equation both +vanish. +<p>Diophantus has only one case, <MATH><I>x</I><SUP>3</SUP>-3<I>x</I><SUP>2</SUP>+3<I>x</I>+1=<I>y</I><SUP>2</SUP></MATH> (VI. 18), +and uses the first method. +<p>2. Equation <MATH><I>Ax</I><SUP>4</SUP>+<I>Bx</I><SUP>3</SUP>+<I>Cx</I><SUP>2</SUP>+<I>Dx</I>+<I>E</I>=<I>y</I><SUP>2</SUP></MATH>, where either <I>A</I> or +<I>E</I> is a square. +<p>If <I>A</I> is a square (=<I>a</I><SUP>2</SUP>), we may assume <MATH><I>y</I>=<I>ax</I><SUP>2</SUP>+<I>B</I>/(2<I>a</I>)<I>x</I>+<I>n</I></MATH>, +determining <I>n</I> so that the term in <I>x</I><SUP>2</SUP> in the resulting equa- +tion may vanish. If <I>E</I> is a square (=<I>e</I><SUP>2</SUP>), we may assume +<MATH><I>y</I>=<I>mx</I><SUP>2</SUP>+<I>D</I>/(2<I>e</I>)<I>x</I>+<I>e</I></MATH>, determining <I>m</I> so that the term in <I>x</I><SUP>2</SUP> in the +resulting equation may vanish. We shall then, in either case, +obtain a simple equation in <I>x</I>. +<p>3. Equation <MATH><I>Ax</I><SUP>4</SUP>+<I>Cx</I><SUP>2</SUP>+<I>E</I>=<I>y</I><SUP>2</SUP></MATH>, but in special cases only where +all the coefficients are squares. +<p>4. Equation <MATH><I>Ax</I><SUP>4</SUP>+<I>E</I>=<I>y</I><SUP>2</SUP></MATH>. +<p>The case occurring in Diophantus is <MATH><I>x</I><SUP>4</SUP>+97=<I>y</I><SUP>2</SUP></MATH> (V. 29). +Diophantus tries one assumption, <MATH><I>y</I>=<I>x</I><SUP>2</SUP>-10</MATH>, and finds that +this gives <MATH><I>x</I><SUP>2</SUP>=3/(20)</MATH>, which leads to no rational result. He +therefore goes back and alters his assumptions so that he +is able to replace the refractory equation by <MATH><I>x</I><SUP>4</SUP>+337=<I>y</I><SUP>2</SUP></MATH>, +and at the same time to find a suitable value for <I>y</I>, namely +<MATH><I>y</I>=<I>x</I><SUP>2</SUP>-25</MATH>, which produces a rational result, <MATH><I>x</I>=(12)/5</MATH>. +<p>5. Equation of sixth degree in the special form +<MATH><I>x</I><SUP>6</SUP>-<I>Ax</I><SUP>3</SUP>+<I>Bx</I>+<I>c</I><SUP>2</SUP>=<I>y</I><SUP>2</SUP></MATH>. +<p>Putting <MATH><I>y</I>=<I>x</I><SUP>3</SUP>+<I>c</I></MATH>, we have <MATH>-<I>Ax</I><SUP>2</SUP>+<I>B</I>=2<I>cx</I><SUP>2</SUP></MATH>, and +<MATH><I>x</I><SUP>2</SUP>=<I>B</I>/(<I>A</I>+2<I>c</I>)</MATH>, which gives a rational solution if <I>B</I>/(<I>A</I>+2<I>c</I>) is +<pb n=475><head>INDETERMINATE EQUATIONS</head> +a square. Where this does not hold (in IV. 18) Diophantus +harks back and replaces the equation <MATH><I>x</I><SUP>6</SUP>-16<I>x</I><SUP>3</SUP>+<I>x</I>+64=<I>y</I><SUP>2</SUP></MATH> +by another, <MATH><I>x</I><SUP>6</SUP>-128<I>x</I><SUP>3</SUP>+<I>x</I>+4096=<I>y</I><SUP>2</SUP></MATH>. +<p>Of expressions which have to be made <I>cubes</I>, we have the +following cases. +<p>1. <MATH><I>Ax</I><SUP>2</SUP>+<I>Bx</I>+<I>C</I>=<I>y</I><SUP>3</SUP></MATH>. +<p>There are only two cases of this. First, in VI. 1, <MATH><I>x</I><SUP>2</SUP>-4<I>x</I>+4</MATH> +has to be made a cube, being already a square. Diophantus +naturally makes <I>x</I>-2 a cube. +<p>Secondly, a peculiar case occurs in VI. 17, where a cube has +to be found exceeding a square by 2. Diophantus assumes +(<I>x</I>-1)<SUP>3</SUP> for the cube and (<I>x</I>+1)<SUP>2</SUP> for the square. This gives +<MATH><I>x</I><SUP>3</SUP>-3<I>x</I><SUP>2</SUP>+3<I>x</I>-1=<I>x</I><SUP>2</SUP>+2<I>x</I>+3</MATH>, +or <MATH><I>x</I><SUP>3</SUP>+<I>x</I>=4<I>x</I><SUP>2</SUP>+4</MATH>. We divide out by <I>x</I><SUP>2</SUP>+1, and <MATH><I>x</I>=4</MATH>. It +seems evident that the assumptions were made with knowledge +and intention. That is, Diophantus knew of the solution 27 +and 25 and deliberately led up to it. It is unlikely that he was +aware of the fact, observed by Fermat, that 27 and 25 are the +only integral numbers satisfying the condition. +<p>2. <MATH><I>Ax</I><SUP>3</SUP>+<I>Bx</I><SUP>2</SUP>+<I>Cx</I>+<I>D</I>=<I>y</I><SUP>3</SUP></MATH>, where either <I>A</I> or <I>D</I> is a cube +number, or both are cube numbers. Where <I>A</I> is a cube (<I>a</I><SUP>3</SUP>), +we have only to assume <MATH><I>y</I>=<I>ax</I>+<I>B</I>/(3<I>a</I><SUP>2</SUP>)</MATH>, and where <I>D</I> is a cube +(<I>d</I><SUP>3</SUP>), <MATH><I>y</I>=<I>C</I>/(3<I>d</I><SUP>2</SUP>)<I>x</I>+<I>d</I></MATH>. Where <MATH><I>A</I>=<I>a</I><SUP>3</SUP></MATH> and <MATH><I>D</I>=<I>d</I><SUP>3</SUP></MATH>, we can use +either assumption, or put <MATH><I>y</I>=<I>ax</I>+<I>d</I></MATH>. Apparently Diophantus +used the last assumption only in this case, for in IV. 27 he +rejects as impossible the equation <MATH>8<I>x</I><SUP>3</SUP>-<I>x</I><SUP>2</SUP>+8<I>x</I>-1=<I>y</I><SUP>3</SUP></MATH>, +because the assumption <MATH><I>y</I>=2<I>x</I>-1</MATH> gives a negative value +<MATH><I>x</I>=-2/(11)</MATH>, whereas either of the above assumptions gives +a rational value. +<p>(2) <I>Double equations</I>. +<p>Here one expression has to be made a square and another +a cube. The cases are mostly very simple, e.g. (VI. 19) +<MATH><BRACE>4<I>x</I>+2=<I>y</I><SUP>3</SUP> +2<I>x</I>+1=<I>z</I><SUP>2</SUP></BRACE></MATH>; +thus <MATH><I>y</I><SUP>3</SUP>=2<I>z</I><SUP>2</SUP></MATH>, and <MATH><I>z</I>=2</MATH>. +<pb n=476><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>More complicated is the case in VI. 21: +<MATH><BRACE>2<I>x</I><SUP>2</SUP>+2<I>x</I>=<I>y</I><SUP>2</SUP> +<I>x</I><SUP>3</SUP>+2<I>x</I><SUP>2</SUP>+<I>x</I>=<I>z</I><SUP>3</SUP></BRACE></MATH>. +<p>Diophantus assumes <MATH><I>y</I>=<I>mx</I></MATH>, whence <MATH><I>x</I>=2/(<I>m</I><SUP>2</SUP>-2)</MATH>, and +<MATH>(2/(<I>m</I><SUP>2</SUP>-2))<SUP>3</SUP>+2(2/(<I>m</I><SUP>2</SUP>-2))<SUP>2</SUP>+2/(<I>m</I><SUP>2</SUP>-2)=<I>z</I><SUP>3</SUP></MATH>, +or <MATH>(2<I>m</I><SUP>4</SUP>)/((<I>m</I><SUP>2</SUP>-2)<SUP>3</SUP>)=<I>z</I><SUP>3</SUP></MATH>. +<p>We have only to make 2<I>m</I><SUP>4</SUP>, or 2<I>m</I>, a cube. +<C>II. Method of Limits.</C> +<p>As Diophantus often has to find a series of numbers in +order of magnitude, and as he does not admit negative +solutions, it is often necessary for him to reject a solution +found in the usual course because it does not satisfy the +necessary conditions; he is then obliged, in many cases, to +find solutions lying <I>within certain limits</I> in place of those +rejected. For example: +<p>1. It is required to find a value of <I>x</I> such that some power of +it, <I>x</I><SUP><I>n</I></SUP>, shall lie between two given numbers, say <I>a</I> and <I>b</I>. +<p>Diophantus multiplies both <I>a</I> and <I>b</I> by 2<SUP><I>n</I></SUP>, 3<SUP><I>n</I></SUP>, and so on, +successively, until some <I>n</I>th power is seen which lies between +the two products. Suppose that <I>c</I><SUP><I>n</I></SUP> lies between <I>ap</I><SUP><I>n</I></SUP> and <I>bp</I><SUP><I>n</I></SUP>; +then we can put <MATH><I>x</I>=<I>c/p</I></MATH>, for <MATH>(<I>c/p</I>)<SUP><I>n</I></SUP></MATH> lies between <I>a</I> and <I>b</I>. +<p>Ex. To find a square between 1 1/4 and 2. Diophantus +multiplies by a square 64; this gives 80 and 128, between +which lies 100. Therefore ((10)/8)<SUP>2</SUP> or (25)/(16) solves the problem +(IV. 31 (2)). +<p>To find a sixth power between 8 and 16. The sixth powers +of 1, 2, 3, 4 are 1, 64, 729, 4096. Multiply 8 and 16 by 64 +and we have 512 and 1024, between which 729 lies; (729)/(64) is +therefore a solution (VI. 21). +<p>2. Sometimes a value of <I>x</I> has to be found which will give +<pb n=477><head>METHOD OF LIMITS</head> +some function of <I>x</I> a value intermediate between the values +of two other functions of <I>x</I>. +<p>Ex. 1. In IV. 25 a value of <I>x</I> is required such that 8/(<I>x</I><SUP>2</SUP>+<I>x</I>) +shall lie between <I>x</I> and <I>x</I>+1. +<p>One part of the condition gives <MATH>8><I>x</I><SUP>3</SUP>+<I>x</I><SUP>2</SUP></MATH>. Diophantus +accordingly assumes <MATH>8=(<I>x</I>+1/3)<SUP>3</SUP>=<I>x</I><SUP>3</SUP>+<I>x</I><SUP>2</SUP>+1/3<I>x</I>+1/(27)</MATH>, which is +<MATH>> <I>x</I><SUP>3</SUP>+<I>x</I><SUP>2</SUP></MATH>. Thus <MATH><I>x</I>+1/3=2</MATH> or <MATH><I>x</I>=5/3</MATH> satisfies one part of +the condition. Incidentally it satisfies the other, namely +<MATH>8/(<I>x</I><SUP>2</SUP>+<I>x</I>)<<I>x</I>+1</MATH>. This is a piece of luck, and Diophantus +is satisfied with it, saying nothing more. +<p>Ex. 2. We have seen how Diophantus concludes that, if +<MATH>1/5 (<I>x</I><SUP>2</SUP>-60)><I>x</I>>1/8(<I>x</I><SUP>2</SUP>-60)</MATH>, +then <I>x</I> is not less than 11 and not greater than 12 (V. 30). +<p>The problem further requires that <I>x</I><SUP>2</SUP>-60 shall be a square. +Assuming <MATH><I>x</I><SUP>2</SUP>-60=(<I>x</I>-<I>m</I>)<SUP>2</SUP></MATH>, we find <MATH><I>x</I>=(<I>m</I><SUP>2</SUP>+60)/2<I>m</I></MATH>. +<p>Since <I>x</I>>11 and<12, says Diophantus, it follows that +<MATH>24<I>m</I>><I>m</I><SUP>2</SUP>+60>22<I>m</I></MATH>; +from which he concludes that <I>m</I> lies between 19 and 21. +Putting <MATH><I>m</I>=20</MATH>, he finds <MATH><I>x</I>=11 1/2</MATH>. +<C>III. Method of approximation to Limits.</C> +<p>Here we have a very distinctive method called by Diophantus +<G>pariso/ths</G> or <G>pariso/thtos a)gwgh/</G>. The object is to solve such +problems as that of finding two or three square numbers the +sum of which is a given number, while each of them either +approximates to one and the same number, or is subject to +limits which may be the same or different. +<p>Two examples will best show the method. +<p>Ex. 1. Divide 13 into two squares each of which > 6 (V. 9). +<p>Take half of 13, i.e. 6 1/2, and find what <I>small</I> fraction 1/<I>x</I><SUP>2</SUP> +added to it will give a square; +thus <MATH>6 1/2+1/(<I>x</I><SUP>2</SUP>)</MATH>, or <MATH>26+1/(<I>y</I><SUP>2</SUP>)</MATH>, must be a square. +<pb n=478><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>Diophantus assumes +<MATH>26+1/<I>y</I><SUP>2</SUP>=(5+1/<I>y</I>)<SUP>2</SUP></MATH>, or <MATH>26<I>y</I><SUP>2</SUP>+1=(5<I>y</I>+1)<SUP>2</SUP></MATH>, +whence +<MATH><I>y</I>=10</MATH>, and <MATH>1/<I>y</I><SUP>2</SUP>=1/(100)</MATH>, i.e. <MATH>1/<I>x</I><SUP>2</SUP>=1/(400)</MATH>; and <MATH>6 1/2+1/(400)=((51)/(20))<SUP>2</SUP></MATH>. +<p>[The assumption of 5+1/<I>y</I> as the side is not haphazard: 5 is +chosen because it is the most suitable as giving the largest +rational value for <I>y</I>.] +<p>We have now, says Diophantus, to divide 13 into two +squares each of which is as nearly as possible equal to ((51)/(20))<SUP>2</SUP>. +<p>Now <MATH>13=3<SUP>2</SUP>+2<SUP>2</SUP></MATH> [it is necessary that the original number +shall be capable of being expressed as the sum of two squares]; +and <MATH>3>(51)/(20) by 9/(20)</MATH>, +while <MATH>2<(51)/(20) by (11)/(20)</MATH>. +<p>But if we took <MATH>3-9/(20), 2+(11)/(20)</MATH> as the sides of two squares, +their sum would be <MATH>2((51)/(20))<SUP>2</SUP>=(5202)/(400)</MATH>, which is > 13. +<p>Accordingly we assume <MATH>3-9<I>x</I>, 2+11<I>x</I></MATH> as the sides of the +required squares (so that <I>x</I> is not exactly 1/20 but near it). +<p>Thus <MATH>(3-9<I>x</I>)<SUP>2</SUP>+(2+11<I>x</I>)<SUP>2</SUP>=13</MATH>, +and we find <MATH><I>x</I>=5/(101)</MATH>. +<p>The sides of the required squares are (257)/(101), (258)/(101). +<p>Ex. 2. Divide 10 into three squares each of which > 3 +(V. 11). +<p>[The original number, here 10, must of course be expressible +as the sum of three squares.] +<p>Take one-third of 10, i.e. 3 1/3, and find what small fraction +1/<I>x</I><SUP>2</SUP> added to it will make a square; i.e. we have to make +<MATH>3 1/3+1/(<I>x</I><SUP>2</SUP>)</MATH> a square, i.e. <MATH>30+9/(<I>x</I><SUP>2</SUP>)</MATH> must be a square, or <MATH>30+1/(<I>y</I><SUP>2</SUP>) +=</MATH> a square, where <MATH>3/<I>x</I>=1/<I>y</I></MATH>. +<p>Diophantus assumes +<MATH>30<I>y</I><SUP>2</SUP>+1=(5<I>y</I>+1)<SUP>2</SUP></MATH>, +the coefficient of <I>y</I>, i.e. 5, being so chosen as to make 1/<I>y</I> as +small as possible; +<pb n=479><head>METHOD OF APPROXIMATION TO LIMITS</head> +therefore <MATH><I>y</I>=2</MATH>, and <MATH>1/<I>x</I><SUP>2</SUP>=1/(36)</MATH>; and <MATH>3 1/3+1/(36)=(121)/(36)</MATH>, a square. +<p>We have now, says Diophantus, to divide 10 into three +squares with sides as near as may be to (11)/6. +<p>Now <MATH>10=9+1=3<SUP>2</SUP>+(3/5)<SUP>2</SUP>+(4/5)<SUP>2</SUP></MATH>. +<p>Bringing 3, 3/5, 4/5 and (11)/6 to a common denominator, we have +(90)/(30), (18)/(30), (24)/(30) and (55)/(30), +and <MATH>3>(55)/(30) by (35)/(30), +3/5<(55)/(30) by (37)/(30), +4/5<(55)/(30) by (31)/(30)</MATH>. +<p>If now we took 3-(35)/(30), 3/5+(37)/(30), 4/5+(31)/(30) as the sides of squares, +the sum of the squares would be 3((11)/6)<SUP>2</SUP> or (363)/(36), which is > 10. +<p>Accordingly we assume as the sides <MATH>3-35<I>x</I>, 3/5+37<I>x</I>, 4/5+31<I>x</I></MATH>, +where <I>x</I> must therefore be not exactly 1/(30) but near it. +<p>Solving <MATH>(3-35<I>x</I>)<SUP>2</SUP>+(3/5+37<I>x</I>)<SUP>2</SUP>+(4/5+31<I>x</I>)<SUP>2</SUP>=10</MATH>, +or <MATH>10-116<I>x</I>+3555<I>x</I><SUP>2</SUP>=10</MATH>, +we find <MATH><I>x</I>=(116)/(3555)</MATH>; +thus the sides of the required squares are (1321)/(711), (1285)/(711), (1288)/(711); +the squares themselves are (1745041)/(505521), (1651225)/(505521), (1658944)/(505521). +<p>Other instances of the application of the method will be +found in V. 10, 12, 13, 14. +<C>Porisms and propositions in the Theory of Numbers.</C> +<p>I. Three propositions are quoted as occurring in the <I>Porisms</I> +(‘We have it in the Porisms that ...’); and some other pro- +positions assumed without proof may very likely have come +from the same collection. The three propositions from the +<I>Porisms</I> are to the following effect. +<p>1. If <I>a</I> is a given number and <I>x</I>, <I>y</I> numbers such that +<MATH><I>x</I>+<I>a</I>=<I>m</I><SUP>2</SUP>, <I>y</I>+<I>a</I>=<I>n</I><SUP>2</SUP></MATH>, then, if <MATH><I>xy</I>+<I>a</I></MATH> is also a square, <I>m</I> and <I>n</I> +differ by unity (V. 3). +<p>[From the first two equations we obtain easily +<MATH><I>xy</I>+<I>a</I>=<I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP>-<I>a</I>(<I>m</I><SUP>2</SUP>+<I>n</I><SUP>2</SUP>-1)+<I>a</I><SUP>2</SUP></MATH>, +and this is obviously a square if <MATH><I>m</I><SUP>2</SUP>+<I>n</I><SUP>2</SUP>-1=2<I>mn</I></MATH>, or +<MATH><I>m</I>-<I>n</I>=±1</MATH>.] +<pb n=480><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>2. If <I>m</I><SUP>2</SUP>, (<I>m</I> + 1)<SUP>2</SUP> be consecutive squares and a third number +be taken equal to <MATH>2{<I>m</I><SUP>2</SUP> + (<I>m</I> + 1)<SUP>2</SUP>} + 2</MATH>, or <MATH>4(<I>m</I><SUP>2</SUP> + <I>m</I> + 1)</MATH>, the +three numbers have the property that the product of any two +<I>plus</I> either the sum of those two or the remaining number +gives a square (V. 5). +<p>[In fact, if <I>X, Y, Z</I> denote the numbers respectively, +<MATH><I>XY</I> + <I>X</I> + <I>Y</I> = (<I>m</I><SUP>2</SUP> + <I>m</I> + 1)<SUP>2</SUP>, <I>XY</I> + <I>Z</I> = (<I>m</I><SUP>2</SUP> + <I>m</I> + 2)<SUP>2</SUP>, +<I>YZ</I> + <I>Y</I> + <I>Z</I> = (2<I>m</I><SUP>2</SUP> + 3<I>m</I> + 3)<SUP>2</SUP>, <I>YZ</I> + <I>X</I> = (2<I>m</I><SUP>2</SUP> + 3<I>m</I> + 2)<SUP>2</SUP>, +<I>ZX</I> + <I>Z</I> + <I>X</I> = (2<I>m</I><SUP>2</SUP> + <I>m</I> + 2)<SUP>2</SUP>, <I>ZX</I> + <I>Y</I> = (2<I>m</I><SUP>2</SUP> + <I>m</I> + 1)<SUP>2</SUP></MATH>.] +<p>3. The difference of any two cubes is also the sum of two +cubes, i.e. can be transformed into the sum of two cubes +(V. 16). +<p>[Diophantus merely states this without proving it or show- +ing how to make the transformation. The subject of the +transformation of sums and differences of cubes was investi- +gated by Vieta, Bachet and Fermat.] +<p>II. Of the many other propositions assumed or implied by +Diophantus which are not referred to the <I>Porisms</I> we may +distinguish two classes. +<p>1. The first class are of two sorts; some are more or less +of the nature of identical formulae, e.g. the facts that the +expressions <MATH>{1/2(<I>a</I> + <I>b</I>)}<SUP>2</SUP> - <I>ab</I></MATH> and <MATH><I>a</I><SUP>2</SUP>(<I>a</I> + 1)<SUP>2</SUP> + <I>a</I><SUP>2</SUP> + (<I>a</I> + 1)<SUP>2</SUP></MATH> are +respectively squares, that <MATH><I>a</I>(<I>a</I><SUP>2</SUP> - <I>a</I>) + <I>a</I> + (<I>a</I><SUP>2</SUP> - <I>a</I>)</MATH> is always a +cube, and that 8 times a triangular number <I>plus</I> 1 gives +a square, i.e. <MATH>8.(1/2)<I>x</I>(<I>x</I> + 1) + 1 = (2<I>x</I> + 1)<SUP>2</SUP></MATH>. Others are of the +same kind as the first two propositions quoted from the +<I>Porisms,</I> e.g. +<p>(1) If <MATH><I>X</I> = <I>a</I><SUP>2</SUP><I>x</I> + 2<I>a, Y</I> = (<I>a</I> + 1)<SUP>2</SUP><I>x</I> + 2(<I>a</I> + 1)</MATH> or, in other +words, if <MATH><I>xX</I> + 1 = (<I>ax</I> + 1)<SUP>2</SUP></MATH> and <MATH><I>xY</I> + 1 = {(<I>a</I> + 1)<I>x</I> + 1}<SUP>2</SUP></MATH>, +then <I>XY</I> + 1 is a square (IV. 20). In fact +<MATH><I>XY</I> + 1 = {<I>a</I>(<I>a</I> + 1)<I>x</I> + (2<I>a</I> + 1)}<SUP>2</SUP></MATH>. +<p>(2) If <MATH><I>X</I> ± <I>a</I> = <I>m</I><SUP>2</SUP>, <I>Y</I> ± <I>a</I> = (<I>m</I> + 1)<SUP>2</SUP></MATH>, and <MATH><I>Z</I> = 2(<I>X</I> + <I>Y</I>) - 1</MATH>, +then <MATH><I>YZ</I> ± <I>a, ZX</I> ± <I>a, XY</I> ± <I>a</I></MATH> are all squares (V. 3, 4). +<pb n=481><head>PORISMS AND PROPOSITIONS ASSUMED</head> +<p>In fact <MATH><I>YZ</I> ± <I>a</I> = {(<I>m</I> + 1) (2<I>m</I> + 1) ∓ 2<I>a</I>}<SUP>2</SUP>, +<I>ZX</I> ± <I>a</I> = {<I>m</I>(2<I>m</I> + 1) ∓ 2<I>a</I>}<SUP>2</SUP>, +<I>XY</I> ± <I>a</I> = {<I>m</I>(<I>m</I> + 1) ∓ <I>a</I>}<SUP>2</SUP></MATH>. +<p>(3) If +<MATH><I>X</I> = <I>m</I><SUP>2</SUP> + 2, <I>Y</I> = (<I>m</I> + 1)<SUP>2</SUP> + 2, <I>Z</I> = 2{<I>m</I><SUP>2</SUP> + (<I>m</I> + 1)<SUP>2</SUP> + 1} + 2</MATH>, +then the six expressions +<MATH><I>YZ</I> - (<I>Y</I> + <I>Z</I>), <I>ZX</I> - (<I>Z</I> + <I>X</I>), <I>XY</I> - (<I>X</I> + <I>Y</I>), +<I>YZ</I> - <I>X, ZX</I> - <I>Y, XY</I> - <I>Z</I></MATH> +are all squares (V. 6). +<p>In fact +<MATH><I>YZ</I> - (<I>Y</I> + <I>Z</I>) = (2<I>m</I><SUP>2</SUP> + 3<I>m</I> + 3)<SUP>2</SUP>, <I>YZ</I> - <I>X</I> = (2<I>m</I><SUP>2</SUP> + 3<I>m</I> + 4)<SUP>2</SUP></MATH>, &c. +<p>2. The second class is much more important, consisting of +propositions in the Theory of Numbers which we find first +stated or assumed in the <I>Arithmetica.</I> It was in explana- +tion or extension of these that Fermat's most famous notes +were written. How far Diophantus possessed scientific proofs +of the theorems which he assumes must remain largely a +matter of speculation. +<C>(<G>a</G>) <I>Theorems on the composition of numbers as the sum +of two squares.</I></C> +<p>(1) Any square number can be resolved into two squares in +any number of ways (II. 8). +<p>(2) Any number which is the sum of two squares can be +resolved into two other squares in any number of ways (II. 9). +<p>(It is implied throughout that the squares may be fractional +as well as integral.) +<p>(3) If there are two whole numbers each of which is the +sum of two squares, the product of the numbers can be +resolved into the sum of two squares in two ways. +<p>In fact <MATH>(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>) (<I>c</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP>) = (<I>ac</I> ± <I>bd</I>)<SUP>2</SUP> + (<I>ad</I> ∓ <I>bc</I>)<SUP>2</SUP></MATH>. +<p>This proposition is used in III. 19, where the problem is +to find four rational right-angled triangles with the same +<pb n=482><head>DIOPHANTUS OF ALEXANDRIA</head> +hypotenuse. The method is this. Form two right-angled +triangles from (<I>a, b</I>) and (<I>c, d</I>) respectively, by which Dio- +phantus means, form the right-angled triangles +<MATH>(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>, <I>a</I><SUP>2</SUP> - <I>b</I><SUP>2</SUP>, 2<I>ab</I>) and (<I>c</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP>, <I>c</I><SUP>2</SUP> - <I>d</I><SUP>2</SUP>, 2<I>cd</I>)</MATH>. +<p>Multiply all the sides in each triangle by the hypotenuse of +the other; we have then two rational right-angled triangles +with the same hypotenuse <MATH>(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>) (<I>c</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP>)</MATH>. +<p>Two others are furnished by the formula above; for we +have only to ‘form two right-angled triangles’ from <MATH>(<I>ac</I> + <I>bd, +ad</I> - <I>bc</I>)</MATH> and from <MATH>(<I>ac</I> - <I>bd, ad</I> + <I>bc</I>)</MATH> respectively. The method +fails if certain relations hold between <I>a, b, c, d.</I> They must +not be such that one number of either pair vanishes, i.e. such +that <MATH><I>ad</I> = <I>bc</I> or <I>ac</I> = <I>bd</I></MATH>, or such that the numbers in either +pair are equal to one another, for then the triangles are +illusory. +<p>In the case taken by Diophantus <MATH><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP> = 2<SUP>2</SUP> + 1<SUP>2</SUP> = 5, +<I>c</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP> = 3<SUP>2</SUP> + 2<SUP>2</SUP> = 13</MATH>, and the four right-angled triangles are +(65, 52, 39), (65, 60, 25), (65, 63, 16) and (65, 56, 33). +<p>On this proposition Fermat has a long and interesting note +as to the number of ways in which a prime number of the +form 4<I>n</I> + 1 and its powers can be (<I>a</I>) the hypotenuse of +a rational right-angled triangle, (<I>b</I>) the sum of two squares. +He also extends theorem (3) above: ‘If a prime number which +is the sum of two squares be multiplied by another prime +number which is also the sum of two squares, the product +will be the sum of two squares in two ways; if the first prime +be multiplied by the square of the second, the product will be +the sum of two squares in three ways; the product of the first +and the cube of the second will be the sum of two squares +in four ways, and so on <I>ad infinitum.</I>’ +<p>Although the hypotenuses selected by Diophantus, 5 and 13, +are prime numbers of the form 4<I>n</I> + 1, it is unlikely that he +was aware that prime numbers of the form 4<I>n</I> + 1 and +numbers arising from the multiplication of such numbers are +the only classes of numbers which are always the sum of two +squares; this was first proved by Euler. +<p>(4) More remarkable is a condition of possibility of solution +prefixed to V. 9, ‘To divide 1 into two parts such that, if +<pb n=483><head>NUMBERS AS THE SUMS OF SQUARES</head> +a given number is added to either part, the result will be a +square.’ The condition is in two parts. There is no doubt as +to the first, ‘The given number must not be odd’ [i.e. no +number of the form 4<I>n</I> + 3 or 4<I>n</I> - 1 can be the sum of two +squares]; the text of the second part is corrupt, but the words +actually found in the text make it quite likely that corrections +made by Hankel and Tannery give the real meaning of the +original, ‘nor must the double of the given number <I>plus</I> 1 be +measured by any prime number which is less by 1 than a +multiple of 4’. This is tolerably near the true condition +stated by Fermat, ‘The given number must not be odd, and +the double of it increased by 1, when divided by the greatest +square which measures it, must not be divisible by a prime +number of the form 4<I>n</I> - 1.’ +<C>(<G>b</G>) <I>On numbers which are the sum of three squares.</I></C> +<p>In V. 11 the number 3<I>a</I> + 1 has to be divisible into three +squares. Diophantus says that <I>a</I> ‘must not be 2 or any +multiple of 8 increased by 2’. That is, ‘<I>a number of the +form 24n</I> + <I>7 cannot be the sum of three squares</I>’. As a matter +of fact, the factor 3 in the 24 is irrelevant here, and Diophantus +might have said that a number of the form 8<I>n</I> + 7 cannot be +the sum of three squares. The latter condition is true, but +does not include <I>all</I> the numbers which cannot be the sum of +three squares. Fermat gives the conditions to which <I>a</I> must be +subject, proving that 3<I>a</I> + 1 cannot be of the form 4<SUP><I>n</I></SUP>(24<I>k</I> + 7) +or 4<SUP><I>n</I></SUP>(8<I>k</I> + 7), where <I>k</I> = 0 or any integer. +<C>(<G>g</G>) <I>Composition of numbers as the sum of four squares.</I></C> +<p>There are three problems, IV. 29, 30 and V. 14, in which it +is required to divide a number into four squares. Diophantus +states no necessary condition in this case, as he does when +it is a question of dividing a number into <I>three</I> or <I>two</I> squares. +Now <I>every number is either a square or the sum of two, three +or four squares</I> (a theorem enunciated by Fermat and proved +by Lagrange who followed up results obtained by Euler), and +this shows that any number can be divided into four squares +(admitting fractional as well as integral squares), since any +square number can be divided into two other squares, integral +<pb n=484><head>DIOPHANTUS OF ALEXANDRIA</head> +or fractional. It is possible, therefore, that Diophantus was +<I>empirically</I> aware of the truth of the theorem of Fermat, but +we cannot be sure of this. +<C>Conspectus of the <I>Arithmetica,</I> with typical solutions.</C> +<p>There seems to be no means of conveying an idea of the +extent of the problems solved by Diophantus except by giving +a conspectus of the whole of the six Books. Fortunately this +can be done by the help of modern notation without occupying +too many pages. +<p>It will be best to classify the propositions according to their +character rather than to give them in Diophantus's order. It +should be premised that <I>x, y, z</I> ... indicating the first, second +and third ... numbers required do not mean that Diophantus +indicates any of them by his unknown (<G>s</G>); he gives his un- +known in each case the signification which is most convenient, +his object being to express all his required numbers at once in +terms of the one unknown (where possible), thereby avoiding the +necessity for eliminations. Where I have occasion to specify +Diophantus's unknown, I shall as a rule call it <G>x</G>, except when +a problem includes a subsidiary problem and it is convenient +to use different letters for the unknown in the original and +subsidiary problems respectively, in order to mark clearly the +distinction between them. When in the equations expressions +are said to be = <I>u</I><SUP>2</SUP>, <I>v</I><SUP>2</SUP>, <I>w</I><SUP>2</SUP>, <I>t</I><SUP>2</SUP> ... this means simply that they +are to be made squares. Given numbers will be indicated by +<I>a, b, c ... m, n</I> ... and will take the place of the numbers used +by Diophantus, which are always specific numbers. +<p>Where the solutions, or particular devices employed, are +specially ingenious or interesting, the methods of solution will +be shortly indicated. The character of the book will be best +appreciated by means of such illustrations. +<p>[The problems marked with an asterisk are probably +spurious.] +<C>(i) Equations of the first degree with one unknown.</C> +<p>I. 7. <MATH><I>x</I> - <I>a</I> = <I>m</I>(<I>x</I> - <I>b</I>)</MATH>. +<p>I. 8. <MATH><I>x</I> + <I>a</I> = <I>m</I>(<I>x</I> + <I>b</I>)</MATH>. +<pb n=485><head>DETERMINATE EQUATIONS</head> +<p>I. 9. <MATH><I>a</I> - <I>x</I> = <I>m</I>(<I>b</I> - <I>x</I>)</MATH>. +<p>I. 10. <MATH><I>x</I> + <I>b</I> = <I>m</I>(<I>a</I> - <I>x</I>)</MATH>. +<p>I. 11. <MATH><I>x</I> + <I>b</I> = <I>m</I>(<I>x</I> - <I>a</I>)</MATH>. +<MATH> +<BRACE><note>(<I>a</I> > <I>b</I>)</note> +I. 39. (<I>a</I> + <I>x</I>)<I>b</I> + (<I>b</I> + <I>x</I>)<I>a</I> = 2(<I>a</I> + <I>b</I>)<I>x</I>, +or (<I>a</I> + <I>b</I>)<I>x</I> + (<I>b</I> + <I>x</I>)<I>a</I> = 2(<I>a</I> + <I>x</I>)<I>b</I>, +or (<I>a</I> + <I>b</I>)<I>x</I> + (<I>a</I> + <I>x</I>)<I>b</I> = 2(<I>b</I> + <I>x</I>)<I>a</I></BRACE></MATH>. +<p>Diophantus states this problem in this form, ‘Given +two numbers (<I>a, b</I>), to find a third number (<I>x</I>) such that +the numbers +<MATH>(<I>a</I> + <I>x</I>)<I>b,</I> (<I>b</I> + <I>x</I>)<I>a,</I> (<I>a</I> + <I>b</I>)<I>x</I></MATH> +are in arithmetical progression.’ +<p>The result is of course different according to the order +of magnitude of the three expressions. If <I>a</I> > <I>b</I> (5 and 3 +are the numbers in Diophantus), then <MATH>(<I>a</I> + <I>x</I>)<I>b</I> < (<I>b</I> + <I>x</I>)<I>a</I></MATH>; +there are consequently three alternatives, since <MATH>(<I>a</I> + <I>x</I>)<I>b</I></MATH> +must be either the least or the middle, and <MATH>(<I>b</I> + <I>x</I>)<I>a</I></MATH> either +the middle or the greatest of the three products. We may +have +<MATH>(<I>a</I> + <I>x</I>)<I>b</I> < (<I>a</I> + <I>b</I>)<I>x</I> < (<I>b</I> + <I>x</I>)<I>a</I>, +or (<I>a</I> + <I>b</I>)<I>x</I> < (<I>a</I> + <I>x</I>)<I>b</I> < (<I>b</I> + <I>x</I>)<I>a</I>, +or (<I>a</I> + <I>x</I>)<I>b</I> < (<I>b</I> + <I>x</I>)<I>a</I> < (<I>a</I> + <I>b</I>)<I>x</I></MATH>, +and the corresponding equations are as set out above. +<C>(ii) Determinate systems of equations of the first degree.</C> +<MATH>I. 1. <I>x</I> + <I>y</I> = <I>a, x</I> - <I>y</I> = <I>b.</I> +<BRACE>I. 2. <I>x</I> + <I>y</I> = <I>a, x</I> = <I>my,</I> +I. 4. <I>x</I> - <I>y</I> = <I>a, x</I> = <I>my.</I></BRACE></MATH> +<p><MATH>I. 3. <I>x</I> + <I>y</I> = <I>a, x</I> = <I>my</I> + <I>b.</I></MATH> +<MATH><BRACE>I. 5. <I>x</I> + <I>y</I> = <I>a,</I> (1/<I>m</I>)<I>x</I> + (1/<I>n</I>)<I>y</I>=<I>b,</I> subject to necessary condition. +I. 6. <I>x</I> + <I>y</I> = <I>a,</I> (1/<I>m</I>)<I>x</I> - (1/<I>n</I>)<I>y</I>=<I>b,</I> ” ” ”</BRACE></MATH> +<pb n=486><head>DIOPHANTUS OF ALEXANDRIA</head> +<MATH><BRACE>I. 12. <I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> = <I>y</I><SUB>1</SUB> + <I>y</I><SUB>2</SUB> = <I>a, x</I><SUB>1</SUB> = <I>my</I><SUB>2</SUB>, <I>y</I><SUB>1</SUB> = <I>nx</I><SUB>2</SUB> (<I>x</I><SUB>1</SUB> > +<I>x</I><SUB>2</SUB>, <I>y</I><SUB>1</SUB> > <I>y</I><SUB>2</SUB>). +<BRACE><note>(<I>x</I><SUB>1</SUB> > <I>x</I><SUB>2</SUB>, <I>y</I><SUB>1</SUB> > <I>y</I><SUB>2</SUB>, <I>z</I><SUB>1</SUB> > <I>z</I><SUB>2</SUB>).</note> +I. 13. <I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> = <I>y</I><SUB>1</SUB> + <I>y</I><SUB>2</SUB> = <I>z</I><SUB>1</SUB> + <I>z</I><SUB>2</SUB> = <I>a</I> +<I>x</I><SUB>1</SUB> = <I>my</I><SUB>2</SUB>, <I>y</I><SUB>1</SUB> = <I>nz</I><SUB>2</SUB>, <I>z</I><SUB>1</SUB> = <I>px</I><SUB>2</SUB></BRACE></BRACE></MATH> +<p>I. 15. <MATH><I>x</I> + <I>a</I> = <I>m</I>(<I>y</I> - <I>a</I>), <I>y</I> + <I>b</I> = <I>n</I>(<I>x</I> - <I>b</I>)</MATH>. +<p>[Diophantus puts <MATH><I>y</I> = <G>x</G> + <I>a</I></MATH>, where <G>x</G> is his unknown.] +<MATH><BRACE>I. 16. <I>y</I> + <I>z</I> = <I>a, z</I> + <I>x</I> = <I>b, x</I> + <I>y</I> =<I>c.</I> [Dioph. puts <G>x</G> = <I>x</I> + <I>y</I> + <I>z</I>.] +I. 17. <I>y</I> + <I>z</I> + <I>w</I> = <I>a, z</I> + <I>w</I> + <I>x</I> = <I>b, w</I> + <I>x</I> + <I>y</I> = <I>c, x</I> + <I>y</I> + <I>z</I> = <I>d.</I> +[<I>x</I> + <I>y</I> + <I>z</I> + <I>w</I> = <G>x</G>.]</BRACE> +<BRACE>I. 18. <I>y</I> + <I>z</I> - <I>x</I> = <I>a, z</I> + <I>x</I> - <I>y</I> = <I>b, x</I> + <I>y</I> - <I>z</I> = <I>c.</I> +[Dioph. puts 2<G>x</G> = <I>x</I> + <I>y</I> + <I>z</I>.] +I. 19. <I>y</I> + <I>z</I> + <I>w</I> - <I>x</I> = <I>a, z</I> + <I>w</I> + <I>x</I> - <I>y</I> = <I>b, w</I> + <I>x</I> + <I>y</I> - <I>z</I> = <I>c,</I> +<I>x</I> + <I>y</I> + <I>z</I> - <I>w</I> = <I>d.</I> +[2<G>x</G> = <I>x</I> + <I>y</I> + <I>z</I> + <I>w.</I>]</BRACE> +<p>I. 20. <I>x</I> + <I>y</I> + <I>z</I> = <I>a, x</I> + <I>y</I> = <I>mz, y</I> + <I>z</I> = <I>nx.</I> +<p>I. 21. <I>x</I> = <I>y</I> + (1/<I>m</I>)<I>z, y</I> = <I>z</I> + (1/<I>n</I>)<I>x, z</I> = <I>a</I> + (1/<I>p</I>)<I>y</I> (where <I>x > y > z</I>), +with necessary condition. +<p>II. 18*. <I>x</I> - ((1/<I>m</I>)<I>x</I> + <I>a</I>) + ((1/<I>p</I>)<I>z</I> + <I>c</I>) = <I>y</I> - ((1/<I>n</I>)<I>y</I> + <I>b</I>) + ((1/<I>m</I>)<I>x</I> + <I>a</I>) += <I>z</I> - ((1/<I>p</I>)<I>z</I> + <I>c</I>) + ((1/<I>n</I>)<I>y</I> + <I>b</I>), <I>x</I> + <I>y</I> + <I>z</I> = <I>a.</I></MATH> +<p>[Solution wanting.] +<C>(iii) Determinate systems of equations reducible to the +first degree.</C> +<p><MATH>I. 26. <I>ax</I> = <G>a</G><SUP>2</SUP>, <I>bx</I> = <G>a</G>. +<p>I. 29. <I>x</I> + <I>y</I> = <I>a, x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>b.</I> [Dioph. puts 2<G>x</G> = <I>x</I> - <I>y.</I>] +<BRACE>I. 31. <I>x</I> = <I>my, x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> + <I>y</I>). +I. 32. <I>x</I> = <I>my, x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> - <I>y</I>). +I. 33. <I>x</I> = <I>my, x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> + <I>y</I>). +I. 34. <I>x</I> = <I>my, x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> - <I>y</I>). +I. 34. Cor. 1. <I>x</I> = <I>my, xy</I> = <I>n</I>(<I>x</I> + <I>y</I>). +Cor. 2. <I>x</I> = <I>my, xy</I> = <I>n</I>(<I>x</I> - <I>y</I>)</BRACE></MATH>. +<pb n=487><head>DETERMINATE EQUATIONS</head> +<MATH><BRACE>I. 35. <I>x</I> = <I>my, y</I><SUP>2</SUP> = <I>nx.</I> +I. 36. <I>x</I> = <I>my, y</I><SUP>2</SUP> = <I>ny</I></BRACE>. +<p>I. 37. <I>x</I> = <I>my, y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> + <I>y</I>). +<p>I. 38. <I>x</I> = <I>my, y</I><SUP>2</SUP> = <I>n</I>(<I>x</I> - <I>y</I>). +<p>I. 38. Cor. <I>x</I> = <I>my, x</I><SUP>2</SUP> = <I>ny.</I> +” <I>x</I> = <I>my, x</I><SUP>2</SUP> = <I>nx.</I> +” <I>x</I> = <I>my, x</I><SUP>2</SUP> = <I>n</I>(<I>x</I> + <I>y</I>). +” <I>x</I> = <I>my, x</I><SUP>2</SUP> = <I>n</I>(<I>x</I> - <I>y</I>). +<p>II. 6*. <I>x</I> - <I>y</I> = <I>a, x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>x</I> - <I>y</I> + <I>b.</I> +<p>IV. 36. <I>yz</I> = <I>m</I>(<I>y</I> + <I>z</I>), <I>zx</I> = <I>n</I>(<I>z</I> + <I>x</I>), <I>xy</I> = <I>p</I>(<I>x</I> + <I>y</I>).</MATH> +<p>[Solved by means of Lemma: see under (vi) Inde- +terminate equations of the first degree.] +<C>(iv) Determinate systems reducible to equations of +second degree.</C> +<MATH><BRACE>I. 27. <I>x</I> + <I>y</I> = <I>a, xy</I> = <I>b.</I> +[Dioph. states the necessary condition, namely that +(1/4)<I>a</I><SUP>2</SUP> - <I>b</I> must be a square, with the words <G>e)/sti de\ tou=to +plasmatiko/n</G>, which no doubt means ‘this is of the +nature of a formula (easily obtained)’. He puts +<I>x</I> - <I>y</I> = 2<G>x</G>.] +I. 30. <I>x</I> - <I>y</I> = <I>a, xy</I> = <I>b</I></BRACE></MATH>. +<p>[Necessary condition (with the same words) <MATH>4<I>b</I> + <I>a</I><SUP>2</SUP> =</MATH> +a square. <MATH><I>x</I> + <I>y</I> is put = 2<G>x</G></MATH>.] +<p>I. 28. <MATH><I>x</I> + <I>y</I> = <I>a, x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>b</I></MATH>. +<p>[Necessary condition <MATH>2<I>b</I> - <I>a</I><SUP>2</SUP> = a square. <I>x</I> - <I>y</I> = 2<G>x</G></MATH>.] +<MATH><BRACE>IV. 1. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> = <I>a, x</I> + <I>y</I> = <I>b.</I> +[Dioph. puts <I>x</I> - <I>y</I> = 2<G>x</G>, whence <I>x</I>=(1/2)<I>b</I> + <G>x</G>, <I>y</I> = (1/2)<I>b</I> - <G>x</G>. +The numbers <I>a, b</I> are so chosen that (<I>a</I> - (1/4)<I>b</I><SUP>3</SUP>)/3<I>b</I> is +a square.] +IV. 2. <I>a</I><SUP>3</SUP> - <I>y</I><SUP>3</SUP> = <I>a, x</I> - <I>y</I> = <I>b.</I> +[<I>x</I> + <I>y</I> = 2<G>x</G>.]</BRACE></MATH> +<pb n=488><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>IV. 15. <MATH>(<I>y</I> + <I>z</I>)<I>x</I> = <I>a,</I> (<I>z</I> + <I>x</I>)<I>y</I> = <I>b,</I> (<I>x</I> + <I>y</I>)<I>z</I> = <I>c</I></MATH>. +<p>[Dioph. takes the third number <I>z</I> as his unknown; +thus <MATH><I>x</I> + <I>y</I> = <I>c/z.</I> +<p>Assume <I>x</I> = <I>p/z, y</I> = <I>q/z.</I> Then +<I>pq/z</I><SUP>2</SUP> + <I>p</I> = <I>a,</I> +<I>pq/z</I><SUP>2</SUP> + <I>q</I> = <I>b.</I></MATH> +<p>These equations are inconsistent unless <MATH><I>p</I> - <I>q</I> = <I>a</I> - <I>b</I></MATH>. +We have therefore to determine <I>p, q</I> by dividing <I>c</I> into +two parts such that their difference <MATH>= <I>a</I> - <I>b</I></MATH> (cf. I. 1). +<p>A very interesting use of the ‘false hypothesis’ +(Diophantus first takes two <I>arbitrary</I> numbers for <I>p, q</I> +such that <MATH><I>p</I> + <I>q</I> = <I>c</I></MATH>, and finds that the values taken have +to be corrected). +<p>The final equation being <MATH><I>pq/z</I><SUP>2</SUP> + <I>p</I> = <I>a</I></MATH>, where <I>p, q</I> are +determined in the way described, <MATH><I>z</I><SUP>2</SUP> = <I>pq</I>/(<I>a</I> - <I>p</I>)</MATH> or +<MATH><I>pq</I>/(<I>b</I> - <I>q</I>)</MATH>, and the numbers <I>a, b, c</I> have to be such that +either of these expressions gives a square.] +<p>IV. 34. <MATH><I>yz</I> + (<I>y</I> + <I>z</I>) = <I>a</I><SUP>2</SUP> - 1, <I>zx</I> + (<I>z</I> + <I>x</I>) = <I>b</I><SUP>2</SUP> - 1, +<I>xy</I> + (<I>x</I> + <I>y</I>) = <I>c</I><SUP>2</SUP> - 1</MATH>. +<p>[Dioph. states as the necessary condition for a rational +solution that each of the three constants to which the +three expressions are to be equal must be some square +diminished by 1. The true condition is seen in our +notation by transforming the equations <MATH><I>yz</I> + (<I>y</I> + <I>z</I>) = <G>a</G>, +<I>zx</I> + (<I>z</I> + <I>x</I>) = <G>b</G>, <I>xy</I> + (<I>x</I> + <I>y</I>) = <G>g</G></MATH> into +<MATH>(<I>y</I> + 1) (<I>z</I> + 1) = <G>a</G> + 1, +(<I>z</I> + 1) (<I>x</I> + 1) = <G>b</G> + 1, +(<I>x</I> + 1) (<I>y</I> + 1) = <G>g</G> + 1,</MATH> +<pb n=489><head>DETERMINATE EQUATIONS</head> +whence <MATH><I>x</I> + 1 = √{((<G>b</G> + 1) (<G>g</G> + 1))/(<G>a</G> + 1)}</MATH> &c.; +and it is only necessary that <MATH>(<G>a</G> + 1) (<G>b</G> + 1) (<G>g</G> + 1)</MATH> should +be a square, not that <I>each</I> of the expressions <MATH><G>a</G> + 1, <G>b</G> + 1, +<G>g</G> + 1</MATH> should be a square. +<p>Dioph. finds in a Lemma (see under (vi) below) a solu- +tion <G>e)n a)ori/stw|</G> (indeterminately) of <MATH><I>xy</I> + (<I>x</I> + <I>y</I>) = <I>k</I></MATH>, +which practically means finding <I>y</I> in terms of <I>x.</I>] +<p>IV. 35. <MATH><I>yz</I> - (<I>y</I> + <I>z</I>) = <I>a</I><SUP>2</SUP> - 1, <I>zx</I> - (<I>z</I> + <I>x</I>) = <I>b</I><SUP>2</SUP> - 1, +<I>xy</I> - (<I>x</I> + <I>y</I>) = <I>c</I><SUP>2</SUP> - 1</MATH>. +<p>[The remarks on the last proposition apply <I>mutatis +mutandis.