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+<pb>
+<C><B>A HISTORY OF
+GREEK MATHEMATICS</B></C>
+<C><B>SIR THOMAS HEATH</B></C>
+<C><B>VOLUME I</B></C>
+<C><B>FROM THALES TO EUCLID</B></C>
+<C><B><I>An independent world,
+Created out of pure intelligence.
+&mdash;Wordsworth</I></B></C>
+<C><B>Dover Publications, Inc.
+New York</B></C>
+<pb>
+<C>Published in Canada by General Publishing Company,
+Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.</C>
+<C>Published in the United Kingdom by Constable and Com-
+pany, Ltd.</C>
+<C>This Dover edition, first published in 1981, is an unabridged
+republication of the work first published in 1921 by the
+Clarendon Press, Oxford. For this edition the errata of the first
+edition have been corrected.</C>
+<C><I>International Standard Book Number: 0-486-24073-8
+Library of Congress Catalog Card Number: 80-70126</I></C>
+<C>Manufactured in the United States of America</C>
+<C>Dover Publications, Inc.</C>
+<C>180 Varick Street</C>
+<C>New York, N.Y. 10014</C>
+<pb>
+<head><B>PREFACE</B></head>
+<p>THE idea may seem quixotic, but it is nevertheless the
+author's confident hope that this book will give a fresh interest
+to the story of Greek mathematics in the eyes both of
+mathematicians and of classical scholars.
+<p>For the mathematician the important consideration is that
+the foundations of mathematics and a great portion of its
+content are Greek. The Greeks laid down the first principles,
+invented the methods <I>ab initio,</I> and fixed the terminology,
+Mathematics in short is a Greek science, whatever new
+developments modern analysis has brought or may bring.
+<p>The interest of the subject for the classical scholar is no
+doubt of a different kind. Greek mathematics reveals an
+important aspect of the Greek genius of which the student of
+Greek culture is apt to lose sight. Most people, when they
+think of the Greek genius, naturally call to mind its master-
+pieces in literature and art with their notes of beauty, truth,
+freedom and humanism. But the Greek, with his insatiable
+desire to know the true meaning of everything in the uni-
+verse and to be able to give a rational explanation of it, was
+just as irresistibly driven to natural science, mathematics, and
+exact reasoning in general or logic. This austere side of the
+Greek genius found perhaps its most complete expression in
+Aristotle. Aristotle would, however, by no means admit that
+mathematics was divorced from aesthetic; he could conceive,
+he said, of nothing more beautiful than the objects of mathe-
+matics. Plato delighted in geometry and in the wonders of
+numbers; <G>a)gewme/trhtos mhdei\s ei)si/tw</G>, said the inscription
+over the door of the Academy. Euclid was a no less typical
+Greek. Indeed, seeing that so much of Greek is mathematics,
+<pb n=vi>
+<head>PREFACE</head>
+it is arguable that, if one would understand the Greek genius
+fully, it would be a good plan to begin with their geometry.
+<p>The story of Greek mathematics has been written before.
+Dr. James Gow did a great service by the publication in 1884
+of his <I>Short History of Greek Mathematics</I>, a scholarly and
+useful work which has held its own and has been quoted with
+respect and appreciation by authorities on the history of
+mathematics in all parts of the world. At the date when he
+wrote, however, Dr. Gow had necessarily to rely upon the
+works of the pioneers Bretschneider, Hankel, Allman, and
+Moritz Cantor (first edition). Since then the subject has been
+very greatly advanced; new texts have been published, im-
+portant new documents have been discovered, and researches
+by scholars and mathematicians in different countries have
+thrown light on many obscure points. It is, therefore, high
+time for the complete story to be rewritten.
+<p>It is true that in recent years a number of attractive
+histories of mathematics have been published in England and
+America, but these have only dealt with Greek mathematics
+as part of the larger subject, and in consequence the writers
+have been precluded, by considerations of space alone, from
+presenting the work of the Greeks in sufficient detail.
+<p>The same remark applies to the German histories of mathe-
+matics, even to the great work of Moritz Cantor, who treats
+of the history of Greek mathematics in about 400 pages of
+vol. i. While no one would wish to disparage so great a
+monument of indefatigable research, it was inevitable that
+a book on such a scale would in time prove to be inadequate,
+and to need correction in details; and the later editions have
+unfortunately failed to take sufficient account of the new
+materials which have become available since the first edition
+saw the light.
+<p>The best history of Greek mathematics which exists at
+present is undoubtedly that of Gino Loria under the title
+<I>Le scienze esatte nell' antica Grecia</I> (second edition 1914,
+<pb n=vii>
+<head>PREFACE</head>
+Ulrico Hoepli, Milano). Professor Loria arranges his material
+in five Books, (1) on pre-Euclidean geometry, (2) on the
+Golden Age of Greek geometry (Euclid to Apollonius), (3) on
+applied mathematics, including astronomy, sphaeric, optics,
+&amp;c., (4) on the Silver Age of Greek geometry, (5) on the
+arithmetic of the Greeks. Within the separate Books the
+arrangement is chronological, under the names of persons or
+schools. I mention these details because they raise the
+question whether, in a history of this kind, it is best to follow
+chronological order or to arrange the material according to
+subjects, and, if the latter, in what sense of the word &lsquo;subject&rsquo;
+and within what limits. As Professor Loria says, his arrange-
+ment is &lsquo;a compromise between arrangement according to
+subjects and a strict adherence to chronological order, each of
+which plans has advantages and disadvantages of its own&rsquo;.
+<p>In this book I have adopted a new arrangement, mainly
+according to subjects, the nature of which and the reasons for
+which will be made clear by an illustration. Take the case of
+a famous problem which plays a great part in the history of
+Greek geometry, the doubling of the cube, or its equivalent,
+the finding of two mean proportionals in continued proportion
+between two given straight lines. Under a chronological
+arrangement this problem comes up afresh on the occasion of
+each new solution. Now it is obvious that, if all the recorded
+solutions are collected together, it is much easier to see the
+relations, amounting in some cases to substantial identity,
+between them, and to get a comprehensive view of the history
+of the problem. I have therefore dealt with this problem in
+a separate section of the chapter devoted to &lsquo;Special Problems&rsquo;,
+and I have followed the same course with the other famous
+problems of squaring the circle and trisecting any angle.
+<p>Similar considerations arise with regard to certain well-
+defined subjects such as conic sections. It would be incon-
+venient to interrupt the account of Menaechmus's solution
+of the problem of the two mean proportionals in order to
+<pb n=viii>
+<head>PREFACE</head>
+consider the way in which he may have discovered the conic
+sections and their fundamental properties. It seems to me
+much better to give the complete story of the origin and
+development of the geometry of the conic sections in one
+place, and this has been done in the chapter on conic sections
+associated with the name of Apollonius of Perga. Similarly
+a chapter has been devoted to algebra (in connexion with
+Diophantus) and another to trigonometry (under Hipparchus,
+Menelaus and Ptolemy).
+<p>At the same time the outstanding personalities of Euclid
+and Archimedes demand chapters to themselves. Euclid, the
+author of the incomparable <I>Elements</I>, wrote on almost all
+the other branches of mathematics known in his day. Archi-
+medes's work, all original and set forth in treatises which are
+models of scientific exposition, perfect in form and style, was
+even wider in its range of subjects. The imperishable and
+unique monuments of the genius of these two men must be
+detached from their surroundings and seen as a whole if we
+would appreciate to the full the pre-eminent place which they
+occupy, and will hold for all time, in the history of science.
+<p>The arrangement which I have adopted necessitates (as does
+any other order of exposition) a certain amount of repetition
+and cross-references; but only in this way can the necessary
+unity be given to the whole narrative.
+<p>One other point should be mentioned. It is a defect in the
+existing histories that, while they state generally the contents
+of, and the main propositions proved in, the great treatises of
+Archimedes and Apollonius, they make little attempt to
+describe the procedure by which the results are obtained.
+I have therefore taken pains, in the most significant cases,
+to show the course of the argument in sufficient detail to
+enable a competent mathematician to grasp the method used
+and to apply it, if he will, to other similar investigations.
+<p>The work was begun in 1913, but the bulk of it was
+written, as a distraction, during the first three years of the
+<pb n=ix>
+<head>PREFACE</head>
+war, the hideous course of which seemed day by day to
+enforce the profound truth conveyed in the answer of Plato
+to the Delians. When they consulted him on the problem set
+them by the Oracle, namely that of duplicating the cube, he
+replied, &lsquo;It must be supposed, not that the god specially
+wished this problem solved, but that he would have the
+Greeks desist from war and wickedness and cultivate the
+Muses, so that, their passions being assuaged by philosophy
+and mathematics, they might live in innocent and mutually
+helpful intercourse with one another&rsquo;.
+<p>Truly
+Greece and her foundations are<lb>
+Built below the tide of war,<lb>
+Based on the cryst&agrave;lline sea<lb>
+Of thought and its eternity.<lb>
+T. L. H.
+<pb>
+<table>
+<caption><B>CONTENTS OF VOL. I</B></caption>
+<tr><td>I. INTRODUCTORY</td><td align=right>PAGES 1-25</td></tr>
+<tr><td>The Greeks and mathematics</td><td align=right>1-3</td></tr>
+<tr><td>Conditions favouring development of philosophy among the Greeks</td><td align=right>3-10</td></tr>
+<tr><td>Meaning and classification of mathematics</td><td align=right>10-18</td></tr>
+<tr><td>(<G>a</G>) Arithmetic and logistic</td><td align=right>13-16</td></tr>
+<tr><td>(<G>b</G>) Geometry and geodaesia</td><td align=right>16</td></tr>
+<tr><td>(<G>g</G>) Physical subjects, mechanics, optics, &amp;c.</td><td align=right>17-18</td></tr>
+<tr><td>Mathematics in Greek education</td><td align=right>18-25</td></tr>
+<tr><td>II. GREEK NUMERICAL NOTATION AND ARITHMETICAL OPERATIONS</td><td align=right>26-64</td></tr>
+<tr><td>The decimal system</td><td align=right>26-27</td></tr>
+<tr><td>Egyptian numerical notation</td><td align=right>27-28</td></tr>
+<tr><td>Babylonian systems</td></tr>
+<tr><td>(<G>a</G>) Decimal. (<G>b</G>) Sexagesimal</td><td align=right>28-29</td></tr>
+<tr><td>Greek numerical notation</td><td align=right>29-45</td></tr>
+<tr><td>(<G>a</G>) The &lsquo;Herodianic&rsquo; signs</td><td align=right>30-31</td></tr>
+<tr><td>(<G>b</G>) The ordinary alphabetic numerals</td><td align=right>31-35</td></tr>
+<tr><td>(<G>g</G>) Mode of writing numbers in the ordinary alphabetic notation</td><td align=right>36-37</td></tr>
+<tr><td>(<G>d</G>) Comparison of the two systems of numerical notation</td><td align=right>37-39</td></tr>
+<tr><td>(<G>e</G>) Notation, for large numbers</td><td align=right>39-41</td></tr>
+<tr><td>(i) Apollonius's &lsquo;tetrads&rsquo;</td><td align=right>40</td></tr>
+<tr><td>(ii) Archimedes's system (by octads)</td><td align=right>40-41</td></tr>
+<tr><td>Fractions</td></tr>
+<tr><td>(<G>a</G>) The Egyptian system</td><td align=right>41-42</td></tr>
+<tr><td>(<G>b</G>) The ordinary Greek form, variously written</td><td align=right>42-44</td></tr>
+<tr><td>(<G>g</G>) Sexagesimal fractions</td><td align=right>44-45</td></tr>
+<tr><td>Practical calculation</td></tr>
+<tr><td>(<G>a</G>) The abacus</td><td align=right>46-52</td></tr>
+<tr><td>(<G>b</G>) Addition and subtraction</td><td align=right>52</td></tr>
+<tr><td>(<G>g</G>) Multiplication</td></tr>
+<tr><td>(i) The Egyptian method</td><td align=right>52-53</td></tr>
+<tr><td>(ii) The Greek method</td><td align=right>53-54</td></tr>
+<tr><td>(iii) Apollonius's continued multiplications</td><td align=right>54-57</td></tr>
+<tr><td>(iv) Examples of ordinary multiplications</td><td align=right>57-58</td></tr>
+<tr><td>(<G>d</G>) Division</td><td align=right>58-60</td></tr>
+<tr><td>(<G>e</G>) Extraction of the square root</td><td align=right>60-63</td></tr>
+<tr><td>(<G>z</G>) Extraction of the cube root</td><td align=right>63-64</td></tr>
+</table>
+<pb n=xii>
+<head>CONTENTS</head>
+<table>
+<tr><td>III. PYTHAGOREAN ARITHMETIC</td><td align=right>PAGES 65-117</td></tr>
+<tr><td>Numbers and the universe</td><td align=right>67-69</td></tr>
+<tr><td>Definitions of the unit and of number</td><td align=right>69-70</td></tr>
+<tr><td>Classification of numbers</td><td align=right>70-74</td></tr>
+<tr><td>&lsquo;Perfect&rsquo; and &lsquo;Friendly&rsquo; numbers</td><td align=right>74-76</td></tr>
+<tr><td>Figured numbers</td></tr>
+<tr><td>(<G>a</G>) Triangular numbers</td><td align=right>76-77</td></tr>
+<tr><td>(<G>b</G>) Square numbers and gnomons</td><td align=right>77</td></tr>
+<tr><td>(<G>g</G>) History of the term &lsquo;gnomon&rsquo;</td><td align=right>78-79</td></tr>
+<tr><td>(<G>d</G>) Gnomons of the polygonal numbers</td><td align=right>79</td></tr>
+<tr><td>(<G>e</G>) Right-angled triangles with sides in rational numbers</td><td align=right>79-82</td></tr>
+<tr><td>(<G>z</G>) Oblong numbers</td><td align=right>82-84</td></tr>
+<tr><td>The theory of proportion and means</td><td align=right>84-90</td></tr>
+<tr><td>(<G>a</G>) Arithmetic, geometric and harmonic means</td><td align=right>85-86</td></tr>
+<tr><td>(<G>b</G>) Seven other means distinguished</td><td align=right>86-89</td></tr>
+<tr><td>(<G>g</G>) Plato on geometric means between two squares or two cubes</td><td align=right>89-90</td></tr>
+<tr><td>(<G>d</G>) A theorem of Archytas</td><td align=right>90</td></tr>
+<tr><td>The &lsquo;irrational&rsquo;</td><td align=right>90-91</td></tr>
+<tr><td>Algebraic equations</td></tr>
+<tr><td>(<G>a</G>) &lsquo;Side-&rsquo; and &lsquo;diameter-&rsquo; numbers, giving successive approximations to &radic;2 (solutions of <MATH>2<I>x</I><SUP>2</SUP> - <I>y</I><SUP>2</SUP> = &plusmn; 1</MATH>)</td><td align=right>91-93</td></tr>
+<tr><td>(<G>b</G>) The <G>e)pa/nqhua</G> (&lsquo;bloom&rsquo;) of Thymaridas</td><td align=right>94-96</td></tr>
+<tr><td>(<G>g</G>) Area of rectangles in relation to perimeter (equation <MATH><I>xy</I> = 2<I>x</I> + <I>y</I></MATH>)</td><td align=right>96-97</td></tr>
+<tr><td>Systematic treatises on arithmetic (theory of numbers)</td><td align=right>97-115</td></tr>
+<tr><td>Nicomachus, <I>Introductio Arithmetica</I></td><td align=right>97-112</td></tr>
+<tr><td>Sum of series of cube numbers</td><td align=right>108-110</td></tr>
+<tr><td>Theon of Smyrna</td><td align=right>112-113</td></tr>
+<tr><td>Iamblichus, Commentary on Nicomachus</td><td align=right>113-115</td></tr>
+<tr><td>The <I>pythmen</I> and the rule of nine or seven</td><td align=right>115-117</td></tr>
+<tr><td>IV. THE EARLIEST GREEK GEOMETRY. THALES</td><td align=right>118-140</td></tr>
+<tr><td>The &lsquo;Summary&rsquo; of Proclus</td><td align=right>118-121</td></tr>
+<tr><td>Tradition as to the origin of geometry</td><td align=right>121-122</td></tr>
+<tr><td>Egyptian geometry, i.e. mensuration</td><td align=right>122-128</td></tr>
+<tr><td>The beginnings of Greek geometry. Thales</td><td align=right>128-139</td></tr>
+<tr><td>(<G>a</G>) Measurement of height of pyramid</td><td align=right>129-130</td></tr>
+<tr><td>(<G>b</G>) Geometrical theorems attributed to Thales</td><td align=right>130-137</td></tr>
+<tr><td>(<G>g</G>) Thales as astronomer</td><td align=right>137-139</td></tr>
+<tr><td>From Thales to Pythagoras</td><td align=right>139-140</td></tr>
+<tr><td>V. PYTHAGOREAN GEOMETRY</td><td align=right>141-169</td></tr>
+<tr><td>Pythagoras</td><td align=right>141-142</td></tr>
+<tr><td>Discoveries attributed to the Pythagoreans</td></tr>
+<tr><td>(<G>a</G>) Equality of sum of angles of any triangle to two right angles</td><td align=right>143-144</td></tr>
+<tr><td>(<G>b</G>) The &lsquo;Theorem of Pythagoras&rsquo;</td><td align=right>144-149</td></tr>
+<tr><td>(<G>g</G>) Application of areas and geometrical algebra (solu-tion of quadratic equations)</td><td align=right>150-154</td></tr>
+<tr><td>(<G>d</G>) The irrational</td><td align=right>154-157</td></tr>
+<tr><td>(<G>e</G>) The five regular solids</td><td align=right>158-162</td></tr>
+<tr><td>(<G>z</G>) Pythagorean astronomy</td><td align=right>162-165</td></tr>
+<tr><td>Recapitulation</td><td align=right>165-169</td></tr>
+</table>
+<pb n=xiii>
+<head>CONTENTS</head>
+<table>
+<tr><td>VI. PROGRESS IN THE ELEMENTS DOWN TO PLATO'S TIME</td><td align=right>PAGES 170-217</td></tr>
+<tr><td>Extract from Proclus's summary</td><td align=right>170-172</td></tr>
+<tr><td>Anaxagoras</td><td align=right>172-174</td></tr>
+<tr><td>Oenopides of Chios</td><td align=right>174-176</td></tr>
+<tr><td>Democritus</td><td align=right>176-181</td></tr>
+<tr><td>Hippias of Elis</td><td align=right>182</td></tr>
+<tr><td>Hippocrates of Chios</td><td align=right>182-202</td></tr>
+<tr><td>(<G>a</G>) Hippocrates's quadrature of lunes</td><td align=right>183-200</td></tr>
+<tr><td>(<G>b</G>) Reduction of the problem of doubling the cube to the finding of two mean proportionals</td><td align=right>200-201</td></tr>
+<tr><td>(<G>g</G>) The Elements as known to Hippocrates</td><td align=right>201-202</td></tr>
+<tr><td>Theodorus of Cyrene</td><td align=right>202-209</td></tr>
+<tr><td>Theaetetus</td><td align=right>209-212</td></tr>
+<tr><td>Archytas</td><td align=right>213-216</td></tr>
+<tr><td>Summary</td><td align=right>216-217</td></tr>
+<tr><td>VII. SPECIAL PROBLEMS</td><td align=right>218-270</td></tr>
+<tr><td>The squaring of the circle</td><td align=right>220-235</td></tr>
+<tr><td>Antiphon</td><td align=right>221-223</td></tr>
+<tr><td>Bryson</td><td align=right>223-225</td></tr>
+<tr><td>Hippias, Dinostratus, Nicomedes, &amp;c.</td><td align=right>225-226</td></tr>
+<tr><td>(<G>a</G>) The quadratrix of Hippias</td><td align=right>226-230</td></tr>
+<tr><td>(<G>b</G>) The spiral of Archimedes</td><td align=right>230-231</td></tr>
+<tr><td>(<G>g</G>) Solutions by Apollonius and Carpus</td><td align=right>231-232</td></tr>
+<tr><td>(<G>d</G>) Approximations to the value of <G>p</G></td><td align=right>232-235</td></tr>
+<tr><td>The trisection of any angle</td><td align=right>235-244</td></tr>
+<tr><td>(<G>a</G>) Reduction to a certain <G>neu=sis</G>, solved by conics</td><td align=right>235-237</td></tr>
+<tr><td>(<G>b</G>) The <G>neu=sis</G> equivalent to a cubic equation</td><td align=right>237-238</td></tr>
+<tr><td>(<G>g</G>) The conchoids of Nicomedes</td><td align=right>238-240</td></tr>
+<tr><td>(<G>d</G>) Another reduction to a <G>neu=sis</G> (Archimedes)</td><td align=right>240-241</td></tr>
+<tr><td>(<G>e</G>) Direct solutions by means of conics (Pappus)</td><td align=right>241-244</td></tr>
+<tr><td>The duplication of the cube, or the problem of the two mean proportionals</td><td align=right>244-270</td></tr>
+<tr><td>(<G>a</G>) History of the problem</td><td align=right>244-246</td></tr>
+<tr><td>(<G>b</G>) Archytas</td><td align=right>246-249</td></tr>
+<tr><td>(<G>g</G>) Eudoxus</td><td align=right>249-251</td></tr>
+<tr><td>(<G>d</G>) Menaechmus</td><td align=right>251-255</td></tr>
+<tr><td>(<G>e</G>) The solution attributed to Plato</td><td align=right>255-258</td></tr>
+<tr><td>(<G>z</G>) Eratosthenes</td><td align=right>258-260</td></tr>
+<tr><td>(<G>h</G>) Nicomedes</td><td align=right>260-262</td></tr>
+<tr><td>(<G>q</G>) Apollonius, Heron, Philon of Byzantium</td><td align=right>262-264</td></tr>
+<tr><td>(<G>i</G>) Diocles and the cissoid</td><td align=right>264-266</td></tr>
+<tr><td>(<G>k</G>) Sporus and Pappus</td><td align=right>266-268</td></tr>
+<tr><td>(<G>l</G>) Approximation to a solution by plane methods only</td><td align=right>268-270</td></tr>
+<tr><td>VIII. ZENO OF ELEA</td><td align=right>271-283</td></tr>
+<tr><td>Zeno's arguments about motion</td><td align=right>273-283</td></tr>
+<tr><td>IX. PLATO</td><td align=right>284-315</td></tr>
+<tr><td>Contributions to the philosophy of mathematics</td><td align=right>288-294</td></tr>
+<tr><td>(<G>a</G>) The hypotheses of mathematics</td><td align=right>289-290</td></tr>
+<tr><td>(<G>b</G>) The two intellectual methods</td><td align=right>290-292</td></tr>
+<tr><td>(<G>g</G>) Definitions</td><td align=right>292-294</td></tr>
+</table>
+<pb n=xiv>
+<head>CONTENTS</head>
+<table>
+<tr><td>IX. CONTINUED</td></tr>
+<tr><td>Summary of the mathematics in Plato</td><td align=right>PAGES 294-308</td></tr>
+<tr><td>(<G>a</G>) Regular and semi-regular solids</td><td align=right>294-295</td></tr>
+<tr><td>(<G>b</G>) The construction of the regular solids</td><td align=right>296-297</td></tr>
+<tr><td>(<G>g</G>) Geometric means between two square numbers or two cubes</td><td align=right>297</td></tr>
+<tr><td>(<G>d</G>) The two geometrical passages in the <I>Meno</I></td><td align=right>297-303</td></tr>
+<tr><td>(<G>e</G>) Plato and the doubling of the cube</td><td align=right>303</td></tr>
+<tr><td>(<G>z</G>) Solution of <MATH><I>x</I><SUP>2</SUP> + <I>y</I><SUP>2</SUP> = <I>z</I><SUP>2</SUP></MATH> in integers</td><td align=right>304</td></tr>
+<tr><td>(<G>h</G>) Incommensurables</td><td align=right>304-305</td></tr>
+<tr><td>(<G>q</G>) The Geometrical Number</td><td align=right>305-308</td></tr>
+<tr><td>Mathematical &lsquo;arts&rsquo;</td><td align=right>308-315</td></tr>
+<tr><td>(<G>a</G>) Optics</td><td align=right>309</td></tr>
+<tr><td>(<G>b</G>) Music</td><td align=right>310</td></tr>
+<tr><td>(<G>g</G>) Astronomy</td><td align=right>310-315</td></tr>
+<tr><td>X. FROM PLATO TO EUCLID</td><td align=right>316-353</td></tr>
+<tr><td>Heraclides of Pontus: astronomical discoveries</td><td align=right>316-317</td></tr>
+<tr><td>Theory of numbers (Speusippus, Xenocrates)</td><td align=right>318-319</td></tr>
+<tr><td>The Elements. Proclus's summary (<I>continued</I>)</td><td align=right>319-321</td></tr>
+<tr><td>Eudoxus</td><td align=right>322-335</td></tr>
+<tr><td>(<G>a</G>) Theory of proportion</td><td align=right>325-327</td></tr>
+<tr><td>(<G>b</G>) The method of exhaustion</td><td align=right>327-329</td></tr>
+<tr><td>(<G>g</G>) Theory of concentric spheres</td><td align=right>329-335</td></tr>
+<tr><td>Aristotle</td><td align=right>335-348</td></tr>
+<tr><td>(<G>a</G>) First principles</td><td align=right>336-338</td></tr>
+<tr><td>(<G>b</G>) Indications of proofs differing from Euclid's</td><td align=right>338-340</td></tr>
+<tr><td>(<G>g</G>) Propositions not found in Euclid</td><td align=right>340-341</td></tr>
+<tr><td>(<G>d</G>) Curves and solids known to Aristotle</td><td align=right>341-342</td></tr>
+<tr><td>(<G>e</G>) The continuous and the infinite</td><td align=right>342-344</td></tr>
+<tr><td>(<G>z</G>) Mechanics</td><td align=right>344-346</td></tr>
+<tr><td>The Aristotclian tract on indivisible lines</td><td align=right>346-348</td></tr>
+<tr><td>Sphaeric</td></tr>
+<tr><td>Autolycus of Pitane</td><td align=right>348-353</td></tr>
+<tr><td>A lost text-book on Sphaeric</td><td align=right>349-350</td></tr>
+<tr><td>Autolycus, <I>On the Moving Sphere</I>: relation to Euclid</td><td align=right>351-352</td></tr>
+<tr><td>Autolycus, <I>On Risings and Settings</I></td><td align=right>352-353</td></tr>
+<tr><td>XI. EUCLID</td><td align=right>354-446</td></tr>
+<tr><td>Date and traditions</td><td align=right>354-357</td></tr>
+<tr><td>Ancient commentaries, criticisms and references</td><td align=right>357-360</td></tr>
+<tr><td>The text of the <I>Elements</I></td><td align=right>360-361</td></tr>
+<tr><td>Latin and Arabic translations</td><td align=right>361-364</td></tr>
+<tr><td>The first printed editions</td><td align=right>364-365</td></tr>
+<tr><td>The study of Euclid in the Middle Ages</td><td align=right>365-369</td></tr>
+<tr><td>The first English editions</td><td align=right>369-370</td></tr>
+<tr><td>Technical terms</td></tr>
+<tr><td>(<G>a</G>) Terms for the formal divisions of a proposition</td><td align=right>370-371</td></tr>
+<tr><td>(<G>b</G>) The <G>diorismo/s</G> or statement of conditions of possi-bility</td><td align=right>371</td></tr>
+<tr><td>(<G>g</G>) Analysis, synthesis, reduction, <I>reductio ad absurdum</I></td><td align=right>371-372</td></tr>
+<tr><td>(<G>d</G>) Case, objection, porism, lemma</td><td align=right>372-373</td></tr>
+<tr><td>Analysis of the <I>Elements</I></td></tr>
+<tr><td>Book I</td><td align=right>373-379</td></tr>
+<tr><td>&quot; II</td><td align=right>379-380</td></tr>
+</table>
+<pb n=xv>
+<head>CONTENTS</head>
+<table>
+<tr><td>Book III</td><td align=right>PAGES 380-383</td></tr>
+<tr><td>&quot; IV</td><td align=right>383-384</td></tr>
+<tr><td>&quot; V</td><td align=right>384-391</td></tr>
+<tr><td>&quot; VI</td><td align=right>391-397</td></tr>
+<tr><td>&quot; VII</td><td align=right>397-399</td></tr>
+<tr><td>&quot; VIII</td><td align=right>399-400</td></tr>
+<tr><td>&quot; IX</td><td align=right>400-402</td></tr>
+<tr><td>&quot; X</td><td align=right>402-412</td></tr>
+<tr><td>&quot; XI</td><td align=right>412-413</td></tr>
+<tr><td>&quot; XII</td><td align=right>413-415</td></tr>
+<tr><td>&quot; XIII</td><td align=right>415-419</td></tr>
+<tr><td>The so-called Books XIV, XV</td><td align=right>419-421</td></tr>
+<tr><td>The <I>Data</I></td><td align=right>421-425</td></tr>
+<tr><td><I>On divisions</I> (<I>of figures</I>)</td><td align=right>425-430</td></tr>
+<tr><td>Lost geometrical works</td></tr>
+<tr><td>(<G>a</G>) The <I>Pseudaria</I></td><td align=right>430-431</td></tr>
+<tr><td>(<G>b</G>) The <I>Porisms</I></td><td align=right>431-438</td></tr>
+<tr><td>(<G>g</G>) The <I>Conics</I></td><td align=right>438-439</td></tr>
+<tr><td>(<G>d</G>) The <I>Surface Loci</I></td><td align=right>439-440</td></tr>
+<tr><td>Applied mathematics</td></tr>
+<tr><td>(<G>a</G>) The <I>Phaenomena</I></td><td align=right>440-441</td></tr>
+<tr><td>(<G>b</G>) <I>Optics</I> and <I>Catoptrica</I></td><td align=right>441-444</td></tr>
+<tr><td>(<G>g</G>) Music</td><td align=right>444-445</td></tr>
+<tr><td>(<G>d</G>) Works on mechanics attributed to Euclid</td><td align=right>445-446</td></tr>
+</table>
+<pb>
+<C>I</C>
+<C>INTRODUCTORY</C>
+<C>The Greeks and mathematics.</C>
+<p>IT is an encouraging sign of the times that more and more
+effort is being directed to promoting a due appreciation and
+a clear understanding of the gifts of the Greeks to mankind.
+What we owe to Greece, what the Greeks have done for
+civilization, aspects of the Greek genius: such are the themes
+of many careful studies which have made a wide appeal and
+will surely produce their effect. In truth all nations, in the
+West at all events, have been to school to the Greeks, in art,
+literature, philosophy, and science, the things which are essen-
+tial to the rational use and enjoyment of human powers and
+activities, the things which make life worth living to a rational
+human being. &lsquo;Of all peoples the Greeks have dreamed the
+dream of life the best.&rsquo; And the Greeks were not merely the
+pioneers in the branches of knowledge which they invented
+and to which they gave names. What they began they carried
+to a height of perfection which has not since been surpassed;
+if there are exceptions, it is only where a few crowded centuries
+were not enough to provide the accumulation of experience
+required, whether for the purpose of correcting hypotheses
+which at first could only be of the nature of guesswork, or of
+suggesting new methods and machinery.
+<p>Of all the manifestations of the Greek genius none is more
+impressive and even awe-inspiring than that which is revealed
+by the history of Greek mathematics. Not only are the range
+and the sum of what the Greek mathematicians actually
+accomplished wonderful in themselves; it is necessary to bear
+in mind that this mass of original work was done in an almost
+incredibly short space of time, and in spite of the comparative
+inadequacy (as it would seem to us) of the only methods at
+their disposal, namely those of pure geometry, supplemented,
+where necessary, by the ordinary arithmetical operations.
+<pb n=2><head>INTRODUCTORY</head>
+Let us, confining ourselves to the main subject of pure
+geometry by way of example, anticipate so far as to mark
+certain definite stages in its development, with the intervals
+separating them. In Thales's time (about 600 B. C.) we find
+the first glimmerings of a theory of geometry, in the theorems
+that a circle is bisected by any diameter, that an isosceles
+triangle has the angles opposite to the equal sides equal, and
+(if Thales really discovered this) that the angle in a semicircle
+is a right angle. Rather more than half a century later
+Pythagoras was taking the first steps towards the theory of
+numbers and continuing the work of making geometry a
+theoretical science; he it was who first made geometry one of
+the subjects of a liberal education. The Pythagoreans, before
+the next century was out (i. e. before, say, 450 B. C.), had practi-
+cally completed the subject-matter of Books I-II, IV, VI (and
+perhaps III) of Euclid's <I>Elements</I>, including all the essentials
+of the &lsquo;geometrical algebra&rsquo; which remained fundamental in
+Greek geometry; the only drawback was that their theory of
+proportion was not applicable to incommensurable but only
+to commensurable magnitudes, so that it proved inadequate
+as soon as the incommensurable came to be discovered.
+In the same fifth century the difficult problems of doubling
+the cube and trisecting any angle, which are beyond the
+geometry of the straight line and circle, were not only mooted
+but solved theoretically, the former problem having been first
+reduced to that of finding two mean proportionals in continued
+proportion (Hippocrates of Chios) and then solved by a
+remarkable construction in three dimensions (Archytas), while
+the latter was solved by means of the curve of Hippias of
+Elis known as the <I>quadratrix</I>; the problem of squaring the
+circle was also attempted, and Hippocrates, as a contribution
+to it, discovered and squared three out of the five lunes which
+can be squared by means of the straight line and circle. In
+the fourth century Eudoxus discovered the great theory of
+proportion expounded in Euclid, Book V, and laid down the
+principles of the <I>method of exhaustion</I> for measuring areas and
+volumes; the conic sections and their fundamental properties
+were discovered by Menaechmus; the theory of irrationals
+(probably discovered, so far as &radic;(2) is concerned, by the
+early Pythagoreans) was generalized by Theaetetus; and the
+<pb n=3><head>THE GREEKS AND MATHEMATICS</head>
+geometry of the sphere was worked out in systematic trea-
+tises. About the end of the century Euclid wrote his
+<I>Elements</I> in thirteen Books. The next century, the third,
+is that of Archimedes, who may be said to have anticipated
+the integral calculus, since, by performing what are practi-
+cally <I>integrations</I>, he found the area of a parabolic segment
+and of a spiral, the surface and volume of a sphere and a
+segment of a sphere, the volume of any segment of the solids
+of revolution of the second degree, the centres of gravity of
+a semicircle, a parabolic segment, any segment of a paraboloid
+of revolution, and any segment of a sphere or spheroid.
+Apollonius of Perga, the &lsquo;great geometer&rsquo;, about 200 B. C.,
+completed the theory of geometrical conics, with specialized
+investigations of normals as maxima and minima leading
+quite easily to the determination of the circle of curvature
+at any point of a conic and of the equation of the evolute of
+the conic, which with us is part of analytical conics. With
+Apollonius the main body of Greek geometry is complete, and
+we may therefore fairly say that four centuries sufficed to
+complete it.
+<p>But some one will say, how did all this come about? What
+special aptitude had the Greeks for mathematics? The answer
+to this question is that their genius for mathematics was
+simply one aspect of their genius for philosophy. Their
+mathematics indeed constituted a large part of their philo-
+sophy down to Plato. Both had the same origin.
+<C>Conditions favouring the development of philosophy
+among the Greeks.</C>
+<p>All men by nature desire to know, says Aristotle.<note>Arist. <I>Metaph.</I> A. 1, 980 a 21.</note> The
+Greeks, beyond any other people of antiquity, possessed the
+love of knowledge for its own sake; with them it amounted
+to an instinct and a passion.<note>Cf. Butcher, <I>Some Aspects of the Greek Genius</I>, 1892, p. 1.</note> We see this first of all in their
+love of adventure. It is characteristic that in the <I>Odyssey</I>
+Odysseus is extolled as the hero who had &lsquo;seen the cities of
+many men and learned their mind&rsquo;,<note><I>Od.</I> i. 3.</note> often even taking his life
+in his hand, out of a pure passion for extending his horizon,
+<pb n=4><head>INTRODUCTORY</head>
+as when he went to see the Cyclopes in order to ascertain &lsquo;what
+sort of people they were, whether violent and savage, with no
+sense of justice, or hospitable and godfearing&rsquo;.<note><I>Od.</I> ix. 174-6.</note> Coming
+nearer to historical times, we find philosophers and statesmen
+travelling in order to benefit by all the wisdom that other
+nations with a longer history had gathered during the cen-
+turies. Thales travelled in Egypt and spent his time with
+the priests. Solon, according to Herodotus,<note>Herodotus, i. 30.</note> travelled &lsquo;to see
+the world&rsquo; (<G>qewri/hs ei(/neken</G>), going to Egypt to the court of
+Amasis, and visiting Croesus at Sardis. At Sardis it was not
+till &lsquo;after he had seen and examined everything&rsquo; that he had
+the famous conversation with Croesus; and Croesus addressed
+him as the Athenian of whose wisdom and peregrinations he
+had heard great accounts, proving that he had covered much
+ground in seeing the world and pursuing philosophy.
+(Herodotus, also a great traveller, is himself an instance of
+the capacity of the Greeks for assimilating anything that
+could be learnt from any other nations whatever; and,
+although in Herodotus's case the object in view was less the
+pursuit of philosophy than the collection of interesting infor-
+mation, yet he exhibits in no less degree the Greek passion
+for seeing things as they are and discerning their meaning
+and mutual relations; &lsquo;he compares his reports, he weighs the
+evidence, he is conscious of his own office as an inquirer after
+truth&rsquo;.) But the same avidity for learning is best of all
+illustrated by the similar tradition with regard to Pythagoras's
+travels. Iamblichus, in his account of the life of Pythagoras,<note>Iamblichus, <I>De vita Pythagorica</I>, cc. 2-4.</note>
+says that Thales, admiring his remarkable ability, communi-
+cated to him all that he knew, but, pleading his own age and
+failing strength, advised him for his better instruction to go
+and study with the Egyptian priests. Pythagoras, visiting
+Sidon on the way, both because it was his birthplace and
+because he properly thought that the passage to Egypt would
+be easier by that route, consorted there with the descendants
+of Mochus, the natural philosopher and prophet, and with the
+other Phoenician hierophants, and was initiated into all
+the rites practised in Biblus, Tyre, and in many parts of
+Syria, a regimen to which he submitted, not out of religious
+<pb n=5><head>DEVELOPMENT OF PHILOSOPHY</head>
+enthusiasm, &lsquo;<I>as you might think</I>&rsquo; (<G>w(s a)/n tis a(plw=s u(pola/boi</G>),
+but much more through love and desire for philosophic
+inquiry, and in order to secure that he should not overlook
+any fragment of knowledge worth acquiring that might lie
+hidden in the mysteries or ceremonies of divine worship;
+then, understanding that what he found in Phoenicia was in
+some sort an offshoot or descendant of the wisdom of the
+priests of Egypt, he concluded that he should acquire learning
+more pure and more sublime by going to the fountain-head in
+Egypt itself.
+<p>&lsquo;There&rsquo;, continues the story, &lsquo;he studied with the priests
+and prophets and instructed himself on every possible topic,
+neglecting no item of the instruction favoured by the best
+judges, no individual man among those who were famous for
+their knowledge, no rite practised in the country wherever it
+was, and leaving no place unexplored where he thought he
+could discover something more. . . . And so he spent 22
+years in the shrines throughout Egypt, pursuing astronomy
+and geometry and, of set purpose and not by fits and starts or
+casually, entering into all the rites of divine worship, until he
+was taken captive by Cambyses's force and carried off to
+Babylon, where again he consorted with the Magi, a willing
+pupil of willing masters. By them he was fully instructed in
+their solemn rites and religious worship, and in their midst he
+attained to the highest eminence in arithmetic, music, and the
+other branches of learning. After twelve years more thus
+spent he returned to Samos, being then about 56 years old.&rsquo;
+<p>Whether these stories are true in their details or not is
+a matter of no consequence. They represent the traditional
+and universal view of the Greeks themselves regarding the
+beginnings of their philosophy, and they reflect throughout
+the Greek spirit and outlook.
+<p>From a scientific point of view a very important advantage
+possessed by the Greeks was their remarkable capacity for
+accurate observation. This is attested throughout all periods,
+by the similes in Homer, by vase-paintings, by the ethno-
+graphic data in Herodotus, by the &lsquo;Hippocratean&rsquo; medical
+books, by the biological treatises of Aristotle, and by the
+history of Greek astronomy in all its stages. To take two
+commonplace examples. Any person who examines the
+under-side of a horse's hoof, which we call a &lsquo;frog&rsquo; and the
+<pb n=6><head>INTRODUCTORY</head>
+Greeks called a &lsquo;swallow&rsquo;, will agree that the latter is
+the more accurate description. Or again, what exactness
+of perception must have been possessed by the architects and
+workmen to whom we owe the pillars which, seen from below,
+appear perfectly straight, but, when measured, are found to
+bulge out (<G>e)/ntasis</G>).
+<p>A still more essential fact is that the Greeks were a race of
+<I>thinkers.</I> It was not enough for them to know the fact (the
+<G>o(/ti</G>); they wanted to know the why and wherefore (the <G>dia\ ti/</G>),
+and they never rested until they were able to give a rational
+explanation, or what appeared to them to be such, of every
+fact or phenomenon. The history of Greek astronomy fur-
+nishes a good example of this, as well as of the fact that no
+visible phenomenon escaped their observation. We read in
+Cleomedes<note>Cleomedes, <I>De motu circulari</I>, ii. 6, pp. 218 sq.</note> that there were stories of extraordinary lunar
+eclipses having been observed which &lsquo;the more ancient of the
+mathematicians&rsquo; had vainly tried to explain; the supposed
+&lsquo;paradoxical&rsquo; case was that in which, while the sun appears
+to be still above the western horizon, the <I>eclipsed</I> moon is
+seen to rise in the east. The phenomenon was seemingly
+inconsistent with the recognized explanation of lunar eclipses
+as caused by the entrance of the moon into the earth's
+shadow; how could this be if both bodies were above the
+horizon at the same time? The &lsquo;more ancient&rsquo; mathemati-
+cians tried to argue that it was possible that a spectator
+standing on an <I>eminence</I> of the spherical earth might see
+along the generators of a <I>cone</I>, i.e. a little downwards on all
+sides instead of merely in the plane of the horizon, and so
+might see both the sun and the moon although the latter was
+in the earth's shadow. Cleomedes denies this, and prefers to
+regard the whole story of such cases as a fiction designed
+merely for the purpose of plaguing astronomers and philoso-
+phers; but it is evident that the cases had actually been
+observed, and that astronomers did not cease to work at the
+problem until they had found the real explanation, namely
+that the phenomenon is due to atmospheric refraction, which
+makes the sun visible to us though it is actually beneath the
+horizon. Cleomedes himself gives this explanation, observing
+that such cases of atmospheric refraction were especially
+<pb n=7><head>DEVELOPMENT OF PHILOSOPHY</head>
+noticeable in the neighbourhood of the Black Sea, and com-
+paring the well-known experiment of the ring at the bottom
+of a jug, where the ring, just out of sight when the jug is
+empty, is brought into view when water is poured in. We do
+not know who the &lsquo;more ancient&rsquo; mathematicians were who
+were first exercised by the &lsquo;paradoxical&rsquo; case; but it seems
+not impossible that it was the observation of this phenomenon,
+and the difficulty of explaining it otherwise, which made
+Anaxagoras and others adhere to the theory that there are
+other bodies besides the earth which sometimes, by their
+interposition, cause lunar eclipses. The story is also a good
+illustration of the fact that, with the Greeks, pure theory
+went hand in hand with observation. Observation gave data
+upon which it was possible to found a theory; but the theory
+had to be modified from time to time to suit observed new
+facts; they had continually in mind the necessity of &lsquo;saving
+the phenomena&rsquo; (to use the stereotyped phrase of Greek
+astronomy). Experiment played the same part in Greek
+medicine and biology.
+<p>Among the different Greek stocks the Ionians who settled
+on the coast of Asia Minor were the most favourably situated
+in respect both of natural gifts and of environment for initiat-
+ing philosophy and theoretical science. When the colonizing
+spirit first arises in a nation and fresh fields for activity and
+development are sought, it is naturally the younger, more
+enterprising and more courageous spirits who volunteer to
+leave their homes and try their fortune in new countries;
+similarly, on the intellectual side, the colonists will be at
+least the equals of those who stay at home, and, being the
+least wedded to traditional and antiquated ideas, they will be
+the most capable of striking out new lines. So it was with
+the Greeks who founded settlements in Asia Minor. The
+geographical position of these settlements, connected with the
+mother country by intervening islands, forming stepping-
+stones as it were from the one to the other, kept them in
+continual touch with the mother country; and at the same
+time their geographical horizon was enormously extended by
+the development of commerce over the whole of the Mediter-
+ranean. The most adventurous seafarers among the Greeks
+of Asia Minor, the Phocaeans, plied their trade successfully
+<pb n=8><head>INTRODUCTORY</head>
+as far as the Pillars of Hercules, after they had explored the
+Adriatic sea, the west coast of Italy, and the coasts of the
+Ligurians and Iberians. They are said to have founded
+Massalia, the most important Greek colony in the western
+countries, as early as 600 B. C. Cyrene, on the Libyan coast,
+was founded in the last third of the seventh century. The
+Milesians had, soon after 800 B. C., made settlements on the
+east coast of the Black Sea (Sinope was founded in 785); the
+first Greek settlements in Sicily were made from Euboea and
+Corinth soon after the middle of the eighth century (Syracuse
+734). The ancient acquaintance of the Greeks with the south
+coast of Asia Minor and with Cyprus, and the establishment
+of close relations with Egypt, in which the Milesians had a
+large share, belongs to the time of the reign of Psammetichus I
+(664-610 B. C.), and many Greeks had settled in that country.
+<p>The free communications thus existing with the whole of
+the known world enabled complete information to be collected
+with regard to the different conditions, customs and beliefs
+prevailing in the various countries and races; and, in parti-
+cular, the Ionian Greeks had the inestimable advantage of
+being in contact, directly and indirectly, with two ancient
+civilizations, the Babylonian and the Egyptian.
+<p>Dealing, at the beginning of the <I>Metaphysics</I>, with the
+evolution of science, Aristotle observes that science was
+preceded by the arts. The arts were invented as the result
+of general notions gathered from experience (which again was
+derived from the exercise of memory); those arts naturally
+came first which are directed to supplying the necessities of
+life, and next came those which look to its amenities. It was
+only when all such arts had been established that the sciences,
+which do not aim at supplying the necessities or amenities
+of life, were in turn discovered, and this happened first in
+the places where men began to have leisure. This is why
+the mathematical arts were founded in Egypt; for there the
+priestly caste was allowed to be at leisure. Aristotle does not
+here mention Babylon; but, such as it was, Babylonian
+science also was the monopoly of the priesthood.
+<p>It is in fact true, as Gomperz says,<note><I>Griechische Denker</I>, i, pp. 36, 37.</note> that the first steps on
+the road of scientific inquiry were, so far as we know from
+<pb n=9><head>DEVELOPMENT OF PHILOSOPHY</head>
+history, never accomplished except where the existence of an
+organized caste of priests and scholars secured the necessary
+industry, with the equally indispensable continuity of tradi-
+tion. But in those very places the first steps were generally
+the last also, because the scientific doctrines so attained tend,
+through their identification with religious prescriptions, to
+become only too easily, like the latter, mere lifeless dogmas.
+It was a fortunate chance for the unhindered spiritual de-
+velopment of the Greek people that, while their predecessors
+in civilization had an organized priesthood, the Greeks never
+had. To begin with, they could exercise with perfect freedom
+their power of unerring eclecticism in the assimilation of every
+kind of lore. &lsquo;It remains their everlasting glory that they
+discovered and made use of the serious scientific elements in
+the confused and complex mass of exact observations and
+superstitious ideas which constitutes the priestly wisdom of
+the East, and threw all the fantastic rubbish on one side.&rsquo;<note>Cumont, <I>Neue Jahrb&uuml;cher</I>, xxiv, 1911, p. 4.</note>
+For the same reason, while using the earlier work of
+Egyptians and Babylonians as a basis, the Greek genius
+could take an independent upward course free from every
+kind of restraint and venture on a flight which was destined
+to carry it to the highest achievements.
+<p>The Greeks then, with their &lsquo;unclouded clearness of mind&rsquo;
+and their freedom of thought, untrammelled by any &lsquo;Bible&rsquo; or
+its equivalent, were alone capable of creating the sciences as
+they did create them, i.e. as living things based on sound first
+principles and capable of indefinite development. It was a
+great boast, but a true one, which the author of the <I>Epinomis</I>
+made when he said, &lsquo;Let us take it as an axiom that, whatever
+the Greeks take from the barbarians, they bring it to fuller
+perfection&rsquo;.<note><I>Epinomis</I>, 987 D.</note> He has been speaking of the extent to which
+the Greeks had been able to explain the relative motions and
+speeds of the sun, moon and planets, while admitting that
+there was still much progress to be made before absolute
+certainty could be achieved. He adds a characteristic sen-
+tence, which is very relevant to the above remarks about the
+Greek's free outlook:
+<p>&lsquo;Let no Greek ever be afraid that we ought not at any time
+to study things divine because we are mortal. We ought to
+<pb n=10><head>INTRODUCTORY</head>
+maintain the very contrary view, namely, that God cannot
+possibly be without intelligence or be ignorant of human
+nature: rather he knows that, when he teaches them, men
+will follow him and learn what they are taught. And he is
+of course perfectly aware that he does teach us, and that we
+learn, the very subject we are now discussing, number and
+counting; if he failed to know this, he would show the
+greatest want of intelligence; the God we speak of would in
+fact not know himself, if he took it amiss that a man capable
+of learning should learn, and if he did not rejoice unreservedly
+with one who became good by divine influence.&rsquo;<note><I>Epinomis</I>, 988 A.</note>
+<p>Nothing could well show more clearly the Greek conviction
+that there could be no opposition between religion and scien-
+tific truth, and therefore that there could be no impiety in the
+pursuit of truth. The passage is a good parallel to the state-
+ment attributed to Plato that <G>qeo\s a)ei\ lewmetrei=</G>.
+<C>Meaning and classification of mathematics.</C>
+<p>The words <G>maqh/mata</G> and <G>maqhmatiko/s</G> do not appear to
+have been definitely appropriated to the special meaning of
+mathematics and mathematicians or things mathematical until
+Aristotle's time. With Plato <G>ma/qhma</G> is quite general, mean-
+ing any subject of instruction or study; he speaks of <G>kala\
+maqh/mata</G>, good subjects of instruction, as of <G>kala\ e)pithdeu/-
+mata</G>, good pursuits, of women's subjects as opposed to men's,
+of the Sophists hawking sound <G>maqh/mata</G>; what, he asks in
+the <I>Republic</I>, are the greatest <G>maqh/mata</G>? and he answers that
+the greatest <G>ma/qhma</G> is the Idea of the Good.<note><I>Republic</I>, vi. 505 A.</note> But in the
+<I>Laws</I> he speaks of <G>tri/a maqh/mata</G>, three subjects, as fit for
+freeborn men, the subjects being arithmetic, the science of
+measurement (geometry), and astronomy<note><I>Laws</I>, vii. 817 E.</note>; and no doubt the
+pre-eminent place given to mathematical subjects in his scheme
+of education would have its effect in encouraging the habit of
+speaking of these subjects exclusively as <G>maqh/mata</G>. The
+Peripatetics, we are told, explained the special use of the
+word in this way; they pointed out that, whereas such things
+as rhetoric and poetry and the whole of popular <G>mousikh/</G> can
+be understood even by one who has not learnt them, the sub-
+jects called by the special name of <G>maqh/mata</G> cannot be known
+<pb n=11><head>CLASSIFICATION OF MATHEMATICS</head>
+by any one who has not first gone through a course of instruc-
+tion in them; they concluded that it was for this reason that
+these studies were called <G>maqhmatikh/</G>.<note>Anatolius in Hultsch's Heron, pp. 276-7 (Heron, vol. iv, Heiberg,
+p. 160. 18-24).</note> The special use of the
+word <G>maqhmatikh/</G> seems actually to have originated in the
+school of Pythagoras. It is said that the esoteric members
+of the school, those who had learnt the theory of know-
+ledge in its most complete form and with all its elaboration
+of detail, were known as <G>maqhmatikoi/</G>, mathematicians (as
+opposed to the <G>a)kousmatikoi/</G>, the exoteric learners who were
+entrusted, not with the inner theory, but only with the prac-
+tical rules of conduct); and, seeing that the Pythagorean
+philosophy was mostly mathematics, the term might easily
+come to be identified with the mathematical subjects as
+distinct from others. According to Anatolius, the followers
+of Pythagoras are said to have applied the term <G>maqhmatikh/</G>
+more particularly to the two subjects of geometry and
+arithmetic, which had previously been known by their own
+separate names only and not by any common designation
+covering both.<note>Heron, ed. Hultsch, p. 277; vol. iv, p. 160. 24-162. 2, Heiberg.</note> There is also an apparently genuine frag-
+ment of Archytas, a Pythagorean and a contemporary and
+friend of Plato, in which the word <G>maqh/mata</G> appears as
+definitely appropriated to mathematical subjects:
+<p>&lsquo;The mathematicians (<G>toi\ peri\ ta\ maqh/mata</G>) seem to me to
+have arrived at correct conclusions, and it is not therefore
+surprising that they have a true conception of the nature of
+each individual thing: for, having reached such correct con-
+clusions regarding the nature of the universe, they were
+bound to see in its true light the nature of particular things
+as well. Thus they have handed down to us clear knowledge
+about the speed of the stars, their risings and settings, and
+about geometry, arithmetic, and sphaeric, and last, not least,
+about music; for these <G>maqh/mata</G> seem to be sisters.&rsquo;<note>Diels, <I>Vorsokratiker</I>, i<SUP>3</SUP>, pp. 330-1.</note>
+<p>This brings us to the Greek classification of the different
+branches of mathematics. Archytas, in the passage quoted,
+specifies the four subjects of the Pythagorean <I>quadrivium</I>,
+geometry, arithmetic, astronomy, and music (for &lsquo;sphaeric&rsquo;
+means astronomy, being the geometry of the sphere con-
+<pb n=12><head>INTRODUCTORY</head>
+sidered solely with reference to the problem of accounting for
+the motions of the heavenly bodies); the same list of subjects
+is attributed to the Pythagoreans by Nicomachus, Theon of
+Smyrna, and Proclus, only in a different order, arithmetic,
+music, geometry, and sphaeric; the idea in this order was
+that arithmetic and music were both concerned with number
+(<G>poso/n</G>), arithmetic with number in itself, music with number
+in relation to something else, while geometry and sphaeric were
+both concerned with magnitude (<G>phli/kon</G>), geometry with mag-
+nitude at rest, sphaeric with magnitude in motion. In Plato's
+curriculum for the education of statesmen the same subjects,
+with the addition of stereometry or solid geometry, appear,
+arithmetic first, then geometry, followed by solid geometry,
+astronomy, and lastly harmonics. The mention of stereometry
+as an independent subject is Plato's own idea; it was, however,
+merely a formal addition to the curriculum, for of course
+solid problems had been investigated earlier, as a part of
+geometry, by the Pythagoreans, Democritus and others.
+Plato's reason for the interpolation was partly logical. Astro-
+nomy treats of the motion of solid bodies. There is therefore
+a gap between plane geometry and astronomy, for, after con-
+sidering plane figures, we ought next to add the third dimen-
+sion and consider solid figures in themselves, before passing
+to the science which deals with such figures in motion. But
+Plato emphasized stereometry for another reason, namely that
+in his opinion it had not been sufficiently studied. &lsquo;The
+properties of solids do not yet seem to have been discovered.&rsquo;
+He adds:
+<p>&lsquo;The reasons for this are two. First, it is because no State
+holds them in honour that these problems, which are difficult,
+are feebly investigated; and, secondly, those who do investi-
+gate them are in need of a superintendent, without whose
+guidance they are not likely to make discoveries. But, to
+begin with, it is difficult to find such a superintendent, and
+then, even supposing him found, as matters now stand, those
+who are inclined to these researches would be prevented by
+their self-conceit from paying any heed to him.&rsquo;<note>Plato, <I>Republic</I>, vii. 528 A-C.</note>
+<p>I have translated <G>w(s nu=n e)/xei</G> (&lsquo;as matters now stand&rsquo;) in
+this passage as meaning &lsquo;in present circumstances&rsquo;, i.e. so
+<pb n=13><head>CLASSIFICATION OF MATHEMATICS</head>
+long as the director has not the authority of the State behind
+him: this seems to be the best interpretation in view of the
+whole context; but it is possible, as a matter of construction,
+to connect the phrase with the preceding words, in which case
+the meaning would be &lsquo;and, even when such a superintendent
+has been found, as is the case at present&rsquo;, and Plato would
+be pointing to some distinguished geometer among his con-
+temporaries as being actually available for the post. If Plato
+intended this, it would presumably be either Archytas or
+Eudoxus whom he had in mind.
+<p>It is again on a logical ground that Plato made harmonics
+or music follow astronomy in his classification. As astronomy
+is the motion of bodies (<G>fora\ ba/qous</G>) and appeals to the eye,
+so there is a harmonious motion (<G>e)narmo/nios fora/</G>), a motion
+according to the laws of harmony, which appeals to the ear.
+In maintaining the sisterhood of music and astronomy Plato
+followed the Pythagorean view (cf. the passage of Archytas
+above quoted and the doctrine of the &lsquo;harmony of the
+spheres&rsquo;).
+<C>(<G>a</G>) <I>Arithmetic and logistic.</I></C>
+<p>By arithmetic Plato meant, not arithmetic in our sense, but
+the science which considers numbers in themselves, in other
+words, what we mean by the Theory of Numbers. He does
+not, however, ignore the art of calculation (arithmetic in our
+sense); he speaks of number and calculation (<G>a)riqmo\n kai\
+logismo/n</G>) and observes that &lsquo;the art of calculation (<G>logistikh/</G>)
+and arithmetic (<G>a)riqmhtikh/</G>) are both concerned with number&rsquo;;
+those who have a natural gift for calculation (<G>oi( fu/sei logi-
+stikoi/</G>) have, generally speaking, a talent for learning of all
+kinds, and even those who are slow are, by practice in it,
+made smarter.<note><I>Republic</I>, vii. 522 C, 525 A, 526 B.</note> But the art of calculation (<G>logistikh/</G>) is only
+preparatory to the true science; those who are to govern the
+city are to get a grasp of <G>logistikh/</G>, not in the popular
+sense with a view to use in trade, but only for the purpose of
+knowledge, until they are able to contemplate the nature of
+number in itself by thought alone.<note><I>Ib.</I> vii. 525 B, C.</note> This distinction between
+<G>a)riqmhtikh/</G> (the theory of numbers) and <G>logistikh/</G> (the art of
+<pb n=14><head>INTRODUCTORY</head>
+calculation) was a fundamental one in Greek mathematics.
+It is found elsewhere in Plato,<note>Cf. <I>Gorgias</I>, 451 B, C; <I>Theaetetus</I>, 145 A with 198 A, &amp;c.</note> and it is clear that it was well
+established in Plato's time. Archytas too has <G>logistikh/</G> in
+the same sense; the art of calculation, he says, seems to be far
+ahead of other arts in relation to wisdom or philosophy, nay
+it seems to make the things of which it chooses to treat even
+clearer than geometry does; moreover, it often succeeds even
+where geometry fails.<note>Diels, <I>Vorsokratiker</I>, i<SUP>3</SUP>, p. 337. 7-11.</note> But it is later writers on the classification
+of mathematics who alone go into any detail of what <G>logistikh/</G>
+included. Geminus in Proclus, Anatolius in the <I>Variae Collec-
+tiones</I> included in Hultsch's Heron, and the scholiast to Plato's
+<I>Charmides</I> are our authorities. Arithmetic, says Geminus,<note>Proclus on Eucl. I, p. 39. 14-20.</note> is
+divided into the theory of linear numbers, the theory of plane
+numbers, and the theory of solid numbers. It investigates,
+in and by themselves, the species of number as they are succes-
+sively evolved from the unit, the formation of plane numbers,
+similar and dissimilar, and the further progression to the third
+dimension. As for the <G>logistiko/s</G>, it is not in and by themselves
+that he considers the properties of numbers but with refer-
+ence to sensible objects; and for this reason he applies to
+them names adapted from the objects measured, calling some
+(numbers) <G>mhli/ths</G> (from <G>mh=lon</G>, a sheep, or <G>mh=lon</G>, an apple,
+more probably the latter) and others <G>fiali/ths</G> (from <G>fia/lh</G>,
+a bowl).<note><I>Ib.</I>, p. 40. 2-5.</note> The scholiast to the <I>Charmides</I> is fuller still:<note>On <I>Charmides</I>, 165 E.</note>
+<p>&lsquo;Logistic is the science which deals with numbered things,
+not numbers; it does not take number in its essence,
+but it presupposes 1 as unit, and the numbered object as
+number, e.g. it regards 3 as a triad, 10 as a decad, and
+applies the theorems of arithmetic to such (particular) cases.
+Thus it is logistic which investigates on the one hand what
+Archimedes called the cattle-problem, and on the other hand
+<I>melites</I> and <I>phialites</I> numbers, the latter relating to bowls,
+the former to flocks (he should probably have said &ldquo;apples&rdquo;);
+in other kinds too it investigates the numbers of sensible
+bodies, treating them as absolute (<G>w(s peri\ telei/wn</G>). Its sub-
+ject-matter is everything that is numbered. Its branches
+include the so-called Greek and Egyptian methods in multi-
+plications and divisions,<note>See Chapter II, pp. 52-60.</note> the additions and decompositions
+<pb n=15><head>ARITHMETIC AND LOGISTIC</head>
+of fractions; which methods it uses to explore the secrets of
+the theory of triangular and polygonal numbers with reference
+to the subject-matter of particular problems.&rsquo;
+<p>The content of <I>logistic</I> is for the most part made fairly
+clear by the scholia just quoted. First, it comprised the
+ordinary arithmetical operations, addition, subtraction, multi-
+plication, division, and the handling of fractions; that is, it
+included the elementary parts of what we now call <I>arithmetic.</I>
+Next, it dealt with problems about such things as sheep
+(or apples), bowls, &amp;c.; and here we have no difficulty in
+recognizing such problems as we find in the arithmetical
+epigrams included in the Greek anthology. Several of them
+are problems of dividing a number of apples or nuts among
+a certain number of persons; others deal with the weights of
+bowls, or of statues and their pedestals, and the like; as a
+rule, they involve the solution of simple equations with one
+unknown, or easy simultaneous equations with two unknowns;
+two are indeterminate equations of the first degree to be solved
+in positive integers. From Plato's allusions to such problems
+it is clear that their origin dates back, at least, to the fifth
+century B. C. The cattle-problem attributed to Archimedes
+is of course a much more difficult problem, involving the
+solution of a &lsquo;Pellian&rsquo; equation in numbers of altogether
+impracticable size. In this problem the sums of two pairs
+of unknowns have to be respectively a square and a tri-
+angular number; the problem would therefore seem to
+correspond to the description of those involving &lsquo;the theory
+of triangular and polygonal numbers&rsquo;. Tannery takes the
+allusion in the last words to be to problems in indeter-
+minate analysis like those of Diophantus's <I>Arithmetica.</I> The
+difficulty is that most of Diophantus's problems refer to num-
+bers such that their sums, differences, &amp;c., are <I>squares</I>, whereas
+the scholiast mentions only triangular and polygonal numbers.
+Tannery takes squares to be included among polygons, or to
+have been accidentally omitted by a copyist. But there is
+only one use in Diophantus's <I>Arithmetica</I> of a triangular
+number (in IV. 38), and none of a polygonal number; nor can
+the <G>trigw/nous</G> of the scholiast refer, as Tannery supposes, to
+right-angled triangles with sides in rational numbers (the
+main subject of Diophantus's Book VI), the use of the mascu-
+<pb n=16><head>INTRODUCTORY</head>
+line showing that only <G>trigw/nous a)riqmou/s</G>, triangular <I>num-
+bers</I>, can be meant. Nevertheless there can, I think, be no
+doubt that Diophantus's <I>Arithmetica</I> belongs to <I>Logistic.</I>
+Why then did Diophantus call his thirteen books <I>Arithmetica</I>?
+The explanation is probably this. Problems of the Diophan-
+tine type, like those of the arithmetical epigrams, had pre-
+viously been enunciated of concrete numbers (numbers of
+apples, bowls, &amp;c.), and one of Diophantus's problems (V. 30)
+is actually in epigram form, and is about measures of wine
+with prices in drachmas. Diophantus then probably saw that
+there was no reason why such problems should refer to
+numbers of any one particular thing rather than another, but
+that they might more conveniently take the form of finding
+numbers <I>in the abstract</I> with certain properties, alone or in
+combination, and therefore that they might claim to be part
+of arithmetic, the abstract science or theory of numbers.
+<p>It should be added that to the distinction between <I>arith-
+metic</I> and <I>logistic</I> there corresponded (up to the time of
+Nicomachus) different methods of treatment. With rare
+exceptions, such as Eratosthenes's <G>ko/skinon</G>, or sieve, a device
+for separating out the successive prime numbers, the theory
+of numbers was only treated in connexion with geometry, and
+for that reason only the geometrical form of proof was used,
+whether the figures took the form of dots marking out squares,
+triangles, gnomons, &amp;c. (as with the early Pythagoreans), or of
+straight lines (as in Euclid VII-IX); even Nicomachus did
+not entirely banish geometrical considerations from his work,
+and in Diophantus's treatise on Polygonal Numbers, of which
+a fragment survives, the geometrical form of proof is used.
+<C>(<G>b</G>) <I>Geometry and geodaesia.</I></C>
+<p>By the time of Aristotle there was separated out from
+geometry a distinct subject, <G>gewdaisi/a</G>, <I>geodesy</I>, or, as we
+should say, <I>mensuration</I>, not confined to land-measuring, but
+covering generally the practical measurement of surfaces and
+volumes, as we learn from Aristotle himself,<note>Arist. <I>Metaph.</I> B. 2, 997 b 26, 31.</note> as well as from
+a passage of Geminus quoted by Proclus.<note>Proclus on Eucl. I, p. 39. 20-40. 2.</note>
+<pb n=17><head>PHYSICAL SUBJECTS AND THEIR BRANCHES</head>
+<C>(<G>g</G>) <I>Physical subjects, mechanics, optics, harmonics,
+astronomy, and their branches.</I></C>
+<p>In applied mathematics Aristotle recognizes optics and
+mechanics in addition to astronomy and harmonics. He calls
+optics, harmonics, and astronomy the <I>more physical</I> (branches)
+of mathematics,<note>Arist. <I>Phys.</I> ii. 2, 194 a 8.</note> and observes that these subjects and mechanics
+depend for the proofs of their propositions upon the pure
+mathematical subjects, optics on geometry, mechanics on
+geometry or stereometry, and harmonics on arithmetic; simi-
+larly, he says, <I>Phaenomena</I> (that is, observational astronomy)
+depend on (theoretical) astronomy.<note>Arist. <I>Anal. Post.</I> i. 9, 76 a 22-5; i. 13, 78 b 35-9.</note>
+<p>The most elaborate classification of mathematics is that given
+by Geminus.<note>Proclus on Eucl. I, p. 38. 8-12.</note> After arithmetic and geometry, which treat of
+non-sensibles, or objects of pure thought, come the branches
+which are concerned with sensible objects, and these are six
+in number, namely mechanics, astronomy, optics, geodesy,
+<I>canonic</I> (<G>kanonikh/</G>), <I>logistic.</I> Anatolius distinguishes the same
+subjects but gives them in the order <I>logistic</I>, geodesy, optics,
+<I>canonic</I>, mechanics, astronomy.<note>See Heron, ed. Hultsch, p. 278; ed. Heiberg, iv, p. 164.</note> <I>Logistic</I> has already been
+discussed. Geodesy too has been described as <I>mensuration</I>,
+the practical measurement of surfaces and volumes; as
+Geminus says, it is the function of geodesy to measure, not
+a cylinder or a cone (as such), but heaps as cones, and tanks
+or pits as cylinders.<note>Proclus on Eucl. I, p. 39. 23-5.</note> <I>Canonic</I> is the theory of the musical
+intervals as expounded in works like Euclid's <G>katatomh\
+kano/nos</G>, <I>Division of the canon.</I>
+<p>Optics is divided by Geminus into three branches.<note><I>Ib.</I>, p. 40. 13-22.</note> (1) The
+first is Optics proper, the business of which is to explain why
+things appear to be of different sizes or different shapes
+according to the way in which they are placed and the
+distances at which they are seen. Euclid's <I>Optics</I> consists
+mainly of propositions of this kind; a circle seen edge-
+wise looks like a straight line (Prop. 22), a cylinder seen by
+one eye appears less than half a cylinder (Prop. 28); if the
+line joining the eye to the centre of a circle is perpendicular
+<pb n=18><head>INTRODUCTORY</head>
+to the plane of the circle, all its diameters will look equal
+(Prop. 34), but if the joining line is neither perpendicular to
+the plane of the circle nor equal to its radius, diameters with
+which it makes unequal angles will appear unequal (Prop. 35);
+if a visible object remains stationary, there exists a locus such
+that, if the eye is placed at any point on it, the object appears
+to be of the same size for every position of the eye (Prop. 38).
+(2) The second branch is <I>Catoptric</I>, or the theory of mirrors,
+exemplified by the <I>Catoptrica</I> of Heron, which contains,
+e.g., the theorem that the angles of incidence and reflexion
+are equal, based on the assumption that the broken line
+connecting the eye and the object reflected is a minimum.
+(3) The third branch is <G>skhnografikh/</G> or, as we might say,
+<I>scene-painting</I>, i.e. applied perspective.
+<p>Under the general term of mechanics Geminus<note>Proclus on Eucl. I, p. 41. 3-18.</note> dis-
+tinguishes (1) <G>o)rganopoii+kh/</G>, the art of making engines of war
+(cf. Archimedes's reputed feats at the siege of Syracuse and
+Heron's <G>belopoii+ka/</G>), (2) <G>qaumatopoii+kh/</G>, the art of making
+<I>wonderful machines</I>, such as those described in Heron's
+<I>Pneumatica</I> and <I>Automatic Theatre</I>, (3) Mechanics proper,
+the theory of centres of gravity, equilibrium, the mechanical
+powers, &amp;c., (4) <I>Sphere-making</I>, the imitation of the move-
+ments of the heavenly bodies; Archimedes is said to have
+made such a sphere or orrery. Last of all,<note><I>Ib.</I>, pp. 41. 19-42. 6.</note> astronomy
+is divided into (1) <G>gnwmonikh/</G>, the art of the gnomon, or the
+measurement of time by means of the various forms of
+sun-dials, such as those enumerated by Vitruvius,<note>Vitruvius, <I>De architectura</I>, ix. 8.</note> (2) <G>metewro-
+skopikh/</G>, which seems to have included, among other things,
+the measurement of the heights at which different stars cross
+the meridian, (3) <G>dioptrikh/</G>, the use of the <I>dioptra</I> for the
+purpose of determining the relative positions of the sun,
+moon, and stars.
+<C>Mathematics in Greek education.<note>Cf. Freeman, <I>Schools of Hellas</I>, especially pp. 100-7, 159.</note></C>
+<p>The elementary or primary stage in Greek education lasted
+till the age of fourteen. The main subjects were letters
+(reading and writing followed by dictation and the study of
+<pb n=19><head>MATHEMATICS IN GREEK EDUCATION</head>
+literature), music and gymnastics; but there is no reasonable
+doubt that practical arithmetic (in our sense), including
+weights and measures, was taught along with these subjects.
+Thus, at the stage of spelling, a common question asked of
+the pupils was, How many letters are there in such and such
+a word, e.g. Socrates, and in what order do they come?<note>Xenophon, <I>Econ.</I> viii. 14.</note> This
+would teach the cardinal and ordinal numbers. In the same
+connexion Xenophon adds, &lsquo;Or take the case of numbers.
+Some one asks, What is twice five?&rsquo;<note>Xenophon, <I>Mem.</I> iv. 4. 7.</note> This indicates that
+counting was a part of learning letters, and that the multipli-
+cation table was a closely connected subject. Then, again,
+there were certain games, played with cubic dice or knuckle-
+bones, to which boys were addicted and which involved some
+degree of arithmetical skill. In the game of knucklebones in
+the <I>Lysis</I> of Plato each boy has a large basket of them, and
+the loser in each game pays so many over to the winner.<note>Plato, <I>Lysis</I>, 206 E; cf. Apollonius Rhodius, iii. 117.</note>
+Plato connects the art of playing this game with mathe-
+matics<note><I>Phaedrus</I>, 274 C-D.</note>; so too he associates <G>pettei/a</G> (games with <G>pessoi/</G>,
+somewhat resembling draughts or chess) with arithmetic in
+general.<note><I>Politicus</I>, 299 E; <I>Laws</I>, 820 C.</note> When in the <I>Laws</I> Plato speaks of three subjects
+fit for freeborn citizens to learn, (1) calculation and the science
+of numbers, (2) mensuration in one, two and three dimen-
+sions, and (3) astronomy in the sense of the knowledge of
+the revolutions of the heavenly bodies and their respective
+periods, he admits that profound and accurate knowledge of
+these subjects is not for people in general but only for a few.<note><I>Laws</I>, 817 E-818 A.</note>
+But it is evident that practical arithmetic was, after letters
+and the lyre, to be a subject for all, so much of arithmetic,
+that is, as is necessary for purposes of war, household
+management, and the work of government. Similarly, enough
+astronomy should be learnt to enable the pupil to understand
+the calendar.<note><I>Ib.</I> 809 C, D.</note> Amusement should be combined with instruc-
+tion so as to make the subjects attractive to boys. Plato was
+much attracted by the Egyptian practice in this matter:<note><I>Ib.</I> 819 A-C.</note>
+<p>&lsquo;Freeborn boys should learn so much of these things as
+vast multitudes of boys in Egypt learn along with their
+<pb n=20><head>INTRODUCTORY</head>
+letters. First there should be calculations specially devised
+as suitable for boys, which they should learn with amusement
+and pleasure, for example, distributions of apples or garlands
+where the same number is divided among more or fewer boys,
+or (distributions) of the competitors in boxing or wrestling
+matches on the plan of drawing pairs with byes, or by taking
+them in consecutive order, or in any of the usual ways<note>The Greek of this clause is,(<G>dianomai\</G>) <G>puktw=n kai\ palaistw=n e)fedrei/as
+te kai\ sullh/xews e)n me/rei kai\ e)fexh=s kai\ w(s pefu/kasi gi/gnesqai</G>. So far as
+I can ascertain, <G>e)n me/rei</G> (by itself) and <G>e)fexh=s</G> have always been taken
+as indicating alternative methods, &lsquo;in turn and in consecutive order&rsquo;.
+But it is impossible to get any satisfactory contrast of meaning between
+&lsquo;in turn&rsquo; and &lsquo;in consecutive order&rsquo;. It is clear to me that we have
+here merely an instance of Plato's habit of changing the order of words
+for effect, and that <G>e)n me/rei</G> must be taken with the genitives <G>e)fedrei/as kai\
+sullh/xews</G>; i.e. we must translate as if we had <G>e)n e)fedrei/as te kai\ sullh/-
+xews me/rei</G>, &lsquo;<I>by way of</I> byes and drawings&rsquo;. This gives a proper distinction
+between (1) drawings with byes and (2) taking competitors in consecutive
+order.</note>; and
+again there should be games with bowls containing gold,
+bronze, and silver (coins?) and the like mixed together,<note>It is difficult to decide between the two possible interpretations
+of the phrase <G>fia/las a(/ma xrusou= kai\ xalkou= kai\ a)rgu/rou kai\ toiou/twn tinw=n
+a)/llwn kerannu/ntes</G>. It may mean &lsquo;taking bowls made of gold, bronze,
+silver and other metals mixed together (in certain proportions)&rsquo; or
+&lsquo;filling bowls with gold, bronze, silver, &amp;c. (<I>sc.</I> objects such as coins)
+mixed together&rsquo;. The latter version seems to agree best with <G>pai/zontes</G>
+(making a game out of the process) and to give the better contrast to
+&lsquo;distributing the bowls <I>as wholes</I>&rsquo; (<G>o(/las pws diadido/ntes</G>).</note> or the
+bowls may be distributed as undivided units; for, as I said,
+by connecting with games the essential operations of practical
+arithmetic, you supply the boy with what will be useful to
+him later in the ordering of armies, marches and campaigns,
+as well as in household management; and in any case you
+make him more useful to himself and more wide awake.
+Then again, by calculating measurements of things which
+have length, breadth, and depth, questions on all of which
+the natural condition of all men is one of ridiculous and dis-
+graceful ignorance, they are enabled to emerge from this
+state.&rsquo;
+<p>It is true that these are Plato's ideas of what elementary
+education <I>should</I> include; but it can hardly be doubted that
+such methods were actually in use in Attica.
+<p>Geometry and astronomy belonged to secondary education,
+which occupied the years between the ages of fourteen and
+eighteen. The pseudo-Platonic <I>Axiochus</I> attributes to Prodi-
+cus a statement that, when a boy gets older, i. e. after he has
+<pb n=21><head>MATHEMATICS IN GREEK EDUCATION</head>
+passed the primary stage under the <I>paidagogos, grammatistes</I>,
+and <I>paidotribes</I>, he comes under the tyranny of the &lsquo;critics&rsquo;,
+the <I>geometers</I>, the tacticians, and a host of other masters.<note><I>Axiochus</I>, 366 E.</note>
+Teles, the philosopher, similarly, mentions arithmetic and
+geometry among the plagues of the lad.<note>Stobaeus, <I>Ecl.</I> iv. 34, 72 (vol. v, p. 848, 19 sq., Wachsmuth and
+Hense).</note> It would appear
+that geometry and astronomy were newly introduced into the
+curriculum in the time of Isocrates. &lsquo;I am so far&rsquo;, he says,<note>See Isocrates, <I>Panathenaicus</I>, &sect;&sect; 26-8 (238 b-d); <G>*peri\ a)ntido/sews</G>,
+&sect;&sect; 261-8.</note>
+&lsquo;from despising the instruction which our ancestors got, that
+I am a supporter of that which has been established in our
+time, I mean geometry, astronomy, and the so-called eristic
+dialogues.&rsquo; Such studies, even if they do no other good,
+keep the young out of mischief, and in Isocrates's opinion no
+other subjects could have been invented more useful and
+more fitting; but they should be abandoned by the time that
+the pupils have reached man's estate. Most people, he says,
+think them idle, since (say they) they are of no use in private
+or public affairs; moreover they are forgotten directly because
+they do not go with us in our daily life and action, nay, they
+are altogether outside everyday needs. He himself, however,
+is far from sharing these views. True, those who specialize in
+such subjects as astronomy and geometry get no good from
+them unless they choose to teach them for a livelihood; and if
+they get too deeply absorbed, they become unpractical and
+incapable of doing ordinary business; but the study of these
+subjects up to the proper point trains a boy to keep his atten-
+tion fixed and not to allow his mind to wander; so, being
+practised in this way and having his wits sharpened, he will be
+capable of learning more important matters with greater ease
+and speed. Isocrates will not give the name of &lsquo;philosophy&rsquo; to
+studies like geometry and astronomy, which are of no imme-
+diate use for producing an orator or man of business; they
+are rather means of training the mind and a preparation for
+philosophy. They are a more manly discipline than the sub-
+jects taught to boys, such as literary study and music, but in
+other respects have the same function in making them quicker
+to learn greater and more important subjects.
+<pb n=22><head>INTRODUCTORY</head>
+<p>It would appear therefore that, notwithstanding the in-
+fluence of Plato, the attitude of cultivated people in general
+towards mathematics was not different in Plato's time from
+what it is to-day.
+<p>We are told that it was one of the early Pythagoreans,
+unnamed, who first taught geometry for money: &lsquo;One of the
+Pythagoreans lost his property, and when this misfortune
+befell him he was allowed to make money by teaching
+geometry.&rsquo;<note>Iamblichus, <I>Vit. Pyth.</I> 89.</note> We may fairly conclude that Hippocrates of
+Chios, the first writer of <I>Elements</I>, who also made himself
+famous by his quadrature of lunes, his reduction of the
+duplication of the cube to the problem of finding two mean
+proportionals, and his proof that the areas of circles are in
+the ratio of the squares on their diameters, also taught for
+money and for a like reason. One version of the story is that
+he was a merchant, but lost all his property through being
+captured by a pirate vessel. He then came to Athens to
+prosecute the offenders and, during a long stay, attended
+lectures, finally attaining such proficiency in geometry that
+he tried to square the circle.<note>Philoponus on Arist. <I>Phys.</I>, p. 327 b 44-8, Brandis.</note> Aristotle has the different
+version that he allowed himself to be defrauded of a large
+sum by custom-house officers at Byzantium, thereby proving,
+in Aristotle's opinion, that, though a good geometer, he was
+stupid and incompetent in the business of ordinary life.<note><I>Eudemian Ethics</I>, H. 14, 1247 a 17.</note>
+<p>We find in the Platonic dialogues one or two glimpses of
+mathematics being taught or discussed in school- or class-
+rooms. In the <I>Erastae</I><note><I>Erastae</I>, 32 A, B.</note> Socrates is represented as going into
+the school of Dionysius (Plato's own schoolmaster<note>Diog. L. iii. 5.</note>) and find-
+ing two lads earnestly arguing some point of astronomy;
+whether it was Anaxagoras or Oenopides whose theories they
+were discussing he could not catch, but they were drawing
+circles and imitating some inclination or other with their
+hands. In Plato's <I>Theaetetus</I><note><I>Theaetetus</I>, 147 D-148 B.</note> we have the story of Theodorus
+lecturing on surds and proving separately, for the square root
+of every non-square number from 3 to 17, that it is incom-
+mensurable with 1, a procedure which set Theaetetus and the
+<pb n=23><head>MATHEMATICS IN GREEK EDUCATION</head>
+younger Socrates thinking whether it was not possible to
+comprehend all such surds under one definition. In these two
+cases we have advanced or selected pupils discussing among
+themselves the subject of lectures they had heard and, in the
+second case, trying to develop a theory of a more general
+character.
+<p>But mathematics was not only taught by regular masters
+in schools; the Sophists, who travelled from place to place
+giving lectures, included mathematics (arithmetic, geometry,
+and astronomy) in their very wide list of subjects. Theo-
+dorus, who was Plato's teacher in mathematics and is
+described by Plato as a master of geometry, astronomy,
+<I>logistic</I> and music (among other subjects), was a pupil of
+Protagoras, the Sophist, of Abdera.<note><I>Theaetetus</I>, 164 E, 168 E.</note> Protagoras himself, if we
+may trust Plato, did not approve of mathematics as part of
+secondary education; for he is made to say that
+<p>&lsquo;the other Sophists maltreat the young, for, at an age when
+the young have escaped the arts, they take them against their
+will and plunge them once more into the arts, teaching them
+the art of calculation, astronomy, geometry, and music&mdash;and
+here he cast a glance at Hippias&mdash;whereas, if any one comes
+to me, he will not be obliged to learn anything except what
+he comes for.&rsquo;<note><I>Protagoras</I>, 318 D, E.</note>
+<p>The Hippias referred to is of course Hippias of Elis, a really
+distinguished mathematician, the inventor of a curve known
+as the <I>quadratrix</I> which, originally intended for the solution
+of the problem of trisecting any angle, also served (as the
+name implies) for squaring the circle. In the <I>Hippias Minor</I><note><I>Hippias Minor</I>, pp. 366 C-368 E.</note>
+there is a description of Hippias's varied accomplishments.
+He claimed, according to this passage, to have gone once to
+the Olympian festival with everything that he wore made by
+himself, ring and seal (engraved), oil-bottle, scraper, shoes,
+clothes, and a Persian girdle of expensive type; he also took
+poems, epics, tragedies, dithyrambs, and all sorts of prose
+works. He was a master of the science of calculation
+(<I>logistic</I>), geometry, astronomy, &lsquo;rhythms and harmonies
+and correct writing&rsquo;. He also had a wonderful system of
+mnemonics enabling him, if he once heard a string of fifty
+<pb n=24><head>INTRODUCTORY</head>
+names, to remember them all. As a detail, we are told that
+he got no fees for his lectures in Sparta, and that the Spartans
+could not endure lectures on astronomy or geometry or
+<I>logistic</I>; it was only a small minority of them who could
+even count; what they liked was history and archaeology.
+<p>The above is almost all that we know of the part played
+by mathematics in the Greek system of education. Plato's
+attitude towards mathematics was, as we have seen, quite
+exceptional; and it was no doubt largely owing to his influence
+and his inspiration that mathematics and astronomy were so
+enormously advanced in his school, and especially by Eudoxus
+of Cnidos and Heraclides of Pontus. But the popular atti-
+tude towards Plato's style of le&cacute;turing was not encouraging.
+There is a story of a lecture of his on &lsquo;The Good&rsquo; which
+Aristotle was fond of telling.<note>Aristoxenus, <I>Harmonica</I>, ii <I>ad init.</I></note> The lecture was attended by
+a great crowd, and &lsquo;every one went there with the idea that
+he would be put in the way of getting one or other of the
+things in human life which are usually accounted good, such
+as Riches, Health, Strength, or, generally, any extraordinary
+gift of fortune. But when they found that Plato discoursed
+about mathematics, arithmetic, geometry, and astronomy, and
+finally declared the One to be the Good, no wonder they were
+altogether taken by surprise; insomuch that in the end some
+of the audience were inclined to scoff at the whole thing, while
+others objected to it altogether.&rsquo; Plato, however, was able to
+pick and choose his pupils, and he could therefore insist on
+compliance with the notice which he is said to have put over
+his porch, &lsquo;Let no one unversed in geometry enter my doors&rsquo;;<note>Tzetzes, <I>Chiliad.</I> viii. 972.</note>
+and similarly Xenocrates, who, after Speusippus, succeeded to
+the headship of the school, could turn away an applicant for
+admission who knew no geometry with the words, &lsquo;Go thy
+way, for thou hast not the means of getting a grip of
+philosophy&rsquo;.<note>Diog. L. iv. 10.</note>
+<p>The usual attitude towards mathematics is illustrated by
+two stories of Pythagoras and Euclid respectively. Pytha-
+goras, we are told,<note>Iamblichus, <I>Vit. Pyth.</I> c. 5.</note> anxious as he was to transplant to his own
+country the system of education which he had seen in opera-
+<pb n=25><head>MATHEMATICS IN GREEK EDUCATION</head>
+tion in Egypt, and the study of mathematics in particular,
+could get none of the Samians to listen to him. He adopted
+therefore this plan of communicating his arithmetic and
+geometry, so that it might not perish with him. Selecting
+a young man who from his behaviour in gymnastic exercises
+seemed adaptable and was withal poor, he promised him that,
+if he would learn arithmetic and geometry systematically, he
+would give him sixpence for each &lsquo;figure&rsquo; (proposition) that he
+mastered. This went on until the youth got interested in
+the subject, when Pythagoras rightly judged that he would
+gladly go on without the sixpence. He therefore hinted
+that he himself was poor and must try to earn his daily bread
+instead of doing mathematics; whereupon the youth, rather
+than give up the study, volunteered to pay sixpence himself
+to Pythagoras for each proposition. We must presumably
+connect with this story the Pythagorean motto, &lsquo;a figure and
+a platform (from which to ascend to the next higher step), not
+a figure and sixpence&rsquo;.<note>Proclus on Eucl. I, p. 84. 16.</note>
+<p>The other story is that of a pupil who began to learn
+geometry with Euclid and asked, when he had learnt one
+proposition, &lsquo;What advantage shall I get by learning these
+things?&rsquo; And Euclid called the slave and said, &lsquo;Give him
+sixpence, since he must needs gain by what he learns.&rsquo;
+<p>We gather that the education of kings in the Macedonian
+period did not include much geometry, whether it was Alex-
+ander who asked Menaechmus, or Ptolemy who asked Euclid,
+for a short-cut to geometry, and got the reply that &lsquo;for travel-
+ling over the country there are royal roads and roads for com-
+mon citizens: but in geometry there is one road for all&rsquo;.<note>Stobaeus, <I>Ecl.</I> ii. 31, 115 (vol. ii, p. 228, 30, Wachsmuth).</note>
+<pb>
+<C>II</C>
+<C>GREEK NUMERICAL NOTATION AND ARITH-
+METICAL OPERATIONS</C>
+<C>The decimal system.</C>
+<p>THE Greeks, from the earliest historical times, followed the
+decimal system of numeration, which had already been
+adopted by civilized peoples all the world over. There are,
+it is true, traces of <I>quinary</I> reckoning (reckoning in terms of
+five) in very early times; thus in Homer <G>pempa/zein</G> (to &lsquo;five&rsquo;)
+is used for &lsquo;to count&rsquo;.<note>Homer, <I>Od</I>. iv. 412.</note> But the counting by fives was pro-
+bably little more than auxiliary to counting by tens; five was
+a natural halting-place between the unit and ten, and the use
+of five times a particular power of ten as a separate category
+intermediate between that power and the next was found
+convenient in the earliest form of numerical symbolism estab-
+lished in Greece, just as it was in the Roman arithmetical
+notation. The reckoning by five does not amount to such a
+variation of the decimal system as that which was in use
+among the Celts and Danes; these peoples had a vigesimal
+system, traces of which are still left in the French <I>quatre-
+vingts, quatre-vingt-treize</I>, &amp;c., and in our <I>score</I>, three-score
+and ten, twenty-one, &amp;c.
+<p>The natural explanation of the origin of the decimal system,
+as well as of the quinary and vigesimal variations, is to
+suppose that they were suggested by the primitive practice of
+reckoning with the fingers, first of one hand, then of both
+together, and after that with the ten toes in addition (making
+up the 20 of the vigesimal system). The subject was mooted
+in the Aristotelian <I>Problems</I>,<note>XV. 3, 910 b 23-911 a 4.</note> where it is asked:
+<p>&lsquo;Why do all men, whether barbarians or Greeks, count up
+to ten, and not up to any other number, such as 2, 3, 4, or 5,
+so that, for example, they do not say one-<I>plus</I>-five (for 6),
+<pb n=27><head>THE DECIMAL SYSTEM</head>
+two-<I>plus</I>-five (for 7), as they say one-<I>plus</I>-ten (<G>e(/ndeka</G>, for 11),
+two-<I>plus</I>-ten (<G>dw/deka</G>, for 12), while on the other hand they
+do not go beyond ten for the first halting-place from which to
+start again repeating the units? For of course any number
+is the next before it <I>plus</I> 1, or the next before that <I>plus</I> 2,
+and so with those preceding numbers; yet men fixed definitely
+on ten as the number to count up to. It cannot have been
+chance; for chance will not account for the same thing being
+done always: what is always and universally done is not due
+to chance but to some natural cause.&rsquo;
+<p>Then, after some fanciful suggestions (e.g. that 10 is a
+&lsquo;perfect number&rsquo;), the author proceeds:
+<p>&lsquo;Or is it because men were born with ten fingers and so,
+because they possess the equivalent of pebbles to the number
+of their own fingers, come to use this number for counting
+everything else as well?&rsquo;
+<p>Evidence for the truth of this latter view is forthcoming in
+the number of cases where the word for 5 is either the same
+as, or connected with, the word for &lsquo;hand&rsquo;. Both the Greek
+<G>xei/r</G> and the Latin <I>manus</I> are used to denote &lsquo;a number&rsquo; (of
+men). The author of the so-called geometry of Bo&euml;tius says,
+moreover, that the ancients called all the numbers below ten
+by the name <I>digits</I> (&lsquo;fingers&rsquo;).<note>Bo&euml;tius, <I>De Inst. Ar.</I>, &amp;c., p. 395. 6-9, Friedlein.</note>
+<p>Before entering on a description of the Greek numeral signs
+it is proper to refer briefly to the systems of notation used
+by their forerunners in civilization, the Egyptians and
+Babylonians.
+<C>Egyptian numerical notation.</C>
+<p>The Egyptians had a purely decimal system, with the signs
+<G>*i</G> for the unit, <FIG> for 10, <FIG> for 100, <FIG> for 1,000, <FIG> for 10,000,
+<FIG> for 100,000. The number of each denomination was
+expressed by repeating the sign that number of times; when
+the number was more than 4 or 5, lateral space was saved by
+arranging them in two or three rows, one above the other.
+The greater denomination came before the smaller. Numbers
+could be written from left to right or from right to left; in
+the latter case the above signs were turned the opposite way.
+The fractions in use were all submultiples or single aliquot
+<pb n=28><head>GREEK NUMERICAL NOTATION</head>
+parts, except 2/3, which had a special sign <FIG> or <FIG>; the
+submultiples were denoted by writing <FIG> over the corre-
+sponding whole number; thus
+<MATH><FIG>=1/23, <FIG>=1/324 <FIG>=1/2190</MATH>.
+<C>Babylonian systems.</C>
+<C>(<G>a</G>) <I>Decimal</I>. (<G>b</G>) <I>Sexagesimal</I>.</C>
+<p>The ancient Babylonians had two systems of numeration.
+The one was purely decimal based on the following signs.
+The simple wedge <FIG> represented the unit, which was repeated
+up to nine times: where there were more than three, they
+were placed in two or three rows, e.g. <MATH><FIG>=4, <FIG>=7</MATH>. 10
+was represented by <FIG>; 11 would therefore be <FIG>. 100 had
+the compound sign <FIG>, and 1000 was expressed as 10 hun-
+dreds, by <FIG>, the prefixed <FIG> (10) being here multiplicative.
+Similarly, the <FIG> was regarded as one sign, and <FIG> de-
+noted not 2000 but 10000, the prefixed <FIG> being again multi-
+plicative. Multiples of 10000 seem to have been expressed
+as multiples of 1000; at least, 120000 seems to be attested
+in the form 100.1000 + 20.1000. The absence of any definite
+unit above 1000 (if it was really absent) must have rendered
+the system very inconvenient as a means of expressing large
+numbers.
+<p>Much more interesting is the second Babylonian system,
+the sexagesimal. This is found in use on the Tables of
+Senkereh, discovered by W. K. Loftus in 1854, which may go
+back as far as the time between 2300 and 1600 B.C. In this
+system numbers above the units (which go from 1 to 59) are
+arranged according to powers of 60. 60 itself was called
+<I>sussu</I> (=<I>soss</I>), 60<SUP>2</SUP> was called <I>sar</I>, and there was a name also
+(<I>ner</I>) for the intermediate number 10.60=600. The multi-
+ples of the several powers of 60, 60<SUP>2</SUP>, 60<SUP>3</SUP>, &amp;c., contained in the
+number to be written down were expressed by means of the
+same wedge-notation as served for the units, and the multi-
+ples were placed in columns side by side, the columns being
+appropriated to the successive powers of 60. The unit-term
+<pb n=29><head>EGYPTIAN AND BABYLONIAN NOTATION</head>
+was followed by similar columns appropriated, in order, to the
+successive submultiples 1/60, 1/60<SUP>2</SUP>, &amp;c., the number of sixtieths,
+&amp;c., being again denoted by the ordinary wedge-numbers.
+Thus <FIG> represents <MATH>44.60<SUP>2</SUP>+26.60+40=160,000;
+<FIG>=27.60<SUP>2</SUP>+21.60+36=98,496</MATH>. Simi-
+larly we find <FIG> representing 30+30/60 and <FIG>
+representing 30+27/60; the latter case also shows that the
+Babylonians, on occasion, used the subtractive plan, for the 27
+is here written 30 <I>minus</I> 3.
+<p>The sexagesimal system only required a definite symbol
+for 0 (indicating the absence of a particular denomination),
+and a fixed arrangement of columns, to become a complete
+position-value system like the Indian. With a sexagesimal
+system 0 would occur comparatively seldom, and the Tables of
+Senkereh do not show a case; but from other sources it
+appears that a gap often indicated a zero, or there was a sign
+used for the purpose, namely <G>c</G>, called the &lsquo;divider&rsquo;. The
+inconvenience of the system was that it required a multipli-
+cation table extending from 1 times 1 to 59 times 59. It had,
+however, the advantage that it furnished an easy means of
+expressing very large numbers. The researches of H. V.
+Hilprecht show that 60<SUP>4</SUP>=12,960,000 played a prominent
+part in Babylonian arithmetic, and he found a table con-
+taining certain quotients of the number <MATH><FIG>
+=60<SUP>8</SUP>+10.60<SUP>7</SUP></MATH>, or 195,955,200,000,000. Since the number of
+units of any denomination are expressed in the purely decimal
+notation, it follows that the latter system preceded the sexa-
+gesimal. What circumstances led to the adoption of 60 as
+the base can only be conjectured, but it may be presumed that
+the authors of the system were fully alive to the convenience
+of a base with so many divisors, combining as it does the
+advantages of 12 and 10.
+<C>Greek numerical notation.</C>
+<p>To return to the Greeks. We find, in Greek inscriptions of
+all dates, instances of numbers and values written out in full;
+but the inconvenience of this longhand, especially in such
+things as accounts, would soon be felt, and efforts would be
+made to devise a scheme for representing numbers more
+<pb n=30><head>GREEK NUMERICAL NOTATION</head>
+concisely by means of conventional signs of some sort. The
+Greeks conceived the original idea of using the letters of the
+ordinary Greek alphabet for this purpose.
+<C>(<G>a</G>) <I>The &lsquo;Herodianic&rsquo; signs</I>.</C>
+<p>There were two main systems of numerical notation in use in
+classical times. The first, known as the Attic system and
+used for cardinal numbers exclusively, consists of the set of
+signs somewhat absurdly called &lsquo;Herodianic&rsquo; because they are
+described in a fragment<note>Printed in the Appendix to Stephanus's <I>Thesaurus</I>, vol. viii.</note> attributed to Herodian, a gram-
+marian of the latter half of the second century A.D. The
+authenticity of the fragment is questioned, but the writer
+says that he has seen the signs used in Solon's laws, where
+the prescribed pecuniary fines were stated in this notation,
+and that they are also to be found in various ancient inscrip-
+tions, decrees and laws. These signs cannot claim to be
+numerals in the proper sense; they are mere compendia or
+abbreviations; for, except in the case of the stroke <G>*i</G> repre-
+senting a unit, the signs are the first letters of the full words
+for the numbers, and all numbers up to 50000 were repre-
+sented by combinations of these signs. <G>*i</G>, representing the
+unit, may be repeated up to four times; <G><*></G> (the first letter of
+<G>pe/nte</G>) stands for 5, <G>*d</G> (the first letter of <G>de/ka</G>) for 10, <G>*h</G>
+(representing <G>e(/katon</G>) for 100, <G>*x</G> (<G>xi/lioi</G>) for 1000, and <G>*m</G>
+(<G>mu/rioi</G>) for 10000. The half-way numbers 50, 500, 5000
+were expressed by combining <G><*></G> (five) with the other signs
+respectively; <G><*></G>, <G><*></G>, <G><*></G>, made up of <G><*></G> (5) and <G>*d</G> (10), = 50;
+<G><*></G>, made up of <G><*></G> and <G>*h</G>,=500; <G><*></G>=5000; and <G><*></G>=50000.
+There are thus six simple and four compound symbols, and all
+other numbers intermediate between those so represented are
+made up by juxtaposition on an additive basis, so that each
+of the simple signs may be repeated not more than four times;
+the higher numbers come before the lower. For example,
+<G><*>*i</G>=6, <G>*d*i*i*i*i</G>=14, <G>*h<*></G>=105, <G>*x*x*x*x<*>*h*h*h*h<*>*d*d*d*d<*>*i*i*i*i</G>
+=4999. Instances of this system of notation are found in
+Attic inscriptions from 454 to about 95 B.C. Outside Attica
+the same system was in use, the precise form of the symbols
+varying with the form of the letters in the local alphabets.
+Thus in Boeotian inscriptions <G><*></G> or <G><*></G>=50, <G><*></G>=100, <G><*></G>=500,
+<pb n=31><head>THE &lsquo;HERODIANIC&rsquo; SIGNS</head>
+<G><*></G>=1000, <G><*></G>=5000; and <G><*><*><*><*><*><*><*>*i*i*i</G>=5823. But,
+in consequence of the political influence of Athens, the Attic
+system, sometimes with unimportant modifications, spread to
+other states.<note>Larfeld, <I>Handbuch der griechischen Epigraphik</I>, vol. i, p. 417.</note>
+<p>In a similar manner compendia were used to denote units
+of coinage or of weight. Thus in Attica <G>*t</G>=<G>ta/lanton</G> (6000
+drachmae), <G>*m</G>=<G>mna=</G> (1000 drachmae), <G>*s</G> or <G><*></G>=<G>stath/r</G>
+(1/3000th of a talent or 2 drachmae), <G><*></G>=<G>draxmh/</G>, <G>*i</G>=<G>o)bolo/s</G>
+(1/6th of a drachma), <G><*></G>=<G>h(miwbe/lion</G> (1/12th of a drachma),
+&c; or <G>*t</G>=<G>tetarthmo/rion</G> (1/4th of an obol or 1/24th of a
+drachma), <G>*x</G>=<G>xalkou=s</G> (1/8th of an obol or 1/48th of a
+drachma). Where a number of one of these units has to be
+expressed, the sign for the unit is written on the left of that
+for the number; thus <G><*><*>*d*i</G>=61 drachmae. The two com-
+pendia for the numeral and the unit are often combined into
+one; e.g. <G><*></G>, <G><*></G>=5 talents, <G><*></G>=50 talents, <G><*></G>=100 talents,
+<G><*></G>=500 talents, <G><*></G>=1000 talents, <G><*></G>=10 minas, <G><*></G>=5 drach-
+mae, <G><*></G>, <G><*></G>, <G><*></G>=10 staters, &amp;c.
+<C>(<G>b</G>) <I>The ordinary alphabetic numerals</I>.</C>
+<p>The second main system, used for all kinds of numerals, is
+that with which we are familiar, namely the alphabetic
+system. The Greeks took their alphabet from the Phoe-
+nicians. The Phoenician alphabet contained 22 letters, and,
+in appropriating the different signs, the Greeks had the
+happy inspiration to use for the vowels, which were not
+written in Phoenician, the signs for certain spirants for which
+the Greeks had no use; Aleph became A, He was used for E,
+Yod for I, and Ayin for O; when, later, the long E was
+differentiated, Cheth was used, <G><*></G> or <G>*h</G>. Similarly they
+utilized superfluous signs for sibilants. Out of Zayin and
+Samech they made the letters <G>*z</G> and <G>*e</G>. The remaining two
+sibilants were Ssade and Shin. From the latter came the
+simple Greek <G>*s</G> (although the name Sigma seems to corre-
+spond to the Semitic Samech, if it is not simply the &lsquo;hissing&rsquo;
+letter, from <G>si/zw</G>). Ssade, a softer sibilant (=<G>ss</G>), also called
+San in early times, was taken over by the Greeks in the
+place it occupied after <G><*></G>, and written in the form <G><*></G> or <G><*></G>.
+The form <G><*></G> (=<G>ss</G>) appearing in inscriptions of Halicarnassus
+<pb n=32><head>GREEK NUMERICAL NOTATION</head>
+(e.g. <G>*(alikarna<*></G>[<G>e/wn</G>]=<G>*(alikarnacce/wn</G>) and Teos ([<G>q</G>]<G>ala/<*>hs</G>;
+cf. <G>qa/laccan</G> in another place) seems to be derived from some
+form of Ssade; this <G><*></G>, after its disappearance from the
+literary alphabet, remained as a numeral, passing through
+the forms <G><*></G>, <G><*></G>, <G><*></G>, <G><*></G>, and <G><*></G> to the fifteenth century form <G><*></G>,
+to which in the second half of the seventeenth century the
+name Sampi was applied (whether as being the San which
+followed Pi or from its resemblance to the cursive form of <G>p</G>).
+The original Greek alphabetalso retained the Phoenician Vau (<G><*></G>)
+in its proper place between E and Z and the Koppa=Qoph (<G>O|</G>)
+immediately before P. The Phoenician alphabet ended with
+T; the Greeks first added <G><*></G>, derived from Vau apparently
+(notwithstanding the retention of <G><*></G>), then the letters <G>*f</G>, <G>*x</G>, <G>*y</G>
+and, still later, <G>*w</G>. The 27 letters used for numerals are
+divided into three sets of nine each; the first nine denote
+the units, 1, 2, 3, &amp;c., up to 9; the second nine the tens, from
+10 to 90; and the third nine the hundreds, from 100 to 900.
+The following is the scheme:
+<table>
+<tr><td><G>*a</G> =1</td><td><G>*i</G>=10</td><td><G>*p</G> =100</td></tr>
+<tr><td><G>*b</G> =2</td><td><G>*k</G>=20</td><td><G>*s</G> =200</td></tr>
+<tr><td><G>*g</G> =3</td><td><G>*l</G>=30</td><td><G>*t</G> =300</td></tr>
+<tr><td><G>*d</G> =4</td><td><G>*m</G>=40</td><td><G>*u</G> =400</td></tr>
+<tr><td><G>*e</G> =5</td><td><G>*n</G>=50</td><td><G>*f</G> =500</td></tr>
+<tr><td><G><*></G>[<G>s</G>]=6</td><td><G>*c</G>=60</td><td><G>*x</G> =600</td></tr>
+<tr><td><G>*z</G> =7</td><td><G>*o</G>=70</td><td><G>*y</G> =700</td></tr>
+<tr><td><G>*h</G> =8</td><td><G>*p</G>=80</td><td><G>*w</G> =800</td></tr>
+<tr><td><G>*q</G> =9</td><td><G>O|</G>=90</td><td><G>*t</G>[<G><*></G>] =900</td></tr>
+</table>
+<p>The sixth sign in the first column (<G><*></G>) is a form of the
+digamma <*>. It came, in the seventh and eighth centuries
+A. D., to be written in the form <G><*></G> and then, from its similarity
+to the cursive <G>s</G> (=<G>st</G>), was called Stigma.
+<p>This use of the letters of the alphabet as numerals was
+original with the Greeks; they did not derive it from the
+Phoenicians, who never used their alphabet for numerical
+purposes but had separate signs for numbers. The earliest
+occurrence of numerals written in this way appears to be in
+a Halicarnassian inscription of date not long after 450 B.C.
+Two caskets from the ruins of a famous mausoleum built at
+Halicarnassus in 351 B.C., which are attributed to the time
+of Mausolus, about 350 B.C., are inscribed with the letters
+<pb n=33><head>THE ORDINARY ALPHABETIC NUMERALS</head>
+<G>*y*n*d</G>=754 and <G>*sO|*g</G>=293. A list of priests of Poseidon
+at Halicarnassus, attributable to a date at least as early as the
+fourth century, is preserved in a copy of the second or first
+century, and this copy, in which the numbers were no doubt
+reproduced from the original list, has the terms of office of the
+several priests stated on the alphabetical system. Again, a
+stone inscription found at Athens and perhaps belonging to
+the middle of the fourth century B.C. has, in five fragments
+of columns, numbers in tens and units expressed on the same
+system, the tens on the right and the units on the left.
+<p>There is a difference of opinion as to the approximate date
+of the actual formulation of the alphabetical system of
+numerals. According to one view, that of Larfeld, it must
+have been introduced much earlier than the date (450 B.C. or
+a little later) of the Halicarnassus inscription, in fact as early
+as the end of the eighth century, the place of its origin being
+Miletus. The argument is briefly this. At the time of the
+invention of the system all the letters from <G>*a</G> to <G>*w</G>, including
+<G><*></G> and <G>O|</G> in their proper places, were still in use, while
+Ssade (<G><*></G>, the double <I>ss</I>) had dropped out; this is why the
+last-named sign (afterwards <G><*></G>) was put at the end. If
+<G><*></G> (=6) and <G>O|</G> (=90) had been no longer in use as letters,
+they too would have been put, like Ssade, at the end. The
+place of origin of the numeral system must have been one in
+which the current alphabet corresponded to the content and
+order of the alphabetic numerals. The order of the signs
+<G>*f</G>, <G>*x</G>, <G>*y</G> shows that it was one of the <I>Eastern</I> group of
+alphabets. These conditions are satisfied by one alphabet,
+and one only, that of Miletus, at a stage which still recognized
+the Vau (<G><*></G>) as well as the Koppa (<G>O|</G>). The <G>O|</G> is found along
+with the so-called complementary letters including <G>*w</G>, the
+latest of all, in the oldest inscriptions of the Milesian colony
+Naucratis (about 650 B.C.); and, although there are no
+extant Milesian inscriptions containing the <G><*></G>, there is at all
+events one very early example of <G><*></G> in Ionic, namely <G>*)aga-
+sile/<*>o</G> (<G>*)agasilh/<*>ou</G>) on a vase in the Boston (U.S.) Museum
+of Fine Arts belonging to the end of the eighth or (at latest)
+the middle of the seventh century. Now, as <G>*w</G> is fully
+established at the date of the earliest inscriptions at Miletus
+(about 700 B.C.) and Naucratis (about 650 B.C.), the earlier
+<pb n=34><head>GREEK NUMERICAL NOTATION</head>
+extension of the alphabet by the letters <G>*f *x *y</G> must have
+taken place not later than 750 B.C. Lastly, the presence in
+the alphabet of the Vau indicates a time which can hardly
+be put later than 700 B.C. The conclusion is that it was
+about this time, if not earlier, that the numerical alphabet
+was invented.
+<p>The other view is that of Keil, who holds that it originated
+in Dorian Caria, perhaps at Halicarnassus itself, about
+550-425 B.C., and that it was artificially put together by
+some one who had the necessary knowledge to enable him
+to fill up his own alphabet, then consisting of twenty-four
+letters only, by taking over <G><*></G> and <G>O|</G> from other alphabets and
+putting them in their proper places, while he completed the
+numeral series by adding <G><*></G> at the end.<note><I>Hermes</I>, 29, 1894, p. 265 sq.</note> Keil urges, as
+against Larfeld, that it is improbable that <G><*></G> and <G>*w</G> ever
+existed together in the Milesian alphabet. Larfeld's answer<note>Larfeld, <I>op. cit.</I>, i, p. 421.</note>
+is that, although <G><*></G> had disappeared from ordinary language
+at Miletus towards the end of the eighth century, we cannot
+say exactly when it disappeared, and even if it was practically
+gone at the time of the formulation of the numerical alphabet,
+it would be in the interest of instruction in schools, where
+Homer was read, to keep the letter as long as possible in the
+official alphabet. On the other hand, Keil's argument is open
+to the objection that, if the Carian inventor could put the
+<G><*></G> and <G>O|</G> into their proper places in the series, he would hardly
+have failed to put the Ssade <G><*></G> in its proper place also, instead
+of at the end, seeing that <G><*></G> is found in Caria itself, namely
+in a Halicarnassus (Lygdamis) inscription of about 453 B.C.,
+and also in Ionic Teos about 476 B.C.<note><I>Ib.</I>, i, p. 358.</note> (see pp. 31-2 above).
+<p>It was a long time before the alphabetic numerals found
+general acceptance. They were not officially used until the
+time of the Ptolemies, when it had become the practice to write,
+in inscriptions and on coins, the year of the reign of the ruler
+for the time being. The conciseness of the signs made them
+particularly suitable for use on coins, where space was limited.
+When coins went about the world, it was desirable that the
+notation should be uniform, instead of depending on local
+alphabets, and it only needed the support of some paramount
+<pb n=35><head>THE ORDINARY ALPHABETIC NUMERALS</head>
+political authority to secure the final triumph of the alphabetic
+system. The alphabetic numerals are found at Alexandria
+on coins of Ptolemy II, Philadelphus, assigned to 266 B.C.
+A coin with the inscription <G>*)alexa/ndrou *k*d</G> (twenty-fourth
+year after Alexander's death) belongs, according to Keil, to
+the end of the third century.<note><I>Hermes</I>, 29, 1894, p. 276 <I>n</I>.</note> A very old Graeco-Egyptian
+papyrus (now at Leyden, No. 397), ascribed to 257 B.C.,
+contains the number <G>kq</G>=29. While in Boeotia the Attic
+system was in use in the middle of the third century, along
+with the corresponding local system, it had to give way about
+200 B.C. to the alphabetic system, as is shown by an inventory
+from the temple of Amphiaraus at Oropus<note>Keil in <I>Hermes</I>, 25, 1890, pp. 614-15.</note>; we have here
+the first official use of the alphabetic system in Greece proper.
+From this time Athens stood alone in retaining the archaic
+system, and had sooner or later to come into line with other
+states. The last certainly attested use of the Attic notation
+in Athens was about 95 B.C.; the alphabetic numerals were
+introduced there some time before 50 B.C., the first example
+belonging to the time of Augustus, and by A.D. 50 they were
+in official use.
+<p>The two systems are found side by side in a number of
+papyrus-rolls found at Herculaneum (including the treatise
+of Philodemus <I>De pietate</I>, so that the rolls cannot be older than
+40 or 50 B.C.); these state on the title page, after the name of
+the author, the number of books in alphabetic numerals, and
+the number of lines in the Attic notation, e.g. <G>*e*r*i*k*o*u*p*o*u <*>
+*r*e*p*i <*> *f*u*s*e*w*s <*> *i*e a)riq</G> . . <G>*x*x*x*h*h</G> (where <G>*i*e</G> = 15 and
+<G>*x*x*x*h*h</G> = 3200), just as we commonly use Roman figures
+to denote <I>Books</I> and Arabic figures for <I>sections</I> or <I>lines</I>.<note>Reference should be made, in passing, to another, <I>quasi</I>-numerical,
+use of the letters of the ordinary alphabet, as current at the time, for
+numbering particular things. As early as the fifth century we find in
+a Locrian bronze-inscription the letters A to <FIG> (including <G><*></G> then and
+there current) used to distinguish the nine paragraphs of the text. At
+the same period the Athenians, instead of following the old plan of
+writing out ordinal numbers in full, adopted the more convenient device
+of denoting them by the letters of the alphabet. In the oldest known
+example <G>o(/ros</G> K indicated &lsquo;boundary stone No. 10&rsquo;; and in the fourth
+century the tickets of the ten panels of jurymen were marked with the
+letters A to K. In like manner the Books in certain works of Aristotle
+(the <I>Ethics, Metaphysics, Politics</I>, and <I>Topics</I>) were at some time
+numbered on the same principle; so too the Alexandrine scholars
+(about 280 B.C.) numbered the twenty-four Books of Homer with the
+letters A to <G>*w</G>. When the number of objects exceeded 24, doubled
+letters served for continuing the series, as AA, BB, &amp;c. For example,
+a large quantity of building-stones have been found; among these are
+stones from the theatre at the Piraeus marked AA, BB, &amp;c., and again
+AA|BB, BB|BB, &amp;c. when necessary. Sometimes the numbering by
+double letters was on a different plan, the letter A denoting the full
+number of the first set of letters (24); thus AP would be <MATH>24+17=41</MATH>.</note>
+<pb n=36><head>GREEK NUMERICAL NOTATION</head>
+<C>(<G>g</G>) <I>Mode of writing numbers in the ordinary alphabetic
+notation</I>.</C>
+<p>Where, in the alphabetical notation, the number to be
+written contained more than one denomination, say, units
+with tens, or with tens and hundreds, the higher numbers
+were, as a rule, put before the lower. This was generally the
+case in European Greece; on the other hand, in the inscrip-
+tions of Asia Minor, the smaller number comes first, i. e. the
+letters are arranged in alphabetical order. Thus 111 may be
+represented either by <G>*p*i*a</G> or by <G>*a*i*p</G>; the arrangement is
+sometimes mixed, as <G>*p*a*i</G>. The custom of writing the numbers
+in descending order became more firmly established in later
+times through the influence of the corresponding Roman
+practice.<note>Larfeld, <I>op. cit.</I>, i, p. 426.</note>
+<p>The alphabetic numerals sufficed in themselves to express
+all numbers from 1 to 999. For thousands (up to 9000) the
+letters were used again with a distinguishing mark; this was
+generally a sloping stroke to the left, e.g. <G>*/a</G> or <G>*<SUB>'</SUB>a</G>=1000,
+but other forms are also found, e.g. the stroke might be
+combined with the letter as <G><*></G>=1000 or again <G>*(a</G>=1000,
+<G>(<*></G>=6000. For tens of thousands the letter <G>*m</G> (<G>mu/rioi</G>) was
+borrowed from the other system, e.g. 2 myriads would be
+<G>*b*m</G>, <G>*m*b</G>, or <FIG>.
+<p>To distinguish letters representing numbers from the
+letters of the surrounding text different devices are used:
+sometimes the number is put between dots <FIG> or:, or separ-
+ated by spaces from the text on both sides of it. In Imperial
+times distinguishing marks, such as a horizontal stroke above
+the letter, become common, e.g. <G>h( boulh\ tw=n &horbar;*x</G>, other
+variations being <G><*></G>, <G><*></G>, <G><*></G> and the like.
+<p>In the cursive writing with which we are familiar the
+<pb n=37><head>ORDINARY ALPHABETIC NOTATION</head>
+orthodox way of distinguishing numerals was by a horizontal
+stroke above each sign or collection of signs; the following
+was therefore the scheme (with <G>s</G> substituted for <G><*></G> repre-
+senting 6, and with <G><*></G>=900 at the end):
+<table>
+<tr><td>units (1 to 9)</td><td><G>&horbar;a</G>, <G>&horbar;b</G>, <G>&horbar;g</G>, <G>&horbar;d</G>, <G>&horbar;e</G>, <G>&horbar;s</G>, <G>&horbar;z</G>, <G>&horbar;h</G>, <G>&horbar;q</G>;</td></tr>
+<tr><td>tens (10 to 90)</td><td><G>&horbar;i</G>, <G>&horbar;k</G>, <G>&horbar;l</G>, <G>&horbar;m</G>, <G>&horbar;n</G>, <G>&horbar;x</G>, <G>&horbar;o</G>, <G>&horbar;p</G>, <G>&horbar;O|</G>;</td></tr>
+<tr><td>hundreds (100 to 900)</td><td><G>&horbar;r</G>, <G>&horbar;s</G>, <G>&horbar;t</G>, <G>&horbar;u</G>, <G>&horbar;f</G>, <G>&horbar;*x</G>, <G>&horbar;*y</G>, <G>&horbar;w</G>, <G>&horbar;<*></G>;</td></tr>
+<tr><td>thousands (1000 to 9000)</td><td><G><SUB>'</SUB>&horbar;a</G>, <G><SUB>'</SUB>&horbar;b</G>, <G><SUB>'</SUB>&horbar;g</G>, <G><SUB>'</SUB>&horbar;d</G>, <G><SUB>'</SUB>&horbar;e</G>, <G><SUB>'</SUB>&horbar;s</G>, <G><SUB>'</SUB>&horbar;z</G>,
+<G><SUB>'</SUB>&horbar;h</G>, <G><SUB>'</SUB>&horbar;q</G>;</td></tr>
+</table>
+(for convenience of printing, the horizontal stroke above the
+sign will hereafter, as a rule, be omitted).
+<C>(<G>d</G>) <I>Comparison of the two systems of numerical notation</I>.</C>
+<p>The relative merits of the two systems of numerical
+notation used by the Greeks have been differently judged.
+It will be observed that the <I>initial</I>-numerals correspond
+closely to the Roman numerals, except that there is no
+formation of numbers by subtraction as <G>*i*x</G>, <G>*x<*></G>, <G>*x<*></G>; thus
+<G>*x*x*x*x<*>*h*h*h*h<*>*d*d*d*d<*>*i*i*i*i</G>=<G>*m*m*m*mDCCCCL*x*x*x*x<*>*i*i*i*i</G>
+as compared with <G>*m*m*m*mC*m*xC*i*x</G>=4999. The absolute
+inconvenience of the Roman system will be readily appreci-
+ated by any one who has tried to read Bo&euml;tius (Bo&euml;tius
+would write the last-mentioned number as <G>&horbar;*i<*></G>.<G><*><*><*><*><*>*xCV*i*i*i*i</G>).
+Yet Cantor<note>Cantor, <I>Gesch. d. Math</I>. I<SUP>3</SUP>, p. 129.</note> draws a comparison between the two systems
+much to the disadvantage of the alphabetic numerals.
+&lsquo;Instead&rsquo;, he says, &lsquo;of an advance we have here to do with
+a decidedly retrograde step, especially so far as its suitability
+for the further development of the numeral system is con-
+cerned. If we compare the older &ldquo;Herodianic&rdquo; numerals
+with the later signs which we have called alphabetic numerals,
+we observe in the latter two drawbacks which do not attach
+to the former. There now had to be more signs, with values
+to be learnt by heart; and to reckon with them required
+a much greater effort of memory. The addition
+<MATH><G>*d*d*d</G>+<G>*d*d*d*d</G>=<G><*>*d*d</G>(30+40=70)</MATH>
+could be coordinated in one act of memory with that of
+<MATH><G>*h*h*h</G>+<G>*h*h*h*h</G>=<G><*>*h*h</G>(300+400=700)</MATH>
+in so far as the sum of 3 and 4 units of the same kind added
+<pb n=38><head>GREEK NUMERICAL NOTATION</head>
+up to 5 and 2 units of the same kind. On the other hand
+<MATH><G>l</G>+<G>m</G>=<G>o</G></MATH> did not at all immediately indicate that <MATH><G>t</G>+<G>u</G>=<G>*y</G></MATH>.
+The new notation had only one advantage over the other,
+namely that it took less space. Consider, for instance, 849,
+which in the &ldquo;Herodianic&rdquo; form is <G><*>*h*h*h*d*d*d*d<*>*i*i*i*i</G>, but
+in the alphabetic system is <G>wmq</G>. The former is more self-
+explanatory and, for reckoning with, has most important
+advantages.&rsquo; Gow follows Cantor, but goes further and says
+that &lsquo;the alphabetical numerals were a fatal mistake and
+hopelessly confined such nascent arithmetical faculty as the
+Greeks may have possessed&rsquo;!<note>Gow, <I>A Short History of Greek Mathematics</I>, p. 46.</note> On the other hand, Tannery,
+holding that the merits of the alphabetic numerals could only
+be tested by using them, practised himself in their use until,
+applying them to the whole of the calculations in Archimedes's
+<I>Measurement of a Circle</I>, he found that the alphabetic nota-
+tion had practical advantages which he had hardly suspected
+before, and that the operations took little longer with Greek
+than with modern numerals.<note>Tannery, <I>M&eacute;moires scientifiques</I> (ed. Heiberg and Zeuthen), i,
+pp. 200-1.</note> Opposite as these two views are,
+they seem to be alike based on a misconception. Surely we do
+not &lsquo;reckon with&rsquo; the numeral <I>signs</I> at all, but with the
+<I>words</I> for the numbers which they represent. For instance,
+in Cantor's illustration, we do not conclude that the <I>figure</I> 3
+and the <I>figure</I> 4 added together make the <I>figure</I> 7; what we
+do is to say &lsquo;three and four are seven&rsquo;. Similarly the Greek
+would not say to himself &lsquo;<G>g</G> and <G>d</G>=<G>z</G>&rsquo; but <G>trei=s kai\ te/ssares
+e(pta/</G>; and, notwithstanding what Cantor says, this <I>would</I>
+indicate the corresponding addition &lsquo;three hundred and four
+hundred are seven hundred&rsquo;, <G>triako/sioi kai\ tetrako/sioi
+e(ptako/sioi</G>, and similarly with multiples of ten or of 1000 or
+10000. Again, in using the multiplication table, we say
+&lsquo;three times four is twelve&rsquo;, or &lsquo;three multiplied by four =
+twelve&rsquo;; the Greek would say <G>tri\s te/ssares</G>, or <G>trei=s e)pi\
+te/ssaras, dw/deka</G>, and this would equally indicate that &lsquo;<I>thirty</I>
+times <I>forty</I> is <I>twelve</I> hundred or one thousand two hundred&rsquo;,
+or that &lsquo;<I>thirty</I> times <I>four</I> hundred is <I>twelve</I> thousand or a
+myriad and two thousand&rsquo; (<G>triakonta/kis tessara/konta xi/lioi
+kai\ diako/sioi</G>, or <G>triakonta/kis tetrako/sioi mu/rioi kai\ disxi/lioi</G>).
+<pb n=39><head>COMPARISON OF THE TWO SYSTEMS</head>
+The truth is that in mental calculation (whether the opera-
+tion be addition, subtraction, multiplication, or division), we
+reckon with the corresponding <I>words</I>, not with the symbols,
+and it does not matter a jot to the calculation how we choose
+to write the figures down. While therefore the alphabetical
+numerals had the advantage over the &lsquo;Herodianic&rsquo; of being
+so concise, their only disadvantage was that there were more
+signs (twenty-seven) the meaning of which had to be com-
+mitted to memory: truly a very slight disadvantage. The
+one real drawback to the alphabetic system was the absence
+of a sign for 0 (zero); for the <G>*o</G> for <G>ou)demi/a</G> or <G>ou)de/n</G> which
+we find in Ptolemy was only used in the notation of sexa-
+gesimal fractions, and not as part of the numeral system. If
+there had been a sign or signs to indicate the absence in
+a number of a particular denomination, e.g. units or tens or
+hundreds, the Greek symbols could have been made to serve
+as a position-value system scarcely less effective than ours.
+For, while the position-values are clear in such a number
+as 7921 (<G><SUB>'</SUB>z<*>ka</G>), it would only be necessary in the case of
+such a number as 7021 to show a blank in the proper place
+by writing, say, <G><SUB>'</SUB>z-ka</G>. Then, following Diophantus's plan
+of separating any number of myriads by a dot from the
+thousands, &amp;c., we could write <G>z<*>ka . <SUB>'</SUB>stpd</G> for 79216384 or
+<G><SUB>'</SUB>z---.-t-d</G> for 70000304, while we could continually add
+sets of four figures to the left, separating each set from the
+next following by means of a dot.
+<C>(<G>e</G>) <I>Notation for large numbers</I>.</C>
+<p>Here too the orthodox way of writing tens of thousands
+was by means of the letter <G>*m</G> with the number of myriads
+above it, e.g. <FIG>=20000, <FIG> <G><SUB>'</SUB>ewoe</G>=71755875 (Aristarchus
+of Samos); another method was to write <G>*m</G> or <FIG> for the
+myriad and to put the number of myriads after it, separated
+by a dot from the remaining thousands, &amp;c., e.g.
+<FIG> <G>rn.<SUB>'</SUB>z<*>pd</G>=1507984
+(Diophantus, IV. 28). Yet another way of expressing myriads
+was to use the symbol representing the number of myriads
+with two dots over it; thus <G>a+<SUB>'</SUB>hfo|b</G>=18592 (Heron, <I>Geo-
+metrica</I>, 17. 33). The word <G>muria/des</G> could, of course, be
+<pb n=40><head>GREEK NUMERICAL NOTATION</head>
+written in full, e.g. <G>muria/des <SUB>'</SUB>bsoh kai\ <*>ib</G>=22780912
+(<I>ib.</I> 17. 34). To express still higher numbers, powers of
+myriads were used; a myriad (10000) was a <I>first myriad</I>
+(<G>prw/th muria/s</G>) to distinguish it from a <I>second myriad</I> (<G>deute/ra
+muria/s</G>) or 10000<SUP>2</SUP>, and so on; the words <G>prw=tai muria/des,
+deu/terai muria/des</G>, &amp;c., could either be written in full or
+expressed by <FIG>, &amp;c., respectively; thus <G>deu/terai muria/des
+is prw=tai</G> (<G>muria/des</G>) <G><SUB>'</SUB>b<*>nh</G> <FIG> <G><SUB>'</SUB>sfx</G>=1629586560 (Dio
+phantus, V. 8), where <FIG>=<G>mona/des</G> (units) is inserted to
+distinguish the <G><SUB>'</SUB>b<*>nh</G>, the number of the &lsquo;first myriads&rsquo;,
+from the <G><SUB>'</SUB>sfx</G> denoting 6560 <I>units</I>.
+<C>(i) Apollonius's &lsquo;tetrads&rsquo;.</C>
+<p>The latter system is the same as that adopted by Apollonius
+in an arithmetical work, now lost, the character of which is,
+however, gathered from the elucidations in Pappus, Book II;
+the only difference is that Apollonius called his <I>tetrads</I> (sets
+of four digits) <G>muria/des a(plai=</G>, <G>diplai=</G>, <G>triplai=</G>, &amp;c., &lsquo;simple
+myriads&rsquo;, &lsquo;double&rsquo;, &lsquo;triple&rsquo;, &amp;c., meaning 10000, 10000<SUP>2</SUP>,
+10000<SUP>3</SUP>, and so on. The abbreviations for these successive
+powers in Pappus are <G>m<SUP>a</SUP></G>, <G>m<SUP>b</SUP></G>, <G>m<SUP><*></SUP></G>, &amp;c.; thus <G>m<SUP><*></SUP> <SUB>'</SUB>euxb kai\ m<SUP>b</SUP> <SUB>'</SUB>gx
+kai\ m<SUP>a</SUP> <SUB>'</SUB>su</G>=5462360064000000. Another, but a less con-
+venient, method of denoting the successive powers of 10000
+is indicated by Nicolas Rhabdas (fourteenth century A.D.)
+who says that, while a pair of dots above the ordinary
+numerals denoted the number of myriads, the &lsquo;double
+myriad&rsquo; was indicated by two pairs of dots one above the other,
+the &lsquo;triple myriad&rsquo; by three pairs of dots, and so on. Thus
+<G><*><SUP>..</SUP></G>=9000000, <G>b<SUP>....</SUP></G>=2(10000)<SUP>2</SUP>, <G>m<SUP>......</SUP></G>=40(10000)<SUP>3</SUP>, and so on.
+<C>(ii) Archimedes's system (by octads).</C>
+<p>Yet another special system invented for the purpose of
+expressing very large numbers is that of Archimedes's
+<I>Psammites</I> or <I>Sand-reckoner</I>. This goes by <I>octads</I>:
+<MATH>10000<SUP>2</SUP>=100000000=10<SUP>8</SUP></MATH>,
+and all the numbers from 1 to 10<SUP>8</SUP> form the <I>first order</I>;
+the last number, 10<SUP>8</SUP>, of the <I>first order</I> is taken as the unit
+of the <I>second order</I>, which consists of all the numbers from
+<pb n=41><head>ARCHIMEDES'S SYSTEM (BY OCTADS)</head>
+10<SUP>8</SUP>, or 100000000, to 10<SUP>16</SUP>, or 100000000<SUP>2</SUP>; similarly 10<SUP>16</SUP> is
+taken as the unit of the <I>third order</I>, which consists of all
+numbers from 10<SUP>16</SUP> to 10<SUP>24</SUP>, and so on, the <I>100000000th order</I>
+consisting of all the numbers from (100000000)<SUP>99999999</SUP> to
+(100000000)<SUP>100000000</SUP>, i.e. from 10<SUP>8.(10<SUP>8</SUP>-1)</SUP> to 10<SUP>8.10<SUP>8</SUP></SUP>. The aggre-
+gate of all the <I>orders</I> up to the 100000000th form the <I>first
+period</I>; that is, if <I>P</I>&equalse;(100000000)<SUP>10<SUP>8</SUP></SUP>, the numbers of the
+<I>first period</I> go from 1 to <I>P.</I> Next, <I>P</I> is the unit of the <I>first
+order</I> of the <I>second period</I>; the <I>first order</I> of the <I>second
+period</I> then consists of all numbers from <I>P</I> up to 100000000 <I>P</I>
+or <I>P</I>.10<SUP>8</SUP>; <I>P</I>.10<SUP>8</SUP> is the unit of the <I>second order</I> (of the
+<I>second period</I>) which ends with (100000000)<SUP>2</SUP> <I>P</I> or <I>P</I>.10<SUP>16</SUP>;
+<I>P</I>.10<SUP>16</SUP> begins the <I>third order</I> of the <I>second period</I>, and so
+on; the <I>100000000th order</I> of the <I>second period</I> consists of
+the numbers from (100000000)<SUP>99999999</SUP> <I>P</I> or <I>P</I>.10<SUP>8.(10<SUP>8</SUP>-1)</SUP> to
+(100000000)<SUP>100000000</SUP> <I>P</I> or <I>P</I>.10<SUP>8.10<SUP>8</SUP></SUP>, i.e. <I>P</I><SUP>2</SUP>. Again, <I>P</I><SUP>2</SUP> is the
+unit of the <I>first order</I> of the <I>third period</I>, and so on. The
+<I>first order</I> of the <I>100000000th period</I> consists of the numbers
+from <I>P</I><SUP>10<SUP>8</SUP>-1</SUP> to <I>P</I><SUP>10<SUP>8</SUP>-1</SUP>.10<SUP>8</SUP>, the <I>second order</I> of the same
+<I>period</I> of the numbers from <I>P</I><SUP>10<SUP>8</SUP>-1</SUP>.10<SUP>8</SUP> to <I>P</I><SUP>10<SUP>8</SUP>-1</SUP>.10<SUP>16</SUP>, and so
+on, the (10<SUP>8</SUP>)th <I>order</I> of the (10<SUP>8</SUP>)th <I>period</I>, or the <I>period</I>
+itself, ending with <I>P</I><SUP>10<SUP>8</SUP>-1</SUP>.10<SUP>8.10<SUP>8</SUP></SUP>, i.e. <I>P</I><SUP>10<SUP>8</SUP></SUP>. The last number
+is described by Archimedes as a &lsquo;myriad-myriad units of the
+myriad-myriadth order of the myriad-myriadth period (<G>ai(
+muriakismuriosta=s perio/dou muriakismuriostw=n a)riqmw=n mu/riai
+muria/des</G>)&rsquo;. This system was, however, a <I>tour de force</I>, and has
+nothing to do with the ordinary Greek numerical notation.