</I> The lemma in this case is the indeterminate +solution of <MATH><I>xy</I> - (<I>x</I> + <I>y</I>) = <I>k</I></MATH>.] +<p>IV. 37. <MATH><I>yz</I> = <I>a</I>(<I>x</I> + <I>y</I> + <I>z</I>), <I>zx</I> = <I>b</I>(<I>x</I> + <I>y</I> + <I>z</I>), <I>xy</I> = <I>c</I>(<I>x</I> + <I>y</I> + <I>z</I>)</MATH>. +<p>[Another interesting case of ‘false hypothesis’. Dioph. +first gives <MATH><I>x</I> + <I>y</I> + <I>z</I></MATH> an <I>arbitrary</I> value, then finds that +the result is not rational, and proceeds to solve the new +problem of finding a value of <MATH><I>x</I> + <I>y</I> + <I>z</I></MATH> to take the place of +the first value. +<p>If <MATH><I>w</I> = <I>x</I> + <I>y</I> + <I>z</I></MATH>, we have <MATH><I>x</I> = <I>cw/y, z</I> = <I>aw/y</I></MATH>, so that +<MATH><I>zx</I> = <I>acw</I><SUP>2</SUP>/<I>y</I><SUP>2</SUP> = <I>bw</I></MATH> by hypothesis; therefore <MATH><I>y</I><SUP>2</SUP> = (<I>ac/b</I>)<I>w</I></MATH>. +<p>For a rational solution this last expression must be +a square. Suppose, therefore, that <MATH><I>w</I> = (<I>ac/b</I>)<G>x</G><SUP>2</SUP></MATH>, and we have +<MATH><I>x</I> + <I>y</I> + <I>z</I> = (<I>ac/b</I>)<G>x</G><SUP>2</SUP>, <I>y</I> = (<I>ac/b</I>)<G>x</G>, <I>z</I> = <I>a</I><G>x</G>, <I>x</I> = <I>c</I><G>x</G></MATH>. +<p>Eliminating <I>x, y, z,</I> we obtain <MATH><G>x</G> = (<I>bc</I> + <I>ca</I> + <I>ab</I>)/<I>ac</I></MATH>, +and +<MATH><I>x</I> = (<I>bc</I> + <I>ca</I> + <I>ab</I>)/<I>a, y</I> = (<I>bc</I> + <I>ca</I> + <I>ab</I>)/<I>b,</I> +<I>z</I> = (<I>bc</I> + <I>ca</I> + <I>ab</I>)/<I>c</I></MATH>.] +Lemma to V. 8. <MATH><I>yz</I> = <I>a</I><SUP>2</SUP>, <I>zx</I> = <I>b</I><SUP>2</SUP>, <I>xy</I> = <I>c</I><SUP>2</SUP></MATH>. +<pb n=490><head>DIOPHANTUS OF ALEXANDRIA</head> +<C>(v) Systems of equations apparently indeterminate but +really reduced, by arbitrary assumptions, to deter- +minate equations of the first degree.</C> +<p><MATH>I. 14. <I>xy</I> = <I>m</I>(<I>x</I> + <I>y</I>). [Value of <I>y</I> arbitrarily assumed.] +<BRACE><note>[<I>x</I> assumed = 2<I>y.</I>]</note> +II. 3*. <I>xy</I> = <I>m</I>(<I>x</I> + <I>y</I>), and <I>xy</I> = <I>m</I>(<I>x</I> - <I>y</I>). +II. 1*. (cf. I. 31). <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> + <I>y</I>). +II. 2*. (cf. I. 34). <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> - <I>y</I>). +II. 4*. (cf. I. 32). <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> - <I>y</I>). +II. 5*. (cf. I. 33). <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> + <I>y</I>).</BRACE> +<p>II. 7*. <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>x</I> - <I>y</I>) + <I>a.</I> [Dioph. assumes <I>x</I> - <I>y</I> = 2.] +<BRACE>I. 22. <I>x</I> - (1/<I>m</I>)<I>x</I> + (1/<I>p</I>)<I>z</I> = <I>y</I> - (1/<I>n</I>)<I>y</I> + (1/<I>m</I>)<I>x</I> = <I>z</I> - (1/<I>p</I>)<I>z</I> + (1/<I>n</I>)<I>y.</I> +[Value of <I>y</I> assumed.] +I. 23. <I>x</I> - (1/<I>m</I>)<I>x</I> + (1/<I>q</I>)<I>w</I> = <I>y</I> - (1/<I>n</I>)<I>y</I> + (1/<I>m</I>)<I>x</I> = <I>z</I> - (1/<I>p</I>)<I>z</I> + (1/<I>n</I>)<I>y</I> += <I>w</I> - (1/<I>q</I>)<I>w</I> + (1/<I>p</I>)<I>z.</I> [Value of <I>y</I> assumed.]</BRACE> +<BRACE>I. 24. <I>x</I> + (1/<I>m</I>)(<I>y</I> + <I>z</I>) = <I>y</I> + (1/<I>n</I>)(<I>z</I> + <I>x</I>) = <I>z</I> + (1/<I>p</I>)(<I>x</I> + <I>y</I>). +[Value of <I>y</I> + <I>z</I> assumed.] +I. 25. <I>x</I> + (1/<I>m</I>)(<I>y</I> + <I>z</I> + <I>w</I>) = <I>y</I> + (1/<I>n</I>)(<I>z</I> + <I>w</I> + <I>x</I>) += <I>z</I> + (1/<I>p</I>)(<I>w</I> + <I>x</I> + <I>y</I>) = <I>w</I> + (1/<I>q</I>)(<I>x</I> + <I>y</I> + <I>z</I>). +[Value of <I>y</I> + <I>z</I> + <I>w</I> assumed.]</BRACE> +<p>II. 17*. (cf. I. 22). <I>x</I> - ((1/<I>m</I>)<I>x</I> + <I>a</I>) + ((1/<I>p</I>)<I>z</I> + <I>c</I>) += <I>y</I> - ((1/<I>n</I>)<I>y</I> + <I>b</I>) + ((1/<I>m</I>)<I>x</I> + <I>a</I>) = <I>z</I> - ((1/<I>p</I>)<I>z</I> + <I>c</I>) + ((1/<I>n</I>)<I>y</I> + <I>b</I>)</MATH>. +<p>[Ratio of <I>x</I> to <I>y</I> assumed.] +<pb n=491><head>INDETERMINATE ANALYSIS</head> +<p>IV. 33. <MATH><I>x</I> + (1/<I>z</I>)<I>y</I> = <I>m</I>(<I>y</I> - (1/<I>z</I>)<I>y</I>), <I>y</I> + (1/<I>z</I>)<I>x</I> = <I>n</I>(<I>x</I> - (1/<I>z</I>)<I>x</I>). +<p>[Dioph. assumes (1/<I>z</I>)<I>y</I> = 1.]</MATH> +<C>(vi) Indeterminate equations of the first degree.</C> +<MATH><BRACE><note>[Solutions <G>e)n a)ori/stw|</G>. +<I>y</I> practically found +in terms of <I>x.</I>]</note> +Lemma to IV. 34. <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>a.</I> +” ” IV. 35. <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>a.</I> +” ” IV. 36. <I>xy</I> = <I>m</I>(<I>x</I> + <I>y</I>).</BRACE></MATH> +<C>(vii) Indeterminate analysis of the second degree.</C> +<MATH><BRACE>II. 8. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>a</I><SUP>2</SUP>. +[<I>y</I><SUP>2</SUP> = <I>a</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP> must be a square = (<I>mx</I> - <I>a</I>)<SUP>2</SUP>, say.] +II. 9. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>. [Put <I>x</I> = <G>x</G> + <I>a, y</I> = <I>m</I><G>x</G> - <I>b.</I>] +II. 10. <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>a.</I> +[Put <I>x</I> = <I>y</I> + <I>m,</I> choosing <I>m</I> such that <I>m</I><SUP>2</SUP> < <I>a.</I>]</BRACE> +<BRACE>II. 11. <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>b</I> = <I>v</I><SUP>2</SUP>. +II. 12. <I>a</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>b</I> - <I>x</I> = <I>v</I><SUP>2</SUP>. +II. 13. <I>x</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>x</I> - <I>b</I> = <I>v</I><SUP>2</SUP></BRACE></MATH>. +<p>[Dioph. solves II. 11 and 13, (1) by means of the +‘double equation’ (see p. 469 above), (2) without a double +equation by putting <MATH><I>x</I> = <G>x</G><SUP>2</SUP> ± <I>a</I></MATH> and equating <MATH>(<G>x</G><SUP>2</SUP> ± <I>a</I>) ± <I>b</I></MATH> +to <MATH>(<G>x</G> - <I>m</I>)<SUP>2</SUP></MATH>. In II. 12 he puts <MATH><I>x</I> = <I>a</I> - <G>x</G><SUP>2</SUP></MATH>.] +<p>II. 14 = III. 21. <MATH><I>x</I> + <I>y</I> = <I>a, x</I> + <I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>z</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP></MATH>. +<p>[Diophantus takes <I>z</I> as the unknown, and puts +<MATH><I>u</I><SUP>2</SUP> = (<I>z</I> + <I>m</I>)<SUP>2</SUP>, <I>v</I><SUP>2</SUP> = (<I>z</I> + <I>n</I>)<SUP>2</SUP></MATH>. Therefore <MATH><I>x</I> = 2<I>mz</I> + <I>m</I><SUP>2</SUP></MATH>, +<MATH><I>y</I> = 2<I>nz</I> + <I>n</I><SUP>2</SUP></MATH>, and <I>z</I> is found, by substitution in the first +equation, to be <MATH>(<I>a</I> - (<I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP>))/(2(<I>m</I> + <I>n</I>))</MATH>. In order that the solution +may be rational, <I>m, n</I> must satisfy a certain condition. +Dioph. takes them such that <MATH><I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> < <I>a</I></MATH>, but it is suffi- +cient, if <I>m</I> > <I>n,</I> that <I>a</I> + <I>mn</I> should be > <I>n</I><SUP>2</SUP>.] +<p>II. 15 = III. 20. <MATH><I>x</I> + <I>y</I> = <I>a, z</I><SUP>2</SUP> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> - <I>y</I> = <I>v</I><SUP>2</SUP></MATH>. +<p>[The solution is similar, and a similar remark applies +to Diophantus's implied condition.] +<pb n=492><head>DIOPHANTUS OF ALEXANDRIA</head> +<p><MATH>II. 16. <I>x</I> = <I>my, a</I><SUP>2</SUP> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>a</I><SUP>2</SUP> + <I>y</I> = <I>v</I><SUP>2</SUP>. +<p>II. 19. <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>y</I><SUP>2</SUP> - <I>z</I><SUP>2</SUP>). +<BRACE>II. 20. <I>x</I><SUP>2</SUP> + <I>y</I> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + <I>x</I> = <I>v</I><SUP>2</SUP>. +[Assume <I>y</I> = 2<I>mx</I> + <I>m</I><SUP>2</SUP>, and one condition is satisfied.] +II. 21. <I>x</I><SUP>2</SUP> - <I>y</I> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - <I>x</I> = <I>v</I><SUP>2</SUP>. +[Assume <I>x</I> = <G>x</G> + <I>m, y</I> = 2<I>m</I><G>x</G> + <I>m</I><SUP>2</SUP>, and one condition +is satisfied.]</BRACE> +<BRACE>II. 22. <I>x</I><SUP>2</SUP> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + (<I>x</I> + <I>y</I>) = <I>v</I><SUP>2</SUP>. +[Put <I>x</I> + <I>y</I> = 2<I>mx</I> + <I>m</I><SUP>2</SUP>.] +II. 23. <I>x</I><SUP>2</SUP> - (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - (<I>x</I> + <I>y</I>) = <I>v</I><SUP>2</SUP>.</BRACE> +<BRACE>II. 24. (<I>x</I> + <I>y</I>)<SUP>2</SUP> + <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I>)<SUP>2</SUP> + <I>y</I> = <I>v</I><SUP>2</SUP>. +[Assume <I>x</I> = (<I>m</I><SUP>2</SUP> - 1)<G>x</G><SUP>2</SUP>, <I>y</I> = (<I>n</I><SUP>2</SUP> - 1)<G>x</G><SUP>2</SUP>, <I>x</I> + <I>y</I> = <G>x</G>.] +II. 25. (<I>x</I> + <I>y</I>)<SUP>2</SUP> - <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I>)<SUP>2</SUP> - <I>y</I> = <I>v</I><SUP>2</SUP>.</BRACE> +<BRACE>II. 26. <I>xy</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>xy</I> + <I>y</I> = <I>v</I><SUP>2</SUP>, <I>u</I> + <I>v</I> = <I>a.</I> +[Put <I>y</I> = <I>m</I><SUP>2</SUP><I>x</I> - 1.] +II. 27. <I>xy</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>xy</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, <I>u</I> + <I>v</I> = <I>a.</I></BRACE> +<BRACE>II. 28. <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>. +II. 29. <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>.</BRACE> +<p>II. 30. <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>v</I><SUP>2</SUP></MATH>. +<p>[Since <MATH><I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> ± 2<I>mn</I></MATH> is a square, assume +<MATH><I>xy</I> = (<I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP>)<G>x</G><SUP>2</SUP> and <I>x</I> + <I>y</I> = 2<I>mn</I><G>x</G><SUP>2</SUP>; +put <I>x</I> = <I>p<G>x</G>, y</I> = <I>q</I><G>x</G>, where <I>pq</I> = <I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP>; then +(<I>p</I> + <I>q</I>)<G>x</G> = 2<I>mn</I><G>x</G><SUP>2</SUP></MATH>.] +<p>II. 31. <MATH><I>xy</I> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>.</MATH> +<p>[Suppose <MATH><I>w</I><SUP>2</SUP> = 2.2<I>m.m</I></MATH>, which is a square, and use +formula <MATH>(2<I>m</I>)<SUP>2</SUP> + <I>m</I><SUP>2</SUP> ± 2.2<I>m.m</I> = a</MATH> square.] +<MATH><BRACE>II. 32. <I>y</I><SUP>2</SUP> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> + <I>y</I> = <I>w</I><SUP>2</SUP>. +[<I>y</I> = <G>x</G>, <I>z</I> = (2<I>a</I><G>x</G> + <I>a</I><SUP>2</SUP>), <I>x</I> = 2<I>b</I>(2<I>a</I><G>x</G> + <I>a</I><SUP>2</SUP>) + <I>b</I><SUP>2</SUP>.] +II. 33. <I>y</I><SUP>2</SUP> - <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> - <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> - <I>y</I> = <I>w</I><SUP>2</SUP>.</BRACE></MATH> +<pb n=493><head>INDETERMINATE ANALYSIS</head> +<MATH><BRACE>II. 34. <I>x</I><SUP>2</SUP> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>v</I><SUP>2</SUP>, +<I>z</I><SUP>2</SUP> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>w</I><SUP>2</SUP>.</MATH> +[Since {(1/2)(<I>m</I> - <I>n</I>)}<SUP>2</SUP> + <I>mn</I> is a square, take any number +separable into two factors (<I>m, n</I>) in three ways. This +gives three values, say, <I>p, q, r</I> for (1/2)(<I>m</I> - <I>n</I>). Put +<I>x</I> = <I>p<G>x</G>, y</I> = <I>q<G>x</G>, z</I> = <I>r</I><G>x</G>, and <I>x</I> + <I>y</I> + <I>z</I> = <I>mn</I><G>x</G><SUP>2</SUP>; therefore +(<I>p</I> + <I>q</I> + <I>r</I>)<G>x</G> = <I>mn</I><G>x</G><SUP>2</SUP>, and <G>x</G> is found.] +II. 35. <MATH><I>x</I><SUP>2</SUP> - (<I>x</I> + <I>y</I> + <I>z</I>) = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - (<I>x</I> + <I>y</I> + <I>z</I>) = <I>v</I><SUP>2</SUP>, +<I>z</I><SUP>2</SUP> - (<I>x</I> + <I>y</I> + <I>z</I>) = <I>w</I><SUP>2</SUP>. +[Use the formula {1/2(<I>m</I> + <I>n</I>)}<SUP>2</SUP> - <I>mn</I> = a square and +proceed similarly.] +III. 1*. (<I>x</I> + <I>y</I> + <I>z</I>) - <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>) - <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>) - <I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>.</BRACE> +<BRACE>III. 2*. (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> + <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> + <I>y</I> = <I>v</I><SUP>2</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> + <I>z</I> = <I>w</I><SUP>2</SUP>. +III. 3*. (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> - <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> - <I>y</I> = <I>v</I><SUP>2</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> - <I>z</I> = <I>w</I><SUP>2</SUP>. +III. 4*. <I>x</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>y</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> = <I>v</I><SUP>2</SUP>, +<I>z</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>2</SUP> = <I>w</I><SUP>2</SUP>.</BRACE> +<p>III. 5. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>y</I> + <I>z</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, +<I>x</I> + <I>y</I> - <I>z</I> = <I>w</I><SUP>2</SUP>.</MATH> +<p>[The first solution of this problem assumes +<MATH><I>t</I><SUP>2</SUP> = <I>x</I> + <I>y</I> + <I>z</I> = (<G>x</G> + 1)<SUP>2</SUP>, <I>w</I><SUP>2</SUP> = 1, <I>u</I><SUP>2</SUP> = <G>x</G><SUP>2</SUP></MATH>, +whence <I>x, y, z</I> are found in terms of <G>x</G>, and <MATH><I>z</I> + <I>x</I> - <I>y</I></MATH> +is then made a square. +<p>The alternative solution, however, is much more ele- +gant, and can be generalized thus. +<p>We have to find <I>x, y, z</I> so that +<MATH><BRACE> +- <I>x</I> + <I>y</I> + <I>z</I> = a square +<I>x</I> - <I>y</I> + <I>z</I> = a square +<I>x</I> + <I>y</I> - <I>z</I> = a square +<I>x</I> + <I>y</I> + <I>z</I> = a square</BRACE></MATH>. +<p>Equate the first three expressions to <I>a</I><SUP>2</SUP>, <I>b</I><SUP>2</SUP>, <I>c</I><SUP>2</SUP>, being +squares such that their sum is also a square = <I>k</I><SUP>2</SUP>, say. +<pb n=494><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>Then, since the sum of the first three expressions is +itself equal to <MATH><I>x</I> + <I>y</I> + <I>z</I></MATH>, we have +<MATH><I>x</I> = (1/2)(<I>b</I><SUP>2</SUP> + <I>c</I><SUP>2</SUP>), <I>y</I> = (1/2)(<I>c</I><SUP>2</SUP> + <I>a</I><SUP>2</SUP>), <I>z</I> = (1/2)(<I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>).] +<p>III. 6. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>. +<p>III. 7. <I>x</I> - <I>y</I> = <I>y</I> - <I>z, y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>. +<BRACE>III. 8. <I>x</I> + <I>y</I> + <I>z</I> + <I>a</I> = <I>t</I><SUP>2</SUP>, <I>y</I> + <I>z</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, +<I>x</I> + <I>y</I> + <I>a</I> = <I>w</I><SUP>2</SUP>. +III. 9. <I>x</I> + <I>y</I> + <I>z</I> - <I>a</I> = <I>t</I><SUP>2</SUP>, <I>y</I> + <I>z</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> - <I>a</I> = <I>v</I><SUP>2</SUP>, +<I>x</I> + <I>y</I> - <I>a</I> = <I>w</I><SUP>2</SUP>.</BRACE> +<p>III. 10. <I>yz</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>a</I> = <I>w</I><SUP>2</SUP>.</MATH> +<p>[Suppose <MATH><I>yz</I> + <I>a</I> = <I>m</I><SUP>2</SUP></MATH>, and let <MATH><I>y</I> = (<I>m</I><SUP>2</SUP> - <I>a</I>)<G>x</G>, <I>z</I> = 1/<G>x</G></MATH>: +also let <MATH><I>zx</I> + <I>a</I> = <I>n</I><SUP>2</SUP></MATH>; therefore <MATH><I>x</I> = (<I>n</I><SUP>2</SUP> - <I>a</I>)<G>x</G></MATH>. +We have therefore to make +<MATH>(<I>m</I><SUP>2</SUP> - <I>a</I>) (<I>n</I><SUP>2</SUP> - <I>a</I>) <G>x</G><SUP>2</SUP> + <I>a</I></MATH> a square. +<p>Diophantus takes <MATH><I>m</I><SUP>2</SUP> = 25, <I>a</I> = 12, <I>n</I><SUP>2</SUP> = 16</MATH>, and +arrives at <MATH>52<G>x</G><SUP>2</SUP> + 12</MATH>, which is to be made a square. +Although <MATH>52.1<SUP>2</SUP> + 12</MATH> is a square, and it follows that any +number of other solutions giving a square are possible +by substituting <MATH>1 + <G>h</G></MATH> for <G>x</G> in the expression, and so on, +Diophantus says that the equation could easily be solved +if 52 was a square, and proceeds to solve the problem of +finding two squares such that each increased by 12 will +give a square, in which case their product also will be +a square. In other words, we have to find <I>m</I><SUP>2</SUP> and <I>n</I><SUP>2</SUP> +such that <MATH><I>m</I><SUP>2</SUP> - <I>a, n</I><SUP>2</SUP> - <I>a</I></MATH> are both squares, which, as he +says, is easy. We have to find two pairs of squares +differing by <I>a.</I> If +<MATH><I>a</I> = <I>pq</I> = <I>p′q′,</I> {(1/2)(<I>p</I> - <I>q</I>)}<SUP>2</SUP> + <I>a</I> = {(1/2)(<I>p</I> + <I>q</I>)}<SUP>2</SUP>, +and {(1/2)(<I>p′</I> - <I>q′</I>)}<SUP>2</SUP> + <I>a</I> = {(1/2)(<I>p′</I> + <I>q′</I>)}<SUP>2</SUP>; +let, then, <I>m</I><SUP>2</SUP> = {(1/2)(<I>p</I> + <I>q</I>)}<SUP>2</SUP>, <I>n</I><SUP>2</SUP> = {(1/2)(<I>p′</I> + <I>q′</I>)}<SUP>2</SUP>.] +<p>III. 11. <I>yz</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> - <I>a</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>a</I> = <I>w</I><SUP>2</SUP>. +<p>[The solution is like that of III. 10 <I>mutatis mutandis.</I>] +<BRACE>III. 12. <I>yz</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> + <I>y</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>z</I> = <I>w</I><SUP>2</SUP>. +III. 13. <I>yz</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>z</I> = <I>w</I><SUP>2</SUP>.</BRACE></MATH> +<pb n=495><head>INDETERMINATE ANALYSIS</head> +<MATH> +<p>III. 14. <I>yz</I> + <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>zx</I> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>. +<p>III. 15. <I>yz</I> + (<I>y</I> + <I>z</I>) = <I>u</I><SUP>2</SUP>, <I>zx</I> + (<I>z</I> + <I>x</I>) = <I>v</I><SUP>2</SUP>, <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>w</I><SUP>2</SUP>. +</MATH> +<p>[<I>Lemma.</I> If <I>a,</I> <I>a</I> + 1 be two consecutive numbers, +<MATH><I>a</I><SUP>2</SUP>(<I>a</I> + 1)<SUP>2</SUP> + <I>a</I><SUP>2</SUP> + (<I>a</I> + 1)<SUP>2</SUP></MATH> is a square. Let +<MATH> +<I>y</I> = <I>m</I><SUP>2</SUP>, <I>z</I> = (<I>m</I> + 1)<SUP>2</SUP>. +<BRACE>Therefore (<I>m</I><SUP>2</SUP> + 2<I>m</I> + 2)<I>x</I> + (<I>m</I> + 1)<SUP>2</SUP> +and (<I>m</I><SUP>2</SUP> + 1)<I>x</I> + <I>m</I><SUP>2</SUP></BRACE> +</MATH> +have to be made squares. This is solved as a double- +equation; in Diophantus's problem <I>m</I> = 2. +<p><I>Second solution.</I> Let <I>x</I> be the first number, <I>m</I> the +second; then (<I>m</I> + 1) <I>x</I> + <I>m</I> is a square = <I>n</I><SUP>2</SUP>, say; there- +fore <MATH><I>x</I> = (<I>n</I><SUP>2</SUP> - <I>m</I>)/(<I>m</I> + 1)</MATH>, while <I>y</I> = <I>m.</I> We have then +<MATH> +<BRACE> +(<I>m</I> + 1)<I>z</I> + <I>m</I> = a square +and ((<I>n</I><SUP>2</SUP> + 1)/(<I>m</I> + 1))<I>z</I> + (<I>n</I><SUP>2</SUP> - <I>m</I>)/(<I>m</I> + 1) = a square</BRACE> +</MATH>. +<p>Diophantus has <I>m</I> = 3, <I>n</I> = 5, so that the expressions +to be made squares are with him +<MATH> +<BRACE> +4<I>z</I> + 3 +(6 1/2)<I>z</I> + 5 1/2</BRACE> +</MATH>. +This is not possible because, of the corresponding coeffi- +cients, neither pair are in the ratio of squares. In order to +substitute, for 6 1/2, 4, coefficients which are in the ratio +of a square to a square he then finds two numbers, say, +<I>p, q</I> to replace 5 1/2, 3 such that <MATH><I>pq</I> + <I>p</I> + <I>q</I></MATH> = a square, and +<MATH>(<I>p</I> + 1)/(<I>q</I> + 1)</MATH> = a square. He assumes <G>x</G> and 4<G>x</G> + 3, +which satisfies the second condition, and then solves for <G>x</G>, +which must satisfy +<MATH> +4<G>x</G><SUP>2</SUP> + 8<G>x</G> + 3 = a square = (2<G>x</G> - 3)<SUP>2</SUP>, say, +which gives <G>x</G> = 3/10, 4<G>x</G> + 3 = 4 1/5. +</MATH> +<p>He then solves, for <I>z,</I> the third number, the double- +equation +<MATH> +<BRACE> +(5 1/5)<I>z</I> + 4 1/5 = square +(13/10)<I>z</I> + 3/10 = square</BRACE> +</MATH>, +<pb n=496><head>DIOPHANTUS OF ALEXANDRIA</head> +after multiplying by 25 and 100 respectively, making +expressions +<MATH> +<BRACE> +130<I>x</I> + 105 +130<I>x</I> + 30</BRACE> +</MATH>. +<p>In the above equations we should only have to make +<I>n</I><SUP>2</SUP> + 1 a square, and then multiply the first by <I>n</I><SUP>2</SUP> + 1 and +the second by (<I>m</I> + 1)<SUP>2</SUP>. +<p>Diophantus, with his notation, was hardly in a position +to solve, as we should, by writing +<MATH> +(<I>y</I> + 1) (<I>z</I> + 1) = <I>a</I><SUP>2</SUP> + 1, +(<I>z</I> + 1) (<I>x</I> + 1) = <I>b</I><SUP>2</SUP> + 1, +(<I>x</I> + 1) (<I>y</I> + 1) = <I>c</I><SUP>2</SUP> + 1, +which gives <I>x</I> + 1 = √{(<I>b</I><SUP>2</SUP> + 1) (<I>c</I><SUP>2</SUP> + 1)/(<I>a</I><SUP>2</SUP> + 1)}, &c.] +<p>III. 16. <I>yz</I> - (<I>y</I> + <I>z</I>) = <I>u</I><SUP>2</SUP>, <I>zx</I> - (<I>z</I> + <I>x</I>) = <I>v</I><SUP>2</SUP>, <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>w</I><SUP>2</SUP>. +</MATH> +<p>[The method is the same <I>mutatis mutandis</I> as the +second of the above solutions.] +<MATH> +<BRACE>III. 17. <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>xy</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>y</I> = <I>w</I><SUP>2</SUP>. +III. 18. <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>xy</I> - <I>x</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>y</I> = <I>w</I><SUP>2</SUP></BRACE>. +<BRACE><note>III. 19. (<I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB>)<SUP>2</SUP> ± <I>x</I><SUB>1</SUB> =</note> +<I>t</I><SUP>2</SUP> +<I>t</I>′<SUP>2</SUP></BRACE> +<BRACE><note>(<I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB>)<SUP>2</SUP> ± <I>x</I><SUB>2</SUB> =</note> +<I>u</I><SUP>2</SUP> +<I>u</I>′<SUP>2</SUP></BRACE> +<BRACE><note>(<I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB>)<SUP>2</SUP> ± <I>x</I><SUB>3</SUB> =</note> +<I>v</I><SUP>2</SUP> +<I>v</I>′<SUP>2</SUP></BRACE> +<BRACE><note>(<I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB>)<SUP>2</SUP> ± <I>x</I><SUB>4</SUB> =</note> +<I>w</I><SUP>2</SUP> +<I>w</I>′<SUP>2</SUP></BRACE> +</MATH> +<p>[Diophantus finds, in the way we have seen (p. 482), +four different rational right-angled triangles with the +same hypotenuse, namely (65, 52, 39), (65, 60, 25), (65, +56, 33), (65, 63, 16), or, what is the same thing, a square +which is divisible into two squares in four different ways; +this will solve the problem, since, if <I>h, p, b</I> be the three +sides of a right-angled triangle, <I>h</I><SUP>2</SUP> ± 2 <I>pb</I> are both squares. +<pb n=497><head>INDETERMINATE ANALYSIS</head> +<MATH> +<p>Put therefore <I>x</I><SUB>1</SUB> + <I>x</I><SUB>2</SUB> + <I>x</I><SUB>3</SUB> + <I>x</I><SUB>4</SUB> = 65<G>x</G>. +and <I>x</I><SUB>1</SUB> = 2.39.52<G>x</G><SUP>2</SUP>, <I>x</I><SUB>2</SUB> = 2.25.60<G>x</G><SUP>2</SUP>, <I>x</I><SUB>3</SUB> = 2.33.56<G>x</G><SUP>2</SUP>, +<I>x</I><SUB>4</SUB> = 2.16.63<G>x</G><SUP>2</SUP>; +this gives 12768<G>x</G><SUP>2</SUP> = 65<G>x</G>, and <G>x</G> = 65/12768.] +<BRACE>IV. 4. <I>x</I><SUP>2</SUP> + <I>y</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>u.</I> +IV. 5. <I>x</I><SUP>2</SUP> + <I>y</I> = <I>u,</I> <I>x</I> + <I>y</I> = <I>u</I><SUP>2</SUP>.</BRACE> +<p>IV. 13. <I>x</I> + 1 = <I>t</I><SUP>2</SUP>, <I>y</I> + 1 = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>y</I> + 1 = <I>v</I><SUP>2</SUP>, <I>y</I> - <I>x</I> + 1 = <I>w</I><SUP>2</SUP>. +</MATH> +<p>[Put <MATH><I>x</I> = (<I>m</I><G>x</G> + 1)<SUP>2</SUP> - 1 = <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>m</I><G>x</G></MATH>; the second and +third conditions require us to find two squares with <I>x</I> as +difference. The difference <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>m</I><G>x</G> is separated into +the factors <I>m</I><SUP>2</SUP><G>x</G> + 2<I>m</I>, <G>x</G>; the square of half the differ- +ence = {(1/2)(<I>m</I><SUP>2</SUP> - 1)<G>x</G> + <I>m</I>}<SUP>2</SUP>. Put this equal to <I>y</I> + 1, so +that <MATH><I>y</I> = (1/4)(<I>m</I><SUP>2</SUP> - 1)<SUP>2</SUP><G>x</G><SUP>2</SUP> + <I>m</I>(<I>m</I><SUP>2</SUP> - 1)<G>x</G> + <I>m</I><SUP>2</SUP> - 1</MATH>, and the +first three conditions are satisfied. The fourth gives +<MATH>(1/4)(<I>m</I><SUP>4</SUP> - 6<I>m</I><SUP>2</SUP> + 1)<G>x</G><SUP>2</SUP> + (<I>m</I><SUP>3</SUP> - 3<I>m</I>)<G>x</G> + <I>m</I><SUP>2</SUP></MATH> = a square, which +we can equate to (<I>n</I><G>x</G> - <I>m</I>)<SUP>2</SUP>.] +<MATH> +<p>IV. 14. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> = (<I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP>) + (<I>y</I><SUP>2</SUP> - <I>z</I><SUP>2</SUP>) + (<I>x</I><SUP>2</SUP> - <I>z</I><SUP>2</SUP>). +(<I>x</I>><I>y</I>><I>z.</I>) +<p>IV. 16. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> + <I>y</I> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + <I>z</I> = <I>v</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>x</I> = <I>w</I><SUP>2</SUP>. +</MATH> +<p>[Put 4<I>m</I><G>x</G> for <I>y,</I> and by means of the factors 2<I>m</I><G>x</G>, 2 +we can satisfy the second condition by making <I>x</I> equal +to half the difference, or <I>m</I><G>x</G> - 1. The third condition +is satisfied by subtracting (4<I>m</I><G>x</G>)<SUP>2</SUP> from some square, say +(4<I>m</I><G>x</G> + 1)<SUP>2</SUP>; therefore <MATH><I>z</I> = 8<I>m</I><G>x</G> + 1</MATH>. By the first con- +dition 13<I>m</I><G>x</G> must be a square. Let it be 169<G>h</G><SUP>2</SUP>; the +numbers are therefore 13<G>h</G><SUP>2</SUP> - 1, 52<G>h</G><SUP>2</SUP>, 104<G>h</G><SUP>2</SUP> + 1, and +the last condition gives <MATH>10816<G>h</G><SUP>4</SUP> + 221<G>h</G><SUP>2</SUP></MATH> = a square, +i.e. <MATH>10816<G>h</G><SUP>2</SUP> + 221 = a square = (104<G>h</G> + 1)<SUP>2</SUP></MATH>, say. This +gives the value of <G>h</G>, and solves the problem.] +<MATH> +<p>IV. 17. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> - <I>y</I> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - <I>z</I> = <I>v</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> - <I>x</I> = <I>w</I><SUP>2</SUP>. +<p>IV. 19. <I>yz</I> + 1 = <I>u</I><SUP>2</SUP>, <I>zx</I> + 1 = <I>v</I><SUP>2</SUP>, <I>xy</I> + 1 = <I>w</I><SUP>2</SUP>. +</MATH> +<p>[We are asked to solve this indeterminately (<G>e)n tw=| +a)ori/stw|</G>). Put for <I>yz</I> some square minus 1, say <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP> ++ 2<I>m</I><G>x</G>; one condition is now satisfied. Put <I>z</I> = <G>x</G>, so +that <MATH><I>y</I> = <I>m</I><SUP>2</SUP><G>x</G> + 2<I>m</I></MATH>. +<pb n=498><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>Similarly we satisfy the second condition by assuming +<MATH><I>zx</I> = <I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>n</I><G>x</G>; therefore <I>x</I> = <I>n</I><SUP>2</SUP><G>x</G> + 2<I>n</I></MATH>. To satisfy the +third condition, we must have +<MATH>(<I>m</I><SUP>2</SUP><I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>mn</I>.―(<I>m</I> + <I>n</I>)<G>x</G> + 4<I>mn</I>) + 1</MATH> a square. +We must therefore have 4<I>mn</I> + 1 a square and also +<MATH><I>mn</I>(<I>m</I> + <I>n</I>) = <I>mn</I>√(4<I>mn</I> + 1)</MATH>. The first condition is +satisfied by <MATH><I>n</I> = <I>m</I> + 1</MATH>, which incidentally satisfies the +second condition also. We put therefore <MATH><I>yz</I> = (<I>m</I><G>x</G> + 1)<SUP>2</SUP> - 1 +and <I>zx</I> = {(<I>m</I> + 1)<G>x</G> + 1}<SUP>2</SUP> - 1</MATH>, and assume that <I>z</I> = <G>x</G>, so that +<MATH><I>y</I> = <I>m</I><SUP>2</SUP><G>x</G> + 2<I>m</I>, <I>x</I> = (<I>m</I> + 1)<SUP>2</SUP><G>x</G> + 2(<I>m</I> + 1)</MATH>, and we have +shown that the third condition is also satisfied. Thus we +have a solution in terms of the undetermined unknown <G>x</G>. +The above is only slightly generalized from Diophantus.] +<MATH> +<p>IV. 20. <I>x</I><SUB>2</SUB><I>x</I><SUB>3</SUB> + 1 = <I>r</I><SUP>2</SUP>, <I>x</I><SUB>3</SUB><I>x</I><SUB>1</SUB> + 1 = <I>s</I><SUP>2</SUP>, <I>x</I><SUB>1</SUB><I>x</I><SUB>2</SUB> + 1 = <I>t</I><SUP>2</SUP>, +<I>x</I><SUB>1</SUB><I>x</I><SUB>4</SUB> + 1 = <I>u</I><SUP>2</SUP>, <I>x</I><SUB>2</SUB><I>x</I><SUB>4</SUB> + 1 = <I>v</I><SUP>2</SUP>, <I>x</I><SUB>3</SUB><I>x</I><SUB>4</SUB> + 1 = <I>w</I><SUP>2</SUP>. +</MATH> +<p>[This proposition depends on the last, <I>x</I><SUB>1</SUB>, <I>x</I><SUB>2</SUB>, <I>x</I><SUB>3</SUB> being +determined as in that proposition. If <I>x</I><SUB>3</SUB> corresponds to <I>z</I> +in that proposition, we satisfy the condition <MATH><I>x</I><SUB>3</SUB><I>x</I><SUB>4</SUB> + 1 = <I>w</I><SUP>2</SUP></MATH> +by putting <MATH><I>x</I><SUB>3</SUB><I>x</I><SUB>4</SUB> = {(<I>m</I> + 2)<G>x</G> + 1}<SUP>2</SUP> - 1</MATH>, and so find <I>x</I><SUB>4</SUB> in +terms of <G>x</G>, after which we have only two conditions more +to satisfy. The condition <I>x</I><SUB>1</SUB><I>x</I><SUB>4</SUB> + 1 = square is auto- +matically satisfied, since +<MATH>{(<I>m</I> + 1)<SUP>2</SUP><G>x</G> + 2(<I>m</I> + 1)} {(<I>m</I> + 2)<SUP>2</SUP><G>x</G> + 2(<I>m</I> + 2)} + 1</MATH> +is a square, and it only remains to satisfy <I>x</I><SUB>2</SUB><I>x</I><SUB>4</SUB> + 1 = square. +That is, +<MATH> +(<I>m</I><SUP>2</SUP><G>x</G> + 2<I>m</I>) {(<I>m</I> + 2)<SUP>2</SUP><G>x</G> + 2(<I>m</I> + 2)} + 1 += <I>m</I><SUP>2</SUP>(<I>m</I> + 2)<SUP>2</SUP><G>x</G><SUP>2</SUP> + 2<I>m</I>(<I>m</I> + 2) (2<I>m</I> + 2)<G>x</G> + 4<I>m</I>(<I>m</I> + 2) + 1 +</MATH> +has to be made a square, which is easy, since the coefficient +of <G>x</G><SUP>2</SUP> is a square. +<p>With Diophantus <I>m</I> = 1, so that <MATH><I>x</I><SUB>1</SUB> = 4<G>x</G> + 4, <I>x</I><SUB>2</SUB> = <G>x</G> + 2, +<I>x</I><SUB>3</SUB> = <G>x</G>, <I>x</I><SUB>4</SUB> = 9<G>x</G> + 6, and 9<G>x</G><SUP>2</SUP> + 24<G>x</G> + 13</MATH> has to be made +a square. He equates this to (3<G>x</G> - 4)<SUP>2</SUP>, giving <G>x</G> = 1/16.] +<MATH> +<p>IV. 21. <I>xz</I> = <I>y</I><SUP>2</SUP>, <I>x</I> - <I>y</I> = <I>u</I><SUP>2</SUP>, <I>x</I> - <I>z</I> = <I>v</I><SUP>2</SUP>, <I>y</I> - <I>z</I> = <I>w</I><SUP>2</SUP>. (<I>x</I>><I>y</I>><I>z</I>) +<BRACE>IV. 22. <I>xyz</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>xyz</I> + <I>y</I> = <I>v</I><SUP>2</SUP>, <I>xyz</I> + <I>z</I> = <I>w</I><SUP>2</SUP>. +IV. 23. <I>xyz</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, <I>xyz</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, <I>xyz</I> - <I>z</I> = <I>w</I><SUP>2</SUP></BRACE>. +</MATH> +<pb n=499><head>INDETERMINATE ANALYSIS</head> +<MATH> +<BRACE>IV. 29. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>w</I><SUP>2</SUP> + <I>x</I> + <I>y</I> + <I>z</I> + <I>w</I> = <I>a</I>. +[Since <I>x</I><SUP>2</SUP> + <I>x</I> + 1/4 is a square, +(<I>x</I><SUP>2</SUP> + <I>x</I>) + (<I>y</I><SUP>2</SUP> + <I>y</I>) + (<I>z</I><SUP>2</SUP> + <I>z</I>) + (<I>w</I><SUP>2</SUP> + <I>w</I>) + 1 +is the sum of four squares, and we only have to separate +<I>a</I> + 1 into four squares.] +IV. 30. <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>w</I><SUP>2</SUP> - (<I>x</I> + <I>y</I> + <I>z</I> + <I>w</I>) = <I>a.</I></BRACE> +<p>IV. 31. <I>x</I> + <I>y</I> = 1, (<I>x</I> + <I>a</I>) (<I>y</I> + <I>b</I>) = <I>u</I><SUP>2</SUP>. +<p>IV. 32. <I>x</I> + <I>y</I> + <I>z</I> = <I>a,</I> <I>xy</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>xy</I> - <I>z</I> = <I>v</I><SUP>2</SUP>. +<p>IV. 39. <I>x</I> - <I>y</I> = <I>m</I>(<I>y</I> - <I>z</I>), <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>. +<p>IV. 40. <I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>m</I>(<I>y</I> - <I>z</I>), <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>. +<BRACE>V. 1. <I>xz</I> = <I>y</I><SUP>2</SUP>, <I>x</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> - <I>a</I> = <I>v</I><SUP>2</SUP>, <I>z</I> - <I>a</I> = <I>w</I><SUP>2</SUP>. +V. 2. <I>xz</I> = <I>y</I><SUP>2</SUP>, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>z</I> + <I>a</I> = <I>w</I><SUP>2</SUP>.</BRACE> +<BRACE>V. 3. <I>x</I> + <I>a</I> = <I>r</I><SUP>2</SUP>, <I>y</I> + <I>a</I> = <I>s</I><SUP>2</SUP>, <I>z</I> + <I>a</I> = <I>t</I><SUP>2</SUP>, +<I>yz</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> + <I>a</I> = <I>w</I><SUP>2</SUP>. +V. 4. <I>x</I> - <I>a</I> = <I>r</I><SUP>2</SUP>, <I>y</I> - <I>a</I> = <I>s</I><SUP>2</SUP>, <I>z</I> - <I>a</I> = <I>t</I><SUP>2</SUP>, +<I>yz</I> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> - <I>a</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>a</I> = <I>w</I><SUP>2</SUP></BRACE>. +</MATH> +<p>[Solved by means of the <I>Porisms</I> that, if <I>a</I> be the +given number, the numbers <I>m</I><SUP>2</SUP> - <I>a</I>, (<I>m</I> + 1)<SUP>2</SUP> - <I>a</I> satisfy +the conditions of V. 3, and the numbers <I>m</I><SUP>2</SUP> + <I>a</I>, +(<I>m</I> + 1)<SUP>2</SUP> + <I>a</I> the conditions of V. 4 (see p. 479 above). The +third number is taken to be <MATH>2{<I>m</I><SUP>2</SUP>∓<I>a</I> + (<I>m</I> + 1)<SUP>2</SUP>∓<I>a</I>} - 1</MATH>, +and the three numbers automatically satisfy two more +conditions (see p. 480 above). It only remains to make +<MATH> +2{<I>m</I><SUP>2</SUP>∓<I>a</I> + (<I>m</I> + 1)<SUP>2</SUP>∓<I>a</I>} - 1±<I>a</I> a square, +or 4<I>m</I><SUP>2</SUP> + 4<I>m</I>∓3<I>a</I> + 1 = a square, +</MATH> +which is easily solved. +<p>With Diophantus <G>x</G> + 3 takes the place of <I>m</I> in V. 3 +and <G>x</G> takes its place in V. 4, while <I>a</I> is 5 in V. 3 and 6 +in V. 4.] +<MATH> +<p>V. 5. <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> = <I>r</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>s</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> = +<I>t</I><SUP>2</SUP>, +<I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>, +<I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP> +</MATH> +<p>[Solved by means of the Porism numbered 2 on p. 480. +<pb n=500><head>DIOPHANTUS OF ALEXANDRIA</head> +<MATH> +<p>V. 6. <I>x</I> - 2 = <I>r</I><SUP>2</SUP>, <I>y</I> - 2 = <I>s</I><SUP>2</SUP>, <I>z</I> - 2 = <I>t</I><SUP>2</SUP>, +<I>yz</I> - <I>y</I> - <I>z</I> = <I>u</I><SUP>2</SUP>, <I>zx</I> - <I>z</I> - <I>x</I> = <I>v</I><SUP>2</SUP>, <I>xy</I> - <I>x</I> - <I>y</I> = <I>w</I><SUP>2</SUP>, +<I>yz</I> - <I>x</I> = <I>u</I>′<SUP>2</SUP>, <I>zx</I> - <I>y</I> = <I>v</I>′<SUP>2</SUP>, <I>xy</I> - <I>z</I> = <I>w</I>′<SUP>2</SUP>. +</MATH> +<p>[Solved by means of the proposition numbered (3) on +p. 481.] +<MATH> +<p>Lemma 1 to V. 7. <I>xy</I> + <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>. +<BRACE><note>V. 7. <I>x</I><SUP>2</SUP> ± (<I>x</I> + <I>y</I> + <I>z</I>) =</note> +<I>u</I><SUP>2</SUP> +<I>u</I>′<SUP>2</SUP>,</BRACE> +<BRACE><I>y</I><SUP>2</SUP>±(<I>x</I> + <I>y</I> + <I>z</I>) =</note> +<I>v</I><SUP>2</SUP> +<I>v</I>′<SUP>2</SUP>,</BRACE> +<BRACE><note><I>z</I><SUP>2</SUP>±(<I>x</I> + <I>y</I> + <I>z</I>) =</note> +<I>w</I><SUP>2</SUP> +<I>w</I>′<SUP>2</SUP>. +</BRACE> +</MATH> +<p>[Solved by means of the subsidiary problem (Lemma 2) +of finding three rational right-angled triangles with +equal area. If <I>m, n</I> satisfy the condition in Lemma 1, +i.e. if <MATH><I>mn</I> + <I>m</I><SUP>2</SUP> + <I>n</I><SUP>2</SUP> = <I>p</I><SUP>2</SUP></MATH>, the triangles are ‘formed’ from +the pairs of numbers (<I>p, m</I>), (<I>p, n</I>), (<I>p, m</I> + <I>n</I>) respec- +tively. Diophantus assumes this, but it is easy to prove. +In his case <I>m</I> = 3, <I>n</I> = 5, so that <I>p</I> = 7. Now, in +a right-angled triangle, (hypotenuse)<SUP>2</SUP>±four times area +is a square. We equate, therefore, <I>x</I> + <I>y</I> + <I>z</I> to four +times the common area multiplied by <G>x</G><SUP>2</SUP>, and the several +numbers <I>x, y, z</I> to the three hypotenuses multiplied by <G>x</G>, +and equate the two values. In Diophantus's case the +triangles are (40, 42, 58), (24, 70, 74) and (15, 112, 113), +and 245<G>x</G> = 3360<G>x</G><SUP>2</SUP>.] +<MATH> +<BRACE><note>V. 8. <I>yz</I>±(<I>x</I> + <I>y</I> + <I>z</I>) =</note> +<I>u</I><SUP>2</SUP> +<I>u</I>′<SUP>2</SUP>,</BRACE> +<BRACE><note><I>zx</I> ± (<I>x</I> + <I>y</I> + <I>z</I>) =</note> +<I>v</I><SUP>2</SUP> +<I>v</I>′<SUP>2</SUP>, +<BRACE><note><I>xy</I> ± (<I>x</I> + <I>y</I> + <I>z</I>) =</note> +<I>w</I><SUP>2</SUP> +<I>w</I>′<SUP>2</SUP> +</BRACE> +</MATH> +<p>[Solved by means of the same three rational right- +angled triangles found in the Lemma to V. 7, together +with the Lemma that we can solve the equations <I>yz</I> = <I>a</I><SUP>2</SUP>, +<I>zx</I> = <I>b</I><SUP>2</SUP>, <I>xy</I> = <I>c</I><SUP>2</SUP>.] +<MATH> +<BRACE>V. 9. (Cf. II. 11). <I>x</I> + <I>y</I> = 1, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>a</I> = <I>v</I><SUP>2</SUP>. +V. 11. <I>x</I> + <I>y</I> + <I>z</I> = 1, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>z</I> + <I>a</I> = <I>w</I><SUP>2</SUP>.</BRACE> +</MATH> +<p>[These are the problems of <G>pariso/thtos a)gwgh/</G> +<pb n=501><head>INDETERMINATE ANALYSIS</head> +described above (pp. 477-9). The problem is ‘to divide +unity into two (or three) parts such that, if one and the +same given number be added to each part, the results are +all squares’.] +<MATH> +<BRACE>V. 10. <I>x</I> + <I>y</I> = 1, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>b</I> = <I>v</I><SUP>2</SUP>. +V. 12. <I>x</I> + <I>y</I> + <I>z</I> = 1, <I>x</I> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>y</I> + <I>b</I> = <I>v</I><SUP>2</SUP>, <I>z</I> + <I>c</I> = <I>w</I><SUP>2</SUP></BRACE>. +</MATH> +<p>[These problems are like the preceding except that +<I>different</I> given numbers are added. The second of the +two problems is not worked out, but the first is worth +reproducing. We must take the particular figures used +by Diophantus, namely <I>a</I> = 2, <I>b</I> = 6. We have then to +divide 9 into two squares such that one of them lies +between 2 and 3. Take two squares lying between 2 +and 3, say 289/144, 361/144. We have then to find a square <G>x</G><SUP>2</SUP> +lying between them; if we can do this, we can make +9 - <G>x</G><SUP>2</SUP> a square, and so solve the problem. +<p>Put <MATH>9 - <G>x</G><SUP>2</SUP> = (3 - <I>m</I><G>x</G>)<SUP>2</SUP></MATH>, say, so that <MATH><G>x</G> = 6<I>m</I>/(<I>m</I><SUP>2</SUP> + 1)</MATH>; +and <I>m</I> has to be determined so that <G>x</G> lies between +17/12 and 19/12. +<p>Therefore <MATH>17/12 < (6<I>m</I>)/(<I>m</I><SUP>2</SUP> + 1) < 19/12</MATH>. +<p>Diophantus, as we have seen, finds <I>a fortiori</I> integral +limits for <I>m</I> by solving these inequalities, making <I>m</I> not +greater than 67/17 and not less than 66/19 (see pp. 463-5 above). +He then takes <I>m</I> = 3 1/2 and puts <MATH>9 - <G>x</G><SUP>2</SUP> = (3 - (3 1/2)<G>x</G>)<SUP>2</SUP></MATH>, +which gives <G>x</G> = 84/53.] +<MATH> +<BRACE>V. 13. <I>x</I> + <I>y</I> + <I>z</I> = <I>a</I>, <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>z</I> + <I>x</I> = <I>v</I><SUP>2</SUP>, <I>x</I> + <I>y</I> = <I>w</I><SUP>2</SUP>. +V. 14. <I>x</I> + <I>y</I> + <I>z</I> + <I>w</I> = <I>a</I>, <I>x</I> + <I>y</I> + <I>z</I> = <I>s</I><SUP>2</SUP>, <I>y</I> + <I>z</I> + <I>w</I> = <I>t</I><SUP>2</SUP>, +<I>z</I> + <I>w</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>w</I> + <I>x</I> + <I>y</I> = <I>v</I><SUP>2</SUP></BRACE>. +</MATH> +<p>[The method is the same.] +<MATH> +<BRACE>V. 21. <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>, +<I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>. +V. 22. <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>, +<I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> - <I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>. +V. 23. <I>x</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> - <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP> <I>z</I><SUP>2</SUP> - +<I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP>.