+<C>Fractions.</C>
+<C>(<G>a</G>) <I>The Egyptian system</I></C>
+<p>We now come to the methods of expressing fractions. A
+fraction may be either a submultiple (an &lsquo;aliquot part&rsquo;, i.e.
+a fraction with numerator unity) or an ordinary proper
+fraction with a number not unity for numerator and a
+greater number for denominator. The Greeks had a pre-
+ference for expressing ordinary proper fractions as the sum
+of two or more submultiples; in this they followed the
+Egyptians, who always expressed fractions in this way, with
+the exception that they had a single sign for 2/3, whereas we
+<pb n=42><head>GREEK NUMERICAL NOTATION</head>
+should have expected them to split it up into 1/2+1/6, as 3/4 was
+split up into 1/2+1/4. The orthodox sign for a submultiple
+was the letter for the corresponding number (the denomi-
+nator) but with an accent instead of a horizontal stroke
+above it; thus <G>g&prime;</G>=1/3, the full expression being <G>g&prime; me/ros</G>=
+<G>tri/ton me/ros</G>, a third part (<G>g&prime;</G> is in fact short for <G>tri/tos</G>, so
+that it is also used for the ordinal number &lsquo;third&rsquo; as well
+as for the fraction 1/3, and similarly with all other accented
+numeral signs); <G>lb&prime;</G>=1/32, <G>rib&prime;</G>=1/112, &amp;c. There were
+special signs for 1/2, namely <G>&angsph;&prime;</G> or <G><*>&prime;</G>,<note>It has been suggested that the forms <G><*></G> and <G><*></G> for 1/2 found in
+inscriptions may perhaps represent half an <G>*o</G>, the sign, at all events
+in Boeotia, for 1 obol.</note> and for 2/3, namely <G>w&prime;</G>.
+When a number of submultiples are written one after the
+other, the sum of them is meant, and similarly when they
+follow a whole number; e.g. <G>&angsph;&prime;d&prime;</G>=1/2 1/4 or 3/4 (Archimedes);
+<MATH><G>kq w&prime; ig&prime; lq&prime;</G>=29 2/3 1/13 1/39=29 2/3+1/13+1/39 or 29 10/13;
+<G>mq&angsph;&prime;iz&prime;ld&prime;na&prime;</G>=49 1/2 1/17 1/34 1/51=49 31/51</MATH>
+(Heron, <I>Geom</I>. 15. 8, 13). But <G>ig&prime; to\ ig&prime;</G> means 1/13th times
+1/13 or 1/169 (<I>ibid.</I> 12. 5), &amp;c. A less orthodox method found
+in later manuscripts was to use two accents and to write,
+e.g., <G>z&Prime;</G> instead of <G>z&prime;</G>, for 1/7. In Diophantus we find a different
+mark in place of the accent; Tannery considers the genuine
+form of it to be &chi;, so that <G>g</G><SUP>&chi;</SUP>=1/3, and so on.
+<C>(<G>b</G>) <I>The ordinary Greek form, variously written</I>.</C>
+<p>An ordinary proper fraction (called by Euclid <G>me/rh</G>, <I>parts</I>,
+in the plural; as meaning a certain number of aliquot parts,
+in contradistinction to <G>me/ros</G>, <I>part</I>, in the singular, which he
+restricts to an aliquot part or submultiple) was expressed in
+various ways. The first was to use the ordinary cardinal
+number for the numerator followed by the accented number
+representing the denominator. Thus we find in Archimedes
+<G>&horbar;i oa&prime;</G>=10/71 and <G>&horbar;<SUB>'</SUB>awlh &horbar;q ia&prime;</G>=1838 9/11: (it should be noted,
+however, that the <G>&horbar;i oa&prime;</G> is a correction from <G>oia</G>, and this
+seems to indicate that the original reading was <FIG>, which
+would accord with Diophantus's and Heron's method of
+writing fractions). The method illustrated by these cases is
+open to objection as likely to lead to confusion, since <G>i oa&prime;</G>
+<pb n=43><head>FRACTIONS</head>
+would naturally mean 10 1/71 and <G>q ia&prime;</G> 9 1/11; the context alone
+shows the true meaning. Another form akin to that just
+mentioned was a little less open to misconstruction; the
+numerator was written in full with the accented numeral
+(for the denominator) following, e.g. <G>du/o me&prime;</G> for 2/45ths
+(Aristarchus of Samos). A better way was to turn the
+aliquot part into an abbreviation for the ordinal number
+with a termination superposed to represent the <I>case</I>, e.g.
+<G>d<SUP>wn</SUP> s</G>=6/4 (Dioph. Lemma to V. 8), <G>n kg<SUP>wn</SUP></G>=50/23 (<I>ibid.</I> I. 23),
+<G>rka<SUP>wn</SUP> <SUB>'</SUB>awld&angsph;&prime;</G>=1834 1/2/121 (<I>ibid.</I> IV. 39), just as <G>g<SUP>os</SUP></G> was
+written for the ordinal <G>tri/tos</G> (cf. <G>to\ s<SUP>on</SUP></G>, the 1/6th part, Dioph.
+IV. 39; <G>ai)/rw ta\ ig<SUP>a</SUP></G> &lsquo;I remove the 13ths&rsquo;, i.e. I multiply up
+by the denominator 13, <I>ibid.</I> IV. 9). But the trouble was
+avoided by each of two other methods.
+<p>(1) The accented letters representing the denominator were
+written twice, along with the cardinal number for the
+numerator. This method is mostly found in the <I>Geometrica</I>
+and other works of Heron: cf. <G>e ig&prime; ig&prime;</G>=5/13, <G>ta\ s z&prime;z&prime;</G>=6/7.
+The fractional signification is often emphasized by adding
+the word <G>lepta/</G> (&lsquo;fractions&rsquo; or &lsquo;fractional parts&rsquo;), e.g. in
+<G>lepta\ ig&prime; ig&prime; ib</G>=12/13 (<I>Geom</I>. 12. 5), and, where the expression
+contains units as well as fractions, the word &lsquo;units&rsquo; (<G>mona/des</G>)
+is generally added, for clearness' sake, to indicate the integral
+number, e.g. <G>mona/des ib kai\ lepta\ ig&prime; ig&prime; ib</G>=12 12/13 (<I>Geom</I>.
+12. 5), <G>mona/des rmd lepta\ ig&prime; ig&prime; sO|q</G>=144 299/13 (<I>Geom</I>. 12. 6).
+Sometimes in Heron fractions are alternatively given in this
+notation and in that of submultiples, e.g. <G>b g&prime; ie&prime; h)/toi b kai\
+b e&prime; e&prime;</G>=&lsquo;2 1/3 1/15 or 2 2/5&rsquo; (<I>Geom</I>. 12. 48); <G>z &angsph;&prime; i&prime; ie&prime; oe&prime; h)/toi
+mona/des z e&prime; e&prime; g kai\ b e&prime; e&prime; tw=n e&prime; e&prime;</G>=&lsquo;7 1/2 1/10 1/15 1/75 or 7 3/5+2/5X1/5&rsquo;,
+i.e. 7 3/5+2/25 (<I>ibid.</I>); <G>h &angsph;&prime; i&prime; ke&prime; h)/toi mona/des h e&prime; e&prime; g kai\ e&prime; to\ e&prime;</G>=
+&lsquo;8 1/2 1/10 1/25 or 8 3/5+1/5X1/5&rsquo;, i.e. 8 3/5+1/25 (<I>ibid.</I> 12. 46). (In
+Hultsch's edition of Heron single accents were used to de-
+note whole numbers and the numerators of fractions, while
+aliquot parts or denominators were represented by double
+accents; thus the last quoted expression was written
+<G>h&prime; <*> i&Prime; ke&Prime; h)/toi mona/des h&prime; e&Prime; e&Prime; g&prime; kai\ e&Prime; to\ e&Prime;</G>.)
+<p>But (2) the most convenient notation of all is that which
+is regularly employed by Diophantus, and occasionally in the
+<I>Metrica</I> of Heron. In this system the numerator of any
+fraction is written in the line, with the denominator <I>above</I> it,
+<pb n=44><head>GREEK NUMERICAL NOTATION</head>
+without accents or other marks (except where the numerator
+or denominator itself contains an accented fraction); the
+method is therefore simply the reverse of ours, but equally
+convenient. In Tannery's edition of Diophantus a line is
+put between the numerator below and the denominator above:
+thus <G>is</G>/<G>rka</G>=121/16. But it is better to omit the horizontal line
+(cf. <G>rkh</G>/<G>r</G>=100/128 in Kenyon's Papyri ii, No. cclxv. 40, and the
+fractions in Sch&ouml;ne's edition of Heron's <I>Metrica</I>). A few
+more instances from Diophantus may be given: <G>fib</G>/<G><SUB>'</SUB>buns</G>=2456/512
+(IV. 28); <G>a.sa</G>/<G><SUB>'</SUB>etnh</G>=5358/10201 (V. 9); <G>rnb</G>/<G>tpq&angsph;&prime;</G>=(389 1/2)/152. The deno-
+minator is rarely found above the numerator, but to the
+right (like an exponent); e.g. <G>&horbar;ie<SUP>d</SUP></G>=15/4 (I. 39). Even in the
+case of a submultiple, where, as we have said, the orthodox
+method was to omit the numerator and simply write the
+denominator with an accent, Diophantus often follows the
+method applicable to other fractions, e.g. he writes <G>fib</G>/<G>a</G> for
+1/512 (IV. 28). Numbers partly integral and partly fractional,
+where the fraction is a submultiple or expressed as the sum
+of submultiples, are written much as we write them, the
+fractions simply following the integer, e.g. <G>a g</G><SUP>&chi;</SUP>=1 1/3;
+<G>b &angsph;&prime; s</G><SUP>&chi;</SUP>=2 1/2 1/6 (Lemma to V. 8); <G>to &angsph;&prime; is</G><SUP>&chi;</SUP>=370 1/2 1/16 (III. 11).
+Complicated fractions in which the numerator and denomi-
+nator are algebraical expressions or large numbers are often
+expressed by writing the numerator first and separating it
+by <G>mori/ou</G> or <G>e)n mori/w|</G> from the denominator; i.e. the fraction
+is expressed as the numerator <I>divided by</I> the denominator:
+thus <FIG><G>rn.<SUB>'</SUB>z<*>pd mori/ou ks.<SUB>'</SUB>brmd</G>=1507984/262144 (IV. 28).
+<C>(<G>g</G>) <I>Sexagesimal fractions</I>.</C>
+<p>Great interest attaches to the system of sexagesimal
+fractions (Babylonian in its origin, as we have seen) which
+was used by the Greeks in astronomical calculations, and
+<pb n=45><head>SEXAGESIMAL FRACTIONS</head>
+appears fully developed in the <I>Syntaxis</I> of Ptolemy. The
+circumference of a circle, and with it the four right angles
+subtended by it at the centre, were divided into 360 parts
+(<G>tmh/mata</G> or <G>moi=rai</G>), as we should say <I>degrees</I>, each <G>moi=ra</G>
+into 60 parts called (<G>prw=ta</G>) <G>e(xhkosta/</G>, (<I>first</I>) <I>sixtieths</I> or
+<I>minutes</I> (<G>lepta/</G>), each of these again into 60 <G>deu/tera e(xhkosta/</G>,
+<I>seconds</I>, and so on. In like manner, the diameter of the
+circle was divided into 120 <G>tmh/mata</G>, <I>segments</I>, and each of
+these segments was divided into sixtieths, each sixtieth
+again into sixty parts, and so on. Thus a convenient
+fractional system was available for arithmetical calculations
+in general; for the unit could be chosen at will, and any
+mixed number could be expressed as so many of those units
+<I>plus</I> so many of the fractions which we should represent
+by 1/60, so many of those which we should write (1/60)<SUP>2</SUP>, (1/60)<SUP>3</SUP>,
+and so on to any extent. The units, <G>tmh/mata</G> or <G>moi=rai</G> (the
+latter often denoted by the abbreviation <G>m</G>&deg;), were written
+first, with the ordinary numeral representing the number
+of them; then came a simple numeral with one accent repre-
+senting that number of <I>first sixtieths</I>, or minutes, then a
+numeral with two accents representing that number of
+<I>second sixtieths</I>, or seconds, and so on. Thus <G>m&deg; b</G>=2&deg;,
+<G>moirw=n mz mb&prime; m&Prime;</G>=47&deg; 42&prime; 40&Prime;. Similarly, <G>tmhma/twn xz
+d&prime; ne&Prime;</G>=67<I>p</I> 4&prime; 55&Prime;, where <I>p</I> denotes the <I>segment</I> (of the
+diameter). Where there was no unit, or no number of
+sixtieths, second sixtieths, &amp;c., the symbol <G>*o</G>, signifying
+<G>ou)demi/a moi=ra, ou)de\n e(xhkosto/n</G>, and the like, was used; thus
+<G>moirw=n *o a&prime; b&Prime; *o&tprime;</G>=0&deg;1&prime;2&Prime;0&tprime;. The system is parallel to
+our system of decimal fractions, with the difference that the
+submultiple is 1/60 instead of 1/10 nor is it much less easy to
+work with, while it furnishes a very speedy way of approxi-
+mating to the values of quantities not expressible in whole
+numbers. For example, in his Table of Chords, Ptolemy says
+that the chord subtending an angle of 120&deg; at the centre is
+(<G>tmhma/twn</G>) <G>rg ne&prime; kg&Prime;</G> or 103<I>p</I> 55&prime; 23&Prime;; this is equivalent
+(since the radius of the circle is 60 <G>tmh/mata</G>) to saying that
+&radic;3=1+43/60+55/60<SUP>2</SUP>+23/60<SUP>3</SUP>, and this works out to 1.7320509 ...,
+which is correct to the seventh decimal place, and exceeds
+the true value by 0.00000003 only.
+<pb n=46><head>GREEK NUMERICAL NOTATION</head>
+<C>Practical calculation.</C>
+<C>(<G>a</G>) <I>The abacus</I>.</C>
+<p>In practical calculation it was open to the Greeks to secure
+the advantages of a position-value system by using the
+abacus. The essence of the abacus was the arrangement of
+it in columns which might be vertical or horizontal, but were
+generally vertical, and pretty certainly so in Greece and
+Egypt; the columns were marked off by lines or in some
+other way and allocated to the successive denominations of
+the numerical system in use, i.e., in the case of the decimal
+system, the units, tens, hundreds, thousands, myriads, and so
+on. The number of units of each denomination was shown in
+each column by means of pebbles, pegs, or the like. When,
+in the process of addition or multiplication, the number of
+pebbles collected in one column becomes sufficient to make
+one or more units of the next higher denomination, the num-
+ber of pebbles representing the complete number of the higher
+units is withdrawn from the column in question and the
+proper number of the higher units added to the next higher
+column. Similarly, in subtraction, when a number of units of
+one denomination has to be subtracted and there are not
+enough pebbles in the particular column to subtract from, one
+pebble from the next higher column is withdrawn and actually
+or mentally resolved into the number of the lower units
+equivalent in value; the latter number of additional pebbles
+increases the number already in the column to a number from
+which the number to be subtracted can actually be withdrawn.
+The details of the columns of the Greek abacus have unfor-
+tunately to be inferred from the corresponding details of the
+Roman abacus, for the only abaci which have been preserved
+and can with certainty be identified as such are Roman.
+There were two kinds; in one of these the marks were
+buttons or knobs which could be moved up and down in each
+column, but could not be taken out of it, while in the other
+kind they were pebbles which could also be moved from one
+column to another. Each column was in two parts, a shorter
+portion at the top containing one button only, which itself
+represented half the number of units necessary to make up
+one of the next higher units, and a longer portion below
+<pb n=47><head>PRACTICAL CALCULATION</head>
+containing one less than half the same number. This arrange-
+ment of the columns in two parts enabled the total number of
+buttons to be economized. The columns represented, so far as
+integral numbers were concerned, units, tens, hundreds, thou-
+sands, &amp;c., and in these cases the one button in the top
+portion of each column represented five units, and there were
+four buttons in the lower portion representing four units.
+But after the columns representing integers came columns
+representing fractions; the first contained buttons represent-
+ing <I>unciae</I>, of which there were 12 to the unit, i.e. fractions
+of 1/(12)th, and in this case the one button in the top portion
+represented 6 <I>unciae</I> or 6/(12)ths, while there were 5 buttons in
+the lower portion (instead of 4), the buttons in the column
+thus representing in all 11 <I>unciae</I> or 12ths. After this column
+there were (in one specimen) three other shorter ones along-
+side the lower portions only of the columns for integers, the
+first representing fractions of 1/(24)th (one button), the second
+fractions of 1/(48)th (one button), and the third fractions of 1/(72)nd
+(two buttons, which of course together made up 1/(36)th).
+<p>The mediaeval writer of the so-called geometry of Bo&euml;tius
+describes another method of indicating in the various columns
+the number of units of each denomination.<note>Bo&euml;tius, <I>De Inst. Ar.</I>, ed. Friedlein, pp. 396 sq.</note> According to him
+&lsquo;abacus&rsquo; was a later name for what was previously called
+<I>mensa Pythagorea</I>, in honour of the Master who had taught
+its use. The method was to put in the columns, not the neces-
+sary number of pebbles or buttons, but the corresponding
+<I>numeral</I>, which might be written in sand spread over the
+surface (in the same way as Greek geometers are said to have
+drawn geometrical figures in sand strewn on boards similarly
+called <G>a)/bax</G> or <G>a)\ba/kion</G>). The figures put in the columns were
+called <I>apices</I>. The first variety of numerals mentioned by the
+writer are rough forms of the Indian figures (a fact which
+proves the late date of the composition); but other forms were
+(1) the first letters of the alphabet (which presumably mean
+the Greek alphabetic numerals) or (2) the ordinary Roman
+figures.
+<p>We should expect the arrangement of the Greek abacus to
+correspond to the Roman, but the actual evidence regarding its
+form and the extent to which it was used is so scanty that
+<pb n=48><head>GREEK NUMERICAL NOTATION</head>
+we may well doubt whether any great use was made of it at
+all. But the use of pebbles to reckon with is attested by
+several writers. In Aristophanes (<I>Wasps</I>, 656-64) Bdelycleon
+tells his father to do an easy sum &lsquo;not with pebbles but with
+fingers&rsquo;, as much as to say, &lsquo;There is no need to use pebbles
+for this sum; you can do it on your fingers.&rsquo; &lsquo;The income
+of the state&rsquo;, he says, &lsquo;is 2000 talents; the yearly payment
+to the 6000 dicasts is only 150 talents.&rsquo; &lsquo;Why&rsquo;, answers the
+old man, &lsquo;we don't get a tenth of the revenue.&rsquo; The calcula-
+tion in this case amounted to multiplying 150 by 10 to show
+that the product is less than 2000. But more to the purpose
+are the following allusions. Herodotus says that, in reckoning
+with pebbles, as in writing, the Greeks move their hand from
+left to right, the Egyptians from right to left<note>Herodotus, ii. c. 36.</note>; this indicates
+that the columns were vertical, facing the reckoner. Diogenes
+Laertius attributes to Solon a statement that those who had
+influence with tyrants were like the pebbles on a reckoning-
+board, because they sometimes stood for more and sometimes
+for less.<note>Diog. L. i. 59.</note> A character in a fourth-century comedy asks for an
+abacus and pebbles to do his accounts.<note>Alexis in Athenaeus, 117 c.</note> But most definite of
+all is a remark of Polybius that &lsquo;These men are really like
+the pebbles on reckoning-boards. For the latter, according
+to the pleasure of the reckoner, have the value, now of a
+<G>xalkou=s</G> (1/8th of an obol or 1/(48)th of a drachma), and the next
+moment of a talent.&rsquo;<note>Polybius, v. 26. 13.</note> The passages of Diogenes Laertius and
+Polybius both indicate that the pebbles were not fixed in the
+columns, but could be transferred from one to another, and
+the latter passage has some significance in relation to the
+Salaminian table presently to be mentioned, because the talent
+and the <G>xalkou=s</G> are actually the extreme denominations on
+one side of the table.
+<p>Two relics other than the Salaminian table may throw
+some light on the subject. First, the so-called Darius-vase
+found at Canosa (Canusium), south-west of Barletta, represents
+a collector of tribute of distressful countenance with a table in
+front of him having pebbles, or (as some maintain) coins, upon
+it and, on the right-hand edge, beginning on the side farthest
+away and written in the direction towards him, the letters
+<pb n=49><head>PRACTICAL CALCULATION</head>
+<G>*m*y*h<*><*>*o<*>*t</G>, while in his left hand he holds a sort of book in
+which, presumably, he has to enter the receipts. Now <G>*m</G>, <G>*y</G>
+(=<G>*x</G>), <G>*h</G>, and <G><*></G> are of course the initial letters of the words
+for 10000, 1000, 100, and 10 respectively. Here therefore we
+have a purely decimal system, without the halfway numbers
+represented by <G><*></G> (=<G>pe/nte</G>, 5) in combination with the other
+initial letters which we find in the &lsquo;Attic&rsquo; system. The sign
+<G><*></G> after <G><*></G> seems to be wrongly written for <G><*></G>, the older sign
+for a drachma, <G>*o</G> stands for the obol, <G><*></G> for the 1/2-obol, and <G>*t</G>
+(<G>tetarthmo/rion</G>) for the 1/4-obol.<note>Keil in <I>Hermes</I>, 29, 1894, pp. 262-3.</note> Except that the fractions of
+the unit (here the drachma) are different from the fractions
+of the Roman unit, this scheme corresponds to the Roman,
+and so far might represent the abacus. Indeed, the decimal
+arrangement corresponds better to the abacus than does the
+Salaminian table with its intermediate &lsquo;Herodianic&rsquo; signs for
+500, 50, and 5 drachmas. Prof. David Eugene Smith is, how-
+ever, clear that any one can see from a critical examination of
+the piece that what is represented is an ordinary money-
+changer or tax-receiver with coins on a table such as one
+might see anywhere in the East to-day, and that the table has
+no resemblance to an abacus.<note><I>Bibliotheca Mathematica</I>, ix<SUB>3</SUB>, p. 193.</note> On the other hand, it is to be
+observed that the open book held by the tax-receiver in his
+left hand has <G>*t*a*l*n</G> on one page and <G>*t<*>*i/*h</G> on the other,
+which would seem to indicate that he was entering totals in
+<I>talents</I> and must therefore presumably have been <I>adding</I> coins
+or pebbles on the table before him.
+<p>There is a second existing monument of the same sort,
+namely a so-called <G>sh/kwma</G> (or arrangement of measures)
+discovered about forty years ago<note>Dumont in <I>Revue arch&eacute;ologique</I>, xxvi (1873), p. 43.</note>; it is a stone tablet with
+fluid measures and has, on the right-hand side, the numerals
+<G>*x<*>*h<*><*><*><*>*t*i<*></G>. The signs are the &lsquo;Herodianic&rsquo;, and they
+include those for 500, 50, and 5 drachmas; <G><*></G> is the sign for
+a drachma, <G>*t</G> evidently stands for some number of obols
+making a fraction of the drachma, i.e. the <G>triw/bolon</G> or 3
+obols, <G>*i</G> for an obol, and <G><*></G> for a 1/2-obol.
+<p>The famous Salaminian table was discovered by Rangab&eacute;,
+who gave a drawing and description of it immediately after-
+<pb n=50><head>GREEK NUMERICAL NOTATION</head>
+wards (1846).<note><I>Revue arch&eacute;ologique</I>, iii. 1846.</note> The table, now broken into two unequal parts,
+is in the Epigraphical Museum at Athens. The facts with
+regard to it are stated, and a photograph of it is satisfactorily
+produced, by Wilhelm Kubitschek.<note><I>Wiener numismatische Zeitschrift</I>, xxxi. 1899, pp. 393-8, with
+Plate xxiv.</note> A representation of it is
+also given by Nagl<note><I>Abh. zur Gesch. d. Math.</I> ix. 1899, plate after p. 357.</note> based on Rangab&eacute;'s description, and the
+sketch of it here appended follows Nagl's drawing. The size
+and material of the table (according to Rangab&eacute;'s measure-
+ments it is 1.5 metres long and 0.75 metre broad) show that
+<FIG>
+it was no ordinary abacus; it may
+have been a fixture intended for
+quasi-public use, such as a banker's
+or money-changer's table, or again
+it may have been a scoring-table
+for some kind of game like <I>tric-
+trac</I> or backgammon. Opinion has
+from the first been divided between
+the two views; it has even been
+suggested that the table was in-
+tended for both purposes. But there
+can be no doubt that it was used
+for some kind of calculation and,
+if it was not actually an abacus, it
+may at least serve to give an idea
+of what the abacus was like. The
+difficulties connected with its in-
+terpretation are easily seen. The
+series of letters on the three sides are the same except
+that two of them go no higher than <G>*x</G> (1000 drachmae),
+but the third has <G><*></G> (5000 drachmae), and <G>*t</G> (the talent or
+6000 drachmae) in addition; <G><*></G> is the sign for a drachma,
+<G>*i</G> for an obol (1/6th of the drachma), <G><*></G> for 1/2-obol, <G>*t</G> for 1/4-obol
+(<G>tetarthmo/rion</G>, Boeckh's suggestion), not 1/3-obol (<G>trithmo/rion</G>,
+Vincent), and <G>*x</G> for 1/8-obol (<G>xalkou=s</G>). It seems to be
+agreed that the four spaces provided between the five shorter
+lines were intended for the fractions of the drachma; the first
+space would require 5 pebbles (one less than the 6 obols
+making up a drachma), the others one each. The longer
+<pb n=51><head>PRACTICAL CALCULATION</head>
+lines would provide the spaces for the drachmae and higher
+denominations. On the assumption that the cross line indi-
+cates the Roman method of having one pebble above it to
+represent 5, and four below it representing units, it is clear
+that, including denominations up to the talent (6000 drachmae),
+only five columns are necessary, namely one for the talent or
+6000 drachmae, and four for 1000, 100, 10 drachmae, and 1
+drachma respectively. But there are actually ten spaces pro-
+vided by the eleven lines. On the theory of the game-board,
+five of the ten on one side (right or left) are supposed to
+belong to each of two players placed facing each other on the
+two longer sides of the table (but, if in playing they had to
+use the shorter columns for the fractions, it is not clear how
+they would make them suffice); the cross on the middle of the
+middle line might in that case serve to mark the separation
+between the lines belonging to the two players, or perhaps all
+the crosses may have the one object of helping the eye to dis-
+tinguish all the columns from one another. On the assump-
+tion that the table is an abacus, a possible explanation of the
+<I>eleven</I> lines is to suppose that they really supply <I>five</I> columns
+only, the odd lines marking the divisions between the columns,
+and the even lines, one in the middle of each column,
+marking where the pebbles should be placed in rows; in this
+case, if the crosses are intended to mark divisions between the
+four pebbles representing units and the one pebble represent-
+ing 5 in each column, the crosses are only required in the last
+three columns (for 100, 10, and 1), because, the highest de-
+nomination being 6000 drachmae, there was no need for a
+division of the 1000-column, which only required five unit-
+pebbles altogether. Nagl, a thorough-going supporter of the
+abacus-theory to the exclusion of the other, goes further and
+shows how the Salaminian table could have been used for the
+special purpose of carrying out a long multiplication; but this
+development seems far-fetched, and there is no evidence of
+such a use.
+<p>The Greeks in fact had little need of the abacus for calcu-
+lations. With their alphabetic numerals they could work out
+their additions, subtractions, multiplications, and divisions
+without the help of any marked columns, in a form little less
+convenient than ours: examples of long multiplications, which
+<pb n=52><head>GREEK NUMERICAL NOTATION</head>
+include addition as the last step in each case, are found in
+Eutocius's commentary on Archimedes's <I>Measurement of
+a Circle</I>. We will take the four arithmetical operations
+separately.
+<C>(<G>b</G>) <I>Addition and Subtraction</I>.</C>
+<p>There is no doubt that, in writing down numbers for the
+purpose of these operations, the Greeks would keep the several
+powers of 10 separate in a manner practically corresponding
+to our system of numerals, the hundreds, thousands, &amp;c., being
+written in separate vertical rows. The following would be
+a typical example of a sum in addition:
+<table>
+<tr><td align=right><G><SUB>'</SUB>aukd</G></td><td>=</td><td align=right>1424</td></tr>
+<tr><td align=right><G>r g</G></td><td></td><td align=right>103</td></tr>
+<tr><td align=right><G><FIG><SUB>'</SUB>bspa</G></td><td></td><td align=right>12281</td></tr>
+<tr><td align=right><G><FIG> l</G></td><td></td><td align=right>30030</td></tr>
+<tr><td align=right><G><FIG><SUB>'</SUB>gwlh</G></td><td></td><td align=right>43838</td></tr>
+</table>
+and the mental part of the work would be the same for the
+Greek as for us.
+<p>Similarly a subtraction would be represented as follows:
+<table>
+<tr><td><G><FIG><SUB>'</SUB>gxls</G></td><td>=</td><td>93636</td></tr>
+<tr><td><G><FIG><SUB>'</SUB>gu q</G></td><td></td><td>23409</td></tr>
+<tr><td><G><FIG> skz</G></td><td></td><td>70227</td></tr>
+</table>
+<C>(<G>g</G>) <I>Multiplication</I>.</C>
+<C>(i) The Egyptian method.</C>
+<p>For carrying out multiplications two things were required.
+The first was a multiplication table. This the Greeks are
+certain to have had from very early times. The Egyptians,
+indeed, seem never to have had such a table. We know from
+the Papyrus Rhind that in order to multiply by any number
+the Egyptians began by successive doubling, thus obtaining
+twice, four times, eight times, sixteen times the multiplicand,
+and so on; they then added such sums of this series of multi-
+ples (including once the multiplicand) as were required. Thus,
+<pb n=53><head>MULTIPLICATION</head>
+to multiply by 13, they did not take 10 times and 3 times
+the multiplicand respectively and add them, but they found
+13 times the multiplicand by adding once and 4 times and 8
+times it, which elements they had obtained by the doubling
+process; similarly they would find 25 times any number by
+adding once and 8 times and 16 times the number.<note>I have been told that there is a method in use to-day (some say in
+Russia, but I have not been able to verify this), which is certainly attractive
+and looks original, but which will immediately be seen to amount simply
+to an elegant practical method of carrying out the Egyptian procedure.
+Write out side by side in successive lines, so as to form two columns,
+(1) the multiplier and multiplicand, (2) half the multiplier (or the
+nearest integer below it if the multiplier is odd) and twice the multi-
+plicand, (3) half (or the nearest integer below the half) of the number
+in the first column of the preceding row and twice the number in the
+second column of the preceding row, and so on, until we have 1 in
+the first column. Then strike out all numbers in the second column
+which are opposite <I>even</I> numbers in the first column, and add all the
+numbers left in the second column. The sum will be the required
+product. Suppose e.g. that 157 is to be multiplied by 83. The rows
+and columns then are:
+<table>
+<tr><td align=right>83</td><td align=right>157</td><td></td></tr>
+<tr><td align=right>41</td><td align=right>314</td><td></td></tr>
+<tr><td align=right>20</td><td align=right><STRIKE>628</STRIKE></td><td></td></tr>
+<tr><td align=right>10</td><td align=right><STRIKE>1256</STRIKE></td><td></td></tr>
+<tr><td align=right>5</td><td align=right>2512</td><td></td></tr>
+<tr><td align=right>2</td><td align=right><STRIKE>5024</STRIKE></td><td></td></tr>
+<tr><td align=right>1</td><td align=right>10048</td><td></td></tr>
+<tr><td></td><td align=right>13031</td><td>= 83 x 157</td></tr>
+</table>
+The explanation is, of course, that, where we take half the preceding
+number in the first column <I>less one</I>, we omit once the figure in the right-
+hand column, so that it must be left in that column to be added in at
+the end; and where we take the exact half of an even number, we
+omit nothing in the right-hand column, but the new line is the <I>exact</I>
+equivalent of the preceding one, which can therefore be struck out.</note> Division
+was performed by the Egyptians in an even more rudimen-
+tary fashion, namely by a tentative back-multiplication begin-
+ning with the same doubling process. But, as we have seen
+(p. 14), the scholiast to the <I>Charmides</I> says that the branches
+of <G>logistikh/</G> include the &lsquo;so-called Greek and Egyptian
+methods in multiplications and divisions&rsquo;.
+<C>(ii) The Greek method.</C>
+<p>The Egyptian method being what we have just described, it
+seems clear that the Greek method, which was different,
+depended on the direct use of a multiplication table. A frag-
+ment of such a multiplication table is preserved on a two-
+leaved wax tablet in the British Museum (Add. MS. 34186).
+<pb n=54><head>GREEK NUMERICAL NOTATION</head>
+It is believed to date from the second century A. D., and it
+probably came from Alexandria or the vicinity. But the
+form of the characters and the mingling of capitals and small
+letters both allow of an earlier date; e.g. there is in the
+Museum a Greek papyrus assigned to the third century B.C.
+in which the numerals are very similar to those on the tablet.<note>David Eugene Smith in <I>Bibliotheca Mathematica</I>, ix<SUB>3</SUB>, pp. 193-5.</note>
+<p>The second requirement is connected with the fact that the
+Greeks began their multiplications by taking the product of
+the highest constituents first, i.e. they proceeded as we should
+if we were to begin our long multiplications from the left
+instead of the right. The only difficulty would be to settle
+the denomination of the products of two high powers of ten.
+With such numbers as the Greeks usually had to multiply
+there would be no trouble; but if, say, the factors were un-
+usually large numbers, e.g. millions multiplied by millions or
+billions, care would be required, and even some rule for
+settling the denomination, or determining the particular
+power or powers of 10 which the product would contain.
+This exceptional necessity was dealt with in the two special
+treatises, by Archimedes and Apollonius respectively, already
+mentioned. The former, the <I>Sand-reckoner</I>, proves that, if
+there be a series of numbers, 1, 10, 10<SUP>2</SUP>, 10<SUP>3</SUP>... 10<SUP><I>m</I></SUP>... 10<SUP><I>n</I></SUP>...,
+then, if 10<SUP><I>m</I></SUP>, 10<SUP><I>n</I></SUP> be any two terms of the series, their product
+10<SUP><I>m</I></SUP>.10<SUP><I>n</I></SUP> will be a term in the same series and will be as many
+terms distant from 10<SUP><I>n</I></SUP> as the term 10<SUP><I>m</I></SUP> is distant from 1;
+also it will be distant from 1 by a number of terms less by
+one than the sum of the numbers of terms by which 10<SUP><I>m</I></SUP> and
+10<SUP><I>n</I></SUP> respectively are distant from 1. This is easily seen to be
+equivalent to the fact that, 10<SUP><I>m</I></SUP> being the (<I>m</I>+1)th term
+beginning with 1, and 10<SUP><I>n</I></SUP> the (<I>n</I>+1)th term beginning
+with 1, the product of the two terms is the (<I>m</I>+<I>n</I>+1)th
+term beginning with 1, and is 10<SUP><I>m</I>+<I>n</I></SUP>.
+<C>(iii) Apollonius's continued multiplications.</C>
+<p>The system of Apollonius deserves a short description.<note>Our authority here is the <I>Synagoge</I> of Pappus, Book ii, pp. 2-28, Hultsch.</note> Its
+object is to give a handy method of finding the continued
+product of any number of factors, each of which is represented
+by a single letter in the Greek numeral notation. It does not
+<pb n=55><head>MULTIPLICATION</head>
+therefore show how to multiply two large numbers each of
+which contains a number of digits (in our notation), that is,
+a certain number of units, a certain number of tens, a certain
+number of hundreds, &amp;c.; it is confined to the multiplication
+of any number of factors each of which is one or other of the
+following: (<I>a</I>) a number of units as 1, 2, 3, ... 9, (<I>b</I>) a number
+of even tens as 10, 20, 30, ... 90, (<I>c</I>) a number of even hundreds
+as 100, 200, 300, ... 900. It does not deal with factors above
+hundreds, e.g. 1000 or 4000; this is because the Greek
+numeral alphabet only went up to 900, the notation begin-
+ning again after that with <G><SUB>'</SUB>a</G>, <G><SUB>'</SUB>b</G>, ... for 1000, 2000, &amp;c. The
+essence of the method is the separate multiplication (1) of the
+<I>bases</I>, <G>puqme/nes</G>, of the several factors, (2) of the powers of ten
+contained in the factors, that is, what we represent by the
+ciphers in each factor. Given a multiple of ten, say 30, 3 is
+the <G>puqmh/n</G> or base, being the same number of units as the
+number contains tens; similarly in a multiple of 100, say 800,
+8 is the base. In multiplying three numbers such as 2, 30,
+800, therefore, Apollonius first multiplies the bases, 2, 3, and 8,
+then finds separately the product of the ten and the hundred,
+and lastly multiplies the two products. The final product has
+to be expressed as a certain number of units less than a
+myriad, then a certain number of myriads, a certain number
+of &lsquo;double myriads&rsquo; (myriads squared), &lsquo;triple myriads&rsquo;
+(myriads cubed), &amp;c., in other words in the form
+<MATH><I>A</I><SUB>0</SUB>+<I>A</I><SUB>1</SUB><I>M</I>+<I>A</I><SUB>2</SUB><I>M</I><SUP>2</SUP>+...</MATH>,
+where <I>M</I> is a myriad or 10<SUP>4</SUP> and <I>A</I><SUB>0</SUB>, <I>A</I><SUB>1</SUB> ... respectively repre-
+sent some number not exceeding 9999.
+<p>No special directions are given for carrying out the multi-
+plication of the <I>bases</I> (digits), or for the multiplication of
+their product into the product of the tens, hundreds, &amp;c.,
+when separately found (directions for the latter multiplica-
+tion may have been contained in propositions missing from
+the mutilated fragment in Pappus). But the method of deal-
+ing with the tens and hundreds (the ciphers in our notation)
+is made the subject of a considerable number of separate
+propositions. Thus in two propositions the factors are all of
+one sort (tens or hundreds), in another we have factors of two
+sorts (a number of factors containing units only multiplied
+<pb n=56><head>GREEK NUMERICAL NOTATION</head>
+by a number of multiples of ten, each less than 100, or by
+multiples of 100, each less than 1000), and so on. In the final
+proposition (25), with which the introductory lemmas close,
+the factors are of all three kinds, some containing units only,
+others being multiples of 10 (less than 100) and a third set
+being multiples of 100 (less than 1000 in each case). As
+Pappus frequently says, the proof is easy &lsquo;in numbers&rsquo;;
+Apollonius himself seems to have proved the propositions by
+means of lines or a diagram in some form. The method is the
+equivalent of taking the indices of all the separate powers of
+ten included in the factors (in which process ten =10<SUP>1</SUP> counts
+as 1, and 100=10<SUP>2</SUP> as 2), adding the indices together, and then
+dividing the sum by 4 to obtain the power of the myriad
+(10000) which the product contains. If the whole number in
+the quotient is <I>n</I>, the product contains (10000)<SUP><I>n</I></SUP> or the
+<I>n</I>-myriad in Apollonius's notation. There will in most cases
+be a remainder left after division by 4, namely 3, 2, or 1: the
+remainder then represents (in our notation) 3, 2, or 1 more
+ciphers, that is, the product is 1000, 100, or 10 times the
+<I>n</I>-myriad, or the 10000<SUP><I>n</I></SUP>, as the case may be.
+<p>We cannot do better than illustrate by the main problem
+which Apollonius sets himself, namely that of multiplying
+together all the numbers represented by the separate letters
+in the hexameter:
+<C><G>*)arte/midos klei=te kra/tos e)/xokon e)nne/a kou=rai</G>.</C>
+<p>The number of letters, and therefore of factors, is 38, of which
+10 are multiples of 100 less than 1000, namely <G>r</G>, <G>t</G>, <G>s</G>, <G>t</G>, <G>r</G>, <G>t</G>,
+<G>s</G>, <G>x</G>, <G>u</G>, <G>r</G> (=100, 300, 200, 300, 100, 300, 200, 600, 400, 100),
+17 are multiples of 10 less than 100, namely <G>m</G>, <G>i</G>, <G>o</G>, <G>k</G>, <G>l</G>, <G>i</G>, <G>k</G>, <G>o</G>, <G>x</G>,
+<G>o</G>, <G>o</G>, <G>n</G>, <G>n</G>, <G>n</G>, <G>k</G>, <G>o</G>, <G>i</G> (=40, 10, 70, 20, 30, 10, 20, 70, 60, 70, 70, 50,
+50, 50, 20, 70, 10), and 11 are numbers of units not exceeding
+9, namely <G>a</G>, <G>e</G>, <G>d</G>, <G>e</G>, <G>e</G>, <G>a</G>, <G>e</G>, <G>e</G>, <G>e</G>, <G>a</G>, <G>a</G> (=1, 5, 4, 5, 5, 1, 5, 5, 5, 1, 1).
+The sum of the indices of powers of ten contained in the
+factors is therefore <MATH>10.2+17.1=37</MATH>. This, when divided by
+4, gives 9 with 1 as remainder. Hence the product of all the
+tens and hundreds, excluding the <I>bases</I> in each, is 10.10000<SUP>9</SUP>.
+<p>We have now, as the second part of the operation, to mul-
+tiply the numbers containing units only by the <I>bases</I> of all the
+other factors, i.e. (beginning with the <I>bases</I>, first of the hun-
+dreds, then of the tens) to multiply together the numbers:
+<pb n=57><head>MULTIPLICATION</head>
+1, 3, 2, 3, 1, 3, 2, 6, 4, 1,
+4, 1, 7, 2, 3, 1, 2, 7, 6, 7, 7, 5, 5, 5, 2, 7, 1,
+and 1, 5, 4, 5, 5, 1, 5, 5, 5, 1, 1.
+<p>The product is at once given in the text as 19 &lsquo;quadruple
+myriads&rsquo;, 6036 &lsquo;triple myriads&rsquo;, and 8480 &lsquo;double myriads&rsquo;, or
+<MATH>19.10000<SUP>4</SUP>+6036.10000<SUP>3</SUP>+8480.10000<SUP>2</SUP></MATH>.
+(The detailed multiplication line by line, which is of course
+perfectly easy, is bracketed by Hultsch as interpolated.)
+<p>Lastly, says Pappus, this product multiplied by the other
+(the product of the tens and hundreds without the <I>bases</I>),
+namely 10.10000<SUP>9</SUP>, as above, gives
+<MATH>196.10000<SUP>13</SUP>+368.10000<SUP>12</SUP>+4800.10000<SUP>11</SUP></MATH>.
+<C>(iv) Examples of ordinary multiplications.</C>
+<p>I shall now illustrate, by examples taken from Eutocius, the
+Greek method of performing long multiplications. It will be
+seen that, as in the case of addition and subtraction, the
+working is essentially the same as ours. The multiplicand is
+written first, and below it is placed the multiplier preceded by
+<G>e)pi/</G> (=&lsquo;by&rsquo; or &lsquo;into&rsquo;). Then the term containing the highest
+power of 10 in the multiplier is taken and multiplied into all
+the terms in the multiplicand, one after the other, first into that
+containing the highest power of 10, then into that containing
+the next highest power of 10, and so on in descending order;
+after which the term containing the next highest power of 10
+in the multiplier is multiplied into all the terms of the multi-
+plicand in the same order; and so on. The same procedure
+is followed where either or both of the numbers to be multi-
+plied contain fractions. Two examples from Eutocius will
+make the whole operation clear.
+<p>(1)
+<table>
+<tr><td></td><td><G><SUB>'</SUB>atna</G></td><td align=right>1351</td><td></td><td></td><td></td><td></td><td></td></tr>
+<tr><td align=right><G>e)pi/</G></td><td><G><SUB>'</SUB>atna</G></td><td align=right>X 1351</td><td></td><td></td><td></td><td></td><td></td></tr>
+<tr><td></td><td><G><FIG><SUB>'</SUB>a</G></td><td align=right>1000000</td><td align=right>300000</td><td align=right>50000</td><td align=right>1000</td><td></td><td></td></tr>
+<tr><td></td><td><G><FIG><SUB>'</SUB>et</G></td><td align=right>300000</td><td align=right>90000</td><td align=right>15000</td><td align=right>300</td><td></td><td></td></tr>
+<tr><td></td><td><G><FIG><SUB>'</SUB>e<SUB>'</SUB>bfn</G></td><td></td><td align=right>50000</td><td align=right>15000</td><td align=right>2500</td><td>50</td><td></td></tr>
+<tr><td></td><td align=right><G><SUB>'</SUB>atna</G></td><td></td><td></td><td align=right>1000</td><td align=right>300</td><td>50</td><td>1</td></tr>
+<tr><td align=right><G>o(mou=</G></td><td><G><FIG><SUB>'</SUB>esa</G></td><td align=right><I>together</I></td><td align=right>1825201.</td><td></td><td></td><td></td><td></td></tr>
+</table>
+<pb n=58><head>GREEK NUMERICAL NOTATION</head>
+<p>(2)
+<table>
+<tr><td></td><td><G><SUB>'</SUB>gig&angsph;d&prime;</G></td><td align=right>3013 1/2 1/4</td><td colspan=2>[=3013 3/4]</td><td></td><td></td></tr>
+<tr><td align=right><G>e)pi\</G></td><td><G><SUB>'</SUB>gig&angsph;d&prime;</G></td><td align=right>X 3013 1/2 1/4</td><td></td><td></td><td></td><td></td></tr>
+<tr><td></td><td><G><FIG><SUB>'</SUB>q<SUB>'</SUB>afyn</G></td><td align=right>9000000</td><td align=right>30000</td><td align=right>9000</td><td align=right>1500</td><td align=right>750</td></tr>
+<tr><td></td><td><G><FIG>rleb&angsph;</G></td><td align=right>30000</td><td align=right>100</td><td align=right>30</td><td align=right>5</td><td align=right>2 1/2</td></tr>
+<tr><td></td><td><G><SUB>'</SUB>qlqa&angsph;&angsph;d&prime;</G></td><td align=right>9000</td><td align=right>30</td><td align=right>9</td><td align=right>1 1/2</td><td align=right>1/2 1/4</td></tr>
+<tr><td></td><td><G><SUB>'</SUB>afea&angsph;d&prime;h&prime;</G></td><td align=right>1500</td><td align=right>5</td><td align=right>1 1/2</td><td align=right>1/4</td><td align=right>1/8</td></tr>
+<tr><td></td><td><G>ynb&angsph;&angsph;d&prime;h&prime;is&prime;</G></td><td align=right>750</td><td align=right>2 1/2</td><td align=right>1/2 1/4</td><td align=right>1/8</td><td align=right>1/(16)</td></tr>
+<tr><td align=right><G>o(mou=</G></td><td><G><FIG><SUB>'</SUB>bxpqis&prime;</G></td><td colspan=4 align=center><I>together</I> 9082689 1/(16).</td><td></td></tr>
+</table>
+<p>The following is one among many instances in which Heron
+works out a multiplication of two numbers involving fractions.
+He has to multiply 4 (33)/(64) by 7 (62)/(64), which he effects as follows
+(<I>Geom</I>. 12. 68):
+<MATH>4.7 = 28,
+4.(62)/(64) = (248)/(64),
+(33)/(64).7 = (231)/(64)
+(33)/(64).(62)/(64) = (2046)/(64).1/(64) = (31)/(64)+(62)/(64).1/(64)</MATH>;
+the result is therefore
+<MATH>28 (510)/(64)+(62)/(64).1/(64) = 28+7 (62)/(64)+(62)/(64).1/(64)
+= 35 (62)/(64)+(62)/(64).1/(64)</MATH>.
+<p>The multiplication of 37&deg;4&prime;55&Prime; (in the sexagesimal system)
+by itself is performed by Theon of Alexandria in his com-
+mentary on Ptolemy's <I>Syntaxis</I> in an exactly similar manner.
+<C>(<G>d</G>) <I>Division</I>.</C>
+<p>The operation of division depends on those of multiplication
+and subtraction, and was performed by the Greeks, <I>mutatis
+mutandis</I>, in the same way as we perform it to-day. Suppose,
+for example, that the process in the first of the above multi-
+plications had to be reversed and <G><FIG><SUB>'</SUB>esa</G> (1825201) had to be
+divided by <G><SUB>'</SUB>atna</G> (1351). The terms involving the successive
+powers of 10 would be mentally kept separate, as in addition
+and subtraction, and the first question would be, how many
+times does one thousand go into one million, allowing for the
+fact that the one thousand has 351 behind it, while the one
+million has 825 thousands behind it. The answer is one
+thousand or <G><SUB>'</SUB>a</G>, and this multiplied by the divisor <G><SUB>'</SUB>atna</G> gives
+<G><FIG><SUB>'</SUB>a</G> which, subtracted from <G><FIG><SUB>'</SUB>esa</G>, leaves <G><FIG><SUB>'</SUB>dsa</G>. This
+<pb n=59><head>DIVISION</head>
+remainder (=474201) has now to be divided by <G><SUB>'</SUB>atna</G> (1351),
+and it would be seen that the latter would go into the former
+<G>t</G> (300) times, but not <G>u</G> (400) times. Multiplying <G><SUB>'</SUB>atna</G> by <G>t</G>,
+we obtain <G><FIG><SUB>'</SUB>et</G> (405300), which, when subtracted from <G><FIG><SUB>'</SUB>dsa</G>
+(474201), leaves <G><FIG><SUB>'</SUB>h<*>a</G> (68901). This has again to be divided
+by <G><SUB>'</SUB>atna</G> and goes <G>n</G> (50) times; multiplying <G><SUB>'</SUB>atna</G> by <G>n</G>, we
+have <G><FIG><SUB>'</SUB>zfn</G> (67550), which, subtracted from <G><FIG><SUB>'</SUB>h<*>a</G> (68901),
+leaves <G><SUB>'</SUB>atna</G> (1351). The last quotient is therefore <G>a</G> (1), and
+the whole quotient is <G><SUB>'</SUB>atna</G> (1351).
+<p>An actual case of long division where both dividend and
+divisor contain sexagesimal fractions is described by Theon.
+The problem is to divide 1515 20&prime;15&Prime; by 25 12&prime; 10&Prime;, and
+Theon's account of the process amounts to the following:
+<table>
+<tr align=center><td>Divisor.</td><td></td><td colspan=2>Dividend.</td><td></td><td>Quotient.</td></tr>
+<tr><td>25 12&prime; 10&Prime;</td><td></td><td>1515</td><td>20&prime;</td><td>15&Prime;</td><td>First term 60</td></tr>
+<tr><td></td><td>25.60</td><td>= 1500</td><td></td><td></td><td></td></tr>
+<tr><td></td><td colspan=2>Remainder 15=</td><td>900&prime;</td><td></td><td></td></tr>
+<tr><td></td><td>Sum</td><td></td><td>920&prime;</td><td></td><td></td></tr>
+<tr><td></td><td>12&prime;.60</td><td>=</td><td>720&prime;</td><td></td><td></td></tr>
+<tr><td></td><td colspan=2 align=center>Remainder</td><td>200&prime;</td><td></td><td></td></tr>
+<tr><td></td><td>10&Prime;.60</td><td>=</td><td>10&prime;</td><td></td><td></td></tr>
+<tr><td></td><td colspan=2 align=center>Remainder</td><td>190&prime;</td><td></td><td>Second term 7&prime;</td></tr>
+<tr><td></td><td>25.7&prime;</td><td>=</td><td>175&prime;</td><td></td><td></td></tr>
+<tr><td></td><td></td><td></td><td>15&prime; =</td><td>900&Prime;</td><td></td></tr>
+<tr><td></td><td></td><td>Sum</td><td></td><td>915&Prime;</td><td></td></tr>
+<tr><td></td><td></td><td>12&prime;.7&prime; =</td><td></td><td>84&Prime;</td><td></td></tr>
+<tr><td></td><td></td><td colspan=2 align=center>Remainder</td><td>831&Prime;</td><td></td></tr>
+<tr><td></td><td></td><td>10&Prime;.7&prime; =</td><td></td><td>1&Prime; 10&tprime;</td><td></td></tr>
+<tr><td></td><td></td><td colspan=2 align=center>Remainder</td><td>829&Prime; 50&tprime;</td><td>Third</td></tr>
+<tr><td></td><td></td><td>25.33&Prime; =</td><td></td><td>825&Prime;</td><td>term 33&tprime;</td></tr>
+<tr><td></td><td></td><td colspan=2 align=center>Remainder</td><td>4&Prime; 50&tprime; =</td><td>290&tprime;</td></tr>
+<tr><td></td><td></td><td>12&prime;.33&Prime; =</td><td></td><td></td><td>396&tprime;</td></tr>
+<tr><td></td><td></td><td></td><td colspan=2 align=center>(<I>too great by</I>)</td><td>106&Prime;</td></tr>
+</table>
+Thus the quotient is something less than 60 7&prime;33&Prime;. It will
+be observed that the difference between this operation of
+<pb n=60><head>GREEK NUMERICAL NOTATION</head>
+Theon's and that of dividing <G><FIG><SUB>'</SUB>esa</G> by <G><SUB>'</SUB>atna</G> as above is that
+Theon makes <I>three</I> subtractions for one term of the quotient,
+whereas the remainder was arrived at in the other case after
+<I>one</I> subtraction. The result is that, though Theon's method
+is quite clear, it is longer, and moreover makes it less easy to
+foresee what will be the proper figure to try in the quotient,
+so that more time would probably be lost in making un-
+successful trials.
+<C>(<G>e</G>) <I>Extraction of the square root</I>.</C>
+<p>We are now in a position to see how the problem of extract-
+ing the square root of a number would be attacked. First, as
+in the case of division, the given whole number would be
+separated into terms containing respectively such and such
+a number of units and of the separate powers of 10. Thus
+there would be so many units, so many tens, so many hun-
+dreds, &amp;c., and it would have to be borne in mind that the
+squares of numbers from 1 to 9 lie between 1 and 99, the
+squares of numbers from 10 to 90 between 100 and 9900, and
+so on. Then the first term of the square root would be some
+number of tens or hundreds or thousands, and so on, and
+would have to be found in much the same way as the first
+term of a quotient in a long division, by trial if necessary.
+If <I>A</I> is the number the square root of which is required, while
+<I>a</I> represents the first term or denomination of the square root,
+and <I>x</I> the next term or denomination to be found, it would be
+necessary to use the identity <MATH>(<I>a</I>+<I>x</I>)<SUP>2</SUP>=<I>a</I><SUP>2</SUP>+2<I>ax</I>+<I>x</I><SUP>2</SUP></MATH> and to
+find <I>x</I> so that 2<I>ax</I>+<I>x</I><SUP>2</SUP> might be somewhat less than the
+remainder <I>A-a</I><SUP>2</SUP>, i.e. we have to divide <I>A-a</I><SUP>2</SUP> by 2<I>a</I>, allowing
+for the fact that not only must 2<I>ax</I> (where <I>x</I> is the quotient)
+but also (2<I>a</I>+<I>x</I>)<I>x</I> be less than <I>A-a</I><SUP>2</SUP>. Thus, by trial, the
+highest possible value of <I>x</I> satisfying the condition would be
+easily found. If that value were <I>b</I>, the further quantity
+2<I>ab</I>+<I>b</I><SUP>2</SUP> would have to be subtracted from the first remainder
+<I>A-a</I><SUP>2</SUP>, and from the second remainder thus left a third term
+or denomination of the square root would have to be found in
+like manner; and so on. That this was the actual procedure
+followed is clear from a simple case given by Theon of Alex-
+andria in his commentary on the <I>Syntaxis</I>. Here the square
+root of 144 is in question, and it is obtained by means of
+<pb n=61><head>EXTRACTION OF THE SQUARE ROOT</head>
+Eucl. II. 4. The highest possible denomination (i.e. power
+of 10) in the square root is 10; 10<SUP>2</SUP> subtracted from 144 leaves
+44, and this must contain, not only twice the product of 10
+and the next term of the square root, but also the square of
+the next term itself. Now twice 1.10 itself produces 20, and
+the division of 44 by 20 suggests 2 as the next term of the
+square root; this turns out to be the exact figure required, since
+<MATH>2.20+2<SUP>2</SUP>=44</MATH>.
+<p>The same procedure is illustrated by Theon's explanation
+of Ptolemy's method of extracting square roots according to
+the sexagesimal system of fractions. The problem is to find
+approximately the square root of 4500 <G>moi=rai</G> or <I>degrees</I>, and
+<FIG>
+a geometrical figure is used which proves beyond doubt the
+essentially Euclidean basis of the whole method. The follow-
+ing arithmetical representation of the purport of the passage,
+when looked at in the light of the figure, will make the
+matter clear. Ptolemy has first found the integral part of
+&radic;(4500) to be 67. Now 67<SUP>2</SUP>=4489, so that the remainder is
+11. Suppose now that the rest of the square root is expressed
+by means of sexagesimal fractions, and that we may therefore
+write
+<MATH>&radic;(4500)=67+<I>x</I>/(60)+<I>y</I>/(60)<SUP>2</SUP>)</MATH>,
+where <I>x, y</I> are yet to be found. Thus <I>x</I> must be such that
+2.67<I>x</I>/60 is somewhat less than 11, or <I>x</I> must be somewhat
+<pb n=62><head>GREEK NUMERICAL NOTATION</head>
+less than (11.60)/(2.67) or (330)/(67), which is at the same time greater than
+4. On trial it turns out that 4 will satisfy the conditions of
+the problem, namely that <MATH>(67+4/(60))<SUP>2</SUP></MATH> must be less than 4500,
+so that a remainder will be left by means of which <I>y</I> can be
+found.
+<p>Now this remainder is <MATH>11-(2.67.4)/(60)-(4/(60))<SUP>2</SUP></MATH>, and this is
+equal to <MATH>(11.60<SUP>2</SUP>-2.67.4.60-16)/(60<SUP>2</SUP>)</MATH> or (7424)/(60<SUP>2</SUP>).
+<p>Thus we must suppose that <MATH>2(67+4/(60))<I>y</I>/(60<SUP>2</SUP>)</MATH> approximates to
+(7424)/(60<SUP>2</SUP>), or that 8048<I>y</I> is approximately equal to 7424.60.
+Therefore <I>y</I> is approximately equal to 55.
+<p>We have then to subtract <MATH>2(67+4/(60))(55)/(60<SUP>2</SUP>)+((55)/(60<SUP>2</SUP>)<SUP>2</SUP></MATH>, or
+<MATH>(442640)/(60<SUP>3</SUP>)+(3025)/(60<SUP>4</SUP>)</MATH>, from the remainder (7424)/(60<SUP>2</SUP>) above found.
+<p>The subtraction of (442640)/(60<SUP>3</SUP>) from (7424)/(60<SUP>2</SUP>) gives (2800)/(60<SUP>3</SUP>) or <MATH>(46)/(60<SUP>2</SUP>)+(40)/(60<SUP>3</SUP>)</MATH>;
+but Theon does not go further and subtract the remaining
+(3025)/(60<SUP>4</SUP>); he merely remarks that the square of (55)/(60<SUP>2</SUP>) approximates
+to <MATH>(46)/(60<SUP>2</SUP>)+(40)/(60<SUP>3</SUP>)</MATH>. As a matter of fact, if we deduct the (3025)/(60<SUP>4</SUP>) from
+(2800)/(60<SUP>3</SUP>), so as to obtain the correct remainder, it is found
+to be (164975)/(60<SUP>4</SUP>).
+<p>Theon's plan does not work conveniently, so far as the
+determination of the first fractional term (the <I>first-sixtieths</I>)
+is concerned, unless the integral term in the square root is
+large relatively to <I>x</I>/(60); if this is not the case, the term (<I>x</I>/(60))<SUP>2</SUP> is
+not comparatively negligible, and the tentative ascertainment
+of <I>x</I> is more difficult. Take the case of &radic;3, the value of which,
+in Ptolemy's Table of Chords, is equal to <MATH>1+(43)/(60)+(55)/(60<SUP>2</SUP>)+(23)/(60<SUP>3</SUP>)</MATH>.
+<pb n=63><head>EXTRACTION OF THE SQUARE ROOT</head>
+If we first found the unit 1 and then tried to find the next
+term by trial, it would probably involve a troublesome amount
+of trials. An alternative method in such a case was to
+multiply the number by 60<SUP>2</SUP>, thus reducing it to second-
+sixtieths, and then, taking the square root, to ascertain the
+number of first-sixtieths in it. Now 3.60<SUP>2</SUP>=10800, and, as
+103<SUP>2</SUP>=10609, the first element in the square root of 3 is
+found in this way to be <MATH>(103)/(60)(=1+(43)/(60))</MATH>. That this was the
+method in such cases is indicated by the fact that, in the Table
+of Chords, each chord is expressed as a certain number of
+first-sixtieths, followed by the second-sixtieths, &amp;c., &radic;3 being
+expressed as <MATH>(103)/(60)+(55)/(60<SUP>2</SUP>)+(23)/(60<SUP>3</SUP>)</MATH>. The same thing is indicated by
+the scholiast to Eucl., Book X, who begins the operation of
+finding the square root of 31 10&prime;36&Prime; by reducing this to
+second-sixtieths; the number of second-sixtieths is 112236,
+which gives, as the number of first-sixtieths in the square
+root, 335, while <MATH>(335)/(60)=5 35&prime;</MATH>. The second-sixtieths in the
+square root can then be found in the same way as in Theon's
+example. Or, as the scholiast says, we can obtain the square
+root as far as the second-sixtieths by reducing the original
+number to fourth-sixtieths, and so on. This would no doubt
+be the way in which the approximate value 2 49&prime;42&Prime;20&tprime;10&prime;&prime;&prime;&prime;
+given by the scholiast for &radic;8 was obtained, and similarly
+with other approximations of his, such as <MATH>&radic;2=1 24&prime;51&Prime;</MATH> and
+<MATH>&radic;(27)=5 11&prime; 46&Prime; 50&tprime;</MATH> (the 50&tprime; should be 10&tprime;).
+<C>(<G>z</G>) <I>Extraction of the cube root</I></C>
+<p>Our method of extracting the cube root of a number depends
+upon the formula <MATH>(<I>a</I>+<I>x</I>)<SUP>3</SUP>=<I>a</I><SUP>3</SUP>+3<I>a</I><SUP>2</SUP><I>x</I>+3<I>ax</I><SUP>2</SUP>+<I>x</I><SUP>3</SUP></MATH>, just as the
+extraction of the square root depends on the formula
+<MATH>(<I>a</I>+<I>x</I>)<SUP>2</SUP>=<I>a</I><SUP>2</SUP>+2<I>ax</I>+<I>x</I><SUP>2</SUP></MATH>. As we have seen, the Greek method
+of extracting the square root was to use the latter (Euclidean)
+formula just as we do; but in no extant Greek writer do we
+find any description of the operation of extracting the cube
+root. It is possible that the Greeks had not much occasion
+for extracting cube roots, or that a table of cubes would
+suffice for most of their purposes. But that they had some
+<pb n=64><head>GREEK NUMERICAL NOTATION</head>
+method is clear from a passage of Heron, where he gives 4 9/(14)
+as an approximation to &radic;<SUP>3</SUP>(100), and shows how he obtains it.<note>Heron, <I>Metrica</I>, iii. c. 20.</note>
+Heron merely gives the working dogmatically, in concrete
+numbers, without explaining its theoretical basis, and we
+cannot be quite certain as to the precise formula underlying
+the operation. The best suggestion which has been made on
+the subject will be given in its proper place, the chapter
+on Heron.
+<pb><C>III</C>
+<C>PYTHAGOREAN ARITHMETIC</C>
+<p>THERE is very little early evidence regarding Pythagoras's
+own achievements, and what there is does not touch his mathe-
+matics. The earliest philosophers and historians who refer
+to him would not be interested in this part of his work.
+Heraclitus speaks of his wide knowledge, but with disparage-
+ment: &lsquo;much learning does not teach wisdom; otherwise
+it would have taught Hesiod and Pythagoras, and again
+Xenophanes and Hecataeus&rsquo;.<note>Diog. L. ix. 1 (Fr. 40 in <I>Vorsokratiker,</I> i<SUP>3</SUP>, p. 86. 1-3).</note> Herodotus alludes to Pytha-
+goras and the Pythagoreans several times; he calls Pythagoras
+&lsquo;the most able philosopher among the Greeks&rsquo; (<G>*(ellh/nwn on)
+tw=| a)sqenesta/tw| sofisth=| *puqago/rh|</G>).<note>Herodotus, iv. 95.</note> In Empedocles he had
+an enthusiastic admirer: &lsquo;But there was among them a man
+of prodigious knowledge who acquired the profoundest wealth
+of understanding and was the greatest master of skilled arts
+of every kind; for, whenever he willed with his whole heart,
+he could with ease discern each and every truth in his ten&mdash;
+nay, twenty&mdash;men's lives.&rsquo;<note>Diog. L. viii. 54 and Porph. <I>V. Pyth.</I> 30 (Fr. 129 in <I>Vors.</I> i<SUP>3</SUP>, p. 272. 15-20).</note>
+<p>Pythagoras himself left no written exposition of his
+doctrines, nor did any of his immediate successors, not even
+Hippasus, about whom the different stories ran (1) that he
+was expelled from the school because he published doctrines
+of Pythagoras, and (2) that he was drowned at sea for
+revealing the construction of the dodecahedron in the sphere
+and claiming it as his own, or (as others have it) for making
+known the discovery of the irrational or incommensurable.
+Nor is the absence of any written record of Pythagorean
+<pb n=66><head>PYTHAGOREAN ARITHMETIC</head>
+doctrines down to the time of Philolaus to be attributed
+to a pledge of secrecy binding the school; at all events, it
+did not apply to their mathematics or their physics; the
+supposed secrecy may even have been invented to explain
+the absence of documents. The fact appears to be that oral
+communication was the tradition of the school, while their
+doctrine would in the main be too abstruse to be understood
+by the generality of people outside.
+<p>In these circumstances it is difficult to disentangle the
+portions of the Pythagorean philosophy which can safely
+be attributed to the founder of the school. Aristotle evi-
+dently felt this difficulty; it is clear that he knew nothing
+for certain of any ethical or physical doctrines going back
+to Pythagoras himself; and when he speaks of the Pytha-
+gorean system, he always refers it to &lsquo;the Pythagoreans&rsquo;,
+sometimes even to &lsquo;the so-called Pythagoreans&rsquo;.
+<p>The earliest direct testimony to the eminence of Pythagoras
+in mathematical studies seems to be that of Aristotle, who in
+his separate book <I>On the Pythagoreans</I>, now lost, wrote that
+<p>&lsquo;Pythagoras, the son of Mnesarchus, first worked at mathe-
+matics and arithmetic, and afterwards, at one time, condescended
+to the wonder-working practised by Pherecydes.&rsquo;<note>Apollonius, <I>Hist. mirabil.</I> 6 (<I>Vors.</I> i<SUP>3</SUP>, p. 29. 5).</note>
+<p>In the <I>Metaphysics</I> he speaks in similar terms of the
+Pythagoreans:
+<p>&lsquo;In the time of these philosophers (Leucippus and
+Democritus) and before them the so-called Pythagoreans
+applied themselves to the study of mathematics, and were
+the first to advance that science; insomuch that, having been
+brought up in it, they thought that its principles must be
+the principles of all existing things.&rsquo;<note>Arist. <I>Metaph.</I> A. 5, 985 b 23.</note>
+<p>It is certain that the Theory of Numbers originated in
+the school of Pythagoras; and, with regard to Pythagoras
+himself, we are told by Aristoxenus that he &lsquo;seems to have
+attached supreme importance to the study of arithmetic,
+which he advanced and took out of the region of commercial
+utility&rsquo;.<note>Stobaeus, <I>Ecl.</I> i. proem. 6 (<I>Vors.</I> i<SUP>3</SUP>, p. 346. 12).</note>
+<pb n=67><head>PYTHAGOREAN ARITHMETIC</head>
+<C>Numbers and the universe.</C>
+<p>We know that Thales (about 624-547 B.C.) and Anaximander
+(born probably in 611/10 B.C.) occupied themselves with
+astronomical phenomena, and, even before their time, the
+principal constellations had been distinguished. Pythagoras
+(about 572-497 B.C. or a little later) seems to have been
+the first Greek to discover that the planets have an inde-
+pendent movement of their own from west to east, i.e. in
+a direction contrary to the daily rotation of the fixed stars;
+or he may have learnt what he knew of the planets from the
+Babylonians. Now any one who was in the habit of intently
+studying the heavens would naturally observe that each
+constellation has two characteristics, the number of the stars
+which compose it and the geometrical figure which they
+form. Here, as a recent writer has remarked,<note>L. Brunschvicg, <I>Les &eacute;tapes de la philosophie math&eacute;matique</I>, 1912, p. 33.</note> we find, if not
+the origin, a striking illustration of the Pythagorean doctrine.
+And, just as the constellations have a number characteristic
+of them respectively, so all known objects have a number;
+as the formula of Philolaus states, &lsquo;all things which can
+be known have number; for it is not possible that without
+number anything can either be conceived or known&rsquo;.<note>Stob. <I>Ecl.</I> i. 21, 7<SUP>b</SUP> (<I>Vors.</I> i<SUP>3</SUP>, p. 310. 8-10).</note>
+<p>This formula, however, does not yet express all the content
+of the Pythagorean doctrine. Not only do all things possess
+numbers; but, in addition, all things <I>are</I> numbers; &lsquo;these
+thinkers&rsquo;, says Aristotle, &lsquo;seem to consider that number is
+the principle both as matter for things and as constituting
+their attributes and permanent states&rsquo;.<note>Aristotle, <I>Metaph.</I> A. 5, 986 a 16.</note> True, Aristotle
+seems to regard the theory as originally based on the analogy
+between the properties of things and of numbers.
+<p>&lsquo;They thought they found in numbers, more than in fire,
+earth, or water, many resemblances to things which are and
+become; thus such and such an attribute of numbers is jus-
+tice, another is soul and mind, another is opportunity, and so
+on; and again they saw in numbers the attributes and ratios
+of the musical scales. Since, then, all other things seemed
+in their whole nature to be assimilated to numbers, while
+numbers seemed to be the first things in the whole of nature,
+<pb n=68><head>PYTHAGOREAN ARITHMETIC</head>
+they supposed the elements of numbers to be the elements
+of all things, and the whole heaven to be a musical scale and
+a number.&rsquo;<note><I>Metaph.</I> A. 5, 985 b 27-986 a 2.</note>
+<p>This passage, with its assertion of &lsquo;resemblances&rsquo; and
+&lsquo;assimilation&rsquo;, suggests numbers as affections, states, or rela-
+tions rather than as substances, and the same is implied by
+the remark that existing things exist by virtue of their
+<I>imitation</I> of numbers.<note><I>Ib.</I> A. 5, 987 b 11.</note> But again we are told that the
+numbers are not separable from the things, but that existing
+things, even perceptible substances, are made up of numbers;
+that the substance of all things is number, that things are
+numbers, that numbers are made up from the unit, and that the
+whole heaven is numbers.<note><I>Ib.</I> N. 3, 1090 a 22-23; M. 7, 1080 b 17; A. 5, 987 a 19, 987 b 27, 986 a 20.</note> Still more definite is the statement
+that the Pythagoreans &lsquo;construct the whole heaven out of
+numbers, but not of <I>monadic</I> numbers, since they suppose the
+units to have magnitude&rsquo;, and that, &lsquo;as we have said before,
+the Pythagoreans assume the numbers to have magnitude&rsquo;.<note><I>Ib.</I> M. 7, 1080 b 18, 32.</note>
+Aristotle points out certain obvious difficulties. On the one
+hand the Pythagoreans speak of &lsquo;this number of which the
+heaven is composed&rsquo;; on the other hand they speak of &lsquo;attri-
+butes of numbers&rsquo; and of numbers as &lsquo;the <I>causes</I> of the things
+which exist and take place in the heaven both from the begin-
+ning and now&rsquo;. Again, according to them, abstractions and
+immaterial things are also numbers, and they place them in
+different regions; for example, in one region they place
+opinion and opportunity, and in another, a little higher up or
+lower down, such things as injustice, sifting, or mixing.
+Is it this same &lsquo;number in the heaven&rsquo; which we must
+assume each of these things to be, or a number other than
+this number?<note><I>Ib.</I> A. 8, 990 a 18-29.</note>
+<p>May we not infer from these scattered remarks of Aristotle
+about the Pythagorean doctrine that &lsquo;the number in the
+heaven&rsquo; is the number of the visible stars, made up of
+units which are material points? And may this not be
+the origin of the theory that all things are numbers, a
+theory which of course would be confirmed when the further
+<pb n=69><head>NUMBERS AND THE UNIVERSE</head>
+capital discovery was made that musical harmonies depend
+on numerical ratios, the octave representing the ratio 2:1
+in length of string, the fifth 3:2 and the fourth 4:3?
+<p>The use by the Pythagoreans of visible points to represent
+the units of a number of a particular form is illustrated by
+the remark of Aristotle that
+<p>&lsquo;Eurytus settled what is the number of what object (e.g.
+this is the number of a man, that of a horse) and imitated
+the shapes of living things by pebbles <I>after the manner of
+those who bring numbers into the forms of triangle or
+square</I>&rsquo;.<note><I>Metaph.</I> N. 5, 1092 b 10.</note>
+<p>They treated the unit, which is a point without position
+(<G>stigmh\ a)/qetos</G>), as a point, and a point as a unit having
+position (<G>mona\s qe/sin e)/xousa</G>).<note><I>Ib.</I> M. 8, 1084 b 25; <I>De an.</I> i. 4, 409 a 6; Proclus on Eucl. I, p. 95. 21.</note>
+<C>Definitions of the unit and of number.</C>
+<p>Aristotle observes that the One is reasonably regarded as
+not being itself a number, because a measure is not the things
+measured, but the measure or the One is the beginning (or
+principle) of number.<note><I>Metaph.</I> N. 1, 1088 a 6.</note> This doctrine may be of Pythagorean
+origin; Nicomachus has it<note>Nicom. <I>Introd. arithm.</I> ii. 6. 3, 7. 3.</note>; Euclid implies it when he says
+that a unit is that by virtue of which each of existing things
+is called one, while a number is &lsquo;the multitude made up of
+units&rsquo;<note>Eucl. VII, Defs. 1, 2.</note>; and the statement was generally accepted. According
+to Iamblichus,<note>Iambl. <I>in Nicom. ar. introd.</I>, p. 11. 2-10.</note> Thymaridas (an ancient Pythagorean, probably
+not later than Plato's time) defined a unit as &lsquo;limiting quan-
+tity&rsquo; (<G>perai/nousa poso/ths</G>) or, as we might say, &lsquo;limit of few-
+ness&rsquo;, while some Pythagoreans called it &lsquo;the confine between
+number and parts&rsquo;, i.e. that which separates multiples
+and submultiples. Chrysippus (third century B.C.) called it
+&lsquo;multitude one&rsquo; (<G>plh=qos e(/n</G>), a definition objected to by
+Iamblichus as a contradiction in terms, but important as an
+attempt to bring 1 into the conception of number.
+<p>The first definition of number is attributed to Thales, who
+defined it as a collection of units (<G>mona/dwn su/sthma</G>), &lsquo;follow-
+<pb n=70><head>PYTHAGOREAN ARITHMETIC</head>
+ing the Egyptian view&rsquo;.<note>Iambl. <I>in Nicom. ar. introd.</I>, p. 10. 8-10.</note> The Pythagoreans &lsquo;made number
+out of one&rsquo;<note>Arist. <I>Metaph.</I> A. 5, 986 a 20.</note> some of them called it &lsquo;a progression of multi-
+tude beginning from a unit and a regression ending in it&rsquo;.<note>Theon of Smyrna, p. 18. 3-5.</note>
+(Stobaeus credits Moderatus, a Neo-Pythagorean of the time
+of Nero, with this definition.<note>Stob. <I>Ecl.</I> i. pr. 8.</note>) Eudoxus defined number as
+a &lsquo;determinate multitude&rsquo; (<G>plh=qos w(risme/non</G>).<note>Iambl. <I>op. cit.</I>, p. 10. 17.</note> Nicoma-
+chus has yet another definition, &lsquo;a flow of quantity made up
+of units&rsquo;<note>Nicom. i. 7. 1.</note> (<G>poso/thtos xu/ma e)k mona/dwn sugkei/menon</G>). Aris-
+totle gives a number of definitions equivalent to one or other
+of those just mentioned, &lsquo;limited multitude&rsquo;,<note><I>Metaph.</I> &utri;. 13, 1020 a 13.</note> &lsquo;multitude (or
+&lsquo;combination&rsquo;) of units&rsquo;,<note><I>Ib.</I> I. 1, 1053 a 30; Z. 13, 1039 a 12.</note> &lsquo;multitude of indivisibles&rsquo;,<note><I>Ib.</I> M. 9, 1085 b 22.</note> &lsquo;several
+ones&rsquo; (<G>e(/na plei/w</G>),<note><I>Phys.</I> iii. 7, 207 b 7.</note> &lsquo;multitude measurable by one&rsquo;,<note><I>Metaph.</I> I. 6, 1057 a 3.</note> &lsquo;multi-
+tude measured&rsquo;, and &lsquo;multitude of measures&rsquo;<note><I>Ib.</I> N. 1, 1088 a 5.</note> (the measure
+being the unit).
+<C>Classification of numbers.</C>
+<p>The distinction between <I>odd</I> (<G>perisso/s</G>) and <I>even</I> (<G>a)/rtios</G>)
+doubtless goes back to Pythagoras. A Philolaus fragment
+says that &lsquo;number is of two special kinds, odd and even, with
+a third, even-odd, arising from a mixture of the two; and of
+each kind there are many forms&rsquo;.<note>Stob. <I>Ecl.</I> i. 21. 7<SUP>c</SUP> (<I>Vors.</I> i<SUP>3</SUP>, p. 310. 11-14).</note> According to Nicomachus,
+the Pythagorean definitions of odd and even were these:
+<p>&lsquo;An <I>even</I> number is that which admits of being divided, by
+one and the same operation, into the greatest and the least
+parts, greatest in size but least in number (i. e. into <I>two halves</I>)
+..., while an <I>odd</I> number is that which cannot be so divided
+but is only divisible into two unequal parts.&rsquo;<note>Nicom. i. 7. 3.</note>
+<p>Nicomachus gives another ancient definition to the effect
+that
+&lsquo;an <I>even</I> number is that which can be divided both into two
+equal parts and into two unequal parts (except the funda-
+mental dyad which can only be divided into two equal parts),
+but, however it is divided, must have its two parts <I>of the same
+kind</I> without part in the other kind (i. e. the two parts are
+<pb n=71><head>CLASSIFICATION OF NUMBERS</head>
+both odd or both even); while an <I>odd</I> number is that which,
+however divided, must in any case fall into two unequal parts,
+and those parts always belonging to the two <I>different</I> kinds
+respectively (i.e. one being odd and one even).&rsquo;<note>Nicom. i. 7. 4.</note>
+<p>In the latter definition we have a trace of the original
+conception of 2 (the dyad) as being, not a number at all, but
+the principle or beginning of the even, just as one was not a
+number but the principle or beginning of number; the defini-
+tion implies that 2 was not originally regarded as an even
+number, the qualification made by Nicomachus with reference
+to the dyad being evidently a later addition to the original
+definition (Plato already speaks of two as even).<note>Plato, <I>Parmenides</I>, 143 D.</note>
+<p>With regard to the term &lsquo;odd-even&rsquo;, it is to be noted that,
+according to Aristotle, the Pythagoreans held that &lsquo;the One
+arises from both kinds (the odd and the even), for it is both
+even and odd&rsquo;.<note>Arist. <I>Metaph.</I> A. 5, 986 a 19.</note> The explanation of this strange view might
+apparently be that the unit, being the principle of all number,
+even as well as odd, cannot itself be odd and must therefore
+be called even-odd. There is, however, another explanation,
+attributed by Theon of Smyrna to Aristotle, to the effect that the
+unit when added to an even number makes an odd number, but
+when added to an odd number makes an even number: which
+could not be the case if it did not partake of both species;
+Theon also mentions Archytas as being in agreement with this
+view.<note>Theon of Smyrna, p. 22. 5-10.</note> But, inasmuch as the fragment of Philolaus speaks of
+&lsquo;many forms&rsquo; of the species odd and even, and &lsquo;a third&rsquo;
+(even-odd) obtained from a combination of them, it seems
+more natural to take &lsquo;even-odd&rsquo; as there meaning, not the
+unit, but the product of an odd and an even number, while, if
+&lsquo;even&rsquo; in the same passage excludes such a number, &lsquo;even&rsquo;
+would appear to be confined to powers of 2, or 2<SUP><I>n</I></SUP>.
+<p>We do not know how far the Pythagoreans advanced
+towards the later elaborate classification of the varieties of
+odd and even numbers. But they presumably had not got
+beyond the point of view of Plato and Euclid. In Plato we
+have the terms &lsquo;even-times even&rsquo; (<G>a)/rtia a)rtia/kis</G>), &lsquo;odd-
+times odd&rsquo; (<G>peritta\ peritta/kis</G>), &lsquo;odd-times even&rsquo; (<G>a)/rtia</G>
+<pb n=72><head>PYTHAGOREAN ARITHMETIC</head>
+<G>peritta/kis</G>) and &lsquo;even-times odd&rsquo; (<G>peritta\ a)rtia/kis</G>), which
+are evidently used in the simple sense of the products of even
+and even, odd and odd, odd and even, and even and odd
+factors respectively.<note>Plato, <I>Parmenides</I>, 143 E.</note> Euclid's classification does not go much
+beyond this; he does not attempt to make the four defini-
+tions mutually exclusive.<note>See Eucl. VII. Defs. 8-10.</note> An &lsquo;odd-times odd&rsquo; number is of
+course any odd number which is not prime; but &lsquo;even-times
+even&rsquo; (&lsquo;a number measured by an even number according to
+an even number&rsquo;) does not exclude &lsquo;even-times odd&rsquo; (&lsquo;a
+number measured by an even number according to an odd
+number&rsquo;); e.g. 24, which is 6 times 4, or 4 times 6, is also
+8 times 3. Euclid did not apparently distinguish, any more
+than Plato, between &lsquo;even-times odd&rsquo; and &lsquo;odd-times even&rsquo;
+(the definition of the latter in the texts of Euclid was pro-
+bably interpolated). The Neo-Pythagoreans improved the
+classification thus. With them the &lsquo;even-times even&rsquo; number
+is that which has its halves even, the halves of the halves
+even, and so on till unity is reached&rsquo;<note>Nicom. i. 8. 4.</note>; in short, it is a number
+of the form 2<SUP><I>n</I></SUP>. The &lsquo;even-odd&rsquo; number (<G>a)rtiope/rittos</G> in one
+word) is such a number as, when once halved, leaves as quo-
+tient an odd number,<note><I>Ib.</I> i. 9. 1.</note> i.e. a number of the form 2 (2<I>m</I>+1).
+The &lsquo;odd-even&rsquo; number (<G>perissa/rtios</G>) is a number such that
+it can be halved twice or more times successively, but the
+quotient left when it can no longer be halved is an odd num-
+ber not unity,<note><I>Ib.</I> i. 10. 1.</note> i.e. it is a number of the form 2<SUP><I>n</I>+1</SUP> (2<I>m</I>+1).
+The &lsquo;odd-times odd&rsquo; number is not defined as such by
+Nicomachus and Iamblichus, but Theon of Smyrna quotes
+a curious use of the term; he says that it was one of the
+names applied to prime numbers (excluding of course 2), for
+these have two odd factors, namely 1 and the number itself.<note>Theon of Smyrna, p. 23. 14-23.</note>
+<p><I>Prime</I> or <I>incomposite</I> numbers (<G>prw=tos kai\ a)su/nqetos</G>) and
+<I>secondary</I> or <I>composite</I> numbers (<G>deu/teros kai\ su/nqetos</G>) are
+distinguished in a fragment of Speusippus based upon works
+of Philolaus.<note><I>Theol. Ar.</I> (Ast), p. 62 (<I>Vors.</I> i<SUP>3</SUP>, p. 304. 5).</note> We are told<note>Iambl. <I>in Nicom.</I>, p. 27. 4.</note> that Thymaridas called a prime
+number <I>rectilinear</I> (<G>eu)qugrammiko/s</G>), the ground being that it
+can only be set out in one dimension<note>Cf. Arist. <I>Metaph.</I> &utri;. 13, 1020 b 3<SUP>'</SUP>, 4.</note> (since the only measure
+<pb n=73><head>CLASSIFICATION OF NUMBERS</head>
+of it, excluding the number itself, is 1); Theon of Smyrna
+gives <I>euthymetric</I> and <I>linear</I> as alternative terms,<note>Theon of Smyrna, p. 23. 12.</note> and the
+latter (<G>grammiko/s</G>) also occurs in the fragment of Speusippus.
+Strictly speaking, the prime number should have been called
+that which is rectilinear or linear <I>only.</I> As we have seen,
+2 was not originally regarded as a prime number, or even as
+a number at all. But Aristotle speaks of the dyad as &lsquo;the
+only even number which is prime,&rsquo;<note>Arist. <I>Topics</I>, q. 2, 157 a 39.</note> showing that this diver-
+gence from early Pythagorean doctrine took place before
+Euclid's time. Euclid defined a prime number as &lsquo;that which
+is measured by a unit alone&rsquo;,<note>Eucl. VII. Def. 11.</note> a composite number as &lsquo;that
+which is measured by some number&rsquo;,<note><I>Ib.</I> Def. 13.</note> while he adds defini-
+tions of numbers &lsquo;prime to one another&rsquo; (&lsquo;those which are
+measured by a unit alone as a common measure&rsquo;) and of
+numbers &lsquo;composite to one another&rsquo; (&lsquo;those which are mea-
+sured by some number as a common measure&rsquo;).<note><I>Ib.</I> Defs. 12, 14.</note> Euclid then,
+as well as Aristotle, includes 2 among prime numbers. Theon
+of Smyrna says that even numbers are not measured by the
+unit alone, except 2, which therefore is odd-<I>like</I> without being
+prime.<note>Theon of Smyrna, p. 24. 7.</note> The Neo-Pythagoreans, Nicomachus and Iamblichus,
+not only exclude 2 from prime numbers, but define composite
+numbers, numbers prime to one another, and numbers com-
+posite to one another as excluding all even numbers; they
+make all these categories subdivisions of <I>odd.</I><note>Nicom. i, cc. 11-13; Iambl. <I>in N<SUP>^</SUP>icom.</I>, pp. 26-8.</note> Their object
+is to divide odd into three classes parallel to the three subdivi-
+sions of even, namely even-even = 2<SUP><I>n</I></SUP>, even-odd = 2 (2<I>m</I>+1)
+and the quasi-intermediate odd-even = 2<SUP><I>n</I>+1</SUP> (2<I>m</I>+1); accord-
+ingly they divide odd numbers into (<I>a</I>) the prime and
+incomposite, which are Euclid's primes excluding 2, (<I>b</I>) the
+secondary and composite, the factors of which must all be not
+only odd but prime numbers, (<I>c</I>) those which are &lsquo;secondary and
+composite in themselves but prime and incomposite to another
+number,&rsquo; e.g. 9 and 25, which are both secondary and com-
+posite but have no common measure except 1. The incon-
+venience of the restriction in (<I>b</I>) is obvious, and there is the
+<pb n=74><head>PYTHAGOREAN ARITHMETIC</head>
+further objection that (<I>b</I>) and (<I>c</I>) overlap, in fact (<I>b</I>) includes
+the whole of (<I>c</I>).
+<C>&lsquo;Perfect&rsquo; and &lsquo;Friendly&rsquo; numbers.</C>
+<p>There is no trace in the fragments of Philolaus, in Plato or
+Aristotle, or anywhere before Euclid, of the <I>perfect</I> number
+(<G>te/leios</G>) in the well-known sense of Euclid's definition
+(VII. Def. 22), a number, namely, which is &lsquo;equal to (the
+sum of) its own parts&rsquo; (i.e. all its factors including 1),
+e.g.
+<MATH>6=1+2+3; 28=1+2+4+7+14;
+496=1+2+4+8+16+31+62+124+248</MATH>.
+The law of the formation of these numbers is proved in
+Eucl. IX. 36, which is to the effect that, if the sum of any
+number of terms of the series 1, 2, 2<SUP>2</SUP>, 2<SUP>3</SUP> .... 2<SUP><I>n</I>-1</SUP>(=<I>S<SUB>n</SUB></I>) is prime,
+then <I>S<SUB>n</SUB></I>.2<SUP><I>n</I>-1</SUP> is a &lsquo;perfect&rsquo; number. Theon of Smyrna<note>Theon of Smyrna, p. 45.</note> and
+Nicomachus<note>Nicom. i. 16, 1-4.</note> both define a &lsquo;perfect&rsquo; number and explain the
+law of its formation; they further distinguish from it two
+other kinds of numbers, (1) <I>over-perfect</I> (<G>u(pertelh/s</G> or <G>u(perte/-
+leios</G>), so called because the sum of all its aliquot parts is
+greater than the number itself, e.g. 12, which is less than
+1+2+3+4+6, (2) <I>defective</I> (<G>e)lliph/s</G>), so called because the
+sum of all its aliquot parts is less than the number itself,
+e.g. 8, which is greater than 1+2+4. Of perfect numbers
+Nicomachus knew four (namely 6, 28, 496, 8128) but no more.
+He says they are formed in &lsquo;ordered&rsquo; fashion, there being one
+among the units (i. e. less than 10), one among the tens (less
+than 100), one among the hundreds (less than 1000), and one
+among the thousands (less than a myriad); he adds that they
+terminate alternately in 6 or 8. They do all terminate in 6 or
+8 (as we can easily prove by means of the formula (2<SUP><I>n</I>-1</SUP>) 2<SUP><I>n</I>-1</SUP>),
+but not alternately, for the fifth and sixth perfect numbers
+both end in 6, and the seventh and eighth both end in 8.
+Iamblichus adds a tentative suggestion that there may (<G>ei)
+tu/xoi</G>) in like manner be one perfect number among the first
+myriads (less than 10000<SUP>2</SUP>), one among the second myriads
+(less than 10000<SUP>3</SUP>), and so on <I>ad infinitum.</I><note>Iambl. <I>in Nicom.</I>, p. 33. 20-23.</note> This is incorrect,
+for the next perfect numbers are as follows:<note>The fifth perfect number may have been known to Iamblichus,
+though he does not give it; it was, however, known, with all its factors,
+in the fifteenth century, as appears from a tract written in German
+which was discovered by Curtze (Cod. lat. Monac. 14908). The first
+eight &lsquo;perfect&rsquo; numbers were calculated by Jean Prestet (d. 1670);
+Fermat (1601-65) had stated, and Euler proved, that 2<SUP>31</SUP>-1 is prime.
+The ninth perfect number was found by P. Seelhoff, <I>Zeitschr. f. Math. u.
+Physik</I>, 1886, pp. 174 sq.) and verified by E. Lucas (<I>Math&eacute;sis</I>, vii, 1887,
+pp. 44-6). The tenth was found by R. E. Powers (<I>Bull. Amer. Math.
+Soc.</I>, 1912, p. 162).</note>
+<pb n=75><head>&lsquo;PERFECT&rsquo;, AND &lsquo;FRIENDLY&rsquo; NUMBERS</head>
+fifth, <MATH>2<SUP>12</SUP> (2<SUP>13</SUP>-1)=33 550 336</MATH>
+sixth, <MATH>2<SUP>16</SUP> (2<SUP>17</SUP>-1)=8 589 869 056</MATH>
+seventh, <MATH>2<SUP>18</SUP> (2<SUP>19</SUP>-1)=137 438 691 328</MATH>
+eighth, <MATH>2<SUP>30</SUP> (2<SUP>31</SUP>-1)=2 305 843 008 139 952 128</MATH>
+ninth, <MATH>2<SUP>60</SUP> (2<SUP>61</SUP>-1)=2 658 455 991 569 831 744 654 692
+615 953 842 176</MATH>
+tenth, <MATH>2<SUP>88</SUP> (2<SUP>89</SUP>-1)</MATH>.
+With these &lsquo;perfect&rsquo; numbers should be compared the so-
+called &lsquo;friendly numbers&rsquo;. Two numbers are &lsquo;friendly&rsquo; when
+each is the sum of all the aliquot parts of the other, e.g. 284 and
+220 (for <MATH>284=1+2+4+5+10+11+20+22+44+55+110</MATH>,
+while <MATH>220=1+2+4+71+142</MATH>). Iamblichus attributes the
+discovery of such numbers to Pythagoras himself, who, being
+asked &lsquo;what is a friend?&rsquo; said &lsquo;<I>Alter ego</I>&rsquo;, and on this analogy
+applied the term &lsquo;friendly&rsquo; to two numbers the aliquot parts
+of either of which make up the other.<note>Iambl. <I>in Nicom.</I>, p. 35. 1-7. The subject of &lsquo;friendly&rsquo; numbers
+was taken up by Euler, who discovered no less than sixty-one pairs of
+such numbers. Descartes and van Schooten had previously found three
+pairs but no more.</note>
+<p>While for Euclid, Theon of Smyrna, and the Neo-Pytha-
+goreans the &lsquo;perfect&rsquo; number was the kind of number above
+described, we are told that the Pythagoreans made 10 the
+perfect number. Aristotle says that this was because they
+found within it such things as the void, proportion, oddness,
+and so on.<note>Arist. <I>Metaph.</I> M. 8, 1084 a 32-4.</note> The reason is explained more in detail by Theon
+of Smyrna<note>Theon of Smyrna, p. 93. 17-94. 9 (<I>Vorsokratiker</I>, i<SUP>3</SUP>, pp. 303-4).</note> and in the fragment of Speusippus. 10 is the
+sum of the numbers 1, 2, 3, 4 forming the <G>tetraktu/s</G> (&lsquo;their
+greatest oath&rsquo;, alternatively called the &lsquo;principle of health&rsquo;<note>Lucian, <I>De lapsu in salutando</I>, 5.</note>).
+These numbers include the ratios corresponding to the musical
+intervals discovered by Pythagoras, namely 4:3 (the fourth),
+<pb n=76><head>PYTHAGOREAN ARITHMETIC</head>
+3:2 (the fifth), and 2:1 (the octave). Speusippus observes
+further that 10 contains in it the &lsquo;linear&rsquo;, &lsquo;plane&rsquo; and &lsquo;solid&rsquo;
+varieties of number; for 1 is a point, 2 is a line,<note>Cf. Arist. <I>Metaph.</I> Z. 10, 1036 b 12.</note> 3 a triangle,
+and 4 a pyramid.<note><I>Theol. Ar.</I> (Ast), p. 62. 17-22.</note>
+<C>Figured numbers.</C>
+<p>This brings us once more to the theory of figured numbers,
+which seems to go back to Pythagoras himself. A point or
+dot is used to represent 1; two dots placed apart represent
+2, and at the same time define the straight line joining the
+two dots; three dots, representing 3, mark out the first
+rectilinear plane figure, a triangle; four dots, one of which is
+outside the plane containing the other three, represent 4 and
+also define the first rectilineal solid figure. It seems clear
+that the oldest Pythagoreans were acquainted with the forma-
+tion of triangular and square numbers by means of pebbles or
+dots<note>Cf. Arist. <I>Metaph.</I> N. 5, 1092 b 12.</note>; and we judge from the account of Speusippus's book,
+<I>On the Pythagorean Numbers</I>, which was based on works of
+Philolaus, that the latter dealt with linear numbers, polygonal
+numbers, and plane and solid numbers of all sorts, as well as
+with the five regular solid figures.<note><I>Theol. Ar.</I> (Ast), p. 61.</note> The varieties of plane
+numbers (triangular, square, oblong, pentagonal, hexagonal,
+and so on), solid numbers (cube, pyramidal, &amp;c.) are all dis-
+cussed, with the methods of their formation, by Nicomachus<note>Nicom. i. 7-11, 13-16, 17.</note>
+and Theon of Smyrna.<note>Theon of Smyrna, pp. 26-42.</note>
+<C>(<G>a</G>) <I>Triangular numbers.</I></C>
+<p>To begin with <I>triangular</I> numbers. It was probably
+Pythagoras who discovered that the sum of any number of
+successive terms of the series of natural numbers 1, 2, 3 ...
+beginning from 1 makes a triangular number. This is obvious
+enough from the following arrangements of rows of points;
+<FIG>
+Thus <MATH>1+2+3+...+<I>n</I>=1/2<I>n</I> (<I>n</I>+1)</MATH> is a triangular number
+<pb n=77><head>FIGURED NUMBERS</head>
+of side <I>n.</I> The particular triangle which has 4 for its side is
+mentioned in a story of Pythagoras by Lucian. Pythagoras
+told some one to count. He said 1, 2, 3, 4, whereon Pytha-
+goras interrupted, &lsquo;Do you see? What you take for 4 is 10,
+a perfect triangle and our oath&rsquo;.<note>Lucian, <G>*bi/wv pra=sis,</G> 4.</note> This connects the know-
+ledge of triangular numbers with true Pythagorean ideas.
+<C>(<G>b</G>) <I>Square numbers and gnomons.</I></C>
+<p>We come now to <I>square</I> numbers. It is easy to see that, if
+we have a number of dots forming and filling
+up a square as in the accompanying figure repre-
+<FIG>
+senting 16, the square of 4, the next higher
+square, the square of 5, can be formed by adding
+a row of dots round two sides of the original
+square, as shown; the number of these dots is
+2.4+1, or 9. This process of forming successive squares can
+be applied throughout, beginning from the first square
+number 1. The successive additions are shown in the annexed
+figure between the successive pairs of straight
+<FIG>
+lines forming right angles; and the successive
+numbers added to the 1 are
+<MATH>3, 5, 7 ... (2<I>n</I>+1)</MATH>,
+that is to say, the successive odd numbers.
+This method of formation shows that the
+sum of any number of successive terms
+of the series of odd numbers 1, 3, 5, 7 ... starting from
+1 is a square number, that, if <I>n</I><SUP>2</SUP> is any square number, the
+addition of the odd number 2<I>n</I>+1 makes it into the next
+square, (<I>n</I>+1)<SUP>2</SUP>, and that the sum of the series of odd num-
+bers <MATH>1+3+5+7+...+(2<I>n</I>+1)=(<I>n</I>+1)<SUP>2</SUP></MATH>, while
+<MATH>1+3+5+7+...+(2<I>n</I>-1)=<I>n</I><SUP>2</SUP></MATH>.
+All this was known to Pythagoras. The odd numbers succes-
+sively added were called <I>gnomons</I>; this is clear from Aristotle's
+allusion to gnomons placed round 1 which now produce different
+figures every time (oblong figures, each dissimilar to the pre-
+ceding one), now preserve one and the same figure (squares)<note>Arist. <I>Phys.</I> iii. 4, 203 a 13-15.</note>;
+the latter is the case with the gnomons now in question.
+<pb n=78><head>PYTHAGOREAN ARITHMETIC</head>
+<C>(<G>g</G>) <I>History of the term &lsquo;gnomon&rsquo;.</I></C>
+<p>It will be noticed that the gnomons shown in the above
+figure correspond in shape to the geometrical gnomons with
+which Euclid, Book II, has made us familiar. The history of
+the word &lsquo;gnomon&rsquo; is interesting. (1) It was originally an
+astronomical instrument for the measuring of time, and con-
+sisted of an upright stick which cast shadows on a plane or
+hemispherical surface. This instrument is said to have been
+introduced into Greece by Anaximander<note>Suidas, <I>s. v.</I></note> and to have come
+from Babylon.<note>Herodotus, ii. 109.</note> Following on this application of the word
+&lsquo;gnomon&rsquo; (a &lsquo;marker&rsquo; or &lsquo;pointer&rsquo;, a means of reading off and
+knowing something), we find Oenopides calling a perpendicular
+let fall on a straight line from an external point a straight line
+drawn &lsquo;<I>gnomon-wise</I>&rsquo; (<G>kata\ gnw/mona</G>).<note>Proclus on Eucl. I, p. 283. 9.</note> Next (2) we find the
+term used of an instrument for drawing right angles, which
+took the form shown in the annexed figure. This seems to
+<FIG>
+be the meaning in Theognis 805, where it is said
+that the envoy sent to consult the oracle at Delphi
+should be &lsquo;straighter than the <G>to/pvos</G> (an instru-
+ment with a stretched string for drawing a circle),
+the <G>sta/qmh</G> (a plumb-line), and the <I>gnomon</I>&rsquo;.
+It was natural that, owing to its shape, the gnomon should
+then be used to describe (3) the figure which remained of
+a square when a smaller square was cut out of it (or the figure
+which, as Aristotle says, when added to a square, preserves
+the shape and makes up a larger square). The term is used
+in a fragment of Philolaus where he says that &lsquo;number makes
+all things knowable and mutually agreeing in the way charac-
+teristic of the <I>gnomon</I>&rsquo;.<note>Boeckh, <I>Philolaos des Pythagoreers Lehren</I>, p. 141; <I>ib.</I>, p. 144; <I>Vors.</I> i<SUP>3</SUP>, p. 313. 15.</note> Presumably, as Boeckh says, the
+connexion between the gnomon and the square to which it is
+added was regarded as symbolical of union and agreement,
+and Philolaus used the idea to explain the knowledge of
+things, making the <I>knowing</I> embrace the <I>known</I> as the
+gnomon does the square.<note>Cf. Scholium No. 11 to Book II in Euclid, ed. Heib., vol. v, p. 225.</note> (4) In Euclid the geometrical
+meaning of the word is further extended (II. Def. 2) to cover
+<pb n=79><head>HISTORY OF THE TERM &lsquo;GNOMON&rsquo;</head>
+the figure similarly related to any parallelogram, instead of
+<FIG>
+a square; it is defined as made up of &lsquo;any
+one whatever of the parallelograms about
+the diameter (diagonal) with the two com-
+plements&rsquo;. Later still (5) Heron of Alex-
+andria defines a <I>gnomon</I> in general as that
+which, when added to anything, number or figure, makes the
+whole similar to that to which it is added.<note>Heron, Def. 58 (Heron, vol. iv, Heib., p. 225).</note>
+<C>(<G>d</G>) <I>Gnomons of the polygonal numbers.</I></C>
+<p>Theon of Smyrna uses the term in this general sense with
+reference to numbers: &lsquo;All the successive numbers which [by
+being successively added] produce triangles or squares <I>or
+polygons</I> are called gnomons.&rsquo;<note>Theon of Smyrna, p. 37. 11-13.</note> From the accompanying
+figures showing successive pentagonal and hexagonal numbers
+it will be seen that the outside rows or gnomons to be succes-
+<FIG>
+sively added after 1 (which is the first pentagon, hexagon, &amp;c.)
+are in the case of the pentagon 4, 7, 10, .. or the terms of an
+arithmetical progression beginning from 1 with common differ-
+ence 3, and in the case of the hexagon 5, 9, 13 .... or the
+terms of an arithmetical progression beginning from 1 with
+common difference 4. In general the successive <I>gnomonic</I>
+numbers for any polygonal number, say of <I>n</I> sides, have
+(<I>n</I>-2) for their common difference.<note><I>Ib.</I>, p. 34. 13-15.</note>
+<C>(<G>e</G>) <I>Right-angled triangles with sides in rational numbers.</I></C>
+<p>To return to Pythagoras. Whether he learnt the fact from
+Egypt or not, Pythagoras was certainly aware that, while
+<MATH>3<SUP>2</SUP>+4<SUP>2</SUP>=5<SUP>2</SUP></MATH>, any triangle with its sides in the ratio of the
+<pb n=80><head>PYTHAGOREAN ARITHMETIC</head>
+numbers 3, 4, 5 is right angled. This fact could not but add
+strength to his conviction that all things were numbers, for it
+established a connexion between numbers and the <I>angles</I> of
+geometrical figures. It would also inevitably lead to an
+attempt to find other square numbers besides 5<SUP>2</SUP> which are
+the sum of two squares, or, in other words, to find other sets
+of three integral numbers which can be made the sides of
+right-angled triangles; and herein we have the beginning of
+the <I>indeterminate analysis</I> which reached so high a stage of
+development in Diophantus. In view of the fact that the
+sum of any number of successive terms of the series of odd
+numbers 1, 3, 5, 7 ... beginning from 1 is a square, it was
+only necessary to pick out of this series the odd numbers
+which are themselves squares; for if we take one of these,
+say 9, the addition of this square to the square which is the sum
+of all the preceding odd numbers makes the square number
+which is the sum of the odd numbers up to the number (9) that
+we have taken. But it would be natural to seek a formula
+which should enable all the three numbers of a set to be imme-
+diately written down, and such a formula is actually attributed
+to Pythagoras.<note>Proclus on Eucl. I, p. 487. 7-21.</note> This formula amounts to the statement that,
+if <I>m</I> be any odd number,
+<MATH><I>m</I><SUP>2</SUP>+(<I>m</I><SUP>2</SUP>-1/2)<SUP>2</SUP>=(<I>m</I><SUP>2</SUP>+1/2)<SUP>2</SUP></MATH>.
+Pythagoras would presumably arrive at this method of forma-
+tion in the following way. Observing that the gnomon put
+round <I>n</I><SUP>2</SUP> is 2<I>n</I>+1, he would only have to make 2<I>n</I>+1 a
+square.
+<p>If we suppose that <MATH>2<I>n</I>+1=<I>m</I><SUP>2</SUP></MATH>,
+we obtain <MATH><I>n</I>=1/2(<I>m</I><SUP>2</SUP>-1)</MATH>,
+and therefore <MATH><I>n</I>+1=1/2(<I>m</I><SUP>2</SUP>+1)</MATH>.
+<p>It follows that
+<MATH><I>m</I><SUP>2</SUP>+(<I>m</I><SUP>2</SUP>-1/2)<SUP>2</SUP>=(<I>m</I><SUP>2</SUP>+1/2)<SUP>2</SUP></MATH>.
+<pb n=81><head>RATIONAL RIGHT-ANGLED TRIANGLES</head>
+<p>Another formula, devised for the same purpose, is attributed
+to Plato,<note>Proclus on Eucl. I, pp. 428. 21-429. 8.</note> namely
+<MATH>(2<I>m</I>)<SUP>2</SUP>+(<I>m</I><SUP>2</SUP>-1)<SUP>2</SUP>=(<I>m</I><SUP>2</SUP>+1)<SUP>2</SUP></MATH>.
+We could obtain this formula from that of Pythagoras by
+doubling the sides of each square in the latter; but it would
+be incomplete if so obtained, for in Pythagoras's formula <I>m</I> is
+necessarily odd, whereas in Plato's it need not be. As Pytha-
+goras's formula was most probably obtained from the gnomons
+of dots, it is tempting to suppose that Plato's was similarly
+<FIG>
+evolved. Consider the square with <I>n</I> dots in its
+side in relation to the next smaller square (<I>n</I>-1)<SUP>2</SUP>
+and the next larger (<I>n</I>+1)<SUP>2</SUP>. Then <I>n</I><SUP>2</SUP> exceeds
+(<I>n</I>-1)<SUP>2</SUP> by the gnomon 2<I>n</I>-1, but falls short of
+(<I>n</I>+1)<SUP>2</SUP> by the gnomon 2<I>n</I>+1. Therefore the
+square (<I>n</I>+1)<SUP>2</SUP> exceeds the square (<I>n</I>-1)<SUP>2</SUP> by
+the sum of the two gnomons 2<I>n</I>-1 and 2<I>n</I>+1, which
+is 4<I>n.</I>
+<p>That is, <MATH>4<I>n</I>+(<I>n</I>-1)<SUP>2</SUP>=(<I>n</I>+1)<SUP>2</SUP></MATH>,
+and, substituting <I>m</I><SUP>2</SUP> for <I>n</I> in order to make 4<I>n</I> a square, we
+obtain the Platonic formula
+<MATH>(2<I>m</I>)<SUP>2</SUP>+(<I>m</I><SUP>2</SUP>-1)<SUP>2</SUP>=(<I>m</I><SUP>2</SUP>+1)<SUP>2</SUP></MATH>.
+<p>The formulae of Pythagoras and Plato supplement each
+other. Euclid's solution (X, Lemma following Prop. 28) is
+more general, amounting to the following.
+<p>If <I>AB</I> be a straight line bisected at <I>C</I> and produced to <I>D</I>,
+then (Eucl. II. 6)
+<MATH><I>AD.DB</I>+<I>CB</I><SUP>2</SUP>=<I>CD</I><SUP>2</SUP></MATH>,
+which we may write thus:
+<MATH><I>uv</I>=<I>c</I><SUP>2</SUP>-<I>b</I><SUP>2</SUP></MATH>,
+where <MATH><I>u</I>=<I>c</I>+<I>b</I>, <I>v</I>=<I>c</I>-<I>b</I></MATH>,
+and consequently
+<MATH><I>c</I>=1/2(<I>u</I>+<I>v</I>), <I>b</I>=1/2(<I>u</I>-<I>v</I>)</MATH>.
+<p>In order that <I>uv</I> may be a square, says Euclid, <I>u</I> and <I>v</I>
+must, if they are not actually squares, be &lsquo;similar plane num-
+bers&rsquo;, and further they must be either both odd or both even
+<pb n=82><head>PYTHAGOREAN ARITHMETIC</head>
+in order that <I>b</I> (and <I>c</I> also) may be a whole number. &lsquo;Similar
+plane&rsquo; numbers are of course numbers which are the product
+of two factors proportional in pairs, as <I>mp.np</I> and <I>mq.nq</I>, or
+<I>mnp</I><SUP>2</SUP> and <I>mnq</I><SUP>2</SUP>. Provided, then, that these numbers are both
+even or both odd,
+<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>+((<I>mnp</I><SUP>2</SUP>-<I>mnq</I><SUP>2</SUP>)/2)<SUP>2</SUP>=((<I>mnp</I><SUP>2</SUP>+<I>mnq</I><SUP>2</SUP>)/2)<SUP>2</SUP></MATH>
+is the solution, which includes both the Pythagorean and the
+Platonic formulae.
+<C>(<G>z</G>) <I>Oblong numbers.</I></C>
+<p>Pythagoras, or the earliest Pythagoreans, having discovered
+that, by adding any number of successive terms (beginning
+from 1) of the series <MATH>1+2+3+...+<I>n</I>=1/2<I>n</I>(<I>n</I>+1)</MATH>, we obtain
+triangular numbers, and that by adding the successive odd
+numbers <MATH>1+3+5+...+(2<I>n</I>-1)=<I>n</I><SUP>2</SUP></MATH> we obtain squares, it
+cannot be doubted that in like manner they summed the
+series of even numbers <MATH>2+4+6+...+2<I>n</I>=<I>n</I>(<I>n</I>+1)</MATH> and
+discovered accordingly that the sum of any number of succes-
+sive terms of the series beginning with 2 was an &lsquo;oblong&rsquo;
+number (<G>e(teromh/khs</G>), with &lsquo;sides&rsquo; or factors differing by 1.
+They would also see that the oblong number is double of
+a triangular number. These facts would be brought out by
+taking two dots representing 2 and then placing round them,
+gnomon-wise and successively, the even numbers 4, 6, &amp;c.,
+thus:
+<FIG>
+The successive oblong numbers are
+<MATH>2.3=6, 3.4=12, 4.5=20..., <I>n</I>(<I>n</I>+1)...</MATH>,
+and it is clear that no two of these numbers are similar, for
+the ratio <I>n</I>:(<I>n</I>+1) is different for all different values of <I>n.</I>
+We may have here an explanation of the Pythagorean identi-
+fication of &lsquo;odd&rsquo; with &lsquo;limit&rsquo; or &lsquo;limited&rsquo; and of &lsquo;even&rsquo; with
+<pb n=83><head>OBLONG NUMBERS</head>
+&lsquo;unlimited&rsquo;<note>Arist. <I>Metaph.</I> A. 5, 986 a 17.</note> (cf. the Pythagorean scheme of ten pairs of
+opposites, where odd, limit and square in one set are opposed
+to even, unlimited and oblong respectively in the other).<note><I>Ib.</I> A. 5, 986 a 23-26.</note> For,
+while the adding of the successive odd numbers as gnomons
+round 1 gives only one form, the square, the addition of the
+successive even numbers to 2 gives a succession of &lsquo;oblong&rsquo;
+numbers all dissimilar in form, that is to say, an infinity of
+forms. This seems to be indicated in the passage of Aristotle's
+<I>Physics</I> where, as an illustration of the view that the even
+is unlimited, he says that, where gnomons are put round 1,
+the resulting figures are in one case always different in
+species, while in the other they always preserve one form<note>Arist. <I>Phys.</I> iii. 4, 203 a 10-15.</note>;
+the one form is of course the square formed by adding the
+odd numbers as gnomons round 1; the words <G>kai\ *xwri/s</G>
+(&lsquo;and in the separate case&rsquo;, as we may perhaps translate)
+imperfectly describe the second case, since in that case
+even numbers are put round 2, not 1, but the meaning
+seems clear.<note>Cf. Plut. (?) Stob. <I>Ecl.</I> i. pr. 10, p. 22. 16 Wachsmuth.</note> It is to be noted that the word <G>e(teromh/khs</G>
+(&lsquo;oblong&rsquo;) is in Theon of Smyrna and Nicomachus limited to
+numbers which are the product of two factors differing by
+unity, while they apply the term <G>promh/khs</G> (&lsquo;prolate&rsquo;, as it
+were) to numbers which are the product of factors differing
+by two or more (Theon makes <G>promh/khs</G> include <G>e(teromh/khs</G>).
+In Plato and Aristotle <G>e(teromh/khs</G> has the wider sense of any
+non-square number with two unequal factors.
+<p>It is obvious that any &lsquo;oblong&rsquo; number <I>n</I>(<I>n</I>+1) is the
+<FIG>
+sum of two equal triangular numbers. Scarcely less obvious
+is the theorem of Theon that any square number is made up
+of two triangular numbers<note>Theon of Smyrna, p. 41. 3-8.</note>; in this case, as is seen from the
+<pb n=84><head>PYTHAGOREAN ARITHMETIC</head>
+figure, the sides of the triangles differ by unity, and of course
+<FIG>
+<MATH>1/2<I>n</I>(<I>n</I>-1)+1/2<I>n</I>(<I>n</I>+1)=<I>n</I><SUP>2</SUP></MATH>.
+<p>Another theorem connecting triangular num-
+bers and squares, namely that 8 times any
+triangular number +1 makes a square, may
+easily go back to the early Pythagoreans. It is
+quoted by Plutarch<note>Plutarch, <I>Plat. Quaest.</I> v. 2. 4, 1003 F.</note> and used by Diophantus,<note>Dioph. IV. 38.</note> and is equi-
+valent to the formula
+<MATH>8.1/2<I>n</I>(<I>n</I>+1)+1=4<I>n</I>(<I>n</I>+1)+1=(2<I>n</I>+1)<SUP>2</SUP></MATH>.
+<p>It may easily have been proved by means of a figure
+<FIG>
+made up of dots in the usual way. Two
+equal triangles make up an oblong figure
+of the form <I>n</I>(<I>n</I>+1), as above. Therefore
+we have to prove that four equal figures
+of this form with one more dot make up
+(2<I>n</I>+1)<SUP>2</SUP>. The annexed figure representing
+7<SUP>2</SUP> shows how it can be divided into four
+&lsquo;oblong&rsquo; figures 3.4 leaving 1 over.
+<p>In addition to Speusippus, Philippus of Opus (fourth
+century), the editor of Plato's <I>Laws</I> and author of the <I>Epi-
+nomis</I>, is said to have written a work on polygonal numbers.<note><G>*biogra/foi</G>, <I>Vitarum scriptores Graeci minores</I>, ed. Westermann, p. 446.</note>
+Hypsicles, who wrote about 170 B.C., is twice mentioned in
+Diophantus's <I>Polygonal Numbers</I> as the author of a &lsquo;defini-
+tion&rsquo; of a polygonal number.
+<C>The theory of proportion and means.</C>
+<p>The &lsquo;summary&rsquo; of Proclus (as to which see the beginning
+of Chapter IV) states (if Friedlein's reading is right) that
+Pythagoras discovered &lsquo;the theory of irrationals (<G>th\n tw=n
+a)lo/gwn pragmatei/an</G>) and the construction of the cosmic
+figures&rsquo; (the five regular solids).<note>Proclus on Eucl. I, p. 65. 19.</note> We are here concerned
+with the first part of this statement in so far as the reading
+<G>a)lo/gwn</G> (&lsquo;irrationals&rsquo;) is disputed. Fabricius seems to have
+been the first to record the variant <G>a)nalo/gwn</G>, which is also
+noted by E. F. August<note>In his edition of the Greek text of Euclid (1824-9), vol. i, p. 290.</note>; Mullach adopted this reading from
+<pb n=85><head>THE THEORY OF PROPORTION AND MEANS</head>
+Fabricius. <G>a)nalo/gwn</G> is not the correct form of the word, but
+the meaning would be &lsquo;proportions&rsquo; or &lsquo;proportionals&rsquo;, and
+the true reading may be either <G>tw=n a)nalogiw=n</G> (&lsquo;proportions&rsquo;),
+or, more probably, <G>tw=n a)na\ lo/gon</G> (&lsquo;proportionals&rsquo;); Diels
+reads <G>tw=n a)na\ lo/gon</G>, and it would seem that there is now
+general agreement that <G>a)lo/gwn</G> is wrong, and that the theory
+which Proclus meant to attribute to Pythagoras is the theory
+of <I>proportion</I> or <I>proportionals</I>, not of irrationals.
+<C>(<G>a</G>) <I>Arithmetic, geometric, and harmonic means.</I></C>
+<p>It is true that we have no positive evidence of the use by
+Pythagoras of proportions in geometry, although he must
+have been conversant with similar figures, which imply some
+theory of proportion. But he discovered the dependence of
+musical intervals on numerical ratios, and the theory of <I>means</I>
+was developed very early in his school with reference to
+the theory of music and arithmetic. We are told that in
+Pythagoras's time there were three means, the arithmetic,
+the geometric, and the subcontrary, and that the name of the
+third (&lsquo;subcontrary&rsquo;) was changed by Archytas and Hippasus
+to &lsquo;harmonic&rsquo;.<note>Iambl. <I>in Nicom.</I>, p. 100. 19-24.</note> A fragment of Archytas's work <I>On Music</I>
+actually defines the three; we have the <I>arithmetic</I> mean
+when, of three terms, the first exceeds the second by the
+same amount as the second exceeds the third; the <I>geometric</I>
+mean when, of the three terms, the first is to the second as
+the second is to the third; the &lsquo;<I>subcontrary</I>, which we call
+<I>harmonic</I>&rsquo;, when the three terms are such that &lsquo;by whatever
+part of itself the first exceeds the second, the second exceeds
+the third by the same part of the third&rsquo;.<note>Porph. <I>in Ptol. Harm.</I>, p. 267 (<I>Vors.</I> i<SUP>3</SUP>, p. 334. 17 sq.).</note> That is, if <I>a, b, c</I>
+are in harmonic progression, and <MATH><I>a</I>=<I>b</I>+<I>a</I>/<I>n</I></MATH>, we must have
+<MATH><I>b</I>=<I>c</I>+<I>c</I>/<I>n</I></MATH>, whence in fact
+<MATH><I>a</I>/<I>c</I>=(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)</MATH>, or <MATH>1/<I>c</I>-1/<I>b</I>=1/<I>b</I>-1/<I>a</I></MATH>.
+Nicomachus too says that the name &lsquo;harmonic mean&rsquo; was
+adopted in accordance with the view of Philolaus about the
+&lsquo;geometrical harmony&rsquo;, a name applied to the cube because
+it has 12 edges, 8 angles, and 6 faces, and 8 is the mean
+<pb n=86><head>PYTHAGOREAN ARITHMETIC</head>
+between 12 and 6 according to the theory of harmonics (<G>kata\
+th\n a(rmonikh/n</G>).<note>Nicom. ii. 26. 2.</note>
+<p>Iamblichus,<note>Iambl. <I>in Nicom.</I>, p. 118. 19sq.</note> after Nicomachus,<note>Nicom. ii. 29.</note> mentions a special &lsquo;most
+perfect proportion&rsquo; consisting of four terms and called
+&lsquo;musical&rsquo;, which, according to tradition, was discovered by
+the Babylonians and was first introduced into Greece by
+Pythagoras. It was used, he says, by many Pythagoreans,
+e.g. (among others) Aristaeus of Croton, Timaeus of Locri,
+Philolaus and Archytas of Tarentum, and finally by Plato
+in the <I>Timaeus</I>, where we are told that the double and triple
+intervals were filled up by two means, one of which exceeds
+and is exceeded by the same part of the extremes (the
+harmonic mean), and the other exceeds and is exceeded by
+the same numerical magnitude (the arithmetic mean).<note>Plato, <I>Timaeus</I>, 36 A.</note> The
+proportion is
+<MATH><I>a</I>:(<I>a</I>+<I>b</I>)/2=(2<I>ab</I>)/(<I>a</I>+<I>b</I>):<I>b</I></MATH>,
+an example being 12:9=8:6.
+<C>(<G>b</G>) <I>Seven other means distinguished.</I></C>
+<p>The theory of means was further developed in the school
+by the gradual addition of seven others to the first three,
+making ten in all. The accounts of the discovery of the
+fourth, fifth, and sixth are not quite consistent. In one place
+Iamblichus says they were added by Eudoxus<note>Iambl. <I>in Nicom.</I>, p. 101. 1-5.</note>; in other
+places he says they were in use by the successors of Plato
+down to Eratosthenes, but that Archytas and Hippasus made
+a beginning with their discovery,<note><I>Ib.</I>, p. 116. 1-4.</note> or that they were part of
+the Archytas and Hippasus tradition.<note><I>Ib.</I>, p. 113, 16-18.</note> The remaining four
+means (the seventh to the tenth) are said to have been added
+by two later Pythagoreans, Myonides and Euphranor.<note><I>Ib.</I>, p. 116. 4-6.</note> From
+a remark of Porphyry it would appear that one of the first
+seven means was discovered by Simus of Posidonia, but
+that the jealousy of other Pythagoreans would have robbed
+him of the credit.<note>Porphyry, <I>Vit. Pyth.</I> 3; <I>Vors.</I> i<SUP>3</SUP>, p. 343. 12-15 and note.</note> The ten means are described by
+<pb n=87><head>THE SEVERAL MEANS DISTINGUISHED</head>
+Nicomachus<note>Nicom. ii. 28.</note> and Pappus<note>Pappus, iii, p. 102.</note>; their accounts only differ as
+regards one of the ten. If <I>a>b>c</I>, the formulae in the third
+column of the following table show the various means.
+<table>
+<tr><td>No. in Nicom.</td><td>No. in Pappus.</td><td>Formulae.</td><td>Equivalent.</td></tr>
+<tr><td align=center>1</td><td align=center>1</td><td><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>a</I>/<I>a</I>=<I>b</I>/<I>b</I>=<I>c</I>/<I>c</I></MATH></td><td><MATH><I>a</I>+<I>c</I>=2<I>b</I></MATH> (arithmetic)</td></tr>
+<tr><td align=center>2</td><td align=center>2</td><td><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>a</I>/<I>b</I>[=<I>b</I>/<I>c</I>]</MATH></td><td><MATH><I>ac</I>=<I>b</I><SUP>2</SUP></MATH> (geometric)</td></tr>
+<tr><td align=center>3</td><td align=center>3</td><td><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>a</I>/<I>c</I></MATH></td><td><MATH>1/<I>a</I>+1/<I>c</I>=2/<I>b</I></MATH> (harmonic)</td></tr>
+<tr><td align=center>4</td><td align=center>4</td><td><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>c</I>/<I>a</I></MATH></td><td><MATH>(<I>a</I><SUP>2</SUP>+<I>c</I><SUP>2</SUP>)/(<I>a</I>+<I>c</I>)=<I>b</I></MATH> (subcontrary to harmonic)</td></tr>
+<tr><td align=center>5</td><td align=center>5</td><td><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>c</I>/<I>b</I></MATH></td>
+<td><MATH><BRACE><note>(subcontrary to geometric)</note><I>a</I>=<I>b</I>+<I>c</I>-<I>c</I><SUP>2</SUP>/<I>b</I><I>c</I>=<I>a</I>+<I>b</I>-<I>a</I><SUP>2</SUP>/<I>b</I></BRACE></MATH></td></tr>
+<tr><td align=center>6</td><td align=center>6</td><td><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>b</I>/<I>a</I></MATH></td></tr>
+<tr><td align=center>7</td><td align=center>(omitted)</td><td><MATH>(<I>a</I>-<I>c</I>)/(<I>b</I>-<I>c</I>)=<I>a</I>/<I>c</I></MATH></td><td><MATH><I>c</I><SUP>2</SUP>=2<I>ac</I>-<I>ab</I></MATH></td></tr>
+<tr><td align=center>8</td><td align=center>9</td><td><MATH>(<I>a</I>-<I>c</I>)/(<I>a</I>-<I>b</I>)=<I>a</I>/<I>c</I></MATH></td><td><MATH><I>a</I><SUP>2</SUP>+<I>c</I><SUP>2</SUP>=<I>a</I>(<I>b</I>+<I>c</I>)</MATH></td></tr>
+<tr><td align=center>9</td><td align=center>10</td><td><MATH>(<I>a</I>-<I>c</I>)/(<I>b</I>-<I>c</I>)=<I>b</I>/<I>c</I></MATH></td><td><MATH><I>b</I><SUP>2</SUP>+<I>c</I><SUP>2</SUP>=<I>c</I>(<I>a</I>+<I>b</I>)</MATH></td></tr>
+<tr><td align=center>10</td><td align=center>7</td><td><MATH>(<I>a</I>-<I>c</I>)/(<I>a</I>-<I>b</I>)=<I>b</I>/<I>c</I></MATH></td><td><MATH><I>a</I>=<I>b</I>+<I>c</I></MATH></td></tr>
+<tr><td align=center>(omitted)</td><td align=center>8</td><td><MATH>(<I>a</I>-<I>c</I>)/(<I>a</I>-<I>b</I>)=<I>a</I>/<I>b</I></MATH></td><td><MATH><I>a</I><SUP>2</SUP>=2<I>ab</I>-<I>bc</I></MATH></td></tr>
+</table>
+<p>The two lists together give <I>five</I> means in addition to the
+first six which are common to both; there would be six more
+(as Theon of Smyrna says<note>Theon of Smyrna, p. 106. 15, p. 116. 3.</note>) were it not that <MATH>(<I>a</I>-<I>c</I>)/(<I>b</I>-<I>c</I>)=<I>a</I>/<I>b</I></MATH> is
+illusory, since it gives <MATH><I>a</I>=<I>b</I></MATH>. Tannery has remarked that
+<pb n=88><head>PYTHAGOREAN ARITHMETIC</head>
+Nos. 4, 5, 6 of the above means give equations of the second
+degree, and he concludes that the geometrical and even the
+arithmetical solution of such equations was known to the dis-
+coverer of these means, say about the time of Plato<note>Tannery, <I>M&eacute;moires scientifiques</I>, i, pp. 92-3.</note>; Hippo-
+crates of Chios, in fact, assumed the geometrical solution of
+a mixed quadratic equation in his quadrature of lunes.
+<p>Pappus has an interesting series of propositions with
+regard to eight out of the ten means defined by him.<note>Pappus, iii, pp. 84-104.</note> He
+observes that if <G>a, b, g</G> be three terms in geometrical pro-
+gression, we can form from these terms three other terms
+<I>a, b, c</I>, being linear functions of <G>a, b, g</G> which satisfy respec-
+tively eight of the above ten relations; that is to say, he
+gives a solution of eight problems in indeterminate analysis
+of the second degree. The solutions are as follows:
+<table>
+<tr align=center><td>No. in Nicom.</td><td>No. in Pappus.</td><td>Formulae.</td><td>Solution in terms of <G>a, b, g.</G></td><td>Smallest solution.</td></tr>
+<tr><td rowspan=2>2</td><td rowspan=2>2</td><td rowspan=2><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>a</I>/<I>b</I>=<I>b</I>/<I>c</I></MATH></td><td><MATH><I>a</I>=<G>a</G>+2<G>b</G>+<G>g</G></MATH></td><td><MATH><I>a</I>=4</MATH></td></tr>
+<tr><td><MATH><I>b</I>=<G>b</G>+<G>g</G></MATH></td><td><MATH><I>b</I>=2</MATH></td></tr>
+<tr><td></td><td></td><td></td><td><MATH><I>c</I>=<G>g</G></MATH></td><td><MATH><I>c</I>=1</MATH></td></tr>
+<tr><td rowspan=2>3</td><td rowspan=2>3</td><td rowspan=2><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>a</I>/<I>c</I></MATH></td><td><MATH><I>a</I>=2<G>a</G>+3<G>b</G>+<G>g</G></MATH></td><td><MATH><I>a</I>=6</MATH></td></tr>
+<tr><td><MATH><I>b</I>=2<G>b</G>+<G>g</G></MATH></td><td><MATH><I>b</I>=3</MATH></td></tr>
+<tr><td></td><td></td><td></td><td><MATH><I>c</I>=<G>b</G>+<G>g</G></MATH></td><td><MATH><I>c</I>=2</MATH></td></tr>
+<tr><td rowspan=2>4</td><td rowspan=2>4</td><td rowspan=2><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>c</I>/<I>a</I></MATH></td><td><MATH><I>a</I>=2<G>a</G>+3<G>b</G>+<G>g</G></MATH></td><td><MATH><I>a</I>=6</MATH></td></tr>
+<tr><td><MATH><I>b</I>=2<G>a</G>+2<G>b</G>+<G>g</G></MATH></td><td><MATH><I>b</I>=5</MATH></td></tr>
+<tr><td></td><td></td><td></td><td><MATH><I>c</I>=<G>b</G>+<G>g</G></MATH></td><td><MATH><I>c</I>=2</MATH></td></tr>
+<tr><td rowspan=2>5</td><td rowspan=2>5</td><td rowspan=2><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>c</I>/<I>b</I></MATH></td><td><MATH><I>a</I>=<G>a</G>+3<G>b</G>+<G>g</G></MATH></td><td><MATH><I>a</I>=5</MATH></td></tr>
+<tr><td><MATH><I>b</I>=<G>a</G>+2<G>b</G>+<G>g</G></MATH></td><td><MATH><I>b</I>=4</MATH></td></tr>
+<tr><td></td><td></td><td></td><td><MATH><I>c</I>=<G>b</G>+<G>g</G></MATH></td><td><MATH><I>c</I>=2</MATH></td></tr>
+<tr><td rowspan=2>6</td><td rowspan=2>6</td><td rowspan=2><MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>b</I>/<I>a</I></MATH></td><td><MATH><I>a</I>=<G>a</G>+3<G>b</G>+2<G>g</G></MATH></td><td><MATH><I>a</I>=6</MATH></td></tr>
+<tr><td><MATH><I>b</I>=<G>a</G>+2<G>b</G>+<G>g</G></MATH></td><td><MATH><I>b</I>=4</MATH></td></tr>
+<tr><td></td><td></td><td></td><td><MATH><I>c</I>=<G>a</G>+<G>b</G>-<G>g</G></MATH></td><td><MATH><I>c</I>=1</MATH></td></tr>
+</table>
+<pb n=89><head>THE SEVERAL MEANS DISTINGUISHED</head>
+<table>
+<tr align=center><td>No. in Nicom.</td><td>No. in Pappus.</td><td>Formulae.</td><td>Solution in terms of <G>a, b, g.</G></td><td>Smallest solution.</td></tr>
+<tr><td rowspan=2>---</td><td rowspan=2>8</td><td rowspan=2><MATH>(<I>a</I>-<I>c</I>)/(<I>a</I>-<I>b</I>)=<I>a</I>/<I>b</I></MATH></td><td><MATH><I>a</I>=2<G>a</G>+3<G>b</G>+<G>g</G></MATH></td><td><MATH><I>a</I>=6</MATH></td></tr>
+<tr><td><MATH><I>b</I>=<G>a</G>+2<G>b</G>+<G>g</G></MATH></td><td><MATH><I>b</I>=4</MATH></td></tr>
+<tr><td></td><td></td><td></td><td><MATH><I>c</I>=2<G>b</G>+<G>g</G></MATH></td><td><MATH><I>c</I>=3</MATH></td></tr>
+<tr><td rowspan=2>8</td><td rowspan=2>9</td><td rowspan=2><MATH>(<I>a</I>-<I>c</I>)/(<I>a</I>-<I>b</I>)=<I>a</I>/<I>c</I></MATH></td><td><MATH><I>a</I>=<G>a</G>+2<G>b</G>+<G>g</G></MATH></td><td><MATH><I>a</I>=4</MATH></td></tr>
+<tr><td><MATH><I>b</I>=<G>a</G>+<G>b</G>+<G>g</G></MATH></td><td><MATH><I>b</I>=3</MATH></td></tr>
+<tr><td></td><td></td><td></td><td><MATH><I>c</I>=<G>b</G>+<G>g</G></MATH></td><td><MATH><I>c</I>=2</MATH></td></tr>
+<tr><td rowspan=2>9</td><td rowspan=2>10</td><td rowspan=2><MATH>(<I>a</I>-<I>c</I>)/(<I>b</I>-<I>c</I>)=<I>b</I>/<I>c</I></MATH></td><td><MATH><I>a</I>=<G>a</G>+<G>b</G>+<G>g</G></MATH></td><td><MATH><I>a</I>=3</MATH></td></tr>
+<tr><td><MATH><I>b</I>=<G>b</G>+<G>g</G></MATH></td><td><MATH><I>b</I>=2</MATH></td></tr>
+<tr><td></td><td></td><td></td><td><MATH><I>c</I>=<G>g</G></MATH></td><td><MATH><I>c</I>=1</MATH></td></tr>
+</table>
+<p>Pappus does not include a corresponding solution for his
+No. 1 and No. 7, and Tannery suggests as the reason for this
+that, the equations in these cases being already linear, there
+is no necessity to assume <MATH><G>ag</G>=<G>b</G><SUP>2</SUP></MATH>, and consequently there is
+one indeterminate too many.<note>Tannery, <I>loc. cit.</I>, pp. 97-8.</note> Pappus does not so much prove
+as verify his results, by transforming the proportion <MATH><G>a</G>/<G>b</G>=<G>b</G>/<G>g</G></MATH>
+in all sorts of ways, <I>componendo, dividendo</I>, &amp;c.