</BRACE> +</MATH> +<p>[Solved by means of right-angled triangles in rational +numbers.] +<pb n=502><head>DIOPHANTUS OF ALEXANDRIA</head> +<MATH> +<BRACE>V. 24. <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> + 1 = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + 1 = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> + 1 = <I>w</I><SUP>2</SUP>. +V. 25. <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> - 1 = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> - 1 = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> - 1 = <I>w</I><SUP>2</SUP>. +V. 26. 1 - <I>y</I><SUP>2</SUP><I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, 1 - <I>z</I><SUP>2</SUP><I>x</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>, 1 - <I>x</I><SUP>2</SUP><I>y</I><SUP>2</SUP> = <I>w</I><SUP>2</SUP></BRACE>. +</MATH> +<p>[These reduce to the preceding set of three problems.] +<MATH> +<BRACE>V. 27. <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> + <I>a</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> + <I>a</I> = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>a</I> = <I>w</I><SUP>2</SUP>. +V. 28. <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> - <I>a</I> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>x</I><SUP>2</SUP> - <I>a</I> = <I>v</I><SUP>2</SUP>, <I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> - <I>a</I> = +<I>w</I><SUP>2</SUP></BRACE>. +<p>V. 30. <I>mx</I> + <I>ny</I> = <I>u</I><SUP>2</SUP>, <I>u</I><SUP>2</SUP> + <I>a</I> = (<I>x</I> + <I>y</I>)<SUP>2</SUP>. +</MATH> +<p>[This problem is enunciated thus. ‘A man buys a +certain number of measures of wine, some at 8 drachmas, +some at 5 drachmas each. He pays for them a <I>square</I> +number of drachmas; and if 60 is added to this number, +the result is a square, the side of which is equal to the +whole number of measures. Find the number bought at +each price.’ +<p>Let <G>x</G> = the whole number of measures; therefore +<G>x</G><SUP>2</SUP> - 60 was the number of drachmas paid, and <G>x</G><SUP>2</SUP> - 60 += a square, say (<G>x</G> - <I>m</I>)<SUP>2</SUP>; hence <MATH><G>x</G> = (<I>m</I><SUP>2</SUP> + 60)/2<I>m</I></MATH>. +<p>Now 1/5 of the price of the five-drachma measures + 1/3 +of that of the eight-drachma measures = <G>x</G>; therefore +<G>x</G><SUP>2</SUP> - 60, the total price, has to be divided into two parts +such that 1/5 of one + 1/8 of the other = <G>x</G>. +<p>We cannot have a real solution of this unless +<MATH> +<G>x</G> > (1/8)(<G>x</G><SUP>2</SUP> - 60) and < (1/5)(<G>x</G><SUP>2</SUP> - 60); +therefore 5<G>x</G> < <G>x</G><SUP>2</SUP> - 60 < 8<G>x</G>. +</MATH> +<p>Diophantus concludes, as we have seen (p. 464 above), +that <G>x</G> is not less than 11 and not greater than 12. +<p>Therefore, from above, since <MATH><G>x</G> = (<I>m</I><SUP>2</SUP> + 60)/2<I>m</I>, +22<I>m</I> < <I>m</I><SUP>2</SUP> + 60 < 24<I>m</I></MATH>; +and Diophantus concludes that <I>m</I> is not less than 19 and +not greater than 21. He therefore puts <I>m</I> = 20. +<p>Therefore <MATH><G>x</G> = (<I>m</I><SUP>2</SUP> + 60)/2<I>m</I> = 11 1/2, <G>x</G><SUP>2</SUP> = 132 1/4, and +<G>x</G><SUP>2</SUP> - 60 = 72 1/4</MATH>. +<p>We have now to divide 72 1/4 into two parts such that +1/5 of one part + 1/8 of the other = 11 1/2. +<pb n=503><head>INDETERMINATE ANALYSIS</head> +<p>Let the first part = 5<I>z</I>; therefore 1/3 (second part) += 11 1/2 - <I>z,</I> or second part = 92 - 8<I>z.</I> +<p>Therefore <MATH>5<I>z</I> + 92 - 8<I>z</I> = 72 1/4, and <I>z</I> = 79/12</MATH>; +therefore the number of five-drachma measures is 79/12 and +the number of eight-drachma measures 59/12.] +<MATH> +<BRACE><note>(see p. 467 +above.)</note> +Lemma 2 to VI. 12. <I>ax</I><SUP>2</SUP> + <I>b</I> = <I>u</I><SUP>2</SUP> (where <I>a</I> + <I>b</I> = <I>c</I><SUP>2</SUP>). +Lemma to VI. 15. <I>ax</I><SUP>2</SUP> - <I>b</I> = <I>u</I><SUP>2</SUP> (where <I>ad</I><SUP>2</SUP> - <I>b</I> = <I>c</I><SUP>2</SUP>). +</BRACE> +<BRACE> +[III. 15]. <I>xy</I> + <I>x</I> + <I>y</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + 1 = (<I>v</I><SUP>2</SUP>/<I>w</I><SUP>2</SUP>)(<I>y</I> + 1). +[III. 16]. <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>u</I><SUP>2</SUP>, <I>x</I> - 1 = (<I>v</I><SUP>2</SUP>/<I>w</I><SUP>2</SUP>)(<I>y</I> - 1). +</BRACE> +<p>[IV. 32]. <I>x</I> + 1 = (<I>u</I><SUP>2</SUP>/<I>v</I><SUP>2</SUP>)(<I>x</I> - 1). +<p>[V. 21]. <I>x</I><SUP>2</SUP> + 1 = <I>u</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> + 1 = <I>v</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + 1 = <I>w</I><SUP>2</SUP>. +</MATH> +<C>(viii) Indeterminate analysis of the third degree.</C> +<MATH> +<p>IV. 3. <I>x</I><SUP>2</SUP><I>y</I> = <I>u, xy</I> = <I>u</I><SUP>3</SUP>. +<BRACE> +IV. 6. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>2</SUP> = <I>u</I><SUP>3</SUP>, <I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>2</SUP>. +IV. 7. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP>, <I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>v</I><SUP>3</SUP>. +</BRACE> +<BRACE> +<BRACE><note>(really reducible +to the second +degree.)</note> +<BRACE> +IV. 8. <I>x</I> + <I>y</I><SUP>3</SUP> = <I>u</I><SUP>3</SUP>, <I>x</I> + <I>y</I> = <I>u.</I> +IV. 9. <I>x</I> + <I>y</I><SUP>3</SUP> = <I>u, x</I> + <I>y</I> = <I>u</I><SUP>3</SUP>. +</BRACE> +<BRACE> +IV. 10. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> = <I>x</I> + <I>y.</I> +<BRACE><note>the same problem.</note> +IV. 11. <I>x</I><SUP>3</SUP> - <I>y</I><SUP>3</SUP> = <I>x</I> - <I>y.</I> +IV. 12. <I>x</I><SUP>3</SUP> + <I>y</I> = <I>y</I><SUP>3</SUP> + <I>x.</I> +</BRACE> +</BRACE> +</BRACE> +</MATH> +<p>[We may give as examples the solutions of IV. 7, +IV. 8, IV. 11. +<p>IV. 7. Since <I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = a cube, suppose <MATH><I>z</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>x</I><SUP>3</SUP></MATH>. +To make <I>x</I><SUP>3</SUP> + <I>y</I><SUP>2</SUP> a square, put <MATH><I>x</I><SUP>3</SUP> = <I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP>, <I>y</I><SUP>2</SUP> = 2 <I>ab</I></MATH>, +which also satisfies <MATH><I>x</I><SUP>3</SUP> - <I>y</I><SUP>2</SUP> = <I>z</I><SUP>2</SUP></MATH>. We have then to make +2<I>ab</I> a square. Let <I>a</I> = <G>x</G>, <I>b</I> = 2<G>x</G>; therefore <MATH><I>a</I><SUP>2</SUP> + <I>b</I><SUP>2</SUP> = 5<G>x</G><SUP>2</SUP></MATH>, +2<I>ab</I> = 4<G>x</G><SUP>2</SUP>, <I>y</I> = 2<G>x</G>, <I>z</I> = <G>x</G>, and we have only to make +5<G>x</G><SUP>2</SUP> a cube. <G>x</G> = 5, and <I>x</I><SUP>3</SUP> = 125, <I>y</I><SUP>2</SUP> = 100, <I>z</I><SUP>2</SUP> = 25. +<pb n=504><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>IV. 8. Suppose <I>x</I> = <G>x</G>, <I>y</I><SUP>3</SUP> = <I>m</I><SUP>3</SUP><G>x</G><SUP>3</SUP>; therefore <MATH><I>u</I> = (<I>m</I> + 1)<G>x</G></MATH> +must be the side of the cube <I>m</I><SUP>3</SUP><G>x</G><SUP>3</SUP> + <G>x</G>, and +<MATH><I>m</I><SUP>3</SUP><G>x</G><SUP>2</SUP> + 1 = (<I>m</I><SUP>3</SUP> + 3<I>m</I><SUP>2</SUP> + 3<I>m</I> + 1)<G>x</G><SUP>2</SUP></MATH>. +To solve this, we must have 3<I>m</I><SUP>2</SUP> + 3<I>m</I> + 1 (the difference +between consecutive cubes) a square. Put +<MATH>3<I>m</I><SUP>2</SUP> + 3<I>m</I> + 1 = (1 - <I>nm</I>)<SUP>2</SUP>, and <I>m</I> = (3 + 2<I>n</I>)/(<I>n</I><SUP>2</SUP> - 3)</MATH>. +<p>IV. 11. Assume <MATH><I>x</I> = (<I>m</I> + 1)<G>x</G>, <I>y</I> = <I>m</I><G>x</G></MATH>, and we have +to make (3<I>m</I><SUP>3</SUP> + 3<I>m</I><SUP>2</SUP> + 1)<G>x</G><SUP>2</SUP> equal to 1, i.e. we have +only to make 3<I>m</I><SUP>2</SUP> + 3<I>m</I> + 1 a square.] +<p>IV. 18. <MATH><I>x</I><SUP>3</SUP> + <I>y</I> = <I>u</I><SUP>3</SUP>, <I>y</I><SUP>2</SUP> + <I>x</I> = <I>v</I><SUP>2</SUP></MATH>. +<p>IV. 24. <MATH><I>x</I> + <I>y</I> = <I>a, xy</I> = <I>u</I><SUP>3</SUP> - <I>u</I></MATH>. +<p>[<MATH><I>y</I> = <I>a</I> - <I>x</I></MATH>; therefore <I>ax</I> - <I>x</I><SUP>2</SUP> has to be made a cube +<I>minus</I> its side, say (<I>mx</I> - 1)<SUP>3</SUP> - (<I>mx</I> - 1). +<p>Therefore <MATH><I>ax</I> - <I>x</I><SUP>2</SUP> = <I>m</I><SUP>3</SUP><I>x</I><SUP>3</SUP> - 3<I>m</I><SUP>2</SUP><I>x</I><SUP>2</SUP> + 2<I>mx</I></MATH>. +To reduce this to a simple equation, we have only to +put <I>m</I> = (1/2)<I>a.</I>] +<p>IV. 25. <MATH><I>x</I> + <I>y</I> + <I>z</I> = <I>a, xyz</I> = {(<I>x</I> - <I>y</I>) + (<I>x</I> - <I>z</I>) + (<I>y</I> - <I>z</I>)}<SUP>3</SUP>. +(<I>x</I> > <I>y</I> > <I>z</I>)</MATH> +<p>[The cube = 8(<I>x</I> - <I>z</I>)<SUP>3</SUP>. Let <MATH><I>x</I> = (<I>m</I> + 1)<G>x</G>, <I>z</I> = <I>m</I><G>x</G></MATH>, so +that <MATH><I>y</I> = 8<G>x</G> / (<I>m</I><SUP>2</SUP> + <I>m</I>)</MATH>, and we have only to contrive that +<MATH>8 / (<I>m</I><SUP>2</SUP> + <I>m</I>)</MATH> lies between <I>m</I> and <I>m</I> + 1. Dioph. takes the +first limit <MATH>8 > <I>m</I><SUP>3</SUP> + <I>m</I><SUP>2</SUP></MATH>, and puts +<MATH>8 = (<I>m</I> + 1/3)<SUP>3</SUP> or <I>m</I><SUP>3</SUP> + <I>m</I><SUP>2</SUP> + 1/3<I>m</I> + 1/27</MATH>, +whence <I>m</I> = 5/3; therefore <I>x</I> = (8/3)<G>x</G>, <I>y</I> = (9/5)<G>x</G>, <I>z</I> = 5/3<G>x</G>. Or, +multiplying by 15, we have <I>x</I> = 40<G>x</G>, <I>y</I> = 27<G>x</G>, <I>z</I> = 25<G>x</G>. +The first equation then gives <G>x</G>.] +<MATH> +<BRACE> +IV. 26. <I>xy</I> + <I>x</I> = <I>u</I><SUP>3</SUP>, <I>xy</I> + <I>y</I> = <I>v</I><SUP>3</SUP>. +IV. 27. <I>xy</I> - <I>x</I> = <I>u</I><SUP>3</SUP>, <I>xy</I> - <I>y</I> = <I>v</I><SUP>3</SUP>. +</BRACE> +<p>IV. 28. <I>xy</I> + (<I>x</I> + <I>y</I>) = <I>u</I><SUP>3</SUP>, <I>xy</I> - (<I>x</I> + <I>y</I>) = <I>v</I><SUP>3</SUP>. +[<I>x</I> + <I>y</I> = 1/2(<I>u</I><SUP>3</SUP> - <I>v</I><SUP>3</SUP>), <I>xy</I> = 1/2(<I>u</I><SUP>3</SUP> + <I>v</I><SUP>3</SUP>); therefore +(<I>x</I> - <I>y</I>)<SUP>2</SUP> = 1/4(<I>u</I><SUP>3</SUP> - <I>v</I><SUP>3</SUP>)<SUP>2</SUP> - 2(<I>u</I><SUP>3</SUP> + <I>v</I><SUP>3</SUP>), +</MATH> +which latter expression has to be made a square. +<pb n=505><head>INDETERMINATE ANALYSIS</head> +<p>Diophantus assumes <MATH><I>u</I> = <G>x</G> + 1, <I>v</I> = <G>x</G> - 1</MATH>, whence +<MATH>1/4(6<G>x</G><SUP>2</SUP> + 2)<SUP>2</SUP> - 2(2<G>x</G><SUP>3</SUP> + 6<G>x</G>)</MATH> +must be a square, or +<MATH>9<G>x</G><SUP>4</SUP> - 4<G>x</G><SUP>3</SUP> + 6<G>x</G><SUP>2</SUP> - 12<G>x</G> + 1 = a square = (3<G>x</G><SUP>2</SUP> - 6<G>x</G> + 1)<SUP>2</SUP></MATH>, say; +therefore 32<G>x</G><SUP>3</SUP> = 36<G>x</G><SUP>2</SUP>, and <G>x</G> = 9/8. Thus <I>u, v</I> are found, +and then <I>x, y.</I> +<p>The second (alternative) solution uses the formula that +<MATH><G>x</G>(<G>x</G><SUP>2</SUP> - <G>x</G>) + (<G>x</G><SUP>2</SUP> - <G>x</G>) + <G>x</G> =</MATH> a cube. Put <MATH><I>x</I> = <G>x</G>, <I>y</I> = <G>x</G><SUP>2</SUP> - <G>x</G></MATH>, +and one condition is satisfied. We then only have to +make <MATH><G>x</G>(<G>x</G><SUP>2</SUP> - <G>x</G>) - <G>x</G> - (<G>x</G><SUP>2</SUP> - <G>x</G>) or <G>x</G><SUP>3</SUP> - 2<G>x</G><SUP>2</SUP></MATH> a cube (less than +<G>x</G><SUP>3</SUP>), i. e. <MATH><G>x</G><SUP>3</SUP> - 2<G>x</G><SUP>2</SUP> = ((1/2)<G>x</G>)<SUP>3</SUP></MATH>, say.] +<p>IV. 38. <MATH>(<I>x</I> + <I>y</I> + <I>z</I>)<I>x</I> = (1/2)<I>u</I>(<I>u</I> + 1), (<I>x</I> + <I>y</I> + <I>z</I>)<I>y</I> = <I>v</I><SUP>2</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>)<I>z</I> = <I>w</I><SUP>3</SUP>, [<I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>]</MATH>. +<p>[Suppose <MATH><I>x</I> + <I>y</I> + <I>z</I> = <G>x</G><SUP>2</SUP></MATH>; then +<MATH><I>x</I> = (<I>u</I>(<I>u</I> + 1))/2<G>x</G><SUP>2</SUP>, <I>y</I> = <I>v</I><SUP>2</SUP>/<G>x</G><SUP>2</SUP>, <I>z</I> = <I>w</I><SUP>3</SUP>/<G>x</G><SUP>2</SUP></MATH>; +therefore <MATH><G>x</G><SUP>4</SUP> = (1/2)<I>u</I>(<I>u</I> + 1) + <I>v</I><SUP>2</SUP> + <I>w</I><SUP>3</SUP></MATH>. +Diophantus puts 8 for <I>w</I><SUP>3</SUP>, but we may take any cube, as +<I>m</I><SUP>3</SUP>; and he assumes <MATH><I>v</I><SUP>2</SUP> = (<G>x</G><SUP>2</SUP> - 1)<SUP>2</SUP></MATH>, for which we might +substitute (<G>x</G><SUP>2</SUP> - <I>n</I><SUP>2</SUP>)<SUP>2</SUP>. We then have the triangular +number <MATH>(1/2)<I>u</I>(<I>u</I> + 1) = 2<I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP> - <I>n</I><SUP>4</SUP> - <I>m</I><SUP>3</SUP></MATH>. Since 8 times a +triangular number <I>plus</I> 1 gives a square, +<MATH>16<I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP> - 8<I>n</I><SUP>4</SUP> - 8<I>m</I><SUP>3</SUP> + 1 = a square = (4<I>n</I><G>x</G> - <I>k</I>)<SUP>2</SUP></MATH>, say, +and the problem is solved.] +<BRACE> +V. 15. <MATH>(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>x</I> = <I>u</I><SUP>3</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>y</I> = <I>v</I><SUP>3</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>z</I> = <I>w</I><SUP>3</SUP></MATH>. +[Let <MATH><I>x</I> + <I>y</I> + <I>z</I> = <G>x</G>, <I>u</I><SUP>3</SUP> = <I>m</I><SUP>3</SUP><G>x</G><SUP>3</SUP>, <I>v</I><SUP>3</SUP> = <I>n</I><SUP>3</SUP><G>x</G><SUP>3</SUP>, <I>w</I><SUP>3</SUP> = +<I>p</I><SUP>3</SUP><G>x</G><SUP>3</SUP></MATH>; therefore <MATH><G>x</G> = {(<I>m</I><SUP>3</SUP> - 1) + (<I>n</I><SUP>3</SUP> - 1) + (<I>p</I><SUP>3</SUP> - 1)}<G>x</G><SUP>3</SUP></MATH>; +and we have to find three cubes <I>m</I><SUP>3</SUP>, <I>n</I><SUP>3</SUP>, <I>p</I><SUP>3</SUP> such that +<MATH><I>m</I><SUP>3</SUP> + <I>n</I><SUP>3</SUP> + <I>p</I><SUP>3</SUP> - 3 =</MATH> a square. Diophantus assumes as +the sides of the cubes (<I>k</I> + 1), (2 - <I>k</I>), 2; this gives +<pb n=506><head>DIOPHANTUS OF ALEXANDRIA</head> +<MATH>9<I>k</I><SUP>2</SUP> - 9<I>k</I> + 14 = a square = (3<I>k</I> - <I>l</I>)<SUP>2</SUP></MATH>, say; and <I>k</I> is found. +Retracing our steps, we find <G>x</G> and therefore <I>x, y, z.</I>] +V. 16. <MATH>(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>x</I> = <I>u</I><SUP>3</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>y</I> = <I>v</I><SUP>3</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>z</I> = <I>w</I><SUP>3</SUP></MATH>. +V. 17. <MATH><I>x</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>u</I><SUP>3</SUP>, <I>y</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>v</I><SUP>3</SUP>, +<I>z</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>w</I><SUP>3</SUP></MATH>. +</BRACE> +<p>V. 18. <MATH><I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>x</I> = <I>u</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>y</I> = <I>v</I><SUP>2</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> + <I>z</I> = <I>w</I><SUP>2</SUP></MATH>. +<p>[Put <MATH><I>x</I> + <I>y</I> + <I>z</I> = <G>x</G><SUP>2</SUP>, <I>x</I> = (<I>p</I><SUP>2</SUP> - 1)<G>x</G><SUP>6</SUP>, <I>y</I> = (<I>q</I><SUP>2</SUP> - 1)<G>x</G><SUP>6</SUP>, +<I>z</I> = (<I>r</I><SUP>2</SUP> - 1)<G>x</G><SUP>6</SUP></MATH>, whence <MATH><G>x</G><SUP>2</SUP> = (<I>p</I><SUP>2</SUP> - 1 + <I>q</I><SUP>2</SUP> - 1 + <I>r</I><SUP>2</SUP> - 1)<G>x</G><SUP>6</SUP></MATH>, so +that <I>p</I><SUP>2</SUP> - 1 + <I>q</I><SUP>2</SUP> - 1 + <I>r</I><SUP>2</SUP> - 1 must be made a fourth +power. Diophantus assumes <MATH><I>p</I><SUP>2</SUP> = (<I>m</I><SUP>2</SUP> - 1)<SUP>2</SUP>, <I>q</I><SUP>2</SUP> = (<I>m</I> + 1)<SUP>2</SUP>, +<I>r</I><SUP>2</SUP> = (<I>m</I> - 1)<SUP>2</SUP></MATH>, since <MATH><I>m</I><SUP>4</SUP> - 2<I>m</I><SUP>2</SUP> + <I>m</I><SUP>2</SUP> + 2<I>m</I> + <I>m</I><SUP>2</SUP> - 2<I>m</I> = <I>m</I><SUP>4</SUP></MATH>.] +<MATH> +<p>V. 19. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>x</I> = <I>u</I><SUP>2</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>y</I> = <I>v</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> - <I>z</I> = <I>w</I><SUP>2</SUP>. +<p>V. 19a. <I>x</I> + <I>y</I> + <I>z</I> = <I>t</I><SUP>2</SUP>, <I>x</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>u</I><SUP>2</SUP>, +<I>y</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>v</I><SUP>2</SUP>, <I>z</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>w</I><SUP>2</SUP>. +<p>V. 19. b, c. <I>x</I> + <I>y</I> + <I>z</I> = <I>a,</I> (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> ± <I>x</I> = <I>u</I><SUP>2</SUP>, +(<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> ± <I>y</I> = <I>v</I><SUP>2</SUP>, (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> ± <I>z</I> = <I>w</I><SUP>2</SUP>. +<p>V. 20. <I>x</I> + <I>y</I> + <I>z</I> = 1/<I>m</I>, <I>x</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>u</I><SUP>2</SUP>, +<I>y</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>v</I><SUP>2</SUP>, <I>z</I> - (<I>x</I> + <I>y</I> + <I>z</I>)<SUP>3</SUP> = <I>w</I><SUP>2</SUP>. +<p>[IV. 8]. <I>x</I> - <I>y</I> = 1, <I>x</I><SUP>3</SUP> - <I>y</I><SUP>3</SUP> = <I>u</I><SUP>2</SUP>. +<p>[IV. 9, 10]. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> = (<I>u</I><SUP>2</SUP>/<I>v</I><SUP>2</SUP>)(<I>x</I> + <I>y</I>). +<p>[IV. 11]. <I>x</I><SUP>3</SUP> - <I>y</I><SUP>3</SUP> = (<I>u</I><SUP>2</SUP>/<I>v</I><SUP>2</SUP>)(<I>x</I> - <I>y</I>). +<p>[V. 15]. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> + <I>z</I><SUP>3</SUP> - 3 = <I>u</I><SUP>2</SUP>. +<p>[V. 16]. 3 - (<I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> + <I>z</I><SUP>3</SUP>) = <I>u</I><SUP>2</SUP>. +<p>[V. 17]. <I>x</I><SUP>3</SUP> + <I>y</I><SUP>3</SUP> + <I>z</I><SUP>3</SUP> + 3 = <I>u</I><SUP>2</SUP>. +</MATH> +<pb n=507><head>INDETERMINATE ANALYSIS</head> +<C>(ix) Indeterminate analysis of the fourth degree.</C> +<p>V. 29. <MATH><I>x</I><SUP>4</SUP> + <I>y</I><SUP>4</SUP> + <I>z</I><SUP>4</SUP> = <I>u</I><SUP>2</SUP></MATH>. +<p>[‘Why’, says Fermat, ‘did not Diophantus seek <I>two</I> +fourth powers such that their sum is a square. This +problem is, in fact, impossible, as by my method I am +able to prove with all rigour.’ No doubt Diophantus +knew this truth empirically. Let <MATH><I>x</I><SUP>2</SUP> = <G>x</G><SUP>2</SUP>, <I>y</I><SUP>2</SUP> = <I>p</I><SUP>2</SUP>, +<I>z</I><SUP>2</SUP> = <I>q</I><SUP>2</SUP></MATH>. Therefore <MATH><G>x</G><SUP>4</SUP> + <I>p</I><SUP>4</SUP> + <I>q</I><SUP>4</SUP> = a square = (<G>x</G><SUP>2</SUP> - <I>r</I>)<SUP>2</SUP></MATH>, say; +therefore <MATH><G>x</G><SUP>2</SUP> = (<I>r</I><SUP>2</SUP> - <I>p</I><SUP>4</SUP> - <I>q</I><SUP>4</SUP>)/2<I>r</I></MATH>, and we have to make +this expression a square. +<p>Diophantus puts <MATH><I>r</I> = <I>p</I><SUP>2</SUP> + 4, <I>q</I><SUP>2</SUP> = 4</MATH>, so that the expres- +sion reduces to <MATH>8<I>p</I><SUP>2</SUP>/(2<I>p</I><SUP>2</SUP> + 8) or 4<I>p</I><SUP>2</SUP>/(<I>p</I><SUP>2</SUP> + 4)</MATH>. To make +this a square, let <MATH><I>p</I><SUP>2</SUP> + 4 = (<I>p</I> + 1)<SUP>2</SUP></MATH>, say; therefore <I>p</I> = 1 1/2, +and <I>p</I><SUP>2</SUP> = 2 1/4, <I>q</I><SUP>2</SUP> = 4, <I>r</I> = 6 1/4; or (multiplying by 4) +<I>p</I><SUP>2</SUP> = 9, <I>q</I><SUP>2</SUP> = 16, <I>r</I> = 25, which solves the problem.] +<p>[V. 18]. <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> + <I>z</I><SUP>2</SUP> - 3 = <I>u</I><SUP>4</SUP></MATH>. +<p>(See above under V. 18.) +<C>(x) Problems of constructing right-angled triangles with +sides in rational numbers and satisfying various +other conditions.</C> +<p>[I shall in all cases call the hypotenuse <I>z,</I> and the +other two sides <I>x, y,</I> so that the condition <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>z</I><SUP>2</SUP></MATH> +applies in all cases, in addition to the other conditions +specified.] +<p>[Lemma to V. 7]. <MATH><I>xy</I> = <I>x</I><SUB>1</SUB><I>y</I><SUB>1</SUB> = <I>x</I><SUB>2</SUB><I>y</I><SUB>2</SUB></MATH>. +<BRACE> +VI. 1. <MATH><I>z</I> - <I>x</I> = <I>u</I><SUP>3</SUP>, <I>z</I> - <I>y</I> = <I>v</I><SUP>3</SUP></MATH>. +[Form a right-angled triangle from <G>x</G>, <I>m,</I> so that +<MATH><I>z</I> = <G>x</G><SUP>2</SUP> + <I>m</I><SUP>2</SUP>, <I>x</I> = 2<I>m</I><G>x</G>, <I>y</I> = <G>x</G><SUP>2</SUP> - <I>m</I><SUP>2</SUP></MATH>; thus <MATH><I>z</I> - <I>y</I> = 2<I>m</I><SUP>2</SUP></MATH>, +and, as this must be a cube, we put <I>m</I> = 2; therefore +<MATH><I>z</I> - <I>x</I> = <G>x</G><SUP>2</SUP> - 4<G>x</G> + 4</MATH> must be a cube, or <G>x</G> - 2 = a cube, +say <I>n</I><SUP>3</SUP>, and <MATH><G>x</G> = <I>n</I><SUP>3</SUP> + 2</MATH>.] +VI. 2. <MATH><I>z</I> + <I>x</I> = <I>u</I><SUP>3</SUP>, <I>z</I> + <I>y</I> = <I>v</I><SUP>3</SUP></MATH>. +</BRACE> +<pb n=508><head>DIOPHANTUS OF ALEXANDRIA</head> +<BRACE> +VI. 3. <MATH>(1/2)<I>xy</I> + <I>a</I> = <I>u</I><SUP>2</SUP></MATH>. +[Suppose the required triangle to be <I>h</I><G>x</G>, <I>p</I><G>x</G>, <I>b</I><G>x</G>; there- +fore (1/2)<I>pb</I><G>x</G><SUP>2</SUP> + <I>a</I> = a square = <I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP>, say, and the ratio of <I>a</I> +to <I>n</I><SUP>2</SUP> - (1/2)<I>pb</I> must be the ratio of a square to a square. +To find <I>n, p, b</I> so as to satisfy this condition, form +a right-angled triangle from <I>m,</I> 1/<I>m</I>, +i. e. <MATH>(<I>m</I><SUP>2</SUP> + 1/<I>m</I><SUP>2</SUP>, 2, <I>m</I><SUP>2</SUP> - 1/<I>m</I><SUP>2</SUP>)</MATH>; +therefore <MATH>(1/2)<I>pb</I> = <I>m</I><SUP>2</SUP> - 1/<I>m</I><SUP>2</SUP></MATH>. Assume <MATH><I>n</I><SUP>2</SUP> = (<I>m</I> + 2<I>a/m</I>)<SUP>2</SUP></MATH>; +therefore <MATH><I>n</I><SUP>2</SUP> - (1/2)<I>pb</I> = 4<I>a</I> + (4<I>a</I><SUP>2</SUP> + 1)/<I>m</I><SUP>2</SUP></MATH>; and <MATH>(4<I>a</I> + (4<I>a</I><SUP>2</SUP> + 1)/<I>m</I><SUP>2</SUP>) / <I>a</I></MATH>, +or <MATH>4<I>a</I><SUP>2</SUP> + (<I>a</I>(4<I>a</I><SUP>2</SUP> + 1))/<I>m</I><SUP>2</SUP></MATH>, has to be made a square. Put +<MATH>4<I>a</I><SUP>2</SUP><I>m</I><SUP>2</SUP> + <I>a</I>(4<I>a</I><SUP>2</SUP> + 1) = (2<I>am</I> + <I>k</I>)<SUP>2</SUP></MATH>, and we have a solution. +Diophantus has <I>a</I> = 5, leading to <MATH>100<I>m</I><SUP>2</SUP> + 505 = a square += (10<I>m</I> + 5)<SUP>2</SUP></MATH>, say, which gives <I>m</I> = 24/5 and <I>n</I> = 413/60. +<I>h, p, b</I> are thus determined in such a way that +<MATH>(1/2)<I>pb</I><G>x</G><SUP>2</SUP> + <I>a</I> = <I>n</I><SUP>2</SUP><G>x</G><SUP>2</SUP></MATH> gives a rational solution.] +VI. 4. <MATH>(1/2)<I>xy</I> - <I>a</I> = <I>u</I><SUP>2</SUP></MATH>. +VI. 5. <MATH><I>a</I> - (1/2)<I>xy</I> = <I>u</I><SUP>2</SUP></MATH>. +</BRACE> +<BRACE> +VI. 6. <MATH>(1/2)<I>xy</I> + <I>x</I> = <I>a</I></MATH>. +[Assume the triangle to be <I>h</I><G>x</G>, <I>p</I><G>x</G>, <I>b</I><G>x</G>, so that +<MATH>(1/2)<I>pb</I><G>x</G><SUP>2</SUP> + <I>p</I><G>x</G> = <I>a</I></MATH>, and for a rational solution of this equa- +tion we must have ((1/2)<I>p</I>)<SUP>2</SUP> + <I>a</I>((1/2)<I>pb</I>) a square. Diophantus +assumes <I>p</I> = 1, <I>b</I> = <I>m,</I> whence (1/2)<I>am</I> + 1/4 or 2<I>am</I> + 1 += a square. +But, since the triangle is rational, <I>m</I><SUP>2</SUP> + 1 = a square. +That is, we have a double equation. Difference +<MATH>= <I>m</I><SUP>2</SUP> - 2<I>am</I> = <I>m</I>(<I>m</I> - 2<I>a</I>)</MATH>. Put +<MATH>2<I>am</I> + 1 = {1/2(<I>m</I> - &hordar;(<I>m</I> - 2<I>a</I>)}<SUP>2</SUP> = <I>a</I><SUP>2</SUP>, and <I>m</I> = (<I>a</I><SUP>2</SUP> - 1)/2<I>a</I></MATH>. +The sides of the auxiliary triangle are thus determined +in such a way that the original equation in <G>x</G> is solved +rationally.] +VI. 7. <MATH>(1/2)<I>xy</I> - <I>x</I> = <I>a</I></MATH>. +</BRACE> +<pb n=509><head>INDETERMINATE ANALYSIS</head> +<MATH> +<BRACE> +VI. 8. (1/2)<I>xy</I> + (<I>x</I> + <I>y</I>) = <I>a.</I> +VI. 9. (1/2)<I>xy</I> - (<I>x</I> + <I>y</I>) = <I>a.</I> +</BRACE> +</MATH> +<p>[With the same assumptions we have in these cases +to make {1/2 (<I>p</I> + <I>b</I>)}<SUP>2</SUP> + <I>a</I> ((1/2)<I>pb</I>) a square. Diophantus +assumes as before 1, <I>m</I> for the values of <I>p, b,</I> and obtains +the double equation +<MATH> +<BRACE> +1/4 (<I>m</I> + 1)<SUP>2</SUP> + (1/2)<I>am</I> = square +<I>m</I><SUP>2</SUP> + 1 = square +</BRACE>, +<BRACE> +<I>m</I><SUP>2</SUP> + (2<I>a</I> + 2)<I>m</I> + 1 = square +or <I>m</I><SUP>2</SUP> + 1 = square +</BRACE> +</MATH>, +solving in the usual way.] +<MATH> +<BRACE> +VI. 10. (1/2)<I>xy</I> + <I>x</I> + <I>z</I> = <I>a.</I> +VI. 11. (1/2)<I>xy</I> - (<I>x</I> + <I>z</I>) = <I>a.</I> +</BRACE> +</MATH> +<p>[In these cases the auxiliary right-angled triangle has +to be found such that +<MATH>{1/2 (<I>h</I> + <I>p</I>)}<SUP>2</SUP> + <I>a</I>((1/2)<I>pb</I>) =</MATH> a square. +<p>Diophantus assumes it formed from 1, <I>m</I> + 1; thus +<MATH>1/4 (<I>h</I> + <I>p</I>)<SUP>2</SUP> = 1/4 {<I>m</I><SUP>2</SUP> + 2<I>m</I> + 2 + <I>m</I><SUP>2</SUP> + 2<I>m</I>}<SUP>2</SUP> = (<I>m</I><SUP>2</SUP> + 2<I>m</I> + 1)<SUP>2</SUP>, +and <I>a</I>((1/2)<I>pb</I>) = <I>a</I> (<I>m</I> + 1) (<I>m</I><SUP>2</SUP> + 2<I>m</I>)</MATH>. +Therefore +<MATH><I>m</I><SUP>4</SUP> + (<I>a</I> + 4)<I>m</I><SUP>3</SUP> + (3<I>a</I> + 6)<I>m</I><SUP>2</SUP> + (2<I>a</I> + 4)<I>m</I> + 1</MATH> += a square +<MATH>= {1 + (<I>a</I> + 2)<I>m</I> - <I>m</I><SUP>2</SUP>}<SUP>2</SUP></MATH>, say; +and <I>m</I> is found.] +<p>Lemma 1 to VI. 12. <MATH><I>x</I> = <I>u</I><SUP>2</SUP>, <I>x</I> - <I>y</I> = <I>v</I><SUP>2</SUP>, (1/2)<I>xy</I> + <I>y</I> = <I>w</I><SUP>2</SUP></MATH>. +<MATH> +<BRACE> +VI. 12. (1/2)<I>xy</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> + <I>y</I> = <I>v</I><SUP>2</SUP>. +VI. 13. (1/2)<I>xy</I> - <I>x</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> - <I>y</I> = <I>v</I><SUP>2</SUP>. +</BRACE> +</MATH> +<p>[These problems and the two following are interesting, +but their solutions run to some length; therefore only +one case can here be given. We will take VI. 12 with +its Lemma 1. +<pb n=510><head>DIOPHANTUS OF ALEXANDRIA</head> +<I>Lemma</I> 1. If a rational right-angled triangle be formed +from <I>m, n,</I> the perpendicular sides are 2<I>mn, m</I><SUP>2</SUP> - <I>n</I><SUP>2</SUP>. +We will suppose the greater of the two to be 2<I>mn.</I> +The first two relations are satisfied by making <I>m</I> = 2<I>n.</I> +Form, therefore, a triangle from <G>x</G>, 2<G>x</G>. The third con- +dition then gives <MATH>6<G>x</G><SUP>4</SUP> + 3<G>x</G><SUP>2</SUP> = a square or 6<G>x</G><SUP>2</SUP> + 3 =</MATH> a +square. One solution is <G>x</G> = 1 (and there are an infinite +number of others to be found by means of it). If <G>x</G> = 1, +the triangle is formed from 1, 2. +<p>VI. 12. Suppose the triangle to be (<I>h</I><G>x</G>, <I>b</I><G>x</G>, <I>p</I><G>x</G>). Then +<MATH>((1/2)<I>pb</I>)<G>x</G><SUP>2</SUP> + <I>p</I><G>x</G> = a square = (<I>k</I><G>x</G>)<SUP>2</SUP></MATH>, say, and <MATH><G>x</G> = <I>p</I>/(<I>k</I><SUP>2</SUP> - (1/2)<I>pb</I>)</MATH>. +This value must be such as to make <MATH>((1/2)<I>pb</I>)<G>x</G><SUP>2</SUP> + <I>b</I><G>x</G></MATH> a square +also. By substitution of the value of <G>x</G> we get +<MATH>{<I>bpk</I><SUP>2</SUP> + (1/2)<I>p</I><SUP>2</SUP><I>b</I>(<I>p</I> - <I>b</I>)}/(<I>k</I><SUP>2</SUP> - (1/2)<I>pb</I>)<SUP>2</SUP></MATH>; +so that <I>bpk</I><SUP>2</SUP> + (1/2)<I>p</I><SUP>2</SUP><I>b</I>(<I>p</I> - <I>b</I>) must be a square; or, if <I>p,</I> +the greater perpendicular, is made a square number, +<I>bk</I><SUP>2</SUP> + (1/2)<I>pb</I>(<I>p</I> - <I>b</I>) has to be made a square. This by +Lemma 2 (see p. 467 above) can be made a square if +<I>b</I> + (1/2)<I>pb</I>(<I>p</I> - <I>b</I>) is a square. <I>How to solve these problems,</I> +says Diophantus, <I>is shown in the Lemmas.</I> It is not +clear how they were applied, but, in fact, his solution +is such as to make <I>p, p</I> - <I>b,</I> and <I>b</I> + (1/2)<I>pb</I> all squares, +namely <I>b</I> = 3, <I>p</I> = 4, <I>h</I> = 5. +<p>Accordingly, putting for the original triangle 3<G>x</G>, 4<G>x</G>, 5<G>x</G>, +we have +<MATH> +<BRACE> +6<G>x</G><SUP>2</SUP> + 4<G>x</G> = a square +6<G>x</G><SUP>2</SUP> + 3<G>x</G> = a square +</BRACE> +</MATH>. +<p>Assuming <MATH>6<G>x</G><SUP>2</SUP> + 4<G>x</G> = <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP></MATH>, we have <MATH><G>x</G> = 4 / (<I>m</I><SUP>2</SUP> - 6)</MATH>, and +the second condition gives +<MATH>96/(<I>m</I><SUP>4</SUP> - 12<I>m</I><SUP>2</SUP> + 36) + 12/(<I>m</I><SUP>2</SUP> - 6) =</MATH> a square, +or <MATH>12<I>m</I><SUP>2</SUP> + 24 =</MATH> a square. +<p>This can be solved, since <I>m</I> = 1 satisfies it (Lemma 2). +<p>A solution is <I>m</I><SUP>2</SUP> = 25, whence <G>x</G> = 4/19.] +<MATH> +<BRACE> +VI. 14. (1/2)<I>xy</I> - <I>z</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> - <I>x</I> = <I>v</I><SUP>2</SUP>. +VI. 15. (1/2)<I>xy</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> + <I>x</I> = <I>v</I><SUP>2</SUP>. +</BRACE> +</MATH> +<pb n=511><head>INDETERMINATE ANALYSIS</head> +<p>[The auxiliary right-angled triangle in this case must +be such that +<MATH><I>m</I><SUP>2</SUP><I>hp</I> - (1/2)<I>pb.p</I>(<I>h</I> - <I>p</I>)</MATH> is a square. +<p>If, says Diophantus (VI. 14), we form a triangle from +the numbers <I>X</I><SUB>1</SUB>, <I>X</I><SUB>2</SUB> and suppose that <I>p</I> = 2<I>X</I><SUB>1</SUB><I>X</I><SUB>2</SUB>, and if +we then divide out by (<I>X</I><SUB>1</SUB> - <I>X</I><SUB>2</SUB>)<SUP>2</SUP>, which is equal to <I>h</I> - <I>p,</I> +we must find a square <MATH><I>k</I><SUP>2</SUP>[= <I>m</I><SUP>2</SUP> √(<I>X</I><SUB>1</SUB> - <I>X</I><SUB>2</SUB>)<SUP>2</SUP>]</MATH> such that +<MATH><I>k</I><SUP>2</SUP><I>hp</I> - (1/2)<I>pb.p</I></MATH> is a square. +<p>The problem, says Diophantus, can be solved if <I>X</I><SUB>1</SUB>, <I>X</I><SUB>2</SUB> +are ‘similar plane numbers’ (numbers such as <I>ab, m</I><SUP>2</SUP>/<I>n</I><SUP>2</SUP> <I>ab</I>). +This is stated without proof, but it can easily be verified +that, if <I>k</I><SUP>2</SUP> = <I>X</I><SUB>1</SUB><I>X</I><SUB>2</SUB>, the expression is a square. Dioph. +takes 4, 1 as the numbers, so that <I>k</I><SUP>2</SUP> = 4. The equation +for <I>m</I> becomes +<MATH>8 . 17<I>m</I><SUP>2</SUP> - 4 . 15 . 8 . 9 =</MATH> a square, +or <MATH>136<I>m</I><SUP>2</SUP> - 4320 =</MATH> a square. +The solution <I>m</I><SUP>2</SUP> = 36 (derived from the fact that +<MATH><I>k</I><SUP>2</SUP> = <I>m</I><SUP>2</SUP>/(<I>X</I><SUB>1</SUB> - <I>X</I><SUB>2</SUB>)<SUP>2</SUP>, or 4 = <I>m</I><SUP>2</SUP>/3<SUP>2</SUP>)</MATH> +satisfies the condition that +<MATH><I>m</I><SUP>2</SUP><I>hp</I> - (1/2)<I>pb.p</I>(<I>h</I> - <I>p</I>)</MATH> is a square.] +<p>VI. 16. <MATH><G>x</G> + <G>h</G> = <I>x,</I> <G>x</G>/<G>h</G> = <I>y</I>/<I>z</I></MATH>. +<p>[To find a rational right-angled triangle such that the +number representing the (portion intercepted within +the triangle of the) bisector of an acute angle is rational. +<FIG> +Let the bisector be 5<G>x</G>, the segment <I>BD</I> of the base 3<G>x</G>, +so that the perpendicular is 4<G>x</G>. +Let <I>CB</I> = 3<I>n.</I> Then <I>AC</I> : <I>AB</I> = <I>CD</I> : <I>DB,</I> +<pb n=512><head>DIOPHANTUS OF ALEXANDRIA</head> +so that <I>AC</I> = 4(<I>n</I> - <G>x</G>). Therefore (Eucl. I. 47) +<MATH>16(<I>n</I><SUP>2</SUP> - 2<I>n</I><G>x</G> + <G>x</G><SUP>2</SUP>) = 16<G>x</G><SUP>2</SUP> + 9<I>n</I><SUP>2</SUP></MATH>, +so that <MATH><G>x</G> = 7<I>n</I><SUP>2</SUP>/32<I>n</I> = (7/32)<I>n</I></MATH>. [Dioph. has <I>n</I> = 1.] +<BRACE> +VI. 17. <MATH>(1/2)<I>xy</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>y</I> + <I>z</I> = <I>v</I><SUP>3</SUP></MATH>. +[Let <G>x</G> be the area (1/2)<I>xy,</I> and let <I>z</I> = <I>k</I><SUP>2</SUP> - <G>x</G>. Since +<I>xy</I> = 2<G>x</G>, suppose <I>x</I> = 2, <I>y</I> = <G>x</G>. Therefore 2 + <I>k</I><SUP>2</SUP> must +be a cube. As we have seen (p. 475), Diophantus +takes (<I>m</I> - 1)<SUP>3</SUP> for the cube and (<I>m</I> + 1)<SUP>2</SUP> for <I>k</I><SUP>2</SUP>, giving +<MATH><I>m</I><SUP>3</SUP> - 3<I>m</I><SUP>2</SUP> + 3<I>m</I> - 1 = <I>m</I><SUP>2</SUP> + 2<I>m</I> + 3</MATH>, whence <I>m</I> = 4. There- +fore <I>k</I> = 5, and we assume <MATH>(1/2)<I>xy</I> = <G>x</G>, <I>z</I> = 25 - <G>x</G></MATH>, with +<I>x</I> = 2, <I>y</I> = <G>x</G> as before. Then we have to make +<MATH>(25 - <G>x</G>)<SUP>2</SUP> = 4 + <G>x</G><SUP>2</SUP></MATH>, and <G>x</G> = 621/50.] +VI. 18. <MATH>(1/2)<I>xy</I> + <I>z</I> = <I>u</I><SUP>3</SUP>, <I>x</I> + <I>y</I> + <I>z</I> = <I>v</I><SUP>2</SUP></MATH>. +</BRACE> +<BRACE> +VI. 19. <MATH>(1/2)<I>xy</I> + <I>x</I> = <I>u</I><SUP>2</SUP>, <I>x</I> + <I>y</I> + <I>z</I> = <I>v</I><SUP>3</SUP></MATH>. +[Here a right-angled triangle is formed from one odd +number, say 2<G>x</G> + 1, according to the Pythagorean for- +mula <MATH><I>m</I><SUP>2</SUP> + {(1/2)(<I>m</I><SUP>2</SUP> - 1)}<SUP>2</SUP> = {(1/2)(<I>m</I><SUP>2</SUP> + 1)}<SUP>2</SUP></MATH>, where <I>m</I> is an +odd number. The sides are therefore 2<G>x</G> + 1, 2<G>x</G><SUP>2</SUP> + 2<G>x</G>, +2<G>x</G><SUP>2</SUP> + 2<G>x</G> + 1. Since the perimeter = a cube, +<MATH>4<G>x</G><SUP>2</SUP> + 6<G>x</G> + 2 = (4<G>x</G> + 2) (<G>x</G> + 1) =</MATH> a cube. +Or, if we divide the sides by <G>x</G> + 1, 4<G>x</G> + 2 has to be +made a cube. +Again <MATH>(1/2)<I>xy</I> + <I>x</I> = ((2<G>x</G><SUP>3</SUP> + 3<G>x</G><SUP>2</SUP> + <G>x</G>)/((<G>x</G> + 1)<SUP>2</SUP>)) + ((2<G>x</G> + 1)/(<G>x</G> + 1)) =</MATH> a square, +which reduces to <MATH>2<G>x</G> + 1 =</MATH> a square. +But 4<G>x</G> + 2 is a cube. We therefore put 8 for the cube, +and <G>x</G> = 1 1/2.] +VI. 20. <MATH>(1/2)<I>xy</I> + <I>x</I> = <I>u</I><SUP>3</SUP>, <I>x</I> + <I>y</I> + <I>z</I> = <I>v</I><SUP>2</SUP></MATH>. +</BRACE> +<p>VI. 21. <MATH><I>x</I> + <I>y</I> + <I>z</I> = <I>u</I><SUP>2</SUP>, (1/2)<I>xy</I> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>v</I><SUP>3</SUP></MATH>. +<p>[Form a right-angled triangle from <G>x</G>, 1, i. e. (2<G>x</G>, <G>x</G><SUP>2</SUP> - 1, +<G>x</G><SUP>2</SUP> + 1). Then 2<G>x</G><SUP>2</SUP> + 2<G>x</G> must be a square, and <G>x</G><SUP>3</SUP> + 2<G>x</G><SUP>2</SUP> + <G>x</G> +<pb n=513><head>INDETERMINATE ANALYSIS</head> +a cube. Put <MATH>2<G>x</G><SUP>2</SUP> + 2<G>x</G> = <I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP></MATH>, so that <G>x</G> = 2/(<I>m</I><SUP>2</SUP> - 2), +and we have to make +<MATH>8/(<I>m</I><SUP>2</SUP> - 2)<SUP>3</SUP> + 8/(<I>m</I><SUP>2</SUP> - 2)<SUP>2</SUP> + 2/(<I>m</I><SUP>2</SUP> - 2), or (2<I>m</I><SUP>4</SUP>)/(<I>m</I><SUP>2</SUP> - 2)<SUP>3</SUP></MATH>, a cube. +Make 2<I>m</I> a cube = <I>n</I><SUP>3</SUP>, so that 2<I>m</I><SUP>4</SUP> = <I>m</I><SUP>3</SUP><I>n</I><SUP>3</SUP>, and +<I>m</I> = (1/2)<I>n</I><SUP>3</SUP>; therefore <MATH><G>x</G> = 8/(<I>n</I><SUP>6</SUP> - 8)</MATH>, and <G>x</G> must be made +greater than 1, in order that <G>x</G><SUP>2</SUP> - 1 may be positive. +<p>Therefore 8 < <I>n</I><SUP>6</SUP> < 16; +this is satisfied by <I>n</I><SUP>6</SUP> = 729/64 or <I>n</I><SUP>3</SUP> = 27/8, and <I>m</I> = 27/16.] +<p>VI. 22. <MATH><I>x</I> + <I>y</I> + <I>z</I> = <I>u</I><SUP>3</SUP>, (1/2)<I>xy</I> + (<I>x</I> + <I>y</I> + <I>z</I>) = <I>v</I><SUP>2</SUP></MATH>. +<p>[(1) First seek a rational right-angled triangle such +that its perimeter and its area are given numbers, +say <I>p, m.</I> +<p>Let the perpendiculars be 1/<G>x</G>, 2<I>m</I><G>x</G>; therefore the hypo- +tenuse = <I>p</I> - (1/<G>x</G>) - 2<I>m</I><G>x</G>, and (Eucl. I. 47) +<MATH>(1/<G>x</G><SUP>2</SUP>) + 4<I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP> + (<I>p</I><SUP>2</SUP> + 4<I>m</I>) - (2<I>p</I>/<G>x</G>) - 4<I>mp</I><G>x</G> = (1/<G>x</G><SUP>2</SUP>) + 4<I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP>, +or <I>p</I><SUP>2</SUP> + 4<I>m</I> = 4<I>mp</I><G>x</G> + (2<I>p</I>/<G>x</G>), +that is, (<I>p</I><SUP>2</SUP> + 4<I>m</I>)<G>x</G> = 4<I>mp</I><G>x</G><SUP>2</SUP> + 2<I>p</I></MATH>. +<p>(2) In order that this may have a rational solution, +<MATH>{(1/2)(<I>p</I><SUP>2</SUP> + 4<I>m</I>)}<SUP>2</SUP> - 8<I>p</I><SUP>2</SUP><I>m</I></MATH> must be a square, +i.e. <MATH>4<I>m</I><SUP>2</SUP> - 6<I>p</I><SUP>2</SUP><I>m</I> + (1/4)<I>p</I><SUP>4</SUP> =</MATH> a square, +<BRACE> +or <MATH><I>m</I><SUP>2</SUP> - (3/2)<I>p</I><SUP>2</SUP><I>m</I> + (1/16)<I>p</I><SUP>4</SUP> =</MATH> a square +Also, by the second condition, <MATH><I>m</I> + <I>p</I> =</MATH> a square +</BRACE>. +<p>To solve this, we must take for <I>p</I> some number which +is both a square and a cube (in order that it may be +possible, by multiplying the second equation by some +square, to make the constant term equal to the constant +<pb n=514><head>DIOPHANTUS OF ALEXANDRIA</head> +term in the first). Diophantus takes <I>p</I> = 64, making +the equations +<BRACE> +<MATH><I>m</I><SUP>2</SUP> - 6144<I>m</I> + 1048576 =</MATH> a square +<MATH><I>m</I> + 64 =</MATH> a square +</BRACE>. +Multiplying the second by 16384, and subtracting the two +expressions, we have as the difference <I>m</I><SUP>2</SUP> - 22528<I>m.</I> +<p>Diophantus observes that, if we take <I>m, m</I> - 22528 as +the factors, we obtain <I>m</I> = 7680, an impossible value for +the area of a right-angled triangle of perimeter <I>p</I> = 64. +<p>We therefore take as factors 11<I>m,</I> (1/11)<I>m</I> - 2048, and, +equating the square of half the difference (= (60/11)<I>m</I> + 1024) +to 16384<I>m</I> + 1048576, we have <I>m</I> = 39424/225. +<p>(3) Returning to the original problem, we have to +substitute this value for <I>m</I> in +<MATH>(64 - (1/<G>x</G>) - 2<I>m</I><G>x</G>)<SUP>2</SUP> = (1/<G>x</G><SUP>2</SUP>) + 4<I>m</I><SUP>2</SUP><G>x</G><SUP>2</SUP></MATH>, +and we obtain +<MATH>78848<G>x</G><SUP>2</SUP> - 8432<G>x</G> + 225 = 0</MATH>, +the solution of which is rational, namely <G>x</G> = 25/448 (or 9/176). +Diophantus naturally takes the first value, though the +second gives the same triangle.] +<p>VI. 23. <MATH><I>z</I><SUP>2</SUP> = <I>u</I><SUP>2</SUP> + <I>u, z</I><SUP>2</SUP>/<I>x</I> = <I>v</I><SUP>3</SUP> + <I>v</I></MATH>. +<p>VI. 24. <MATH><I>z</I> = <I>u</I><SUP>3</SUP> + <I>u, x</I> = <I>v</I><SUP>3</SUP> - <I>v, y</I> = <I>w</I><SUP>3</SUP></MATH>. +<p>[VI. 6, 7]. <MATH>((1/2)<I>x</I>)<SUP>2</SUP> + (1/2)<I>mxy</I> = <I>u</I><SUP>2</SUP></MATH>. +<p>[VI. 8, 9]. <MATH>{(1/2)(<I>x</I> + <I>y</I>)}<SUP>2</SUP> + (1/2)<I>mxy</I> = <I>u</I><SUP>2</SUP></MATH>. +<p>[VI. 10, 11]. <MATH>{(1/2)(<I>z</I> + <I>x</I>)}<SUP>2</SUP> + (1/2)<I>mxy</I> = <I>u</I><SUP>2</SUP></MATH>. +<p>[VI. 12.] <MATH><I>y</I> + (<I>x</I> - <I>y</I>) . (1/2)<I>xy</I> = <I>u</I><SUP>2</SUP>, <I>x</I> = <I>v</I><SUP>2</SUP>. (<I>x</I> > <I>y.