+<C>(<G>g</G>) <I>Plato on geometric means between two squares
+or two cubes.</I></C>
+<p>It is well known that the mathematics in Plato's <I>Timaeus</I>
+is essentially Pythagorean. It is therefore <I>a priori</I> probable
+that Plato <G>puqagori/zei</G> in the passage<note>Plato, <I>Timaeus</I>, 32 A, B.</note> where he says that
+between two <I>planes</I> one mean suffices, but to connect two
+<I>solids</I> two means are necessary. By <I>planes</I> and <I>solids</I> he
+really means square and cube numbers, and his remark is
+equivalent to stating that, if <I>p</I><SUP>2</SUP>, <I>q</I><SUP>2</SUP> are two square numbers,
+<MATH><I>p</I><SUP>2</SUP>:<I>pq</I>=<I>pq:q</I><SUP>2</SUP></MATH>,
+while, if <I>p</I><SUP>3</SUP>, <I>q</I><SUP>3</SUP> are two cube numbers,
+<MATH><I>p</I><SUP>3</SUP>:<I>p</I><SUP>2</SUP><I>q</I>=<I>p</I><SUP>2</SUP><I>q:pq</I><SUP>2</SUP>=<I>pq</I><SUP>2</SUP>:<I>q</I><SUP>3</SUP></MATH>,
+the means being of course means in continued geometric pro-
+portion. Euclid proves the properties for square and cube
+<pb n=90><head>PYTHAGOREAN ARITHMETIC</head>
+numbers in VIII. 11, 12, and for similar plane and solid num-
+bers in VIII. 18, 19. Nicomachus quotes the substance of
+Plato's remark as a &lsquo;Platonic theorem&rsquo;, adding in explanation
+the equivalent of Eucl. VIII. 11, 12.<note>Nicom. ii. 24. 6, 7.</note>
+<C>(<G>d</G>) <I>A theorem of Archytas.</I></C>
+<p>Another interesting theorem relative to geometric means
+evidently goes back to the Pythagoreans. If we have two
+numbers in the ratio known as <G>e)pimo/rios</G>, or <I>superparticularis</I>,
+i.e. the ratio of <I>n</I>+1 to <I>n</I>, there can be no number which is
+a mean proportional between them. The theorem is Prop. 3 of
+Euclid's <I>Sectio Canonis</I>,<note><I>Musici Scriptores Graeci</I>, ed. Jan, pp. 148-66; Euclid, vol. viii, ed.
+Heiberg and Menge, p. 162.</note> and Bo&euml;tius has preserved a proof
+of it by Archytas, which is substantially identical with that of
+Euclid.<note>Bo&euml;tius, <I>De Inst. Musica</I>, iii. 11 (pp. 285-6, ed. Friedlein); see <I>Biblio-
+theca Mathematica</I>, vi<SUB>3</SUB>, 1905/6, p. 227.</note> The proof will be given later (pp. 215-16). So far as
+this chapter is concerned, the importance of the proposition lies
+in the fact that it implies the existence, at least as early
+as the date of Archytas (about 430-365 B.C.), of an <I>Elements
+of Arithmetic</I> in the form which we call Euclidean; and no
+doubt text-books of the sort existed even before Archytas,
+which probably Archytas himself and others after him im-
+proved and developed in their turn.
+<C>The &lsquo;irrational&rsquo;.</C>
+<p>We mentioned above the dictum of Proclus (if the reading
+<G>a)lo/gwn</G> is right) that Pythagoras discovered the theory, or
+study, of <I>irrationals.</I> This subject was regarded by the
+Greeks as belonging to geometry rather than arithmetic.
+The irrationals in Euclid, Book X, are straight lines or areas,
+and Proclus mentions as special topics in geometry matters
+relating (1) to <I>positions</I> (for numbers have no position), (2) to
+<I>contacts</I> (for tangency is between <I>continuous</I> things), and (3)
+to <I>irrational straight lines</I> (for where there is division <I>ad
+infinitum</I>, there also is the irrational).<note>Proclus on Eucl. I, p. 60. 12-16.</note> I shall therefore
+postpone to Chapter V on the Pythagorean geometry the
+question of the date of the discovery of the theory of irra-
+tionals. But it is certain that the incommensurability of the
+<pb n=91><head>THE &lsquo;IRRATIONAL&rsquo;</head>
+diagonal of a square with its side, that is, the &lsquo;irrationality&rsquo;
+of &radic;2, was discovered in the school of Pythagoras, and it is
+more appropriate to deal with this particular case here, both
+because the traditional proof of the fact depends on the
+elementary theory of numbers, and because the Pythagoreans
+invented a method of obtaining an infinite series of arith-
+metical ratios approaching more and more closely to the value
+of &radic;2.
+<p>The actual method by which the Pythagoreans proved the
+fact that &radic;2 is incommensurable with 1 was doubtless that
+indicated by Aristotle, a <I>reductio ad absurdum</I> showing that,
+if the diagonal of a square is commensurable with its side, it
+will follow that the same number is both odd and even.<note>Arist. <I>Anal. pr.</I> i. 23, 41 a 26-7.</note> This
+is evidently the proof interpolated in the texts of Euclid as
+X. 117, which is in substance as follows:
+<p>Suppose <I>AC</I>, the diagonal of a square, to be commensur-
+able with <I>AB</I>, its side; let <G>a</G>:<G>b</G> be their ratio expressed in
+the smallest possible numbers.
+<p>Then <G>a</G>><G>b</G>, and therefore <G>a</G> is necessarily > 1.
+<p>Now <MATH><I>AC</I><SUP>2</SUP>:<I>AB</I><SUP>2</SUP>=<G>a</G><SUP>2</SUP>:<G>b</G><SUP>2</SUP>;</MATH>
+and, since <MATH><I>AC</I><SUP>2</SUP>=2<I>AB</I><SUP>2</SUP>, <G>a</G><SUP>2</SUP>=2<G>b</G><SUP>2</SUP></MATH>.
+<p>Hence <G>a</G><SUP>2</SUP>, and therefore <G>a</G>, is even.
+<p>Since <G>a</G>:<G>b</G> is in its lowest terms, it follows that <G>b</G> must
+be <I>odd.</I>
+<p>Let <MATH><G>a</G>=2<G>g</G></MATH>; therefore <MATH>4<G>g</G><SUP>2</SUP>=2<G>b</G><SUP>2</SUP></MATH>, or <MATH>2<G>g</G><SUP>2</SUP>=<G>b</G><SUP>2</SUP></MATH>, so that <G>b</G><SUP>2</SUP>,
+and therefore <G>b</G>, is <I>even.</I>
+<p>But <G>b</G> was also <I>odd</I>: which is impossible.
+<p>Therefore the diagonal <I>AC</I> cannot be commensurable with
+the side <I>AB.</I>
+<C>Algebraic equations.</C>
+<C>(<G>a</G>) <I>&lsquo;Side-&rsquo; and &lsquo;diameter-&rsquo; numbers, giving successive
+approximations to</I> &radic;2.</C>
+<p>The Pythagorean method of finding any number of succes-
+sive approximations to the value of &radic;2 amounts to finding
+all the integral solutions of the indeterminate equations
+<MATH>2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>=&plusmn;1</MATH>,
+the solutions being successive pairs of what were called <I>side-</I>
+<pb n=92><head>PYTHAGOREAN ARITHMETIC</head>
+and <I>diameter-</I> (diagonal-) <I>numbers</I> respectively. The law of
+formation of these numbers is explained by Theon of Smyrna,
+and is as follows.<note>Theon of Smyrna, pp. 43, 44.</note> The unit, being the beginning of all things,
+must be potentially both a side and a diameter. Consequently
+we begin with two units, the one being the first <I>side</I>, which we
+will call <I>a</I><SUB>1</SUB>, the other being the first <I>diameter</I>, which we will
+call <I>d</I><SUB>1</SUB>.
+<p>The second side and diameter (<I>a</I><SUB>2</SUB>, <I>d</I><SUB>2</SUB>) are formed from the
+first, the third side and diameter (<I>a</I><SUB>3</SUB>, <I>d</I><SUB>3</SUB>) from the second, and
+so on, as follows:
+<MATH><I>a</I><SUB>2</SUB>=<I>a</I><SUB>1</SUB>+<I>d</I><SUB>1</SUB>, <I>d</I><SUB>2</SUB>=2<I>a</I><SUB>1</SUB>+<I>d</I><SUB>1</SUB>,
+<I>a</I><SUB>3</SUB>=<I>a</I><SUB>2</SUB>+<I>d</I><SUB>2</SUB>, <I>d</I><SUB>3</SUB>=2<I>a</I><SUB>2</SUB>+<I>d</I><SUB>2</SUB>,
+. . . . . . . . . .
+<I>a</I><SUB><I>n</I>+1</SUB>=<I>a<SUB>n</SUB></I>+<I>d<SUB>n</SUB></I>, <I>d</I><SUB><I>n</I>+1</SUB>=2<I>a<SUB>n</SUB></I>+<I>d<SUB>n</SUB></I></MATH>.
+<p>Since <MATH><I>a</I><SUB>1</SUB>=<I>d</I><SUB>1</SUB>=1</MATH>, it follows that
+<MATH><I>a</I><SUB>2</SUB>=1+1=2, <I>d</I><SUB>2</SUB>=2.1+1=3,
+<I>a</I><SUB>3</SUB>=2+3=5, <I>d</I><SUB>3</SUB>=2.2+3=7,
+<I>a</I><SUB>4</SUB>=5+7=12, <I>d</I><SUB>4</SUB>=2.5+7=17</MATH>,
+and so on.
+<p>Theon states, with reference to these numbers, the general
+proposition that
+<MATH><I>d<SUB>n</SUB></I><SUP>2</SUP>=2<I>a<SUB>n</SUB></I><SUP>2</SUP>&plusmn;1</MATH>,
+and he observes (1) that the signs alternate as successive <I>d</I>'s
+and <I>a</I>'s are taken, <I>d</I><SUB>1</SUB><SUP>2</SUP>-2<I>a</I><SUB>1</SUB><SUP>2</SUP> being equal to -1, <I>d</I><SUB>2</SUB><SUP>2</SUP>-2<I>a</I><SUB>2</SUB><SUP>2</SUP>
+equal to +1, <I>d</I><SUB>3</SUB><SUP>2</SUP>-2<I>a</I><SUB>3</SUB><SUP>2</SUP> equal to -1, and so on, while (2) the
+sum of the squares of <I>all</I> the <I>d</I>'s will be double of the squares
+of <I>all</I> the <I>a</I>'s. [If the number of successive terms in each
+series is finite, it is of course necessary that the number should
+be even.]
+<p>The properties stated depend on the truth of the following
+identity
+<MATH>(2<I>x</I>+<I>y</I>)<SUP>2</SUP>-2(<I>x</I>+<I>y</I>)<SUP>2</SUP>=2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>;</MATH>
+for, if <I>x, y</I> be numbers which satisfy one of the two equations
+<MATH>2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>=&plusmn;1</MATH>,
+the formula (if true) gives us two higher numbers, <I>x</I>+<I>y</I> and
+2<I>x</I>+<I>y</I>, which satisfy the other of the two equations.
+<p>Not only is the identity true, but we know from Proclus
+<pb n=93><head>&lsquo;SIDE-&rsquo; AND &lsquo;DIAMETER-&rsquo; NUMBERS</head>
+how it was proved.<note>Proclus, <I>Comm. on Rep. of Plato</I>, ed. Kroll, vol. ii, 1901, cc. 23 and
+27, pp. 24, 25, and 27-9.</note> Observing that &lsquo;it is proved by him
+(Euclid) graphically (<G>grammikw=s</G>) in the Second Book of the
+<FIG>
+Elements&rsquo;, Proclus adds the enunciation of Eucl. II. 10.
+This proposition proves that, if <I>AB</I> is bisected at <I>C</I> and pro-
+duced to <I>D</I>, then
+<MATH><I>AD</I><SUP>2</SUP>+<I>DB</I><SUP>2</SUP>=2<I>AC</I><SUP>2</SUP>+2<I>CD</I><SUP>2</SUP>;</MATH>
+and, if <MATH><I>AC</I>=<I>CB</I>=<I>x</I></MATH> and <MATH><I>BD</I>=<I>y</I></MATH>, this gives
+<MATH>(2<I>x</I>+<I>y</I>)<SUP>2</SUP>+<I>y</I><SUP>2</SUP>=2<I>x</I><SUP>2</SUP>+2(<I>x</I>+<I>y</I>)<SUP>2</SUP></MATH>,
+or <MATH>(2<I>x</I>+<I>y</I>)<SUP>2</SUP>-2(<I>x</I>+<I>y</I>)<SUP>2</SUP>=2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP></MATH>,
+which is the formula required.
+<p>We can of course prove the property of consecutive side-
+and diameter- numbers algebraically thus:
+<MATH><I>d<SUB>n</SUB></I><SUP>2</SUP>-2<I>a<SUB>n</SUB></I><SUP>2</SUP>=(2<I>a</I><SUB><I>n</I>-1</SUB>+<I>d</I><SUB><I>n</I>-1</SUB>)<SUP>2</SUP>-2(<I>a</I><SUB><I>n</I>-1</SUB>+<I>d</I><SUB><I>n</I>-1</SUB>)<SUP>2</SUP>
+=2<I>a</I><SUB><I>n</I>-1</SUB><SUP>2</SUP>-<I>d</I><SUB><I>n</I>-1</SUB><SUP>2</SUP>
+=-(<I>d</I><SUB><I>n</I>-1</SUB><SUP>2</SUP>-2<I>a</I><SUB><I>n</I>-1</SUB><SUP>2</SUP>)
+=+(<I>d</I><SUB><I>n</I>-2</SUB><SUP>2</SUP>-2<I>a</I><SUB><I>n</I>-2</SUB><SUP>2</SUP>)</MATH>, in like manner;
+and so on.
+<p>In the famous passage of the <I>Republic</I> (546 C) dealing with
+the geometrical number Plato distinguishes between the
+&lsquo;irrational diameter of 5&rsquo;, i.e. the diagonal of a square having
+5 for its side, or &radic;(50), and what he calls the &lsquo;rational
+diameter&rsquo; of 5. The square of the &lsquo;rational diameter&rsquo; is less
+by 1 than the square of the &lsquo;irrational diameter&rsquo;, and is there-
+fore 49, so that the &lsquo;rational diameter&rsquo; is 7; that is, Plato
+refers to the fact that <MATH>2.5<SUP>2</SUP>-7<SUP>2</SUP>=1</MATH>, and he has in mind the
+particular pair of side- and diameter- numbers, 5 and 7, which
+must therefore have been known before his time. As the proof
+of the property of these numbers in general is found, as Proclus
+says, in the geometrical theorem of Eucl. II. 10, it is a fair
+inference that that theorem is Pythagorean, and was prob-
+ably invented for the special purpose.
+<pb n=94><head>PYTHAGOREAN ARITHMETIC</head>
+<C>(<G>b</G>) <I>The</I> <G>e)pa/nqhma</G> (&lsquo;<I>bloom</I>&rsquo;) <I>of Thymaridas.</I></C>
+<p>Thymaridas of Paros, an ancient Pythagorean already
+mentioned (p. 69), was the author of a rule for solving a
+certain set of <I>n</I> simultaneous simple equations connecting <I>n</I>
+unknown quantities. The rule was evidently well known, for
+it was called by the special name of <G>e)pa/nqhm(a</G>, the &lsquo;flower&rsquo; or
+&lsquo;bloom&rsquo; of Thymaridas.<note>Iambl. <I>in Nicom.</I>, p. 62. 18 sq.</note> (The term <G>e)pa/nqhma</G> is not, how-
+ever, confined to the particular proposition now in question;
+Iamblichus speaks of <G>e)panqh/mata</G> of the <I>Introductio arith-
+metica</I>, &lsquo;arithmetical <G>e)panqh/mata</G>&rsquo; and <G>e)panqh/mata</G> of par-
+ticular numbers.) The rule is stated in general terms and no
+symbols are used, but the content is pure algebra. The known
+or determined quantities (<G>w(risme/non</G>) are distinguished from
+the undetermined or unknown (<G>a)o/riston</G>), the term for the
+latter being the very word used by Diophantus in the expres-
+sion <G>plh=qos mona/dwn a)o/riston</G>, &lsquo;an undefined or undetermined
+number of units&rsquo;, by which he describes his <G>a)riqmo/s</G> or un-
+known quantity (=<I>x</I>). The rule is very obscurely worded,
+but it states in effect that, if we have the following <I>n</I> equa-
+tions connecting <I>n</I> unknown quantities <I>x</I>, <I>x</I><SUB>1</SUB>, <I>x</I><SUB>2</SUB>...<I>x</I><SUB><I>n</I>-1</SUB>,
+namely
+<MATH><I>x</I>+<I>x</I><SUB>1</SUB>+<I>x</I><SUB>2</SUB>+...+<I>x</I><SUB><I>n</I>-1</SUB>=<I>s</I>,
+<I>x</I>+<I>x</I><SUB>1</SUB>=<I>a</I><SUB>1</SUB>,
+<I>x</I>+<I>x</I><SUB>2</SUB>=<I>a</I><SUB>2</SUB>
+. . . .
+<I>x</I>+<I>x</I><SUB><I>n</I>-1</SUB>=<I>a</I><SUB><I>n</I>-1</SUB></MATH>,
+the solution is given by
+<MATH><I>x</I>=((<I>a</I><SUB>1</SUB>+<I>a</I><SUB>2</SUB>+...+<I>a</I><SUB><I>n</I>-1</SUB>)-<I>s</I>)/(<I>n</I>-2)</MATH>.
+<p>Iamblichus, our informant on this subject, goes on to show
+that other types of equations can be reduced to this, so that
+the rule does not &lsquo;leave us in the lurch&rsquo; in those cases either.<note><I>Ib.</I>, p. 63. 16.</note>
+He gives as an instance the indeterminate problem represented
+by the following three linear equations between four unknown
+quantities:
+<MATH><I>x</I>+<I>y</I>=<I>a</I>(<I>z</I>+<I>u</I>),
+<I>x</I>+<I>z</I>=<I>b</I>(<I>u</I>+<I>y</I>),
+<I>x</I>+<I>u</I>=<I>c</I>(<I>y</I>+<I>z</I>)</MATH>.
+<pb n=95><head>THE <G>*)e*p*a*n*q*h*m*a</G> (&lsquo;BLOOM&rsquo;) OF THYMARIDAS</head>
+<p>From these equations we obtain
+<MATH><I>x</I>+<I>y</I>+<I>z</I>+<I>u</I>=(<I>a</I>+1)(<I>z</I>+<I>u</I>)=(<I>b</I>+1)(<I>u</I>+<I>y</I>)=(<I>c</I>+1)(<I>y</I>+<I>z</I>)</MATH>.
+<p>If now <I>x, y, z, u</I> are all to be integers, <I>x</I>+<I>y</I>+<I>z</I>+<I>u</I> must
+contain <MATH><I>a</I>+1, <I>b</I>+1, <I>c</I>+1</MATH> as factors. If <I>L</I> be the least common
+multiple of <MATH><I>a</I>+1, <I>b</I>+1, <I>c</I>+1</MATH>, we can put <MATH><I>x</I>+<I>y</I>+<I>z</I>+<I>u</I>=<I>L</I></MATH>, and
+we obtain from the above equations in pairs
+<MATH><I>x</I>+<I>y</I>=(<I>a</I>/(<I>a</I>+1))<I>L</I>,
+<I>x</I>+<I>z</I>=(<I>b</I>/(<I>b</I>+1))<I>L</I>,
+<I>x</I>+<I>u</I>=(<I>c</I>/(<I>c</I>+1))<I>L</I></MATH>,
+while <MATH><I>x</I>+<I>y</I>+<I>z</I>+<I>u</I>=<I>L</I></MATH>.
+<p>These equations are of the type to which Thymaridas's rule
+applies, and, since the number of unknown quantities (and
+equations) is 4, <I>n</I>-2 is in this case 2, and
+<MATH><I>x</I>=(<I>L</I>(<I>a</I>/(<I>a</I>+1)+<I>b</I>/(<I>b</I>+1)+<I>c</I>/(<I>c</I>+1))-<I>L</I>)/2</MATH>
+<p>The numerator is integral, but it may be an odd number, in
+which case, in order that <I>x</I> may be integral, we must take 2<I>L</I>
+instead of <I>L</I> as the value of <MATH><I>x</I>+<I>y</I>+<I>z</I>+<I>u</I></MATH>.
+<p>Iamblichus has the particular case where <I>a</I>=2, <I>b</I>=3, <I>c</I>=4.
+<I>L</I> is thus 3.4.5=60, and the numerator of the expression for
+<I>x</I> becomes 133-60, or 73, an odd number; he has therefore
+to put 2<I>L</I> or 120 in place of <I>L</I>, and so obtains <MATH><I>x</I>=73, <I>y</I>=7,
+<I>z</I>=17, <I>u</I>=23</MATH>.
+<p>Iamblichus goes on to apply the method to the equations
+<MATH><I>x</I>+<I>y</I>=3/2(<I>z</I>+<I>u</I>),
+<I>x</I>+<I>z</I>=4/3(<I>u</I>+<I>y</I>),
+<I>x</I>+<I>u</I>=5/4(<I>y</I>+<I>z</I>)</MATH>,
+<pb n=96><head>PYTHAGOREAN ARITHMETIC</head>
+which give
+<MATH><I>x</I>+<I>y</I>+<I>z</I>+<I>u</I>=5/2(<I>z</I>+<I>u</I>=7/3(<I>u</I>+<I>y</I>)=9/4(<I>y</I>+<I>z</I>)</MATH>.
+<p>Therefore
+<MATH><I>x</I>+<I>y</I>+<I>z</I>+<I>u</I>=5/3(<I>x</I>+<I>y</I>)=7/4(<I>x</I>+<I>z</I>)=9/5(<I>x</I>+<I>u</I>)</MATH>.
+<p>In this case we take <I>L</I>, the least common multiple of 5, 7, 9,
+or 315, and put
+<MATH><I>x</I>+<I>y</I>+<I>z</I>+<I>u</I>=<I>L</I>=315,
+<I>x</I>+<I>y</I>=3/5<I>L</I>=189,
+<I>x</I>+<I>z</I>=4/7<I>L</I>=180,
+<I>x</I>+<I>u</I>=5/9<I>L</I>=175</MATH>,
+whence <MATH><I>x</I>=(544-315)/2=229/2</MATH>.
+<p>In order that <I>x</I> may be integral, we have to take 2<I>L</I>, or 630,
+instead of <I>L</I>, or 315, and the solution is <MATH><I>x</I>=229, <I>y</I>=149,
+<I>z</I>=131, <I>u</I>=121</MATH>.
+<C>(<G>g</G>) <I>Area of rectangles in relation to perimeter</I>.</C>
+<p>Sluse,<note><I>&OElig;uvres compl&egrave;tes de C. Huygens</I>, pp. 64, 260.</note> in letters to Huygens dated Oct. 4, 1657, and Oct. 25,
+1658, alludes to a property of the numbers 16 and 18 of
+which he had read somewhere in Plutarch that it was known
+to the Pythagoreans, namely that each of these numbers
+represents the perimeter as well as the area of a rectangle;
+for 4.4=2.4+2.4 and 3.6=2.3+2.6. I have not found the
+passage of Plutarch, but the property of 16 is mentioned in the
+<I>Theologumena Arithmetices</I>, where it is said that 16 is the only
+square the area of which is equal to its perimeter, the peri-
+meter of smaller squares being greater, and that of all larger
+squares being less, than the area.<note><I>Theol. Ar.</I>, pp. 10, 23 (Ast).</note> We do not know whether
+the Pythagoreans proved that 16 and 18 were the only num-
+bers having the property in question; but it is likely enough
+that they did, for the proof amounts to finding the integral
+<pb n=97><head>TREATISES ON ARITHMETIC</head>
+solutions of <MATH><I>xy</I>=2(<I>x</I>+<I>y</I>)</MATH>. This is easy, for the equation is
+equivalent to <MATH>(<I>x</I>-2)(<I>y</I>-2)=4</MATH>, and we have only to equate
+<I>x</I>-2 and <I>y</I>-2 to the respective factors of 4. Since 4 is only
+divisible into integral factors in two ways, as 2.2 or as 1.4,
+we get, as the only possible solutions for <I>x, y</I>, (4, 4) or (3, 6).
+<C>Systematic treatises on arithmetic (theory of
+numbers).</C>
+<p>It will be convenient to include in this chapter some
+account of the arithmetic of the later Pythagoreans, begin-
+ning with NICOMACHUS. If any systematic treatises on
+arithmetic were ever written between Euclid (Books VII-IX)
+and Nicomachus, none have survived. Nicomachus, of
+Gerasa, probably the Gerasa in Judaea east of the river
+Jordan, flourished about 100 A.D., for, on the one hand, in
+a work of his entitled the <I>Enchiridion Harmonices</I> there is
+an allusion to Thrasyllus, who arranged the Platonic dialogues,
+wrote on music, and was the astrologer-friend of Tiberius; on
+the other hand, the <I>Introductio Arithmetica</I> of Nicomachus
+was translated into Latin by Apuleius of Madaura under the
+Antonines. Besides the <G>*)ariqmhtikh\ ei)sagwgh/</G>, Nicomachus
+is said to have written another treatise on the theology or the
+mystic properties of numbers, called <G>*qeologou/mena a)riqmh-
+tikh=s</G>, in two Books. The curious farrago which has come
+down to us under that title and which was edited by Ast<note><I>Theologumena arithmeticae. Accedit Nicomachi Geraseni Institutio
+arithmetica</I>, ed. Ast, Leipzig, 1817.</note> is,
+however, certainly not by Nicomachus; for among the authors
+from whom it gives extracts is Anatolius, Bishop of Laodicaea
+(A.D. 270); but it contains quotations from Nicomachus which
+appear to come from the genuine work. It is possible that
+Nicomachus also wrote an <I>Introduction to Geometry</I>, since in
+one place he says, with regard to certain solid numbers, that
+they have been specially treated &lsquo;in the geometrical intro-
+duction, being more appropriate to the theory of magnitude&rsquo;<note>Nicom. <I>Arithm</I>. ii. 6. 1.</note>;
+but this geometrical introduction may not necessarily have
+been a work of his own.
+<p>It is a very far cry from Euclid to Nicomachus. In the
+<pb n=98><head>PYTHAGOREAN ARITHMETIC</head>
+<I>Introductio arithmetica</I> we find the form of exposition
+entirely changed. Numbers are represented in Euclid by
+straight lines with letters attached, a system which has the
+advantage that, as in algebraical notation, we can work with
+numbers in general without the necessity of giving them
+specific values; in Nicomachus numbers are no longer de-
+noted by straight lines, so that, when different undetermined
+numbers have to be distinguished, this has to be done by
+circumlocution, which makes the propositions cumbrous and
+hard to follow, and it is necessary, after each proposition
+has been stated, to illustrate it by examples in concrete
+numbers. Further, there are no longer any proofs in the
+proper sense of the word; when a general proposition has been
+enunciated, Nicomachus regards it as sufficient to show that
+it is true in particular instances; sometimes we are left to
+infer the general proposition by induction from particular
+cases which are alone given. Occasionally the author makes
+a quite absurd remark through failure to distinguish between
+the general and the particular case, as when, after he has
+defined the mean which is &lsquo;subcontrary to the harmonic&rsquo; as
+being determined by the relation <MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>c</I>/<I>a</I></MATH>, where <MATH><I>a</I>><I>b</I>><I>c</I></MATH>,
+and has given 6, 5, 3 as an illustration, he goes on to observe
+that it is a property peculiar to this mean that the product of
+the greatest and middle terms is double of the product of the
+middle and least,<note>Nicom. ii. 28. 3.</note> simply because this happens to be true in
+the particular case! Probably Nicomachus, who was not
+really a mathematician, intended his <I>Introduction</I> to be, not
+a scientific treatise, but a popular treatment of the subject
+calculated to awaken in the beginner an interest in the theory
+of numbers by making him acquainted with the most note-
+worthy results obtained up to date; for proofs of most of his
+propositions he could refer to Euclid and doubtless to other
+treatises now lost. The style of the book confirms this hypo-
+thesis; it is rhetorical and highly coloured; the properties of
+numbers are made to appear marvellous and even miraculous;
+the most obvious relations between them are stated in turgid
+language very tiresome to read. It was the mystic rather
+than the mathematical side of the theory of numbers that
+<pb n=99><head>NICOMACHUS</head>
+interested Nicomachus. If the verbiage is eliminated, the
+mathematical content can be stated in quite a small com-
+pass. Little or nothing in the book is original, and, except
+for certain definitions and refinements of classification, the
+essence of it evidently goes back to the early Pythagoreans.
+Its success is difficult to explain except on the hypothesis that
+it was at first read by philosophers rather than mathemati-
+cians (Pappus evidently despised it), and afterwards became
+generally popular at a time when there were no mathemati-
+cians left, but only philosophers who incidentally took an
+interest in mathematics. But a success it undoubtedly was;
+this is proved by the number of versions or commentaries
+which appeared in ancient times. Besides the Latin transla-
+tion by Apuleius of Madaura (born about A.D. 125), of which
+no trace remains, there was the version of Bo&euml;tius (born about
+480, died 524 A.D.); and the commentators include Iamblichus
+(fourth century), Heronas,<note><I>v.</I> Eutoc. <I>in Archim.</I> (ed. Heib. iii, p. 120. 22).</note> Asclepius of Tralles (sixth century),
+Joannes Philoponus, Proclus.<note><I>v.</I> Suidas.</note> The commentary of Iamblichus
+has been published,<note>The latest edition is Pistelli's (Teubner, 1894).</note> as also that of Philoponus,<note>Ed. Hoche, Heft 1, Leipzig, 1864, Heft 2, Berlin, 1867.</note> while that of
+Asclepius is said to be extant in MSS. When (the pseudo-)
+Lucian in his <I>Philopatris</I> (c. 12) makes Critias say to Triephon
+&lsquo;you calculate like Nicomachus&rsquo;, we have an indication that
+the book was well known, although the remark may be less a
+compliment than a laugh at Pythagorean subtleties.<note>Triephon tells Critias to swear by the Trinity (&lsquo;One (proceeding) from
+Three and Three from One&rsquo;), and Critias replies, &lsquo;You would have me
+learn to calculate, for your oath is mere arithmetic and you calculate
+like Nicomachus of Gerasa. I do not know what you mean by your
+&ldquo;One-Three and Three-One&rdquo;; I suppose you don't mean the <G>tetraktu/s</G>
+of Pythagoras or the <G>o)gdoa/s</G> or the <G>triaka/s</G>?&rsquo;</note>
+<p>Book I of the <I>Introductio</I>, after a philosophical prelude
+(cc. 1-6), consists principally of definitions and laws of forma-
+tion. Numbers, odd and even, are first dealt with (c. 7); then
+comes the subdivision of even into three kinds (1) evenly-even,
+of the form 2<SUP><I>n</I></SUP>, (2) even-odd, of the form 2(2<I>n</I>+1), and (3)
+odd-even, of the form 2<SUP><I>m</I>+1</SUP>(2<I>n</I>+1), the last-named occupying
+a sort of intermediate position in that it partakes of the
+character of both the others. The odd is next divided into
+three kinds: (1) &lsquo;prime and incomposite&rsquo;, (2) &lsquo;secondary and
+<pb n=100><head>PYTHAGOREAN ARITHMETIC</head>
+composite&rsquo;, a product of prime factors (excluding 2, which is
+even and not regarded as prime), and (3) &lsquo;that which is in itself
+secondary and composite but in relation to another is prime and
+incomposite&rsquo;, e.g. 9 in relation to 25, which again is a sort of
+intermediate class between the two others (cc. 11-13); the
+defects of this classification have already been noted (pp. 73-4).
+In c. 13 we have these different classes of odd numbers ex-
+hibited in a description of Eratosthenes's &lsquo;sieve&rsquo; (<G>ko/skinon</G>), an
+appropriately named device for finding prime numbers. The
+method is this. We set out the series of odd numbers begin-
+ning from 3.
+3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, ......
+<p>Now 3 is a prime number, but multiples of 3 are not; these
+multiples, 9, 15 ... are got by passing over two numbers at
+a time beginning from 3; we therefore strike out these num-
+bers as not being prime. Similarly 5 is a prime number, but
+by passing over four numbers at a time, beginning from 5, we
+get multiples of 5, namely 15, 25 ...; we accordingly strike
+out all these multiples of 5. In general, if <I>n</I> be a prime num-
+ber, its multiples appearing in the series are found by passing
+over <I>n</I>-1 terms at a time, beginning from <I>n</I>; and we can
+strike out all these multiples. When we have gone far enough
+with this process, the numbers which are still left will be
+primes. Clearly, however, in order to make sure that the
+odd number 2<I>n</I>+1 in the series is prime, we should have to
+try all the prime divisors between 3 and &radic;(2<I>n</I>+1); it is
+obvious, therefore, that this primitive empirical method would
+be hopeless as a practical means of obtaining prime numbers
+of any considerable size.
+<p>The same c. 13 contains the rule for finding whether two
+given numbers are prime to one another; it is the method of
+Eucl. VII. 1, equivalent to our rule for finding the greatest
+common measure, but Nicomachus expresses the whole thing
+in words, making no use of any straight lines or symbols to
+represent the numbers. If there is a common measure greater
+than unity, the process gives it; if there is none, i.e. if 1 is
+left as the last remainder, the numbers are prime to one
+another.
+<p>The next chapters (cc. 14-16) are on <I>over-perfect</I> (<G>u(pertelh/s</G>),
+<pb n=101><head>NICOMACHUS</head>
+<I>deficient</I> (<G>e)lliph/s</G>), and <I>perfect</I> (<G>te/leios</G>) numbers respectively.
+The definitions, the law of formation of perfect numbers,
+and Nicomachus's observations thereon have been given above
+(p. 74).
+<p>Next comes (cc. 17-23) the elaborate classification of
+numerical ratios greater than unity, with their counterparts
+which are less than unity. There are five categories of each,
+and under each category there is (<I>a</I>) the general name, (<I>b</I>) the
+particular names corresponding to the particular numbers
+taken.
+<p>The enumeration is tedious, but, for purposes of reference,
+is given in the following table:&mdash;
+<table width=100%>
+<tr><th colspan=2 width=60%>RATIOS GREATER THAN UNITY</th><th colspan=2 width=40%>RATIOS LESS THAN UNITY</th></tr>
+<tr><td width=20%>1. (a)</td><td width=30%>General</td><td width=20%>1. (a)</td><td width=30%>General</td></tr>
+<tr><td></td><td><G>pollapla/sios</G>, multiple</td><td></td><td><G>u(popollapla/sios</G>, submultiple</td></tr>
+<tr><td></td><td>(multiplex)</td><td></td><td>(submultiplex)</td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;&nbsp;(b)</td><td>Particular</td><td>&nbsp;&nbsp;&nbsp;&nbsp;(b)</td><td>Particular</td></tr>
+<tr><td></td><td><G>dipla/sios</G>, double</td><td></td><td><G>u(podipla/sios</G>, one half</td></tr>
+<tr><td></td><td>(duplus)</td><td></td><td>(subduplus)</td></tr>
+<tr><td></td><td><G>tripla/sios</G>, triple</td><td></td><td><G>u(potripla/sios</G>, one third</td></tr>
+<tr><td></td><td>(triplus)</td><td></td><td>(subtriplus)</td></tr>
+<tr><td></td><td>&nbsp;&nbsp;&amp;c.</td><td></td><td>&nbsp;&nbsp;&amp;c.</td></tr>
+<tr><td>2. (a)</td><td>General</td><td>2. (a)</td><td>General</td></tr>
+<tr><td></td><td><BRACE><LABLE>a number which is of the form 1+1/<I>n</I> or (<I>n</I>+1)/<I>n</I>, where <I>n</I> is any integer.</LABLE><G>e)pimo/rios</G> (superparticularis)</BRACE></td><td></td><td><BRACE><LABLE>the fraction <I>n</I>/(<I>n</I>+1),
+where <I>n</I> is any integer.</LABLE><G>u(pepimo/rios</G> (subsuperparticularis)</BRACE></td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;&nbsp;(b)</td><td>Particular</td><td>&nbsp;&nbsp;&nbsp;&nbsp;(b)</td><td>Particular</td></tr>
+<tr><td></td><td>According to the value of</td><td></td><td><G>u(fhmio/lios</G> =2/3</td></tr>
+<tr><td></td><td align=center><I>n</I>, we have the names</td><td></td><td>(subsesquialter)</td></tr>
+<tr><td></td><td><G>h(mio/lios</G> =1 1/2</td><td></td><td><G>u(pepi/tritos</G> =3/4</td></tr>
+<tr><td></td><td>(sesquialter)</td><td></td><td>(subsesquitertius)</td></tr>
+<tr><td></td><td><G>e)pi/tritos</G>) =1 1/3</td><td></td><td><G>u(pepite/tartos</G> =4/5</td></tr>
+<tr><td></td><td>(sesquitertius)</td><td></td><td>(subsesquiquartus)</td></tr>
+<tr><td></td><td><G>e)pite/tartos</G> =1 1/4</td><td></td><td>&nbsp;&nbsp;&amp;c.</td></tr>
+<tr><td></td><td>(sesquiquartus)</td><td></td><td></td></tr>
+<tr><td></td><td>&nbsp;&nbsp;&amp;c.</td><td></td><td></td></tr>
+</table>
+<pb n=102><head>PYTHAGOREAN ARITHMETIC</head>
+<table>
+<tr><th colspan=2>RATIOS GREATER THAN UNITY&nbsp;</th><th colspan=2>RATIOS LESS THAN UNITY</th></tr>
+<tr><td>3. (a) General</td><td>3. (a) General</td></tr>
+<tr><td align=right><BRACE><LABLE>which exceeds 1 by twice, thrice, or more times a submultiple, and which therefore may be represented by 1+<I>m</I>/(<I>m</I>+<I>n</I>) or (2<I>m</I>+<I>n</I>)/(<I>m</I>+<I>n</I>).</LABLE><G>e)pimerh/s</G>
+(superpartiens)</BRACE></td><td align=right><BRACE><LABLE>which is of the form (<I>m</I>+<I>n</I>)/(2<I>m</I>+<I>n</I>).</LABLE><G>u(pepimerh/s</G> (subsuperpartiens)</BRACE></td></tr>
+<tr><td>&nbsp;&nbsp;(b) Particular</td><td></td></tr>
+<tr><td>The formation of the names for the series of particular <I>superpartientes</I> follows three different plans.</td><td></td></tr>
+<tr><td>Thus, of numbers of the form 1+<I>m</I>/(<I>m</I>+1),</td><td></td></tr>
+<tr><td align=center><MATH><BRACE><note>1 2/3</note><G>e)pidimerh/s</G> (superbipartiens) or <G>e)pidi/tritos</G> (superbitertius) or <G>disepi/tritos</G></BRACE></MATH></td><td>The corresponding names are not specified in Nicomachus.</td></tr>
+<tr><td align=center><MATH><BRACE><note>1 3/4</note><G>e)pitrimerh/s</G> (supertripartiens) or <G>e)pitrite/tartos</G> (supertriquartus) or <G>trisepite/tartos</G></BRACE></MATH></td><td></td></tr>
+<tr><td align=center><MATH><BRACE><note>1 4/5 is</note><G>e)pitetramerh/s</G> (superquadripartiens) or <G>e)pitetra/pemptos</G> (superquadriquintus) or <G>tetrakisepi/pemptos</G></BRACE></MATH></td><td></td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;&amp;c.</td><td></td></tr>
+<tr><td>As regards the first name in each case we note that, with <G>e)pidimerh/s</G> we must understand <G>tri/twn</G>; with <G>e)pitrimerh/s</G>, <G>teta/rtwn</G>, and so on.</td><td></td></tr>
+</table>
+<pb n=103><head>NICOMACHUS</head>
+<table width=100%>
+<tr><th width=30%>RATIOS GREATER THAN UNITY</th><th width=70%>RATIOS LESS THAN UNITY</th></tr>
+<tr><td>Where the more general form 1+<I>m</I>/(<I>m</I>+<I>n</I>), instead of 1+<I>m</I>/(<I>m</I>+1), has to be expressed, Nicomachus uses terms following the <I>third</I> plan of formation above, e.g.</td><td></td></tr>
+<tr><td align=center>1 3/5=<G>trisepi/pemptos</G></td><td></td></tr>
+<tr><td align=center>1 4/7=<G>tetrakisefe/bdomos</G></td><td></td></tr>
+<tr><td align=center>1 5/9=<G>pentakisepe/natos</G></td><td></td></tr>
+<tr><td>and so on, although he might have used the second and called these ratios <G>e)pitri/pemptos</G>, &amp;c.</td><td></td></tr>
+<tr><td width=30%>4. (a) General</td><td width=70% align=center>4. (a) General</td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;<G>pollaplasiepimo/rios</G></td><td align=center>&nbsp;&nbsp;&nbsp;<G>u(popollaplasiepimo/rios</G></td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;(multiplex superparticularis)</td><td align=center>&nbsp;&nbsp;&nbsp;(submultiplex superparticularis)</td></tr>
+<tr><td>This contains a certain <I>multiple</I> plus a certain submultiple (instead of 1 plus a submultiple) and is therefore of the form <I>m</I>+1/<I>n</I> (instead of the 1+1/<I>n</I> of the <G>e)pimo/rios</G>) or
+(<I>mn</I>+1)/<I>n</I>.</td><td align=center>of the form <I>n</I>/(<I>mn</I>+1).</td></tr>
+<tr><td>&nbsp;&nbsp;(b) Particular</td><td></td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;2 1/2=<G>diplasiefh/misus</G></td><td align=center>&nbsp;&nbsp;The corresponding particular</td></tr>
+<tr><td align=right>(duplex sesquialter)</td><td align=center>names do not seem to occur in</td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;2 1/3=<G>diplasiepi/tritos</G></td><td align=center>Nicomachus, but Bo&euml;tius has</td></tr>
+<tr><td align=right>(duplex sesquitertius)&nbsp;</td><td align=center>them, e.g. subduplex sesquialter,</td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;3 1/5=<G>triplasiepi/pemptos</G></td><td align=center>subduplex sesquiquartus.</td></tr>
+<tr><td align=right>(triplex sesquiquintus)&nbsp;</td><td></td></tr>
+<tr><td>&nbsp;&nbsp;&amp;c.</td><td></td></tr>
+<tr><td>5. (a) General</td><td align=center>5. (a) General</td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;<G>pollaplasiepimerh/s</G></td><td align=center>&nbsp;&nbsp;&nbsp;<G>u(popollaplasiepimerh/s</G></td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;(multiplex superpartiens).</td><td align=center>(submultiplex superpartiens),</td></tr>
+<tr><td>This is related to <G>e)pimerh/s</G> [(3) above] in the same way as <G>pollaplasiepimo/rios</G> to <G>e)pimo/rios</G>; that is to say, it is of the form <I>p</I>+<I>m</I>/(<I>m</I>+<I>n</I>) or
+((<I>p</I>+1)<I>m</I>+<I>n</I>)/(<I>m</I>+<I>n</I>).</td><td align=center>a fraction of the form (<I>m</I>+<I>n</I>)/((<I>p</I>+1)<I>m</I>+<I>n</I>).</td></tr>
+</table>
+<pb n=104><head>PYTHAGOREAN ARITHMETIC</head>
+<table>
+<tr><th align=right>RATIOS GREATER THAN UNITY&nbsp;</th><th align=center>RATIOS LESS THAN UNITY</th></tr>
+<tr><td>&nbsp;&nbsp;(b) Particular</td><td></td></tr>
+<tr><td>&nbsp;&nbsp;These names are only given for cases where <I>n</I>=1; they follow the first form of the names for particular <G>e)pimerei=s</G>, e.g.</td>
+<td>&nbsp;&nbsp;Corresponding names not found in Nicomachus; but Bo&euml;tius has <I>subduplex superbipartiens</I>,
+&amp;c.</td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;2 2/3=<G>diplasiepidimerh/s</G></td><td></td></tr>
+<tr><td align=right>(duplex superbipartiens)</td><td></td></tr>
+<tr><td>&nbsp;&nbsp;&nbsp;&amp;c.</td><td></td></tr>
+</table>
+<p>In c. 23 Nicomachus shows how these various ratios can be
+got from one another by means of a certain rule. Suppose
+that
+<I>a, b, c</I>
+are three numbers such that <I>a</I>:<I>b</I>=<I>b</I>:<I>c</I>=one of the ratios
+described; we form the three numbers
+<MATH><I>a, a</I>+<I>b, a</I>+2<I>b</I>+<I>c</I></MATH>
+and also the three numbers
+<MATH><I>c, c</I>+<I>b</I>, <I>c</I>+2<I>b</I>+<I>a</I></MATH>
+Two illustrations may be given. If <I>a</I>=<I>b</I>=<I>c</I>=1, repeated
+application of the first formula gives (1, 2, 4), then (1, 3, 9),
+then (1, 4, 16), and so on, showing the successive multiples.
+Applying the second formula to (1, 2, 4), we get (4, 6, 9) where
+the ratio is 3/2; similarly from (1, 3, 9) we get (9, 12, 16) where
+the ratio is 4/3, and so on; that is, from the <G>pollapla/sioi</G> we
+get the <G>e)pimo/rioi</G>. Again from (9, 6, 4), where the ratio is
+of the latter kind, we get by the first formula (9, 15, 25),
+giving the ratio 1 2/3, an <G>e)pimerh/s</G>, and by the second formula
+(4, 10, 25), giving the ratio 2 1/2, a <G>pollaplasiepimo/rios</G>. And
+so on.
+<p>Book II begins with two chapters showing how, by a con-
+verse process, three terms in continued proportion with any
+one of the above forms as common ratio can be reduced to
+three equal terms. If
+<I>a, b, c</I>
+<pb n=105><head>NICOMACHUS</head>
+are the original terms, <I>a</I> being the smallest, we take three
+terms of the form
+<MATH><I>a, b</I>-<I>a</I>, {<I>c</I>-<I>a</I>-2(<I>b</I>-<I>a</I>)}=<I>c</I>+<I>a</I>-2<I>b</I></MATH>,
+then apply the same rule to these three, and so on.
+<p>In cc. 3-4 it is pointed out that, if
+<MATH>1, <I>r</I>, <I>r</I><SUP>2</SUP>..., <I>r</I><SUP><I>n</I></SUP>...</MATH>
+be a geometrical progression, and if
+<MATH><G>r</G><SUB><I>n</I></SUB>=<I>r</I><SUP><I>n</I>-1</SUP>+<I>r</I><SUP><I>n</I></SUP></MATH>,
+then <MATH><G>r</G><SUB><I>n</I></SUB>/<I>r</I><SUP><I>n</I></SUP>=(<I>r</I>+1)/<I>r</I></MATH>, an <G>e)pimo/rios</G> ratio,
+and similarly, if <MATH><G>r/</G><SUB><I>n</I></SUB>=<G>r</G><SUB><I>n</I>-1</SUB>+<G>r</G><SUB><I>n</I></SUB>,
+<G>r/</G><SUB><I>n</I></SUB>/<G>r</G><SUB><I>n</I></SUB>=(<I>r</I>+1)/<I>r</I></MATH>;
+and so on.
+<p>If we set out in rows numbers formed in this way,
+<table>
+<tr><td>1,</td><td><I>r</I>,</td><td><I>r</I><SUP>2</SUP>,</td><td><I>r</I><SUP>3</SUP>...</td><td><I>r</I><SUP><I>n</I></SUP></td></tr>
+<tr><td></td><td><I>r</I>+1,</td><td><I>r</I><SUP>2</SUP>+<I>r</I>,</td><td><I>r</I><SUP>3</SUP>+<I>r</I><SUP>2</SUP>...</td><td><I>r</I><SUP><I>n</I></SUP>+<I>r</I><SUP><I>n</I>-1</SUP></td></tr>
+<tr><td></td><td></td><td><I>r</I><SUP>2</SUP>+2<I>r</I>+1,</td><td><I>r</I><SUP>3</SUP>+2<I>r</I><SUP>2</SUP>+<I>r</I>...</td><td><I>r</I><SUP><I>n</I></SUP>+2<I>r</I><SUP><I>n</I>-1</SUP>+<I>r</I><SUP><I>n</I>-2</SUP></td></tr>
+<tr><td></td><td></td><td></td><td><I>r</I><SUP>3</SUP>+3<I>r</I><SUP>2</SUP>+3<I>r</I>+1...</td><td><I>r</I><SUP><I>n</I></SUP>+3<I>r</I><SUP><I>n</I>-1</SUP>+3<I>r</I><SUP><I>n</I>-2</SUP>+<I>r</I><SUP><I>n</I>-3</SUP></td></tr>
+<tr><td></td><td></td><td></td><td></td><td align=center>.</td></tr>
+<tr><td></td><td></td><td></td><td></td><td align=center>.</td></tr>
+<tr><td></td><td></td><td></td><td></td><td align=center>.</td></tr>
+<tr><td></td><td></td><td></td><td></td><td><I>r</I><SUP><I>n</I></SUP>+<I>nr</I><SUP><I>n</I>-1</SUP>+(<I>n</I>(<I>n</I>-1))/2<I>r</I><SUP><I>n</I>-2</SUP>+...+1,</td></tr>
+</table>
+the vertical rows are successive numbers in the ratio <I>r</I>/(<I>r</I>+1),
+while diagonally we have the geometrical series 1, <I>r</I>+1,
+(<I>r</I>+1)<SUP>2</SUP>, (<I>r</I>+1)<SUP>3</SUP>....
+<p>Next follows the theory of polygonal numbers. It is pre-
+faced by an explanation of the quasi-geometrical way of
+representing numbers by means of dots or <I>a</I>'s. Any number
+from 2 onwards can be represented as a <I>line</I>; the <I>plane</I> num-
+bers begin with 3, which is the first number that can be
+represented in the form of a <I>triangle</I>; after triangles follow
+squares, pentagons, hexagons, &amp;c. (c. 7). Triangles (c. 8) arise
+by adding any number of successive terms, beginning with 1,
+of the series of natural numbers
+1, 2, 3, ... <I>n</I>, ....
+<pb n=106><head>PYTHAGOREAN ARITHMETIC</head>
+The <I>gnomons</I> of triangles are therefore the successive natural
+numbers. Squares (c. 9) are obtained by adding any number
+of successive terms of the series of odd numbers, beginning
+with 1, or
+<MATH>1, 3, 5, ...2<I>n</I>-1,....</MATH>
+The <I>gnomons</I> of squares are the successive odd numbers.
+Similarly the <I>gnomons</I> of pentagonal numbers (c. 10) are the
+numbers forming an arithmetical progression with 3 as com-
+mon difference, or
+<MATH>1, 4, 7, ... 1+(<I>n</I>-1) 3, ...</MATH>;
+and generally (c. 11) the gnomons of polygonal numbers of <I>a</I>
+sides are
+<MATH>1, 1+(<I>a</I>-2), 1+2(<I>a</I>-2),...1+(<I>r</I>-1)(<I>a</I>-2),...</MATH>
+and the <I>a</I>-gonal number with side <I>n</I> is
+<MATH>1+1+(<I>a</I>-2)+1+2(<I>a</I>-2)+...+1+(<I>n</I>-1)(<I>a</I>-2)
+=<I>n</I>+1/2<I>n</I>(<I>n</I>-1)(<I>a</I>-2)</MATH>
+The general formula is not given by Nicomachus, who con-
+tents himself with writing down a certain number of poly-
+gonal numbers of each species up to heptagons.
+<p>After mentioning (c. 12) that any square is the sum of two
+successive triangular numbers, i.e.
+<MATH><I>n</I><SUP>2</SUP>=1/2(<I>n</I>-1)<I>n</I>+1/2<I>n</I>(<I>n</I>+1)</MATH>,
+and that an <I>a</I>-gonal number of side <I>n</I> is the sum of an
+(<I>a</I>-1)-gonal number of side <I>n</I> plus a triangular number of
+side <I>n</I>-1, i.e.
+<MATH><I>n</I>+1/2<I>n</I>(<I>n</I>-1)(<I>a</I>-2)=<I>n</I>+1/2<I>n</I>(<I>n</I>-1)(<I>a</I>-3)+1/2<I>n</I>(<I>n</I>-1)</MATH>,
+he passes (c. 13) to the first <I>solid</I> number, the <I>pyramid.</I> The
+base of the pyramid may be a triangular, a square, or any
+polygonal number. If the base has the side <I>n</I>, the pyramid is
+formed by similar and similarly situated polygons placed
+successively upon it, each of which has 1 less in its side than
+that which precedes it; it ends of course in a unit at the top,
+the unit being &lsquo;potentially&rsquo; any polygonal number. Nico-
+machus mentions the first triangular pyramids as being 1, 4,
+10, 20, 35, 56, 84, and (c. 14) explains the formation of the
+series of pyramids with square bases, but he gives no general
+<pb n=107><head>NICOMACHUS</head>
+formula or summation. An <I>a</I>-gonal number with <I>n</I> in its
+side being
+<MATH><I>n</I>+1/2<I>n</I>(<I>n</I>-1)(<I>a</I>-2)</MATH>,
+it follows that the pyramid with that polygonal number for
+base is
+<MATH>1+2+3+...+<I>n</I>+1/2(<I>a</I>-2){1.2+2.3+...+(<I>n</I>-1)<I>n</I>}
+=(<I>n</I>(<I>n</I>+1))/2+(<I>a</I>-2)/2.((<I>n</I>-1)<I>n</I>(<I>n</I>+1))/3</MATH>.
+<p>A pyramid is <G>ko/louros</G>, <I>truncated</I>, when the unit is cut off
+the top, <G>diko/louros</G>, <I>twice-truncated</I>, when the unit and the
+next layer is cut off, <G>triko/louros</G>, <I>thrice-truncated</I>, when three
+layers are cut off, and so on (c. 14).
+<p>Other solid numbers are then classified (cc. 15-17): <I>cubes</I>,
+which are the product of three equal numbers; <I>scalene</I> num-
+bers, which are the product of three numbers all unequal,
+and which are alternatively called <I>wedges</I> (<G>sfhni/skoi</G>), <I>stakes</I>
+(<G>sfhki/skoi</G>), or <I>altars</I> (<G>bwmi/skoi</G>). The latter three names are
+in reality inappropriate to mere products of three unequal
+factors, since the figure which could properly be called by
+these names should <I>taper</I>, i.e. should have the plane face at
+the top less than the base. We shall find when we come to
+the chapter on Heron's mensuration that true (geometrical)
+<G>bwmi/skoi</G> and <G>sfhni/skoi</G> have there to be measured in which
+the top rectangular face is in fact smaller than the rectangular
+base parallel to it. Iamblichus too indicates the true nature
+of <G>bwmi/skoi</G> and <G>sfhni/skoi</G> when he says that they have not
+only their dimensions but also their faces and angles unequal,
+and that, while the <G>plinqi/s</G> or <G>doki/s</G> corresponds to the paral-
+lelogram, the <G>sfhni/skos</G> corresponds to the trapezium.<note>Iambl. <I>in Nicom.</I>, p. 93. 18, 94. 1-3.</note> The
+use, therefore, of the terms in question as alternatives to <I>scalene</I>
+appears to be due to a misapprehension. Other varieties of
+solid numbers are <I>parallelepipeds</I>, in which there are faces
+which are <G>e(teromh/keis</G> (oblong) or of the form <I>n</I>(<I>n</I>+1), so
+that two factors differ by unity; <I>beams</I> (<G>doki/des</G>) or <I>columns</I>
+(<G>sthli/des</G>, Iamblichus) of the form <I>m</I><SUP>2</SUP>(<I>m</I>+<I>n</I>); <I>tiles</I> (<G>plinqi/des</G>)
+of the form <I>m</I><SUP>2</SUP>(<I>m</I>-<I>n</I>). Cubes, the last digit (the units) of
+which are the same as the last digit in the side, are <I>spherical</I>
+<pb n=108><head>PYTHAGOREAN ARITHMETIC</head>
+(<G>sfairikoi/</G>) or <I>recurring</I> (<G>a)pokatastatikoi/</G>); these sides and
+cubes end in 1, 5, or 6, and, as the squares end in the same
+digits, the squares are called <I>circular</I> (<G>kuklikoi/</G>).
+<p><I>Oblong</I> numbers (<G>e(teromh/keis</G>) are, as we have seen, of the
+form <I>m</I>(<I>m</I>+1); <I>prolate</I> numbers (<G>promh/keis</G>) of the form
+<I>m</I>(<I>m</I>+<I>n</I>) where <I>n</I>>1 (c. 18). Some simple relations between
+oblong numbers, squares, and triangular numbers are given
+(cc. 19-20). If <I>h</I><SUB><I>n</I></SUB> represents the oblong number <I>n</I>(<I>n</I>+1), and
+<I>t</I><SUB><I>n</I></SUB> the triangular number 1/2<I>n</I>(<I>n</I>+1) of side <I>n</I>, we have, for
+example,
+<MATH><I>h</I><SUB><I>n</I></SUB>/<I>n</I><SUP>2</SUP>=(<I>n</I>+1)/<I>n</I>, <I>h</I><SUB><I>n</I></SUB>-<I>n</I><SUP>2</SUP>=<I>n</I>, <I>n</I><SUP>2</SUP>/<I>h</I><SUB><I>n</I>-1</SUB>=<I>n</I>/(<I>n</I>-1),
+<I>n</I><SUP>2</SUP>/<I>h</I><SUB><I>n</I></SUB>=<I>h</I><SUB><I>n</I></SUB>/(<I>n</I>+1)<SUP>2</SUP>, <I>n</I><SUP>2</SUP>+(<I>n</I>+1)<SUP>2</SUP>+2<I>h</I><SUB><I>n</I></SUB>=(2<I>n</I>+1)<SUP>2</SUP>,
+<I>n</I><SUP>2</SUP>+<I>h</I><SUB><I>n</I></SUB>=<I>t</I><SUB>2<I>n</I></SUB>, <I>h</I><SUB><I>n</I></SUB>+(<I>n</I>+1)<SUP>2</SUP>=<I>t</I><SUB>2<I>n</I>+1</SUB></MATH>,
+<MATH><I>n</I><SUP>2</SUP>&plusmn;<I>n</I>=<BRACE><I>h</I><SUB><I>n</I></SUB>
+<I>h</I><SUB><I>n</I>-1</SUB></BRACE></MATH>,
+all of which formulae are easily verified.
+<C><I>Sum of series of cube numbers.</I></C>
+<p>C. 20 ends with an interesting statement about cubes. If,
+says Nicomachus, we set out the series of odd numbers
+1, 3, 5, 7, 9, 11, 13, 15, 17, 19,...
+the first (1) is a cube, the sum of the next <I>two</I> (3+5) is a
+cube, the sum of the next <I>three</I> (7+9+11) is a cube, and so on.
+We can prove this law by assuming that <I>n</I><SUP>3</SUP> is equal to the
+sum of <I>n</I> odd numbers beginning with 2<I>x</I>+1 and ending
+with 2<I>x</I>+2<I>n</I>-1. The sum is (2<I>x</I>+<I>n</I>)<I>n</I>; since therefore
+<MATH>(2<I>x</I>+<I>n</I>)<I>n</I>=<I>n</I><SUP>3</SUP></MATH>,
+<MATH><I>x</I>=1/2(<I>n</I><SUP>2</SUP>-<I>n</I>)</MATH>,
+and the formula is
+<MATH>(<I>n</I><SUP>2</SUP>-<I>n</I>+1)+(<I>n</I><SUP>2</SUP>-<I>n</I>+3)+...+(<I>n</I><SUP>2</SUP>+<I>n</I>-1)=<I>n</I><SUP>3</SUP></MATH>.
+<p>By putting successively <I>n</I>=1, 2, 3...<I>r</I>, &amp;c., in this formula
+and adding the results we find that
+<MATH>1<SUP>3</SUP>+2<SUP>3</SUP>+3<SUP>3</SUP>+...+<I>r</I><SUP>3</SUP>=1+(3+5)+(7+9+11)+...+(...<I>r</I><SUP>2</SUP>+<I>r</I>-1)</MATH>.
+<p>The number of terms in this series of odd numbers is clearly
+<MATH>1+2+3+...+<I>r</I> or 1/2<I>r</I>(<I>r</I>+1)</MATH>.
+<p>Therefore <MATH>1<SUP>3</SUP>+2<SUP>3</SUP>+3<SUP>3</SUP>+...+<I>r</I><SUP>3</SUP>=1/4<I>r</I>(<I>r</I>+1)(1+<I>r</I><SUP>2</SUP>+<I>r</I>-1)
+={1/2<I>r</I>(<I>r</I>+1)}<SUP>2</SUP></MATH>.
+<pb n=109><head>SUM OF SERIES OF CUBE NUMBERS</head>
+<p>Nicomachus does not give this formula, but it was known
+to the Roman <I>agrimensores</I>, and it would be strange if
+Nicomachus was not aware of it. It may have been dis-
+covered by the same mathematician who found out the
+proposition actually stated by Nicomachus, which probably
+belongs to a much earlier time. For the Greeks were from
+the time of the early Pythagoreans accustomed to summing
+the series of odd numbers by placing 3, 5, 7, &amp;c., successively
+as gnomons round 1; they knew that the result, whatever
+the number of gnomons, was always a square, and that, if the
+number of gnomons added to 1 is (say) <I>r</I>, the sum (including
+the 1) is (<I>r</I>+1)<SUP>2</SUP>. Hence, when it was once discovered that
+the first cube after 1, i.e. 2<SUP>3</SUP>, is 3+5, the second, or 3<SUP>3</SUP>, is
+7+9+11, the third, or 4<SUP>3</SUP>, is 13+15+17+19, and so on, they
+were in a position to sum the series 1<SUP>3</SUP>+2<SUP>3</SUP>+3<SUP>3</SUP>+...+<I>r</I><SUP>3</SUP>;
+for it was only necessary to find out how many terms of the
+series 1+3+5+... this sum of cubes includes. The number
+of terms being clearly 1+2+3+...+<I>r</I>, the number of
+gnomons (including the 1 itself) is 1/2<I>r</I>(<I>r</I>+1); hence the sum
+of them all (including the 1), which is equal to
+<MATH>1<SUP>3</SUP>+2<SUP>3</SUP>+3<SUP>3</SUP>+...+<I>r</I><SUP>3</SUP></MATH>,
+is <MATH>{1/2<I>r</I>(<I>r</I>+1)}<SUP>2</SUP></MATH>. Fortunately we possess a piece of evidence
+which makes it highly probable that the Greeks actually
+dealt with the problem in this way. Alkarkh&imacr;, the Arabian
+algebraist of the tenth-eleventh century, wrote an algebra
+under the title <I>Al-Fakhr&imacr;.</I> It would seem that there were at
+the time two schools in Arabia which were opposed to one
+another in that one favoured Greek, and the other Indian,
+methods. Alkarkh&imacr; was one of those who followed Greek
+models almost exclusively, and he has a proof of the theorem
+now in question by means of a figure with gnomons drawn
+in it, furnishing an excellent example of the geometrical
+algebra which is so distinctively Greek.
+<p>Let <I>AB</I> be the side of a square <I>AC</I>; let
+<MATH><I>AB</I>=1+2+...+<I>n</I>=1/2<I>n</I>(<I>n</I>+1)</MATH>,
+and suppose <I>BB</I>&prime;=<I>n</I>, <I>B</I>&prime;<I>B</I>&Prime;=<I>n</I>-1, <I>B</I>&Prime;<I>B</I>&tprime;=<I>n</I>-2, and so on.
+Draw the squares on <I>AB</I>&prime;, <I>AB</I>&Prime;... forming the gnomons
+shown in the figure.
+<pb n=110><head>PYTHAGOREAN ARITHMETIC</head>
+<FIG>
+<p>Then the gnomon
+<MATH><I>BC</I>&prime;<I>D</I>=<I>BB</I>&prime;.<I>BC</I>+<I>DD</I>&prime;.<I>C</I>&prime;<I>D</I>&prime;
+=<I>BB</I>&prime;(<I>BC</I>+<I>C</I>&prime;<I>D</I>&prime;)</MATH>.
+<p>Now <MATH><I>BC</I>=1/2<I>n</I>(<I>n</I>+1)</MATH>,
+<MATH><I>C</I>&prime;<I>D</I>&prime;=1+2+3+...+(<I>n</I>-1)=1/2<I>n</I>(<I>n</I>-1), <I>BB</I>&prime;=<I>n</I></MATH>;
+therefore (gnomon <I>BC</I>&prime;<I>D</I>)=<I>n</I>.<I>n</I><SUP>2</SUP>=<I>n</I><SUP>3</SUP>.
+<p>Similarly (gnomon <I>B</I>&prime;<I>C</I>&Prime;<I>D</I>&prime;)=(<I>n</I>-1)<SUP>3</SUP>, and so on.
+<p>Therefore 1<SUP>3</SUP>+2<SUP>3</SUP>+...+<I>n</I><SUP>3</SUP>=the sum of the gnomons round
+the small square at <I>A</I> which has 1 for its side <I>plus</I> that small
+square; that is,
+<MATH>1<SUP>3</SUP>+2<SUP>3</SUP>+3<SUP>3</SUP>+...+<I>n</I><SUP>3</SUP>=square <I>AC</I>={1/2<I>n</I>(<I>n</I>+1)}<SUP>2</SUP></MATH>.
+<p>It is easy to see that the first gnomon about the small
+square at <I>A</I> is 3+5=2<SUP>3</SUP>, the next gnomon is <MATH>7+9+11=3<SUP>3</SUP></MATH>,
+and so on.
+<p>The demonstration therefore hangs together with the
+theorem stated by Nicomachus. Two alternatives are possible.
+Alkarkh&imacr; may have devised the proof himself in the Greek
+manner, following the hint supplied by Nicomachus's theorem.
+Or he may have found the whole proof set out in some
+Greek treatise now lost and reproduced it. Whichever alter-
+native is the true one, we can hardly doubt the Greek origin
+of the summation of the series of cubes.
+<p>Nicomachus passes to the theory of arithmetical proportion
+and the various <I>means</I> (cc. 21-9), a description of which has
+already been given (p. 87 above). There are a few more
+propositions to be mentioned under this head. If <MATH><I>a</I>-<I>b</I>=<I>b</I>-<I>c</I></MATH>,
+so that <I>a, b, c</I> are in arithmetical progression, then (c. 23. 6)
+<MATH><I>b</I><SUP>2</SUP>-<I>ac</I>=(<I>a</I>-<I>b</I>)<SUP>2</SUP>=(<I>b</I>-<I>c</I>)<SUP>2</SUP></MATH>,
+<pb n=111><head>NICOMACHUS</head>
+a fact which, according to Nicomachus, was not generally
+known. Bo&euml;tius<note>Bo&euml;tius, <I>Inst. Ar.</I> ii. c. 43.</note> mentions this proposition which, if we
+take <MATH><I>a</I>+<I>d, a, a</I>-<I>d</I></MATH> as the three terms in arithmetical pro-
+gression, may be written <MATH><I>a</I><SUP>2</SUP>=(<I>a</I>+<I>d</I>)(<I>a</I>-<I>d</I>)+<I>d</I><SUP>2</SUP></MATH>. This is
+presumably the origin of the <I>regula Nicomachi</I> quoted by
+one Ocreatus (? O'Creat), the author of a tract, <I>Prologus in
+Helceph</I>, written in the twelfth or thirteenth century<note>See <I>Abh. zur Gesch. d. Math</I>. 3, 1880, p. 134.</note>
+(&lsquo;Helceph&rsquo; or &lsquo;Helcep&rsquo; is evidently equivalent to <I>Algo-
+rismus</I>; may it perhaps be meant for the <I>Al-K&amacr;f&imacr;</I> of
+Alkarkh&imacr;?). The object of the <I>regula</I> is to find the square
+of a number containing a single digit. If <I>d</I>=10-<I>a</I>, or
+<I>a</I>+<I>d</I>=10, the rule is represented by the formula
+<MATH><I>a</I><SUP>2</SUP>=10(<I>a</I>-<I>d</I>)+<I>d</I><SUP>2</SUP></MATH>,
+so that the calculation of <I>a</I><SUP>2</SUP> is made to depend on that of <I>d</I><SUP>2</SUP>
+which is easier to evaluate if <I>d</I><<I>a</I>.
+<p>Again (c. 24. 3, 4), if <I>a, b, c</I> be three terms in descending
+geometrical progression, <I>r</I> being the common ratio (<I>a/b</I> or <I>b/c</I>),
+then
+<MATH>(<I>a</I>-<I>b</I>)/(<I>b</I>-<I>c</I>)=<I>a</I>/<I>b</I>=<I>b</I>/<I>c</I></MATH>
+and <MATH>(<I>a</I>-<I>b</I>)=(<I>r</I>-1)<I>b</I>, (<I>b</I>-<I>c</I>)=(<I>r</I>-1)<I>c</I>,
+(<I>a</I>-<I>b</I>)-(<I>b</I>-<I>c</I>)=(<I>r</I>-1)(<I>b</I>-<I>c</I>)</MATH>.
+<p>It follows that
+<MATH><I>b</I>=<I>a</I>-<I>b</I>(<I>r</I>-1)=<I>c</I>+<I>c</I>(<I>r</I>-1)</MATH>.
+<p>This is the property of three terms in geometrical pro-
+gression which corresponds to the property of three terms
+<I>a, b, c</I> of a harmonical progression
+<MATH><I>b</I>=<I>a</I>-<I>a</I>/<I>n</I>=<I>c</I>+<I>c</I>/<I>n</I></MATH>,
+from which we derive
+<MATH><I>n</I>=(<I>a</I>+<I>c</I>)/(<I>a</I>-<I>c</I>)</MATH>.
+<p>If <I>a, b, c</I> are in descending order, Nicomachus observes
+(c. 25) that <I>a</I>/<I>b</I><=><I>b</I>/<I>c</I> according as <I>a, b, c</I> are in arith-
+metical, geometrical, or harmonical progression.
+<pb n=112><head>PYTHAGOREAN ARITHMETIC</head>
+<p>The &lsquo;Platonic theorem&rsquo; (c. 24. 6) about the number of
+possible means (geometric) between two square numbers and
+between two cube numbers respectively has already been
+mentioned (pp. 89, 90), as also the &lsquo;most perfect proportion&rsquo;
+(p. 86).
+<p>THEON OF SMYRNA was the author of a book purporting
+to be a manual of mathematical subjects such as a student
+would require to enable him to understand Plato. A fuller
+account of this work will be given later; at present we are
+only concerned with the arithmetical portion. This gives the
+elementary theory of numbers on much the same lines as
+we find it in Nicomachus, though less systematically. We
+can here pass over the things which are common to Theon
+and Nicomachus and confine ourselves to what is peculiar to
+the former. The important things are two. One is the
+theory of side- and diameter-numbers invented by the Pytha-
+goreans for the purpose of finding the successive integral
+solutions of the equations <MATH>2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>=&plusmn;1</MATH>; as to this see
+pp. 91-3 above. The other is an explanation of the limited
+number of forms which square numbers may have.<note>Theon of Smyrna, p. 35. 17-36. 2.</note> If <I>m</I><SUP>2</SUP> is
+a square number, says Theon, either <I>m</I><SUP>2</SUP> or <I>m</I><SUP>2</SUP>-1 is divisible
+by 3, and again either <I>m</I><SUP>2</SUP> or <I>m</I><SUP>2</SUP>-1 is divisible by 4: which
+is equivalent to saying that a square number cannot be of
+any of the following forms, <MATH>3<I>n</I>+2, 4<I>n</I>+2, 4<I>n</I>+3</MATH>. Again, he
+says, for any square number <I>m</I><SUP>2</SUP>, <I>one</I> of the following alterna-
+tives must hold:
+<MATH>(1) (<I>m</I><SUP>2</SUP>-1)/3, <I>m</I><SUP>2</SUP>/4 both integral (e.g. <I>m</I><SUP>2</SUP>=4),
+(2) (<I>m</I><SUP>2</SUP>-1)/4, <I>m</I><SUP>2</SUP>/3 both integral (e.g. <I>m</I><SUP>2</SUP>=9),
+(3) <I>m</I><SUP>2</SUP>/3, <I>m</I><SUP>2</SUP>/4 both integral (e.g. <I>m</I><SUP>2</SUP>=36),
+(4) (<I>m</I><SUP>2</SUP>-1)/3, (<I>m</I><SUP>2</SUP>-1)/4 both integral (e.g. <I>m</I><SUP>2</SUP>=25)</MATH>.
+<pb n=113><head>ARITHMETIC IN THEON OF SMYRNA</head>
+Iamblichus states the same facts in a slightly different form.<note>Iambl. <I>in Nicom.</I>, p. 90. 6-11.</note>
+The truth of these statements can be seen in the following
+way.<note>Cf. Loria, <I>Le scienze esatte nell</I>' <I>antica Grecia</I>, p. 834.</note> Since any number <I>m</I> must have one of the following
+forms
+<MATH>6<I>k</I>, 6<I>k</I>&plusmn;1, 6<I>k</I>&plusmn;2, 6<I>k</I>&plusmn;3</MATH>,
+any square <I>m</I><SUP>2</SUP> must have one or other of the forms
+<MATH>36<I>k</I><SUP>2</SUP>, 36<I>k</I><SUP>2</SUP>&plusmn;12<I>k</I>+1, 36<I>k</I><SUP>2</SUP>&plusmn;24<I>k</I>+4, 36<I>k</I><SUP>2</SUP>&plusmn;36<I>k</I>+9</MATH>.
+For squares of the first type <I>m</I><SUP>2</SUP>/3 and <I>m</I><SUP>2</SUP>/4 are both integral,
+for those of the second type (<I>m</I><SUP>2</SUP>-1)/3, (<I>m</I><SUP>2</SUP>-1)/4 are both integral,
+for those of the third type (<I>m</I><SUP>2</SUP>-1)/3 and <I>m</I><SUP>2</SUP>/4 are both integral,
+and for those of the fourth type <I>m</I><SUP>2</SUP>/3 and (<I>m</I><SUP>2</SUP>-1)/4 are both
+integral; which agrees with Theon's statement. Again, if
+the four forms of squares be divided by 3 or 4, the remainder
+is always either 0 or 1; so that, as Theon says, no square can
+be of the form 3<I>n</I>+2, 4<I>n</I>+2, or 4<I>n</I>+3. We can hardly
+doubt that these discoveries were also Pythagorean.
+<p>IAMBLICHUS, born at Chalcis in Coele-Syria, was a pupil of
+Anatolius and Porphyry, and belongs to the first half of the
+fourth century A.D. He wrote nine Books on the Pythagorean
+Sect, the titles of which were as follows: I. On the Life of
+Pythagoras; II. Exhortation to philosophy (<G>*protreptiko\s
+e)pi\ filosofi/an</G>); III. On mathematical science in general;
+IV. On Nicomachus's <I>Introductio Arithmetica</I>; V. On arith-
+metical science in physics; VI. On arithmetical science in
+ethics; VII. On arithmetical science in theology; VIII. On
+the Pythagorean geometry; IX. On the Pythagorean music.
+The first four of these books survive and are accessible in
+modern editions; the other five are lost, though extracts
+from VII. are doubtless contained in the <I>Theologumena
+arithmetices.</I> Book IV. on Nicomachus's <I>Introductio</I> is that
+which concerns us here; and the few things requiring notice
+are the following. The first is the view of a square number
+<pb n=114><head>PYTHAGOREAN ARITHMETIC</head>
+as a race-course (<G>di/aulos</G>)<note>Iambl. <I>in Nicom.</I>, p. 75. 25-77. 4.</note> formed of successive numbers
+from 1 (as <I>start</I>, <G>u(/splhx</G>) up to <I>n</I>, the side of the square,
+which is the turning-point (<G>kampth/r</G>), and then back again
+through (<I>n</I>-1), (<I>n</I>-2), &amp;c., to 1 (the <I>goal</I>, <G>nu/ssa</G>), thus:
+<MATH>1+2+3+4... (<I>n</I>-1)+<I>n</I>
+1+2+3+4...(<I>n</I>-2)+(<I>n</I>-1)+<I>n</I></MATH>.
+This is of course equivalent to the proposition that <I>n</I><SUP>2</SUP> is the
+sum of the two triangular numbers 1/2<I>n</I>(<I>n</I>+1) and 1/2(<I>n</I>-1)<I>n</I>
+with sides <I>n</I> and <I>n</I>-1 respectively. Similarly Iamblichus
+points out<note><I>Ib.</I>, pp. 77. 4-80. 9.</note> that the <I>oblong</I> number
+<MATH><I>n</I>(<I>n</I>-1)=(1+2+3+...+<I>n</I>)+(<I>n</I>-2+<I>n</I>-3+...+3+2)</MATH>.
+He observes that it was on this principle that, after 10,
+which was called the <I>unit of the second course</I> (<G>deuterw-
+doume/nh mona/s</G>), the Pythagoreans regarded 100=10.10 as
+the <I>unit of the third course</I> (<G>triwdoume/nh mona/s</G>), 1000=10<SUP>3</SUP>
+as the <I>unit of the fourth course</I> (<G>tetrwdoume/nh mona/s</G>), and
+so on,<note><I>Ib.</I>, pp. 88. 15-90. 2.</note> since
+<MATH>1+2+3+...+10+9+8+...+2+1=10.10,
+10+20+30+...+100+90+80+...+20+10=10<SUP>3</SUP>,
+100+200+300+...+1000+900+...+200+100=10<SUP>4</SUP></MATH>,
+and so on. Iamblichus sees herein the special virtue of 10:
+but of course the same formulae would hold in any scale
+of notation as well as the decimal.
+<p>In connexion with this Pythagorean decimal terminology
+Iamblichus gives a proposition of the greatest interest.<note><I>Ib.</I>, pp. 103. 10-104. 13.</note>
+Suppose we have any three consecutive numbers the greatest
+of which is divisible by 3. Take the sum of the three
+numbers; this will consist of a certain number of units,
+a certain number of tens, a certain number of hundreds, and
+so on. Now take the units in the said sum as they are, then
+as many units as there are tens in the sum, as many units as
+there are hundreds, and so on, and add all the units so
+obtained together (i.e. add the <I>digits</I> of the sum expressed
+in our decimal notation). Apply the same procedure to the
+<pb n=115><head>IAMBLICHUS</head>
+result, and so on. Then, says Iamblichus, <I>the final result
+will be the number</I> 6. E.g. take the numbers 10, 11, 12; the
+sum is 33. Add the digits, and the result is 6. Take
+994, 995, 996: the sum is 2985; the sum of the digits is 24;
+and the sum of the digits of 24 is again 6. The truth of the
+general proposition is seen in this way.<note>Loria, <I>op. cit.</I>, pp. 841-2.</note>
+<p>Let <MATH><I>N</I>=<I>n</I><SUB>0</SUB>+10<I>n</I><SUB>1</SUB>+10<SUP>2</SUP><I>n</I><SUB>2</SUB>+...</MATH>
+be a number written in the decimal notation. Let <I>S</I>(<I>N</I>)
+represent the sum of its digits, <I>S</I><SUP>(2)</SUP>(<I>N</I>) the sum of the digits
+of <I>S</I>(<I>N</I>) and so on.
+<p>Now <MATH><I>N</I>-<I>S</I>(<I>N</I>)=9(<I>n</I><SUB>1</SUB>+11<I>n</I><SUB>2</SUB>+111<I>n</I><SUB>3</SUB>+...)</MATH>,
+whence <MATH><I>N</I>&equals3;<I>S</I>(<I>N</I>)</MATH> (mod. 9).
+Similarly <MATH><I>S</I>(<I>N</I>)&equals3;<I>S</I><SUP>(2)</SUP><I>N</I></MATH> (mod. 9).
+.
+.
+.
+<p>Let <MATH><I>S</I><SUP>(<I>k</I>-1)</SUP>(<I>N</I>)&equals3;<I>S</I><SUP>(<I>k</I>)</SUP><I>N</I></MATH> (mod. 9)
+be the last possible relation of this kind; <I>S</I><SUP>(<I>k</I>)</SUP><I>N</I> will be a
+number <I>N</I>&prime;<02>9.
+<p>Adding the congruences, we obtain
+<MATH><I>N</I>&equals3;<I>N</I>&prime;</MATH> (mod. 9), while <MATH><I>N</I>&prime;<02>9</MATH>.
+<p>Now, if we have three consecutive numbers the greatest
+of which is divisible by 3, we can put for their sum
+<MATH><I>N</I>=(3<I>p</I>+1)+(3<I>p</I>+2)+(3<I>p</I>+3)=9<I>p</I>+6</MATH>,
+and the above congruence becomes
+<MATH>9<I>p</I>+6&equals3;<I>N</I>&prime;</MATH> (mod. 9),
+so that <MATH><I>N</I>&prime;&equals3;6</MATH> (mod. 9);
+and, since <MATH><I>N</I>&prime;<02>9</MATH>, <I>N</I>&prime; can only be equal to 6.
+<p>This addition of the digits of a number expressed in our
+notation has an important parallel in a passage of the
+<I>Refutation of all Heresies</I> by saint Hippolytus,<note>Hippolytus, <I>Refut.</I> iv, c. 14.</note> where there
+is a description of a method of foretelling future events
+called the &lsquo;Pythagorean calculus&rsquo;. Those, he says, who
+claim to predict events by means of calculations with numbers,
+letters and names use the principle of the <I>pythmen</I> or <I>base</I>,
+<pb n=116><head>PYTHAGOREAN ARITHMETIC</head>
+that is, what we call a digit of a number expressed in our
+decimal notation; for the Greeks, in the case of any number
+above 9, the <I>pythmen</I> was the same number of units as the
+alphabetical numeral contains tens, hundreds, thousands, &amp;c.
+Thus the <I>pythmen</I> of 700 (<G>y</G> in Greek) is 7 (<G>z</G>); that of
+<G><SUB>'</SUB>s</G> (6000) is <G>s</G> (6), and so on. The method then proceeded
+to find the <I>pythmen</I> of a certain name, say <G>*)agame/mnwn</G>.
+Taking the <I>pythmenes</I> of all the letters and adding them,
+we have
+<MATH>1+3+1+4+5+4+5+8+5=36</MATH>.
+Take the <I>pythmenes</I> of 36, namely 3 and 6, and their sum is
+9. The <I>pythmen</I> of <G>*)agame/mnwn</G> is therefore 9. Next take
+the name <G>*(/ektwr</G>; the <I>pythmenes</I> are 5, 2, 3, 8, 1, the sum of
+which is 19; the <I>pythmenes</I> of 19 are 1, 9; the sum of 1 and
+9 is 10, the pythmen of which is 1. The <I>pythmen</I> of <G>*(/ektwr</G>
+is therefore 1. &lsquo;It is easier&rsquo;, says Hippolytus, &lsquo;to proceed
+thus. Finding the <I>pythmenes</I> of the letters, we obtain, in the
+case of <G>*(/ektwr</G>, 19 as their sum. Divide this by 9 and note
+the remainder: thus, if I divide 19 by 9, the remainder is 1,
+for nine times 2 is 18, and 1 is left, which will accordingly
+be the <I>pythmen</I> of the name <G>*(/ektwr</G>.&rsquo; Again, take the name
+<G>*pa/troklos</G>. The sum of the <I>pythmenes</I> is
+<MATH>8+1+3+1+7+2+3+7+2=34</MATH>:
+and 3+4=7, so that 7 is the <I>pythmen</I> of <G>*pa/troklos</G>.
+&lsquo;Those then who calculate by the <I>rule of nine</I> take one-ninth
+of the sum of the <I>pythmenes</I> and then determine the sum of
+the <I>pythmenes</I> in the remainder. Those on the other hand
+who follow the &ldquo;rule of seven&rdquo; divide by 7. Thus the sum
+of the <I>pythmenes</I> in <G>*pa/troklos</G> was found to be 34. This,
+divided by 7, gives 4, and since 7 times 4 is 28, the remainder
+is 6....&rsquo; &lsquo;It is necessary to observe that, if the division
+gives an integral quotient (without remainder),... the
+<I>pythmen</I> is the number 9 itself&rsquo; (that is, if the <I>rule of nine</I> is
+followed). And so on.
+<p>Two things emerge from this fragment. (1) The use of the
+<I>pythmen</I> was not appearing for the first time when Apollonius
+framed his system for expressing and multiplying large
+numbers; it originated much earlier, with the Pythagoreans.
+<pb n=117><head>IAMBLICHUS</head>
+(2) The method of calculating the <I>pythmen</I> is like the opera-
+tion of &lsquo;casting out nines&rsquo; in the proof which goes by that
+name, where we take the sum of the digits of a number and
+divide by 9 to get the remainder. The method of verification
+by &lsquo;casting out nines&rsquo; came to us from the Arabs, who may,
+as Avicenna and Maximus Planudes tell us, have got it from
+the Indians; but the above evidence shows that, at all events,
+the elements from which it was built up lay ready to hand
+in the Pythagorean arithmetic.
+<pb>
+<C>IV</C>
+<C>THE EARLIEST GREEK GEOMETRY. THALES</C>
+<C>The &lsquo;Summary&rsquo; of Proclus.</C>
+<p>WE shall often, in the course of this history, have occasion
+to quote from the so-called &lsquo;Summary&rsquo; of Proclus, which has
+already been cited in the preceding chapter. Occupying a
+few pages (65-70) of Proclus's <I>Commentary on Euclid</I>, Book I,
+it reviews, in the briefest possible outline, the course of Greek
+geometry from the earliest times to Euclid, with special refer-
+ence to the evolution of the Elements. At one time it was
+often called the &lsquo;Eudemian summary&rsquo;, on the assumption
+that it was an extract from the great <I>History of Geometry</I> in
+four Books by Eudemus, the pupil of Aristotle. But a perusal
+of the summary itself is sufficient to show that it cannot
+have been written by Eudemus; the most that can be said is
+that, down to a certain sentence, it was probably based, more
+or less directly, upon data appearing in Eudemus's <I>History.</I>
+At the sentence in question there is a break in the narrative,
+as follows:
+<p>&lsquo;Those who have compiled histories bring the development
+of this science up to this point. Not much younger than
+these is Euclid, who put together the Elements, collecting
+many of the theorems of Eudoxus, perfecting many others by
+Theaetetus, and bringing to irrefragable demonstration the
+propositions which had only been somewhat loosely proved by
+his predecessors.&rsquo;
+<p>Since Euclid was later than Eudemus, it is impossible that
+Eudemus can have written this; while the description &lsquo;those
+who have compiled histories&rsquo;, and who by implication were
+a little older than Euclid, suits Eudemus excellently. Yet the
+style of the summary after the break does not show any
+such change from that of the earlier portion as to suggest
+<pb n=119><head>THE &lsquo;SUMMARY&rsquo; OF PROCLUS</head>
+different authorship. The author of the earlier portion fre-
+quently refers to the question of the origin of the Elements of
+Geometry in a way in which no one would be likely to write
+who was not later than Euclid; and it seems to be the same
+hand which, in the second portion, connects the Elements of
+Euclid with the work of Eudoxus and Theaetetus. Indeed
+the author, whoever he was, seems to have compiled the sum-
+mary with one main object in view, namely, to trace the origin
+and growth of the Elements of Geometry; consequently he
+omits to refer to certain famous discoveries in geometry such
+as the solutions of the problem of the duplication of the cube,
+doubtless because they did not belong to the Elements. In
+two cases he alludes to such discoveries, as it were in paren-
+thesis, in order to recall to the mind of the reader a current
+association of the name of a particular geometer with a par-
+ticular discovery. Thus he mentions Hippocrates of Chios as
+a famous geometer for the particular reason that he was the
+first to write Elements, and he adds to his name, for the pur-
+pose of identification, &lsquo;the discoverer of the quadrature of the
+lune&rsquo;. Similarly, when he says of Pythagoras &lsquo;(he it was)
+who&rsquo; (<G>o(\s dh\</G> . . .) &lsquo;discovered the theory of irrationals [or
+&ldquo;proportions&rdquo;] and the construction of the cosmic figures&rsquo;,
+he seems to be alluding, entirely on his own account, to a
+popular tradition to that effect. If the summary is the work
+of one author, who was it? Tannery answers that it was
+Geminus; but this seems highly improbable, for the extracts
+from Geminus's work which we possess suggest that the
+subjects therein discussed were of a different kind; they seem
+rather to have been general questions relating to the philoso-
+phy and content of mathematics, and even Tannery admits
+that historical details could only have come incidentally into
+the work.
+<p>Could the author have been Proclus himself? This again
+seems, on the whole, improbable. In favour of the authorship
+of Proclus are the facts (1) that the question of the origin of
+the Elements is kept prominent and (2) that there is no men-
+tion of Democritus, whom Eudemus would not have ignored,
+while a follower of Plato such as Proclus might have done
+him this injustice, following the example of Plato himself, who
+was an opponent of Democritus, never once mentions him, and
+<pb n=120><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+is said to have wished to burn all his writings. On the other
+hand (1) the style of the summary is not such as to point
+to Proclus as the author; (2) if he wrote it, it is hardly
+conceivable that he would have passed over in silence the dis-
+covery of the analytical method, &lsquo;the finest&rsquo;, as he says else-
+where, of the traditional methods in geometry, &lsquo;which Plato is
+said to have communicated to Leodamas&rsquo;. Nor (3) is it
+easy to suppose that Proclus would have spoken in the
+detached way that the author does of Euclid whose <I>Elements</I>
+was the subject of his whole commentary: &lsquo;Not much younger
+than these is Euclid, who compiled the Elements . . .&rsquo;. &lsquo;This
+man lived in the time of the first Ptolemy . . .&rsquo;. On the whole,
+therefore, it would seem probable that the body of the sum-
+mary was taken by Proclus from a compendium made by some
+writer later than Eudemus, though the earlier portion was
+based, directly or indirectly, upon notices in Eudemus's <I>History.</I>
+But the prelude with which the summary is introduced may
+well have been written, or at all events expanded, by Proclus
+himself, for it is in his manner to bring in &lsquo;the inspired
+Aristotle&rsquo; (<G>o( daimo/nios *)aristote/lhs</G>)&mdash;as he calls him here and
+elsewhere&mdash;and the transition to the story of the Egyptian
+origin of geometry may also be his:
+<p>&lsquo;Since, then, we have to consider the beginnings of the arts
+and sciences with reference to the particular cycle [of the
+series postulated by Aristotle] through which the universe is
+at present passing, <I>we say</I> that, according to most accounts,
+geometry was first discovered in Egypt, having had its origin
+in the measurement of areas. For this was a necessity for the
+Egyptians owing to the rising of the Nile which effaced the
+proper boundaries of everybody's lands.&rsquo;
+<p>The next sentences also may well be due to Proclus:
+<p>&lsquo;And it is in no way surprising that the discovery of this as
+well as the other sciences had its beginning in practical needs,
+seeing that everything that is in the course of becoming pro-
+gresses from the imperfect to the perfect. Thus the transition
+from sensation to reasoning and from reasoning to under-
+standing is only natural.&rsquo;
+<p>These sentences look like reflections by Proclus, and the
+transition to the summary proper follows, in the words:
+<p>&lsquo;Accordingly, just as exact arithmetic began among the
+<pb n=121><head>ORIGIN OF GEOMETRY</head>
+Phoenicians owing to its use in commerce and contracts, so
+geometry was discovered in Egypt for the reason aforesaid.&rsquo;
+<C>Tradition as to the origin of geometry.</C>
+<p>Many Greek writers besides Proclus give a similar account
+of the origin of geometry. Herodotus says that Sesostris
+(Ramses II, <I>circa</I> 1300 B.C.) distributed the land among all the
+Egyptians in equal rectangular plots, on which he levied an
+annual tax; when therefore the river swept away a portion
+of a plot and the owner applied for a corresponding reduction
+in the tax, surveyors had to be sent down to certify what the
+reduction in the area had been. &lsquo;This, in my opinion (<G>doke/ei
+moi</G>)&rsquo;, he continues, &lsquo;was the origin of geometry, which then
+passed into Greece.&rsquo;<note>Herodotus ii. 109.</note> The same story, a little amplified, is
+repeated by other writers, Heron of Alexandria,<note>Heron, <I>Geom.</I> c. 2, p. 176, Heib.</note> Diodorus
+Siculus,<note>Diod. Sic. i. 69, 81.</note> and Strabo.<note>Strabo xvii. c. 3.</note> True, all these statements (even if that
+in Proclus was taken directly from Eudemus's <I>History of
+Geometry</I>) may all be founded on the passage of Herodotus,
+and Herodotus may have stated as his own inference what he
+was told in Egypt; for Diodorus gives it as an Egyptian
+tradition that geometry and astronomy were the discoveries
+of Egypt, and says that the Egyptian priests claimed Solon,
+Pythagoras, Plato, Democritus, Oenopides of Chios, and
+Eudoxus as their pupils. But the Egyptian claim to the
+discoveries was never disputed by the Greeks. In Plato's
+<I>Phaedrus</I> Socrates is made to say that he had heard that the
+Egyptian god Theuth was the first to invent arithmetic, the
+science of calculation, geometry, and astronomy.<note>Plato, <I>Phaedrus</I> 274 c.</note> Similarly
+Aristotle says that the mathematical arts first took shape in
+Egypt, though he gives as the reason, not the practical need
+which arose for a scientific method of measuring land, but the
+fact that in Egypt there was a leisured class, the priests, who
+could spare time for such things.<note>Arist. <I>Metaph.</I> A. 1, 981 b 23.</note> Democritus boasted that no
+one of his time had excelled him &lsquo;in making lines into figures
+and proving their properties, not even the so-called <I>Harpe-
+donaptae</I> in Egypt&rsquo;.<note>Clem. <I>Strom.</I> i. 15. 69 (<I>Vorsokratiker</I>, ii<SUP>3</SUP>, p. 123. 5-7).</note> This word, compounded of two Greek
+words, <G>a(rpedo/nh</G> and <G>a(/ptein</G>, means &lsquo;rope-stretchers&rsquo; or &lsquo;rope-
+<pb n=122><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+fasteners&rsquo;; and, while it is clear from the passage that the
+persons referred to were clever geometers, the word reveals a
+characteristic <I>modus operandi.</I> The Egyptians were ex-
+tremely careful about the orientation of their temples, and
+the use of ropes and pegs for marking out the limits,
+e.g. corners, of the sacred precincts is portrayed in all
+pictures of the laying of foundation stones of temples.<note>Brugsch, <I>Steininschrift und Bibelwort</I>, 2nd ed., p. 36.</note> The
+operation of &lsquo;rope-stretching&rsquo; is mentioned in an inscription on
+leather in the Berlin Museum as having been in use as early
+as Amenemhat I (say 2300 B.C.).<note>D&uuml;michen, <I>Denderatempel</I>, p. 33.</note> Now it was the practice
+of ancient Indian and probably also of Chinese geometers
+to make, for instance, a right angle by stretching a rope
+divided into three lengths in the ratio of the sides of a right-
+angled triangle in rational numbers, e.g. 3, 4, 5, in such a way
+that the three portions formed a triangle, when of course a right
+angle would be formed at the point where the two smaller
+sides meet. There seems to be no doubt that the Egyptians
+knew that the triangle (3, 4, 5), the sides of which are so
+related that the square on the greatest side is equal to the
+sum of the squares on the other two, is right-angled; if this
+is so, they were acquainted with at least one case of the
+famous proposition of Pythagoras.
+<C>Egyptian geometry, i.e. mensuration.</C>
+<p>We might suppose, from Aristotle's remark about the
+Egyptian priests being the first to cultivate mathematics
+because they had leisure, that their geometry would have
+advanced beyond the purely practical stage to something
+more like a theory or science of geometry. But the docu-
+ments which have survived do not give any ground for this
+supposition; the art of geometry in the hands of the priests
+never seems to have advanced beyond mere routine. The
+most important available source of information about Egyptian
+mathematics is the Papyrus Rhind, written probably about
+1700 B.C. but copied from an original of the time of King
+Amenemhat III (Twelfth Dynasty), say 2200 B.C. The geo-
+metry in this &lsquo;guide for calculation, a means of ascertaining
+everything, of elucidating all obscurities, all mysteries, all
+<pb n=123><head>EGYPTIAN GEOMETRY</head>
+difficulties&rsquo;, as it calls itself, is rough <I>mensuration.</I> The
+following are the cases dealt with which concern us here.
+(1) There is the <I>rectangle</I>, the area of which is of course
+obtained by multiplying together the numbers representing
+the sides. (2) The measure of a <I>triangle</I> is given as the pro-
+duct of half the base into the <I>side.</I> And here there is a differ-
+ence of opinion as to the kind of triangle measured. Eisenlohr
+and Cantor, taking the diagram to represent an <I>isosceles</I> tri-
+angle rather inaccurately drawn, have to assume error on
+the part of the writer in making the area 1/2<I>ab</I> instead of
+<MATH>1/2<I>a</I>&radic;(<I>b</I><SUP>2</SUP>-1/4<I>a</I><SUP>2</SUP>)</MATH> where <I>a</I> is the base and <I>b</I> the &lsquo;side&rsquo;, an error
+which of course becomes less serious as <I>a</I> becomes smaller
+relatively to <I>b</I> (in the case taken <I>a</I>=4, <I>b</I>=10, and the area
+as given according to the rule, i.e. 20, is not greatly different
+from the true value 19.5959). But other authorities take the
+triangle to be <I>right-angled</I> and <I>b</I> to be the side perpendicular
+to the base, their argument being that the triangle as drawn
+is not a worse representation of a right-angled triangle than
+other triangles purporting to be right-angled which are found
+in other manuscripts, and indeed is a better representation of
+a right-angled triangle than it is of an isosceles triangle, while
+the number representing the side is shown in the figure along-
+side one only of the sides, namely that adjacent to the angle
+which the more nearly represents a right angle. The advan-
+tage of this interpretation is that the rule is then correct
+instead of being more inaccurate than one would expect from
+a people who had expert land surveyors to measure land for
+the purpose of assessing it to tax. The same doubt arises
+with reference to (3) the formula for the area of a trapezium,
+namely <MATH>1/2(<I>a</I>+<I>c</I>)x<I>b</I></MATH>, where <I>a, c</I> are the base and the opposite
+parallel side respectively, while <I>b</I> is the &lsquo;side&rsquo;, i.e. one of the
+non-parallel sides. In this case the figure seems to have been
+intended to be isosceles, whereas the formula is only accurate
+if <I>b</I>, one of the non-parallel sides, is at right angles to the base,
+in which case of course the side opposite to <I>b</I> is not at right
+angles to the base. As the parallel sides (6, 4) in the case
+taken are short relatively to the &lsquo;side&rsquo; (20), the angles at the
+base are not far short of being right angles, and it is possible
+that one of them, adjacent to the particular side which is
+marked 20, was intended to be right. The hypothesis that
+<pb n=124><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+the triangles and trapezia are isosceles, and that the formulae
+are therefore crude and inaccurate, was thought to be con-
+firmed by the evidence of inscriptions on the Temple of Horus
+at Edfu. This temple was planned out in 237 B.C.; the in-
+scriptions which refer to the assignment of plots of ground to
+the priests belong to the reign of Ptolemy XI, Alexander I
+(107-88 B.C.). From so much of these inscriptions as were
+published by Lepsius<note>&lsquo;Ueber eine hieroglyphische Inschrift am Tempel von Edfu&rsquo; (<I>Abh.
+der Berliner Akad.</I>, 1855, pp. 69-114).</note> we gather that <MATH>1/2(<I>a</I>+<I>c</I>).1/2(<I>b</I>+<I>d</I>)</MATH> was a
+formula for the area of a quadrilateral the sides of which in
+order are <I>a, b, c, d.</I> Some of the quadrilateral figures are
+evidently trapezia with the non-parallel sides equal; others are
+not, although they are commonly not far from being rectangles
+or isosceles trapezia. Examples are &lsquo;16 to 15 and 4 to 3 1/2 make
+58 1/8&rsquo; (i.e. <MATH>1/2(16+15)x1/2(4+3 1/2)=58 1/8</MATH>); &lsquo;9 1/2 to 10 1/2 and 24 1/2 1/8 to
+22 1/2 1/8 make 236 1/4&rsquo;; &lsquo;22 to 23 and 4 to 4 make 90&rsquo;, and so on.
+Triangles are not made the subject of a separate formula, but
+are regarded as cases of quadrilaterals in which the length of
+one side is zero. Thus the triangle 5, 17, 17 is described as a
+figure with sides &lsquo;0 to 5 and 17 to 17&rsquo;, the area being accord-
+ingly <MATH>1/2(0+5).1/2(17+17)</MATH> or 42 1/2; 0 is expressed by hieroglyphs
+meaning the word Nen. It is remarkable enough that the use
+of a formula so inaccurate should have lasted till 200 years or
+so after Euclid had lived and taught in Egypt; there is also
+a case of its use in the <I>Liber Geeponicus</I> formerly attributed to
+Heron,<note>Heron, ed. Hultsch, p. 212. 15-20 (Heron, <I>Geom.</I> c. 6. 2, Heib.).</note> the quadrilateral having two opposite sides parallel
+and the pairs of opposite sides being (32, 30) and (18, 16). But
+it is right to add that, in the rest of the Edfu inscriptions
+published later by Brugsch, there are cases where the inaccu-
+rate formula is not used, and it is suggested that what is being
+attempted in these cases is an approximation to the square
+root of a non-square number.<note>M. Simon, <I>Gesch. d. Math. im Altertum</I>, p. 48.</note>
+<p>We come now (4) to the mensuration of circles as found
+in the Papyrus Rhind. If <I>d</I> is the diameter, the area is
+given as <MATH><BRACE>(1-1/9)<I>d</I></BRACE><SUP>2</SUP></MATH> or 64/81<I>d</I><SUP>2</SUP>. As this is the corresponding
+figure to 1/4<G>p</G><I>d</I><SUP>2</SUP>, it follows that the value of <G>p</G> is taken as
+<MATH>256/81=(16/9)<SUP>2</SUP></MATH>, or 3.16, very nearly. A somewhat different
+value for <G>p</G> has been inferred from measurements of certain
+<pb n=125><head>EGYPTIAN GEOMETRY</head>
+heaps of grain or of spaces which they fill. Unfortunately
+the shape of these spaces or heaps cannot be determined with
+certainty. The word in the Papyrus Rhind is <I>shaa</I>; it is
+evident that it ordinarily means a rectangular parallelepiped,
+but it can also be applied to a figure with a circular base,
+e.g. a cylinder, or a figure resembling a thimble, i.e. with
+a rounded top. There is a measurement of a mass of corn
+apparently of the latter sort in one of the Kahu&ndot; papyri.<note>Griffith, <I>Kahu&ndot; Papyri</I>, Pt. I, Plate 8.</note>
+The figure shows a circle with 1365 1/3 as the content of the
+heap written within it, and with 12 and 8 written above and
+to the left of the circle respectively. The calculation is done
+in this way. 12 is taken and 1/3 of it added; this gives 16;
+16 is squared, which gives 256, and finally 256 is multiplied
+by 2/3 of 8, which gives 1365 1/3. If for the original figures
+12 and 8 we write <I>h</I> and <I>k</I> respectively, the formula used for
+the content is <MATH>(4/3<I>h</I>)<SUP>2</SUP>.2/3<I>k.</I></MATH> Griffith took 12 to be the height
+of the figure and 8 to be the diameter of the base. But
+according to another interpretation,<note>Simon, <I>l. c.</I></note> 12 is simply 3/2 of 8, and
+the figure to be measured is a hemisphere with diameter
+8 ells. If this is so, the formula makes the content of a
+hemisphere of diameter <I>k</I> to be <MATH>(4/3.3/2<I>k</I>)<SUP>2</SUP>.2/3<I>k</I></MATH> or 8/3<I>k</I><SUP>3</SUP>. Com-
+paring this with the true volume of the hemisphere, <MATH>2/3.1/8<G>p</G><I>k</I><SUP>3</SUP></MATH>
+or <MATH>1/12<G>p</G><I>k</I><SUP>3</SUP>=134.041</MATH> cubic ells, we see that the result 1365 1/3
+obtained by the formula must be expressed in 1/10ths of a cubic
+ell: consequently for 1/12<G>p</G> the formula substitutes 8/30, so that
+the formula gives 3.2 in place of <G>p</G>, a value different from the
+3.16 of Ahmes. Borchardt suggests that the formula for the
+measurement of a hemisphere was got by repeated practical
+measurements of heaps of corn built up as nearly as possible
+in that form, in which case the inaccuracy in the figure for <G>p</G>
+is not surprising. With this problem from the Kahu&ndot; papyri
+must be compared No. 43 from the Papyrus Rhind. A curious
+feature in the measurements of stores or heaps of corn in
+the Papyrus Rhind is the fact, not as yet satisfactorily ex-
+plained, that the area of the base (square or circular) is first
+found and is then regularly multiplied, not into the &lsquo;height&rsquo;
+itself, but into 3/2 times the height. But in No. 43 the calcula-
+tion is different and more parallel to the case in the Kahu&ndot;
+papyrus. The problem is to find the content of a space round
+<pb n=126><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+in form &lsquo;9 in height and 6 in breadth&rsquo;. The word <I>qa</I>, here
+translated &lsquo;height&rsquo;, is apparently used in other documents
+for &lsquo;length&rsquo; or &lsquo;greatest dimension&rsquo;, and must in this case
+mean the diameter of the base, while the &lsquo;breadth&rsquo; is the
+height in our sense. If we denote the diameter of the circular
+base by <I>k</I>, and the height by <I>h</I>, the formula used in this
+problem for finding the volume is <MATH>(4/3.8/9<I>k</I>)<SUP>2</SUP>.2/3<I>h</I></MATH>. Here it is
+not 3/2<I>h</I>, but 2/3<I>h</I>, which is taken as the last factor of the
+product. Eisenlohr suggests that the analogy of the formula
+for a hemisphere, <MATH><G>p</G><I>r</I><SUP>2</SUP>.2/3<I>r</I></MATH>, may have operated to make the
+calculator take 2/3 of the height, although the height is not
+in the particular case the same as the radius of the base, but
+different. But there remains the difficulty that (4/3)<SUP>2</SUP> or 16/9
+times the area of the circle of diameter <I>k</I> is taken instead
+of the area itself. As to this Eisenlohr can only suggest that
+the circle of diameter <I>k</I> which was accessible for measurement
+was not the real or mean circular section, and that allowance
+had to be made for this, or that the base was not a circle of
+diameter <I>k</I> but an <I>ellipse</I> with 16/9<I>k</I> and <I>k</I> as major and minor
+axes. But such explanations can hardly be applied to the
+factor (4/3)<SUP>2</SUP> in the Kahu&ndot; case <I>if</I> the latter is really the case
+of a hemispherical space as suggested. Whatever the true
+explanation may be, it is clear that these rules of measure-
+ment must have been empirical and that there was little or
+no geometry about them.
+<p>Much more important geometrically are certain calculations
+with reference to the proportions of pyramids (Nos. 56-9 of
+<FIG>
+the Papyrus Rhind) and a monu-
+ment (No. 60). In the case
+of the pyramid two lines in the
+figure are distinguished, (1)
+<I>ukha-thebt</I>, which is evidently
+some line in the base, and
+(2) <I>pir-em-us</I> or <I>per-em-us</I>
+(&lsquo;height&rsquo;), a word from which
+the name <G>purami/s</G> may have
+been derived.<note>Another view is that the words <G>purami/s</G> and <G>puramou=s</G>, meaning a kind
+of cake made from roasted wheat and honey, are derived from <G>puroi/</G>,
+&lsquo;wheat&rsquo;, and are thus of purely Greek origin.</note> The object of
+<pb n=127><head>MEASUREMENT OF PYRAMIDS</head>
+the problems is to find a certain relation called <I>se-qe&tdot;</I>,
+literally &lsquo;that which makes the nature&rsquo;, i.e. that which
+determines the proportions of the pyramid. The relation
+<MATH><I>se-qe&tdot;</I>=(1/2<I>ukha-thebt</I>)/<I>piremus</I></MATH>. In the case of the monument we have
+two other names for lines in the figure, (1) <I>senti</I>, &lsquo;foundation&rsquo;,
+or base, (2) <I>qay en &hdot;eru</I>, &lsquo;vertical length&rsquo;, or height; the
+same term <I>se-qe&tdot;</I> is used for the relation <MATH>(1/2<I>senti</I>)/(<I>qay en &hdot;eru</I>)</MATH> or
+the same inverted. Eisenlohr and Cantor took the lines
+(1) and (2) in the case of the pyramid to be different from
+the lines (1) and (2) called by different names in the monument.
+Suppose <I>ABCD</I> to be the square base of a pyramid, <I>E</I> its
+centre, <I>H</I> the vertex, and <I>F</I> the middle point of the side <I>AD</I>
+of the base. According to Eisenlohr and Cantor the <I>ukha-
+thebt</I> is the diagonal, say <I>AC</I>, of the base, and the <I>pir-em-us</I>
+is the <I>edge</I>, as <I>AH.</I> On this assumption the <I>se-qe&tdot;</I>
+<MATH>=<I>AE</I>/<I>AH</I>=cos <I>HAE</I></MATH>.
+In the case of the monument they took the <I>senti</I> to be the
+side of the base, as <I>AB</I>, the <I>qay en &hdot;eru</I> to be the height of
+the pyramid <I>EH</I>, and the <I>se-qe&tdot;</I> to be the ratio of <I>EH</I> to
+1/2<I>AB</I> or of <I>EH</I> to <I>EF</I>, i.e. the <I>tangent</I> of the angle <I>HFE</I>
+which is the slope of the faces of the pyramid. According
+to Eisenlohr and Cantor, therefore, the one term <I>se-qe&tdot;</I> was
+used in two different senses, namely, in Nos. 56-9 for cos <I>HAE</I>
+and in No. 60 for tan <I>HFE.</I> Borchardt has, however, proved
+that the <I>se-qe&tdot;</I> in all the cases has one meaning, and represents
+the <I>cotangent</I> of the slope of the faces of the pyramid,
+i. e. cot <I>HFE</I> or the ratio of <I>FE</I> to <I>EH.</I> There is no difficulty
+in the use of the different words <I>ukha-thebt</I> and <I>senti</I> to
+express the same thing, namely, the side of the base, and
+of the different words <I>per-em-us</I> and <I>qay en &hdot;eru</I> in the same
+sense of &lsquo;height&rsquo;; such synonyms are common in Egypt, and,
+moreover, the word <I>mer</I> used of the pyramids is different
+from the word <I>&adot;n</I> for the monument. Again, it is clear that,
+while the <I>slope</I>, the angle <I>HFE</I>, is what the builder would
+want to know, the cosine of the angle <I>HAE</I>, formed by the
+<I>edge</I> with the plane of the base, would be of no direct use
+<pb n=128><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+to him. But, lastly, the <I>se-qe&tdot;</I> in No. 56 is 18/25 and, if <I>se-qe&tdot;</I>
+is taken in the sense of cot <I>HFE</I>, this gives for the angle
+<I>HFE</I> the value of 54&deg;14&prime;16&Prime;, which is <I>precisely</I>, to the
+seconds, the slope of the lower half of the southern stone
+pyramid of Daksh&umacr;r; in Nos. 57-9 the <I>se-qe&tdot;</I>, 3/4, is the co-
+tangent of an angle of 53&deg;7&prime;48&Prime;, which again is exactly the
+slope of the second pyramid of Gizeh as measured by Flinders
+Petrie; and the <I>se-qe&tdot;</I> in No. 60, which is 1/4, is the cotangent
+of an angle of 75&deg;57&prime;50&Prime;, corresponding exactly to the slope
+of the Mastaba-tombs of the Ancient Empire and of the
+sides of the M&emacr;d&umacr;m pyramid.<note>Flinders Petrie, <I>Pyramids and Temples of Gizeh</I>, p. 162.</note>
+<p>These measurements of <I>se-qe&tdot;</I> indicate at all events a rule-
+of-thumb use of geometrical proportion, and connect themselves
+naturally enough with the story of Thales's method of measuring
+the heights of pyramids.
+<C>The beginnings of Greek geometry.</C>
+<p>At the beginning of the summary of Proclus we are told
+that THALES (624-547 B. C.)
+&lsquo;first went to Egypt and thence introduced this study
+(geometry) into Greece. He discovered many propositions
+himself, and instructed his successors in the principles under-
+lying many others, his method of attack being in some cases
+more general (i. e. more theoretical or scientific), in others
+more empirical (<G>ai)sqhtikw/teron</G>, more in the nature of simple
+inspection or observation).&rsquo;<note>Proclus on Eucl. I, p. 65. 7-11.</note>
+<p>With Thales, therefore, geometry first becomes a deductive
+science depending on general propositions; this agrees with
+what Plutarch says of him as one of the Seven Wise Men:
+<p>&lsquo;he was apparently the only one of these whose wisdom
+stepped, in speculation, beyond the limits of practical utility:
+the rest acquired the reputation of wisdom in politics.&rsquo;<note>Plutarch, <I>Solon</I>, c. 3.</note>
+<p>(Not that Thales was inferior to the others in political
+wisdom. Two stories illustrate the contrary. He tried to
+save Ionia by urging the separate states to form a federation
+<pb n=129><head>MEASUREMENT OF PYRAMIDS</head>
+with a capital at Teos, that being the most central place in
+Ionia. And when Croesus sent envoys to Miletus to propose
+an alliance, Thales dissuaded his fellow-citizens from accepting
+the proposal, with the result that, when Cyrus conquered, the
+city was saved.)
+<C>(<G>a</G>) <I>Measurement of height of pyramid.</I></C>
+<p>The accounts of Thales's method of measuring the heights
+of pyramids vary. The earliest and simplest version is that
+of Hieronymus, a pupil of Aristotle, quoted by Diogenes
+Laertius:
+<p>&lsquo;Hieronymus says that he even succeeded in measuring the
+pyramids by observation of the length of their shadow at
+the moment when our shadows are equal to our own height.&rsquo;<note>Diog. L. i. 27.</note>
+<p>Pliny says that
+<p>&lsquo;Thales discovered how to obtain the height of pyramids
+and all other similar objects, namely, by measuring the
+shadow of the object at the time when a body and its shadow
+are equal in length.&rsquo;<note><I>N. H.</I> xxxvi. 12 (17).</note>
+<p>Plutarch embellishes the story by making Niloxenus say
+to Thales:
+<p>&lsquo;Among other feats of yours, he (Amasis) was particularly
+pleased with your measurement of the pyramid, when, without
+trouble or the assistance of any instrument, you merely set
+up a stick at the extremity of the shadow cast by the
+pyramid and, having thus made two triangles by the impact
+of the sun's rays, you showed that the pyramid has to the
+stick the same ratio which the shadow has to the shadow.&rsquo;<note>Plut. <I>Conv. sept. sap.</I> 2, p. 147 A.</note>
+<p>The first of these versions is evidently the original one and,
+as the procedure assumed in it is more elementary than the
+more general method indicated by Plutarch, the first version
+seems to be the more probable. Thales could not have failed
+to observe that, at the time when the shadow of a particular
+object is equal to its height, the same relation holds for all
+other objects casting a shadow; this he would probably
+infer by induction, after making actual measurements in a
+<pb n=130><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+considerable number of cases at a time when he found the
+length of the shadow of one object to be equal to its height.
+But, even if Thales used the more general method indicated
+by Plutarch, that method does not, any more than the Egyptian
+<I>se-qet</I> calculations, imply any general theory of similar tri-
+angles or proportions; the solution is itself a <I>se-qe&tdot;</I> calculation,
+just like that in No. 57 of Ahmes's handbook. In the latter
+problem the base and the <I>se-qe&tdot;</I> are given, and we have to
+find the height. So in Thales's problem we get a certain
+<I>se-qe&tdot;</I> by dividing the measured length of the shadow of the
+stick by the length of the stick itself; we then only require
+to know the distance between the point of the shadow corre-
+sponding to the apex of the pyramid and the centre of the
+base of the pyramid in order to determine the height; the
+only difficulty would be to measure or estimate the distance
+from the apex of the shadow to the centre of the base.
+<C>(<G>b</G>) <I>Geometrical theorems attributed to Thales.</I></C>
+<p>The following are the general theorems in elementary
+geometry attributed to Thales.
+<p>(1) He is said to have been the first to demonstrate that
+a circle is bisected by its diameter.<note>Proclus on Eucl. I, p. 157. 10.</note>
+<p>(2) Tradition credited him with the first statement of the
+theorem (Eucl. I. 5) that the angles at the base of any
+isosceles triangle are equal, although he used the more archaic
+term &lsquo;similar&rsquo; instead of &lsquo;equal&rsquo;.<note><I>Ib.</I>, pp. 250. 20-251. 2.</note>
+<p>(3) The proposition (Eucl. I. 15) that, if two straight lines
+cut one another, the vertical and opposite angles are equal
+was discovered, though not scientifically proved, by Thales.
+Eudemus is quoted as the authority for this.<note><I>Ib.</I>, p. 299. 1-5.</note>
+<p>(4) Eudemus in his History of Geometry referred to Thales
+the theorem of Eucl. I. 26 that, if two triangles have two
+angles and one side respectively equal, the triangles are equal
+in all respects.
+<p>&lsquo;For he (Eudemus) says that the method by which Thales
+showed how to find the distances of ships from the shore
+necessarily involves the use of this theorem.&rsquo;<note><I>Ib.</I>, p. 352. 14-18.</note>
+<pb n=131><head>GEOMETRICAL THEOREMS</head>
+<p>(5) &lsquo;Pamphile says that Thales, who learnt geometry from
+the Egyptians, was the first to describe on a circle a triangle
+(which shall be) right-angled (<G>katagra/yai ku/klou to\ tri/gwnon
+o)rqogw/nion</G>), and that he sacrificed an ox (on the strength of
+the discovery). Others, however, including Apollodorus the
+calculator, say that it was Pythagoras.&rsquo;<note>Diog. L. i. 24, 25.</note>
+<p>The natural interpretation of Pamphile's words is to suppose
+that she attributed to Thales the discovery that the angle
+in a semicircle is a right angle.
+<p>Taking these propositions in order, we may observe that,
+when Thales is said to have &lsquo;demonstrated&rsquo; (<G>a)podei=xai</G>) that
+a circle is bisected by its diameter, whereas he only &lsquo;stated&rsquo;
+the theorem about the isosceles triangle and &lsquo;discovered&rsquo;,
+without scientifically proving, the equality of vertically
+opposite angles, the word &lsquo;demonstrated&rsquo; must not be taken
+too literally. Even Euclid did not &lsquo;demonstrate&rsquo; that a circle
+is bisected by its diameter, but merely stated the fact in
+<FIG>
+I. Def. 17. Thales therefore probably
+observed rather than proved the property;
+and it may, as Cantor says, have been
+suggested by the appearance of certain
+figures of circles divided into a number
+of equal sectors by 2, 4, or 6 diameters
+such as are found on Egyptian monu-
+ments or represented on vessels brought
+by Asiatic tributary kings in the time of the eighteenth
+dynasty.<note>Cantor, <I>Gesch. d. Math.</I> i<SUP>3</SUP>, pp. 109, 140.</note>
+<p>It has been suggested that the use of the word &lsquo;similar&rsquo; to
+describe the equal angles of an isosceles triangle indicates that
+Thales did not yet conceive of an angle as a magnitude, but
+as a <I>figure</I> having a certain <I>shape</I>, a view which would agree
+closely with the idea of the Egyptian <I>se-qe&tdot;</I>, &lsquo;that which
+makes the nature&rsquo;, in the sense of determining a similar or
+the same inclination in the faces of pyramids.
+<p>With regard to (4), the theorem of Eucl. I. 26, it will be
+observed that Eudemus only inferred that this theorem was
+known to Thales from the fact that it is necessary to Thales's
+determination of the distance of a ship from the shore.
+Unfortunately the method used can only be conjectured.
+<pb n=132><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+The most usual supposition is that Thales, observing the ship
+from the top of a tower on the sea-shore, used the practical
+equivalent of the proportionality of the sides of two similar
+right-angled triangles, one small and one large. Suppose <I>B</I>
+to be the base of the tower, <I>C</I> the ship. It was only necessary
+<FIG>
+for a man standing at the top of the
+tower to have an instrument with
+two legs forming a right angle, to
+place it with one leg <I>DA</I> vertical and
+in a straight line with <I>B</I>, and the
+other leg <I>DE</I> in the direction of the
+ship, to take any point <I>A</I> on <I>DA</I>,
+and then to mark on <I>DE</I> the point <I>E</I>
+where the line of sight from <I>A</I> to <I>C</I> cuts the leg <I>DE.</I> Then
+<I>AD</I> (=<I>l</I>, say) and <I>DE</I> (=<I>m</I>, say) can be actually measured,
+as also the height <I>BD</I> (= <I>h</I>, say) from <I>D</I> to the foot of the
+tower, and, by similar triangles,
+<MATH><I>BC</I>=(<I>h</I>+<I>l</I>).<I>m</I>/<I>l</I></MATH>.
+The objection to this solution is that it does not depend
+directly on Eucl. I. 26, as Eudemus implies. Tannery<note>Tannery, <I>La g&eacute;om&eacute;trie grecque</I>, pp. 90-1.</note> there-
+fore favours the hypothesis of a solution on the lines followed
+by the Roman agrimensor Marcus Junius Nipsus in his
+<FIG>
+<I>fluminis varatio.</I>&mdash;To find the distance from
+<I>A</I> to an inaccessible point <I>B.</I> Measure from <I>A</I>,
+along a straight line at right angles to <I>AB</I>,
+a distance <I>AC</I>, and bisect it at <I>D.</I> From <I>C</I>, on
+the side of <I>AC</I> remote from <I>B</I>, draw <I>CE</I> at
+right angles to <I>AC</I>, and let <I>E</I> be the point on
+it which is in a straight line with <I>B</I> and <I>D.</I>
+Then clearly, by Eucl. I. 26, <I>CE</I> is equal to
+<I>AB</I>; and <I>CE</I> can be measured, so that <I>AB</I>
+is known.
+<p>This hypothesis is open to a different objec-
+tion, namely that, as a rule, it would be
+difficult, in the supposed case, to get a sufficient amount of
+free and level space for the construction and measurements.
+<p>I have elsewhere<note><I>The Thirteen Books of Euclid's Elements</I>, vol. i, p. 305.</note> suggested a still simpler method free
+<pb n=133><head>DISTANCE OF A SHIP AT SEA</head>
+from this objection, and depending equally directly on Eucl.
+I. 26. If the observer was placed on the top of a tower, he
+had only to use a rough instrument made of a straight stick
+and a cross-piece fastened to it so as to be capable of turning
+about the fastening (say a nail) so that it could form any
+angle with the stick and would remain where it was put.
+Then the natural thing would be to fix the stick upright (by
+means of a plumb-line) and direct the cross-piece towards the
+ship. Next, leaving the cross-piece at the angle so found,
+he would turn the stick round, while keeping it vertical, until
+the cross-piece pointed to some visible object on the shore,
+which would be mentally noted; after this it would only
+be necessary to measure the distance of the object from the
+foot of the tower, which distance would, by Eucl. I. 26, be
+equal to the distance of the ship. It appears that this precise
+method is found in so many practical geometries of the first
+century of printing that it must be assumed to have long
+been a common expedient. There is a story that one of
+Napoleon's engineers won the Imperial favour by quickly
+measuring, in precisely this way, the width of a stream that
+blocked the progress of the army.<note>David Eugene Smith, <I>The Teaching of Geometry</I>, pp. 172-3.</note>
+<p>There is even more difficulty about the dictum of Pamphile
+implying that Thales first discovered the fact that the angle
+in a semicircle is a right angle. Pamphile lived in the reign
+of Nero (A. D. 54-68), and is therefore a late authority. The
+date of Apollodorus the &lsquo;calculator&rsquo; or arithmetician is not
+known, but he is given as only one of several authorities who
+attributed the proposition to Pythagoras. Again, the story
+of the sacrifice of an ox by Thales on the occasion of his
+discovery is suspiciously like that told in the distich of
+Apollodorus &lsquo;when Pythagoras discovered that famous pro-
+position, on the strength of which he offered a splendid
+sacrifice of oxen&rsquo;. But, in quoting the distich of Apollodorus,
+Plutarch expresses doubt whether the discovery so celebrated
+was that of the theorem of the square of the hypotenuse or
+the solution of the problem of &lsquo;application of areas&rsquo;<note>Plutarch, <I>Non posse suaviter vivi secundum Epicurum</I>, c. 11, p. 1094 B.</note>; there
+is nothing about the discovery of the fact of the angle in
+a semicircle being a right angle. It may therefore be that
+<pb n=134><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+Diogenes Laertius was mistaken in bringing Apollodorus into
+the story now in question at all; the mere mention of the
+sacrifice in Pamphile's account would naturally recall Apollo-
+dorus's lines about Pythagoras, and Diogenes may have
+forgotten that they referred to a different proposition.
+<p>But, even if the story of Pamphile is accepted, there are
+difficulties of substance. As Allman pointed out, if Thales
+<FIG>
+knew that the angle in a semicircle
+is a right angle, he was in a position
+at once to infer that the sum of the
+angles of any <I>right-angled</I> triangle is
+equal to two right angles. For suppose
+that <I>BC</I> is the diameter of the semi-
+circle, <I>O</I> the centre, and <I>A</I> a point on
+the semicircle; we are then supposed
+to know that the angle <I>BAC</I> is a right angle. Joining <I>OA</I>,
+we form two isosceles triangles <I>OAB, OAC</I>; and Thales
+knows that the base angles in each of these triangles are
+equal. Consequently the sum of the angles <I>OAB, OAC</I> is
+equal to the sum of the angles <I>OBA, OCA.</I> The former sum
+is known to be a right angle; therefore the second sum is
+also a right angle, and the three angles of the triangle <I>ABC</I>
+are together equal to twice the said sum, i.e. to two right
+angles.
+<p>Next it would easily be seen that <I>any</I> triangle can be
+divided into two right-angled triangles by drawing a perpen-
+<FIG>
+dicular <I>AD</I> from a vertex <I>A</I> to the
+opposite side <I>BC.</I> Then the three
+angles of each of the right-angled
+triangles <I>ABD, ADC</I> are together equal
+to two right angles. By adding together
+the three angles of both triangles we
+find that the sum of the three angles of the triangle <I>ABC</I>
+together with the angles <I>ADB, ADC</I> is equal to four right
+angles; and, the sum of the latter two angles being two
+right angles, it follows that the sum of the remaining angles,
+the angles at <I>A, B, C</I>, is equal to two right angles. And <I>ABC</I>
+is <I>any</I> triangle.
+<p>Now Euclid in III. 31 proves that the angle in a semicircle
+is a right angle by means of the general theorem of I. 32
+<pb n=135><head>THE ANGLE IN A SEMICIRCLE</head>
+that the sum of the angles of any triangle is equal to two
+right angles; but if Thales was aware of the truth of the
+latter general proposition and proved the proposition about
+the semicircle in this way, by means of it, how did Eudemus
+come to credit the Pythagoreans, not only with the general
+proof, but with the <I>discovery</I>, of the theorem that the angles
+of any triangle are together equal to two right angles?<note>Proclus on Eucl. I, p. 379. 2-5.</note>
+<p>Cantor, who supposes that Thales proved his proposition
+after the manner of Euclid III. 31, i.e. by means of the general
+theorem of I. 32, suggests that Thales arrived at the truth of
+the latter, not by a general proof like that attributed by
+Eudemus to the Pythagoreans, but by an argument following
+the steps indicated by Geminus. Geminus says that
+<p>&lsquo;the <I>ancients</I> investigated the theorem of the two right
+angles in each individual species of triangle, first in the equi-
+lateral, then in the isosceles, and afterwards in the scalene
+triangle, but later geometers demonstrated the general theorem
+that in <I>any</I> triangle the three interior angles are equal to two
+right angles&rsquo;.<note>See Eutocius, Comm. on <I>Conics</I> of Apollonius (vol. ii, p. 170, Heib.).</note>
+<p>The &lsquo;later geometers&rsquo; being the Pythagoreans, it is assumed
+that the &lsquo;ancients&rsquo; may be Thales and his contemporaries.
+As regards the equilateral triangle, the fact might be suggested
+by the observation that six such triangles arranged round one
+point as common vertex would fill up the space round that
+point; whence it follows that each angle is one-sixth of four
+right angles, and three such angles make up two right angles.