</I>)</MATH> +<p>[VI. 14, 15]. <MATH><I>u</I><SUP>2</SUP><I>zx</I> - (1/2)<I>xy</I> . <I>x</I>(<I>z</I> - <I>x</I>) = <I>v</I><SUP>2</SUP>. (<I>u</I><SUP>2</SUP> < or > (1/2)<I>xy.</I>)</MATH> +<C>The treatise on Polygonal Numbers.</C> +<p>The subject of Polygonal Numbers on which Diophantus +also wrote is, as we have seen, an old one, going back to the +<pb n=515><head>THE TREATISE ON POLYGONAL NUMBERS</head> +Pythagoreans, while Philippus of Opus and Speusippus carried +on the tradition. Hypsicles (about 170 B.C.) is twice men- +tioned by Diophantus as the author of a ‘definition’ of +a polygonal number which, although it does not in terms +mention any polygonal number beyond the pentagonal, +amounts to saying that the <I>n</I>th <I>a</I>-gon (1 counting as the +first) is +<MATH>(1/2)<I>n</I> {2 + (<I>n</I> - 1) (<I>a</I> - 2)}</MATH>. +Theon of Smyrna, Nicomachus and Iamblichus all devote +some space to polygonal numbers. Nicomachus in particular +gives various rules for transforming triangles into squares, +squares into pentagons, &c. +<p>1. If we put two consecutive triangles together, we get a square. +In fact +<MATH>(1/2)(<I>n</I> - 1)<I>n</I> + (1/2)<I>n</I>(<I>n</I> + 1) = <I>n</I><SUP>2</SUP></MATH>. +<p>2. A pentagon is obtained from a square by adding to it +a triangle the side of which is 1 less than that of the square; +similarly a hexagon from a pentagon by adding a triangle +the side of which is 1 less than that of the pentagon, and so on. +<p>In fact +<MATH>(1/2)<I>n</I> {2 + (<I>n</I> - 1) (<I>a</I> - 2)} + (1/2)(<I>n</I> - 1)<I>n</I> += (1/2)<I>n</I>[2 + (<I>n</I> - 1) {(<I>a</I> + 1) - 2}]</MATH>. +<p>3. Nicomachus sets out the first triangles, squares, pentagons, +hexagons and heptagons in a diagram thus: +<table> +<tr> +<td>Triangles</td> +<td>1</td> +<td>3</td> +<td> 6</td> +<td>10</td> +<td>15</td> +<td>21</td> +<td> 28</td> +<td> 36</td> +<td> 45</td> +<td> 55,</td> +</tr> +<tr> +<td>Squares</td> +<td>1</td> +<td>4</td> +<td> 9</td> +<td>16</td> +<td>25</td> +<td>36</td> +<td> 49</td> +<td> 64</td> +<td> 81</td> +<td>100,</td> +</tr> +<tr> +<td>Pentagons</td> +<td>1</td> +<td>5</td> +<td>12</td> +<td>22</td> +<td>35</td> +<td>51</td> +<td> 70</td> +<td> 92</td> +<td>117</td> +<td>145,</td> +</tr> +<tr> +<td>Hexagons</td> +<td>1</td> +<td>6</td> +<td>15</td> +<td>28</td> +<td>45</td> +<td>66</td> +<td> 91</td> +<td>120</td> +<td>153</td> +<td>190,</td> +</tr> +<tr> +<td>Heptagons</td> +<td>1</td> +<td>7</td> +<td>18</td> +<td>34</td> +<td>55</td> +<td>81</td> +<td>112</td> +<td>148</td> +<td>189</td> +<td>235,</td> +</tr> +</table> +and observes that: +<p>Each polygon is equal to the polygon immediately above it +in the diagram <I>plus</I> the triangle with 1 less in its side, i. e. the +triangle in the preceding column. +<pb n=516><head>DIOPHANTUS OF ALEXANDRIA</head> +<p>4. The vertical columns are in arithmetical progression, the +common difference being the triangle in the preceding column. +<p>Plutarch, a contemporary of Nicomachus, mentions another +method of transforming triangles into squares. <I>Every tri- +angular number taken eight times and then increased by 1 +gives a square.</I> +<p>In fact, <MATH>8.(1/2)<I>n</I>(<I>n</I> + 1) + 1 = (2<I>n</I> + 1)<SUP>2</SUP></MATH>. +<p>Only a fragment of Diophantus's treatise <I>On Polygonal +Numbers</I> survives. Its character is entirely different from +that of the <I>Arithmetica.</I> The method of proof is strictly +geometrical, and has the disadvantage, therefore, of being long +and involved. He begins with some preliminary propositions +of which two may be mentioned. Prop. 3 proves that, if <I>a</I> be +the first and <I>l</I> the last term in an arithmetical progression +of <I>n</I> terms, and if <I>s</I> is the sum of the terms, <MATH>2<I>s</I> = <I>n</I>(<I>l</I> + <I>a</I>)</MATH>. +Prop. 4 proves that, if 1, 1 + <I>b,</I> 1 + 2<I>b,</I> ... 1 + (<I>n</I> - 1)<I>b</I> be an +A. P., and <I>s</I> the sum of the terms, +<MATH>2<I>s</I> = <I>n</I> {2 + (<I>n</I> - 1)<I>b</I>}</MATH>. +<p>The main result obtained in the fragment as we have it +is a generalization of the formula <MATH>8.(1/2)<I>n</I>(<I>n</I> + 1) + 1 = (2<I>n</I> + 1)<SUP>2</SUP></MATH>. +Prop. 5 proves the fact stated in Hypsicles's definition and also +(the generalization referred to) that +<MATH>8<I>P</I>(<I>a</I> - 2) + (<I>a</I> - 4)<SUP>2</SUP> =</MATH> a square, +where <I>P</I> is any polygonal number with <I>a</I> angles. +<p>It is also proved that, if <I>P</I> be the <I>n</I>th <I>a</I>-gonal number +(1 being the first), +<MATH>8<I>P</I>(<I>a</I> - 2) + (<I>a</I> - 4)<SUP>2</SUP> = {2 + (2<I>n</I> - 1) (<I>a</I> - 2)}<SUP>2</SUP></MATH>. +<p>Diophantus deduces rules as follows. +<p>1. <I>To find the number from its side.</I> +<MATH><I>P</I> = ({2 + (2<I>n</I> - 1) (<I>a</I> - 2)}<SUP>2</SUP> - (<I>a</I> - 4)<SUP>2</SUP>)/(8(<I>a</I> - 2))</MATH>. +<p>2. <I>To find the side from the number.</I> +<MATH><I>n</I> = (1/2)((√{8<I>P</I>(<I>a</I> - 2) + (<I>a</I> - 4)<SUP>2</SUP>} - 2)/(<I>a</I> - 2) + 1)</MATH>. +<pb n=517><head>THE TREATISE ON POLYGONAL NUMBERS</head> +<p>The last proposition, which breaks off in the middle, is: +<p><I>Given a number, to find in how many ways it can be +polygonal.</I> +<p>The proposition begins in a way which suggests that +Diophantus first proved geometrically that, if +<MATH>8<I>P</I>(<I>a</I> - 2) + (<I>a</I> - 4)<SUP>2</SUP> = {2 + (2<I>n</I> - 1) (<I>a</I> - 2)}<SUP>2</SUP></MATH>, +then <MATH>2<I>P</I> = <I>n</I>{2 + (<I>n</I> - 1) (<I>a</I> - 2)}</MATH>. +Wertheim. (in his edition of Diophantus) has suggested a +restoration of the complete proof of this proposition, and +I have shown (in my edition) how the proof can be made +shorter. Wertheim adds an investigation of the main pro- +blem, but no doubt opinions will continue to differ as to +whether Diophantus actually solved it. +<pb> +<C>XXI +COMMENTATORS AND BYZANTINES</C> +<p>WE have come to the last stage of Greek mathematics; it +only remains to include in a last chapter references to com- +mentators of more or less note who contributed nothing +original but have preserved, among observations and explana- +tions obvious or trivial from a mathematical point of view, +valuable extracts from works which have perished, or +historical allusions which, in the absence of original docu- +ments, are precious in proportion to their rarity. Nor must +it be forgotten that in several cases we probably owe to the +commentators the fact that the masterpieces of the great +mathematicians have survived, wholly or partly, in the +original Greek or at all. This may have been the case even +with the works of Archimedes on which Eutocius wrote com- +mentaries.. It was no doubt these commentaries which +aroused in the school of Isidorus of Miletus (the colleague +of Anthemius as architect of Saint Sophia at Constantinople) +a new interest in the works of Archimedes and caused them +to be sought out in the various libraries or wherever they had +lain hid. This revived interest apparently had the effect of +evoking new versions of the famous works commented upon +in a form more convenient for the student, with the Doric +dialect of the original eliminated; this translation of the +Doric into the more familiar dialect was systematically +carried out in those books only which Eutocius commented +on, and it is these versions which alone survive. Again, +Eutocius's commentary on Apollonius's <I>Conics</I> is extant for +the first four Books, and it is probably owing to their having +been commented on by Eutocius, as well as to their being +more elementary than the rest, that these four Books alone +<pb n=519><head>SERENUS</head> +survive in Greek. Tannery, as we have seen, conjectured +that, in like manner, the first six of the thirteen Books of +Diophantus's <I>Arithmetica</I> survive because Hypatia wrote +commentaries on these Books only and did not reach the +others. +<p>The first writer who calls for notice in this chapter is one +who was rather more than a commentator in so far as he +wrote a couple of treatises to supplement the <I>Conics</I> of +Apollonius, I mean SERENUS. Serenus came from Antinoeia +or Antinoupolis, a city in Egypt founded by Hadrian (A.D. +117-38). His date is uncertain, but he most probably be- +longed to the fourth century A.D., and came between Pappus +and Theon of Alexandria. He tells us himself that he wrote +a commentary on the <I>Conics</I> of Apollonius.<note>Serenus, <I>Opuscula,</I> ed. Heiberg, p. 52. 25-6.</note> This has +perished and, apart from a certain proposition ‘of Serenus +the philosopher, from the Lemmas’ preserved in certain manu- +scripts of Theon of Smyrna (to the effect that, if a number of +rectilineal angles be subtended at a point on a diameter of a +circle which is not the centre, by equal arcs of that circle, the +angle nearer to the centre is always less than the angle more +remote), we have only the two small treatises by him entitled +<I>On the Section of a Cylinder</I> and <I>On the Section of a Cone.</I> +These works came to be connected, from the seventh century +onwards, with the <I>Conics</I> of Apollonius, on account of the +affinity of the subjects, and this no doubt accounts for their +survival. They were translated into Latin by Commandinus +in 1566; the first Greek text was brought out by Halley along +with his Apollonius (Oxford 1710), and we now have the +definitive text edited by Heiberg (Teubner 1896). +<C>(<G>a</G>) <I>On the Section of a Cylinder.</I></C> +<p>The occasion and the object of the tract <I>On the Section of +a Cylinder</I> are stated in the preface. Serenus observes that +many persons who were students of geometry were under the +erroneous impression that the oblique section of a cylinder +was different from the oblique section of a cone known as an +ellipse, whereas it is of course the same curve. Hence he +thinks it necessary to establish, by a regular geometrical +<pb n=520><head>COMMENTATORS AND BYZANTINES</head> +proof, that the said oblique sections cutting all the generators +are equally ellipses whether they are sections of a cylinder or +of a cone. He begins with ‘a more general definition’ of a +cylinder to include any oblique circular cylinder. ‘If in two +equal and parallel circles which remain fixed the diameters, +while remaining parallel to one another throughout, are moved +round in the planes of the circles about the centres, which +remain fixed, and if they carry round with them the straight line +joining their extremities on the same side until they bring it +back again to the same place, let the surface described by the +straight line so carried round be called a <I>cylindrical surface.</I>’ +The <I>cylinder</I> is the figure contained by the parallel circles and +the cylindrical surface intercepted by them; the parallel +circles are the <I>bases,</I> the <I>axis</I> is the straight line drawn +through their centres; the generating straight line in any +position is a <I>side.</I> Thirty-three propositions follow. Of these +Prop. 6 proves the existence in an oblique cylinder of the +parallel circular sections subcontrary to the series of which +the bases are two, Prop. 9 that the section by any plane not +parallel to that of the bases or of one of the subcontrary +sections but cutting all the generators is not a circle; the +next propositions lead up to the main results, namely those in +Props. 14 and 16, where the said section is proved to have the +property of the ellipse which we write in the form +<MATH><I>QV</I><SUP>2</SUP>:<I>PV.P</I>′<I>V</I>=<I>CD</I><SUP>2</SUP>:<I>CP</I><SUP>2</SUP></MATH>, +and in Prop. 17, where the property is put in the Apollonian +form involving the <I>latus rectum,</I> <MATH><I>QV</I><SUP>2</SUP>=<I>PV.VR</I></MATH> (see figure +on p. 137 above), which is expressed by saying that the square +on the semi-ordinate is equal to the rectangle applied to the +<I>latus rectum PL,</I> having the abscissa <I>PV</I> as breadth and falling +short by a rectangle similar to the rectangle contained by the +diameter <I>PP′</I> and the <I>latus rectum PL</I> (which is determined +by the condition <MATH><I>PL.PP′</I>=<I>DD′</I><SUP>2</SUP></MATH> and is drawn at right angles +to <I>PV</I>). Prop. 18 proves the corresponding property with +reference to the conjugate diameter <I>DD′</I> and the correspond- +ing <I>latus rectum,</I> and Prop. 19 gives the main property in the +form <MATH><I>QV</I><SUP>2</SUP>:<I>PV.P′V</I>=<I>Q′V′</I><SUP>2</SUP>:<I>PV′.P′V′</I></MATH>. Then comes the +proposition that ‘it is possible to exhibit a cone and a cylinder +which are alike cut in one and the same ellipse’ (Prop. 20). +<pb n=521><head>SERENUS</head> +Serenus then solves such problems as these: Given a cone +(or cylinder) and an ellipse on it, to find the cylinder (cone) +which is cut in the same ellipse as the cone (cylinder) +(Props. 21, 22); given a cone (cylinder), to find a cylinder +(cone) and to cut both by one and the same plane so that the +sections thus made shall be similar ellipses (Props. 23, 24). +Props. 27, 28 deal with similar elliptic sections of a scalene +cylinder and cone; there are two pairs of infinite sets of these +similar to any one given section, the first pair being those +which are parallel and subcontrary respectively to the given +section, the other pair subcontrary to one another but not to +either of the other sets and having the conjugate diameter +occupying the corresponding place to the transverse in the +other sets, and vice versa. +<p>In the propositions (29-33) from this point to the end of +the book Serenus deals with what is really an optical pro- +blem. It is introduced by a remark about a certain geometer, +Peithon by name, who wrote a tract on the subject of +parallels. Peithon, not being satisfied with Euclid's treat- +ment of parallels, thought to define parallels by means of an +illustration, observing that parallels are such lines as are +shown on a wall or a roof by the shadow of a pillar with +a light behind it. This definition, it appears, was generally +ridiculed; and Serenus seeks to rehabilitate Peithon, who +was his friend, by showing that his statement is after all +mathematically sound. He therefore proves, with regard to +the cylinder, that, if any number of rays from a point outside +the cylinder are drawn touching it on both sides, all the rays +pass through the sides of a parallelogram (a section of the +cylinder parallel to the axis)—Prop. 29—and if they are +produced farther to meet any other plane parallel to that +of the parallelogram the points in which they meet the plane +will lie on two parallel lines (Prop. 30); he adds that the lines +will not <I>seem</I> parallel (<I>vide</I> Euclid's <I>Optics,</I> Prop. 6). The +problem about the rays touching the surface of a cylinder +suggests the similar one about any number of rays from an +external point touching the surface of a <I>cone</I>; these meet the +surface in points on a triangular section of the cone (Prop. 32) +and, if produced to meet a plane parallel to that of the +triangle, meet that plane in points forming a similar triangle +<pb n=522><head>COMMENTATORS AND BYZANTINES</head> +(Prop. 33). Prop. 31 preceding these propositions is a par- +ticular case of the constancy of the anharmonic ratio of a +pencil of four rays. If two sides <I>AB, AC</I> of a triangle meet +a transversal through <I>D,</I> an external point, in <I>E, F</I> and another +ray <I>AG</I> between <I>AB</I> and <I>AC</I> cuts <I>DEF</I> in a point <I>G</I> such +that <MATH><I>ED</I>:<I>DF</I>=<I>EG</I>:<I>GF</I></MATH>, then any other transversal through +<I>D</I> meeting <I>AB, AG, AC</I> in <I>K, L, M</I> is also divided harmoni- +cally, i.e. <MATH><I>KD</I>:<I>DM</I>=<I>KL</I>:<I>LM</I></MATH>. To prove the succeeding pro- +positions, 32 and 33, Serenus uses this proposition and a +reciprocal of it combined with the harmonic property of the +pole and polar with reference to an ellipse. +<C>(<G>b</G>) <I>On the Section of a Cone.</I></C> +<p>The treatise <I>On the Section of a Cone</I> is even less important, +although Serenus claims originality for it. It deals mainly +with the areas of triangular sections of right or scalene cones +made by planes passing through the vertex and either through +the axis or not through the axis, showing when the area of +a certain triangle of a particular class is a maximum, under +what conditions two triangles of a class may be equal in area, +and so on, and solving in some easy cases the problem of +finding triangular sections of given area. This sort of investi- +gation occupies Props. 1-57 of the work, these propositions +including various lemmas required for the proofs of the +substantive theorems. Props. 58-69 constitute a separate +section of the book dealing with the volumes of right cones +in relation to their heights, their bases and the areas of the +triangular sections through the axis. +<p>The essence of the first portion of the book up to Prop. 57 +is best shown by means of modern notation. We will call <I>h</I> +the height of a right cone, <I>r</I> the radius of the base; in the +case of an oblique cone, let <I>p</I> be the perpendicular from the +vertex to the plane of the base, <I>d</I> the distance of the foot of +this perpendicular from the centre of the base, <I>r</I> the radius +of the base. +<p>Consider first the right cone, and let 2<I>x</I> be the base of any +triangular section through the vertex, while of course 2<I>r</I> is +the base of the triangular section through the axis. Then, if +<I>A</I> be the area of the triangular section with base 2<I>x,</I> +<MATH><I>A</I>=<I>x</I>√(<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP>)</MATH>. +<pb n=523><head>SERENUS</head> +<p>Observing that the sum of <I>x</I><SUP>2</SUP> and <MATH><I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP></MATH> is constant, we +see that <I>A</I><SUP>2</SUP>, and therefore <I>A,</I> is a maximum when +<MATH><I>x</I><SUP>2</SUP>=<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP></MATH>, or <MATH><I>x</I><SUP>2</SUP>=1/2(<I>r</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP>)</MATH>; +and, since <I>x</I> is not greater than <I>r,</I> it follows that, for a real +value of <I>x</I> (other than <I>r</I>), <I>h</I> is less than <I>r,</I> or the cone is obtuse- +angled. When <I>h</I> is not less than <I>r,</I> the maximum triangle is +the triangle through the axis and vice versa (Props. 5, 8); +when <MATH><I>h</I>=<I>r</I></MATH>, the maximum triangle is also right-angled +(Prop. 13). +<p>If the triangle with base 2<I>c</I> is equal to the triangle through +the axis, <MATH><I>h</I><SUP>2</SUP><I>r</I><SUP>2</SUP>=<I>c</I><SUP>2</SUP>(<I>r</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP>)</MATH>, or +<MATH>(<I>r</I><SUP>2</SUP>-<I>c</I><SUP>2</SUP>)(<I>c</I><SUP>2</SUP>-<I>h</I><SUP>2</SUP>)=0</MATH>, and, +since <MATH><I>c</I> < <I>r</I>, <I>h</I>=<I>c</I></MATH>, so that <MATH><I>h</I> < <I>r</I></MATH> (Prop. 10). If <I>x</I> lies between <I>r</I> +and <I>c</I> in this case, <MATH>(<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>)(<I>x</I><SUP>2</SUP>-<I>h</I><SUP>2</SUP>) > 0</MATH> or +<MATH><I>x</I><SUP>2</SUP>(<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>h</I><SUP>2</SUP>) > <I>h</I><SUP>2</SUP><I>r</I><SUP>2</SUP></MATH>, +and the triangle with base 2<I>x</I> is greater than either of the +equal triangles with bases 2<I>r,</I> 2<I>c,</I> or 2<I>h</I> (Prop. 11). +<p>In the case of the scalene cone Serenus compares individual +triangular sections belonging to one of three classes with other +sections of the same class as regards their area. The classes +are: +<p>(1) axial triangles, including all sections through the axis; +<p>(2) isosceles sections, i.e. the sections the bases of which are +perpendicular to the projection of the axis of the cone on the +plane of the base; +<p>(3) a set of triangular sections the bases of which are (<I>a</I>) the +diameter of the circular base which passes through the foot of +the perpendicular from the vertex to the plane of the base, and +(<I>b</I>) the chords of the circular base parallel to that diameter. +<p>After two preliminary propositions (15, 16) and some +lemmas, Serenus compares the areas of the first class of +triangles through the axis. If, as we said, <I>p</I> is the perpen- +dicular from the vertex to the plane of the base, <I>d</I> the distance +of the foot of this perpendicular from the centre of the base, +and <G>q</G> the angle which the base of any axial triangle with area +<I>A</I> makes with the base of the axial triangle passing through +<I>p</I> the perpendicular, +<MATH><I>A</I>=<I>r</I>√(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>sin<SUP>2</SUP><G>q</G>)</MATH>. +<p>This area is a minimum when <G>q</G>=0, and increases with <G>q</G> +<pb n=524><head>COMMENTATORS AND BYZANTINES</head> +until <MATH><G>Q</G>=1/2<G>p</G></MATH> when it is a maximum, the triangle being then +isosceles (Prop. 24). +<p>In Prop. 29 Serenus takes up the third class of sections with +bases parallel to <I>d.</I> If the base of such a section is 2<I>x,</I> +<MATH><I>A</I>=<I>x</I>√(<I>r</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>+<I>p</I><SUP>2</SUP>)</MATH> +and, as in the case of the right cone, we must have for a real +maximum value +<MATH><I>x</I><SUP>2</SUP>=1/2(<I>r</I><SUP>2</SUP>+<I>p</I><SUP>2</SUP>)</MATH>, while <MATH><I>x</I><<I>r</I></MATH>, +so that, for a real value of <I>x</I> other than <I>r, p</I> must be less than +<I>r,</I> and, if <I>p</I> is not less than <I>r,</I> the maximum triangle is that +which is perpendicular to the base of the cone and has 2<I>r</I> for +its base (Prop. 29). If <MATH><I>p</I> < <I>r</I></MATH>, the triangle in question is not +the maximum of the set of triangles (Prop. 30). +<p>Coming now to the isosceles sections (2), we may suppose +2<G>q</G> to be the angle subtended at the centre of the base by the +base of the section in the direction away from the projection +of the vertex. Then +<MATH><I>A</I>=<I>r</I> sin<G>q</G>√{<I>p</I><SUP>2</SUP>+(<I>d</I>+<I>r</I>cos<G>q</G>)<SUP>2</SUP>}</MATH>. +<p>If <I>A</I><SUB>0</SUB> be the area of the isosceles triangle through the axis, +we have +<MATH><I>A</I><SUB>0</SUB><SUP>2</SUP>-<I>A</I><SUP>2</SUP>=<I>r</I><SUP>2</SUP>(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>)-<I>r</I><SUP>2</SUP>sin<SUP>2</SUP><G>q</G>(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>+<I>r</I><SUP>2</SUP>cos<SUP>2</SUP><G>q</G>+ +2<I>dr</I>cos<G>q</G>) +=<I>r</I><SUP>2</SUP>(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>)cos<SUP>2</SUP><G>q</G>-<I>r</I><SUP>4</SUP>sin<SUP>2</SUP><G>q</G>cos<SUP>2</SUP><G>q</G>-2<I>dr</I><SUP>3</SUP>cos<G>q</G>sin<SUP>2</SUP><G>q</G></MATH>. +<p>If <MATH><I>A</I>=<I>A</I><SUB>0</SUB></MATH>, we must have for triangles on the side of the +centre of the base of the cone towards the vertex of the cone +(since cos <G>q</G> is negative for such triangles) +<MATH><I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP><<I>r</I><SUP>2</SUP>sin<SUP>2</SUP><G>q</G></MATH>, and <I>a fortiori</I> <MATH><I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP><<I>r</I><SUP>2</SUP></MATH> (Prop. 35). +<p>If <MATH><I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>≥<I>r</I><SUP>2</SUP></MATH>, <I>A</I><SUB>0</SUB> is always greater than <I>A,</I> so that <I>A</I><SUB>0</SUB> is the +maximum isosceles triangle of the set (Props. 31, 32). +<p>If <I>A</I> is the area of any one of the isosceles triangles with +bases on the side of the centre of the base of the cone away +from the projection of the vertex, cos <G>q</G> is positive and <I>A</I><SUB>0</SUB> is +proved to be neither the minimum nor the maximum triangle +of this set of triangles (Props. 36, 40-4). +<p>In Prop. 45 Serenus returns to the set of triangular sections +through the axis, proving that the feet of the perpendiculars +from the vertex of the cone on their bases all lie on a circle +the diameter of which is the straight line joining the centre of +<pb n=525><head>SERENUS</head> +the base of the cone to the projection of the vertex on its +plane; the areas of the axial triangles are therefore propor- +tional to the generators of the cone with the said circle as +base and the same vertex as the original cone. Prop. 50 is to +the effect that, if the axis of the cone is equal to the radius of +the base, the least axial triangle is a mean proportional +between the greatest axial triangle and the isosceles triangular +section perpendicular to the base; that is, with the above nota- +tion, if <MATH><I>r</I>=√(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>)</MATH>, then <MATH><I>r</I>√(<I>p</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>):<I>rp</I>=<I>rp</I>:<I>p</I>√(<I>r</I><SUP>2</SUP>-<I>d</I><SUP>2</SUP>)</MATH>, +which is indeed obvious. +<p>Prop. 57 is interesting because of the lemmas leading to it. +It proves that the greater axial triangle in a scalene cone has +the greater perimeter, and conversely. This is proved by +means of the lemma (Prop. 54), applied to the variable sides +of axial triangles, that if <MATH><I>a</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP>+<I>c</I><SUP>2</SUP></MATH> and <MATH><I>a</I> > <I>b</I>≥<I>c</I> > <I>d</I></MATH>, +then <MATH><I>a</I>+<I>d</I> < <I>b</I>+<I>c</I></MATH> (<I>a, d</I> are the sides other than the base of one +axial triangle, and <I>b, c</I> those of the other axial triangle com- +pared with it; and if <I>ABC, ADE</I> be two axial triangles and +<I>O</I> the centre of the base, <MATH><I>BA</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>=<I>DA</I><SUP>2</SUP>+<I>AE</I><SUP>2</SUP></MATH> because each +of these sums is equal to <MATH>2<I>AO</I><SUP>2</SUP>+2<I>BO</I><SUP>2</SUP></MATH>, Prop. 17). This proposi- +tion again depends on the lemma (Props. 52, 53) that, if +straight lines be ‘inflected’ from the ends of the base of +a segment of a circle to the curve (i.e. if we join the ends +of the base to any point on the curve) the line (i.e. the sum of +the chords) is greatest when the point taken is the middle +point of the arc, and diminishes as the point is taken farther +and farther from that point. +<p>Let <I>B</I> be the middle point of the +arc of the segment <I>ABC, D, E</I> any +other points on the curve towards +<I>C</I>; I say that +<FIG> +<MATH><I>AB</I>+<I>BC</I> > <I>AD</I>+<I>DC</I> > <I>AE</I>+<I>EC</I></MATH>. +<p>With <I>B</I> as centre and <I>BA</I> as radius +describe a circle, and produce <I>AB, +AD, AE</I> to meet this circle in <I>F, G, +H.</I> Join <I>FC, GC, HC.</I> +<p>Since <MATH><I>AB</I>=<I>BC</I>=<I>BF</I></MATH>, we have <MATH><I>AF</I>=<I>AB</I>+<I>BC</I></MATH>. Also the +angles <I>BFC, BCF</I> are equal, and each of them is half of +the angle <I>ABC.</I> +<pb n=526><head>COMMENTATORS AND BYZANTINES</head> +<p>Again <MATH>∠<I>AGC</I>=∠<I>AFC</I>=1/2∠<I>ABC</I>=1/2∠<I>ADC</I></MATH>; +therefore the angles <I>DGC, DCG</I> are equal and <MATH><I>DG</I>=<I>DC</I></MATH>; +therefore <MATH><I>AG</I>=<I>AD</I>+<I>DC</I></MATH>. +<p>Similarly <MATH><I>EH</I>=<I>EC</I></MATH> and <MATH><I>AH</I>=<I>AE</I>+<I>EC</I></MATH>. +<p>But, by Eucl. III. 7 or 15, <MATH><I>AF</I> > <I>AG</I> > <I>AH</I></MATH>, and so on; +therefore <MATH><I>AB</I>+<I>BC</I> > <I>AD</I>+<I>DC</I> > <I>AE</I>+<I>EC</I></MATH>, and so on. +<p>In the particular case where the segment <I>ABC</I> is a semi- +circle <MATH><I>AB</I><SUP>2</SUP>+<I>BC</I><SUP>2</SUP>=<I>AC</I><SUP>2</SUP>=<I>AD</I><SUP>2</SUP>+<I>DC</I><SUP>2</SUP></MATH>, &c., and the result of +Prop. 57 follows. +<p>Props. 58-69 are propositions of this sort: In equal right +cones the triangular sections through the axis are reciprocally +proportional to their bases and conversely (Props. 58, 59); +right cones of equal height have to one another the ratio +duplicate of that of their axial triangles (Prop. 62); right +cones which are reciprocally proportional to their bases have +axial triangles which are to one another reciprocally in the +triplicate ratio of their bases and conversely (Props. 66, 67); +and so on. +<p>THEON OF ALEXANDRIA lived towards the end of the fourth +century A.D. Suidas places him in the reign of Theodosius I +(379-95); he tells us himself that he observed a solar eclipse +at Alexandria in the year 365, and his notes on the chrono- +logical tables of Ptolemy extend down to 372. +<C>Commentary on the <I>Syntaxis.</I></C> +<p>We have already seen him as the author of a commentary +on Ptolemy's <I>Syntaxis</I> in eleven Books. This commentary is +not calculated to give us a very high opinion of Theon's +mathematical calibre, but it is valuable for several historical +notices that it gives, and we are indebted to it for a useful +account of the Greek method of operating with sexagesimal +fractions, which is illustrated by examples of multiplication, +division, and the extraction of the square root of a non-square +number by way of approximation. These illustrations of +numerical calculation have already been given above (vol. i, +<pb n=527><head>THEON OF ALEXANDRIA</head> +pp. 58-63). Of the historical notices we may mention the +following. (1) Theon mentions the treatise of Menelaus <I>On +Chords in a Circle,</I> i.e. Menelaus's Table of Chords, which came +between the similar Tables of Hipparchus and Ptolemy. (2) A +quotation from Diophantus furnishes incidentally a lower limit +for the date of the <I>Arithmetica.</I> (3) It is in the commentary +on Ptolemy that Theon tells us that the second part of Euclid +VI. 33 relating to <I>sectors</I> in equal circles was inserted by him- +self in his edition of the <I>Elements,</I> a notice which is of capital +importance in that it enables the Theonine manuscripts of +Euclid to be distinguished from the ante-Theonine, and is +therefore the key to the question how far the genuine text +of Euclid was altered in Theon's edition. (4) As we have +seen (pp. 207 sq.), Theon, à propos of an allusion of Ptolemy +to the theory of isoperimetric figures, has preserved for us +several propositions from the treatise by Zenodorus on that +subject. +<C>Theon's edition of Euclid's <I>Elements.</I></C> +<p>We are able to judge of the character of Theon's edition of +Euclid by a comparison between the Theonine manuscripts +and the famous Vatican MS. 190, which contains an earlier +edition than Theon's, together with certain fragments of +ancient papyri. It appears that, while Theon took some +trouble to follow older manuscripts, it was not so much his +object to get the most authoritative text as to make what he +considered improvements of one sort or other. (1) He made +alterations where he found, or thought he found, mistakes in +the original; while he tried to remove some real blots, he +altered other passages too hastily when a little more considera- +tion would have shown that Euclid's words are right or could +be excused, and offer no difficulty to an intelligent reader. +(2) He made emendations intended to improve the form or +diction of Euclid; in general they were prompted by a desire +to eliminate anything which was out of the common in expres- +sion or in form, in order to reduce the language to one and the +same standard or norm. (3) He bestowed, however, most +attention upon additions designed to supplement or explain +the original; (<I>a</I>) he interpolated whole propositions where he +thought them necessary or useful, e.g. the addition to VI. 33 +<pb n=528><head>COMMENTATORS AND BYZANTINES</head> +already referred to, a second case to VI. 27, a porism or corollary +to II. 4, a second porism to III. 16, the proposition VII. 22, +a lemma after X. 12, besides alternative proofs here and there; +(<I>b</I>) he added words for the purpose of making smoother and +clearer, or more precise, things which Euclid had expressed +with unusual brevity, harshness, or carelessness; (<I>c</I>) he sup- +plied intermediate steps where Euclid's argument seemed too +difficult to follow. In short, while making only inconsider- +able additions to the content of the <I>Elements,</I> he endeavoured +to remove difficulties that might be felt by learners in study- +ing the book, as a modern editor might do in editing a classical +text-book for use in schools; and there is no doubt that his +edition was approved by his pupils at Alexandria for whom it +was written, as well as by later Greeks, who used it almost +exclusively, with the result that the more ancient text is only +preserved complete in one manuscript. +<C>Edition of the <I>Optics</I> of Euclid.</C> +<p>In addition to the <I>Elements,</I> Theon edited the <I>Optics</I> of +Euclid; Theon's recension as well as the genuine work is +included by Heiberg in his edition. It is possible that the +<I>Catoptrica</I> included by Heiberg in the same volume is also by +Theon. +<p>Next to Theon should be mentioned his daughter HYPATIA, +who is mentioned by Theon himself as having assisted in the +revision of the commentary on Ptolemy. This learned lady +is said to have been mistress of the whole of pagan science, +especially of philosophy and medicine, and by her eloquence +and authority to have attained such influence that Christianity +considered itself threatened, and she was put to death by +a fanatical mob in March 415. According to Suidas she wrote +commentaries on Diophantus, on the Astronomical Canon (of +Ptolemy) and on the Conics of Apollonius. These works +have not survived, but it has been conjectured (by Tannery) +that the remarks of Psellus (eleventh century) at the begin- +ning of his letter about Diophantus, Anatolius, and the +Egyptian method of arithmetical reckoning were taken bodily +from some manuscript of Diophantus containing an ancient +and systematic commentary which may very well have been +that of Hypatia. Possibly her commentary may have extended +<pb n=529><head>HYPATIA. PORPHYRY</head> +only to the first six Books, in which case the fact that Hypatia +wrote a commentary on them may account for the survival of +these Books while the rest of the thirteen were first forgotten +and then lost. +<p>It will be convenient to take next the series of Neo- +Platonist commentators. It does not appear that Ammonius +Saccas (about A.D. 175-250), the founder of Neo-Platonism, or +his pupil Plotinus (A.D. 204-69), who first expounded the +doctrines in systematic form, had any special connexion with +mathematics, but PORPHYRY (about 232-304), the disciple of +Plotinus and the reviser and editor of his works, appears to +have written a commentary on the <I>Elements.</I> This we gather +from Proclus, who quotes from Porphyry comments on Eucl. +I. 14 and 26 and alternative proofs of I. 18, 20. It is possible +that Porphyry's work may have been used later by Pappus in +writing his own commentary, and Proclus may have got his +references from Pappus, but the form of these references sug- +gests that he had direct access to the original commentary of +Porphyry. +<p>IAMBLICHUS (died about A.D. 330) was the author of a com- +mentary on the <I>Introductio arithmetica</I> of Nicomachus, and +of other works which have already been mentioned. He was +a pupil of Porphyry as well as of Anatolius, also a disciple of +Porphyry. +<p>But the most important of the Neo-Platonists to the his- +torian of mathematics is PROCLUS (A.D. 410-85). Proclus +received his early training at Alexandria, where Olympio- +dorus was his instructor in the works of Aristotle, and +mathematics was taught him by one Heron (of course a +different Heron from the ‘<I>mechanicus</I> Hero’ of the <I>Metrica,</I> +&c.). He afterwards went to Athens, where he learnt the +Neo-Platonic philosophy from Plutarch, the grandson of Nes- +torius, and from his pupil Syrianus, and became one of its +most prominent exponents. He speaks everywhere with the +highest respect of his masters, and was in turn regarded with +extravagant veneration by his contemporaries, as we learn +from Marinus, his pupil and biographer. On the death of +Syrianus he was put at the head of the Neo-Platonic school. +He was a man of untiring industry, as is shown by the +<pb n=530><head>COMMENTATORS AND BYZANTINES</head> +number of books which he wrote, including a large number of +commentaries, mostly on the dialogues of Plato (e.g. the +<I>Timaeus,</I> the <I>Republic,</I> the <I>Parmenides,</I> the <I>Cratylus</I>). He +was an acute dialectician and pre-eminent among his contem- +poraries in the range of his learning; he was a competent +mathematician; he was even a poet. At the same time he +was a believer in all sorts of myths and mysteries, and +a devout worshipper of divinities both Greek and Oriental. +He was much more a philosopher than a mathematician. In +his commentary on the <I>Timaeus,</I> when referring to the ques- +tion whether the sun occupies a middle place among the +planets, he speaks as no real mathematician could have +spoken, rejecting the view of Hipparchus and Ptolemy because +<G>o( qeourgo/s</G> (<I>sc.</I> the Chaldean, says Zeller) thinks otherwise, +‘whom it is not lawful to disbelieve’. Martin observes too, +rather neatly, that ‘for Proclus the Elements of Euclid had +the good fortune not to be contradicted either by the Chaldean +Oracles or by the speculations of Pythagoreans old and new’. +<C>Commentary on Euclid, Book I.</C> +<p>For us the most important work of Proclus is his commen- +tary on Euclid, Book I, because it is one of the main sources +of our information as to the history of elementary geometry. +Its great value arises mainly from the fact that Proclus had +access to a number of historical and critical works which are +now lost except for fragments preserved by Proclus and +others. +<C>(<G>a</G>) <I>Sources of the Commentary.</I></C> +<p>The historical work the loss of which is most deeply to be +deplored is the <I>History of Geometry</I> by Eudemus. There +appears to be no reason to doubt that the work of Eudemus +was accessible to Proclus at first hand. For the later writers +Simplicius and Eutocius refer to it in terms such as leave no +doubt that <I>they</I> had it before them. Simplicius, quoting +Eudemus as the best authority on Hippocrates's quadratures +of lunes, says he will set out what Eudemus says ‘word for +word’, adding only a little explanation in the shape of refer- +ences to Euclid's <I>Elements</I> ‘owing to the <I>memorandum-like +style of Eudemus,</I> who sets out his explanations in the abbre- +<pb n=531><head>PROCLUS</head> +viated form usual with ancient writers. Now in the second +book of the history of geometry he writes as follows’.