+Again, suppose that in either an equilateral or an isosceles
+<FIG>
+triangle the vertical angle is bisected by a straight line meet-
+ing the base, and that the rectangle of which the bisector and
+one half of the base are adjacent sides is completed; the
+rectangle is double of the half of the original triangle, and the
+angles of the half-triangle are together equal to half the sum
+<pb n=136><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+of the angles of the rectangle, i.e. are equal to two right
+angles; and it immediately follows that the sum of the angles
+of the original equilateral or isosceles triangle is equal to two
+right angles. The same thing is easily proved of any triangle
+<FIG>
+by dividing it into two right-angled
+triangles and completing the rectangles
+which are their doubles respectively, as
+in the figure. But the fact that a proof
+on these lines is just as easy in the case
+of the general triangle as it is for the
+equilateral and isosceles triangles throws doubt on the whole
+procedure; and we are led to question whether there is any
+foundation for Geminus's account at all. Aristotle has a re-
+mark that
+<p>&lsquo;even if one should prove, with reference to each (sort of)
+triangle, the equilateral, scalene, and isosceles, separately, that
+each has its angles equal to two right angles, either by one
+proof or by different proofs, he does not yet know that <I>the
+triangle</I>, i.e. the triangle <I>in general</I>, has its angles equal to
+two right angles, except in a sophistical sense, even though
+there exists no triangle other than triangles of the kinds
+mentioned. For he knows it not <I>qu&acirc;</I> triangle, nor of <I>every</I>
+triangle, except in a numerical sense; he does not know it
+<I>notionally</I> of every triangle, even though there be actually no
+triangle which he does not know&rsquo;.<note>Arist. <I>Anal. Post.</I> i. 5, 74 a 25 sq.</note>
+<p>It may well be that Geminus was misled into taking for
+a historical fact what Aristotle gives only as a hypothetical
+illustration, and that the exact stages by which the proposi-
+tion was first proved were not those indicated by Geminus.
+<p>Could Thales have arrived at his proposition about the
+semicircle without assuming, or even knowing, that the sum
+of the angles of <I>any</I> triangle is equal to two right angles? It
+<FIG>
+seems possible, and in the following way.
+Many propositions were doubtless first
+discovered by drawing all sorts of figures
+and lines in them, and observing <I>apparent</I>
+relations of equality, &amp;c., between parts.
+It would, for example, be very natural
+to draw a rectangle, a figure with four right angles (which, it
+<pb n=137><head>THE ANGLE IN A SEMICIRCLE</head>
+would be found, could be drawn in practice), and to put in the
+two diagonals. The equality of the opposite sides would
+doubtless, in the first beginnings of geometry, be assumed as
+obvious, or verified by measurement. If then it was <I>assumed</I>
+that a rectangle is a figure with all its angles right angles and
+each side equal to its opposite, it would be easy to deduce
+certain consequences. Take first the two triangles <I>ADC, BCD.</I>
+Since by hypothesis <I>AD</I>=<I>BC</I> and <I>CD</I> is common, the two
+triangles have the sides <I>AD, DC</I> respectively equal to the sides
+<I>BC, CD</I>, and the included angles, being right angles, are equal;
+therefore the triangles <I>ADC, BCD</I> are equal in all respects
+(cf. Eucl. I. 4), and accordingly the angles <I>ACD</I> (i.e. <I>OCD</I>) and
+<I>BDC</I> (i.e. <I>ODC</I>) are equal, whence (by the converse of Eucl. I. 5,
+known to Thales) <I>OD</I>=<I>OC.</I> Similarly by means of the
+equality of <I>AB, CD</I> we prove the equality of <I>OB, OC.</I> Conse-
+quently <I>OB, OC, OD</I> (and <I>OA</I>) are all equal. It follows that
+a circle with centre <I>O</I> and radius <I>OA</I> passes through <I>B, C, D</I>
+also; since <I>AO, OC</I> are in a straight line, <I>AC</I> is a diameter of
+the circle, and the angle <I>ABC</I>, by hypothesis a right angle, is
+an &lsquo;angle in a semicircle&rsquo;. It would then appear that, given
+any right angle as <I>ABC</I> standing on <I>AC</I> as base, it was only
+necessary to bisect <I>AC</I> at <I>O</I>, and <I>O</I> would then be the centre of
+a semicircle on <I>AC</I> as diameter and passing through <I>B.</I> The
+construction indicated would be the construction of a circle
+about the right-angled triangle <I>ABC</I>, which seems to corre-
+spond well enough to Pamphile's phrase about &lsquo;describing on
+(i.e. in) a circle a triangle (which shall be) right angled&rsquo;.
+<C>(<G>g</G>) <I>Thales as astronomer.</I></C>
+<p>Thales was also the first Greek astronomer. Every one
+knows the story of his falling into a well when star-gazing,
+and being rallied by &lsquo;a clever and pretty maidservant from
+Thrace&rsquo; for being so eager to know what goes on in the
+heavens that he could not see what was straight in front
+of him, nay, at his very feet. But he was not merely a star-
+gazer. There is good evidence that he predicted a solar eclipse
+which took place on May 28, 585 B. C. We can conjecture
+the basis of this prediction. The Babylonians, as the result
+of observations continued through centuries, had discovered
+the period of 223 lunations after which eclipses recur; and
+<pb n=138><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+this period was doubtless known to Thales, either directly or
+through the Egyptians as intermediaries. Thales, however,
+cannot have known the <I>cause</I> of eclipses; he could not have
+given the true explanation of <I>lunar</I> eclipses (as the <I>Doxo-
+graphi</I> say he did) because he held that the earth is a circular
+disc floating on the water like a log; and, if he had correctly
+accounted for <I>solar</I> eclipses, it is impossible that all the
+succeeding Ionian philosophers should, one after another, have
+put forward the fanciful explanations which we find recorded.
+<p>Thales's other achievements in astronomy can be very
+shortly stated. Eudemus attributed to him the discovery of
+&lsquo;the fact that the period of the sun with reference to the
+solstices is not always the same&rsquo;<note>See Theon of Smyrna, p. 198. 17.</note>; the vague phrase seems
+to mean that he discovered the inequality of the length of
+the four astronomical seasons, that is, the four parts of the
+&lsquo;tropical&rsquo; year as divided by the solstices and equinoxes.
+Eudemus presumably referred to the written works by Thales
+<I>On the Solstice</I> and <I>On the Equinoxes</I> mentioned by Diogenes
+Laertius.<note>Diog. L. i. 23.</note> He knew of the division of the year into 365 days,
+which he probably learnt from Egypt.
+<p>Thales observed of the Hyades that there were two of
+them, one north and the other south. He used the Little
+Bear as a means of finding the pole, and advised the Greeks
+to sail by the Little Bear, as the Phoenicians did, in preference
+to their own practice of sailing by the Great Bear. This
+instruction was probably noted in the handbook under the
+title of <I>Nautical Astronomy</I>, attributed by some to Thales
+and by others to Phocus of Samos.
+<p>It became the habit of the <I>Doxographi</I> to assign to Thales,
+in common with other astronomers in each case, a number
+of discoveries not made till later. The following is the list,
+with the names of the astronomers to whom the respective
+discoveries may with most certainty be attributed: (1) the
+fact that the moon takes its light from the sun (Anaxagoras
+and possibly Parmenides); (2) the sphericity of the earth
+(Pythagoras); (3) the division of the heavenly sphere into
+five zones (Pythagoras and Parmenides); (4) the obliquity
+of the ecliptic (Oenopides of Chios); (5) the estimate of the
+<pb n=139><head>THALES AS ASTRONOMER</head>
+sun's diameter as 1/720th part of the sun's circle (Aristarchus
+of Samos).
+<C>From Thales to Pythagoras.</C>
+<p>We are completely in the dark as to the progress of geometry
+between the times of Thales and Pythagoras. ANAXIMANDER
+(born about 611/10 B.C.) put forward some daring and original
+hypotheses in astronomy. According to him the earth is
+a short cylinder with two bases (on one of which we live) and
+of depth equal to one-third of the diameter of either base.
+It is suspended freely in the middle of the universe without
+support, being kept there in equilibrium by virtue of its
+equidistance from the extremities and from the other heavenly
+bodies all round. The sun, moon, and stars are enclosed in
+opaque rings of compressed air concentric with the earth and
+filled with fire; what we see is the fire shining through vents
+(like gas-jets, as it were). The sun's ring is 27 or 28 times, the
+moon's ring 19 times, as large as the earth, i.e. the sun's
+and moon's distances are estimated in terms (as we may
+suppose) of the radius of the circular face of the earth; the
+fixed stars and the planets are nearer to the earth than
+the sun and moon. This is the first speculation on record
+about sizes and distances. Anaximander is also said to have
+introduced the <I>gnomon</I> (or sun-dial with a vertical needle)
+into Greece and to have shown on it the solstices, the times,
+the seasons, and the equinox<note>Euseb. <I>Praep. Evang.</I> x. 14. 11 (<I>Vors.</I> i<SUP>3</SUP>, p. 14. 28).</note> (according to Herodotus<note>Hdt. ii. 109.</note> the
+Greeks learnt the use of the <I>gnomon</I> from the Babylonians).
+He is also credited, like Thales before him, with having
+constructed a sphere to represent the heavens.<note>Diog. L. ii. 2.</note> But Anaxi-
+mander has yet another claim to undying fame. He was the
+first who ventured to draw a map of the inhabited earth.
+The Egyptians had drawn maps before, but only of particular
+districts; Anaximander boldly planned out the whole world
+with &lsquo;the circumference of the earth and sea&rsquo;.<note>Diog. L. <I>l. c.</I></note> This work
+involved of course an attempt to estimate the dimensions of
+the earth, though we have no information as to his results.
+It is clear, therefore, that Anaximander was something of
+<pb n=140><head>THE EARLIEST GREEK GEOMETRY. THALES</head>
+a mathematician; but whether he contributed anything to
+geometry as such is uncertain. True, Suidas says that he
+&lsquo;introduced the gnomon and generally set forth a sketch
+or outline of geometry&rsquo; (<G>o(/lws gewmetri/as u(potu/pwsin e)/deixen</G>);
+but it may be that &lsquo;geometry&rsquo; is here used in its literal sense
+of earth-measurement, and that the reference is only to the
+famous map.
+<p>&lsquo;Next to Thales, Ameristus, a brother of the poet Stesichorus,
+is mentioned as having engaged in the study of geometry;
+and from what Hippias of Elis says it appears that he acquired
+a reputation for geometry.&rsquo;<note>Proclus on Eucl. I, p. 65. 11-15.</note>
+<p>Stesichorus the poet lived about 630-550 B.C. The brother
+therefore would probably be nearly contemporary with Thales.
+We know nothing of him except from the passage of Proclus,
+and even his name is uncertain. In Friedlein's edition of
+Proclus it is given as Mamercus, after a later hand in cod.
+Monac. 427; Suidas has it as Mamertinus (<I>s.v.</I> Stesichorus);
+Heiberg in his edition of Heron's <I>Definitions</I> writes Mamertius,
+noting <G>*marme/tios</G> as the reading of Cod. Paris. Gr. 2385.
+<pb>
+<C>V</C>
+<C>PYTHAGOREAN GEOMETRY</C>
+<p>The special service rendered by PYTHAGORAS to geometry is
+thus described in the Proclus summary:
+<p>&lsquo;After these (Thales and Ameristus or Mamercus) Pythagoras
+transformed the study of geometry into a liberal education,
+examining the principles of the science from the beginning
+and probing the theorems in an immaterial and intellectual
+manner: he it was who discovered the theory of irrationals&rsquo;
+(or &lsquo;proportions&rsquo;) &lsquo;and the construction of the cosmic figures&rsquo;.<note>Proclus on Eucl. I, p. 65. 15-21.</note>
+<p>These supposed discoveries will claim our attention pre-
+sently; the rest of the description agrees with another
+passage about the Pythagoreans:
+<p>&lsquo;Herein&rsquo;, says Proclus, &lsquo;I emulate the Pythagoreans who
+even had a conventional phrase to express what I mean,
+&ldquo;a figure and a platform, not a figure and sixpence&rdquo;, by
+which they implied that the geometry which is deserving of
+study is that which, at each new theorem, sets up a platform to
+ascend by, and lifts the soul on high instead of allowing it
+to go down among sensible objects and so become subser-
+vient to the common needs of this mortal life&rsquo;.<note><I>Ib.</I>, p. 84. 15-22.</note>
+<p>In like manner we are told that &lsquo;Pythagoras used defini-
+tions on account of the mathematical nature of the subject&rsquo;,<note>Favorinus in Diog. L. viii. 25.</note>
+which again implies that he took the first steps towards the
+systematization of geometry as a subject in itself.
+<p>A comparatively early authority, Callimachus (about 250 B.C.),
+is quoted by Diodorus as having said that Pythagoras dis-
+covered some geometrical problems himself and was the first
+to introduce others from Egypt into Greece.<note>Diodorus x. 6. 4 (<I>Vors.</I> i<SUP>3</SUP>, p. 346. 23).</note> Diodorus gives
+what appear to be five verses of Callimachus <I>minus</I> a few words;
+<pb n=142><head>PYTHAGOREAN GEOMETRY</head>
+a longer fragment including the same passage is now available
+(though the text is still deficient) in the Oxyrhynchus Papyri.<note><I>Oxyrhynchus Papyri</I>, Pt. vii, p. 33 (Hunt).</note>
+The story is that one Bathycles, an Arcadian, bequeathed a
+cup to be given to the best of the Seven Wise Men. The cup
+first went to Thales, and then, after going the round of the
+others, was given to him a second time. We are told that
+Bathycles's son brought the cup to Thales, and that (presum-
+ably on the occasion of the first presentation)
+<p>&lsquo;by a happy chance he found . . . the old man scraping the
+ground and drawing the figure discovered by the Phrygian
+Euphorbus (= Pythagoras), who was the first of men to draw
+even scalene triangles and a circle . . ., and who prescribed
+abstinence from animal food&rsquo;.
+<p>Notwithstanding the anachronism, the &lsquo;figure discovered by
+Euphorbus&rsquo; is presumably the famous proposition about the
+squares on the sides of a right-angled triangle. In Diodorus's
+quotation the words after &lsquo;scalene triangles&rsquo; are <G>ku/klon e(pta-
+mh/kh</G>(<G>e(ptamh/ke</G>&rsquo; Hunt), which seems unintelligible unless the
+&lsquo;seven-lengthed circle&rsquo; can be taken as meaning the &lsquo;lengths of
+seven circles&rsquo; (in the sense of the seven independent orbits
+of the sun, moon, and planets) or the circle (the zodiac) com-
+prehending them all.<note>The papyrus has an accent over the <G>e</G> and to the right of the
+accent, above the uncertain <G>p</G>, the appearance of a <G>l</G> in dark ink,
+<G>l</G>
+thus <G>kaikuklone/p</G>, a reading which is not yet satisfactorily explained.
+Diels (<I>Vorsokratiker</I>, i<SUP>3</SUP>, p. 7) considers that the accent over the <G>e</G> is fatal
+to the reading <G>e(ptamh/kh</G>, and conjectures <G>kai\ ku/klon e(/l(ika) kh)di/dace
+nhsteu/ein</G> instead of Hunt's <G>kai\ ku/klon e(p</G>[<G>tamh/ke', h)de\ nhsteu/ein</G>] and
+Diodorus's <G>kai\ ku/klon e(ptamh/kh di/dace nhsteu/ein</G>. But <G>ku/klon e(/lika</G>, &lsquo;twisted
+(or curved) circle&rsquo;, is very indefinite. It may have been suggested to
+Diels by Hermesianax's lines (Athenaeus xiii. 599 A) attributing to
+Pythagoras the &lsquo;refinements of the geometry of spirals&rsquo; (<G>e(li/kwn komya\
+gewmetri/hs</G>). One naturally thinks of Plato's dictum (<I>Timaeus</I> 39 A, B)
+about the circles of the sun, moon, and planets being twisted into spirals
+by the combination of their own motion with that of the daily rotation;
+but this can hardly be the meaning here. A more satisfactory sense
+would be secured if we could imagine the circle to be the circle described
+about the &lsquo;scalene&rsquo; (right-angled) triangle, i.e. if we could take the
+reference to be to the discovery of the fact that the angle in a semi-
+circle is a right angle, a discovery which, as we have seen, was alterna-
+tively ascribed to Thales and Pythagoras.</note>
+<p>But it is time to pass on to the propositions in geometry
+which are definitely attributed to the Pythagoreans.
+<pb n=143><head>PYTHAGOREAN GEOMETRY</head>
+<C>Discoveries attributed to the Pythagoreans.</C>
+<C>(<G>a</G>) <I>Equality of the sum of the three angles of a triangle
+to two right angles.</I></C>
+<p>We have seen that Thales, if he really discovered that the
+angle in a semicircle is a right angle, was in a position, first,
+to show that in any right-angled triangle the sum of the three
+angles is equal to two right angles, and then, by drawing the
+perpendicular from a vertex of any triangle to the opposite
+side and so dividing the triangle into two right-angled
+triangles, to prove that the sum of the three angles of any
+triangle whatever is equal to two right angles. If this method
+of passing from the particular case of a right-angled triangle to
+that of any triangle did not occur to Thales, it is at any rate
+hardly likely to have escaped Pythagoras. But all that we know
+for certain is that Eudemus referred to the Pythagoreans
+the discovery of the general theorem that in any triangle
+the sum of the interior angles is equal to two right angles.<note>Proclus on Eucl. I, p. 397. 2.</note>
+Eudemus goes on to tell us how they proved it. The method
+differs slightly from that of Euclid, but depends, equally with
+Euclid's proof, on the properties of parallels; it can therefore
+only have been evolved at a time when those properties were
+already known.
+<p>Let <I>ABC</I> be any triangle; through <I>A</I> draw <I>DE</I> parallel
+to <I>BC</I>.
+<FIG>
+<p>Then, since <I>BC, DE</I> are parallel, the
+alternate angles <I>DAB, ABC</I> are equal.
+<p>Similarly the alternate angles <I>EAC,
+ACB</I> are equal.
+<p>Therefore the sum of the angles <I>ABC,
+ACB</I> is equal to the sum of the angles <I>DAB, EAC</I>.
+<p>Add to each sum the angle <I>BAC</I>; therefore the sum of the
+three angles <I>ABC, ACB, BAC</I>, i.e. the three angles of the
+triangle, is equal to the sum of the angles <I>DAB, BAC, CAE</I>,
+i.e. to two right angles.
+<p>We need not hesitate to credit the Pythagoreans with the
+more general propositions about the angles of any polygon,
+<pb n=144><head>PYTHAGOREAN GEOMETRY</head>
+namely (1) that, if <I>n</I> be the number of the sides or angles, the
+interior angles of the polygon are together equal to 2<I>n</I> - 4
+right angles, and (2) that the exterior angles of the polygon
+(being the supplements of the interior angles respectively)
+are together equal to four right angles. The propositions are
+interdependent, and Aristotle twice quotes the latter.<note><I>An. Post.</I> i. 24, 85 b 38; <I>ib.</I> ii. 17, 99 a 19.</note> The
+Pythagoreans also discovered that the only three regular
+polygons the angles of which, if placed together round a com-
+mon point as vertex, just fill up the space (four right angles)
+round the point are the equilateral triangle, the square, and
+the regular hexagon.
+<C>(<G>b</G>) <I>The &lsquo;Theorem of Pythagoras&rsquo</I>; (= Eucl. I. 47).</C>
+<p>Though this is the proposition universally associated by
+tradition with the name of Pythagoras, no really trustworthy
+evidence exists that it was actually discovered by him. The
+comparatively late writers who attribute it to him add the
+story that he sacrificed an ox to celebrate his discovery.
+Plutarch<note>Plutarch, <I>Non posse suaviter vivi secundum Epicurum</I>, c. 11, p. 1094 B.</note> (born about A.D. 46), Athenaeus<note>Athenaeus x. 418 F.</note> (about A.D. 200),
+and Diogenes Laertius<note>Diog. L. viii. 12, i. 25.</note> (A.D. 200 or later) all quote the verses
+of Apollodorus the &lsquo;calculator&rsquo; already referred to (p. 133).
+But Apollodorus speaks of the &lsquo;famous theorem&rsquo;, or perhaps
+&lsquo;figure&rsquo; (<G>gra/mma</G>), the discovery of which was the occa-
+sion of the sacrifice, without saying what the theorem was.
+Apollodorus is otherwise unknown; he may have been earlier
+than Cicero, for Cicero<note>Cicero, <I>De nat. deor.</I> iii. 36, 88.</note> tells the story in the same form
+without specifying what geometrical discovery was meant,
+and merely adds that he does not believe in the sacrifice,
+because the Pythagorean ritual forbade sacrifices in which
+blood was shed. Vitruvius<note>Vitruvius, <I>De architectura</I>, ix. pref.</note> (first century B.C.) connects the
+sacrifice with the discovery of the property of the particular
+triangle 3, 4, 5. Plutarch, in quoting Apollodorus, questions
+whether the theorem about the square of the hypotenuse was
+meant, or the problem of the application of an area, while in
+another place<note>Plutarch, <I>Quaest. conviv.</I> viii. 2, 4, p. 720 A.</note> he says that the occasion of the sacrifice was
+<pb n=145><head>THE &lsquo;THEOREM OF PYTHAGORAS&rsquo;</head>
+the solution of the problem, &lsquo;given two figures, to <I>apply</I>
+a third which shall be equal to the one and similar to
+the other&rsquo;, and he adds that this problem is unquestionably
+finer than the theorem about the square on the hypotenuse.
+But Athenaeus and Porphyry<note>Porphyry, <I>Vit. Pyth.</I> 36.</note> (A.D. 233-304) connect the
+sacrifice with the latter proposition; so does Diogenes Laertius
+in one place. We come lastly to Proclus, who is very cautious,
+mentioning the story but declining to commit himself to
+the view that it was Pythagoras or even any single person
+who made the discovery:
+<p>&lsquo;If we listen to those who wish to recount ancient history,
+we may find some of them referring this theorem to Pytha-
+goras, and saying that he sacrificed an ox in honour of his
+discovery. But for my part, while I admire <I>those who</I> first
+observed the truth of this theorem, I marvel more at the
+writer of the Elements, not only because he made it fast by a
+most lucid demonstration, but because he compelled assent to
+the still more general theorem by the irrefutable arguments of
+science in the sixth book.&rsquo;
+<p>It is possible that all these authorities may have built upon
+the verses of Apollodorus; but it is remarkable that, although
+in the verses themselves the particular theorem is not speci-
+fied, there is practical unanimity in attributing to Pythagoras
+the theorem of Eucl. I. 47. Even in Plutarch's observations
+expressing doubt about the particular occasion of the sacrifice
+there is nothing to suggest that he had any hesitation in
+accepting as discoveries of Pythagoras <I>both</I> the theorem of the
+square on the hypotenuse and the problem of the application
+of an area. Like Hankel,<note>Hankel, <I>Zur Geschichte der Math. in Alterthum und Mittelalter</I>, p. 97.</note> therefore, I would not go so far as
+to deny to Pythagoras the credit of the discovery of our pro-
+position; nay, I like to believe that tradition is right, and that
+it was really his.
+<p>True, the discovery is also claimed for India.<note>B&uuml;rk in the <I>Zeitschrift der morgenl&auml;nd. Gesellschaft</I>, lv, 1901, pp. 543-91; lvi, 1902, pp. 327-91.</note> The work
+relied on is the <I>&Amacr;pastamba-&Sacute;ulba-S&umacr;tra</I>, the date of which is
+put at least as early as the fifth or fourth century B.C., while
+it is remarked that the matter of it must have been much
+<pb n=146><head>PYTHAGOREAN GEOMETRY</head>
+older than the book itself; thus one of the constructions for
+right angles, using cords of lengths 15, 36, 39 (= 5, 12, 13), was
+known at the time of the <I>T&amacr;ittir&imacr;ya Samhit&amacr;</I> and the <I>Sata-
+patha Br&amacr;hmana</I>, still older works belonging to the eighth
+century B.C. at latest. A feature of the <I>&Amacr;pastamba-&Sacute;ulba-
+S&umacr;tra</I> is the construction of right angles in this way by means
+of cords of lengths equal to the three sides of certain rational
+right-angled triangles (or, as &Amacr;pastamba calls them, rational
+rectangles, i.e. those in which the diagonals as well as the
+sides are rational). The rational right-angled triangles actually
+used are (3, 4, 5), (5, 12, 13), (8, 15, 17), (12, 35, 37). There is
+a proposition stating the theorem of Eucl. I. 47 as a fact in
+general terms, but without proof, and there are rules based
+upon it for constructing a square equal to (1) the sum of two
+given squares and (2) the difference of two squares. But
+certain considerations suggest doubts as to whether the
+proposition had been established by any proof applicable to
+all cases. Thus &Amacr;pastamba mentions only seven rational
+right-angled triangles, really reducible to the above-mentioned
+four (one other, 7, 24, 25, appears, it is true, in the B&amacr;udh&amacr;-
+yana &Sacute;. S., supposed to be older than &Amacr;pastamba); he had no
+general rule such as that attributed to Pythagoras for forming
+any number of rational right-angled triangles; he refers to
+his seven in the words &lsquo;so many <I>recognizable</I> constructions
+are there&rsquo;, implying that he knew of no other such triangles.
+On the other hand, the truth of the theorem was recognized in
+the case of the isosceles right-angled triangle; there is even
+a construction for &radic;2, or the length of the diagonal of a square
+with side unity, which is constructed as <MATH>(1+1/3+1/(3.4)-1/(3.4.34))</MATH>
+of the side, and is then used with the side for the purpose of
+drawing the square on the side: the length taken is of course
+an approximation to &radic;2 derived from the consideration that
+<MATH>2.12<SUP>2</SUP>=288=17<SUP>2</SUP>-1</MATH>; but the author does not say anything
+which suggests any knowledge on his part that the approxi-
+mate value is not exact. Having drawn by means of the
+approximate value of the diagonal an inaccurate square, he
+proceeds to use it to construct a square with area equal to
+three times the original square, or, in other words, to con-
+struct &radic;3, which is therefore only approximately found.
+<pb n=147><head>THE &lsquo;THEOREM OF PYTHAGORAS&rsquo;</head>
+Thus the theorem is enunciated and used as if it were of
+general application; there is, however, no sign of any general
+proof; there is nothing in fact to show that the assumption of
+its universal truth was founded on anything better than an
+imperfect induction from a certain number of cases, discovered
+empirically, of triangles with sides in the ratios of whole
+numbers in which the property (1) that the square on the
+longest side is equal to the sum of the squares on the other
+two was found to be always accompanied by the property
+(2) that the latter two sides include a right angle. But, even
+if the Indians had actually attained to a scientific proof of
+the general theorem, there is no evidence or probability that
+the Greeks obtained it from India; the subject was doubtless
+developed quite independently in the two countries.
+<p>The next question is, how was the theorem proved by
+Pythagoras or the Pythagoreans? Vitruvius says that
+Pythagoras first discovered the triangle (3, 4, 5), and doubtless
+the theorem was first suggested by the discovery that this
+triangle is right-angled; but this discovery probably came
+to Greece from Egypt. Then a very simple construction
+would show that the theorem is true of an <I>isosceles</I> right-
+angled triangle. Two possible lines are suggested on which
+the general proof may have been developed. One is that of
+decomposing square and rectangular areas into squares, rect-
+angles and triangles, and piecing them together again after
+the manner of Eucl., Book II; the isosceles right-angled
+triangle gives the most obvious case of this method. The
+other line is one depending upon proportions; and we have
+good reason for supposing that Pythagoras developed a theory
+of proportion. That theory was applicable to commensurable
+magnitudes only; but this would not be any obstacle to the
+use of the method so long as the existence of the incom-
+mensurable or irrational remained undiscovered. From
+Proclus's remark that, while he admired those who first
+noticed the truth of the theorem, he admired Euclid still
+more for his most clear proof of it and for the irrefutable
+demonstration of the extension of the theorem in Book VI,
+it is natural to conclude that Euclid's proof in I. 47 was new,
+though this is not quite certain. Now VI. 31 could be proved
+at once by using I. 47 along with VI. 22; but Euclid proves
+<pb n=148><head>PYTHAGOREAN GEOMETRY</head>
+it independently of I. 47 by means of proportions. This
+seems to suggest that he proved I. 47 by the methods of
+Book I instead of by proportions in order to get the proposi-
+tion into Book I instead of Book VI, to which it must have
+been relegated if the proof by proportions had been used.
+If, on the other hand, Pythagoras had proved it by means
+of the methods of Books I and II, it would hardly have been
+necessary for Euclid to devise a new proof of I. 47. Hence
+it would appear most probable that Pythagoras would prove
+the proposition by means of his (imperfect) theory of pro-
+portions. The proof may have taken one of three different
+shapes.
+<FIG>
+<p>(1) If <I>ABC</I> is a triangle right-
+angled at <I>A</I>, and <I>AD</I> is perpen-
+dicular to <I>BC</I>, the triangles <I>DBA,
+DAC</I> are both similar to the tri-
+angle <I>ABC</I>.
+<p>It follows from the theorems of
+Eucl. VI. 4 and 17 that
+<MATH><I>BA</I><SUP>2</SUP>=<I>BD.BC</I></MATH>,
+<MATH><I>AC</I><SUP>2</SUP>=<I>CD.BC</I></MATH>,
+whence, by addition, <MATH><I>BA</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>=<I>BC</I><SUP>2</SUP></MATH>.
+<p>It will be observed that this proof is <I>in substance</I> identical
+with that of Eucl. I. 47, the difference being that the latter
+uses the relations between parallelograms and triangles on
+the same base and between the same parallels instead of
+proportions. The probability is that it was this particular
+proof by proportions which suggested to Euclid the method
+of I. 47; but the transformation of the proof depending on
+proportions into one based on Book I only (which was abso-
+lutely required under Euclid's arrangement of the <I>Elements</I>)
+was a stroke of genius.
+<p>(2) It would be observed that, in the similar triangles
+<I>DBA, DAC, ABC</I>, the corresponding sides opposite to the
+right angle in each case are <I>BA, AC, BC</I>.
+<p>The triangles therefore are in the duplicate ratios of these
+sides, and so are the squares on the latter.
+<p>But of the triangles two, namely <I>DBA, DAC</I>, make up the
+third, <I>ABC</I>.
+<pb n=149><head>THE &lsquo;THEOREM OF PYTHAGORAS&rsquo;</head>
+<p>The same must therefore be the case with the squares, or
+<MATH><I>BA</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>=<I>BC</I><SUP>2</SUP></MATH>.
+<p>(3) The method of VI. 31 might have been followed
+exactly, with squares taking the place of any similar recti-
+lineal figures. Since the triangles <I>DBA, ABC</I> are similar,
+<MATH><I>BD</I>:<I>AB</I>=<I>AB</I>:<I>BC</I></MATH>,
+or <I>BD, AB, BC</I> are three proportionals, whence
+<MATH><I>AB</I><SUP>2</SUP>:<I>BC</I><SUP>2</SUP>=<I>BD</I><SUP>2</SUP>:<I>AB</I><SUP>2</SUP>=<I>BD</I>:<I>BC</I></MATH>.
+<p>Similarly, <MATH><I>AC</I><SUP>2</SUP>:<I>BC</I><SUP>2</SUP>=<I>CD</I>:<I>BC</I></MATH>.
+<p>Therefore <MATH>(<I>BA</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>):<I>BC</I><SUP>2</SUP>=(<I>BD</I>+<I>DC</I>):<I>BC</I>. [V. 24]
+=1</MATH>.
+<p>If, on the other hand, the proposition was originally proved
+by the methods of Euclid, Books I, II alone (which, as I have
+said, seems the less probable supposition), the suggestion of
+<FIG>
+Bretschneider and Hankel seems to be the best. According
+to this we are to suppose, first, a figure like that of Eucl.
+II. 4, representing a larger square, of side (<I>a</I>+<I>b</I>), divided
+into two smaller squares of sides <I>a, b</I> respectively, and
+two complements, being two equal rectangles with <I>a, b</I> as
+sides.
+<p>Then, dividing each complementary rectangle into two
+equal triangles, we dispose the four triangles round another
+square of side <I>a</I>+<I>b</I> in the manner shown in the second figure.
+<p>Deducting the four triangles from the original square in
+each case we get, in the first figure, two squares <I>a</I><SUP>2</SUP> and <I>b</I><SUP>2</SUP>
+and, in the second figure, one square on <I>c</I>, the diagonal of the
+rectangle (<I>a, b</I>) or the hypotenuse of the right-angled triangle
+in which <I>a, b</I> are the sides about the right angle. It follows
+that <MATH><I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>=<I>c</I><SUP>2</SUP></MATH>.
+<pb n=150><head>PYTHAGOREAN GEOMETRY</head>
+<C>(<G>g</G>) <I>Application of areas and geometrical algebra.</I></C>
+<p>We have seen that, in connexion with the story of the
+sacrifice of an ox, Plutarch attributes to Pythagoras himself
+the discovery of the problem of the application of an area
+or, as he says in another place, the problem &lsquo;Given two
+figures, to &ldquo;apply&rdquo; a third figure which shall be equal to the
+one, and similar to the other (of the given figures).&rsquo; The
+latter problem (= Eucl. VI. 25) is, strictly speaking, not so
+much a case of <I>applying</I> an area as of <I>constructing</I> a figure,
+because the base is not given in length; but it depends
+directly upon the simplest case of &lsquo;application of areas&rsquo;,
+namely the problem, solved in Eucl. I. 44, 45, of applying
+to a given straight line as base a parallelogram containing
+a given angle and equal in area to a given triangle or
+rectilineal figure. The method of application of areas is
+fundamental in Greek geometry and requires detailed notice.
+We shall see that in its general form it is equivalent to the
+geometrical solution of a mixed quadratic equation, and it is
+therefore an essential part of what has been appropriately
+called <I>geometrical algebra</I>.
+<p>It is certain that the theory of application of areas
+originated with the Pythagoreans, if not with Pythagoras
+himself. We have this on the authority of Eudemus, quoted
+in the following passage of Proclus:
+<p>&lsquo;These things, says Eudemus, are ancient, being discoveries
+of the Muse of the Pythagoreans, I mean the <I>application of
+areas</I> (<G>parabolh\ tw=n xwri/wn</G>), their <I>exceeding</I> (<G>u(perbolh/</G>) and
+their <I>falling short</I> (<G>e)/lleiyis</G>). It was from the Pythagoreans
+that later geometers [i.e. Apollonius of Perga] took the
+names, which they then transferred to the so-called <I>conic</I>
+lines (curves), calling one of these a <I>parabola</I> (application),
+another a <I>hyperbola</I> (exceeding), and the third an <I>ellipse</I>
+(falling short), whereas those god-like men of old saw the
+things signified by these names in the construction, in a plane,
+of areas upon a given finite straight line. For, when you
+have a straight line set out, and lay the given area exactly
+alongside the whole of the straight line, they say that you
+<I>apply</I> the said area; when, however, you make the length of
+the area greater than the straight line, it is said to <I>exceed</I>,
+and, when you make it less, in which case after the area has
+been drawn there is some part of the straight line extending
+<pb n=151><head>APPLICATION OF AREAS</head>
+beyond it, it is said to <I>fall short</I>. Euclid, too, in the sixth
+book speaks in this way both of exceeding and falling short;
+but in this place (I. 44) he needed the <I>application</I> simply, as
+he sought to apply to a given straight line an area equal
+to a given triangle, in order that we might have in our
+power, not only the <I>construction</I> (<G>su/stasis</G>) of a parallelogram
+equal to a given triangle, but also the application of it to
+a limited straight line.&rsquo;<note>Proclus on Eucl. I, pp. 419. 15-420. 12.</note>
+<p>The general form of the problem involving <I>application</I>
+with <I>exceeding</I> or <I>falling short</I> is the following:
+<p>&lsquo;To apply to a given straight line a rectangle (or, more
+generally, a parallelogram) equal to a given rectilineal figure,
+and (1) <I>exceeding</I> or (2) <I>falling short</I> by a square figure (or,
+in the more general case, by a parallelogram similar to a given
+parallelogram).&rsquo;
+<p>The most general form, shown by the words in brackets,
+is found in Eucl. VI. 28, 29, which are equivalent to the
+geometrical solution of the quadratic equations
+<MATH><I>ax</I>&plusmn;(<I>b</I>/<I>c</I>)<I>x</I><SUP>2</SUP>=<I>C</I>/<I>m</I></MATH>,
+and VI. 27 gives the condition of possibility of a solution
+when the sign is negative and the parallelogram <I>falls short</I>.
+This general case of course requires the use of proportions;
+but the simpler case where the area applied is a rectangle,
+and the form of the portion which overlaps or falls short
+is a square, can be solved by means of Book II only. The
+proposition II. 11 is the geometrical solution of the particular
+quadratic equation
+<MATH><I>a</I>(<I>a</I>-<I>x</I>)=<I>x</I><SUP>2</SUP></MATH>,
+or <MATH><I>x</I><SUP>2</SUP>+<I>ax</I>=<I>a</I><SUP>2</SUP></MATH>.
+The propositions II. 5 and 6 are in the form of theorems.
+Taking, e.g., the figure of the former proposition, and sup-
+posing <MATH><I>AB</I>=<I>a</I>, <I>BD</I>=<I>x</I></MATH>, we have
+<MATH><I>ax</I>-<I>x</I><SUP>2</SUP>=rectangle <I>AH</I>
+=gnomon <I>NOP</I></MATH>.
+If, then, the area of the gnomon is given (= <I>b</I><SUP>2</SUP>, say, for any
+area can be transformed into the equivalent square by means
+of the problems of Eucl. I. 45 and II. 14), the solution of the
+equation <MATH><I>ax</I>-<I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP></MATH>
+<pb n=152><head>PYTHAGOREAN GEOMETRY</head>
+would be, in the language of application of areas, &lsquo;To a given
+straight line (<I>a</I>) to apply a rectangle which shall be equal
+to a given square (<I>b</I><SUP>2</SUP>) and shall fall short by a square figure.&rsquo;
+<FIG>
+<p>As the Pythagoreans solved the somewhat similar equation
+in II. 11, they cannot have failed to solve this one, as well as
+the equations corresponding to II. 6. For in the present case
+it is only necessary to draw <I>CQ</I> at right angles to <I>AB</I> from
+its middle point <I>C</I>, to make <I>CQ</I> equal to <I>b</I>, and then, with
+centre <I>Q</I> and radius equal to <I>CB</I>, or 1/2<I>a</I>, to draw a circle
+cutting <I>QC</I> produced in <I>R</I> and <I>CB</I> in <I>D</I> (<I>b</I><SUP>2</SUP> must be not
+greater than 1/2<I>a</I><SUP>2</SUP>; otherwise a solution is impossible).
+<p>Then the determination of the point <I>D</I> constitutes the
+solution of the quadratic.
+<p>For, by the proposition II. 5,
+<MATH><I>AD.DB</I>+<I>CD</I><SUP>2</SUP>=<I>CB</I><SUP>2</SUP>
+=<I>QD</I><SUP>2</SUP>=<I>QC</I><SUP>2</SUP>+<I>CD</I><SUP>2</SUP></MATH>;
+therefore <MATH><I>AD.DB</I>=<I>QC</I><SUP>2</SUP></MATH>,
+or <MATH><I>ax</I>-<I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP></MATH>.
+<p>Similarly II. 6 enables us to solve the equations
+<MATH><I>ax</I>+<I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP></MATH>,
+and <MATH><I>x</I><SUP>2</SUP>-<I>ax</I>=<I>b</I><SUP>2</SUP></MATH>;
+<FIG>
+the first equation corresponding to <I>AB</I>=<I>a</I>, <I>BD</I>=<I>x</I> and the
+second to <I>AB</I>=<I>a</I>, <I>AD</I>=<I>x</I>, in the figure of the proposition.
+<p>The application of the theory to conics by Apollonius will
+be described when we come to deal with his treatise.
+<p>One great feature of Book II of Euclid's <I>Elements</I> is the
+use of the <I>gnomon</I> (Props. 5 to 8), which is undoubtedly
+Pythagorean and is connected, as we have seen, with the
+<pb n=153><head>APPLICATION OF AREAS</head>
+application of areas. The whole of Book II, with the latter
+section of Book I from Prop. 42 onwards, may be said to deal
+with the transformation of areas into equivalent areas of
+different shape or composition by means of &lsquo;application&rsquo;
+and the use of the theorem of I. 47. Eucl. II. 9 and 10 are
+special cases which are very useful in geometry generally, but
+were also employed by the Pythagoreans for the specific purpose
+of proving the property of &lsquo;side-&rsquo; and &lsquo;diameter-&rsquo; numbers,
+the object of which was clearly to develop a series of closer
+and closer approximations to the value of &radic;2 (see p. 93 <I>ante</I>).
+<p>The <I>geometrical algebra</I>, therefore, as we find it in Euclid,
+Books I and II, was Pythagorean. It was of course confined
+to problems not involving expressions above the second degree.
+Subject to this, it was an effective substitute for modern
+algebra. The product of two linear factors was a rect-
+angle, and Book II of Euclid made it possible to <I>multiply</I>
+two factors with any number of linear terms in each; the
+compression of the result into a single product (rectangle)
+followed by means of the <I>application</I>-theorem (Eucl. I. 44).
+That theorem itself corresponds to <I>dividing</I> the product of
+any two linear factors by a third linear expression. To trans-
+form any area into a square, we have only to turn the area
+into a rectangle (as in Eucl. I. 45), and then find a square
+equal to that rectangle by the method of Eucl. II. 14; the
+latter problem then is equivalent to the <I>extraction of the square
+root</I>. And we have seen that the theorems of Eucl. II. 5, 6
+enable mixed quadratic equations of certain types to be solved
+so far as their roots are real. In cases where a quadratic
+equation has one or both roots negative, the Greeks would
+transform it into one having a positive root or roots (by the
+equivalent of substituting -<I>x</I> for <I>x</I>); thus, where one root is
+positive and one negative, they would solve the problem in
+two parts by taking two cases.
+<p>The other great engine of the Greek geometrical algebra,
+namely the method of proportions, was not in its full extent
+available to the Pythagoreans because their theory of pro-
+portion was only applicable to commensurable magnitudes
+(Eudoxus was the first to establish the general theory, applic-
+able to commensurables and incommensurables alike, which
+we find in Eucl. V, VI). Yet it cannot be doubted that they
+<pb n=154><head>PYTHAGOREAN GEOMETRY</head>
+used the method quite freely before the discovery of the irra-
+tional showed them that they were building on an insecure
+and inadequate foundation.
+<C>(<G>d</G>) <I>The irrational.</I></C>
+<p>To return to the sentence about Pythagoras in the summary
+of Proclus already quoted more than once (pp. 84, 90, 141).
+Even if the reading <G>a)lo/gwn</G> were right and Proclus really
+meant to attribute to Pythagoras the discovery of &lsquo;the theory,
+or study, of irrationals&rsquo;, it would be necessary to consider the
+authority for this statement, and how far it is supported by
+other evidence. We note that it occurs in a relative sentence
+<G>o(\s dh\</G> . . ., which has the appearance of being inserted in paren-
+thesis by the compiler of the summary rather than copied from
+his original source; and the shortened form of the first part
+of the same summary published in the <I>Variae collectiones</I> of
+Hultsch's Heron, and now included by Heiberg in Heron's
+<I>Definitions</I>,<note>Heron, vol. iv, ed. Heib., p. 108.</note> contains no such parenthesis. Other authorities
+attribute the discovery of the theory of the irrational not to
+Pythagoras but to the Pythagoreans. A scholium to Euclid,
+Book X, says that
+<p>&lsquo;the Pythagoreans were the first to address themselves to the
+investigation of commensurability, having discovered it as the
+result of their observation of numbers; for, while the unit is
+a common measure of all numbers, they were unable to find
+a common measure of all magnitudes, . . . because all magni-
+tudes are divisible <I>ad infinitum</I> and never leave a magnitude
+which is too small to admit of further division, but that
+remainder is equally divisible <I>ad infinitum</I>,&rsquo;
+<p>and so on. The scholiast adds the legend that
+<p>&lsquo;the first of the Pythagoreans who made public the investiga-
+tion of these matters perished in a shipwreck&rsquo;.<note>Euclid, ed. Heib., vol. v, pp. 415, 417.</note>
+<p>Another commentary on Eucl. X discovered by Woepcke in
+an Arabic translation and believed, with good reason, to be
+part of the commentary of Pappus, says that the theory of
+irrational magnitudes &lsquo;had its origin in the school of Pytha-
+goras&rsquo;. Again, it is impossible that Pythagoras himself should
+have discovered a &lsquo;theory&rsquo; or &lsquo;study&rsquo; of irrationals in any
+<pb n=155><head>THE IRRATIONAL</head>
+proper sense. We are told in the <I>Theaetetus</I><note>Plato, <I>Theaetetus</I>, 147 D sq.</note> that Theodorus
+of Cyrene (a pupil of Protagoras and the teacher of Plato)
+proved the irrationality of &radic;3, &radic;5, &amp;c., up to &radic;17, and this
+must have been at a date not much, if anything, earlier than
+400 B.C.; while it was Theaetetus who, inspired by Theodorus's
+investigation of these particular &lsquo;roots&rsquo; (or surds), was the
+first to generalize the theory, seeking terms to cover all such
+incommensurables; this is confirmed by the continuation of
+the passage from Pappus's commentary, which says that the
+theory was
+<p>&lsquo;considerably developed by Theaetetus the Athenian, who
+gave proof, in this part of mathematics as in others, of ability
+which has been justly admired . . . As for the exact dis-
+tinctions of the above-named magnitudes and the rigorous
+demonstrations of the propositions to which this theory gives
+rise, I believe that they were chiefly established by this
+mathematician&rsquo;.
+<p>It follows from all this that, if Pythagoras discovered any-
+thing about irrationals, it was not any &lsquo;theory&rsquo; of irrationals
+but, at the most, some particular case of incommensurability.
+Now the passage which states that Theodorus proved that
+&radic;3, &radic;5, &amp;c. are incommensurable says nothing of &radic;2. The
+reason is, no doubt, that the incommensurability of &radic;2 had
+been proved earlier, and everything points to the probability
+that this was the first case to be discovered. But, if Pytha-
+goras discovered even this, it is difficult to see how the theory
+that number is the essence of all existing things, or that all
+things are made of number, could have held its ground for
+any length of time. The evidence suggests the conclusion
+that geometry developed itself for some time on the basis of
+the numerical theory of proportion which was inapplicable to
+any but commensurable magnitudes, and that it received an
+unexpected blow later by reason of the discovery of the irra-
+tional. The inconvenience of this state of things, which
+involved the restriction or abandonment of the use of propor-
+tions as a method pending the discovery of the generalized
+theory by Eudoxus, may account for the idea of the existence
+of the irrational having been kept secret, and of punishment
+having overtaken the first person who divulged it.
+<pb n=156><head>PYTHAGOREAN GEOMETRY</head>
+<p>If then it was not Pythagoras but some Pythagorean who
+discovered the irrationality of &radic;2, at what date are we to
+suppose the discovery to have been made? A recent writer<note>H. Vogt in <I>Bibliotheca mathematica</I>, x<SUB>3</SUB>, 1910, pp. 97-155 (cf. ix<SUB>3</SUB>,
+p. 190 sq.).</note>
+on the subject holds that it was the <I>later</I> Pythagoreans who
+made the discovery, not much before 410 B.C. It is impos-
+sible, he argues, that fifty or a hundred years would elapse
+between the discovery of the irrationality of &radic;2 and the like
+discovery by Theodorus (about 410 or 400 B.C.) about the other
+surds &radic;3, &radic;5, &amp;c. It is difficult to meet this argument
+except by the supposition that, in the interval, the thoughts
+of geometers had been taken up by other famous problems,
+such as the quadrature of the circle and the duplication of the
+cube (itself equivalent to finding &radic;<SUP>3</SUP>2). Another argument is
+based on the passage in the <I>Laws</I> where the Athenian stranger
+speaks of the shameful ignorance of the generality of Greeks,
+who are not aware that it is not all geometrical magnitudes
+that are commensurable with one another; the speaker adds
+that it was only &lsquo;late&rsquo; (<G>o)ye/ pote</G>) that he himself learnt the
+truth.<note>Plato, <I>Laws</I>, 819 D-820 C.</note> Even if we knew for certain whether &lsquo;late&rsquo; means
+&lsquo;late in the day&rsquo; or &lsquo;late in life&rsquo;, the expression would not
+help much towards determining the date of the first discovery
+of the irrationality of &radic;2; for the language of the passage is
+that of rhetorical exaggeration (Plato speaks of men who are
+unacquainted with the existence of the irrational as more
+comparable to swine than to human beings). Moreover, the
+irrational appears in the <I>Republic</I> as something well known,
+and precisely with reference to &radic;2; for the expressions &lsquo;the
+rational diameter of (the square the side of which is) 5&rsquo;
+[= the approximation &radic;(49) or 7] and the &lsquo;irrational
+(<G>a)/rrhtos</G>) diameter of 5&rsquo; [= &radic;(50)] are used without any word
+of explanation.<note>Plato, <I>Republic</I>, vii. 546 D.</note>
+<p>Further, we have a well-authenticated title of a work by
+Democritus (born 470 or 460 B.C.), <G>peri\ a)lo/gwn grammw=n kai\
+nastw=n ab</G>, &lsquo;two books on irrational lines and solids&rsquo; (<G>nasto/n</G>
+is <G>plh=res</G>, &lsquo;full&rsquo;, as opposed to <G>keno/n</G>. &lsquo;void&rsquo;, and Democritus
+called his &lsquo;first bodies&rsquo; <G>nasta/</G>). Of the contents of this work
+we are not informed; the recent writer already mentioned
+<pb n=157><head>THE IRRATIONAL</head>
+suggests that <G>a)/logos</G> does not here mean irrational or incom-
+mensurable at all, but that the book was an attempt to con-
+nect the atomic theory with continuous magnitudes (lines)
+through &lsquo;indivisible lines&rsquo; (cf. the Aristotelian treatise <I>On
+indivisible lines</I>), and that Democritus meant to say that,
+since any two lines are alike made up of an infinite number
+of the (indivisible) elements, they cannot be said to have any
+expressible ratio to one another, that is, he would regard them
+as &lsquo;having no ratio&rsquo;! It is, however, impossible to suppose
+that a mathematician of the calibre of Democritus could have
+denied that any two lines can have a ratio to one another;
+moreover, on this view, since no two straight lines would have
+a ratio to one another, <G>a)/logoi grammai/</G> would not be a <I>class</I> of
+lines, but <I>all</I> lines, and the title would lose all point. But
+indeed, as we shall see, it is also on other grounds inconceiv-
+able that Democritus should have been an upholder of &lsquo;indi-
+visible lines&rsquo; at all. I do not attach any importance to the
+further argument used in support of the interpretation in
+question, namely that <G>a)/logos</G> in the sense of &lsquo;irrational&rsquo; is
+not found in any other writer before Aristotle, and that
+Plato uses the words <G>a)/rrhtos</G> and <G>a)su/mmetros</G> only. The
+latter statement is not even strictly true, for Plato does in
+fact use the word <G>a)/logoi</G> specifically of <G>grammai/</G> in the passage
+of the <I>Republic</I> where he speaks of youths not being <G>a)/logoi
+w(/sper grammai/</G>, &lsquo;irrational like lines&rsquo;.<note>Plato, <I>Republic</I>, 534 D.</note> Poor as the joke is,
+it proves that <G>a)/logoi grammai/</G> was a recognized technical
+term, and the remark looks like a sly reference to the very
+treatise of Democritus of which we are speaking. I think
+there is no reason to doubt that the book was on &lsquo;irrationals&rsquo;
+in the technical sense. We know from other sources that
+Democritus was already on the track of infinitesimals in
+geometry; and nothing is more likely than that he would
+write on the kindred subject of irrationals.
+<p>I see therefore no reason to doubt that the irrationality
+of &radic;2 was discovered by some Pythagorean at a date appre-
+ciably earlier than that of Democritus; and indeed the simple
+proof of it indicated by Aristotle and set out in the propo-
+sition interpolated at the end of Euclid's Book X seems
+appropriate to an early stage in the development of geometry.
+<pb n=158><head>PYTHAGOREAN GEOMETRY</head>
+<C>(<G>e</G>) <I>The five regular solids.</I></C>
+<p>The same parenthetical sentence in Proclus which attributes
+to Pythagoras the discovery of the theory of irrationals
+(or proportions) also states that he discovered the &lsquo;putting
+together (<G>su/stasis</G>) of the cosmic figures&rsquo; (the five regular
+solids). As usual, there has been controversy as to the sense
+in which this phrase is to be taken, and as to the possibility
+of Pythagoras having done what is attributed to him, in any
+sense of the words. I do not attach importance to the
+argument that, whereas Plato, presumably &lsquo;Pythagorizing&rsquo;,
+assigns the first four solids to the four elements, earth, fire,
+air, and water, Empedocles and not Pythagoras was the
+first to declare these four elements to be the material princi-
+ples from which the universe was evolved; nor do I think
+it follows that, because the elements are four, only the first
+four solids had been discovered at the time when the four
+elements came to be recognized, and that the dodecahedron
+must therefore have been discovered later. I see no reason
+why all five should not have been discovered by the early
+Pythagoreans before any question of identifying them with
+the elements arose. The fragment of Philolaus, indeed, says
+that
+<p>&lsquo;there are five bodies in the sphere, the fire, water, earth,
+and air in the sphere, and the vessel of the sphere itself
+making the fifth&rsquo;,<note>Stobaeus, <I>Ecl.</I> I, proem. 3 (p. 18. 5 Wachsmuth); Diels, <I>Vors.</I> i<SUP>3</SUP>,
+p. 314. The Greek of the last phrase is <G>kai\ o(\ ta=s sfai/ras o(lka/s, pe/mpton</G>,
+but <G>o(lka/s</G> is scarcely an appropriate word, and von Wilamowitz (<I>Platon</I>,
+vol. ii, 1919, pp. 91-2) proposes <G>o( ta=s sfai/ras o(lko/s</G>, taking <G>o(lko/s</G> (which
+implies &lsquo;winding&rsquo;) as <I>volumen.</I> We might then translate by &lsquo;the spherical
+envelope&rsquo;.</note>
+<p>but as this is only to be understood of the <I>elements</I> in the
+sphere of the universe, not of the solid figures, in accordance
+with Diels's translation, it would appear that Plato in the
+<I>Timaeus</I><note><I>Timaeus</I>, 53 C-55 C.</note> is the earliest authority for the allocation, and
+it may very well be due to Plato himself (were not the solids
+called the &lsquo;Platonic figures&rsquo;?), although put into the mouth
+of a Pythagorean. At the same time, the fact that the
+<I>Timaeus</I> is fundamentally Pythagorean may have induced
+A&euml;tius's authority (probably Theophrastus) to conclude too
+<pb n=159><head>THE FIVE REGULAR SOLIDS</head>
+hastily that &lsquo;here, too, Plato Pythagorizes&rsquo;, and to say dog-
+matically on the faith of this that
+<p>&lsquo;<I>Pythagoras</I>, seeing that there are five solid figures, which
+are also called the mathematical figures, says that the earth
+arose from the cube, fire from the pyramid, air from the
+octahedron, water from the icosahedron, and the sphere of
+the universe from the dodecahedron.&rsquo;<note>A&euml;t. ii. 6. 5 (<I>Vors.</I> i<SUP>3</SUP>, p. 306. 3-7).</note>
+<p>It may, I think, be conceded that Pythagoras or the early
+Pythagoreans would hardly be able to &lsquo;construct&rsquo; the five
+regular solids in the sense of a complete theoretical construc-
+tion such as we find in Eucl. XIII; and it is possible that
+Theaetetus was the first to give these constructions, whether
+<G>e)/graye</G> in Suidas's notice means that &lsquo;he was the first to
+<I>construct</I>&rsquo; or &lsquo;to <I>write upon</I> the five solids so called&rsquo;. But
+there is no reason why the Pythagoreans should not have
+&lsquo;put together&rsquo; the five figures in the manner in which Plato
+puts them together in the <I>Timaeus</I>, namely, by bringing
+a certain number of angles of equilateral triangles, squares,
+or pentagons severally together at one point so as to make
+a solid angle, and then completing all the solid angles in that
+way. That the early Pythagoreans should have discovered
+the five regular solids in this elementary way agrees well
+with what we know of their having put angles of certain
+regular figures round a point and shown that only three
+kinds of such angles would fill up the space in one plane
+round the point.<note>Proclus on Eucl. I, pp. 304. 11-305. 3.</note> How elementary the construction still was
+in Plato's hands may be inferred from the fact that he argues
+that only three of the elements are transformable into one
+another because only three of the solids are made from
+equilateral triangles; these triangles, when present in suffi-
+cient numbers in given regular solids, can be separated again
+and redistributed so as to form regular solids of a different
+number of faces, as if the solids were really hollow shells
+bounded by the triangular faces as planes or laminae (Aris-
+totle criticizes this in <I>De caelo</I>, iii. 1)! We may indeed treat
+Plato's elementary method as an indication that this was
+actually the method employed by the earliest Pythagoreans.
+<pb n=160><head>PYTHAGOREAN GEOMETRY</head>
+<p>Putting together squares three by three, forming eight
+solid angles, and equilateral triangles three by three, four by
+four, or five by five, forming four, six, or twelve solid angles
+respectively, we readily form a cube, a tetrahedron, an octa-
+hedron, or an icosahedron, but the fifth regular solid, the
+dodecahedron, requires a new element, the regular pentagon.
+True, if we form the angle of an icosahedron by putting
+together five equilateral triangles, the bases of those triangles
+when put together form a regular pentagon; but Pythagoras
+or the Pythagoreans would require a theoretical construction.
+What is the evidence that the early Pythagoreans could have
+constructed and did construct pentagons? That they did
+construct them seems established by the story of Hippasus,
+<p>&lsquo;who was a Pythagorean but, owing to his being the first
+to publish and write down the (construction of the) sphere
+with (<G>e)k</G>, from) the twelve pentagons, perished by shipwreck
+for his impiety, but received credit for the discovery, whereas
+it really belonged to HIM (<G>e)kei/nou tou= a)ndro/s</G>), for it is thus
+that they refer to Pythagoras, and they do not call him by
+his name.&rsquo;<note>Iambl. <I>Vit. Pyth.</I> 88, <I>de c. math. scient.</I> c. 25, p. 77. 18-24.</note>
+<p>The connexion of Hippasus's name with the subject can
+hardly be an invention, and the story probably points to
+a positive achievement by him, while of course the Pytha-
+goreans' jealousy for the Master accounts for the reflection
+upon Hippasus and the moral. Besides, there is evidence for
+the very early existence of dodecahedra in actual fact. In
+1885 there was discovered on Monte Loffa (Colli Euganei,
+near Padua) a regular dodecahedron of Etruscan origin, which
+is held to date from the first half of the first millennium B.C.<note>F. Lindemann, &lsquo;Zur Geschichte der Polyeder und der Zahlzeichen&rsquo;
+(<I>Sitzungsber. der K. Bay. Akad. der Wiss.</I> xxvi. 1897, pp. 625-768).</note>
+Again, it appears that there are extant no less than twenty-six
+objects of dodecahedral form which are of Celtic origin.<note>L. Hugo in <I>Comptes rendus</I> of the Paris Acad. of Sciences, lxiii, 1873,
+pp. 420-1; lxvii, 1875, pp. 433, 472; lxxxi, 1879, p. 332.</note> It
+may therefore be that Pythagoras or the Pythagoreans had
+seen dodecahedra of this kind, and that their merit was to
+have treated them as mathematical objects and brought
+them into their theoretical geometry. Could they then have
+<pb n=161><head>THE FIVE REGULAR SOLIDS</head>
+constructed the regular pentagon? The answer must, I think,
+be yes. If <I>ABCDE</I> be a regular pentagon, and <I>AC, AD, CE</I>
+be joined, it is easy to prove, from the (Pythagorean) proposi-
+tions about the sum of the internal angles of a polygon and
+<FIG>
+the sum of the angles of a triangle, that each of the angles
+<I>BAC, DAE, ECD</I> is 2/5ths of a right angle, whence, in the
+triangle <I>ACD</I>, the angle <I>CAD</I> is 2/5ths of a right angle, and
+each of the base angles <I>ACD, ADC</I> is 4/5ths of a right angle
+or double of the vertical angle <I>CAD</I>; and from these facts
+it easily follows that, if <I>CE</I> and <I>AD</I> meet in <I>F, CDF</I> is an
+isosceles triangle equiangular, and therefore similar, to <I>ACD</I>,
+and also that <MATH><I>AF</I> = <I>FC</I> = <I>CD.</I></MATH> Now, since the triangles
+<I>ACD, CDF</I> are similar,
+<MATH><I>AC</I>:<I>CD</I> = <I>CD</I>:<I>DF</I></MATH>,
+or <MATH><I>AD</I>:<I>AF</I> = <I>AF</I>:<I>FD</I></MATH>;
+that is, if <I>AD</I> is given, the length of <I>AF</I>, or <I>CD</I>, is found by
+dividing <I>AD</I> at <I>F</I> in &lsquo;extreme and mean ratio&rsquo; by Eucl. II. 11.
+This last problem is a particular case of the problem of
+&lsquo;application of areas&rsquo;, and therefore was obviously within
+the power of the Pythagoreans. This method of constructing
+a pentagon is, of course, that taught in Eucl. IV. 10, 11. If
+further evidence is wanted of the interest of the early Pytha-
+goreans in the regular pentagon, it is furnished by the fact,
+attested by Lucian and the scholiast to the <I>Clouds</I> of Aristo-
+phanes, that the &lsquo;triple interwoven triangle, the pentagram&rsquo;,
+i. e. the star-pentagon, was used by the Pythagoreans as a
+symbol of recognition between the members of the same school,
+and was called by them Health.<note>Lucian, <I>Pro lapsu in salut.</I> &sect; 5 (vol. i, pp. 447-8, Jacobitz); schol. on
+<I>Clouds</I> 609.</note> Now it will be seen from the
+separate diagram of the star-pentagon above that it actually
+<pb n=162><head>PYTHAGOREAN GEOMETRY</head>
+shows the equal sides of the five isosceles triangles of the type
+referred to and also the points at which they are divided in
+extreme and mean ratio. (I should perhaps add that the
+pentagram is said to be found on the vase of Aristonophus
+found at Caere and supposed to belong to the seventh
+century B.C., while the finds at Mycenae include ornaments of
+pentagonal form.)
+<p>It would be easy to conclude that the dodecahedron is in-
+scribable in a sphere, and to find the centre of it, without
+constructing both in the elaborate manner of Eucl. XIII. 17
+and working out the relation between an edge of the dodeca-
+hedron and the radius of the sphere, as is there done: an
+investigation probably due to Theaetetus. It is right to
+mention here the remark in scholium No. 1 to Eucl. XIII
+that the book is about
+<p>&lsquo;the five so-called Platonic figures, which, however, do not
+belong to Plato, three of the five being due to the Pytha-
+goreans, namely the cube, the pyramid, and the dodeca-
+hedron, while the octahedron and icosahedron are due to
+Theaetetus&rsquo;.<note>Heiberg's Euclid, vol. v, p. 654.</note>
+<p>This statement (taken probably from Geminus) may per-
+haps rest on the fact that Theaetetus was the first to write
+at any length about the two last-mentioned solids, as he was
+probably the first to construct all five theoretically and in-
+vestigate fully their relations to one another and the circum-
+scribing spheres.
+<C>(<G>z</G>) <I>Pythagorean astronomy.</I></C>
+<p>Pythagoras and the Pythagoreans occupy an important place
+in the history of astronomy. (1) Pythagoras was one of the first
+to maintain that the universe and the earth are spherical
+in form. It is uncertain what led Pythagoras to conclude
+that the earth is a sphere. One suggestion is that he inferred
+it from the roundness of the shadow cast by the earth in
+eclipses of the moon. But it is certain that Anaxagoras was
+the first to suggest this, the true, explanation of eclipses.
+The most likely supposition is that Pythagoras's ground was
+purely mathematical, or mathematico-aesthetical; that is, he
+<pb n=163><head>PYTHAGOREAN ASTRONOMY</head>
+attributed spherical shape to the earth (as to the universe)
+for the simple reason that the sphere is the most beautiful
+of solid figures. For the same reason Pythagoras would
+surely hold that the sun, the moon, and the other heavenly
+bodies are also spherical in shape. (2) Pythagoras is credited
+with having observed the identity of the Morning and the
+Evening Stars. (3) It is probable that he was the first to
+state the view (attributed to Alcmaeon and &lsquo;some of the
+mathematicians&rsquo;) that the planets as well as the sun and
+moon have a motion of their own from west to east opposite
+to and independent of the daily rotation of the sphere of the
+fixed stars from east to west.<note>A&euml;t. ii. 16. 2, 3 (<I>Vors.</I> i<SUP>3</SUP>, p. 132. 15).</note> Hermesianax, one of the older
+generation of Alexandrine poets (about 300 B.C.), is quoted as
+saying:
+<p>&lsquo;What inspiration laid forceful hold on Pythagoras when
+he discovered the subtle geometry of (the heavenly) spirals
+and compressed in a small sphere the whole of the circle which
+the aether embraces.&rsquo;<note>See Athenaeus, xiii. 599 A.</note>
+<p>This would seem to imply the construction of a sphere
+on which were represented the circles described by the sun,
+moon and planets together with the daily revolution of the
+heavenly sphere; but of course Hermesianax is not altogether
+a trustworthy authority.
+<p>It is improbable that Pythagoras himself was responsible
+for the astronomical system known as the Pythagorean, in
+which the earth was deposed from its place at rest in the
+centre of the universe, and became a &lsquo;planet&rsquo;, like the sun,
+the moon and the other planets, revolving about the central
+fire. For Pythagoras the earth was still at the centre, while
+about it there moved (<I>a</I>) the sphere of the fixed stars revolv-
+ing daily from east to west, the axis of rotation being a
+straight line through the centre of the earth, (<I>b</I>) the sun,
+moon and planets moving in independent circular orbits in
+a sense opposite to that of the daily rotation, i.e. from west
+to east.