<note>Simplicius on Arist. <I>Phys.</I>, p. 60. 28, Diels.</note> In +like manner Eutocius speaks of the paralogisms handed down +in connexion with the attempts of Hippocrates and Antiphon +to square the circle, ‘with which I imagine that all persons +are accurately acquainted who have <I>examined</I> (<G>e)peskemme/nous</G>) +the geometrical history of Eudemus and know the <I>Ceria +Aristotelica</I>’.<note>Archimedes, ed. Heib., vol. iii, p. 228. 17-19.</note> +<p>The references by Proclus to Eudemus by name are not +indeed numerous; they are five in number; but on the other +hand he gives at least as many other historical data which can +with great probability be attributed to Eudemus. +<p>Proclus was even more indebted to Geminus, from whom +he borrows long extracts, often mentioning him by name—there +are some eighteen such references—but often omitting +to do so. We are able to form a tolerably certain judge- +ment as to the origin of the latter class of passages on the +strength of the similarity of the subjects treated and the views +expressed to those found in the acknowledged extracts. As +we have seen, the work of Geminus mainly cited seems to +have borne the title <I>The Doctrine</I> or <I>Theory of the Mathematics,</I> +which was a very comprehensive work dealing, in a portion of +it, with the ‘classification of mathematics’. +<p>We have already discussed the question of the authorship +of the famous historical summary given by Proclus. It is +divided, as every one knows, into two distinct parts between +which comes the remark, ‘Those who compiled histories +bring the development of this science up to this point. Not +much younger than these is Euclid, who’, &c. The ultimate +source at any rate of the early part of the summary must +presumably have been the great work of Eudemus above +mentioned. +<p>It is evident that Proclus had before him the original works +of Plato, Aristotle, Archimedes and Plotinus, the <G>*summikta/</G> of +Porphyry and the works of his master Syrianus, as well as a +group of works representing the Pythagorean tradition on its +mystic, as distinct from its mathematical, side, from Philo- +laus downwards, and comprisinǵ the more or less apocryphal +<pb n=532><head>COMMENTATORS AND BYZANTINES</head> +<G>i(ero\s lo/gos</G> of Pythagoras, the <I>Oracles</I> (<G>lo/gia</G>) and Orphic +verses. +<p>The following will be a convenient summary of the other +works used by Proclus, and will at the same time give an +indication of the historical value of his commentary on +Euclid, Book I: +<p>Eudemus: <I>History of Geometry.</I> +<p>Geminus: <I>The Theory of the Mathematical Sciences.</I> +<p>Heron: <I>Commentary on the Elements of Euclid.</I> +<p>Porphyry: ” ” ” +<p>Pappus: ” ” ” +<p>Apollonius of Perga: A work relating to elementary +geometry. +<p>Ptolemy: <I>On the parallel-postulate.</I> +<p>Posidonius: A book controverting Zeno of Sidon. +<p>Carpus: <I>Astronomy.</I> +<p>Syrianus: A discussion on the <I>angle.</I> +<C>(<G>b</G>) <I>Character of the Commentary.</I></C> +<p>We know that in the Neo-Platonic school the pupils learnt +mathematics; and it is clear that Proclus taught this subject, +and that this was the origin of his commentary. Many +passages show him as a master speaking to scholars; in one +place he speaks of ‘my hearers’.<note>Proclus on Eucl. I, p. 210. 19.</note> Further, the pupils whom +he was addressing were <I>beginners</I> in mathematics; thus in one +passage he says that he omits ‘for the present’ to speak of the +discoveries of those who employed the curves of Nicomedes +and Hippias for trisecting an angle, and of those who used the +Archimedean spiral for dividing an angle in a given ratio, +because these things would be ‘too difficult for beginners’.<note><I>Ib.,</I> p. 272. 12.</note> +But there are signs that the commentary was revised and +re-edited for a larger public; he speaks for instance in one +place of ‘those who will come across his work’.<note><I>Ib.,</I> p. 84. 9.</note> There are +also passages, e.g. passages about the cylindrical helix, con- +choids and cissoids, which would not have been understood by +the beginners to whom he lectured. +<pb n=533><head>PROCLUS</head> +<p>The commentary opens with two Prologues. The first is +on mathematics in general and its relation to, and use in, +philosophy, from which Proclus passes to the classification of +mathematics. Prologue II deals with geometry generally and +its subject-matter according to Plato, Aristotle and others. +After this section comes the famous summary (pp. 64-8) +ending with a eulogium of Euclid, with particular reference +to the admirable discretion shown in the selection of the pro- +positions which should constitute the <I>Elements</I> of geometry, +the ordering of the whole subject-matter, the exactness and +the conclusiveness of the demonstrations, and the power with +which every question is handled. Generalities follow, such as +the discussion of the nature of <I>elements</I>, the distinction between +theorems and problems according to different authorities, and +finally a division of Book I into three main sections, (1) the +construction and properties of triangles and their parts and +the comparison between triangles in respect of their angles +and sides, (2) the properties of parallels and parallelograms +and their construction from certain data, and (3) the bringing +of triangles and parallelograms into relation as regards area. +<p>Coming to the Book itself, Proclus deals historically and +critically with all the definitions, postulates and axioms in +order. The notes on the postulates and axioms are preceded +by a general discussion of the principles of geometry, hypo- +theses, postulates and axioms, and their relation to one +another; here as usual Proclus quotes the opinions of all the +important authorities. Again, when he comes to Prop. 1, he +discusses once more the difference between theorems and +problems, then sets out and explains the formal divisions of +a proposition, the <I>enunciation</I> (<G>pro/tasis</G>), the <I>setting-out</I> +(<G>e)/kqesis</G>), the <I>definition</I> or <I>specification</I> (<G>diorismo/s</G>), the <I>con- +struction</I> (<G>kataskeuh/</G>), the <I>proof</I> (<G>a)po/deixis</G>), the <I>conclusion</I> +(<G>sumpe/rasma</G>), and finally a number of other technical terms, +e.g. things said to be <I>given</I>, in the various senses of this term, +the <I>lemma</I>, the <I>case</I>, the <I>porism</I> in its two senses, the <I>objection</I> +(<G>e)/nstasis</G>), the <I>reduction</I> of a problem, <I>reductio ad absurdum, +analysis</I> and <I>synthesis.</I> +<p>In his comments on the separate propositions Proclus +generally proceeds in this way: first he gives explanations +regarding Euclid's proofs, secondly he gives a few different +<pb n=534><head>COMMENTATORS AND BYZANTINES</head> +cases, mainly for the sake of practice, and thirdly he addresses +himself to refuting objections which cavillers had taken or +might take to particular propositions or arguments. He does +not seem to have had any notion of correcting or improving +Euclid; only in one place does he propose anything of his +own to get over a difficulty which he finds in Euclid; this is +where he tries to prove the parallel-postulate, after giving +Ptolemy's attempt to prove it and pointing out objections to +Ptolemy's proof. +<p>The book is evidently almost entirely a compilation, though +a compilation ‘in the better sense of the term’. The <I>onus +probandi</I> is on any one who shall assert that anything in it is +Proclus's own; very few things can with certainty be said to +be so. Instances are (1) remarks on certain things which he +quotes from Pappus, since Pappus was the last of the com- +mentators whose works he seems to have used, (2) a defence +of Geminus against Carpus, who criticized Geminus's view of +the difference between theorems and problems, and perhaps +(3) criticisms of certain attempts by Apollonius to improve on +Euclid's proofs and constructions; but the only substantial +example is (4) the attempted proof of the parallel-postulate, +based on an ‘axiom’ to the effect that, ‘if from one point two +straight lines forming an angle be produced <I>ad infinitum</I>, the +distance between them when so produced <I>ad infinitum</I> exceeds +any finite magnitude (i. e. length)’, an assumption which +purports to be the equivalent of a statement in Aristotle.<note><I>De caelo</I>, i. 5, 271 b 28-30.</note> +Philoponus says that Proclus as well as Ptolemy wrote a whole +book on the parallel-postulate.<note>Philoponus on <I>Anal. Post.</I> i. 10, p. 214 a 9-12, Brandis.</note> +<p>It is still not quite certain whether Proclus continued his +commentaries beyond Book I. He certainly intended to do so, +for, speaking of the trisection of an angle by means of certain +curves, he says, ‘we may perhaps more appropriately examine +these things on the third Book, where the writer of the +Elements bisects a given circumference’, and again, after +saying that of all parallelograms which have the same peri- +meter the square is the greatest ‘and the rhomboid least of +all’, he adds, ‘But this we will prove in another place, for it +is more appropriate to the discussion of the hypotheses of the +<pb n=535><head>PROCLUS</head> +second Book’. But at the time when the commentary on +Book I was written he was evidently uncertain whether he +would be able to continue it, for at the end he says, ‘For my +part, if I should be able to discuss the other Books in the +same way, I should give thanks to the gods; but, if other +cares should draw me away, I beg those who are attracted by +this subject to complete the exposition of the other Books as +well, following the same method and addressing themselves +throughout to the deeper and more sharply defined questions +involved’.<note>Proclus on Eucl. I, p. 432. 9-15.</note> Wachsmuth, finding a Vatican manuscript contain- +ing a collection of scholia on Books I, II, V, VI, X, headed <G>*ei)s ta\ +*eu)klei/dou stoixei=a prolambano/mena e)k tw=n *pro/klou spora/dhn +kai\ kat' e)pitomh/n</G>, and seeing that the scholia on Book I were +extracts from the extant commentary of Proclus, concluded +that those on the other Books were also from Proclus; but +the <G>pro</G>- in <G>prolambano/mena</G> rather suggests that only the +scholia to Book I are from Proclus. Heiberg found and +published in 1903 a scholium to X. 9, in which Proclus is +expressly quoted as the authority, but he does not regard +this circumstance as conclusive. On the other hand, Heiberg +has noted two facts which go against the view that Proclus +wrote on the later Books: (1) the scholiast's copy of +Proclus was not much better than our manuscripts; in +particular, it had the same lacunae in the notes to I. 36, +37, and I. 41-3; this makes it improbable that the scholiast +had further commentaries of Proclus which have vanished +for us; (2) there is no trace in the scholia of the notes +which Proclus promised in the passages already referred to. +All, therefore, that we can say is that, while the Wachsmuth +scholia <I>may</I> be extracts from Proclus, it is on the whole +improbable. +<C><I>Hypotyposis of Astronomical Hypotheses.</I></C> +<p>Another extant work of Proclus which should be referred +to is his <I>Hypotyposis of Astronomical Hypotheses</I>, a sort of +readable and easy introduction to the astronomical system +of Hipparchus and Ptolemy. It has been well edited by +Manitius (Teubner, 1909). Three things may be noted as +<pb n=536><head>COMMENTATORS AND BYZANTINES</head> +regards this work. It contains<note>Proclus, <I>Hypotyposis</I>, c. 4, pp. 120-22.</note> a description of the method +of measuring the sun's apparent diameter by means of +Heron's water-clock, which, by comparison with the corre- +sponding description in Theon's commentary to the <I>Syntaxis</I> +of Ptolemy, is seen to have a common source with it. That +source is Pappus, and, inasmuch as Proclus has a figure (repro- +duced by Manitius in his text from one set of manuscripts) +corresponding to the description, while the text of Theon has +no figure, it is clear that Proclus drew directly on Pappus, +who doubtless gave, in his account of the procedure, a figure +taken from Heron's own work on water-clocks. A simple +proof of the equivalence of the epicycle and eccentric hypo- +theses is quoted by Proclus from one Hilarius of Antioch.<note><I>Ib.</I>, c. 3, pp. 76, 17 sq.</note> +An interesting passage is that in chap. 4 (p. 130, 18) where +Sosigenes the Peripatetic is said to have recorded in his work +‘on reacting spheres’ that an <I>annular</I> eclipse of the sun is +sometimes observed at times of perigee; this is, so far as +I know, the only allusion in ancient times to annular eclipses, +and Proclus himself questions the correctness of Sosigenes's +statement. +<C>Commentary on the <I>Republic.</I></C> +<p>The commentary of Proclus on the <I>Republic</I> contains some +passages of great interest to the historian of mathematics. +The most important is that<note><I>Procli Diadochi in Platonis Rempublicam Commentarii</I>, ed. Kroll, +vol. ii, p. 27.</note> in which Proclus indicates that +Props. 9, 10 of Euclid, Book II, are Pythagorean proposi- +tions invented for the purpose of proving geometrically the +fundamental property of the series of ‘side-’ and ‘diameter-’ +numbers, giving successive approximations to the value of +√2 (see vol. i, p. 93). The explanation<note><I>Ib.</I>, vol. ii, pp. 36-42.</note> of the passage in +Plato about the Geometrical Number is defective and dis- +appointing, but it contains an interesting reference to one +Paterius, of date presumably intermediate between Nestorius +and Proclus. Paterius is said to have made a calculation, in +units and submultiples, of the lengths of different segments of +<pb n=537><head>PROCLUS. MARINUS</head> +straight lines in a figure formed by taking a triangle with +<FIG> +sides 3, 4, 5 as <I>ABC</I>, then drawing +<I>BD</I> from the right angle <I>B</I> perpen- +dicular to <I>AC</I>, and lastly drawing +perpendiculars <I>DE, DF</I> to <I>AB, BC.</I> +A diagram in the text with the +lengths of the segments shown along- +side them in the usual numerical +notation shows that Paterius obtained from the data <I>AB</I>=3, +<I>BC</I>=4, <I>CA</I>=5 the following: +<MATH><I>DC</I>=<G>ge</G>′=3 1/5</MATH> +<MATH><I>BD</I>=<G>bg</G>′<G>ie</G>′=2 1/3 1/15 [=2 2/5]</MATH> +<MATH><I>AD</I>=<G>asd</G>′<G>k</G>′=1 1/2 1/4 1/20 [=1 4/5]</MATH> +<MATH><I>FC</I>=<G>bsk</G>′<G>r</G>′=2 1/2 1/20 1/100 [=2 14/25]</MATH> +<MATH><I>FB</I>=<G>ag</G>′<G>ie</G>′<G>ke</G>′=1 1/3 1/15 1/25 [=1 11/25]</MATH> +<MATH><I>BE</I>=<G>asg</G>′<G>ie</G>′<G>n</G>′=1 1/2 1/3 1/15 1/50 [=1 23/25]</MATH> +<MATH><I>EA</I>=<G>aie</G>′<G>oe</G>′=1 1/15 1/75 [=1 2/25]</MATH>. +<p>This is an example of the Egyptian method of stating frac- +tions preceding by some three or four centuries the exposition +of the same method in the papyrus of Akhmīm. +<p>MARINUS of Neapolis, the pupil and biographer of Proclus, +wrote a commentary or rather introduction to the <I>Data</I> of +Euclid.<note>See Heiberg and Menge's Euclid, vol. vi, pp. 234-56.</note> It is mainly taken up with a discussion of the +question <G>ti/ to\ dedome/non</G>, what is meant by <I>given</I>? There +were apparently many different definitions of the term <I>given</I> +by earlier and later authorities. Of those who tried to define +it in the simplest way by means of a single <I>differentia</I>, three +are mentioned by name. Apollonius in his work on <G>neu/seis</G> +and his ‘general treatise’ (presumably that on elementary +geometry) described the <I>given</I> as <I>assigned</I> or <I>fixed</I> (<G>tetag- +me/non</G>), Diodorus called it <I>known</I> (<G>gnw/rimon</G>); others regarded +it as <I>rational</I> (<G>r(hto/n</G>) and Ptolemy is classed with these, rather +oddly, because ‘he called those things given the measure of +which is given either exactly or approximately’. Others +<pb n=538><head>COMMENTATORS AND BYZANTINES</head> +combined two of these ideas and called it <I>assigned</I> or <I>fixed</I> +and <I>procurable</I> or capable of being found (<G>po/rimon</G>); others +‘fixed and known’, and a third class ‘known and procurable’. +These various views are then discussed at length. +<p>DOMNINUS of Larissa, a pupil of Syrianus at the same time +as Proclus, wrote a <I>Manual of Introductory Arithmetic</I> <G>e)gxei- +ri/dion a)riqmhtikh=s ei)sagwgh=s</G>, which was edited by Boissonade<note><I>Anecdota Graeca</I>, vol. iv, pp. 413-29.</note> +and is the subject of two articles by Tannery,<note><I>Mémoires scientifiques</I>, vol. ii, nos. 35, 40.</note> who also left +a translation of it, with prolegomena, which has since been +published.<note><I>Revue des études grecques</I>, 1906, pp. 359-82; <I>Mémoires scientifiques</I>, +vol. iii, pp. 256-81.</note> It is a sketch of the elements of the theory of +numbers, very concise and well arranged, and is interesting +because it indicates a serious attempt at a reaction against the +<I>Introductio arithmetica</I> of Nicomachus and a return to the +doctrine of Euclid. Besides Euclid, Nicomachus and Theon +of Smyrna, Domninus seems to have used another source, +now lost, which was also drawn upon by Iamblichus. At the +end of this work Domninus foreshadows a more complete +treatise on the theory of numbers under the title <I>Elements of +Arithmetic</I> (<G>a)riqmhtikh\ stoixei/wsis</G>), but whether this was +ever written or not we do not know. Another tract +attributed to Domninus <G>pw=s e)/sti lo/gon e)k lo/gou a)felei=n</G> +(how a ratio can be taken out of a ratio) has been published +with a translation by Ruelle<note><I>Revue de Philologie</I>, 1883, p. 83 sq.</note>; if it is not by Domninus, it +probably belongs to the same period. +<p>A most honourable place in our history must be reserved +for SIMPLICIUS, who has been rightly called ‘the excellent +Simplicius, the Aristotle-commentator, to whom the world can +never be grateful enough for the preservation of the frag- +ments of Parmenides, Empedocles, Anaxagoras, Melissus, +Theophrastus and others’ (v. Wilamowitz-Möllendorff). He +lived in the first half of the sixth century and was a pupil, +first of Ammonius of Alexandria, and then of Damascius, +the last head of the Platonic school at Athens. When in the +year 529 the Emperor Justinian, in his zeal to eradicate +paganism, issued an edict forbidding the teaching of philo- +<pb n=539><head>DOMNINUS. SIMPLICIUS</head> +sophy at Athens, the last members of the school, including +Damascius and Simplicius, migrated to Persia, but returned +about 533 to Athens, where Simplicius continued to teach for +some time though the school remained closed. +<C><I>Extracts from Eudemus.</I></C> +<p>To Simplicius we owe two long extracts of capital impor- +tance for the history of mathematics and astronomy. The +first is his account, based upon and to a large extent quoted +textually from Eudemus's <I>History of Geometry</I>, of the attempt +by Antiphon to square the circle and of the quadratures of +lunes by Hippocrates of Chios. It is contained in Simplicius's +commentary on Aristotle's <I>Physics</I>,<note>Simpl. <I>in Phys.</I>, pp. 54-69, ed. Diels.</note> and has been the subject +of a considerable literature extending from 1870, the date +when Bretschneider first called attention to it, to the latest +critical edition with translation and notes by Rudio (Teubner, +1907). It has already been discussed (vol. i, pp. 183-99). +<p>The second, and not less important, of the two passages is +that containing the elaborate and detailed account of the +system of concentric spheres, as first invented by Eudoxus for +explaining the apparent motion of the sun, moon, and planets, +and of the modifications made by Callippus and Aristotle. It +is contained in the commentary on Aristotle's <I>De caelo</I><note>Simpl. on Arist. <I>De caelo</I>, p. 488. 18-24 and pp. 493-506, ed. Heiberg.</note>; +Simplicius quotes largely from Sosigenes the Peripatetic +(second century A.D.), observing that he in his turn drew +from Eudemus, who dealt with the subject in the second +book of his <I>History of Astronomy.</I> It is this passage of +Simplicius which, along with a passage in Aristotle's <I>Meta- +physics</I>,<note><I>Metaph.</I> <*>. 8, 1073 b 17-1074 a 14.</note> enabled Schiaparelli to reconstruct Eudoxus's system +(see vol. i, pp. 329-34). Nor must it be forgotten that it is in +Simplicius's commentary on the <I>Physics</I><note>Simpl. <I>in Phys.</I>, pp. 291-2, ed. Diels.</note> that the extract +from Geminus's summary of the <I>Meteorologica</I> of Posidonius +occurs which was used by Schiaparelli to support his view +that it was Heraclides of Pontus, not Aristarchus of Samos, +who first propounded the heliocentric hypothesis. +<p>Simplicius also wrote a commentary on Euclid's <I>Elements</I>, +Book I, from which an-Nairīzī, the Arabian commentator, +<pb n=540><head>COMMENTATORS AND BYZANTINES</head> +made valuable extracts, including the account of the attempt of +‘Aganis’ to prove the parallel-postulate (see pp. 228-30 above). +<p>Contemporary with Simplicius, or somewhat earlier, was +EUTOCIUS, the commentator on Archimedes and Apollonius. +As he dedicated the commentary on Book I <I>On the Sphere +and Cylinder</I> to Ammonius (a pupil of Proclus and teacher +of Simplicius), who can hardly have been alive after A.D. 510, +Eutocius was probably born about A.D. 480. His date used +to be put some fifty years later because, at the end of the com- +mentaries on Book II <I>On the Sphere and Cylinder</I> and on +the <I>Measurement of a Circle</I>, there is a note to the effect that +‘the edition was revised by Isidorus of Miletus, the mechanical +engineer, <I>our teacher</I>’. But, in view of the relation to Ammo- +nius, it is impossible that Eutocius can have been a pupil of +Isidorus, who was younger than Anthemius of Tralles, the +architect of Saint Sophia at Constantinople in 532, whose +work was continued by Isidorus after Anthemius's death +about A.D. 534. Moreover, it was to Anthemius that Eutocius +dedicated, separately, the commentaries on the first four +Books of Apollonius's <I>Conics</I>, addressing Anthemius as ‘my +dear friend’. Hence we conclude that Eutocius was an elder +contemporary of Anthemius, and that the reference to Isidorus +is by an editor of Eutocius's commentaries who was a pupil of +Isidorus. For a like reason, the reference in the commentary +on Book II <I>On the Sphere and Cylinder</I><note>Archimedes, ed. Heiberg, vol. iii, p. 84. 8-11.</note> to a <G>diabh/ths</G> +invented by Isidorus ‘our teacher’ for drawing a parabola +must be considered to be an interpolation by the same editor. +<p>Eutocius's commentaries on Archimedes apparently ex- +tended only to the three works, <I>On the Sphere and Cylinder, +Measurement of a Circle</I> and <I>Plane Equilibriums</I>, and those +on the <I>Conics</I> of Apollonius to the first four Books only. +We are indebted to these commentaries for many valuable +historical notes. Those deserving special mention here are +(1) the account of the solutions of the problem of the duplica- +tion of the cube, or the finding of two mean proportionals, +by ‘Plato’, Heron, Philon, Apollonius, Diocles, Pappus, +Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes, (2) +the fragment discovered by Eutocius himself containing the +<pb n=541><head>EUTOCIUS. ANTHEMIUS</head> +missing solution, promised by Archimedes in <I>On the Sphere +and Cylinder</I>, II. 4, of the auxiliary problem amounting +to the solution by means of conics of the cubic equation +<MATH>(<I>a</I>-<I>x</I>)<I>x</I><SUP>2</SUP>=<I>bc</I><SUP>2</SUP></MATH>, (3) the solutions (<I>a</I>) by Diocles of the original +problem of II. 4 without bringing in the cubic, (<I>b</I>) by Diony- +sodorus of the auxiliary cubic equation. +<p>ANTHEMIUS of Tralles, the architect, mentioned above, was +himself an able mathematician, as is seen from a fragment of +a work of his, <I>On Burning-mirrors.</I> This is a document of +considerable importance for the history of conic sections. +Originally edited by L. Dupuy in 1777, it was reprinted in +Westermann's <G>*paradoxogra/foi</G> (<I>Scriptores rerum mirabilium +Graeci</I>), 1839, pp. 149-58. The first and third portions of +the fragment are those which interest us.<note>See <I>Bibliotheca mathematica</I>, vii<SUB>3</SUB>, 1907, pp. 225-33.</note> The first gives +a solution of the problem, To contrive that a ray of the sun +(admitted through a small hole or window) shall fall in a +given spot, without moving away at any hour and season. +This is contrived by constructing an elliptical mirror one focus +of which is at the point where the ray of the sun is admitted +while the other is at the point to which the ray is required +to be reflected at all times. Let <I>B</I> be the hole, <I>A</I> the point +to which reflection must always take place, <I>BA</I> being in the +meridian and parallel to the horizon. Let <I>BC</I> be at right +angles to <I>BA</I>, so that <I>CB</I> is an equinoctial ray; and let <I>BD</I> be +the ray at the summer solstice, <I>BE</I> a winter ray. +<p>Take <I>F</I> at a convenient distance on <I>BE</I> and measure <I>FQ</I> +equal to <I>FA.</I> Draw <I>HFG</I> through <I>F</I> bisecting the angle +<I>AFQ</I>, and let <I>BG</I> be the straight line bisecting the angle <I>EBC</I> +between the winter and the equinoctial rays. Then clearly, +since <I>FG</I> bisects the angle <I>QFA</I>, if we have a plane mirror in +the position <I>HFG</I>, the ray <I>BFE</I> entering at <I>B</I> will be reflected +to <I>A.</I> +<p>To get the equinoctial ray similarly reflected to <I>A</I>, join <I>GA</I>, +and with <I>G</I> as centre and <I>GA</I> as radius draw a circle meeting +<I>BC</I> in <I>K.</I> Bisect the angle <I>KGA</I> by the straight line <I>GLM</I> +meeting <I>BK</I> in <I>L</I> and terminated at <I>M</I>, a point on the bisector +of the angle <I>CBD.</I> Then <I>LM</I> bisects the angle <I>KLA</I> also, and +<I>KL</I>=<I>LA</I>, and <I>KM</I>=<I>MA</I>. If then <I>GLM</I> is a plane mirror, +the ray <I>BL</I> will be reflected to <I>A.</I> +<pb n=542><head>COMMENTATORS AND BYZANTINES</head> +<p>By taking the point <I>N</I> on <I>BD</I> such that <I>MN</I>=<I>MA</I>, and +bisecting the angle <I>NMA</I> by the straight line <I>MOP</I> meeting +<I>BD</I> in <I>O</I>, we find that, if <I>MOP</I> is a plane mirror, the ray <I>BO</I> +is reflected to <I>A.</I> +<p>Similarly, by continually bisecting angles and making more +mirrors, we can get any number of other points of impact. Mak- +ing the mirrors so short as to form a continuous curve, we get +the curve containing all points such that the sum of the distances +of each of them from <I>A</I> and <I>B</I> is constant and equal to <I>BQ, BK</I>, +or <I>BN.</I> ‘If then’, says Anthemius, ‘we stretch a string passed +<FIG> +round the points <I>A, B</I>, and through the first point taken on the +rays which are to be reflected, the said curve will be described, +which is part of the so-called “ellipse”, with reference to +which (i.e. by the revolution of which round <I>BA</I>) the surface +of impact of the said mirror has to be constructed.’ +<p>We have here apparently the first mention of the construc- +tion of an ellipse by means of a string stretched tight round +the foci. Anthemius's construction depends upon two pro- +positions proved by Apollonius (1) that the sum of the focal +distances of any point on the ellipse is constant, (2) that the +focal distances of any point make equal angles with the +tangent at that point, and also (3) upon a proposition not +found in Apollonius, namely that the straight line joining +<pb n=543><head>ANTHEMIUS</head> +the focus to the intersection of two tangents bisects the angle +between the straight lines joining the focus to the two points +of contact respectively. +<p>In the third portion of the fragment Anthemius proves that +parallel rays can be reflected to one single point from a para- +bolic mirror of which the point is the focus. The <I>directrix</I> is +used in the construction, which follows, <I>mutatis mutandis</I>, the +same course as the above construction in the case of the ellipse. +<p>As to the supposition of Heiberg that Anthemius may also +be the author of the <I>Fragmentum mathematicum Bobiense</I>, see +above (p. 203). +<C><I>The Papyrus of Akhmīm.</I></C> +<p>Next in chronological order must apparently be placed +the Papyrus of Akhmīm, a manual of calculation written +in Greek, which was found in the metropolis of Akhmīm, +the ancient Panopolis, and is now in the Musée du +Gizeh. It was edited by J. Baillet<note><I>Mémoires publiés par les membres de la Mission archéologique française +au Caire</I>, vol. ix, part 1, pp. 1-89.</note> in 1892. Accord- +ing to the editor, it was written between the sixth and +ninth centuries by a Christian. It is interesting because +it preserves the Egyptian method of reckoning, with proper +fractions written as the sum of primary fractions or sub- +multiples, a method which survived alongside the Greek and +was employed, and even exclusively taught, in the East. The +advantage of this papyrus, as compared with Ahmes's, is that +we can gather the formulae used for the decomposition of +ordinary proper fractions into sums of submultiples. The +formulae for decomposing a proper fraction into the sum of +two submultiples may be shown thus: +(1) <MATH><I>a</I>/(<I>bc</I>)=1/(<I>c</I>.(<I>b</I>+<I>c</I>)/<I>a</I>)+1/(<I>b</I>.(<I>b</I>+<I>c</I>)/<I>a</I>)</MATH>. +Examples <MATH>2/11=1/6 1/66, 3/110=1/70 1/77, 18/323=1/34 1/38</MATH>. +(2) <MATH><I>a</I>/(<I>bc</I>)=1/(<I>c</I>.(<I>b</I>+<I>mc</I>)/<I>a</I>)+1/(<I>b</I>.((<I>b</I>+<I>mc</I>)/<I>a</I>).(1/<I>m</I>)</MATH>. +<pb n=544><head>COMMENTATORS AND BYZANTINES</head> +Ex. <MATH>7/176=1/(11((16+3.11)/7))+1/(16((16+3.11)/7)1/3)=1/77+3/112</MATH>; +and again <MATH>3/112=1/(7((16+2.7)/3))+1/(16((16+2.7)/3)1/2)=1/70 1/80</MATH> +(3) <MATH><I>a</I>/(<I>cdf</I>)=1/(<I>c</I>.((<I>cd</I>+<I>df</I>)/<I>a</I>)+1/(<I>f</I>.((<I>cd</I>+<I>df</I>)/<I>a</I>))</MATH>. +Example. +<MATH>28/1320=28/(10.12.11)=1/(10.((120+132)/28))+1/(11.((120+132)/28))=1/90 1/99</MATH>. +<p>The object is, of course, to choose the factors of the denomi- +nator, and the multiplier <I>m</I> in (2), in such a way as to make +the two denominators on the right-hand side integral. +<p>When the fraction has to be decomposed into a sum of three +or more submultiples, we take out an obvious submultiple +first, then if necessary a second, until one of the formulae +will separate what remains into two submultiples. Or we +take out a part which is not a submultiple but which can be +divided into two submultiples by one of the formulae. +<p>For example, to decompose 31/616. The factors of 616 are 8.77 +or 7.88. Take out 1/88, and <MATH>31/616=1/88 24/616=1/88 3/77=1/88 1/77 2/77</MATH>; +and <MATH>2/77=1/63 1/99</MATH> by formula (1), so that <MATH>31/616=1/63 1/77 1/88 1/99</MATH>. +<p>Take 239/6460. The factors of 6460 are 85.76 or 95.68. Take +out 1/85, and <MATH>239/6460=1/85 163/6460</MATH>. Again take out 1/95, and we have +1/85 1/95 95/6460 or 1/85 1/95 1/68. The actual problem here is to find +(1/323)rd of 11 1/2 1/3 1/10 1/60, which latter expression reduces to +(1/20).239. +<p>The sort of problems solved in the book are (1) the division +of a number into parts in the proportion of certain given +numbers, (2) the solution of simple equations such as this: +From a certain treasure we take away (1/13)th, then from the +remainder (1/17)th of that remainder, and we find 150 units left; +what was the treasure? <MATH>[{<I>x</I>-(1/<I>a</I>)<I>x</I>-1/<I>b</I>(<I>x</I>-(1/<I>a</I>)<I>x</I>)-...}=<I>R</I>]</MATH>. +<pb n=545><head>THE PAPYRUS OF AKHMĪM. PSELLUS</head> +(3) subtractions such as: From 2/3 subtract 1/10 1/11 1/20 1/22 1/30 1/33 +1/40 1/44 1/50 1/55 1/60 1/66 1/70 1/77 1/88 1/90 1/99 1/100 1/110. Answer, 1/10 1/50. +<p>The book ends with long tables of results obtained (1) by +multiplying successive numbers, tens, hundreds and thousands +up to 10,000 by 2/3, 1/3, 1/4, 1/5, 1/6, &c., up to 1/10, (2) by multiplying +all the successive numbers 1, 2, 3 ... <I>n</I> by 1/<I>n</I>, where <I>n</I> is succes- +sively 11, 12, ... and 20; the results are all arranged as the +sums of integers and submultiples. +<p>The <I>Geodaesia</I> of a Byzantine author formerly called, with- +out any authority, ‘Heron the Younger’ was translated into +Latin by Barocius in 1572, and the Greek text was published +with a French translation by Vincent.<note><I>Notices et extraits</I>, xix, pt. 2, Paris, 1858.</note> The place of the +author's observations was the hippodrome at Constantinople, +and the date apparently about 938. The treatise was modelled +on Heron of Alexandria, especially the <I>Dioptra</I>, while some +measurements of areas and volumes are taken from the +<I>Metrica.</I> +<p>MICHAEL PSELLUS lived in the latter part of the eleventh +century, since his latest work bears the date 1092. Though +he was called ‘first of philosophers’, it cannot be said that +what survives of his mathematics suits this title. Xylander +edited in 1556 the Greek text, with a Latin translation, of +a book purporting to be by Psellus on the four mathematical +sciences, arithmetic, music, geometry and astronomy, but it is +evident that it cannot be entirely Psellus's own work, since +the astronomical portion is dated 1008. The arithmetic con- +tains no more than the names and classification of numbers +and ratios. The geometry has the extraordinary remark that, +while opinions differed as to how to find the area of a circle, +the method which found most favour was to take the area as +the geometric mean between the inscribed and circumscribed +squares; this gives <G>p</G>=√8=2.8284271! The only thing of +Psellus which has any value for us is the letter published by +Tannery in his edition of Diophantus.<note>Diophantus, vol. ii, pp. 37-42.</note> In this letter Psellus +says that both Diophantus and Anatolius (Bishop of Laodicea +about A.D. 280) wrote on the Egyptian method of reckoning, +<pb n=546><head>COMMENTATORS AND BYZANTINES</head> +and that Anatolius's account, which was different and more +succinct, was dedicated to Diophantus (this enables us to +determine Diophantus's date approximately). He also notes +the difference between the Diophantine and Egyptian names +for the successive powers of <G>a)riqmo/s</G>: the next power after +the fourth (<G>dunamodu/namis</G>=<I>x</I><SUP>4</SUP>), i.e. <I>x</I><SUP>5</SUP>, the Egyptians called +‘the first undescribed’ (<G>a)/logos prw=tos</G>) or the ‘fifth number’; +the sixth, <I>x</I><SUP>6</SUP>, they apparently (like Diophantus) called the +cube-cube; but with them the seventh, <I>x</I><SUP>7</SUP>, was the ‘second +undescribed’ or the ‘seventh number’, the eighth (<I>x</I><SUP>8</SUP>) was the +‘quadruple square’ (<G>tetraplh= du/namis</G>), the ninth (<I>x</I><SUP>9</SUP>) the +‘extended cube’ (<G>ku/bos e)xelikto/s</G>). Tannery conjectures that +all these remarks were taken direct from an old commentary +on Diophantus now lost, probably Hypatia's. +<p>GEORGIUS PACHYMERES (1242-1310) was the author of a +work on the Quadrivium (<G>*su/ntagma tw=n tessa/rwn maqhma/twn</G> +or <G>*tetra/biblon</G>). The arithmetical portion contains, besides +excerpts from Nicomachus and Euclid, a paraphrase of Dio- +phantus, Book I, which Tannery published in his edition of +Diophantus<note>Diophantus, vol. ii, pp. 78-122.</note>; the musical section with part of the preface was +published by Vincent,<note><I>Notices et extraits</I>, xvii, 1858, pp. 362-533.</note> and some fragments from Book IV by +Martin in his edition of the <I>Astronomy</I> of Theon of Smyrna. +<p>MAXIMUS PLANUDES, a monk from Nicomedia, was the +envoy of the Emperor Andronicus II at Venice in the year +1297, and lived probably from about 1260 to 1310. He +wrote scholia on the first two Books of Diophantus, which +are extant and are included in Tannery's edition of Dio- +phantus.<note>Diophantus, vol. ii, pp. 125-255.</note> They contain nothing of particular interest except +a number of conspectuses of the working-out of problems of +Diophantus written in Diophantus's own notation but with +steps in separate lines, and with abbreviations on the left of +words indicating the operations (e.g. <G>e)/kq</G>.=<G>e)/kqesis, tetr</G>.= +<G>tetragwnismo/s, su/nq</G>.=<G>su/nqesis</G>, &c.); the result is to make +the work almost as easy to follow as it is in our notation. +<p>Another work of Planudes is called <G>*yhfofori/a kat' *)indou/s</G>, +or <I>Arithmetic after the Indian method</I>, and was edited as <I>Das</I> +<pb n=547><head>PSELLUS. PACHYMERES. PLANUDES</head> +<I>Rechenbuch des Maximus Planudes</I> in Greek by Gerhardt +(Halle, 1865) and in a German translation by H. Waeschke +(Halle, 1878). There was, however, an earlier book under the +similar title <G>*)arxh\ th=s mega/lhs kai\ *)indikh=s yhfifori/as</G> (<I>sic</I>), +written in 1252, which is extant in the Paris MS. Suppl. Gr. +387; and Planudes seems to have raided this work. He +begins with an account of the symbols which, he says, were +<p>‘invented by certain distinguished astronomers for the most +convenient and accurate expression of numbers. There are +nine of these symbols (our 1, 2, 3, 4, 5, 6, 7, 8, 9), to which is +added another called <I>Tzifra</I> (cypher), written 0 and denoting +zero. The nine signs as well as this one are Indian.’ +<p>But this is, of course, not the first occurrence of the Indian +numerals; they were known, except the zero, to Gerbert +(Pope Sylvester II) in the tenth century, and were used by +Leonardo of Pisa in his <I>Liber abaci</I> (written in 1202 and +rewritten in 1228). Planudes used the Persian form of the +numerals, differing in this from the writer of the treatise of +1252 referred to, who used the form then current in Italy. +It scarcely belongs to Greek mathematics to give an account +of Planudes's methods of subtraction, multiplication, &c. +<C><I>Extraction of the square root.</I></C> +<p>As regards the extraction of the square root, he claims to +have invented a method different from the Indian method +and from that of Theon. It does not appear, however, that +there was anything new about it. Let us try to see in what +the supposed new method consisted. +<p>Planudes describes fully the method of extracting the +square root of a number with several digits, a method which +is essentially the same as ours. This appears to be what he +refers to later on as ‘the Indian method’. Then he tells us +how to find a first approximation to the root when the number +is not a complete square. +<p>‘Take the square root of the next lower actual square +number, and double it: then, from the number the square root +of which is required, subtract the next lower square number +so found, and to the remainder (as numerator) give as de- +nominator the double of the square root already found.’ +<pb n=548><head>COMMENTATORS AND BYZANTINES</head> +<p>The example given is √(18). Since <MATH>4<SUP>2</SUP>=16</MATH> is the next +lower square, the approximate square root is 4+2/2.4 or 4 1/4. +The formula used is, therefore, <MATH>√(<I>a</I><SUP>2</SUP>+<I>b</I>)=<I>a</I>+<I>b</I>/(2<I>a</I>)</MATH> approxi- +mately. (An example in larger numbers is +<MATH>√(1690196789)=41112+245/82224</MATH> approximately.) +Planudes multiplies 4 1/4 by itself and obtains 18 1/16, which +shows that the value 4 1/4 is not accurate. He adds that he will +explain later a method which is more exact and nearer the +truth, a method ‘which I claim as a discovery made by me +with the help of God’. Then, coming to the method which he +claims to have discovered, Planudes applies it to √6. The +object is to develop this in units and sexagesimal fractions. +Planudes begins by multiplying the 6 by 3600, making 21600 +second-sixtieths, and finds the square root of 21600 to lie +between 146 and 147. Writing the 146′ as 2 26′, he proceeds +to find the rest of the approximate square root (2 26′ 58″ 9‴) +by the same procedure as that used by Theon in extracting +the square root of 4500 and 2 28′ respectively. The differ- +ence is that in neither of the latter cases does Theon multiply +by 3600 so as to reduce the units to second-sixtieths, but he +begins by taking the approximate square root of 2, viz. 1, just +as he does that of 4500 (viz. 67). It is, then, the multiplication +by 3600, or the reduction to second-sixtieths to start with, that +constitutes the difference from Theon's method, and this must +therefore be what Planudes takes credit for as a new dis- +covery. In such a case as √(2 28′) or √3, Theon's method +has the inconvenience that the number of <I>minutes</I> in the +second term (34′ in the one case and 43′ in the other) cannot +be found without some trouble, a difficulty which is avoided +by Planudes's expedient. Therefore the method of Planudes +had its advantage in such a case. But the discovery was not +new. For it will be remembered that Ptolemy (and doubtless +Hipparchus before him) expressed the chord in a circle sub- +tending an angle of 120° at the centre (in terms of 120th parts +of the diameter) as 103<SUP><I>p</I></SUP> 55′ 23″, which indicates that the first +step in calculating √3 was to multiply it by 3600, making +10800, the nearest square below which is 103<SUP>2</SUP> (=10609). In +<pb n=549><head>PLANUDES. MOSCHOPOULOS</head> +the scholia to Eucl., Book X, the same method is applied. +Examples have been given above (vol. i, p. 63). The supposed +new method was therefore not only already known to the +scholiast, but goes back, in all probability, to Hipparchus. +<C><I>Two problems.</I></C> +<p>Two problems given at the end of the Manual of Planudes +are worth mention. The first is stated thus: ‘A certain man +finding himself at the point of death had his desk or safe +brought to him and divided his money among his sons with +the following words, “I wish to divide my money equally +between my sons: the first shall have one piece and (1/7)th of the +rest, the second 2 and (1/7)th of the remainder, the third 3 and +(1/7)th of the remainder.” At this point the father died without +getting to the end either of his money or the enumeration of +his sons. I wish to know how many sons he had and how +much money.’ The solution is given as (<I>n</I>-1)<SUP>2</SUP> for the number +of coins to be divided and (<I>n</I>-1) for the number of his sons; +or rather this is how it might be stated, for Planudes takes +<I>n</I>=7 arbitrarily. Comparing the shares of the first two we +must clearly have +<MATH>1+1/<I>n</I>(<I>x</I>-1)=2+1/<I>n</I>{<I>x</I>-(1+(<I>x</I>-1)/<I>n</I>+2)}</MATH>, +which gives <MATH><I>x</I>=(<I>n</I>-1)<SUP>2</SUP></MATH>; therefore each of (<I>n</I>-1) sons received +(<I>n</I>-1). +<p>The other problem is one which we have already met with, +that of finding two rectangles of equal perimeter such that +the area of one of them is a given multiple of the area of +the other. If <I>n</I> is the given multiple, the rectangles are +(<I>n</I><SUP>2</SUP>-1, <I>n</I><SUP>3</SUP>-<I>n</I><SUP>2</SUP>) and (<I>n</I>-1, <I>n</I><SUP>3</SUP>-<I>n</I>) respectively. Planudes +states the solution correctly, but how he obtained it is not clear. +<p>We find also in the Manual of Planudes the ‘proof by nine’ +(i.e. by casting out nines), with a statement that it was dis- +covered by the Indians and transmitted to us through the +Arabs. +<p>MANUEL MOSCHOPOULOS, a pupil and friend of Maximus +Planudes, lived apparently under the Emperor Andronicus II +(1282-1328) and perhaps under his predecessor Michael VIII +(1261-82) also. A man of wide learning, he wrote (at the +<pb n=550><head>COMMENTATORS AND BYZANTINES</head> +instance of Nicolas Rhabdas, presently to be mentioned) a +treatise on <I>magic squares</I>; he showed, that is, how the num- +bers 1, 2, 3 ... <I>n</I><SUP>2</SUP> could be placed in the <I>n</I><SUP>2</SUP> compartments of +a square, divided like a chess-board into <I>n</I><SUP>2</SUP> small squares, in such +a way that the sum of the numbers in each horizontal and +each vertical row of compartments, as well as in the rows +forming the diagonals, is always the same, namely 1/2<I>n</I>(<I>n</I><SUP>2</SUP>+1). +Moschopoulos gives rules of procedure for the cases in which +<MATH><I>n</I>=2<I>m</I>+1</MATH> and <MATH><I>n</I>=4<I>m</I></MATH> respectively, and these only, in the +treatise as we have it; he promises to give the case where +<MATH><I>n</I>=4<I>m</I>+2</MATH> also, but does not seem to have done so, as the +two manuscripts used by Tannery have after the first two cases +the words <G>te/los tou= au)tou=</G>. The treatise was translated by +De la Hire,<note><I>Mém. de l'Acad. Royale des Sciences</I>, 1705.</note> edited by S. Günther,<note><I>Vermischte Untersuchungen zur Gesch. d. Math.</I>, Leipzig, 1876.</note> and finally edited in an +improved text with translation by Tannery.<note>‘Le traité de Manuel Moschopoulos sur les carrés magiques’ in +<I>Annuaire de l'Association pour l'encouragement des études grecques</I>, xx, +1886, pp. 88-118.</note> +<p>The work of Moschopoulos was dedicated to Nicolas Arta- +vasdus, called RHABDAS, a person of some importance in the +history of Greek arithmetic. He edited, with some additions +of his own, the Manual of Planudes; this edition exists in +the Paris MS. 2428. But he is also the author of two letters +which have been edited by Tannery in the Greek text with +French translation.<note>‘Notices sur les deux lettres arithmétiques de Nicolas Rhabdas’ in +<I>Notices et extraits des manuscrits de la Bibliothèque Nationale</I>, xxxii, pt. 1, +1886, pp. 121-252.</note> The date of Rhabdas is roughly fixed +by means of a calculation of the date of Easter ‘in the current +year’ contained in one of the letters, which shows that its +date was 1341. It is remarkable that each of the two letters +has a preface which (except for the words <G>th\n dh/lwsin tw=n e)n +toi=s a)riqmoi=s zhthma/twn</G> and the name or title of the person +to whom it is addressed) copies word for word the first thir- +teen lines of the preface to Diophantus's <I>Arithmetica</I>, a piece +of plagiarism which, if it does not say much for the literary +resource of Rhabdas, may indicate that he had studied Dio- +phantus. The first of the two letters has the heading ‘A con- +cise and most clear exposition of the science of calculation +written at Byzantium of Constantine, by Nicolas Artavasdus +<pb n=551><head>MOSCHOPOULOS. RHABDAS</head> +of Smyrna, arithmetician and geometer, <G>tou= *(pabda=</G>, at the +instance of the most revered Master of Requests, Georgius +Chatzyces, and most easy for those who desire to study it.’ +A long passage, called <G>e)/kfrasis tou= daktulikou= me/trou</G>, deals +with a method of finger-notation, in which the fingers of each +hand held in different positions are made to represent num- +bers.<note>A similar description occurs in the works of the Venerable Bede +(‘De computo vel loquela digitorum’, forming chapter i of <I>De temporum +ratione</I>), where expressions are also quoted from St. Jerome (d. 420 A.D.) +as showing that he too was acquainted with the system (<I>The Miscellaneous +Works of the Venerable Bede</I>, ed. J. A. Giles, vol. vi, 1843, pp. 141-3).</note> The fingers of the left hand serve to represent all the +units and tens, those of the right all the hundreds and +thousands up to 9000; ‘for numbers above these it is neces- +sary to use writing, the hands not sufficing to represent such +numbers.’ The numbers begin with the little fingers of each +hand; if we call the thumb and the fingers after it the 1st, +2nd, 3rd, 4th, and 5th fingers in the German style, the succes- +sive signs may be thus described, premising that, where fingers +are not either bent or ‘half-closed’ (<G>klino/menoi</G>) or ‘closed’ +(<G>sustello/menoi</G>), they are supposed to be held out straight +(<G>e)kteino/menoi</G>). +<p>(<I>a</I>) <I>On the left hand</I>: +<p>for 1, half-close the 5th finger only; +<p>” 2, ” ” 4th and 5th fingers only; +<p>” 3, ” ” 3rd, 4th and 5th fingers only; +<p>” 4, ” ” 3rd and 4th fingers only; +<p>” 5, ” ” 3rd finger only; +<p>” 6, ” ” 4th ” ” +<p>” 7, close the 5th finger only; +<p>” 8, ” ” 4th and 5th fingers only; +<p>” 9, ” ” 3rd, 4th and 5th fingers only. +<p>(<I>b</I>) The same operations on the <I>right hand</I> give the <I>thou- +sands</I>, from 1000 to 9000. +<p>(<I>c</I>) <I>On the left hand</I>: +<p>for 10, apply the tip of the forefinger to the first joint of +the thumb so that the resulting figure resembles <G>s</G>; +<pb n=552><head>COMMENTATORS AND BYZANTINES</head> +<p>for 20, stretch out the forefinger straight and vertical, +keep fingers 3, 4, 5 together but separate from it +and inclined slightly to the palm; in this position +touch the forefinger with the thumb; +<p>” 30, join the tips of the forefinger and thumb; +<p>” 40, place the thumb on the knuckle of the forefinger +behind, making a figure like the letter <G>*g</G>; +<p>” 50, make a like figure with the thumb on the knuckle +of the forefinger <I>inside</I>; +<p>” 60, place the thumb inside the forefinger as for 50 and +bring the forefinger down over the thumb, touch- +ing the ball of it; +<p>” 70, rest the forefinger round the tip of the thumb, +making a curve like a spiral; +<p>” 80, fingers 3, 4, 5 being held together and inclined +at an angle to the palm, put the thumb across the +palm to touch the third phalanx of the middle +finger (3) and in this position bend the forefinger +above the first joint of the thumb; +<p>” 90, close the forefinger only as completely as possible. +<p>(<I>d</I>) The same operations on the <I>right hand</I> give the <I>hun- +dreds</I>, from 100 to 900. +<p>The first letter also contains tables for addition and sub- +traction and for multiplication and division; as these are said +to be the ‘invention of Palamedes’, we must suppose that +such tables were in use from a remote antiquity. Lastly, the +first letter contains a statement which, though applied to +particular numbers, expresses a theorem to the effect that +<MATH>(<I>a</I><SUB>0</SUB>+10<I>a</I><SUB>1</SUB>+...+10<I><SUP>m</SUP>a<SUB>m</SUB></I>)(<I>b</I><SUB>0</SUB>+10<I>b</I><SUB>1</SUB>+...+10<I><SUP>n</SUP>b<SUB>n</SUB></I>) +is not>10<SUP>(<I>m</I>+<I>n</I>+2)</SUP></MATH>, +where <I>a</I><SUB>0</SUB>, <I>a</I><SUB>1</SUB> ... <I>b</I><SUB>0</SUB>, <I>b</I><SUB>1</SUB> ... are any numbers from 0 to 9. +<p>In the second letter of Rhabdas we find simple algebraical +problems of the same sort as those of the <I>Anthologia Graeca</I> +and the Papyrus of Akhmīm. Thus there are five problems +leading to equations of the type +<MATH><I>x</I>/<I>m</I>+<I>x</I>/<I>n</I>+...=<I>a</I></MATH>. +<pb n=553><head>RHABDAS</head> +<p>Rhabdas solves the equation <MATH><I>x</I>/<I>m</I>+<I>x</I>/<I>n</I>=<I>a</I></MATH>, practically as we +should, by multiplying up to get rid of fractions, whence he +obtains <MATH><I>x</I>=<I>mna</I>/(<I>m</I>+<I>n</I>)</MATH>. Again he solves the simultaneous +equations <MATH><I>x</I>+<I>y</I>=<I>a</I>, <I>mx</I>=<I>ny</I></MATH>; also the pair of equations +<MATH><I>x</I>+<I>y</I>/<I>m</I>=<I>y</I>+<I>x</I>/<I>n</I>=<I>a</I></MATH>. +Of course, <I>m, n, a</I> ... have particular numerical values in +all cases. +<C><I>Rhabdas's Rule for approximating to the square root of +a non-square number.</I></C> +<p>We find in Rhabdas the equivalent of the Heronian formula +for the approximation to the square root of a non-square +number <MATH><I>A</I>=<I>a</I><SUP>2</SUP>+<I>b</I></MATH>, namely +<MATH><G>a</G>=<I>a</I>+<I>b</I>/(2<I>a</I>)</MATH>; +he further observes that, if <G>a</G> be an approximation by excess, +then <MATH><G>a</G><SUB>1</SUB>=<I>A</I>/<G>a</G></MATH> is an approximation by defect, and 1/2(<G>a</G>+<G>a</G><SUB>1</SUB>) +is an approximation nearer than either. This last form is of +course exactly Heron's formula <MATH><G>a</G>=1/2(<I>a</I>+<I>A</I>/<I>a</I>)</MATH>. The formula +was also known to Barlaam (presently to be mentioned), who +also indicates that the procedure can be continued indefinitely. +<p>It should here be added that there is interesting evidence +of the Greek methods of approximating to square roots in two +documents published by Heiberg in 1899.<note>‘Byzantinische Analekten’ in <I>Abh. zur Gesch. d. Math.</I> ix. Heft, 1899, +pp. 163 sqq.</note> The first of +these documents (from a manuscript of the fifteenth century +at Vienna) gives the approximate square root of certain non- +square numbers from 2 to 147 in integers and proper fractions. +The numerals are the Greek alphabetic numerals, but they are +given place-value like our numerals: thus <G>ah</G>=18, <G>adz</G>=147, +<MATH>(<G>ag</G)>/(<G>bh</G)>=13/28</MATH>, and so on: 0 is indicated by <*> or, sometimes, by. +All these square roots, such as √(21)=4 21/36, √(35)=5 11/12, +√(112)=10 49/84, and so on, can be obtained (either exactly or, +in a few cases, by neglecting or adding a small fraction in the +<pb n=554><head>COMMENTATORS AND BYZANTINES</head> +numerator of the fractional part of the root) in one or other +of the following ways: +<p>(1) by taking the nearest square to the given number <I>A</I>, +say <I>a</I><SUP>2</SUP>, and using the Heronian formulae +<MATH><G>a</G><SUB>1</SUB>=1/2(<I>a</I>+<I>A</I>/<I>a</I>), <G>a</G><SUB>2</SUB>=1/2(<G>a</G><SUB>1</SUB>+<I>A</I>/<G>a</G><SUB>1</SUB>)</MATH>, &c.; +<p>(2) by using one or other of the following approximations, +where +<MATH><I>a</I><SUP>2</SUP><<I>A</I><(<I>a</I>+1)<SUP>2</SUP>, and <I>A</I>=<I>a</I><SUP>2</SUP>+<I>b</I>=(<I>a</I>+1)<SUP>2</SUP>-<I>c</I></MATH>, +namely, <MATH><I>a</I>+<I>b</I>/(2<I>a</I>), <I>a</I>+<I>b</I>/((2<I>a</I>)+<I>b</I>/(2<I>a</I>))</MATH>, +<MATH>(<I>a</I>+1)-<I>c</I>/2(<I>a</I>+1), (<I>a</I>+1)-<I>c</I>/(2(<I>a</I>+1)-<I>c</I>/2(<I>a</I>+1))</MATH>, +or a combination of two of these with +<p>(3) the formula that, if <MATH><I>a</I>/<I>b</I><<I>c</I>/<I>d</I></MATH>, then +<MATH><I>a</I>/<I>b</I><(<I>ma</I>+<I>nc</I>)/(<I>mb</I>+<I>nd</I>)<<I>c</I>/<I>d</I></MATH>. +<p>It is clear that it is impossible to deny to the Greeks the +knowledge of these simple formulae. +<p>Three more names and we have done. +<p>IOANNES PEDIASIMUS, also called Galenus, was Keeper of the +Seal to the Patriarch of Constantinople in the reign of +Andronicus III (1328-41). Besides literary works of his, +some notes on difficult points in arithmetic and a treatise on +the duplication of the cube by him are said to exist in manu- +scripts. His <I>Geometry</I>, which was edited by Friedlein in 1866, +follows very closely the mensuration of Heron. +<p>BARLAAM, a monk of Calabria, was abbot at Constantinople +and later Bishop of Geraci in the neighbourhood of Naples; +he died in 1348. He wrote, in Greek, arithmetical demon- +strations of propositions in Euclid, Book II,<note>Edited with Latin translation by Dasypodius in 1564, and included +in Heiberg and Menge's Euclid, vol. v, <I>ad fin.</I></note> and a <I>Logistic</I> in +six Books, a laborious manual of calculation in whole numbers, +<pb n=555><head>PEDIASIMUS. BARLAAM. ARGYRUS</head> +ordinary fractions and sexagesimal fractions (printed at +Strassburg in 1592 and at Paris in 1600). Barlaam, as we +have seen, knew the Heronian formulae for finding successive +approximations to square roots, and was aware that they could +be indefinitely continued. +<p>ISAAC ARGYRUS, a monk, who lived before 1368, was one of +a number of Byzantine translators of Persian astronomical +works. In mathematics he wrote a <I>Geodaesia</I> and scholia to +the first six Books of Euclid's <I>Elements.</I> The former is con- +tained in the Paris MS. 2428 and is called ‘a method of +geodesy or the measurement of surfaces, exact and shortened’; +the introductory letter addressed to one Colybos is followed +by a compilation of extracts from the <I>Geometrica</I> and <I>Stereo- +metrica</I> of Heron. He is apparently the author of some +further additions to Rhabdas's revision of the Manual of +Planudes contained in the same manuscript. A short tract +of his ‘On the discovery of the square roots of non-rational +square numbers’ is mentioned as contained in two other manu- +scripts at Venice and Rome respectively (Codd. Marcianus Gr. +333 and Vaticanus Gr. 1058), where it is followed by a table +of the square roots of all numbers from 1 to 102 in sexa- +gesimal fractions (e.g. √2=1 24′ 51″ 48‴, √3=1 43′ 56″ 0‴).<note>Heiberg, ‘Byzantinische Analekten’, in <I>Abh. zur Gesch. d. Math.</I> ix, +pp. 169-70.</note> +<pb> +<C><B>APPENDIX</B></C> +<C><I>On Archimedes's proof of the subtangent-property of +a spiral.</I></C> +<p>THE section of the treatise <I>On Spirals</I> from Prop. 3 to +Prop. 20 is an elaborate series of propositions leading up +to the proof of the fundamental property of the subtangent +corresponding to the tangent at any point on any turn of the +spiral. Libri, doubtless with this series of propositions in +mind, remarks (<I>Histoire des sciences mathématiques en Italie,</I> +i, p. 31) that ‘Après vingt siècles de travaux et de décou- +vertes, les intelligences les plus puissantes viennent encore +échouer contre la synthèse difficile du <I>Traité des Spirales</I> +d'Archimède.’ There is no foundation for this statement, +which seems to be a too hasty generalization from a dictum, +apparently of Fontenelle, in the <I>Histoire de l'Académie des +Sciences pour l'année 1704</I> (p. 42 of the edition of 1722), +who says of the proofs of Archimedes in the work <I>On +Spirals</I>: ‘Elles sont si longues, et si difficiles à embrasser, +que, comme on l'a pû voir dans la Préface de l'Analyse des +Infiniment petits, M. Bouillaud a avoué qu'il ne les avoit +jamais bien entendues, et que Viète les a injustement soup- +çonnées de paralogisme, parce qu'il n'avoit pû non plus +parvenir à les bien entendre. Mais toutes les preuves qu'on +peut donner de leur difficulté et de leur obscurité tournent +à la gloire d'Archimède; car quelle vigueur d'esprit, quelle +quantité de vûes différentes, quelle opiniâtreté de travail n'a- +t-il pas fallu pour lier et pour disposer un raisonnement que +quelques-uns de nos plus grands géomètres ne peuvent suivre, +tout lié et tout disposé qu'il est?’ +<p>P. Tannery has observed<note><I>Bulletin +des sciences mathématiques,</I> 1895, Part i, pp. 265-71.</note> that, as a matter of fact, no +mathematicians of real authority who have applied or ex- +tended Archimedes's methods (such men as Huygens, Pascal, +Roberval and Fermat, who alone could have expressed an +opinion worth having), have ever complained of the +<pb n=557><head>APPENDIX</head> +‘obscurity’ of Archimedes; while, as regards Vieta, he has +shown that the statement quoted is based on an entire mis- +apprehension, and that, so far from suspecting a fallacy in +Archimedes's proofs, Vieta made a special study of the treatise +<I>On Spirals</I> and had the greatest admiration for that work. +<p>But, as in many cases in Greek geometry where the analy- +sis is omitted or even (as Wallis was tempted to suppose) of +set purpose hidden, the reading of the completed synthetical +proof leaves a certain impression of mystery; for there is +nothing in it to show <I>why</I> Archimedes should have taken +precisely this line of argument, or how he evolved it. It is +a fact that, as Pappus said, the subtangent-property can be +established by purely ‘plane’ methods, without recourse to +a ‘solid’ <G>neu=sis</G> (whether actually solved or merely assumed +capable of being solved). If, then, Archimedes chose the more +difficult method which we actually find him employing, it is +scarcely possible to assign any reason except his definite +predilection for the form of proof by <I>reductio ad absurdum</I> +based ultimately on his famous ‘Lemma’ or Axiom. +<p>It seems worth while to re-examine the whole question of +the discovery and proof of the property, and to see how +Archimedes's argument compares with an easier ‘plane’ proof +suggested by the figures of some of the very propositions +proved by Archimedes in the treatise. +<p>In the first place, we may be sure that the property was +not discovered by the steps leading to the proof as it stands. +I cannot but think that Archimedes divined the result by an +argument corresponding to our use of the differential calculus +for determining tangents. He must have considered the +instantaneous direction of the motion of the point <I>P</I> describ- +ing the spiral, using for this purpose the parallelogram of +velocities. The motion of <I>P</I> is compounded of two motions, +one along <I>OP</I> and the other at right angles to it. Comparing +the distances traversed in an instant of time in the two direc- +tions, we see that, corresponding to a small increase in the +radius vector <I>r,</I> we have a small distance traversed perpen- +dicularly to it, a tiny arc of a circle of radius <I>r</I> subtended by +the angle representing the simultaneous small increase of the +angle <G>q</G> (<I>AOP</I>). Now <I>r</I> has a constant ratio to <G>q</G> which we call +<I>a</I> (when <G>q</G> is the circular measure of the angle <G>q</G>). Consequently +<pb n=558><head>APPENDIX</head> +the small increases of <I>r</I> and <G>q</G> are in that same ratio. There- +fore what we call the tangent of the angle <I>OPT</I> is <I>r/a,</I> +i.e. <I>OT/r</I> = <I>r/a</I>; and <I>OT</I> = <I>r</I><SUP>2</SUP>/<I>a,</I> or <I>r</I><G>q</G>, that is, the arc of a +circle of radius <I>r</I> subtended by the angle <G>q</G>. +<p>To <I>prove</I> this result Archimedes would doubtless begin by +an <I>analysis</I> of the following sort. Having drawn <I>OT</I> perpen- +dicular to <I>OP</I> and of length equal to the arc <I>ASP,</I> he had to +prove that the straight line joining <I>P</I> to <I>T</I> is the tangent +at <I>P.</I> He would evidently take the line of trying to show +that, if <I>any</I> radius vector to the spiral is drawn, as <I>OQ′,</I> on +either side of <I>OP, Q′</I> is always on the side of <I>TP</I> towards <I>O,</I> +or, if <I>OQ′</I> meets <I>TP</I> in <I>F, OQ′</I> is always less than <I>OF.</I> Suppose +<FIG> +that in the above figure <I>OR′</I> is any radius vector between <I>OP</I> +and <I>OS</I> on the ‘backward’ side of <I>OP,</I> and that <I>OR′</I> meets the +circle with radius <I>OP</I> in <I>R,</I> the tangent to it at <I>P</I> in <I>G,</I> the +spiral in <I>R′,</I> and <I>TP</I> in <I>F′.</I> We have to prove that <I>R, R′</I> lie +on opposite sides of <I>F′,</I> i.e. that <I>RR′</I> > <I>RF′</I>; and again, sup- +posing that <I>any</I> radius vector <I>OQ′</I> on the ‘forward’ side of +<I>OP</I> meets the circle with radius <I>OP</I> in <I>Q,</I> the spiral in <I>Q′</I> and +<I>TP</I> produced in <I>F,</I> we have to prove that <I>QQ′</I> < <I>QF.</I> +<p>Archimedes then had to prove that +(1) <I>F′R : RO</I> < <I>RR′ : RO,</I> and +(2) <I>FQ : QO</I> > <I>QQ′ : QO.</I> +<p>Now (1) is equivalent to +<I>F′R : RO</I> < (arc <I>RP</I>) : (arc <I>ASP</I>), since <I>RO</I> = <I>PO.</I> +<pb n=559><head>APPENDIX</head> +<p>But (arc <I>ASP</I>) = <I>OT,</I> by hypothesis; +therefore it was necessary to prove, <I>alternando,</I> that +(3) <I>F′R</I> : (arc <I>RP</I>) < <I>RO : OT,</I> or <I>PO : OT,</I> +i.e. < <I>PM : MO,</I> where <I>OM</I> is perpendicular to <I>SP.</I> +<p>Similarly, in order to satisfy (2), it was necessary to +prove that +(4) <I>FQ</I> : (arc <I>PQ</I>) > <I>PM : MO.</I> +<p>Now, as a matter of fact, (3) is <I>a fortiori</I> satisfied if +<I>F′R : (chord RP)</I> < <I>PM : MO</I>; +but in the case of (4) we cannot substitute the <I>chord PQ</I> for +the arc <I>PQ,</I> and we have to substitute <I>PG′,</I> where <I>G′</I> is the +<FIG> +<CAP>FIG. 1.</CAP> +point in which the tangent at <I>P</I> to +the circle meets <I>OQ</I> produced; for +of course <I>PG′</I> > (arc <I>PQ</I>), so that (4) +is <I>a fortiori</I> satisfied if +<I>FQ : PG′</I> > <I>PM : MO.</I> +<p>It is remarkable that Archimedes +uses for his proof of the two cases Prop. +8 and Prop. 7 respectively, and makes +no use of Props 6 and 9, whereas +the above argument points precisely to the use of the figures +of the two latter propositions only. +<p>For in the figure of Prop: 6 (Fig. 1), if <I>OFP</I> is any radius +cutting <I>AB</I> in <I>F,</I> and if <I>PB</I> produced cuts <I>OT,</I> the parallel to +<I>AB</I> through <I>O,</I> in <I>H,</I> it is obvious, by parallels, that +<I>PF</I> : (chord <I>PB</I>) = <I>OP : PH.</I> +<p>Also <I>PH</I> becomes greater the farther <I>P</I> moves from <I>B</I> +towards <I>A,</I> so that the ratio <I>PF : PB</I> diminishes continually, +while it is always less than <I>OB : BT</I> (where <I>BT</I> is the tangent +at <I>B</I> and meets <I>OH</I> in <I>T</I>), i.e. always less than <I>BM : MO.</I> +<p>Hence the relation (3) is always satisfied for any point <I>R′</I> of +the spiral on the ‘backward’ side of <I>P.</I> +<p>But (3) is equivalent to (1), from which it follows that <I>F′R</I> +is always less than <I>RR′,</I> so that <I>R′</I> always lies on the side +of <I>TP</I> towards <I>O.</I> +<pb n=560><head>APPENDIX</head> +<p>Next, for the point <I>Q′</I> on the ‘forward’ side of the spiral +from <I>P,</I> suppose that in the figure of Prop. 9 or Prop. 7 (Fig. 2) +any radius <I>OP</I> of the circle meets <I>AB produced</I> in <I>F,</I> and +<FIG> +<CAP>FIG. 2.</CAP> +the tangent at <I>B</I> in <I>G</I>; and draw <I>BPH, BGT</I> meeting <I>OT,</I> the +parallel through <I>O</I> to <I>AB,</I> in <I>H, T.</I> +<table> +<tr> +<td>Then</td> +<td><I>PF : BG</I></td> +<td>> <I>FG : BG,</I> since <I>PF</I> > <I>FG,</I></td> +</tr> +<tr> +<td></td> +<td></td> +<td>> <I>OG : GT,</I> by parallels,</td> +</tr> +<tr> +<td></td> +<td></td> +<td>> <I>OB : BT, a fortiori,</I></td> +</tr> +<tr> +<td></td> +<td></td> +<td>> <I>BM : MO</I>;</td> +</tr> +</table> +and obviously, as <I>P</I> moves away from <I>B</I> towards <I>OT,</I> i.e. as <I>G</I> +moves away from <I>B</I> along <I>BT,</I> the ratio <I>OG : GT</I> increases +continually, while, as shown, <I>PF : BG</I> is always > <I>BM : MO,</I> +and, <I>a fortiori,</I> +<I>PF</I> : (arc <I>PB</I>) > <I>BM : MO.</I> +<p>That is, (4) is always satisfied for any point <I>Q′</I> of the spiral +‘forward’ of <I>P,</I> so that (2) is also satisfied, and <I>QQ′</I> is always +less than <I>QF.</I> +<p>It will be observed that no <G>neu=sis</G>, and nothing beyond +‘plane’ methods, is required in the above proof, and Pappus's +criticism of Archimedes's proof is therefore justified. +<p>Let us now consider for a moment what Archimedes actually +does. In Prop. 8, which he uses to prove our proposition in +the ‘backward’ case (<I>R′, R, F′</I>), he shows that, if <I>PO : OV</I> +is any ratio whatever less than <I>PO : OT</I> or <I>PM : MO,</I> we can +find points <I>F′, G</I> corresponding to any ratio <I>PO : OV′</I> where +<I>OT</I> < <I>OV′</I> < <I>OV,</I> i.e. we can find a point <I>F′</I> corresponding to +a ratio still nearer to <I>PO : OT</I> than <I>PO : OV</I> is. This proves +that the ratio <I>RF′ : PG,</I> while it is always less than <I>PM : MO,</I> +<pb n=561><head>APPENDIX</head> +approaches that ratio without limit as <I>R</I> approach<*>s <I>P.</I> But +the proof does not enable us to say that <I>RF′ : (chord PR),</I> +which is > <I>RF′ : PG,</I> is also always less than <I>PM : MO.</I> At +first sight, therefore, it would seem that the proof must fail. +Not so, however; Archimedes is nevertheless able to prove +that, if <I>PV</I> and not <I>PT</I> is the tangent at <I>P</I> to the spiral, an +absurdity follows. For his proof establishes that, if <I>PV</I> is the +tangent and <I>OF′</I> is drawn as in the proposition, then +<I>F′O : RO</I> < <I>OR′ : OP,</I> +or <I>F′O</I> < <I>OR′,</I> ‘which is impossible’. Why this is impossible +does not appear in Props. 18 and 20, but it follows from the +argument in Prop. 13, which proves that a tangent to the spiral +cannot meet the curve again, and in fact that the spiral is +everywhere concave towards the origin. +<p>Similar remarks apply to the proof by Archimedes of the +impossibility of the other alternative supposition (that the tan- +gent at <I>P</I> meets <I>OT</I> at a point <I>U</I> nearer to <I>O</I> than <I>T</I> is). +<p>Archimedes's proof is therefore in both parts perfectly valid, +in spite of any appearances to the contrary. The only draw- +back that can be urged seems to be that, if we assume the +tangent to cut <I>OT</I> at a point <I>very near</I> to <I>T</I> on either side, +Archimedes's construction brings us perilously near to infini- +tesimals, and the proof may appear to hang, as it were, on +a thread, albeit a thread strong enough to carry it. But it is +remarkable that he should have elaborated such a difficult +proof by means of Props. 7, 8 (including the ‘solid’ <G>neu=sis</G> of +Prop. 8), when the figures of Props. 6 and 7 (or 9) themselves +suggest the direct proof above given, which is independent of +any <G>neu=sis</G>. +<p>P. Tannery,<note>Tannery, +<I>Mémoires scientifiques,</I> i, 1912, pp. 300-16.</note> in a paper on Pappus's criticism of the proof as +unnecessarily involving ‘solid’ methods, has given another +proof of the subtangent-property based on ‘plane’ methods +only; but I prefer the method which I have given above +because it corresponds more closely to the preliminary proposi- +tions actually given by Archimedes. +<pb> +<pb> +<C><B>INDEX OF GREEK WORDS</B></C> +<C>[The pages are those of the first volume except where otherwise stated.]</C> +<p><G>a)/bac, a)ba/kion</G> 47. +<p><G>a)gewme/trhtos, -on</G>: <G>a)gewme/trhtos mh- +dei\s ei)si/tw</G> (Plato) iii. 355. +<p><G>a)dieci/thtos, -on</G>, that cannot be gone +through, i.e. infinite 343. +<p><G>a)du/natos, -on</G> ii. 462: <G>a)pagwgh\ ei)s +a)du/naton</G>, &c. 372. +<p><G>ai)/thma</G>, postulate 373. +<p><G>a)kousmatikoi/</G> 11. +<p><G>a)/logos, -on</G>, irrational 84, 90: <G>peri\ +a)lo/gwn grammw=n kai\ nastw=n</G> (Demo- +critus) 156-7, 181: <G>a)/logoi w(/sper +grammai/</G> (Plato) 157. +<p><G>a)na/lhmma</G> ii. 287. +<p><G>a)na/logon</G>, proportional: used as ad- +jective 85. +<p><G>a)naluo/menos</G> (<G>to/pos</G>), <I>Treasury of +Analysis</I> 421-2, ii. 399, 400, ii. +426. +<p><G>a)na/palin</G>, inversely 385: <G>a)na/palin +lu/sis</G> ii. 400. +<p><G>a)nastre/yanti</G> (<G>a)nastre/fw</G>), <I>conver- +tendo</I> 386. +<p><G>a)nastrofh/</G>, conversion <I>ib.</I> +<p><G>a)nastrofiko\s</G> (<G>to/pos</G>), a class of locus +ii. 185. +<p><G>*)avaforiko/s</G> by Hypsicles 419, ii. 213. +<p><G>a)neli/ttein</G> ii. 244. +<p><G>a)/cwn</G>, axis 341. +<p><G>a)o/ristos, -on</G>, undefined: <G>plh=qos mo- +na/dwn a)o/riston</G> (= unknown, <I>x</I>) +94, ii. 456: <G>e)n a)ori/stw|</G> ii. 489, 491. +<p><G>a)pagwgh/</G>, <I>reduction</I> 372: <G>a)p. ei)s +a)du/naton</G>, <I>reductio ad absurdum</I> +372. +<p><G>a)po/deicis</G>, <I>proof</I> 370, ii. 533. +<p><G>a)pokatastatiko/s, -h/, -o/n</G>, recurring 108. +<p><G>a)po/stasis</G>, distance or <I>dimension</I> +305 <I>n.,</I> or <I>interval</I> 306 <I>n.</I> +<p><G>a)/rbhlos</G>, ‘shoemaker's knife’ ii. 23, +ii. 101-2, ii. 371-7. +<p><G>a)riqmhtikh/</G>, theory of numbers, opp. +to <G>logistikh/</G> 13-16. +<p><G>a)riqmhtiko/s, -h/, -o/n</G>: <G>a)riqmhtikh\ ei)sa- +gwgh/</G> of Nicomachus 97. +<p><G>a)riqmo/s</G>, number: definitions of +‘number’ 69-70: in Diophantus, +used for unknown quantity (<I>x</I>) +94, ii. 456. +<p><G>a)riqmosto/n</G>: reciprocal of <G>a)riqmo/s</G> +(=<I>x</I>) in Diophantus ii. 458. +<p><G>a(rpedona/ptai</G>, ‘rope-stretchers’ 121- +2, 178. +<p><G>a)/rrhtos, -on</G>, irrational 157. +<p><G>a)rtia/kis a)/rtios</G>, <I>even-times-even</I> 71, +with Neo-Pythagoreans =2<SUP><I>u</I></SUP>, 72. +<p><G>a)rtia/kis peritto/s</G>, <I>even-times-odd</I> 72. +<p><G>a)rtiope/rittos</G>, <I>even-odd,</I> restricted +by Neo-Pythagoreans to form +2(2<I>m</I> + 1), 72. +<p><G>a)/rtios, -a, on</G>, <I>even</I> 70. +<p><G>*)arxai/</G>, a lost work of Archimedes +ii. 81. +<p><G>*)astroqesi/ai</G> of Eratosthenes ii. 109. +<p><G>a)strola/bon o)/rganon</G> of Hipparchus +ii. 256. +<p><G>a)su/mmetros, -on</G>, incommensurable +157. +<p><G>a)su/mptwtos, -on</G>, non-secant ii. 227. +<p><G>a)su/nqetos, -on</G>, incomposite 72. +<p><G>a)/tomos, -on</G>, indivisible 181: Aristo- +telian <G>peri\ a)to/mwn grammw=n</G> 157, +346-8. +<p><G>a)/topos, -on</G>, absurd ii. 462. +<p><G>au)ca/nein</G>: <G>tri\s au)chqei/s</G> (Plato) 306-7. +<p><G>au)/ch, tri/th</G>, 297: <G>ku/bwn au)/ch</G>, 297. +<p><G>au)/chsis</G> 305-6 <I>n.</I> +<p><G>au)tomatopoihtikh/</G> ii. 308. +<p><G>a(yi/s</G>, segment of circle less than a +semicircle ii. 314. +<p><G>*baroulko/s</G> of Heron ii. 309, ii. 346-7. +<p><G>*belopoii+ka/</G> of Heron 18, ii. 298, ii. +302, ii. 308-9. +<p><G>bia/zein</G>: <G>bebiasme/nos</G>, forced or un- +natural ii. 362. +<pb n=564><head>INDEX OF GREEK WORDS</head> +<p><G>bwmi/skos</G>, ‘little altar’, properly a +wedge-shaped solid ii. 319, ii. 333: +measurement of(Heron), ii. 332-3: +(=<G>sfhni/skos</G>) of a certain kind +of solid number 107, ii. 240, ii. +315. +<p><G>gewdaisi/a</G>=mensuration 16. +<p><G>*gewmetrou/mena</G> of Heron ii. 318, ii. +453. +<p><G>glwxi/s</G> (arrow-head), Pythagorean +name for angle 166. +<p><G>gnwmonikh/</G> 18. +<p><G>gnw/mwn</G>, <I>gnomon</I>, q.v.: <G>kata\ gnw/mona</G> +=perpendicular 78, 175. +<p><G>gnw/rimos, -on</G>, known: <G>gnw/rimon</G>, an +alternative term for <G>dedome/non</G>, +<I>given</I> ii. 537. +<p><G>gnwri/mws</G>, ‘in the recognized manner’ +ii. 79. +<p><G>gra/mma</G>, ‘figure’ or proposition, of +theorem of Eucl. I. 47, 144. +<p><G>grammh/</G>: <G>dia\</G> or <G>e)k tw=n grammw=n</G> of +theoretical proof ii. 257, 258. +<p><G>grammiko/s, -h/, -o/n</G>, linear: used of +prime numbers 73: <G>grammikai\ +e)pista/seis</G>, ‘Considerations on +Curves’, by Demetrius ii. 359: +<G>grammikw=s</G>, graphically 93. +<p><G>gra/fein</G>, to draw or write on 159, +173: also to <I>prove</I> 203 <I>n.</I>, 339. +<p><G>dedome/nos, -h, -on</G>, <I>given</I>: senses of, +ii. 537-8. +<p><G>deiknu/nai</G>, to prove 328. +<p><G>dei=n</G>: <G>dei= dh/</G> 371. +<p><G>deu/teros</G>, <I>secondary</I>: of composite +numbers 72: <G>deute/ra muria/s</G> (= +10,000<SUP>2</SUP>) 40. +<p><G>diabh/ths</G>, compasses 308, ii. 540. +<p><G>diairei=n</G>: <G>dielo/nti</G>, <I>separando</I> or <I>divi- +dendo</I> (in transformation of ratios) +386. +<p><G>diai/resis</G>: <G>lo/gou</G>, <I>separation</I> of a +ratio 386: <G>peri\ diaire/sewn bibli/on</G>, +<I>On divisions</I> (<I>of figures</I>), by +Euclid 425. +<p><G>dia/stasis</G>, dimension: <G>peri\ diasta/- +sews</G>, a work of Ptolemy ii. 295. +<p><G>dia/sthma</G>, interval 215: distance +239. +<p><G>di/aulos</G>, ‘race-course’: representa- +tions of square and oblong num- +bers as sums of terms 114. +<p><G>dido/nai</G>: <G>dedome/non</G>, <I>given</I>, senses ii. +537-8. +<p><G>diecodiko\s</G> (<G>to/pos</G>), a species of locus +ii. 185. +<p><G>dii+sta/nai</G>: <G>e)f) e(\n diestw/s</G>, extended +one way ii. 428. +<p><G>diko/louros, -on</G>, twice-truncated 107. +<p><G>di/optra</G>, dioptra, q.v. +<p><G>dioptrikh/</G> 18. +<p><G>diori/zein</G>: <G>diwrisme/nh tomh/</G>, <I>Deter- +minate Section</I>, by Apollonius +ii. 180. +<p><G>diorismo/s</G>, definition, delimitation: +two senses (1) a constituent part +of a theorem or problem 370, +(2) a statement of conditions of +possibility of a problem 303, 319- +20, 371, 377, 395, 396, 428, ii. 45- +6, ii. 129-32, ii. 168, ii. 230. +<p><G>diploi+so/ths</G>, double-equation (Dio- +phantus) ii. 468. +<p><G>diplou=s, -h=, -ou=n</G>: <G>diplh= muria/s</G> = +10,000<SUP>2</SUP> (Apollonius) 40: <G>diplh= +i)so/ths, diplh= i)/swsis</G>, double-equa- +tion (Diophantus) ii. 468. +<p><G>doki/s</G>, <I>beam</I>, a class of solid number +107, ii. 240. +<p><G>doko/s</G> = <G>doki/s</G> ii. 315. +<p><G>draxmh/</G>, sign for, 31, 49, 50. +<p><G>du/namis</G>: incommensurable side of +square containing a non-square +number of units of area 203-4: +square or square root 209 <I>n.</I>, +297: square of unknown quantity +(= <I>x</I><SUP>2</SUP>) (Diophantus) ii. 457-8: +<G>duna/mei</G>, ‘in square’ 187, 308: +<G>tetraplh= du/namis</G> = eighth power +(Egypt) ii. 546; <I>power</I> in +mechanics 445. +<p><G>dunamodu/namis</G>, square - square = +fourth power (Heron) ii. 458: +fourth power of unknown (Dio- +phantus) ii. 458, ii. 546. +<p><G>dunamo/kubos</G>, square-cube, = fifth +power of unknown (Diophantus) +ii. 458. +<p><G>dunamosto/n, dunamodunamosto/n</G>, &c., +reciprocals of powers of unknown +(Diophantus) ii. 458. +<p><G>du/nasqai</G>, to be equivalent ‘in square’ +to, i. e. to be the side of a square +equal to (a given area): <G>duname/nh</G> +305-6 <I>n.</I> +<p><G>dunasteuome/nh</G>, opp. to <G>duname/nh</G> +305-6 <I>n.</I> +<p><G>ei)=dos</G>, ‘figure’ of a conic ii. +139: ‘species’ = particular power +<pb n=565><head>INDEX OF GREEK WORDS</head> +of unknown, or term, in an equa- +tion (Diophantus) ii. 460. +<p><G>ei(=s, mi/a, e(/n</G>, one: <G>e(/na plei/w</G>, ‘several +ones’ (definition of ‘number’) +70. +<p><G>ei)shgei=sqai</G>, to introduce or explain +213. +<p><G>e)/kqesis</G>, <I>setting-out</I> 370, ii. 533. +<p><G>*)ekpeta/smata</G> of Democritus 178, +181. +<p><G>e(kth/moros</G> (<G>ku/klos</G>) ii. 288. +<p><G>e)/lleiyis</G>, <I>falling-short</I> (in application +of areas), name given to <I>ellipse</I> by +Apollonius 150, ii. 138. +<p><G>e)lliph/s, -e/s</G>, <I>defective</I> (of numbers), +contrasted with <I>perfect</I> 74, 101: +<G>*y e)llipe\s ka/tw neu=on</G> ii. 459. +<p><G>e)nalla/c</G>, alternately (in proportions) +385. +<p><G>e)/nnoia</G>, notion: <G>koinai\ e)/nnoiai</G>, com- +mon notions = axioms 336. +<p><G>e)/nstasis</G>, <I>objection</I> 372, ii. 311, ii. 533. +<p><G>e)/ntasis</G>, bulging out 6. +<p><G>e)celigmo/s</G> ii. 234. +<p><G>e)ch/ghsis</G>, elucidation ii. 223, ii. +231-2. +<p><G>e(chkosto/n</G>, or <G>prw=ton e(c.</G>, a 60th (= +a <I>minute</I>), <G>deu/teron e(c.</G>, a <I>second</I>, +&c. 45. +<p><G>e)pa/nqhma</G>, (‘bloom’) of Thymaridas: +a system of linear equations solved +94. +<p><G>e)pafh/</G>, contact: <G>*)epafai/</G>, <I>Contacts</I> +or <I>Tangencies</I>, by Apollonius ii. +181. +<p><G>e)pi/</G>, on: <G>to\ shmei=on e)f) w(=|</G> (or <G>ou(=</G>) <I>K,</I> +archaic for ‘the point <I>K</I>’ 199: +<G>h( e)f) h(=|</G> <I>AB</I>, ‘the straight line +<I>AB</I>’ <I>ib.</I> +<p><G>e)pimerh/s</G>, <I>superpartiens</I>, += ratio 1 +(―(<I>m</I> + <I>n</I>)), 102. +<p><G>e)pimo/rios</G>, <I>superparticularis</I> = ratio +of form (<I>n</I> + 1)/<I>n</I>, 90, 101: <G>e)pi- +mo/rion dia/sthma</G> 215. +<p><G>e)pipedometrika/</G> ii. 453. +<p><G>e)pishmasi/ai</G>, weather indications +177 <I>n.</I>, ii. 234. +<p><G>e)pi/tritos</G> = ratio 4/3, 101: <G>e)pi/tritos +puqmh/n</G> (Plato) 306-7. +<p><G>e)/sxatos</G>: <G>ta\ e)/sxata</G>, extremities +293. +<p><G>e(teromh/khs, -es</G>, <I>oblong</I>; of numbers +of form <I>m</I>(<I>m</I> + 1), 82, 108. +<p><G>eu)qugrammiko\s</G> (<G>a)riqmo/s</G>) = <I>prime</I> 72. +<p><G>e)fektiko/s</G>, a class of locus ii. 185, +ii. 193. +<p><G>e)fo/dion</G>, <I>Method</I> ii. 246. +<p><G>zugo/n</G>, lever or balance: <G>peri\ zugw=n</G>, +a work of Archimedes ii. 23-4, +ii. 351. +<p><G>h(mio/lios, -a, -on</G>, ratio of 3/2, 101. +<p><G>h(miwbe/lion</G>, (1/2).obol, sign for, 31, 49, 50. +<p><G>qaumatopoii+kh/</G> 18. +<p><G>qeologou/mena a)riqmhtikh=s</G> 97. +<p><G>qe/sis</G>, position: <G>para\ qe/sei|</G> (<I>sc.</I> <G>dedo- +me/nhn</G>), parallel to a straight line +given in position ii. 193: <G>pro\s +qe/sei eu)qei/ais</G>, on straight lines +given in position ii. 426. +<p><G>qureo/s</G>, <I>shield</I>, old name for ellipse +439, ii. 111, ii. 125. +<p><G>i)/llesqai</G>: <G>i)llome/nhn</G> used by Plato of +the earth 314-15. +<p><G>i)sa/kis i)/sos</G>, equal an equal number +of times, or equal multiplied by +equal 204. +<p><G>i)so/metros, -on</G>, of equal contour: +<G>peri\ i)some/trwn sxhma/twn</G>, by Zeno- +dorus ii. 207, ii. 390. +<p><G>i)so/pleuros, -on</G>, equilateral: of +square number (Plato) 204. +<p><G>i)sorropi/a</G>, equilibrium : <G>peri\ i)sorro- +piw=n</G>, work by Archimedes ii. 24, +ii. 351. +<p><G>i)/sos</G>, equal: <G>di) i)/sou</G>, <I>ex aequali</I> (in +proportions) 386 : <G>di) i)/sou e)n te- +taragme/nh| a)nalogi/a|</G> 386. +<p><G>i)so/ths</G> or <G>i)/swsis</G>, equation ii. 468. +<p><G>i(stori/a</G>, inquiry, Pythagoras's name +for geometry 166. +<p><G>i)sxu/s</G>, power (in mechanics) 445. +<p><G>kampth/r</G>, turning-point in race- +course 114. +<p><G>kampu/los, -h, -on</G>, curved 249, 341. +<p><G>kanonikh/</G>, <I>Canonic</I>, q.v. +<p><G>kanw/n</G>, ruler 239: <I>Table</I> (astron.), +<G>*proxei/rwn kano/nwn dia/tasis kai\ +yhfofori/a</G>, work by Ptolemy ii. +293: <I>canon</I> (in music), v. <G>*katatomh/. +katagra/fein</G>: <I>to inscribe in</I> or <I>on</I> (c. +gen.) 131. +<p><G>*kata/logoi</G>, work by Eratosthenes ii. +108. +<p><G>kataskeua/zein</G> 193 <I>n.</I> +<p><G>kataskeuh/</G>, <I>construction</I> (constituent +part of proposition) 370, ii. 533. +<pb n=566><head>INDEX OF GREEK WORDS</head> +<p><G>*katasterismoi/</G>, work by Eratosthe- +nes ii. 108. +<p><G>*katatomh\ kano/nos</G>, <I>Sectio canonis,</I> +attributed to Euclid 17, 444. +<p><G>katono/macis tw=n a)riqmw=n</G>, naming of +numbers (Archimedes) ii. 23. +<p><G>katoptrikh/</G>, theory of mirrors 18. +<p><G>kentrobarika/</G>, problems on centre of +gravity ii. 24, ii. 350. +<p><G>ke/ntron</G>, centre: <G>h( e)k tou= ke/ntrou</G> = +radius 381. +<p><G>keratoeidh\s</G> (<G>gwni/a</G>) 178, 382. +<p><G>*khri/a</G> of Sporus 234. +<p><G>kla/ein</G>, inflect: <G>kekla/sqai</G> 337. +<p><G>kogxoeidh\s grammh/</G>, <I>conchoid</I> 238. +<p><G>koilogw/nion</G> ii. 211. +<p><G>ko/louros, -on</G>, truncated ii. 333: (of +pyramidal number) 107. +<p><G>ko/skinon</G>, <I>sieve</I> (of Eratosthenes) 16, +100, ii. 105. +<p><G>koxloeidh\s grammh/</G>, <I>cochloid</I> 238. +<p><G>kubo/kubos</G>, cube-cube, = sixth power +of unknown (Diophantus) ii. 458. +<p><G>kubokubosto/n</G>, reciprocal of <G>kubo/- +kubos</G> ii. 458. +<p><G>ku/bos</G>, cube: <G>ku/bwn au)/ch</G> (Plato) +297: cube of unknown (Dio- +phantus) ii. 458: <G>ku/bos e)celikto/s</G> += ninth power of unknown +(Egyptian) ii. 546. +<p><G>kuklikh\ qewri/a</G>, <I>De motu circulari,</I> +by Cleomedes ii. 235. +<p><G>kukliko/s, -h/, -o/n</G>, <I>circular,</I> used of +square numbers ending in 5 or 6, +108. +<p><G>lei/pein</G>: forms used to express <I>minus,</I> +and sign for (Diophantus), ii. 459. +<p><G>lei=yis</G>, <I>wanting</I> (Diophantus): <G>lei/yei</G> += <I>minus</I> ii. 459. +<p><G>le/cis</G>: <G>kata\ le/cin</G>, word for word +183. +<p><G>lepto/n</G>, a <I>fraction</I> (Heron) 43: = a +<I>minute</I> (Ptolemy) 45. +<p><G>lh=mma</G>, <I>lemma</I> 373. +<p><G>logismo/s</G>, calculation 13. +<p><G>logistikh/</G>, art of calculation, opp. +to <G>a)riqmhtikh/</G> 13-16, 53. +<p><G>lo/gos</G>, ratio: <G>lo/gou a)potomh/</G>, <I>sectio +rationis,</I> by Apollonius ii. 175. +<p><G>maqh/mata</G>, subjects of instruction +10-11: term first appropriated +to mathematics by Pythagoreans +11: <G>peri\ tw=n maqhma/twn</G>, a work +by Protagoras 179. +<p><G>maqhmatiko/s, -h/, -o/n</G>: <G>maqhmatikoi/</G> in +Pythagorean school, opp. to +<G>a)kousmatikoi/</G> 11: <G>*maqhmatikh\ su/n- +tacis</G> of Ptolemy ii. 273-4: <G>maqh- +matika/, ta\</G> (Plato) 288. +<p><G>meqo/rion</G>, boundary ii. 449. +<p><G>mei/ouron proeskarifeume/non</G> (Heron), +curtailed and pared in front (cf. +scarify), of a long, narrow, tri- +angular prism (Heib.) ii. 319. +<p><G>me/ros</G>: <G>me/rh</G>, <I>parts</I> (= proper frac- +tion) dist. from <G>me/ros</G> (aliquot +part) 42 (cf. p. 294). +<p><G>mesola/bon</G>, <I>mean-finder</I> (of Erato- +sthenes) ii. 104. ii. 359. +<p><G>mete/wros, -on</G>: <G>peri\ metew/rwn</G>, work +by Posidonius ii. 219, ii. 231-2. +<p><G>metewroskopikh/</G> 18. +<p><G>*metrh/seis</G>, <I>Mensurae</I> (Heronian) ii. +319. +<p><G>mh=kos</G>, length: used by Plato of side +of square containing a square +number of units of area 204. +<p><G>mhli/ths</G> (<G>a)riqmo/s</G>), term for problems +about numbers of apples (e.g.) 14, +ii. 442. +<p><G>*mikro\s a)stronomou/menos</G> (<G>to/pos</G>), <I>Little +Astronomy</I> ii. 273. +<p><G>mna=</G>, mina (= 1000 drachmae): <G>*m</G> +stands for, 31. +<p><G>moi=ra</G>, fraction: 1/360th of circum- +ference or a <I>degree</I> 45, 61: <G>moi=ra +topikh/, xronikh/</G> (in Hypsicles) ii. +214. +<p><G>mona/s</G>, monad or unit 43: definitions +of, 69: <G>mona/dwn su/sthma</G> = number, +69: <G>deuterwdoume/nh mona/s</G> = 10, +<G>triwdoume/rh m.</G> = 100, &c. (Iambl.) +114: <G>mona\s qe/sin e)/xousa</G> = point +69, 283. +<p><G>mo/rion</G>, part or fraction: <G>mori/ou</G> or +<G>e)n mori/w|</G> = divided by (Diophan- +tus) 44. +<p><G>muria/s</G> (with or without <G>prw/th</G> or +<G>a(plh=</G>) myriad (10,000), <G>m. deute/ra</G> +or <G>diplh=</G> 10,000<SUP>2</SUP>, &c. 40. +<p><G>nasto/n</G> (solid?) 156, 178. +<p><G>neu/ein</G>, to verge (towards) 196, 239, +337, ii. 65. +<p><G>neu=sis</G>, <I>inclinatio</I> or ‘verging’, a +type of problem 235-41, 260, ii. +199, ii. 385: <G>neu/seis</G> in Archi- +medes ii 65-8: two books of +<G>neu/seis</G> by Apollonius ii. 189-92 +ii. 401, ii. 412-13. +<pb n=567><head>INDEX OF GREEK WORDS</head> +<p><G>nu/ssa</G>, goal or end of race-course +114. +<p><G>o)bole/s</G>, obol: sign for, 31, 49, 50. +<p><G>*)olumpioni=kai</G>, work by Eratosthenes +ii. 109. +<p><G>o)/nuc</G>, a wedge-shaped figure ii. 319, +ii. 333. +<p><G>o)rganopoii+kh/</G> 18. +<p><G>o)/rqios, -a, -on</G>, right or perpendi- +cular: <G>o)rqi/a pleura/</G>, <I>latus rectum</I> +ii. 139: <G>o)rqi/a dia/metros</G>, ‘erect +diameter’, in double hyperbola, +ii. 134. +<p><G>o(ri/zein</G>: <G>w(risme/nos</G>, defined, i. e. de- +terminate 94, 340. +<p><G>o(ri/zwn</G> (<G>ku/klos</G>), dividing circle: +<I>horizon</I> (Eucl.) 351. 352. +<p><G>o(/ros</G>, (1) definition 373: (2) limit +or boundary 293: (3) <I>term</I> (in a +proportion) 306 <I>n.</I> +<p><G>ou)demi/a</G> or <G>ou)de/n</G>, sign for (O), 39, 45. +<p><G>pa= bw= kai\ kinw= ta\n ga=n</G>, saying of +Archimedes ii. 18. +<p><G>par' h(\n du/nantai</G> (<G>ai( katago/menai tetag- +me/nws</G>), expression for <I>parameter</I> +of ordinates ii. 139. +<p><G>parabolh/</G>, application: <G>p. tw=n xwri/wn</G>, +application of areas 150: <G>ta\ e)k +th=s parabolh=s gino/mena shmei=a</G>, the +foci of a central conic, ii. 156: +<I>parabola</I> (the conic) 150, ii. 138. +<p><G>*paradocogra/foi</G> ii. 541. +<p><G>para/docos grammh/</G>, paradoxical curve +(of Menelaus) ii. 260-1, ii. 360. +<p><G>para/phgma</G> 177, ii. 234. +<p><G>paraspa=n</G>, to pull awry: <G>parespa- +sme/nos</G> ii. 398. +<p><G>pariso/ths</G>, nearness to equality, ap- +proximation: <G>pariso/thtos a)gwgh/</G> +(Diophantus) ii. 477, ii. 500. +<p><G>pe/lekus</G>, axe-shaped figure ii. 315. +<p><G>pempa/zein</G>, to ‘five’ (= count) 26. +<p><G>pe/ntaqlos</G> 176, ii. 104. +<p><G>perai/nousa poso/ths</G> = unit, 69. +<p><G>pe/ras</G>, limit or extremity 293: +limiting surface 166: <G>pe/ras sug- +klei=on</G>, definition of figure ii. 221. +<p><G>perissa/rtios</G>, <I>odd-even</I>: with Neo- +Pythagoreans is of form +2<SUP><I>n</I>+1</SUP> (2<I>m</I>+1), 72. +<p><G>perisso/s, -h/, -o/n</G>, <I>odd,</I> q.v. +<p><G>pettei/a</G> 19. +<p><G>phli/kos, -h, -on</G>, how great (of mag- +nitude) 12. +<p><G>phliko/ths</G>, size 384. +<p><G>pla/gios, -a, -on</G>, transverse: <G>plagi/a +dia/metros</G> or <G>pleura/</G> ii. 139. +<p><G>plasmatiko/s, -o/n</G>, (easily) formable +ii. 487. +<p><G>*platwniko/s</G>, a work by Eratosthenes +ii. 104. +<p><G>plh=qos</G>, multitude: <G>plh=qos e(/n</G> = unit, +69: <G>plh=qos w(risme/non</G> = number, +70: <G>plh=qos mora/dwn a)o/riston</G>, def. +of unknown ‘quantity’ 94, ii. +456. +<p><G>plinqi/s</G>, a <I>brick,</I> a solid number of +a certain form 107, ii. 240, ii. +315. +<p><G>pollaplasiepimerh/s</G>, <I>multiplex super- +partiens,</I> = ratio of form +<I>p</I> + <I>m</I>/(<I>m</I>+<I>n</I>), 103. +<p><G>pollaplasiepimo/rios</G>, <I>multiplex su- +perparticularis,</I> = ratio of form +<I>m</I> + 1/<I>n,</I> 103. +<p><G>pollapla/sios, -a, -on</G>, multiple 101. +<p><G>polu/spastos</G>, a compound pulley ii. +18. +<p><G>po/rimos, -on</G> (<G>pori/zein</G>), procurable: +one sense of <G>dedomenos</G> ii. 538. +<p><G>po/risma</G>, porism: (1) = corollary, +(2) a certain type of proposition +372-3, ii. 533. +<p><G>poso/n</G>, quantity, of number, 12. +<p><G>poso/ths</G>, quantity 69, 70: number +defined as <G>poso/thtos xu/ma e)k mona/- +dwn sugkei/menon</G> 70. +<p><G>promh/khs</G>, <I>prolate</I> (= oblong) 203: +but distinguished from <G>e(teromh/khs</G> +83, 108. +<p><G>prosagw/gion</G> 309. +<p><G>pro/tasis</G> = <I>enunciation</I> 370, ii. +533. +<p><G>prw=tos</G>, <I>prime</I> 72. +<p><G>ptw=sis</G>, <I>case</I> 372. +<p><G>puqmh/n</G>, base; = digit 55-7, 115-17: +<G>e)pi/tritos piqmh/n</G> 306-7. +<p><G>purami/s</G>, pyramid 126. +<p><G>pu/reion, pu/rion</G>, burning mirror: +<G>peri\ purei/wn</G>, work by Diocles +264, ii. 200; <G>peri\ tou= puri/ou</G>, by +Apollonius ii. 194. +<p><G>r(hto/s, -h/, -o/n</G>, rational: used in sense +of ‘given’ ii. 537. +<p><G>r(oph/</G>: <G>peri\ r(opw=n</G>, a mechanical +work by Ptolemy ii. 295. +<pb n=568><head>INDEX OF GREEK WORDS</head> +<p><G>sa/linon</G> of Archimedes ii. 23, ii. +103. +<p><G>sh/kwma</G> 49. +<p><G>ska/fh</G>, a form of sun-dial ii. 1, ii. 4. +<p><G>skhnografikh/</G>, scene-painting 18, ii. +224. +<p><G>*sofi/a</G>, nickname of Democritus 176. +<p><G>spei=ra</G>, <I>spire</I> or <I>tore</I> ii. 117: varie- +ties of (<G>diexh/s, sunexh/s, e)mpepleg- +me/nh</G> or <G>e)palla/ttouta</G>), ii. 204. +<p><G>sta/qmh</G>, plumb-line 78, 309. +<p><G>stath/r</G>, sign for, 31. +<p><G>stereometri/a</G>, solid geometry 12-13. +<p><G>stereometrou/mena</G> ii. 453. +<p><G>sthli/s</G>, column, a class of solid +number. 107. +<p><G>stigmh/</G>, point 69: <G>stigmh\ a)/qetos</G> = +unit, 69. +<p><G>stoixeiw/ths, -o(</G>, the writer of Ele- +ments (<G>stoixei=on</G>), used of Euclid +357. +<p><G>stroggu/los, -on</G>, round or circular +293. +<p><G>sumpe/rasma</G>, <I>conclusion</I> (of proposi- +tion) 370, ii. 533. +<p><G>su/nqesis</G> (<G>lo/gou</G>), composition (of a +ratio) 385. +<p><G>su/ntacis</G>, collection: <G>*mega/lh su/n- +tacis</G> of Ptolemy 348, called +<G>*maqhmatikh\ su/ntacis</G> ii. 273. +<p><G>suntiqe/nai</G>: <G>sunqe/nti</G> = <I>componendo</I> +(in proportion) 385. +<p><G>su/stasis</G>, construction 151, 158. +<p><G>sfairiko/s, -h/, -o/n</G>, spherical: used of +cube numbers ending in 5 or 6, +107-8. +<p><G>sfhki/skos</G>, <I>stake,</I> a form of solid +number, 107. +<p><G>sfhni/skos</G>, <I>wedge,</I> a solid of a certain +form, measurement of, ii. 332-3: +a solid number, 107, ii. 315, ii. +319. +<p><G>sxe/sis</G>, relation 384. +<p><G>sxhmatopoiei=n</G>, to form a figure ii. +226. +<p><G>*ta/lanton</G>, sign for (T), 31, 50. +<p><G>tara/ssein</G>: (<G>di' i)/sou</G>) <G>e)n tetaragme/nh| +a)nalogi/a|</G>, <I>in disturbed proportion</I> +386. +<p><G>ta/ssein</G>: <G>tetagme/non</G>, <I>assigned = da- +tum</I> ii. 192, ii. 537: <G>ai( katago/menai +tetagme/nws</G> (<G>eu)qei=ai</G>), (straight +lines) drawn <I>ordinate-wise</I> = or- +dinates ii. 139: <G>tetagme/nws kat- +h=xqai</G> ii. 134. +<p><G>ta/xos</G>, speed: <G>peri\ taxw=n</G>, work by +Eudoxus 329. +<p><G>te/leios, -a, -on</G>, perfect: <G>te/leios a)riq- +mo/s</G> 74, 101. +<p><G>tetarthmo/rion</G>, 1/4 of obol, sign for, 31, +49, 50. +<p><G>tetragwni/zein</G>, to square: <G>h( tetragw- +ni/zousa</G> (<G>grammh/</G>), the <I>quadratrix</I> +225, ii. 359. +<p><G>tetragwnismo/s</G>, <I>squaring</I> 173. +<p><G>tetraktu/s</G> 75, 99 <I>n.,</I> 313, ii. 241. +<p><G>tetraplh= du/namis</G> = 8th power of +unknown (Egyptian term) ii. +546. +<p><G>tmh=ma</G>, segment: used of lunes as +well as segments of circles 184: +segments or sectors 187-9: <G>tmh/- +mata</G> = 1/360th parts of circum- +ference and 1/120th parts of +diameter of circle (Ptolemy) 45. +<p><G>tomeu/s</G>, shoemaker's knife, term for +<I>sector</I> of circle 381. +<p><G>tomh/</G>, section: <G>ta\ peri\ th\n tomh/n</G> +(Proclus) 324-5. +<p><G>to/pos</G>, locus: classifications of loci +218-19, ii. 185: <G>to/poi pro\s gram- +mai=s, to/poi pro\s e)pifanei/ais</G> (<G>-a|</G>) +218-19, 439: <G>to/poi pro\s meso/thtas</G> +ii. 105: <G>to/pos a)naluo/menos</G>, <I>Trea- +sury of Analysis,</I> q. v. +<p><G>to/rnos</G>, circle-drawer 78, 308. +<p><G>tri/gwnos a)riqmo/s</G>, triangular number, +15-16. +<p><G>triko/louros</G>, thrice-truncated 107. +<p><G>tri/pleuron</G>, <I>three-side,</I> Menelaus's +term for spherical triangle ii. +262. +<p><G>triw/bolon</G>, sign for, 49. +<p><G>u(/dria w(roskopei=a</G>, water-clocks ii. +309. +<p><G>u(/parcis</G>, forthcoming: <I>positive</I> term, +dist. from negative (<G>lei=yis</G>) ii. +459. +<p><G>u(pepimerh/s</G>, <I>subsuperpartiens,</I> reci- +procal of <G>e)pimerh/s</G> 102. +<p><G>u(pepimo/rios</G>, <I>subsuperparticularis,</I> re- +ciprocal of <G>e)pimo/rios</G> 101. +<p><G>u(perbolh/</G>, <I>exceeding</I> (in application +of areas): name given to <I>hyper- +bola</I> 150, ii. 138. +<p><G>u(perte/leios, u(pertelh/s</G>, <I>over-perfect</I> +(number) 74, 100. +<p><G>(*upoqe/seis tw=n planwme/nwn</G>, work by +Ptolemy ii. 293. +<p><G>u(popollapla/sios, u(popollaplasiepi-</G> +<pb n=569><head>INDEX OF GREEK WORDS</head> +<G>merh/s, u(popollaplasiepimo/rios</G>, &c. +101-3. +<p><G>u(potei/nein</G>, subtend 193 <I>n.</I> +<p><G>u(/splhc</G>, starting-point (of race- +course) 114. +<p><G>*fa/seis a)planw=n a)ste/rwn</G>, work by +Ptolemy, ii. 293. +<p><G>fiali/ths</G> (<G>a)riqmo/s</G>), (number) of bowls +(in simple algebraical problems) +14, ii. 442. +<p><G>*filokali/a</G>, by Geminus ii. 223. +<p><G>xalkou=s</G> ((1/8)th of obol), sign for, 31: +48, 50. +<p><G>xei/r</G>, <I>manus,</I> in sense of number of +men 27. +<p><G>xeiroba/llistra</G> ii. 309. +<p><G>xroia/</G>, colour or skin: Pythagorean +name for surface 166, 293. +<p><G>*xronografi/ai</G>, work by Eratosthenes +ii. 109. +<p><G>xrw=ma</G>, colour (in relation to sur- +face) 293. +<p><G>xwri/on</G>, area 300 <I>n.</I>: <G>xwri/ou a)potomh/</G>, +<I>sectio spatii,</I> by Apollonius ii. +179. +<p><G>*yhfofori/a kat) *)indou/s</G> ii. 546. +<p><G>*)wkuto/kion</G> of Apollonius 234, ii. 194, +ii. 253. +<pb> +<C><B>ENGLISH INDEX</B> +[The pages are those of the first volume except where otherwise stated.]</C> +<p>Abacus 46-8. +<p>‘Abdelmelik al-Shīrāzī ii. 128. +<p>Abraham Echellensis ii. 127. +<p>Abū Bekr Mu&hdot;. b. al-&Hdot;asan al- +Karkhī, <I>see</I> al-Karkhī. +<p>Abū 'l Fat&hdot; al-I⋅fahānī ii. 127. +<p>Abū 'l Wafā al-Būzjānī ii. 328, ii. +450, ii. 453. +<p>Abū Nasr Man⋅ūr ii. 262. +<p><I>Achilles</I> of Zeno 275-6, 278-80. +<p>Adam, James, 305-7, 313. +<p>Addition in Greek notation 52. +<p>Adrastus ii. 241, 243, 244. +<p>Aëtius 158-9, 163, ii. 2. +<p>‘Aganis’: attempt to prove paral- +lel-postulate 358, ii. 228-30. +<p>Agatharchus 174. +<p>Ahmes (Papyrus Rhind) 125, 130, +ii. 441. +<p>Akhmīm, Papyrus of, ii. 543-5. +<p>Albertus Pius ii. 26. +<p>Al-Chāzinī ii. 260-1. +<p>Alexander the ‘Aetolian’ ii. 242. +<p>Alexander Aphrodisiensis 184, 185, +186, 222, 223, ii. 223, ii. 231. +<p>Alexeieff, ii. 324-5 <I>n.</I> +<p><I>Al-Fakhri,</I> by al-Karkhī 109, ii. +449-50. +<p>Algebra: beginnings in Egypt ii. +440: <I>hau</I>-calculations ii. 440-1: +Pythagorean, 91-7: <I>epanthema</I> of +Thymaridas 94-6. +<p>Algebra, geometrical, 150-4: ap- +plication of areas (q.v.) 150-3: +scope of geometrical algebra +153-4: method of proportion <I>ib.</I> +<p>Al-&Hdot;ajjāj, translator of Euclid, +362: of Ptolemy ii. 274. +<p>Alhazen, problem of, ii. 294. +<p><I>Al-Kāfī</I> of al-Karkhī 111. +<p>Al-Karkhī: on sum of +1<SUP>3</SUP>+2<SUP>3</SUP> +...+ <I>n</I><SUP>3</SUP> +109-10, 111, ii. 51, ii. 449. +<p>Allman, G. J. 134, 183. +<p><I>Almagest</I> ii. 274. +<p>Alphabet, Greek: derived from +Phoenician, 31-2: Milesian, 33-4: +<I>quasi</I>-numerical use of alphabet, +35-6 <I>n.</I> +<p>Alphabetic numerals 31-40, 42-4. +<p>Amasis 4, 129. +<p>Amenemhat I 122, III 122. +<p>Ameristus 140, 141, 171. +<p>Amyclas (better Amyntas) 320-1. +<p>Amyntas 320-1. +<p><I>Analemma</I> of Ptolemy ii. 286-92: +of Diodorus ii. 287. +<p>Analysis: already used by Pytha- +goreans 168: supposed invention +by Plato 291-2: absent from +Euclid's <I>Elements</I> 371-2: defined +by Pappus ii. 400. +<p>Anatolius 11, 14, 97, ii. 448, ii. 545-6. +<p>Anaxagoras: explanation of eclipses +7, 162, 172: moon borrows light +from sun 138, 172, ii. 244: cen- +trifugal force and centripetal +tendency 172-3: geometry 170: +tried to square circle 173, 220: +on perspective 174: in <I>Erastae</I> +22, 174. +<p>Anaximander 67, 177: introduced +<I>gnomon</I> 78, 139, 140: astronomy +139, ii. 244: distances of sun and +moon 139: first map of inhabited +earth <I>ib.</I> +<p>Anaximenes ii. 244. +<p>Anchor-ring, <I>see</I> Tore. +<p>Anderson, Alex., ii. 190. +<p>Angelo Poliziano ii. 26. +<p>Angle ‘<I>of</I> a segment’ and ‘<I>of</I> a +semicircle’ 179: ‘angle of con- +tact’ 178-9, ii. 202. +<p>Anharmonic property, of arcs of +great circles ii. 269-70: of straight +lines ii. 270, ii. 420-1. +<pb n=571><head>ENGLISH INDEX</head> +<p>Anthemius of Tralles 243, ii. 194, +ii. 200-3, ii. 518, ii. 540, ii. +541-3. +<p>Antiphon 184, 219, 221-2, 224, +271. +<p><I>Āpastamba-Śulba-Sūtra</I> 145-6. +<p>Apelt, E. F. 330. +<p>Apelt, O. 181 <I>n.,</I> 182. +<p><I>Apices</I> 47. +<p>Apollodorus, author of <I>Chronica,</I> +176. +<p>Apollodorus <G>o( logistiko/s</G>: distich of, +131, 133, 134, 144, 145. +<p>Apollonius of Perga ii. 1, ii. 126. +<p>Arithmetic: <G>w)kuto/kion</G> 234, ii. +194, ii. 253 (approximation to +<G>p</G>, <I>ib.</I>), ‘tetrads’ 40, continued +multiplications 54-7. +<p>Astronomy ii. 195-6: A. and +Tycho Brahe 317, ii. 196: on +epicycles and eccentrics ii. 195-6, +ii. 243: trigonometry ii. 253. +<p><I>Conics</I> ii. 126-75: text ii. 126- +8, Arabic translations ii. 127, +profaces ii. 128-32, characteris- +tics ii. 132-3: conics obtained +from oblique cone ii. 134-8, +prime property equivalent to +Cartesian equation (oblique axes) +ii. 139, new names, <I>parabola,</I> &c. +150, 167, ii. 138, transformation +of coordinates ii. 141-7, tangents +ii. 140-1, asymptotes ii. 148-9, +rectangles under segments of in- +tersecting chords ii. 152-3, har- +monic properties ii. 154-5, focal +properties (central conics) ii. 156- +7, normals as maxima and mini- +ma ii. 159-67, construction of +normals ii. 166-7, number of +normals through point ii. 163-4, +propositions giving evolute ii. +164-5. +<p><I>On contacts</I> ii. 181-5 (lemmas +to, ii. 416-17), three-circle pro- +blem ii. 182-5. +<p><I>Sectio rationis</I> ii. 175-9 (lemmas +to, ii. 404-5). +<p><I>Sectio spatii</I> ii. 179-80, ii. 337, +ii. 339. +<p><I>Determinate section</I> ii. 180-1 +(lemmas to, ii. 405-12). +<p>Comparison of dodecahedron +and icosahedron 419-20, ii. 192. +<p>Duplication of cube 262-3, ii. +194. +<p>‘General treatise’ ii. 192-3, ii. +253: on Book I of Euclid 358. +<p><G>neu/seis</G> ii. 68, ii. 189-92 (lemmas +to, ii. 412-16), rhombus-problem +ii. 190-2, square - problem ii. +412-13. +<p><I>Plane Loci</I> ii. 185-9 (lemmas to, +ii. 417-19). +<p><I>On cochlias</I> 232, ii. 193, ‘sister +of cochloid’ 225, 231-2, <I>On irra- +tionals</I> ii. 193, <I>On the burning- +mirror</I> ii. 194, ii. 200-1. +<p>Application of areas 150-3: method +attributed to Pythagoras 150, +equivalent to solution of general +quadratic 150-2, 394-6. +<p>Approximations to √2 (by means +of ‘side-’ and ‘diameter-’ num- +bers) 91-3, (Indian) 146: to √3 +(Ptolemy) 45, 62-3, (Archimedes) +ii. 51-2: to <G>p</G> 232-5, ii. 194, ii. +253: to surds (Heron) ii. 323-6, +cf. ii. 547-9, ii. 553-4: to cube +root (Heron) ii. 341-2. +<p>Apuleius of Madaura 97, 99. +<p>Archibald, R. C. 425 <I>n.</I> +<p>Archimedes 3, 52, 54, 180, 199, 202, +203 <I>n.,</I> 213, 217, 224-5, 229, 234, +272, ii. 1. +<p>Traditions ii. 16-17, engines ii. +17, mechanics ii. 18, general +estimate ii. 19-20. +<p>Works: character of, ii. 20-2, +works extant ii. 22-3, lost ii. 23- +5, 103; text ii. 25-7, MSS. ii. 26, +editions ii. 27: <I>The Method</I> ii. 20, +21, 22, 27-34, ii. 246, ii. 317-18: +<I>On the Sphere and Cylinder</I> ii. 34- +50: <I>Measurement of a circle</I> ii. 50- +6, ii. 253: <I>On Conoids and Sphe- +roids</I> ii. 56-64: <I>On Spirals</I> 230-1, +ii. 64-75 (cf. ii. 377-9), ii. 556-61: +<I>Sand-reckoner</I> ii. 81-5: <I>Quadra- +ture of Parabola</I> ii. 85-91: me- +chanical works, titles ii. 23-4, +<I>Plane equilibriums</I> ii. 75-81: <I>On +Floating Bodies</I> ii. 91-7, problem +of crown ii. 92-4: <I>Liber assump- +torum</I> ii. 101-3: Cattle-problem +14, 15, ii. 23, ii. 97-8, ii. 447: +<I>Catoptrica</I> 444, ii. 24. +<p>Arithmetic: octads 40-1, frac- +tions 42, value of <G>p</G> 232-3, 234, +ii. 50-6: approximations to √3 +ii. 51-2. +<p>Astronomy ii. 17-18, sphere- +<pb n=572><head>ENGLISH INDEX</head> +making ii. 18, on Aristarchus's +hypothesis ii. 3-4. +<p>Conics, propositions in, 438-9, +ii. 122-6. +<p>Cubic equation solved by conics +ii. 45-6. +<p>On Democritus 180, 327, +equality of angles of incidence +and reflection ii. 353-4, integral +calculus anticipated ii. 41-2, 61, +62-3, 74, 89-90: Lemma or Axiom +of A. 326-8, ii. 35: <G>neu/seis</G> in, ii. +65-8 (Pappus on, ii. 68): on semi- +regular solids ii. 98-101: triangle, +area in terms of sides ii. 103: +trisection of any angle 240-1. +<p>Archytas 2, 170, 212-16, ii. 1: on +<G>maqh/mata</G> 11, on <I>logistic</I> 14, on 1 +as odd-even 71: on means 85, 86: +no mean proportional between <I>n</I> +and <I>n</I> + 1, 90, 215: on music 214: +mechanics 213: solution of pro- +blem of two mean proportionals +214, 219, 245, 246-9, 334, ii. 261. +<p>Argyrus, Isaac, 224 <I>n.,</I> ii. 555. +<p>Aristaeus: comparison of five regu- +lar solids 420: <I>Solid Loci</I> (conics) +438, ii. 116, 118-19. +<p>Aristaeus of Croton 86. +<p>Aristarchus of Samos 43, 139, ii. 1- +15, ii. 251: date ii. 2: <G>ska/fh</G> of, +ii. 1: anticipated Copernicus ii. +2-3: other hypotheses ii. 3, 4: +treatise <I>On sizes and distances of +Sun and Moon</I> ii. 1, 3, 4-15, tri- +gonometrical purpose ii. 5: num- +bers in, 39: fractions in, 43. +<p>Aristonophus, vase of, 162. +<p>Aristophanes 48, 161, 220. +<p>Aristotelian treatise on indivisible +lines 157, 346-8. +<p>Aristotherus 348. +<p>Aristotle 5, 120, 121: on origin of +science 8: on mathematical sub- +jects 16-17: on first principles, de- +finitions, postulates, axioms 336-8. +<p>Arithmetic: reckoning by tens +26-7, why 1 is odd-even 71: 2 +even and prime 73: on Pytha- +goreans and numbers 67-9: on +the gnomon 77-8, 83. +<p>Astronomy: Pythagorean sys- +tem 164-5, on hypothesis of con- +centric spheres 329, 335, ii. 244, +on Plato's view about the earth +314-15. +<p>On the continuous and infinite +342-3: proof of incommensura- +bility of diagonal 91: on principle +of exhaustion 340: on Zeno's +paradoxes 272, 275-7, 278-9, 282: +on Hippocrates 22: encomium on +Democritus 176. +<p>Geometry: illustrations from, +335, 336, 338-40, on parallels +339, proofs differing from Euclid's +338-9, propositions not in Euclid +340, on quadratures 184-5, 221, +223, 224 <I>n.,</I> 271, on quadrature +by lunes (Hippocrates) 184-5, +198-9: on Plato and regular +solids 159: curves and solids in +A. 341. +<p>Mechanics 344-6, 445-6: paral- +lelogram of velocities 346: ‘Aris- +totle's wheel’ ii. 347-8. +<p>Aristoxenus 24 <I>n.,</I> 66. +<p>Arithmetic (1) = theory of numbers +(opp. to <G>logistikh/</G>) 13-16: early +‘Elements of Arithmetic’ 90, 216: +systematic treatises, Nicomachus +<I>Introd. Ar.</I> 97-112, Theon of +Smyrna 112-3, Iam blichus, Comm. +on Nicomachus 113-15, Domninus +ii. 538. (2) Practical arithmetic: +originated with Phoenicians 120- +1, in primary education 19-20. +<p>Arithmetic mean, defined 85. +<p><I>Arithmetica</I> of Diophantus 15-16, +ii. 449-514. +<p>Arithmetical operations: <I>see</I> Addi- +tion, Subtraction, &c. +<p><I>Arrow</I> of Zeno 276, 280-1. +<p>Āryabha⃛⃛a 234. +<p>Asclepius of Tralles 99. +<p>Astronomy in elementary education +19: as secondary subject 20-1. +<p>Athelhard of Bath, first translator +of Euclid 362-4. +<p>Athenaeus 144, 145. +<p>Athenaeus of Cyzicus 320-1. +<p>‘Attic’ (or ‘Herodianic’) numerals +30-1. +<p>August, E. F. 299, 302, 361. +<p>Autolycus of Pitane 348: works +<I>On the moving Sphere</I> 348-52, <I>On +Risings and Settings</I> 352-3: rela- +tion to Euclid 351-2. +<p>Auverus, C. ii. 26. +<p>Axioms: Aristotle on, 336: =<I>Com- +mon Notions</I> in Euclid 376: Axiom +of Archimedes 326-8, ii. 35. +<pb n=573><head>ENGLISH INDEX</head> +<p>Babylonians: civilization of, 8, 9: +system of numerals 28-9: sexa- +gesimal fractions 29: ‘perfect +proportion’ 86. +<p>Bachet, editor of Diophantus ii. +454-5, ii. 480. +<p>Bacon, Roger: on Euclid 367-8. +<p>Baillet, J. ii. 543. +<p>Baldi, B. ii. 308. +<p>Barlaam ii. 324 <I>n.,</I> ii. 554-5. +<p>Barocius ii. 545. +<p>Barrow, I., edition of Euclid, 369- +70: on Book V 384. +<p>Bathycles 142. +<p>Bāudhāyana Ś. S. 146. +<p>Baynard, D. ii. 128. +<p>Benecke, A. 298, 302-3. +<p>Benedetti, G. B. 344, 446. +<p>Bertrand, J. ii. 324 <I>n.</I> +<p>Bessarion ii. 27. +<p>Besthorn, R. O. 362, ii. 310. +<p>Billingsley, Sir H. 369. +<p>Björnbo, A. A. 197 <I>n.,</I> 363, ii. 262. +<p>Blass, C. 298. +<p>Blass, F. 182. +<p>Boeckh, A. 50, 78, 315. +<p>Boëtius 37, 47, 90: translation of +Euclid 359. +<p>Boissonade ii. 538. +<p>Bombelli, Rafael, ii. 454. +<p>Borchardt, L. 125, 127. +<p>Borelli, G. A. ii. 127. +<p>Bouillaud (Bullialdus) ii. 238, ii. +556. +<p>Braunmühl, A. von, ii. 268-9 <I>n.,</I> ii. +288, ii. 291. +<p>Breton (de Champ), P. 436, ii. 360. +<p>Bretschneider, C. A. 149, 183, 324-5, +ii. 539. +<p>Brochard, V. 276-7, 279 <I>n.,</I> 282. +<p>Brougham, Lord, 436. +<p>Brugsch, H. K. 124. +<p>Bryson 219, 223-5. +<p>Burnet, J. 203 <I>n.,</I> 285, 314-15. +<p>Butcher, S. H. 299, 300. +<p>Buzengeiger ii. 324 <I>n.</I> +<p>Cajori, F. 283 <I>n.</I> +<p>Calculation, practical: the abacus +46-8, addition and subtraction +52, multiplication (i) Egyptian +52-3 (Russian ? 53 <I>n.</I>), (ii) Greek +53-8, division 58-60, extraction +of square root 60-3, of cube root +63-4, ii. 341-2. +<p>Callimachus 141-2. +<p>Callippus: Great Year 177: system of +concentric spheres 329, 335, ii. 244. +<p>Cambyses 5. +<p>Camerarius, Joachim, ii. 274. +<p>Camerer, J. G. ii. 360. +<p>Campanus, translator of Euclid +363-4. +<p><I>Canonic</I> = theory of musical inter- +vals 17. +<p>Cantor, G. 279. +<p>Cantor, M. 37-8, 123, 127, 131, 135, +182, ii. 203, ii. 207. +<p>Carpus of Antioch 225, 232, ii. +359. +<p><I>Case</I> (<G>ptw=sis</G>) 372, ii. 533. +<p>Cassini ii. 206. +<p>Casting out nines 115-17, ii. 549. +<p><I>Catoptric,</I> theory of mirrors 18. +<p><I>Catoptrica</I>: treatises by Euclid (?) +442, by Theon (?) 444, by Archi- +medes 444, and Heron 444, ii. 294, +ii. 310, ii. 352-4. +<p>Cattle-problem of Archimedes 14, +15, ii. 23, ii. 97-8, ii. 447. +<p>Cavalieri, B. 180, ii. 20. +<p>Censorinus 177. +<p>Centre of gravity: definitions ii. +302, ii. 350-1, ii. 430. +<p><I>Ceria Aristotelica</I> ii. 531. +<p>Chalcidius ii. 242, 244. +<p>Chaldaeans: measurement of angles +by <I>ells</I> ii. 215-16: order of planets +ii. 242. +<p>Charmandrus ii. 359. +<p>Chasles, M. ii. 19, 20: on Porisms +435-7, ii. 419. +<p>Chords, Tables of, 45, ii. 257, ii. +259-60. +<p>Chrysippus 179: definition of unit 69. +<p>Cicero 144, 359, ii. 17, 19. +<p>Circle: division into degrees ii. 214- +15: squaring of, 173, 220-35, +Antiphon 221-2, Bryson 223-4, +by Archimedes's spiral 225, 230- +1, Nicomedes, Dinostratus, and +quadratrix 225-9, Apollonius +225, Carpus 225; approximations +to <G>p</G> 124, 232-5, ii. 194, ii. 253, +ii. 545. +<p>Cissoid of Diocles 264-6. +<p>Clausen, Th. 200. +<p>Cleanthes ii. 2. +<p>Cleomedes: ‘paradoxical’ eclipse 6: +<I>De motu circulari</I> ii. 235-8, 244. +<p>Cleonides 444. +<p><I>Cochlias</I> 232, ii. 193. +<pb n=574><head>ENGLISH INDEX</head> +<p><I>Cochloids</I> 238-40: ‘sister of coch- +loid’ 225, 231-2. +<p>Coins and weights, notation for, 31. +<p>Columella ii. 303. +<p>Commandinus, F., translator of +Euclid, 365, 425, Apollonius ii. +127, <I>Analemma</I> of Ptolemy ii. +287, <I>Planispherium</I> ii. 292, Heron's +<I>Pneumatica</I> ii. 308, Pappus ii. 360, +Serenus ii. 519. +<p>Conchoid of Nicomedes 238-40. +<p><I>Conclusion</I> 370, ii. 533. +<p>Cone: Democritus on, 179-80, ii. +110: volume of, 176, 180, 217, +327, 413, ii. 21, ii. 332: volume +of frustum ii. 334: division of +frustum in given ratio ii. 340-3. +<p>Conic sections: discovered by Me- +naechmus 252-3, ii. 110-16: Eu- +clid's <I>Conics</I> and Aristaeus's <I>Solid +Loci</I> 438, ii. 116-19: propositions +included in Euclid's <I>Conics</I> ii. +121-2 (focus-directrix property +243-4, ii. 119-21), conics in Archi- +medes ii. 122-6: names due to +Apollonius 150, ii. 138: Apollo- +nius's <I>Conics</I> ii. 126-75: conics +in <I>Fragmentum Bobiense</I> ii. 200- +203: in Anthemius ii. 541-3. +<p>Conon of Samos ii. 16, ii. 359. +<p><I>Construction</I> 370, ii. 533. +<p><I>Conversion</I> of ratio (<I>convertendo</I>) 386. +<p>Cook-Wilson, J. 300 <I>n.,</I> ii. 370. +<p>Counter-earth 164. +<p>Croesus 4, 129. +<p>Ctesibius 213: relation to Philon +and Heron ii. 298-302. +<p>Cube: called ‘geometrical har- +mony’ (Philolaus) 85-6. +<p>Cube, duplication of: history of +problem 244-6: reduction by Hip- +pocrates to problem of two mean +proportionals 2, 183, 200, 245: +solutions, by Archytas 246-9, Eu- +doxus 249-51, Menaechmus 251- +5, ‘Plato’ 255-8, Eratosthenes +258-60, Nicomedes 260-2, Apol- +lonius, Philon, Heron 262-4, Dio- +cles 264-6, Sporus and Pappus +266-8: approximation by plane +method 268-70. +<p>Cube root, extraction of, 63-4: +Heron's case ii. 341-2. +<p>Cubic equations, solved by conics, +237-8, ii. 45-6, ii. 46; particular +case in Diophantus ii. 465, ii. 512. +<p>Curtze, M. 75 <I>n.,</I> ii. 309. +<p>Cyrus 129. +<p><I>Dactylus,</I> 1/24th of ell, ii. 216. +<p>Damastes of Sigeum 177. +<p>Damianus ii. 294. +<p>Darius-vase 48-9. +<p>D'Armagnac, G. ii. 26. +<p>Dasypodius ii. 554 <I>n.</I> +<p>De la Hire ii. 550. +<p><I>De levi et ponderoso</I> 445-6. +<p>Decagon inscribed in circle, side of, +416: area of, ii. 328. +<p>Dee, John, 369, 425. +<p>Definitions: Pythagorean 166: in +Plato 289, 292-4: Aristotle on, +337: in Euclid 373: <I>Definitions</I> +of Heron, ii. 314-16. +<p>Demetrius of Alexandria ii. 260, ii. +359. +<p>Democritus of Abdera 12, 119, 121, +182: date 176, travels 177: Aris- +totle's encomium 176: list of +works (1) astronomical 177, (2) +mathematical 178: on irrational +lines and solids 156-7, 181: on +angle of contact 178-9: on cir- +cular sections of cone 179-80, ii. +110: first discovered volume of +cone and pyramid 176, 180, 217, +ii. 21: atoms mathematically di- +visible <I>ad inf.</I> 181: <G>*)ekpeta/smata</G> +178, 181: on perspective 174: on +Great Year 177. +<p>Dercyllides ii. 244. +<p>Descartes 75 <I>n.,</I> 279. +<p>Dicaearchus ii. 242. +<p><I>Dichotomy</I> of Zeno 275, 278 80. +<p>Diels, H., 142 <I>n.,</I> 176, 178, 184, 188. +<p>Digamma: from Phoenician Vau +32: signs for, <I>ib.</I> +<p><I>Digit</I> 27. +<p>Dinostratus 225, 229, 320-1, ii. 359. +<p>Diocles: inventor of cissoid 264-6: +solution of Archimedes <I>On Sph. +and Cyl.</I> II. 4, ii. 47-8: on burn- +ing-mirrors ii. 200-3. +<p>Diodorus (math.): on parallel-pos- +tulate 358: <I>Analemma</I> of, ii. 287, +ii. 359. +<p>Diodorus Siculus 121, 141, 142, 176. +<p>Diogenes Laertius 144, 145, 177, 291. +<p>Dionysius, Plato's master, 22. +<p>Dionysius, a friend of Heron, ii. 306. +<p>Dionysodorus ii. 46, ii. 218-19, ii. +334-5. +<p>Diophantus of Alexandria: date ii. +<pb n=575><head>ENGLISH INDEX</head> +448: works and editions ii. 448- +55: <I>Arithmetica</I> 15-16: fractions +in, 42-4: notation and definitions +ii. 455-61: signs for unknown (<I>x</I>) +and powers ii. 456-9, for <I>minus</I> +ii. 459: methods ii. 462-79: de- +terminate equations ii. 462-5, +484-90: indeterminate analysis +ii. 466-76, 491-514: ‘Porisms’ ii. +449, 450, 451, ii. 479-80: propo- +sitions in theory of numbers ii. +481-4: conspectus of <I>Arithmetica</I> +ii. 484-514: <I>On Polygonal Num- +bers</I> 16, 84, ii. 514-17: ‘<I>Moriastica</I>’ +ii. 449. +<p>Dioptra 18, ii. 256: Heron's <I>Dioptra</I> +ii. 345-6. +<p>Division: Egyptian method 53, +Greek 58-60: example with sexa- +gesimal fractions (Theon of Alex- +andria) 59-60. +<p><I>Divisions (of Figures), On,</I> by Euclid +425-30: similar problems in +Heron ii. 336-40. +<p>Dodecagon, area of, ii. 328. +<p>Dodecahedron: discovery attributed +to Pythagoras or Pythagoreans +65, 141, 158-60, 162: early occur- +rence 160: inscribed in sphere +(Euclid) 418-19, (Pappus) ii. 369: +Apollonius on, 419-20: volume +of, ii. 335. +<p>Domninus ii. 538. +<p>Dositheus ii. 34. +<p>Duhem, P. 446. +<p>Dupuis, J. ii. 239. +<p>Earth: measurements of, ii. 82, +(Eratosthenes) ii. 106-7, (Posido- +nius) ii. 220. +<p>Ecliptic: obliquity discovered by +Oenopides 174, ii. 244: estimate +of inclination (Eratosthenes, Pto- +lemy) ii. 107-8. +<p>Ecphantus 317, ii. 2. +<p>Edfu, Temple of Horus 124. +<p>Egypt: priests 4-5, 8-9: relations +with Greece 8; origin of geometry +in, 120-2: orientation of temples +122. +<p>Egyptian mathematics: numeral +system 27-8, fractions 28, multi- +plication, &c. 14-15, 52-3: geo- +metry (mensuration) 122-8: tri- +angle (3, 4, 5) right-angled 122, +147: value of <G>p</G> 124, 125: measure- +ment of pyramids 126-8: maps +(regional) 139: algebra in Papyrus +Rhind, &c. ii. 440-1. +<p>Eisenlohr, A. 123, 126, 127. +<p>Eisenmann, H. J. ii. 360. +<p>Elements: as known to Pytha- +goreans 166-8: progress in, down +to Plato 170-1, 175-6, 201-2, 209- +13, 216-17: writers of Elements, +Hippocrates of Chios 170-1, 201- +2, Leon, Theudius 320-1: other +contributors to, Leodamas, Ar- +chytas 170, 212-13, Theaetetus +209-12, 354, Hermotimus of Colo- +phon 320, Eudoxus 320, 323-9, +354: <I>Elements</I> of Euclid 357-419: +the so-called ‘Books XIV, XV’ +419-21. +<p><I>Ell,</I> as measure of angles ii. 215-16. +<p>Empedocles: on Pythagoras 65. +<p>Eneström, G. ii. 341-2. +<p>Enneagon: Heron's measurement +of side ii. 259, of area ii. 328-9. +<p><I>Epanthema</I> of Thymaridas (system +of simple equations) 94: other +types reduced to, 94-6. +<p>Equations: simple, in Papyrus +Rhind, &c. ii. 441: in <I>epanthema</I> +of Thymaridas and in lamblichus +94-6: in Greek anthology ii. +441-3: indeterminate, <I>see</I> Inde- +terminate Analysis: <I>see also</I> +Quadratic, Cubic. +<p>Eratosthenes ii. 1, 16: date, &c. +ii. 104: <I>sieve</I> (<G>ko/skinon</G>) for finding +primes 16, 100, ii. 105: on dupli- +cation of cube 244-6, 251, 258-60: +the <I>Platonicus</I> ii. 104-5: <I>On Means</I> +ii. 105-6, ii. 359: <I>Measurement of +earth</I> ii. 106-7, ii. 242, ii. 346: +astronomy ii. 107-9: chronology +and <I>Geographica</I> ii. 109: on <I>Octaë- +teris ib.</I> +<p>Erycinus ii. 359, 365-8. +<p>Euclid 2-3, 93, 131: date, &c. 354- +6: stories of, 25, 354, 357: rela- +tion to predecessors 354, 357: +Pappus on, 356-7. +<p>Arithmetic: classification and +definitions of numbers 72-3, 397, +‘perfect’ numbers 74, 402: for- +mula for right-angled triangles +in rational numbers 81-2, 405. +<p><I>Conics</I> 438-9, ii. 121-2, focus- +directrix property ii. 119-21: on +ellipse 439, ii. 111, ii. 125. +<pb n=576><head>ENGLISH INDEX</head> +<p><I>Data</I> 421-5, <I>Divisions (of +figures)</I> 425-30, ii. 336, 339. +<p><I>Elements</I>: text 360-1, Theon's +edition 358, 360, ii. 527-8, trans- +lation by Boëtius 359, Arabic +translations 362, ancient com- +mentaries 358-9, <I>editio princeps</I> +of Greek text 360, Greek texts of +Gregory, Peyrard, August, Hei- +berg 360-1: Latin translations, +Athelhard 362-3, Gherard 363, +Campanus 363-4, Commandinus +365: first printed editions, Rat- +dolt 364-5, Zamberti 365: first +introduction into England 363: +first English editions, Billingsley, +&c. 369-70: Euclid in Middle +Ages 365-9, at Universities 368- +9: analysis of, 373-419: arrange- +ment of postulates and axioms +361: I. 47, how originally proved +147-9: parallel-postulate 358, +375, ii. 227-30, ii. 295-7, ii. 534: +so-called ‘Books XIV, XV’ 419- +21. +<p>Mechanics 445-6: Music 444- +5, <I>Sectio canonis</I> 17, 90, 215, +444-5: <I>Optics</I> 17-18, 441-4: <I>Phae- +nomena</I> 349, 351-2, 440-1, ii. +249: <I>Porisms</I> 431-8, lemmas to, +ii. 419-24: <I>Pseudaria</I> 430-1: <I>Sur- +face-Loci</I> 243-4, 439-40, lemmas +to, ii. 119-21, ii. 425-6. +<p>Eudemus 201, 209, 222: <I>History of +Geometry</I> 118, 119, 120, 130, 131, +135, 150, 171: on Hippocrates's +lunes 173, 182, 183-98: <I>History +of Astronomy</I> 174, 329, ii. 244. +<p>Eudoxus 24, 118, 119, 121, 320, +322-4: new theory of proportion +(that of Eucl. V. ii) 2, 153, 216, +325-7: discovered method of ex- +haustion 2, 176, 202, 206, 217, +222, 326, 327-9: problem of two +mean proportionals 245, 246, 249- +51: discovered three new means +86: ‘general theorems’ 323-4: +<I>On speeds,</I> theory of concentric +spheres 329-34, ii. 244: <I>Phaeno- +mena</I> and <I>Mirror</I> 322. +<p>Eugenius Siculus, Admiral, ii. 293. +<p>Euler, L. 75 <I>n.,</I> ii. 482, ii. 483. +<p>Euphorbus (= Pythagoras) 142. +<p>Eurytus 69. +<p>Eutocius 52, 57-8, ii. 25, ii. 45, ii. +126, ii. 518, ii. 540-1. +<p>Exhaustion, method of, 2, 176, 202, +217, 222, 326, 327-9: develop- +ment of, by Archimedes 224, ii. +35-6. +<p>False hypothesis: Egyptian use ii. +441: in Diophantus ii. 488, 489. +<p>Fermat, P. 75 <I>n.,</I> ii. 20, ii. 185, ii. +454, ii. 480, ii. 481-4: on Porisms +435. +<p>Fontenelle ii. 556. +<p>Fractions: Egyptian (submultiples +except 2/3) 27-8, 41: Greek sys- +tems 42-4: Greek notation <I>ib.</I>: +sexagesimal fractions, Babylo- +nian 29, in Greek 44-5. +<p>‘Friendly’ numbers 75. +<p>Galilei 344, 446. +<p><I>Geëponicus, Liber,</I> 124, ii. 309, ii. +318, ii. 344. +<p>Geminus 119, ii. 222-34: on arith- +metic and logistic 14: on divi- +sions of optics, &c. 17-18: on +original steps in proof of Eucl. I +32, 135-6: on parallels 358: +attempt to prove parallel-postu- +late ii. 227-30: on original way +of producing the three conics +ii. 111: encyclopaedic work on +mathematics ii. 223-31: on Posi- +donius's <I>Meteorologica</I> ii. 231-2: +<I>Introduction to Phaenomena</I> ii. +232-4. +<p>Geodesy (<G>gewdaisi/a</G>) = mensuration +(as distinct from geometry) 16-17. +<p>Geometric mean, defined (Archytas) +85: one mean between two +squares (or similar numbers), two +between cubes (or similar solid +numbers) 89-90, 112, 201, 297, +400: no rational mean between +consecutive numbers 90, 215. +<p>‘Geometrical harmony’ (Philolaus's +name for cube) 85-6. +<p>Geometry: origin in Egypt 120-2: +geometry in secondary education +20-1. +<p>Georgius Pachymeres ii. 453, ii. +546. +<p>Gerbert (Pope Sylvester II) 365-7: +geometry of, 366: ii. 547. +<p>Gerhardt, C. J. ii. 360, ii. 547. +<p>Gherard of Cremona, translator of +Euclid and an-Nairīzī 363, 367, +ii. 309: of Menelaus ii. 252, ii. 262. +<pb n=577><head>ENGLISH INDEX</head> +<p>Ghetaldi, Marino, ii. 190. +<p>Girard, Albert, 435, ii. 455. +<p><I>Gnomon</I>: history of term 78-9: +gnomons of square numbers 77- +8, of oblong numbers 82-3, of +polygonal numbers 79: in appli- +cation of areas 151-2: use by +al-Karkhī 109-10: in Euclid 379: +sun-dial with vertical needle 139. +<p>Gomperz, Th. 176. +<p>Govi, G. ii. 293 <I>n.</I> +<p>Gow, J. 38. +<p>Great Year, of Oenopides 174-5, +of Callippus and Democritus 177. +<p>Gregory, D. 360-1, 440, 441, ii. 127. +<p>Griffith, F. Ll. 125. +<p>Günther, S. ii. 325 <I>n.,</I> ii. 550. +<p>Guldin's theorem, anticipated by +Pappus ii. 403. +<p>Halicarnassus inscriptions 32 - 3, +34. +<p>Halley, E., editions of Apollonius's +<I>Conics</I> ii. 127-8, and <I>Sectio ratio- +nis</I> ii. 175, 179, of Menelaus ii. +252, ii. 262, of extracts from +Pappus ii. 360, of Serenus ii. 519. +<p>Halma, editor of Ptolemy ii. 274, +275. +<p>Hammer-Jensen, I. ii. 300 <I>n.,</I> ii. +304 <I>n.</I> +<p>Hankel, H. 145, 149, 288, 369, ii. +483. +<p>Hardy, G. H. 280. +<p>Harmonic mean (originally ‘sub- +contrary’) 85. +<p><I>Harpedonaptae,</I> ‘rope-stretchers’ +121-2, 178. +<p>Hārūn ar-Rashīd 362. +<p><I>Hau</I> - calculations (Egyptian) ii. +440-1. +<p>Hecataeus of Miletus 65, 177. +<p>Heiben, J. L. 233 <I>n.</I> +<p>Heiberg, J. L. 184, 187 <I>n.,</I> 188, +192 <I>n.,</I> 196-7 <I>n.,</I> 315, 361, ii. 203, +ii. 309, 310, 316, 318, 319, ii. 519, +ii. 535, 543, 553, 555 <I>n.</I> +<p><I>Helceph</I> 111. +<p>Hendecagon in a circle (Heron) ii. +259, ii. 329. +<p>Henry, C. ii. 453. +<p>Heptagon in a circle, ii. 103: +Heron's measurement of, ii. 328. +<p>Heraclides of Pontus 24, ii. 231-2: +discovered rotation of earth about +axis 316-17, ii. 2-3, and that Venus +and Mercury revolve about sun +312, 317, ii. 2, ii. 244. +<p>Heraclitus of Ephesus 65. +<p>Heraclitus, mathematician ii. 192, +ii. 359, ii. 412. +<p>Hermannus Secundus ii. 292. +<p>Hermesianax 142 <I>n.,</I> 163. +<p>Hermodorus ii. 359. +<p>Hermotimus of Colophon 320-1: +Elements and Loci <I>ib.,</I> 354. +<p>‘Herodianic’ (or ‘Attic’) numerals +30-1. +<p>Herodotus 4, 5, 48, 65, 121, 139. +<p>Heron of Alexandria 121, ii. 198, +ii. 259: controversies on date ii. +298-307: relation to Ctesibius +and Philon ii. 298-302, to Pappus +ii. 299-300, to Posidonius and +Vitruvius ii. 302-3, to <I>agrimen- +sores</I> ii. 303, to Ptolemy ii. 303-6. +<p>Arithmetic: fractions 42-4, mul- +tiplications 58, approximation to +surds ii. 51, ii. 323-6, approxima- +tion to cube root 64, ii. 341-2, +quadratic equations ii. 344, in- +determinate problems ii. 344, +444-7. +<p>Character of works ii. 307-8: +list of treatises ii. 308-10. +<p>Geometry ii. 310-14, <I>Definitions</I> +ii. 314-16: comm. on Euclid's +<I>Elements</I> 358, ii. 310-14: proof of +formula for area of triangle in +terms of sides ii. 321-3: duplica- +tion of cube 262-3. +<p><I>Metrica</I> ii. 320-43: (1) mensu- +ration ii. 316-35: triangles ii. +320-3, quadrilaterals ii. 326, +regular polygons ii. 326-9, circle +and segments ii. 329-31: volumes +ii. 331-5, <G>bwmi/skos</G> ii. 332-3, frus- +tum of cone, sphere and segment +ii. 334, <I>tore</I> ii. 334-5, five regular +solids ii. 335. (2) divisions of +figures ii. 336-43, of frustum of +cone ii. 342-3. +<p><I>Mechanics</I> ii. 346-52: on Ar- +chimedes's mechanical works ii. +23-4, on centre of gravity ii. 350-1, +352. +<p><I>Belopoeïca</I> 18, ii. 308-9, <I>Catop- +trica</I> 18, ii. 294, ii. 310, ii. 352-4. +<p><I>Dioptra</I> ii. 345-6, <I>Pneumatica</I> +and <I>Automata</I> 18, ii. 308, 310. +<p><I>On Water-clocks</I> ii. 429, ii. 536. +<p>Heron, teacher of Proclus ii. 529. +<pb n=578><head>ENGLISH INDEX</head> +<p>‘Heron the Younger’ ii. 545. +<p>Heronas 99. +<p>Hicetas 317. +<p>Hierius 268, ii. 359. +<p>Hieronymus 129. +<p>Hilāl b. Abī Hilāl al-&Hdot;im⋅ī ii. 127. +<p>Hiller, E. ii. 239. +<p>Hilprecht, H. V. 29. +<p>Hipparchus ii. 3, 18, 198, 216, 218: +date, &c. ii. 253: work ii. 254-6: +on epicycles and eccentrics ii. +243, ii. 255: discovery of preces- +sion ii.254: on mean lunar month +ii. 254-5: catalogue of stars ii. +255: geography ii. 256: trigono- +metry ii. 257-60, ii. 270. +<p>Hippasus 65, 85, 86, 214: construc- +tion of ‘twelve pentagons in +sphere’ 160. +<p>Hippias of Elis: taught mathe- +matics 23: varied accomplish- +ments <I>ib.</I>, lectures in Sparta 24: +inventor of <I>quadratrix</I> 2, 171, 182, +219, 225-6. +<p>Hippocrates of Chios 2, 182, 211: +taught for money 22: first writer +of Elements 119, 170, 171: ele- +ments as known to, 201-2: +assumes <G>neu=sis</G> equivalent to solu- +tion of quadratic equation 88, +195-6: on quadratures of lunes +170, 171, 173, 182, 183-99, 220, +221: proved theorem of Eucl.XII +2, 187, 328: reduced duplication +of cube to problem of finding +two mean proportionals 2, 183, +200, 245. +<p>Hippolytus: on <G>puqme/nes</G> (bases) and +‘rule of nine’ and ‘seven’ 115-16. +<p><I>Hippopede</I> of Eudoxus 333-4. +<p>Homer 5. +<p>‘Horizon’: use in technical sense by +Euclid 352. +<p>Horsley, Samuel, ii. 190, ii. 360. +<p>Hultsch, F. 204, 230, 349, 350, ii. 51, +ii. 308, ii. 318, 319, ii. 361. +<p>Hunrath, K. ii. 51. +<p>Hunt, A. S. 142. +<p>Hypatia ii. 449, ii. 519, ii. 528-9. +<p>Hypotenuse, theorem of square on, +142, 144-9: Proclus on discovery +of, 145: supposed Indian origin +145-6. +<p>Hypsicles: author of so-called Book +XIV of Eucl. 419-20, ii. 192: de- +finition of ‘polygonal number’ 84, +ii. 213, ii. 515: <G>*)anaforiko/s</G> ii. +213-18, first Greek division of +zodiac circle into 360 parts ii. 214. +<p>Iamblichus 4, 69, 72, 73, 74, 75, 86, +107, ii. 515, 529: on <G>e)pa/nqhmt</G> of +Thymaridas, &c. 94-6: works +113-14: comm. on Nicomachus +113-15: squares and oblong num- +bers as ‘race-courses’ 114: pro- +perty of sum of numbers 3<I>n</I>-2, +3<I>n</I>-1, 3<I>n</I> 114-15. +<p>Ibn al-Haitham, on burning-mirrors +ii. 201: ii. 453. +<p>Icosahedron 159: discovery attri- +buted to Theaetetus 162: volume +of, ii. 335. +<p>Incommensurable, discovery of, 65, +90-1, 154: proof of incommensu- +rability of diagonal of square 91. +<p>Indeterminate analysis: first cases, +right-angled triangles in rational +numbers 80, 81, ‘side-’ and ‘dia- +meter-’ numbers 91-3, ii. 536: +rectangles with area and peri- +meter numerically equal 96-7: +indeterminate equations, first +degree ii. 443, second degree ii. +443-4 (<I>see also</I> Diophantus), in +Heronian collections ii. 344, ii. +444-7. +<p>India: rational right-angled tri- +angles in, 145-6: approximation +to √2, 146. +<p>Indian Table of Sines ii. 253. +<p>Irrational: discovered by Pythago- +reans 65, 90-1, 154, and with +reference to √2, 155, 168: Demo- +crituson, 156-7,181: Theodorus on, +203-9: extensions by Theaetetus +209-12, Euclid 402-11, Apollonius +ii. 193. +<p>Isaac Argyrus 224 <I>n.</I>, ii. 555. +<p>Is&hdot;āq b. Hunain, translator of +Euclid 362, of Menelaus ii. 261, +and Ptolemy ii. 274. +<p>Isidorus Hispalensis 365. +<p>Isidorus of Miletus 421, ii. 25, ii. +518, ii. 540. +<p>Isocrates: on mathematics in edu- +cation 21. +<p>Isoperimetric figures ii. 206-13, ii. +390-4. +<p>Jacob b. Machir ii. 252, ii. 262. +<p>Jacobus Cremonensis ii. 26-7. +<pb n=579><head>ENGLISH INDEX</head> +<p>Jan, C. 444. +<p>Joachim Camerarius ii. 274. +<p>Joachim, H. H. 348 <I>n.</I> +<p>Johannes de Sacrobosco 368. +<p>Jordanus Nemorarius ii. 328. +<p>Jourdain, P. E. B. 283 <I>n.</I> +<p>Kahu&ndot; Papyri 125, 126. +<p>Kant 173. +<p>Keil, B. 34-5. +<p>Kepler ii. 20, ii. 99. +<p>Köchly, H. A. T. ii. 309. +<p>Koppa (<G>*|o</G>for90)=Phoenician Qoph +32. +<p>Kubitschek, W. 50. +<p>Lagrange ii. 483. +<p>Laird, A. G. 306 <I>n.</I> +<p>Laplace 173. +<p>Larfeld, W. 31 <I>n.</I>, 33-4. +<p>Lawson 436. +<p>Leibniz 279, ii. 20. +<p><I>Lemma</I> 373, ii. 533. +<p>Leodamas of Thasos 120, 170, 212, +291, 319. +<p>Leon 319. +<p>Leon (of Constantinople) ii. 25. +<p>Leonardo of Pisa 367, 426, ii. 547. +<p>Lepsius, C. R. 124. +<p>Leucippus 181. +<p>Libri, G. ii. 556. +<p>‘Linear’ (of numbers) 73. +<p>‘Linear’ loci and problems 218-19. +<p>Lines, classification of, ii. 226. +<p>Livy ii. 18. +<p>Loci: classification of,218-19, plane, +solid, linear 218: loci on surfaces +219: ‘solid loci’ ii. 116-19. +<p>Loftus, W. K. 28. +<p>Logistic (opp. to ‘arithmetic’ +science of calculation 13-16, 23, +53. +<p><I>Logistica speciosa</I> and <I>numerosa</I> +(Vieta) ii. 456. +<p>Loria, G. iv-v, 350 <I>n.</I>, ii. 293 <I>n.</I> +<p>Luca Paciuolo 367, ii. 324 <I>n.</I> +<p>Lucas, E. 75 <I>n.</I> +<p>Lucian 75 <I>n.</I>, 77, 99, 161, ii. 18. +<p>Lucretius 177. +<p>Magic squares ii. 550. +<p>Magnus, <I>Logistica</I> 234-5. +<p>Mamercus or Mamertius 140, 141, +171. +<p>al-Ma'mūn, Caliph 362. +<p>al-Man⋅ūr, Caliph 362. +<p><I>Manus</I>, for number 27. +<p>Marinus 444, ii. 192, ii. 537-8. +<p>Martianus Capella 359, 365. +<p>Martin, T. H. ii. 238, ii. 546. +<p>Maslama b. A&hdot;mad al-Majrī⃛ī ii. +292. +<p>Massalia 8. +<p>Mastaba tombs 128. +<p>Mathematics: meaning 10-11, clas- +sification of subjects 11-18: +branches of applied mathematics +17-18: mathematics in Greek +education 18-25. +<p>Maurolycus ii. 262. +<p>Means: arithmetic, geometric, and +subcontrary (harmonic) known +in Pythagoras's time 85: defined +by Archytas <I>ib.</I>: fourth, fifth, and +sixth discovered, perhaps by Eu- +doxus 86, four more by Myonides +and Euphranor 86: ten means +in Nicomachus and Pappus 87-9, +Pappus's propositions 88-9: no +rational geom. mean between suc- +cessive numbers (Archytas) 90, +215. +<p>Mechanics, divisions of, 18: writers +on, Archytas 213, Aristotle 344-6, +445-6, Archimedes ii. 18, ii. 23-4, +ii. 75-81, Ptolemy ii. 295, Heron ii. +346-52, Pappus ii. 427-34. +<p>Megethion ii. 360. +<p>Memus, Johannes Baptista, ii. 127. +<p>Menaechmus 2, 25, 251-2, 320-1: +discoverer of conic sections 251- +3, ii. 110-16: solved problem of +two mean proportionals 245, 246, +251-5: on ‘problems’ 318. +<p>Menelaus of Alexandria ii. 198, ii. +252-3: date, &c. ii. 260-1: Table of +Chords ii. 257: <I>Sphaerica</I> ii. 261- +73: Menelaus's theorem ii. 266- +8, 270: anharmonic property ii. +269: <G>para/doxos</G> curve ii. 260-1. +<p><I>Mensa Pythagorea</I> 47. +<p>Mensuration: in primary education +19: in Egypt 122-8: in Heron ii. +316-35. +<p>Meton 220. +<p>Metrodorus ii. 442. +<p><I>Minus</I>, sign for, in Diophantus ii. +459-60. +<p>Mochus 4. +<p>Moschopoulos, Manuel, ii. 549-50. +<p>Mu&hdot;ammad Bagdadinus 425. +<p>Multiplication: Egyptian method +<pb n=580><head>ENGLISH INDEX</head> +52-3, Greek 53-4, ‘Russian’ 53 <I>n.</I>: +examples from Eutocius, Heron, +Theon 57-8: Apollonius's con- +tinued multiplications 54-7. +<p>Multiplication Table 53. +<p><I>Murran</I>, an angular measure ii. 215. +<p>Musical intervals and numerical +ratios 69, 75-6, 85, 165. +<p>Myriads, ‘first’, ‘second’, &c., nota- +tion for, 39-40. +<p>Nagl, A. 50. +<p>an-Nairīzī: comm. on Euclid 363, +ii. 224, ii. 228-30, ii. 309-10. +<p>Na⋅īraddīn a⃛-&Tdot;ūsī: version of Eu- +clid 362, of Apollonius's <I>Conics</I> +ii. 127: of Ptolemy ii. 275. +<p>Naucratis inscriptions 33. +<p>Nemesius 441. +<p>Neoclides 319. +<p><I>Ner</I> (Babylonian) (=600) 28, ii. +215. +<p>Nesselmann, G. H. F. ii. 450-1, ii. +455-6. +<p>Newton 370, ii. 20, ii. 182. +<p>Nicolas Rhabdas 40, ii. 324 <I>n.</I>, ii. +550-3. +<p>Nicomachus of Gerasa 12, 69, 70, +72, 73, 74, 76, 83, 85, 86, ii. 238, +ii. 515: works of, 97: <I>Introductio +arithmetica</I>: character of treatise +98-9, contents 99-112, classifica- +tion of numbers 99-100: on ‘per- +fect’ numbers 74, 100-1: on ten +means 87: on a ‘Platonic’ theo- +rem 297: sum of series of +natural cubes 109-10. +<p>Nicomedes 225-6, ii. 199: cochloids +or conchoids 238-40: duplica- +tion of cube 260-2. +<p>Niloxenus 129. +<p>Nine, rule of, 115-16: casting out +nines ii. 549. +<p>Nipsus, M. Junius, 132. +<p>Nix, L. ii. 128, 131, ii. 309. +<p>Noël, G. 282. +<p>Number: defined, by Thales 69, by +Moderatus, Eudoxus, Nicoma- +chus, Aristotle 70: classification +of numbers 70-4: ‘perfect’, +‘over-perfect’ and ‘defective’ +numbers 74-5, ‘friendly’ 75, +figured 76-9: ‘oblong’, ‘prolate’ +82-3, 108, 114, similar plane and +solid numbers 81-2, 90, solid +numbers classified 106-8: ‘the +number in the heaven’ (Pytha- +gorean) 68, ‘number’ of an object +69. +<p>Numerals: systems of, decimal, qui- +nary, vigesimal 26: origin of +decimal system 26-7: Egyptian +27-8; Babylonian systems (1) +&ddot;ecimal 28, (2) sexagesimal 28-9: +Greek (1) ‘Attic’ or ‘Herodianic’ +30-1: (2) alphabetic system, +original in Greece 31-7, how +evolved 31-2, date of introduc- +tion 33-5, mode of writing 36-7, +comparison of two systems 37-9: +notation for large numbers, Apol- +lonius's tetrads 40, Archimedes's +octads 40-1. +<p>Nymphodorus 213. +<p>‘Oblong’ numbers 82-3, 108, 114: +gnomons of, 82-3. +<p>Ocreatus, 111. +<p>Octads, of Archimedes 40-1. +<p>Octagon, regular, area of, ii. 328. +<p>Octahedron 159, 160, 162: volume +of, ii. 335. +<p>‘Odd’ number defined 70-1: 1 +called ‘odd-even’ 71: ‘odd-even’, +‘odd-times-odd’, &c., numbers +71-4. +<p>Oenopides of Chios 22, 121: dis- +covered obliquity of ecliptic 138, +174, ii. 244: Great Year of, 174-5: +called perpendicular <I>gnomon-wise</I> +78, 175: two propositions in ele- +mentary geometry 175. +<p>Olympiodorus 444. +<p>One, the principle of number 69. +<p>Oppermann ii. 324 <I>n.</I> +<p>Optics: divisions of, 17-18: of Euclid +441-4: of Ptolemy ii. 293-4. +<p>Oval of Cassini ii. 206. +<p>Oxyrhynchus Papyri 142. +<p>Pamphile, 131, 133, 134. +<p>Pandrosion ii. 360. +<p>Pappus (<I>see also</I> Table of Contents, +under Chap. XIX) ii. 17-18, ii. 175, +180, 181, 182, 183, 185, 186, 187, +188, 189, 190, ii. 207, 211, 212, +213, ii. 262, ii. 337, ii. 355-439: +on Apollonius's tetrads 40, on +Apollonius's continued multi- +plications 54-7: on ten means +87-9: on mechanical works of +Archimedes ii. 23-4: on conics +<pb n=581><head>ENGLISH INDEX</head> +of Euclid and Apollonius 438, +proof of focus-directrix property +ii. 120-1: commentary on Euclid +358, ii. 356-7, on Book X 154-5, +209, 211, ii. 193: commentary on +Euclid's <I>Data</I> 421-2, ii. 357, on +Diodorus's <I>Analemma</I> ii. 287, +scholia on <I>Syntaxis</I> ii. 274: on +classification of problems and +loci (plane, solid, linear) 218-19, +ii.117-18, criticism on Archimedes +and Apollonius 288, ii. 68, ii. 167: +on surface-loci 439-40, ii. 425-6: +on Euclid's <I>Porisms</I> 431-3, 436-7, +ii. 270, ii. 419-24: on ‘Treasury +of Analysis’ 421, 422, 439, ii. 399- +427: on <I>cochloids</I> 238-9: on <I>quad- +ratrix</I> 229-30, ii. 379-80, con- +structions for, ii. 380-2: on dupli- +cation of cube 266-8, 268-70: on +trisection of any angle 241-3, +ii. 385-6, <G>neu=sis</G> with regard to +parallelogram 236-7: on isoperi- +metry (cf. Zenodorus) ii. 207, ii. +211-12, ii. 390-4. +<p>‘Paradoxes’ of Erycinus ii. 365-8. +<p>Parallelogram of velocities 346, ii. +348-9. +<p><I>Parapegma</I> of Democritus 177. +<p>Parmenides 138. +<p>Paterius ii. 536-7. +<p>Patricius ii. 318, 319. +<p>Pebbles, for calculation 46, 48. +<p>Pentagon, regular: construction +Pythagorean 160-2; area of, ii. 327. +<p>Pentagram, Pythagorean 161-2. +<p>‘Perfect’ numbers 74-5: list of +first ten <I>ib.</I>: contrasted with +‘over-perfect’ and ‘defective’ +<I>ib.</I>: 10 with Pythagoreans 75. +<p>‘Perfect’ proportion 86. +<p>Pericles 172. +<p>Pericles, a mathematician ii. 360. +<p>Perseus 226: spiric sections ii. +203-6. +<p>‘Phaenomena’ = observational as- +tronomy 17: 322, 349. +<p>Philippus of Opus 354: works by, +321: on polygonal numbers 84, +ii. 515: astronomy 321. +<p>Philolaus 67, 72, 76, 78, 86, 158, +ii. 1: on odd, even, and even-odd +numbers 70-1: Pythagorean non- +geocentric astronomy attributed +to, 163-4. +<p>Philon of Byzantium 213: duplica- +tion of cube 262-3: Philon, Ctesi- +bius and Heron ii. 298-302. +<p>Philon of Gadara 234. +<p>Philon of Tyana ii. 260. +<p>Philoponus, Joannes, 99, 223, 224 <I>n.</I> +<p>Phocaeans 7. +<p>Phocus of Samos 138. +<p>Phoenician alphabet, how treated +by Greeks 31-2: arithmetic ori- +ginated with Phoenicians 120-1. +<p>‘Piremus’ or ‘peremus’ in pyramid +126, 127. +<p>‘Plane’ loci 218. +<p>‘Plane’ problems 218-19. +<p><I>Planisphaerium</I> of Ptolemy ii. 292-3. +<p>Planudes, Maximus, 117, ii. 453, ii. +546-9. +<p>Plato 19, 22, 24, 121, 142 <I>n.</I>, 170, 176: +<G>*qeo\s a)ei\gewmetrei=</G> 10: <G>mhdei\sa)gewme/- +trhtos ei)si/tw</G> iii, 24, 355: on educa- +tion in mathematics 19-20, 284: +on mathematical ‘arts’, measure- +ment and weighing 308, instru- +ments for, 308-9, principle of +lever 309: on optics 309, 441: +on music 310: Plato's astronomy +310-15: on arithmetic and logistic +13-14: classification of numbers, +odd, even, &c. 71-2, 292: on +number 5040, 294: the Geometri- +cal Number, 305-8: on arithme- +tical problems 15, ii. 442: on +geometry 286-8, constructions +alien to true geometry <I>ib.</I>: on- +tology of mathematics 288-9: +hypotheses of mathematics 289- +90: two intellectual methods +290-2: supposed discovery of +mathematical analysis, 120, 212- +13, 291-2: definitions of various +species of numbers 292, figure +292-3, line and straight line 293, +circle and sphere 293-4: on +points and indivisible lines 293: +formula for rational right-angled +triangles 81, 304: ‘rational’ and +‘irrational diameter of 5’ 93, +306-7: Plato and the irrational +156, 203-5, 304: on solid geo- +metry 12-13, 303: on regular and +semi-regular solids 294-7: Plato +and duplication of cube 245-6, +255, 287-8, 303: on geometric +means between two squares and +two cubes respectively 89, 112, +<pb n=582><head>ENGLISH INDEX</head> +201, 297, 400: on ‘perfect’ pro- +portion 86: a proposition in +proportion 294: two geometrical +passages in <I>Meno</I> 297-303: pro- +positions ‘on the <I>section</I>’ 304, +324-5. +<p>‘Platonic’ figures (the regular +solids) 158, 162, 294-5, 296-7. +<p>Playfair, John, 436. +<p>Pliny 129, ii. 207. +<p>Plutarch 84, 96, 128, 129, 130, 133, +144, 145, 167, 179, ii. 2, 3, ii. 516: +on Archimedes ii. 17-18. +<p>Point: defined as a ‘unit having +position’ 69, 166: Plato on points +293. +<p>Polybius 48, ii. 17 <I>n.</I>, ii. 207. +<p>Polygon: propositions about sum +of exterior or interior angles 144: +measurement of regular polygons +ii. 326-9. +<p>Polygonal numbers 15, 76, 79, ii. +213, ii. 514-17. +<p>Polyhedra, <I>see</I> Solids. +<p><I>Porism</I> (1) = corollary 372: (2) a +certain type of proposition 373, +431-8: <I>Porisms</I> of Euclid, <I>see</I> +Euclid: of Diophantus, <I>see</I> Dio- +phantus. +<p>Porphyry 145: commentary on Eu- +clid's <I>Elements</I> 358, ii. 529. +<p>Poselger, F. T. ii. 455. +<p>Posidonius ii. 219-22: definitions +ii. 221, 226; on parallels 358, ii. +228: <I>versus</I> Zeno of Sidon ii. +221-2: <I>Meteorologica</I> ii. 219: +measurement of earth ii. 220: on +size of sun ii. 108, ii. 220-1. +<p>Postulates: Aristotle on, 336: in +Euclid 336, 374-5: in Archimedes +336, ii. 75. +<p>Powers, R. E. 75 <I>n.</I> +<p>Prestet, Jean, 75 <I>n.</I> +<p>Prime numbers and numbers prime +to one another 72-3: defined 73: +2 prime with Euclid and Aristotle, +not Theon of Smyrna and Neo- +Pythagoreans <I>ib.</I> +<p>Problems: classification 218-19: +plane and solid ii. 117-18: pro- +blems and theorems 318, 431, ii. +533. +<p>Proclus 12, 99, 175, 183, 213, 224 <I>n.</I>, +ii. 529-37: <I>Comm. on Eucl. I.</I> ii. +530-5: sources ii. 530-2: ‘sum- +mary’ 118-21, 170, object of, 170- +1: on discoveries of Pythagoras +84-5, 90, 119, 141, 154: on Euclid +I. 47, 145, 147: attempt to prove +parallel-postulate 358, ii. 534: on +loci 219: on porisms 433-4: on +Euclid's music 444: comm. on +<I>Republic</I> 92-3, ii. 536-7: <I>Hypoty- +posis of astronomical hypotheses</I> +ii. 535-6. +<p>Prodicus, on secondary education +20-1. +<p><I>Prolate</I>, of numbers 108, 204. +<p><I>Proof</I> 370, ii. 533. +<p>Proportion: theory discovered by +Pythagoras 84-5, but his theory +numerical and applicable to com- +mensurables only 153, 155, 167: +def. of numerical proportion 190: +the ‘perfect’ proportion 86: +Euclid's universally applicable +theory due to Eudoxus 153, 155, +216, 325-7. +<p>Proposition, geometrical: formal +divisions of, 370-1. +<p>Protagoras 202: on mathematics +23, 179. +<p>Prou, V. ii. 309. +<p><I>Psammites</I> or <I>Sand-reckoner</I> of Archi- +medes 40, ii. 3, ii. 81-5. +<p>Psellus, Michael, 223-4 <I>n.</I>, ii. 453, +ii. 545-6. +<p><I>Pseudaria</I> of Euclid 430-1. +<p>Pseudo-Boëtius 47. +<p>Pseudo-Eratosthenes: letter on du- +plication of cube 244-5. +<p>Ptolemies: coins of, with alphabetic +numerals 34-5: Ptolemy I, story +of, 354. +<p>Ptolemy, Claudius, 181, ii. 198, ii. +216, ii. 218, ii. 273-97: sexa- +gesimal fractions 44-5, approxi- +mation to <G>p</G> 233: attempt to prove +parallel-postulate 358, ii. 295-7: +<I>Syntaxis</I> ii. 273-86, commentaries +and editions ii. 274-5, contents +of, ii. 275-6, trigonometry in, ii. +276-86, 290-1, Table of Chords +ii. 259, ii. 283-4, on obliquity of +ecliptic ii. 107-8: <I>Analemma</I> +ii. 286-92: <I>Planispherium</I> ii. 292- +3, <I>Optics</I> ii. 293-4, other works ii. +293: <G>peri\ r(opw=n</G> ii. 295: <G>peri\ dia- +sta/sews</G> <I>ib.</I> +<p>Pyramids: origin of name 126: +measurements of, in Rhind Papy- +rus 126-8: pyramids of Dakshūr, +<pb n=583><head>ENGLISH INDEX</head> +Gizeh, and Mēdūm 128: measure- +ment of height by Thales 129-30; +volume of pyramid 176, 180, 217, +ii. 21, &c., volume of frustum ii. +334. +<p>Pythagoras 65-6, 121, 131, 133, 138: +travels 4-5, story of bribed pupil +24-5: motto 25, 141: Heraclitus, +Empedocles and Herodotus on, +65: Proclus on discoveries of, 84- +5, 90, 119, 141, 154: made mathe- +matics a part of liberal education +141, called geometry ‘inquiry’ +166, used definitions 166: arith- +metic (theory of numbers) 66-80, +figured numbers 76-9: gnomons +77, 79: ‘friendly’ numbers 75: +formula for right-angled tri- +angles in rational numbers 79- +80: founded theory of proportion +84-5, introduced ‘perfect’ pro- +portion 86: discovered depen- +dence of musical intervals on +numerical ratios 69, 75-6, 85, +165: astronomy 162-3, earth +spherical <I>ib.</I>, independent move- +ment of planets 67, 163: Theorem +of Pythagoras 142, 144-9, how +discovered? 147-9, general proof, +how developed <I>ib.</I>, Pappus's ex- +tension ii. 369-71. +<p>Pythagoreans 2, 11, 220: <I>quadri- +vium</I> 11: a Pythagorean first +taught for money 22: first to +advance mathematics 66: ‘all +things are numbers’ 67-9: ‘num- +ber’ of an object 69, ‘number in +the heaven’ 68: figured numbers +69: definition of unit 69: 1 is +odd-even 71: classification of +numbers 72-4: ‘friendly’ num- +bers 75: 10 the ‘perfect’ number +75: oblong numbers 82-3, 108, +114: side-and diameter-numbers +giving approximations to √2, 91- +3: first cases of indeterminate +analysis 80, 91, 96-7: sum of +angles of triangle = 2 <I>R</I>, 135, +143: geometrical theorems attri- +buted to, 143-54: invented appli- +cation of areas and geometrical +algebra 150-4: discovered the in- +commensurable 65, 90-1, 154, +with reference to √2 155, 168: +theory of proportion only ap- +plicable to commensurables 153, +155, 167, 216: construction of +regular pentagon 160-2: astro- +nomical system (non-geocentric) +163-5: definitions 166: on order +of planets ii. 242. +<p><I>Qay en &hdot;eru</I>, height (of pyramid) +127. +<p>Quadratic equation: solved by Py- +thagorean application of areas +150-2, 167, 394-6, 422-3<I>:</I> nu- +merical solutions ii. 344, ii. 448, +ii. 463-5. +<p><I>Quadratrix</I> 2, 23, 171, 182, 218, 219, +225-30, ii. 379-82. +<p><I>Quadrivium</I> of Pythagoreans 11. +<p>Quinary system of numerals 26. +<p>Quintilian ii. 207. +<p>Qus⃛ā b. Lūqā, translator of Euclid +362, ii. 453. +<p>Rangabé, A. R. 49-50. +<p>Ratdolt, Erhard, first edition of +Euclid 364-5. +<p><I>Reductio ad absurdum</I> 372: already +used by Pythagoreans 168. +<p><I>Reduction</I> (of a problem) 372. +<p>Reflection: equality of angles of +incidence and reflection 442, ii. +294, ii. 353-4. +<p>Refraction 6-7, 444: first attempt +at a law (Ptolemy) ii. 294. +<p>Regiomontanus 369, ii. 27, ii. 453-4. +<p><I>Regula Nicomachi</I> 111. +<p>Rhabdas, Nicolas, 40, ii. 324 <I>n.</I>, ii. +550-3. +<p>Rhind Papyrus: mensuration in, +122-8: algebra in, ii. 440-1. +<p>Right-angled triangle: inscribed +by Thales in circle 131: theorem +of Eucl. I. 47, attributed to +Pythagoras 142, 144-5, supposed +Indian origin of, 145-6. +<p>Right-angled triangles in rational +numbers: Pythagoras's formula +80, Plato's 81, Euclid's 81-2, +405: triangle (3, 4, 5) known to +Egyptians 122: Indian examples +146: Diophantus's problems on, +ii. 507-14. +<p>Robertson, Abram, ii. 27. +<p>Rodet, L. 234. +<p>Rodolphus Pius ii. 26. +<p>Roomen, A. van, ii. 182. +<p>Rudio, F. 173, 184, 187-91, ii. +539. +<pb n=584><head>ENGLISH INDEX</head> +<p>Rudolph of Bruges ii. 292. +<p>Ruelle, Ch. Em. ii. 538. +<p>Rüstow, F. W. ii. 309. +<p>Ruler-and - compasses restriction +175-6. +<p>Sachs, Eva, 209 <I>n.</I> +<p>Salaminian table 48, 50-1. +<p><I>Salinon</I> ii. 23, ii. 103. +<p>Sampi (<*> = 900) derived from +Ssade q.v. +<p><I>Sar</I> (Babylonian for 60<SUP>2</SUP>) 28, ii. 215. +<p><I>Satapatha Brāhma&ndot;a</I>, 146. +<p>Savile, Sir H., on Euclid 360, 369. +<p><I>Scalene</I>: of triangles 142: of certain +solid numbers 107: of an odd +number (Plato) 292: of an oblique +cone ii. 134. +<p>Schiaparelli, G. 317, 330, ii. 539. +<p>Schmidt, W. ii. 308, 309, 310. +<p>Schöne, H. ii. 308. +<p>Schöne, R. ii. 308, 317. +<p>Scholiast to <I>Charmides</I> 14, 53. +<p>Schooten, F. van, 75 <I>n.</I>, ii. 185. +<p>Schulz, O. ii. 455. +<p>Scopinas ii. 1. +<p>Secondary numbers 72. +<p><I>Sectio canonis</I> 17, 215, 444. +<p>Seelhoff, P. 75 <I>n.</I> +<p>Seleucus ii. 3. +<p>Semicircle: angle in, is right +(Thales) 131, 133-7. +<p>Senkereh, Tables 28, 29. +<p><I>Senti</I>, base (of pyramid) 127. +<p><I>Se-qet</I>, ‘that which makes the nature’ +(of pyramid) = cotangent of angle +of slope 127-8, 130, 131. +<p>Serenus ii. 519-26: <I>On section of +cylinder</I> ii. 519-22, <I>On section of +cone</I> ii. 522-6. +<p>Sesostris (Ramses II) 121. +<p>Sexagesimal system of numerals +and fractions 28-9: sexagesimal +fractions in Greek 44-5, 59, 61-3, +233, ii. 277-83. +<p>Sextius 220. +<p>Sicily 8. +<p>‘Side-’ and ‘diameter-numbers’ 91- +3, 112, 153, 308, 380, ii. 536. +<p>Simon, M. 200. +<p>Simplicius: extract from Eudemus +on Hippocrates's quadrature of +lunes 171, 182-99: on Antiphon +221-2: on Eudoxus's theory of +concentric spheres 329: commen- +tary on Euclid 358, ii. 539-40: on +mechanical works of Archimedes +ii. 24: ii. 538-40. +<p>Simson, R., edition of Euclid's +<I>Elements</I> 365, 369, and of Euclid's +<I>Data</I> 421: on Euclid's <I>Porisms</I> +435-6: restoration of <I>Plane Loci</I> +of Apollonius ii. 185, ii. 360. +<p>Simus of Posidonia 86. +<p>Sines, Tables of, ii. 253, ii. 259-60. +<p><I>Sinus rectus, sinus versus</I> 367. +<p>Sluse, R. F. de, 96. +<p>Smith, D. E. 49, 133 <I>n.</I> +<p>‘Solid’ loci and problems 218, ii. +117-18: <I>Solid Loci</I> of Aristaeus +438, ii. 118-19. +<p>‘Solid’ numbers, classified 106-8. +<p>Solids, Five regular: discovery at- +tributed to Pythagoras or Pytha- +goreans 84, 141, 158-60, 168, +alternatively (as regards octahe- +dron and icosahedron) to Theae- +tetus 162: all five investigated +by Theaetetus 159, 162, 212, 217: +Plato on, 158-60: Euclid's con- +structions for, 415-19: Pappus's +constructions ii. 368-9: content +of, ii. 335, ii. 395-6. +<p>Solon 4, 48. +<p>Sophists: taught mathematics 23. +<p>Sosigenes 316, 329. +<p><I>Soss = sussu</I> = 60 (Babylonian) 28, +ii. 215. +<p>Speusippus 72, 73, 75, ii. 515: on +Pythagorean numbers 76, 318: +on the five regular solids 318: on +<I>theorems ib.</I> +<p><I>Sphaeric</I> 11-12: treatises on, by Au- +tolycus and Euclid 348-52, 440- +1: earlier text-book presupposed +in Autolycus 349-50: <I>Sphaerica</I> +of Theodosius ii. 245, 246-52, of +Menelaus ii. 252-3, 260, 261-73. +<p>Sphere-making 18: Archimedes on, +ii. 17-18. +<p>Spiric sections ii. 203-6. +<p>Sporus 226: criticisms on <I>quadra- +trix</I> 229-30: <G>khri/a</G> 234: duplica- +tion of cube 266-8. +<p>Square root, extraction of, 60-3: +ex. in sexagesimal fractions +(Theon) 61-2, (scholiast to Eu- +clid) 63: method of approxima- +ting to surds ii. 51-2, ii. 323-6, +ii. 547-9, ii. 553-4. +<p>Square numbers 69: formation by +adding successive gnomons (odd +<pb n=585><head>ENGLISH INDEX</head> +numbers) 77: any square is sum +of two triangular numbers 83-4: +8 times a triangular number ++1 = square, 84, ii. 516. +<p><I>Ssade</I>, Phoenician sibilant (signs +<*> <*> <*> <*>) became <*> (900) 32. +<p>‘Stadium,’ 1/60th of 30°, ii. 215. +<p><I>Stadium</I> of Zeno 276-7, 281-3. +<p>Star-pentagon, or <I>pentagram</I>, of +Pythagoreans 161-2. +<p>Stereographic projection (Ptolemy) +ii. 292. +<p>Stevin, S. ii. 455. +<p>‘Stigma,’ name for numeral <G>s</G>, +originally <*> (digamma) 32. +<p>Strabo 121, ii. 107, ii. 220. +<p>Strato ii. 1. +<p>Subcontrary (= harmonic) mean, +defined 85. +<p>Subtraction in Greek notation 52. +<p>Surds: Theodorus on, 22-3, 155-6, +203-9, 304: Theaetetus's general- +ization 203-4, 205, 209, 304: see +also ‘Approximations’. +<p><I>Surface-Loci</I> 219, ii. 380-5: Euclid's +439-40, ii. 119, ii. 425-6. +<p><I>Sūrya-Siddhānta</I> ii. 253. +<p><I>Sussu = soss</I> (Babylonian for 60) 28, +ii. 215. +<p>Synesius of Cyrene ii. 293. +<p>Synthesis 371-2: defined by Pappus +ii. 400. +<p>Syracuse 8. +<p>Table of Chords 45, ii. 259-60, ii. +283. +<p><I>Tāittirīya Samhitā</I> 146. +<p>Tannery, P. 15, 44, 87, 89, 119, 132, +180, 182, 184, 188, 196 <I>n.</I>, 232, +279, 326, 440, ii. 51, ii. 105, ii. +204-5, ii. 215, ii. 218, ii. 253, +ii. 317, ii. 453, ii. 483, ii. 519, +ii. 538, ii. 545, 546, ii. 550, ii. 556, +ii. 561. +<p>Teles on secondary education 21. +<p>Teos inscription 32, 34. +<p>Tetrads of Apollonius 40. +<p>Tetrahedron: construction 416, ii. +368: volume of, ii. 335. +<p>Thābit b. Qurra: translator of Eu- +clid 362, 363: of Archimedes's +<I>Liber assumptorum</I> ii. 22: of +Apollonius's <I>Conics</I> V-VII ii. 127: +of Menelaus's <I>Elements of Geo- +metry</I> ii. 260: of Ptolemy ii. +274-5. +<p>Thales 2, 4, 67: one of Seven Wise +Men 128, 142: introduced geo- +metry into Greece 128: geometri- +cal theorems attributed to, 130- +7: measurement of height of +pyramid 129-30, and of distance +of ship from shore 131-3: defini- +tion of number 69: astronomy +137-9, ii. 244: predicted solar +eclipse 137-8. +<p>Theaetetus 2, 119, 170: on surds +22-3, 155, 203-4, 205, 209, 304: +investigated regular solids 159, +162, 212, 217: on irrationals 209- +12, 216-17. +<p>Themistius 221, 223, 224. +<p>Theodorus of Cyrene: taught mathe- +matics 22-3: on surds 22-3, 155- +6, 203-9, 304. +<p>Theodosius ii. 245-6: <I>Sphaerica</I> 349- +50, ii. 246-52: other works ii. +246: no trigonometry in, ii. 250. +<p><I>Theologumena arithmetices</I> 96, 97, +318. +<p>Theon of Alexandria: examples of +multiplication and division 58, +59-60: extraction of square root +61-3: edition of Euclid's <I>Elements</I> +360-1, ii. 527-8: of <I>Optics</I> 441, +ii. 528: <I>Catoptrica ib.</I>: commen- +tary on <I>Syntaxis</I> 58, 60, ii. 274, +ii. 526-7. +<p>Theon of Smyrna 12, 72, 73, 74, 75, +76, 79, 83, 87, ii. 515: treatise +of, ii. 238-44: on ‘side-’ and +‘diameter-numbers’ 91-3, 112: +forms of numbers which cannot +be squares 112-13. +<p>Theophrastus 158, 163: on Plato's +view of the earth 315. +<p>Theudius 320-1. +<p>Theuth, Egyptian god, reputed in- +ventor of mathematics 121. +<p>Thévenot, M. ii. 308. +<p>Thrasyllus 97, 176, 177, ii. 241, ii. +243. +<p>Thucydides ii. 207. +<p>Thymaridas: definition of unit 69: +‘rectilinear’ = prime numbers +72: <G>e)pa/nqhma</G>, a system of simple +equations solved 94. +<p>Timaeus of Locri 86. +<p>Tittel ii. 300, 301, 304. +<p><I>Tore</I> (or anchor-ring): use by Ar- +chytas 219, 247-9: sections of +(Perseus), ii. 203-6: volume of +<pb n=586><head>ENGLISH INDEX</head> +(Dionysodorus and Heron), ii. +218-19, ii. 334-5. +<p>Torelli, J. ii. 27. +<p>Transversal: Menelaus's theorem +for spherical and plane triangles +ii. 266-70: lemmas relating to +quadrilateral and transversal +(Pappus) ii. 419-20. +<p>‘Treasury of Analysis’ 421, 422, +439, ii. 399-427. +<p>Triangle: theorem about sum of +angles Pythagorean 135, 143, +Geminus and Aristotle on, 135-6. +<p>Triangle, spherical: called <G>tri/pleu- +ron</G> (Menelaus) ii. 262: proposi- +tions analogous to Euclid's on +plane triangles ii. 262-5: sum of +angles greater than two right +angles ii. 264. +<p>Triangular numbers 15, 69: forma- +tion 76-7: 8 times triangular +number +1 = a square 84, ii. +516. +<p>Trigonometry ii 5, ii. 198, ii. 257-9, +ii. 265-73, ii. 276-86, ii. 290-1. +<p>Trisection of any angle: solutions +235-44: Pappus on, ii. 385-6. +<p>Tschirnhausen, E. W. v., 200. +<p>Tycho Brahe 317, ii. 2, ii. 196. +<p>Tzifra (=0) ii. 547. +<p><I>Ukha-thebt</I> (side of base in pyramid) +126, 127. +<p>Unit: definitions (Pythagoreans, +Euclid, Thymaridas, Chrysippus) +69. +<p>Usener, H. 184, 188. +<p>Valla, G.: translator of extracts +from Euclid 365, and from Archi- +medes ii. 26. +<p>Venatorius, Thomas Gechauff: <I>ed. +princeps</I> of Archimedes ii. 27. +<p>Venturi, G. ii. 308. +<p>Vieta 200, 223, ii. 182, ii. 456, ii. 480, +ii. 557. +<p>Vigesimal system (of numerals) 26. +<p>Vincent, A. J. H. 50, 436, ii. 308, +ii. 545, ii. 546. +<p>Vitruvius 18, 147, 174, 213, ii. 1, +ii. 245: Vitruvius and Heron, +ii. 302-3. +<p>Viviani, V. ii. 261. +<p>Vogt, H., 156 <I>n.</I>, 203 <I>n.</I> +<p>Wescher, C. ii. 309. +<p>Wilamowitz - Moellendorff, U. v., +158 <I>n.</I>, 245, ii. 128. +<p>Xenocrates 24, 319: works on +Numbers 319: upheld ‘indivisible +lines’ 181. +<p>Xenophon, on arithmetic in educa- +tion 19. +<p>Xylander (W. Holzmann) ii. 454-5, +ii. 545. +<p>Ya&hdot;yā b. Khālid b. Barmak ii. 274. +<p>Zamberti, B., translator of Euclid +365, 441. +<p>Zeno of Elea 271-3: arguments on +motion 273-83. +<p>Zeno of Sidon on Eucl. I. 1, 359, ii. +221-2. +<p>Zenodorus ii. 207-13. +<p>Zero in Babylonian notation 29: +<*> in Ptolemy 39, 45. +<p>Zeuthen, H. G. 190, 206-9, 210-11, +398, 437, ii. 52, ii. 105, ii. 203, +ii. 290-1, ii. 405, ii. 444. +<p>Zodiac circle: obliquity discovered +by Oenopides 138, 174, ii. 244.