+<p>The later Pythagorean system is attributed by A&euml;tius
+(probably on the authority of Theophrastus) to Philolaus, and
+<pb n=164><head>PYTHAGOREAN GEOMETRY</head>
+may be described thus. The universe is spherical in shape
+and finite in size. Outside it is infinite void which enables
+the universe to breathe, as it were. At the centre is the
+central fire, the Hearth of the Universe, called by various
+names, the Tower or Watch-tower of Zeus, the Throne of
+Zeus, the House of Zeus, the Mother of the Gods, the Altar,
+Bond and Measure of Nature. In this central fire is located
+the governing principle, the force which directs the movement
+and activity of the universe. In the universe there revolve
+in circles about the central fire the following bodies. Nearest
+to the central fire revolves the counter-earth, which always
+accompanies the earth, the orbit of the earth coming next to
+that of the counter-earth; next to the earth, reckoning in
+order from the centre outwards, comes the moon, next to the
+moon the sun, next to the sun the five planets, and last of
+all, outside the orbits of the five planets, the sphere of the
+fixed stars. The counter-earth, which accompanies the earth
+and revolves in a smaller orbit, is not seen by us because
+the hemisphere of the earth on which we live is turned away
+from the counter-earth (the analogy of the moon which
+always turns one side towards us may have suggested this);
+this involves, incidentally, a rotation of the earth about its
+axis completed in the same time as it takes the earth to
+complete a revolution about the central fire. As the latter
+revolution of the earth was held to produce day and night,
+it is a natural inference that the earth was supposed to
+complete one revolution round the central fire in a day and
+a night, or in twenty-four hours. This motion on the part of
+the earth with our hemisphere always turned outwards would,
+of course, be equivalent, as an explanation of phenomena,
+to a rotation of the earth about a fixed axis, but for the
+parallax consequent on the earth describing a circle in space
+with radius greater than its own radius; this parallax, if we
+may trust Aristotle,<note>Arist. <I>De caelo</I>, ii. 13, 293 b 25-30.</note> the Pythagoreans boldly asserted to be
+negligible. The superfluous thing in this system is the
+introduction of the counter-earth. Aristotle says in one
+place that its object was to bring up the number of the
+moving bodies to ten, the perfect number according to
+<pb n=165><head>PYTHAGOREAN ASTRONOMY</head>
+the Pythagoreans<note>Arist. <I>Metaph.</I> A. 5, 986 a 8-12.</note>; but he hints at the truer explanation in
+another passage where he says that eclipses of the moon
+were considered to be due sometimes to the interposition
+of the earth, sometimes to the interposition of the counter-
+earth (to say nothing of other bodies of the same sort
+assumed by &lsquo;some&rsquo; in order to explain why there appear
+to be more lunar eclipses than solar)<note>Arist. <I>De caelo</I>, ii. 13, 293 b 21-5.</note>; we may therefore
+take it that the counter-earth was invented for the purpose
+of explaining eclipses of the moon and their frequency.
+<C>Recapitulation.</C>
+<p>The astronomical systems of Pythagoras and the Pytha-
+goreans illustrate the purely mathematical character of their
+physical speculations; the heavenly bodies are all spheres,
+the most perfect of solid figures, and they move in circles;
+there is no question raised of <I>forces</I> causing the respective
+movements; astronomy is pure mathematics, it is geometry,
+combined with arithmetic and harmony. The capital dis-
+covery by Pythagoras of the dependence of musical intervals
+on numerical proportions led, with his successors, to the
+doctrine of the &lsquo;harmony of the spheres&rsquo;. As the ratio
+2:1 between the lengths of strings of the same substance
+and at the same tension corresponds to the octave, the
+ratio 3:2 to the fifth, and the ratio 4:3 to the fourth, it
+was held that bodies moving in space produce sounds, that
+those which move more quickly give a higher note than those
+which move more slowly, while those move most quickly which
+move at the greatest distance; the sounds therefore pro-
+duced by the heavenly bodies, depending on their distances
+(i.e. the size of their orbits), combine to produce a harmony;
+&lsquo;the whole heaven is number and harmony&rsquo;.<note>Arist. <I>Metaph.</I> A. 5, 986 a 2.</note>
+<p>We have seen too how, with the Pythagoreans, the theory
+of numbers, or &lsquo;arithmetic&rsquo;, goes hand in hand with geometry;
+numbers are represented by dots or lines forming geometrical
+figures; the species of numbers often take their names from
+their geometrical analogues, while their properties are proved
+by geometry. The Pythagorean mathematics, therefore, is all
+one science, and their science is all mathematics.
+<pb n=166><head>PYTHAGOREAN GEOMETRY</head>
+<p>It is this identification of mathematics (and of geometry
+in particular) with science in general, and their pursuit of it
+for its own sake, which led to the extraordinary advance of
+the subject in the Pythagorean school. It was the great merit
+of Pythagoras himself (apart from any particular geometrical
+or arithmetical theorems which he discovered) that he was the
+first to take this view of mathematics; it is characteristic of
+him that, as we are told, &lsquo;geometry was called by Pythagoras
+<I>inquiry</I> or <I>science</I>&rsquo; (<G>e)kalei=to de\ h( gewmetri/a pro\s *puqago/rou
+i(stori/a</G>).<note>Iambl. <I>Vit. Pyth.</I> 89.</note> Not only did he make geometry a liberal educa-
+tion; he was the first to attempt to explore it down to its
+first principles; as part of the scientific basis which he sought
+to lay down he &lsquo;used definitions&rsquo;. A point was, according to
+the Pythagoreans, a &lsquo;unit having position&rsquo;<note>Proclus on Eucl. I, p. 95. 21.</note>; and, if their
+method of regarding a line, a surface, a solid, and an angle
+does not amount to a definition, it at least shows that they
+had reached a clear idea of the <I>differentiae</I>, as when they said
+that 1 was a point, 2 a line, 3 a triangle, and 4 a pyramid.
+A surface they called <G>xroia/</G>, &lsquo;colour&rsquo;; this was their way of
+describing the superficial appearance, the idea being, as
+Aristotle says, that the colour is either in the limiting surface
+(<G>pe/ras</G>) or is the <G>pe/ras</G>,<note>Arist. <I>De sensu</I>, 3, 439 a 31.</note> so that the meaning intended to be
+conveyed is precisely that intended by Euclid's definition
+(XI. Def. 2) that &lsquo;the limit of a solid is a surface&rsquo;. An angle
+they called <G>glwxi/s</G>, a &lsquo;point&rsquo; (as of an arrow) made by a line
+broken or bent back at one point.<note>Heron, Def. 15.</note>
+<p>The positive achievements of the Pythagorean school in
+geometry, and the immense advance made by them, will be
+seen from the following summary.
+<p>1. They were acquainted with the properties of parallel
+lines, which they used for the purpose of establishing by
+a general proof the proposition that the sum of the three
+angles of any triangle is equal to two right angles. This
+latter proposition they again used to establish the well-known
+theorems about the sums of the exterior and interior angles,
+respectively, of any polygon.
+<p>2. They originated the subject of equivalent areas, the
+transformation of an area of one form into another of different
+<pb n=167><head>RECAPITULATION</head>
+form and, in particular, the whole method of <I>application of
+areas</I>, constituting a <I>geometrical algebru</I>, whereby they effected
+the equivalent of the algebraical processes of addition, sub-
+traction, multiplication, division, squaring, extraction of the
+square root, and finally the complete solution of the mixed
+quadratic equation <MATH><I>x</I><SUP>2</SUP>&plusmn;<I>px</I>&plusmn;<I>q</I> = 0</MATH>, so far as its roots are real.
+Expressed in terms of Euclid, this means the whole content of
+Book I. 35-48 and Book II. The method of <I>application of
+areas</I> is one of the most fundamental in the whole of later
+Greek geometry; it takes its place by the side of the powerful
+method of proportions; moreover, it is the starting point of
+Apollonius's theory of conics, and the three fundamental
+terms, <I>parabole, ellipsis</I>, and <I>hyperbole</I> used to describe the
+three separate problems in &lsquo;application&rsquo; were actually em-
+ployed by Apollonius to denote the three conics, names
+which, of course, are those which we use to-day. Nor was
+the use of the geometrical algebra for solving <I>numerical</I>
+problems unknown to the Pythagoreans; this is proved by
+the fact that the theorems of Eucl. II. 9, 10 were invented
+for the purpose of finding successive integral solutions of the
+indeterminate equations
+<MATH>2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP> = &plusmn; 1</MATH>.
+<p>3. They had a theory of proportion pretty fully developed.
+We know nothing of the form in which it was expounded;
+all we know is that it took no account of incommensurable
+magnitudes. Hence we conclude that it was a numerical
+theory, a theory on the same lines as that contained in
+Book VII of Euclid's <I>Elements.</I>
+<p>They were aware of the properties of similar figures.
+This is clear from the fact that they must be assumed
+to have solved the problem, which was, according to
+Plutarch, attributed to Pythagoras himself, of describing a
+figure which shall be similar to one given figure and equal in
+area to another given figure This implies a knowledge of
+the proposition that similar figures (triangles or polygons) are
+to one another in the duplicate ratio of corresponding sides
+(Eucl. VI. 19, 20). As the problem is solved in Eucl. VI. 25,
+we assume that, subject to the qualification that their
+theorems about similarity, &amp;c., were only established of figures
+<pb n=168><head>PYTHAGOREAN GEOMETRY.</head>
+in which corresponding elements are commensurable, they had
+theorems corresponding to a great part of Eucl., Book VI.
+<p>Again, they knew how to cut a straight line in extreme and
+mean ratio (Eucl. VI. 30); this problem was presumably
+solved by the method used in Eucl. II. 11, rather than by that
+of Eucl. VI. 30, which depends on the solution of a problem
+in the application of areas more general than the methods of
+Book II enable us to solve, the problem namely of Eucl.
+VI. 29.
+<p>4. They had discovered, or were aware of the existence of,
+the five regular solids. These they may have constructed
+empirically by putting together squares, equilateral triangles,
+and pentagons. This implies that they could construct a
+regular pentagon and, as this construction depends upon the
+construction of an isosceles triangle in which each of the base
+angles is double of the vertical angle, and this again on the
+cutting of a line in extreme and mean ratio, we may fairly
+assume that this was the way in which the construction of
+the regular pentagon was actually evolved. It would follow
+that the solution of problems by <I>analysis</I> was already prac-
+tised by the Pythagoreans, notwithstanding that the discovery
+of the analytical method is attributed by Proclus to Plato.
+As the particular construction is practically given in Eucl. IV.
+10, 11, we may assume that the content of Eucl. IV was also
+partly Pythagorean.
+<p>5. They discovered the existence of the irrational in the
+sense that they proved the incommensurability of the diagonal
+of a square with reference to its side; in other words, they
+proved the irrationality of &radic;2. As a proof of this is referred
+to by Aristotle in terms which correspond to the method
+used in a proposition interpolated in Euclid, Book X, we
+may conclude that this proof is ancient, and therefore that it
+was probably the proof used by the discoverers of the proposi-
+tion. The method is to prove that, if the diagonal of a square
+is commensurable with the side, then the same number must
+be both odd and even; here then we probably have an early
+Pythagorean use of the method of <I>reductio ad absurdum.</I>
+<p>Not only did the Pythagoreans discover the irrationality
+of &radic;2; they showed, as we have seen, how to approximate
+as closely as we please to its numerical value.
+<pb n=169><head>RECAPITULATION</head>
+<p>After the discovery of this one case of irrationality, it
+would be obvious that propositions theretofore proved by
+means of the numerical theory of proportion, which was
+inapplicable to incommensurable magnitudes, were only par-
+tially proved. Accordingly, pending the discovery of a theory
+of proportion applicable to incommensurable as well as com-
+mensurable magnitudes, there would be an inducement to
+substitute, where possible, for proofs employing the theory of
+proportions other proofs independent of that theory. This
+substitution is carried rather far in Euclid, Books I-IV; it
+does not follow that the Pythagoreans remodelled their proofs
+to the same extent as Euclid felt bound to do.
+<pb>
+<C>VI</C>
+<C>PROGRESS IN THE ELEMENTS DOWN TO
+PLATO'S TIME</C>
+<p>IN tracing the further progress in the Elements which took
+place down to the time of Plato, we do not get much assistance
+from the summary of Proclus. The passage in which he
+states the succession of geometers from Pythagoras to Plato
+and his contemporaries runs as follows:
+<p>&lsquo;After him [Pythagoras] Anaxagoras of Clazomenae dealt
+with many questions in geometry, and so did Oenopides of
+Chios, who was a little younger than Anaxagoras; Plato
+himself alludes, in the <I>Rivals,</I> to both of them as having
+acquired a reputation for mathematics. After them came
+Hippocrates of Chios, the discoverer of the quadrature of
+the lune, and Theodorus of Cyrene, both of whom became
+distinguished geometers; Hippocrates indeed was the first
+of whom it is recorded that he actually compiled Elements.
+Plato, who came next to them, caused mathematics in general
+and geometry in particular to make a very great advance,
+owing to his own zeal for these studies; for every one knows
+that he even filled his writings with mathematical discourses
+and strove on every occasion to arouse enthusiasm for mathe-
+matics in those who took up philosophy. At this time too
+lived Leodamas of Thasos, Archytas of Taras, and Theaetetus
+of Athens, by whom the number of theorems was increased
+and a further advance was made towards a more scientific
+grouping of them.&rsquo;<note>Proclus on Eucl. I, p. 65. 21-66. 18.</note>
+<p>It will be seen that we have here little more than a list of
+names of persons who advanced, or were distinguished in,
+geometry. There is no mention of specific discoveries made
+by particular geometers, except that the work of Hippocrates
+on the squaring of certain lunes is incidentally alluded to,
+rather as a means of identifying Hippocrates than as a de-
+tail relevant to the subject in hand. It would appear that
+<pb n=171><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+the whole summary was directed to the one object of trac-
+ing progress in the Elements, particularly with reference
+to improvements of method in the direction of greater
+generality and more scientific order and treatment; hence
+only those writers are here mentioned who contributed to this
+development. Hippocrates comes into the list, not because
+of his lunes, but because he was a distinguished geometer
+and was the first to write Elements. Hippias of Elis, on the
+other hand, though he belongs to the period covered by the
+extract, is omitted, presumably because his great discovery,
+that of the curve known as the <I>quadratrix,</I> does not belong
+to elementary geometry; Hippias is, however, mentioned in
+two other places by Proclus in connexion with the quadratrix,<note>Proclus on Eucl. I, p. 272. 7, p. 356. 11.</note>
+and once more as authority for the geometrical achievements
+of Ameristus (or Mamercus or Mamertius).<note><I>Ib.,</I> p. 65. 14.</note> Less justice is
+done to Democritus, who is neither mentioned here nor else-
+where in the commentary; the omission here of the name
+of Democritus is one of the arguments for the view that
+this part of the summary is not quoted from the <I>History
+of Geometry</I> by Eudemus (who would not have been likely to
+omit so accomplished a mathematician as Democritus), but
+is the work either of an intermediary or of Proclus himself,
+based indeed upon data from Eudemus's history, but limited to
+particulars relevant to the object of the commentary, that
+is to say, the elucidation of Euclid and the story of the growth
+of the Elements.
+<p>There are, it is true, elsewhere in Proclus's commentary
+a very few cases in which particular propositions in Euclid,
+Book I, are attributed to individual geometers, e.g. those
+which Thales is said to have discovered. Two propositions
+presently to be mentioned are in like manner put to the
+account of Oenopides; but except for these details about
+Oenopides we have to look elsewhere for evidence of the
+growth of the Elements in the period now under notice.
+Fortunately we possess a document of capital importance,
+from this point of view, in the fragment of Eudemus on
+Hippocrates's quadrature of lunes preserved in Simplicius's
+commentary on the <I>Physics</I> of Aristotle.<note>Simpl. <I>in Arist. Phys.</I> pp. 54-69 Diels.</note> This fragment will
+<pb n=172><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+be described below. Meantime we will take the names men-
+tioned by Proclus in their order.
+<p>ANAXAGORAS (about 500-428 B.C.) was born at Clazomenae
+in the neighbourhood of Smyrna. He neglected his posses-
+sions, which were considerable, in order to devote himself
+to science. Some one once asked him what was the object
+of being born, to which he replied, &lsquo;The investigation of sun,
+moon and heaven.&rsquo; He was apparently the first philosopher
+to take up his abode at Athens, where he enjoyed the friend-
+ship of Pericles. When Pericles became unpopular shortly
+before the outbreak of the Peloponnesian War, he was attacked
+through his friends, and Anaxagoras was accused of impiety
+for holding that the sun was a red-hot stone and the moon
+earth. According to one account he was fined five talents
+and banished; another account says that he was kept in
+prison and that it was intended to put him to death, but
+that Pericles obtained his release; he went and lived at
+Lampsacus till his death.
+<p>Little or nothing is known of Anaxagoras's achievements
+in mathematics proper, though it is credible enough that
+he was a good mathematician. But in astronomy he made
+one epoch-making discovery, besides putting forward some
+remarkably original theories about the evolution of the
+universe. We owe to him the first clear recognition of the
+fact that the moon does not shine by its own light but
+receives its light from the sun; this discovery enabled him
+to give the true explanation of lunar and solar eclipses,
+though as regards the former (perhaps in order to explain
+their greater frequency) he erroneously supposed that there
+were other opaque and invisible bodies &lsquo;below the moon&rsquo;
+which, as well as the earth, sometimes by their interposition
+caused eclipses of the moon. A word should be added about
+his cosmology on account of the fruitful ideas which it con-
+tained. According to him the formation of the world began
+with a vortex set up, in a portion of the mixed mass in which
+&lsquo;all things were together&rsquo;, by Mind (<G>nou=s</G>). This rotatory
+movement began in the centre and then gradually spread,
+taking in wider and wider circles. The first effect was to
+separate two great masses, one consisting of the rare, hot,
+light, dry, called the &lsquo;aether&rsquo;, the other of the opposite
+<pb n=173><head>ANAXAGORAS</head>
+categories and called &lsquo;air&rsquo;. The aether took the outer, the
+air the inner place. From the air were next separated clouds,
+water, earth and stones. The dense, the moist, the dark and
+cold, and all the heaviest things, collected in the centre as the
+result of the circular motion, and it was from these elements
+when consolidated that the earth was formed; but after this,
+in consequence of the violence of the whirling motion, the
+surrounding fiery aether tore stones away from the earth and
+kindled them into stars. Taking this in conjunction with
+the remark that stones &lsquo;rush outwards more than water&rsquo;,
+we see that Anaxagoras conceived the idea of a <I>centrifugal</I>
+force as well as that of concentration brought about by the
+motion of the vortex, and that he assumed a series of pro-
+jections or &lsquo;whirlings-off&rsquo; of precisely the same kind as the
+theory of Kant and Laplace assumed for the formation of
+the solar system. At the same time he held that one of the
+heavenly bodies might break away and fall (this may account
+for the story that he prophesied the fall of the meteoric stone
+at Aegospotami in 468/7 B.C.), a <I>centripetal</I> tendency being
+here recognized.
+<p>In mathematics we are told that Anaxagoras &lsquo;while in
+prison wrote (or drew, <G>e)/grafe</G>) the squaring of the circle&rsquo;.<note>Plutarch, <I>De exil.</I> 17, 607 F.</note>
+But we have no means of judging what this amounted to.
+Rudio translates <G>e)/grafe</G> as &lsquo;zeichnete&rsquo;, &lsquo;drew&rsquo;, observing that
+he probably knew the Egyptian rule for squaring, and simply
+drew on the sand a square as nearly as he could equal to the
+area of a circle.<note>Rudio, <I>Der Bericht des Simplicius &uuml;ber die Quadraturen des Antiphon
+und Hippokrates,</I> 1907, p. 92, 93.</note> It is clear to me that this cannot be right,
+but that the word means &lsquo;wrote upon&rsquo; in the sense that he
+tried to work out theoretically the problem in question. For
+the same word occurs (in the passive) in the extract from
+Eudemus about Hippocrates: &lsquo;The squarings of the lunes ...
+were first written (or proved) by Hippocrates and were found
+to be correctly expounded&rsquo;,<note>Simpl. <I>in Phys.,</I> p. 61. 1-3 Diels; Rudio, <I>op. cit.,</I> pp. 46. 22-48. 4.</note> where the context shows that
+<G>e)gra/fhsan</G> cannot merely mean &lsquo;were drawn&rsquo;. Besides,
+<G>tetragwnismo/s</G>, <I>squaring,</I> is a process or operation, and you
+cannot, properly speaking, &lsquo;draw&rsquo; a process, though you can
+&lsquo;describe&rsquo; it or prove its correctness.
+<pb n=174><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>Vitruvius tells us that one Agatharchus was the first to paint
+stage-scenes at Athens, at the time when Aeschylus was
+having his tragedies performed, and that he left a treatise on
+the subject which was afterwards a guide to Democritus and
+Anaxagoras, who discussed the same problem, namely that of
+painting objects on a plane surface in such a way as to make
+some of the things depicted appear to be in the background
+while others appeared to stand out in the foreground, so that
+you seemed, e.g., to have real buildings before you; in other
+words, Anaxagoras and Democritus both wrote treatises on
+perspective.<note>Vitruvius, <I>De architectura,</I> vii. praef. 11.</note>
+<p>There is not much to be gathered from the passage in
+the <I>Rivals</I> to which Proclus refers. Socrates, on entering the
+school of Dionysius, finds two lads disputing a certain point,
+something about Anaxagoras or Oenopides, he was not certain
+which; but they appeared to be drawing circles, and to be
+imitating certain inclinations by placing their hands at an
+angle.<note>Plato, <I>Erastae</I> 132 A, B.</note> Now this description suggests that what the lads
+were trying to represent was the circles of the equator and
+the zodiac or ecliptic; and we know that in fact Eudemus
+in his <I>History of Astronomy</I> attributed to Oenopides the dis-
+covery of &lsquo;the cincture of the zodiac circle&rsquo;,<note>Theon of Smyrna, p. 198. 14.</note> which must mean
+the discovery of the obliquity of the ecliptic. It would prob-
+ably be unsafe to conclude that Anaxagoras was also credited
+with the same discovery, but it certainly seems to be suggested
+that Anaxagoras had to some extent touched the mathematics
+of astronomy.
+<p>OENOPIDES OF CHIOS was primarily an astronomer. This
+is shown not only by the reference of Eudemus just cited, but
+by a remark of Proclus in connexion with one of two proposi-
+tions in elementary geometry attributed to him.<note>Proclus on Eucl. I, p. 283. 7-8.</note> Eudemus
+is quoted as saying that he not only discovered the obliquity
+of the ecliptic, but also the period of a Great Year. Accord-
+ing to Diodorus the Egyptian priests claimed that it was from
+them that Oenopides learned that the sun moves in an inclined
+orbit and in a sense opposite to the motion of the fixed stars.
+It does not appear that Oenopides made any measurement of
+<pb n=175><head>OENOPIDES OF CHIOS</head>
+the obliquity of the ecliptic. The duration of the Great Year
+he is said to have put at 59 years, while he made the length
+of the year itself to be 365 22/59 days. His Great Year clearly
+had reference to the sun and moon only; he merely sought to
+find the least integral number of complete years which would
+contain an exact number of lunar months. Starting, probably,
+with 365 days as the length of a year and 29 1/2 days as the
+length of a lunar month, approximate values known before
+his time, he would see that twice 29 1/2, or 59, years would con-
+tain twice 365, or 730, lunar months. He may then, from his
+knowledge of the calendar, have obtained 21,557 as the num-
+ber of days in 730 months, for 21,557 when divided by 59 gives
+365 22/59 as the number of days in the year.
+<p>Of Oenopides's geometry we have no details, except that
+Proclus attributes to him two propositions in Eucl. Bk. I. Of
+I. 12 (&lsquo;to draw a perpendicular to a given straight line from
+a point outside it&rsquo;) Proclus says:
+<p>&lsquo;This problem was first investigated by Oenopides, who
+thought it useful for astronomy. He, however, calls the per-
+pendicular in the archaic manner (a straight line drawn)
+<I>gnomon-wise</I> (<G>kata\ gnw/mona</G>), because the gnomon is also at
+right angles to the horizon.&rsquo;<note>Proclus on Eucl. I, p. 283. 7-8.</note>
+<p>On I. 23 (&lsquo;on a given straight line and at a given point on
+it to construct a rectilineal angle equal to a given rectilineal
+angle&rsquo;) Proclus remarks that this problem is &lsquo;rather the dis-
+covery of Oenopides, as Eudemus says&rsquo;.<note>Proclus on Eucl. I, p. 333. 5.</note> It is clear that the
+geometrical reputation of Oenopides could not have rested on
+the mere solution of such simple problems as these. Nor, of
+course, could he have been the first to draw a perpendicular in
+practice; the point may be that he was the first to solve the
+problem by means of the ruler and compasses only, whereas
+presumably, in earlier days, perpendiculars would be drawn
+by means of a set square or a right-angled triangle originally
+constructed, say, with sides proportional to 3, 4, 5. Similarly
+Oenopides may have been the first to give the theoretical,
+rather than the practical, construction for the problem of I. 23
+which we find in Euclid. It may therefore be that Oenopides's
+significance lay in improvements of method from the point of
+view of theory; he may, for example, have been the first to
+<pb n=176><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+lay down the restriction of the means permissible in construc-
+tions to the ruler and compasses which became a canon of
+Greek geometry for all &lsquo;plane&rsquo; constructions, i.e. for all
+problems involving the equivalent of the solution of algebraical
+equations of degree not higher than the second.
+<p>DEMOCRITUS, as mathematician, may be said to have at last
+come into his own. In the <I>Method</I> of Archimedes, happily
+discovered in 1906, we are told that Democritus was the first
+to state the important propositions that the volume of a cone
+is one third of that of a cylinder having the same base and
+equal height, and that the volume of a pyramid is one third of
+that of a prism having the same base and equal height; that is
+to say, Democritus enunciated these propositions some fifty
+years or more before they were first scientifically proved by
+Eudoxus.
+<p>Democritus came from Abdera, and, according to his own
+account, was young when Anaxagoras was old. Apollodorus
+placed his birth in Ol. 80 (= 460-457 B.C.), while according
+to Thrasyllus he was born in Ol. 77. 3 (= 470/69 B.C.), being
+one year older than Socrates. He lived to a great age, 90
+according to Diodorus, 104, 108, 109 according to other
+authorities. He was indeed, as Thrasyllus called him,
+<G>pe/ntaqlos</G> in philosophy<note>Diog. L. ix. 37 (<I>Vors.</I> ii<SUP>3</SUP>, p. 11. 24-30).</note>; there was no subject to which he
+did not notably contribute, from mathematics and physics on
+the one hand to ethics and poetics on the other; he even went
+by the name of &lsquo;Wisdom&rsquo; (<G>*sofi/a</G>).<note>Clem. <I>Strom.</I> vi. 32 (<I>Vors.</I> ii<SUP>3</SUP>, p. 16. 28).</note> Plato, of course, ignores
+him throughout his dialogues, and is said to have wished to
+burn all his works; Aristotle, on the other hand, pays
+handsome tribute to his genius, observing, e.g., that on the
+subject of change and growth no one save Democritus had
+observed anything except superficially; whereas Democritus
+seemed to have thought of everything.<note>Arist. <I>De gen. et corr.</I> i. 2, 315 a 35.</note> He could say
+of himself (the fragment is, it is true, considered by Diels
+to be spurious, while Gomperz held it to be genuine), &lsquo;Of
+all my contemporaries I have covered the most ground in
+my travels, making the most exhaustive inquiries the while;
+I have seen the most climates and countries and listened to
+<pb n=177><head>DEMOCRITUS</head>
+the greatest number of learned men&rsquo;.<note>Clement, <I>Strom.</I> i. 15, 69 (<I>Vors.</I> ii<SUP>3</SUP>, p. 123. 3).</note> His travels lasted for
+five years, and he is said to have visited Egypt, Persia and
+Babylon, where he consorted with the priests and magi; some
+say that he went to India and Aethiopia also. Well might
+he undertake the compilation of a geographical survey of
+the earth as, after Anaximander, Hecataeus of Miletus and
+Damastes of Sigeum had done. In his lifetime his fame was
+far from world-wide: &lsquo;I came to Athens&rsquo;, he says, &lsquo;and no
+one knew me.&rsquo;<note>Diog. L. ix. 36 (<I>Vors.</I> ii<SUP>3</SUP>, p. 11. 22).</note>
+<p>A long list of his writings is preserved in Diogenes Laertius,
+the authority being Thrasyllus. In astronomy he wrote,
+among other works, a book <I>On the Planets,</I> and another <I>On
+the Great Year or Astronomy</I> including a <I>parapegma</I><note>The <I>parapegma</I> was a posted record, a kind of almanac, giving, for
+a series of years, the movements of the sun, the dates of the phases of
+the moon, the risings and settings of certain stars, besides <G>e)pishmasi/ai</G>
+or weather indications; many details from Democritus's <I>parapegma</I>
+are preserved in the Calendar at the end of Geminus's <I>Isagoge</I> and in
+Ptolemy.</note> (or
+calendar). Democritus made the order of the heavenly bodies,
+reckoning outwards from the earth, the following: Moon,
+Venus, Sun, the other planets, the fixed stars. Lucretius<note>Lucretius, v. 621 sqq.</note> has
+preserved an interesting explanation which he gave of the
+reason why the sun takes a year to describe the full circle of
+the zodiac, while the moon completes its circle in a month.
+The nearer any body is to the earth (and therefore the farther
+from the sphere of the fixed stars) the less swiftly can it be
+carried round by the revolution of the heaven. Now the
+moon is nearer than the sun, and the sun than the signs of
+the zodiac; therefore the moon seems to get round faster than
+the sun because, while the sun, being lower and therefore
+slower than the signs, is left behind by them, the moon,.
+being still lower and therefore slower still, is still more left
+behind. Democritus's Great Year is described by Censorinus<note><I>De die natali,</I> 18. 8.</note>
+as 82 (LXXXII) years including 28 intercalary months, the
+latter number being the same as that included by Callippus in
+his cycle of 76 years; it is therefore probable that LXXXII
+is an incorrect reading for LXXVII (77).
+<p>As regards his mathematics we have first the statement in
+<pb n=178><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+the continuation of the fragment of doubtful authenticity
+already quoted that
+<p>&lsquo;in the putting together of lines, with the necessary proof, no
+one has yet surpassed me, not even the so-called <I>harpedon-
+aptae</I> (rope-stretchers) of Egypt&rsquo;.
+<p>This does not tell us much, except that it indicates that
+the &lsquo;rope-stretchers&rsquo;, whose original function was land-
+measuring or practical geometry, had by Democritus's time
+advanced some way in theoretical geometry (a fact which the
+surviving documents, such as the book of Ahmes, with their
+merely practical rules, would not have enabled us to infer).
+However, there is no reasonable doubt that in geometry
+Democritus was fully abreast of the knowledge of his day;
+this is fully confirmed by the titles of treatises by him and
+from other sources. The titles of the works classed as mathe-
+matical are (besides the astronomical works above mentioned):
+<p>1. <I>On a difference of opinion</I> (<G>gnw/mhs</G>: <I>v. l.</I> <G>gnw/monos</G>, gno-
+mon), <I>or on the contact of a circle and a sphere;</I>
+<p>2. <I>On Geometry;</I>
+<p>3. <I>Geometricorum</I> (?I, II);
+<p>4. <I>Numbers;</I>
+<p>5. <I>On irrational lines and solids</I> (<G>nastw=n</G>, atoms?);
+<p>6. <G>*)ekpeta/smata</G>.
+<p>As regards the first of these works I think that the
+attempts to extract a sense out of Cobet's reading <G>gnw/monos</G>
+(on a difference of a gnomon) have failed, and that <G>gnw/mhs</G>
+(Diels) is better. But &lsquo;On a difference of opinion&rsquo; seems
+scarcely determinative enough, if this was really an alternative
+title to the book. We know that there were controversies in
+ancient times about the nature of the &lsquo;angle of contact&rsquo; (the
+&lsquo;angle&rsquo; formed, at the point of contact, between an arc of
+a circle and the tangent to it, which angle was called by the
+special name <I>hornlike,</I> <G>keratoeidh/s</G>), and the &lsquo;angle&rsquo; comple-
+mentary to it (the &lsquo;angle of a semicircle&rsquo;).<note>Proclus on Eucl. I, pp. 121. 24-122. 6.</note> The question was
+whether the &lsquo;hornlike angle&rsquo; was a magnitude comparable
+with the rectilineal angle, i.e. whether by being multiplied
+a sufficient number of times it could be made to exceed a
+<pb n=179><head>DEMOCRITUS</head>
+given rectilineal angle. Euclid proved (in III. 16) that the
+&lsquo;angle of contact&rsquo; is less than any rectilineal angle, thereby
+setting the question at rest. This is the only reference in
+Euclid to this angle and the &lsquo;angle <I>of</I> a semicircle&rsquo;, although
+he defines the &lsquo;angle <I>of</I> a segment&rsquo; in III, Def. 7, and has
+statements about the angles <I>of</I> segments in III. 31. But we
+know from a passage of Aristotle that before his time &lsquo;angles
+<I>of</I> segments&rsquo; came into geometrical text-books as elements in
+figures which could be used in the proofs of propositions<note>Arist. <I>Anal. Pr.</I> i. 24, 41 b 13-22.</note>;
+thus e.g. the equality of the two angles <I>of</I> a segment
+(assumed as known) was used to prove the theorem of
+Eucl. I. 5. Euclid abandoned the use of all such angles in
+proofs, and the references to them above mentioned are only
+survivals. The controversies doubtless arose long before his
+time, and such a question as the nature of the contact of
+a circle with its tangent would probably have a fascination
+for Democritus, who, as we shall see, broached other questions
+involving infinitesimals. As, therefore, the questions of the
+nature of the contact of a circle with its tangent and of the
+character of the &lsquo;hornlike&rsquo; angle are obviously connected,
+I prefer to read <G>gwni/hs</G> (&lsquo;of an angle&rsquo;) instead of <G>gnw/mhs</G>; this
+would give the perfectly comprehensible title, &lsquo;<I>On a difference
+in an angle, or on the contact of a circle and a sphere</I>&rsquo;. We
+know from Aristotle that Protagoras, who wrote a book on
+mathematics, <G>peri\ tw=n maqhma/twn</G>, used against the geometers
+the argument that no such straight lines and circles as
+they assume exist in nature, and that (e.g.) a material circle
+does not in actual fact touch a ruler at one point only<note>Arist. <I>Metaph.</I> B. 2, 998 a 2.</note>; and
+it seems probable that Democritus's work was directed against
+this sort of attack on geometry.
+<p>We know nothing of the contents of Democritus's book
+<I>On Geometry</I> or of his <I>Geometrica.</I> One or other of these
+works may possibly have contained the famous dilemma about
+sections of a cone parallel to the base and very close together,
+which Plutarch gives on the authority of Chrysippus.<note>Plutarch, <I>De comm. not. adv. Stoicos,</I> xxxix. 3.</note>
+<p>&lsquo;If&rsquo;, said Democritus, &lsquo;a cone were cut by a plane parallel
+to the base [by which is clearly meant a plane indefinitely
+<pb n=180><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+near to the base], what must we think of the surfaces forming
+the sections? Are they equal or unequal? For, if they are
+unequal, they will make the cone irregular as having many
+indentations, like steps, and unevennesses; but, if they are
+equal, the sections will be equal, and the cone will appear to
+have the property of the cylinder and to be made up of equal,
+not unequal, circles, which is very absurd.&rsquo;
+<p>The phrase &lsquo;<I>made up</I> of equal ... circles&rsquo; shows that
+Democritus already had the idea of a solid being the sum of
+an infinite number of parallel planes, or indefinitely thin
+laminae, indefinitely near together: a most important an-
+ticipation of the same thought which led to such fruitful
+results in Archimedes. This idea may be at the root of the
+argument by which Democritus satisfied himself of the truth
+of the two propositions attributed to him by Archimedes,
+namely that a cone is one third part of the cylinder, and
+a pyramid one third of the prism, which has the same base
+and equal height. For it seems probable that Democritus
+would notice that, if two pyramids having the same height
+and equal triangular bases are respectively cut by planes
+parallel to the base and dividing the heights in the same
+ratio, the corresponding sections of the two pyramids are
+equal, whence he would infer that the pyramids are equal as
+being the sum of the same infinite number of equal plane
+sections or indefinitely thin laminae. (This would be a par-
+ticular anticipation of Cavalieri's proposition that the areal or
+solid content of two figures is equal if two sections of them
+taken at the same height, whatever the height may be, always
+give equal straight lines or equal surfaces respectively.) And
+Democritus would of course see that the three pyramids into
+which a prism on the same base and of equal height with the
+original pyramid is divided (as in Eucl. XII. 7) satisfy this
+test of equality, so that the pyramid would be one third part
+of the prism. The extension to a pyramid with a polygonal
+base would be easy. And Democritus may have stated the
+proposition for the cone (of course without an absolute proof)
+as a natural inference from the result of increasing indefinitely
+the number of sides in a regular polygon forming the base of
+a pyramid.
+<p>Tannery notes the interesting fact that the order in the list
+<pb n=181><head>DEMOCRITUS</head>
+of Democritus's works of the treatises <I>On Geometry, Geometrica,
+Numbers,</I> and <I>On irrational lines and solids</I> corresponds to
+the order of the separate sections of Euclid's <I>Elements,</I> Books
+I-VI (plane geometry), Books VII-IX (on numbers), and
+Book X (on irrationals). With regard to the work <I>On irra-
+tional lines and solids</I> it is to be observed that, inasmuch as
+his investigation of the cone had brought Democritus con-
+sciously face to face with infinitesimals, there is nothing
+surprising in his having written on irrationals; on the con-
+trary, the subject is one in which he would be likely to take
+special interest. It is useless to speculate on what the treatise
+actually contained; but of one thing we may be sure, namely
+that the <G>a)/logoi grammai/</G>, &lsquo;irrational lines&rsquo;, were not <G>a)/tomoi
+grammai/</G>, &lsquo;<I>indivisible</I> lines&rsquo;.<note>On this cf. O. Apelt, <I>Beitr&auml;ge zur Geschichte der griechischen Philo-
+sophie,</I> 1891, p. 265 sq.</note> Democritus was too good a
+mathematician to have anything to do with such a theory.
+We do not know what answer he gave to his puzzle about the
+cone; but his statement of the dilemma shows that he was
+fully alive to the difficulties connected with the conception of
+the continuous as illustrated by the particular case, and he
+cannot have solved it, in a sense analogous to his physical
+theory of atoms, by assuming indivisible lines, for this would
+have involved the inference that the consecutive parallel
+sections of the cone are <I>unequal,</I> in which case the surface
+would (as he said) be discontinuous, forming steps, as it were.
+Besides, we are told by Simplicius that, according to Demo-
+critus himself, his atoms were, in a mathematical sense
+divisible further and in fact <I>ad infinitum,</I><note>Simpl. <I>in Phys.,</I> p. 83. 5.</note> while the scholia
+to Aristotle's <I>De caelo</I> implicitly deny to Democritus any
+theory of indivisible lines: &lsquo;of those who have maintained
+the existence of indivisibles, some, as for example Leucippus
+and Democritus, believe in indivisible bodies, others, like
+Xenocrates, in indivisible lines&rsquo;.<note>Scholia in Arist., p. 469 b 14, Brandis.</note>
+<p>With reference to the <G>*)ekpeta/smata</G> it is to be noted that
+this word is explained in Ptolemy's <I>Geography</I> as the projec-
+tion of the armillary sphere upon a plane.<note>Ptolemy, <I>Geogr.</I> vii. 7.</note> This work and
+that <I>On irrational lines</I> would hardly belong to elementary
+geometry.
+<pb n=182><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>HIPPIAS OF ELIS, the famous sophist already mentioned (pp. 2,
+23-4), was nearly contemporary with Socrates and Prodicus,
+and was probably born about 460 B.C. Chronologically, there-
+fore, his place would be here, but the only particular discovery
+attributed to him is that of the curve afterwards known as
+the <I>quadratrix,</I> and the <I>quadratrix</I> does not come within the
+scope of the <I>Elements.</I> It was used first for trisecting any
+rectilineal angle or, more generally, for dividing it in any
+ratio whatever, and secondly for squaring the circle, or rather
+for finding the length of any arc of a circle; and these prob-
+lems are not what the Greeks called &lsquo;plane&rsquo; problems, i.e.
+they cannot be solved by means of the ruler and compasses.
+It is true that some have denied that the Hippias who
+invented the <I>quadratrix</I> can have been Hippias of Elis;
+Blass<note>Fleckeisen's <I>Jahrbuch,</I> cv, p. 28.</note> and Apelt<note><I>Beitr&auml;ge zur Gesch. d. gr. Philosophie,</I> p. 379.</note> were of this opinion, Apelt arguing that at
+the time of Hippias geometry had not got far beyond the
+theorem of Pythagoras. To show how wide of the mark this
+last statement is we have only to think of the achievements
+of Democritus. We know, too, that Hippias the sophist
+specialized in mathematics, and I agree with Cantor and
+Tannery that there is no reason to doubt that it was he who
+discovered the <I>quadratrix.</I> This curve will be best described
+when we come to deal with the problem of squaring the circle
+(Chapter VII); here we need only remark that it implies the
+proposition that the lengths of arcs in a circle are proportional
+to the angles subtended by them at the centre (Eucl. VI. 33).
+<p>The most important name from the point of view of this
+chapter is HIPPOCRATES OF CHIOS. He is indeed the first
+person of whom it is recorded that he compiled a book of
+Elements. This is lost, but Simplicius has preserved in his
+commentary on the <I>Physics</I> of Aristotle a fragment from
+Eudemus's <I>History of Geometry</I> giving an account of Hippo-
+crates's quadratures of certain &lsquo;lunules&rsquo; or lunes.<note>Simpl. <I>in Phys.,</I> pp. 60. 22-68. 32, Diels.</note> This is one
+of the most precious sources for the history of Greek geometry
+before Euclid; and, as the methods, with one slight apparent
+exception, are those of the straight line and circle, we can
+form a good idea of the progress which had been made in the
+Elements up to Hippocrates's time.
+<pb n=183><head>HIPPOCRATES OF CHIOS</head>
+<p>It would appear that Hippocrates was in Athens during
+a considerable portion of the second half of the fifth century,
+perhaps from 450 to 430 B.C. We have quoted the story that
+what brought him there was a suit to recover a large sum
+which he had lost, in the course of his trading operations,
+through falling in with pirates; he is said to have remained
+in Athens on this account a long time, during which he con-
+sorted with the philosophers and reached such a degree of
+proficiency in geometry that he tried to discover a method of
+squaring the circle.<note>Philop. <I>in Phys.,</I> p. 31. 3, Vitelli.</note> This is of course an allusion to the
+quadratures of lunes.
+<p>Another important discovery is attributed to Hippocrates.
+He was the first to observe that the problem of doubling the
+cube is reducible to that of finding two mean proportionals in
+continued proportion between two straight lines.<note>Pseudo-Eratosthenes to King Ptolemy in Eutoc. on Archimedes (vol.
+iii, p. 88, Heib.).</note> The effect
+of this was, as Proclus says, that thenceforward people
+addressed themselves (exclusively) to the equivalent problem
+of finding two mean proportionals between two straight lines.<note>Proclus on Eucl. I, p. 213. 5.</note>
+<C>(<G>a</G>) <I>Hippocrates's quadrature of lunes.</I></C>
+<p>I will now give the details of the extract from Eudemus on
+the subject of Hippocrates's quadrature of lunes, which (as
+I have indicated) I place in this chapter because it belongs
+to elementary &lsquo;plane&rsquo; geometry. Simplicius says he will
+quote Eudemus &lsquo;word for word&rsquo; (<G>kata\ le/xin</G>) except for a few
+additions taken from Euclid's <I>Elements,</I> which he will insert
+for clearness' sake, and which are indeed necessitated by the
+summary (memorandum-like) style of Eudemus, whose form
+of statement is condensed, &lsquo;in accordance with ancient prac-
+tice&rsquo;. We have therefore in the first place to distinguish
+between what is textually quoted from Eudemus and what
+Simplicius has added. To Bretschneider<note>Bretschneider, <I>Die Geometrie und die Geometer vor Euklides,</I> 1870,
+pp. 100-21.</note> belongs the credit of
+having called attention to the importance of the passage of
+Simplicius to the historian of mathematics; Allman<note><I>Hermathena,</I> iv, pp. 180-228; <I>Greek Geometry from Thales to Euclid,</I>
+pp. 64-75.</note> was the
+first to attempt the task of distinguishing between the actual
+<pb n=184><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+extracts from Eudemus and Simplicius's amplifications; then
+came the critical text of Simplicius's commentary on the
+<I>Physics</I> edited by Diels (1882), who, with the help of Usener,
+separated out, and marked by spacing, the portions which they
+regarded as Eudemus's own. Tannery,<note>Tannery, <I>M&eacute;moires scientifiques,</I> vol. i, 1912, pp. 339-70, esp. pp.
+347-66.</note> who had contributed
+to the preface of Diels some critical observations, edited
+(in 1883), with a translation and notes, what he judged to be
+Eudemian (omitting the rest). Heiberg<note><I>Philologus,</I> 43, pp. 336-44.</note> reviewed the whole
+question in 1884; and finally Rudio,<note>Rudio, <I>Der Bericht des Simplicius &uuml;ber die Quadraturen des Antiphon
+und Hippokrates</I> (Teubner, 1907).</note> after giving in the
+<I>Bibliotheca Mathematica</I> of 1902 a translation of the whole
+passage of Simplicius with elaborate notes, which again he
+followed up by other articles in the same journal and elsewhere
+in 1903 and 1905, has edited the Greek text, with a transla-
+tion, introduction, notes, and appendices, and summed up the
+whole controversy.
+<p>The occasion of the whole disquisition in Simplicius's com-
+mentary is a remark by Aristotle that there is no obligation
+on the part of the exponent of a particular subject to refute
+a fallacy connected with it unless the author of the fallacy
+has based his argument on the admitted principles lying at
+the root of the subject in question. &lsquo;Thus&rsquo;, he says, &lsquo;it is for
+the geometer to refute the (supposed) quadrature of a circle by
+means of segments (<G>tmhma/twn</G>), but it is not the business of the
+geometer to refute the argument of Antiphon.&rsquo;<note>Arist. <I>Phys.</I> i. 2, 185 a 14-17.</note> Alexander
+took the remark to refer to Hippocrates's attempted quadra-
+ture by means of <I>lunes</I> (although in that case <G>tmh=ma</G> is used
+by Aristotle, not in the technical sense of a <I>segment,</I> but with
+the non-technical meaning of any portion cut out of a figure).
+This, probable enough in itself (for in another place Aristotle
+uses the same word <G>tmh=ma</G> to denote a <I>sector</I> of a circle<note>Arist. <I>De cuelo,</I> ii. 8, 290 a 4.</note>), is
+made practically certain by two other allusions in Aristotle,
+one to a proof that a circle together with certain lunes is
+equal to a rectilineal figure,<note><I>Anal. Pr.</I> ii. 25, 69 a 32.</note> and the other to &lsquo;the (fallacy) of
+Hippocrates or the quadrature by means of the lunes&rsquo;.<note><I>Soph. El.</I> 11, 171 b 15.</note> The
+<pb n=185><head>HIPPOCRATES'S QUADRATURE OF LUNES</head>
+two expressions separated by &lsquo;or&rsquo; may no doubt refer not to
+one but to two different fallacies. But if &lsquo;the quadrature by
+means of lunes&rsquo; is different from Hippocrates's quadratures of
+lunes, it must apparently be some quadrature like the second
+quoted by Alexander (not by Eudemus), and the fallacy attri-
+buted to Hippocrates must be the quadrature of a certain lune
+<I>plus</I> a circle (which in itself contains no fallacy at all). It seems
+more likely that the two expressions refer to one thing, and that
+this is the argument of Hippocrates's tract taken as a whole.
+<p>The passage of Alexander which Simplicius reproduces
+before passing to the extract from Eudemus contains two
+simple cases of quadrature, of a lune, and of lunes <I>plus</I> a semi-
+circle respectively, with an erroneous inference from these
+cases that a circle is thereby squared. It is evident that this
+account does not represent Hippocrates's own argument, for he
+would not have been capable of committing so obvious an
+error; Alexander must have drawn his information, not from
+Eudemus, but from some other source. Simplicius recognizes
+this, for, after giving the alternative account extracted from
+Eudemus, he says that we must trust Eudemus's account rather
+than the other, since Eudemus was &lsquo;nearer the times&rsquo; (of
+Hippocrates).
+<p>The two quadratures given by Alexander are as follows.
+<p>1. Suppose that <I>AB</I> is the diameter of a circle, <I>D</I> its centre,
+and <I>AC, CB</I> sides of a square
+inscribed in it.
+<p>On <I>AC</I> as diameter describe
+the semicircle <I>AEC.</I> Join <I>CD.</I>
+<FIG>
+<p>Now, since
+<MATH><I>AB</I><SUP>2</SUP>=2<I>AC</I><SUP>2</SUP></MATH>,
+and circles (and therefore semi-
+circles) are to one another as the squares on their diameters,
+<MATH>(semicircle <I>ACB</I>)=2(semicircle <I>AEC</I>)</MATH>.
+<p>But <MATH>(semicircle <I>ACB</I>)=2(quadrant <I>ADC</I>)</MATH>;
+therefore <MATH>(semicircle <I>AEC</I>)=(quadrant <I>ADC</I>)</MATH>.
+<p>If now we subtract the common part, the segment <I>AFC,</I>
+we have <MATH>(lune <I>AECF</I>)=&utri;<I>ADC</I></MATH>,
+and the lune is &lsquo;squared&rsquo;.
+<pb n=186><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>2. Next take three consecutive sides <I>CE, EF, FD</I> of a regular
+hexagon inscribed in a circle of diameter <I>CD.</I> Also take <I>AB</I>
+equal to the radius of the circle and therefore equal to each of
+the sides.
+<p>On <I>AB, CE, EF, FD</I> as diameters describe semicircles (in
+the last three cases outwards with reference to the circle).
+<p>Then, since
+<MATH><I>CD</I><SUP>2</SUP>=4<I>AB</I><SUP>2</SUP>=<I>AB</I><SUP>2</SUP>+<I>CE</I><SUP>2</SUP>+<I>EF</I><SUP>2</SUP>+<I>FD</I><SUP>2</SUP></MATH>,
+and circles are to one another as the squares on their
+diameters,
+<MATH>semicircle <I>CEFD</I>)=4 (semicircle <I>ALB</I>)
+=(sum of semicircles <I>ALB, CGE, EHF, FKD</I>)</MATH>.
+<FIG>
+<p>Subtracting from each side the sum of the small segments
+on <I>CE, EF, FD,</I> we have
+<MATH>(trapezium <I>CEFD</I>)=(sum of three lunes)+(semicircle <I>ALB</I>)</MATH>.
+<p>The author goes on to say that, subtracting the rectilineal
+figure equal to the three lunes (&lsquo;for a rectilineal figure was
+proved equal to a lune&rsquo;), we get a rectilineal figure equal
+to the semicircle <I>ALB,</I> &lsquo;and so the circle will have been
+squared&rsquo;.
+<p>This conclusion is obviously false, and, as Alexander says,
+the fallacy is in taking what was proved only of the lune on
+the side of the inscribed square, namely that it can be squared,
+to be true of the lunes on the sides of an inscribed regular
+hexagon. It is impossible that Hippocrates (one of the ablest
+of geometers) could have made such a blunder. We turn there-
+fore to Eudemus's account, which has every appearance of
+beginning at the beginning of Hippocrates's work and pro-
+ceeding in his order.
+<pb n=187><head>HIPPOCRATES'S QUADRATURE OF LUNES</head>
+<p>It is important from the point of view of this chapter to
+preserve the phraseology of Eudemus, which throws light
+on the question how far the technical terms of Euclidean
+geometry were already used by Eudemus (if not by Hippo-
+crates) in their technical sense. I shall therefore translate
+literally so much as can safely be attributed to Eudemus
+himself, except in purely geometrical work, where I shall use
+modern symbols.
+<p>&lsquo;The quadratures of lunes, which were considered to belong
+to an uncommon class of propositions on account of the
+close relation (of lunes) to the circle, were first investigated
+by Hippocrates, and his exposition was thought to be in
+correct form<note><G>kata\ tro/pon</G> (&lsquo;werthvolle Abhandlung&rsquo;, Heib.).</note>; we will therefore deal with them at length
+and describe them. He started with, and laid down as the
+first of the theorems useful for his purpose, the proposition
+that similar segments of circles have the same ratio to one
+another as the squares on their bases have [lit. as their bases
+in square, <G>duna/mei</G>]. And this he proved by first showing
+that the squares on the diameters have the same ratio as the
+circles. For, as the circles are to one another, so also are
+similar segments of them. For similar segments are those
+which are the same part of the circles respectively, as for
+instance a semicircle is similar to a semicircle, and a third
+part of a circle to a third part [here, Rudio argues, the word
+<I>segments</I>, <G>tmh/mata</G>, would seem to be used in the sense of
+<I>sectors</I>]. It is for this reason also (<G>dio\ kai\</G>) that similar
+segments contain equal angles [here &lsquo;segments&rsquo; are certainly
+segments in the usual sense]. The angles of all semicircles
+are right, those of segments greater than a semicircle are less
+than right angles and are less in proportion as the segments
+are greater than semicircles, while those of segments less than
+a semicircle are greater than right angles and are greater in
+proportion as the segments are less than semicircles.&rsquo;
+<p>I have put the last sentences of this quotation in dotted
+brackets because it is matter of controversy whether they
+belong to the original extract from Eudemus or were added by
+Simplicius.
+<p>I think I shall bring out the issues arising out of this
+passage into the clearest relief if I take as my starting-point
+the interpretation of it by Rudio, the editor of the latest
+<pb n=188><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+edition of the whole extract. Whereas Diels, Usener, Tannery,
+and Heiberg had all seen in the sentences &lsquo;For, as the circles
+are to one another . . . less than semicircles&rsquo; an addition by
+Simplicius, like the phrase just preceding (not quoted above),
+&lsquo;a proposition which Euclid placed second in his twelfth book
+with the enunciation &ldquo;Circles are to one another as the squares
+on their diameters&rdquo;&rsquo;, Rudio maintains that the sentences are
+wholly Eudemian, because &lsquo;For, as the circles are to one
+another, so are the similar segments&rsquo; is obviously connected
+with the proposition that similar segments are as the squares
+on their bases a few lines back. Assuming, then, that the
+sentences are Eudemian, Rudio bases his next argument on
+the sentence defining similar segments, &lsquo;For similar segments
+are those which are the same part of the circles: thus a semi-
+circle is similar to a semicircle, and a third part (of one circle)
+to a third part (of another circle)&rsquo;. He argues that a &lsquo;segment&rsquo;
+in the proper sense which is one third, one fourth, &amp;c., of the
+circle is not a conception likely to have been introduced into
+Hippocrates's discussion, because it cannot be visualized by
+actual construction, and so would not have conveyed any clear
+idea. On the other hand, if we divide the four right angles
+about the centre of a circle into 3, 4, or <I>n</I> equal parts by
+means of 3, 4, or <I>n</I> radii, we have an obvious division of the
+circle into equal parts which would occur to any one; that is,
+any one would understand the expression one third or one
+fourth part of a circle if the parts were <I>sectors</I> and not
+segments. (The use of the word <G>tmh=ma</G> in the sense of sector
+is not impossible in itself at a date when mathematical
+terminology was not finally fixed; indeed it means &lsquo;sector&rsquo;
+in one passage of Aristotle.<note>Arist. <I>De caelo</I>, ii. 8, 290 a 4.</note>) Hence Rudio will have it that
+&lsquo;similar segments&rsquo; in the second and third places in our passage
+are &lsquo;similar <I>sectors</I>&rsquo;. But the &lsquo;similar segments&rsquo; in the funda-
+mental proposition of Hippocrates enunciated just before are
+certainly segments in the proper sense; so are those in the
+next sentence which says that similar segments contain equal
+angles. There is, therefore, the very great difficulty that,
+under Rudio's interpretation, the word <G>tmh/mata</G> used in
+successive sentences means, first segments, then sectors, and
+then segments again. However, assuming this to be so, Rudio
+<pb n=189><head>HIPPOCRATES'S QUADRATURE OF LUNES</head>
+is able to make the argument hang together, in the following
+way. The next sentence says, &lsquo;For this reason also (<G>dio\ kai\</G>)
+similar segments contain equal angles&rsquo;; therefore this must be
+inferred from the fact that similar sectors are the same part
+of the respective circles. The intermediate steps are not given
+in the text; but, since the similar sectors are the same part
+of the circles, they contain equal angles, and it follows that the
+angles in the segments which form part of the sectors are
+equal, since they are the supplements of the halves of the
+angles of the sectors respectively (this inference presupposes
+that Hippocrates knew the theorems of Eucl. III. 20-22, which
+is indeed clear from other passages in the Eudemus extract).
+Assuming this to be the line of argument, Rudio infers that in
+Hippocrates's time similar segments were not defined as in
+Euclid (namely as segments containing equal angles) but were
+regarded as the segments belonging to &lsquo;similar <I>sectors</I>&rsquo;, which
+would thus be the prior conception. Similar sectors would
+be sectors having their angles equal. The sequence of ideas,
+then, leading up to Hippocrates's proposition would be this.
+Circles are to one another as the squares on their diameters or
+radii. Similar sectors, having their angles equal, are to one
+another as the whole circles to which they belong. (Euclid has
+not this proposition, but it is included in Theon's addition to
+VI. 33, and would be known long before Euclid's time.)
+Hence similar sectors are as the squares on the radii. But
+so are the triangles formed by joining the extremities of the
+bounding radii in each sector. Therefore (cf. Eucl. V. 19)
+the differences between the sectors and the corresponding
+triangles respectively, i.e. the corresponding <I>segments</I>, are in
+the same ratio as (1) the similar sectors, or (2) the similar
+triangles, and therefore are as the squares on the radii.
+<p>We could no doubt accept this version subject to three <I>ifs</I>,
+(1) if the passage is Eudemian, (2) if we could suppose
+<G>tmh/mata</G> to be used in different senses in consecutive sentences
+without a word of explanation, (3) if the omission of the step
+between the definition of similar &lsquo;segments&rsquo; and the inference
+that the angles in similar segments are equal could be put
+down to Eudemus's &lsquo;summary&rsquo; style. The second of these
+<I>ifs</I> is the crucial one; and, after full reflection, I feel bound
+to agree with the great scholars who have held that this
+<pb n=190><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+hypothesis is impossible; indeed the canons of literary criti-
+cism seem to exclude it altogether. If this is so, the whole
+of Rudio's elaborate structure falls to the ground.
+<p>We can now consider the whole question <I>ab initio.</I> First,
+are the sentences in question the words of Eudemus or of
+Simplicius? On the one hand, I think the-whole paragraph
+would be much more like the &lsquo;summary&rsquo; manner of Eudemus
+if it stopped at &lsquo;have the same ratio as the circles&rsquo;, i.e. if the
+sentences were not there at all. Taken together, they are
+long and yet obscurely argued, while the last sentence is
+really otiose, and, I should have said, quite unworthy of
+Eudemus. On the other hand, I do not see that Simplicius
+had any sufficient motive for interpolating such an explana-
+tion: he might have added the words &lsquo;for, as the circles are
+to one another, so also are similar segments of them&rsquo;, but
+there was no need for him to define similar segments; <I>he</I>
+must have been familiar enough with the term and its
+meaning to take it for granted that his readers would know
+them too. I think, therefore, that the sentences, down to &lsquo;the
+same part of the circles respectively&rsquo; at any rate, may be
+from Eudemus. In these sentences, then, can &lsquo;segments&rsquo; mean
+segments in the proper sense (and not sectors) after all?
+The argument that it cannot rests on the assumption that the
+Greeks of Hippocrates's day would not be likely to speak of
+a segment which was one third of the whole circle if they
+did not see their way to visualize it by actual construction.
+But, though the idea would be of no use to <I>us</I>, it does not
+follow that their point of view would be the same as ours.
+On the contrary, I agree with Zeuthen that Hippocrates may
+well have said, of segments of circles which are in the same
+ratio as the circles, that they are &lsquo;the same part&rsquo; of the circles
+respectively, for this is (in an incomplete form, it is true) the
+language of the definition of proportion in the only theory of
+proportion (the numerical) then known (cf. Eucl. VII. Def. 20,
+&lsquo;Numbers are proportional when the first is the same multiple,
+or the same part, or the same parts, of the second that the
+third is of the fourth&rsquo;, i.e. the two equal ratios are of one
+of the following forms <I>m</I>, 1/<I>n</I> or <I>m/n</I> where <I>m, n</I> are integers);
+the illustrations, namely the semicircles and the segments
+<pb n=191><head>HIPPOCRATES'S QUADRATURE OF LUNES</head>
+which are one third of the circles respectively, are from this
+point of view quite harmless.
+<p>Only the transition to the view of similar segments as
+segments &lsquo;containing equal angles&rsquo; remains to be explained.
+And here we are in the dark, because we do not know how, for
+instance, Hippocrates would have <I>drawn</I> a segment in one
+given circle which should be &lsquo;the same part&rsquo; of that circle
+that a given segment of another given circle is of that circle.
+(If e.g. he had used the proportionality of the parts into which
+the bases of the two similar segments divide the diameters
+of the circles which bisect them perpendicularly, he could,
+by means of the sectors to which the segments belong, have
+proved that the segments, like the sectors, are in the ratio
+of the circles, just as Rudio supposes him to have done; and
+the equality of the angles in the segments would have followed
+as in Rudio's proof.)
+<p>As it is, I cannot feel certain that the sentence <G>dio\ kai\ ktl</G>.
+&lsquo;this is the reason why similar segments contain equal angles&rsquo;
+is not an addition by Simplicius. Although Hippocrates was
+fully aware of the fact, he need not have stated it in this
+place, and Simplicius may have inserted the sentence in order
+to bring Hippocrates's view of similar segments into relation
+with Euclid's definition. The sentence which follows about
+&lsquo;angles of&rsquo; semicircles and &lsquo;angles of&rsquo; segments, greater or
+less than semicircles, is out of place, to say the least, and can
+hardly come from Eudemus.
+<p>We resume Eudemus's account.
+<p>&lsquo;After proving this, he proceeded to show in what way it
+was possible to square a lune the outer circumference of which
+is that of a semicircle. This he effected by circumscribing
+a semicircle about an isosceles right-angled triangle and
+(circumscribing) about the base [=describing on the base]
+a segment of a circle similar to those cut off by the sides.&rsquo;
+[This is the problem of Eucl. III. 33,
+and involves the knowledge that similar
+segments contain equal angles.]
+<FIG>
+<p>&lsquo;Then, since the segment about the
+base is equal to the sum of those about
+the sides, it follows that, when the part
+of the triangle above the segment about the base is added
+to both alike, the lune will be equal to the triangle.
+<pb n=192><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>&lsquo;Therefore the lune, having been proved equal to the triangle,
+can be squared.
+<p>&lsquo;In this way, assuming that the outer circumference of
+the lune is that of a semicircle, Hippocrates easily squared
+the lune.
+<p>&lsquo;Next after this he assumes (an outer circumference) greater
+than a semicircle (obtained) by constructing a trapezium in
+which three sides are equal to one another, while one, the
+greater of the parallel sides, is such that the square on it is
+triple of the square on each one of the other sides, and then
+comprehending the trapezium in a circle and circumscribing
+about (=describing on) its greatest side a segment similar
+to those cut off from the circle by
+the three equal sides.&rsquo;
+<FIG>
+<p>[Simplicius here inserts an easy
+proof that a circle <I>can</I> be circum-
+scribed about the trapezium.<note>Heiberg (<I>Philologus</I>, 43, p. 340) thinks that the words <G>kai\ o(/ti me\n
+perilhfqh/setai ku/klw| to\ trape/zion dei/xeis</G> [<G>ou(/tws</G>] <G>dixotomh/sas ta\s tou= trapezi/ou
+gwni/as</G> (&lsquo;Now, that the trapezium can be comprehended in a circle you
+can prove by bisecting the angles of the trapezium&rsquo;) <I>may</I> (without <G>ou(/tws</G>&mdash;
+F omits it) be Eudemus's own. For <G>o(/ti me\n</G> ... forms a natural contrast
+to <G>o(/ti de\ mei=zon</G> . . . in the next paragraph. Also cf. p. 65. 9 Diels, <G>tou/twn
+ou=)n ou(/tws e)xo/ntwn to\ trape/zio/n fhmi e)f) ou(=</G> <I>EKBH</I> <G>perilh/yetai ku/klos</G>.</note>]
+<p>&lsquo;That the said segment [bounded
+by the outer circumference <I>BACD</I>
+in the figure] is greater than a
+semicircle is clear, if a diagonal
+be drawn in the trapezium.
+<p>&lsquo;For this diagonal [say <I>BC</I>],
+subtending two sides [<I>BA, AC</I>] of
+the trapezium, is such that the
+square on it is greater than double
+the square on one of the remain-
+ing sides.&rsquo;
+<p>[This follows from the fact that, <I>AC</I> being parallel to
+<I>BD</I> but less than it, <I>BA</I> and <I>DC</I> will meet, if produced, in
+a point <I>F.</I> Then, in the isosceles triangle <I>FAC</I>, the angle
+<I>FAC</I> is less than a right angle, so that the angle <I>BAC</I> is
+obtuse.]
+<p>&lsquo;Therefore the square on [<I>BD</I>] the greatest side of the trape-
+zium [=3 <I>CD</I><SUP>2</SUP> by hypothesis] is less than the sum of the
+squares on the diagonal [<I>BC</I>] and that one of the other sides
+<pb n=193><head>HIPPOCRATES'S QUADRATURE OF LUNES</head>
+[<I>CD</I>] which is subtended<note>Observe the curious use of <G>u(potei/nein</G>, stretch under, subtend. The
+third side of a triangle is said to be &lsquo;subtended&rsquo; by the other two
+together.</note> by the said (greatest) side [<I>BD</I>]
+together with the diagonal [<I>BC</I>]&rsquo; [i.e. <MATH><I>BD</I><SUP>2</SUP><<I>BC</I><SUP>2</SUP>+<I>CD</I><SUP>2</SUP></MATH>].
+<p>&lsquo;Therefore the angle standing on the greater side of the
+trapezium [&angle;<I>BCD</I>] is acute.
+<p>&lsquo;Therefore the segment in which the said angle is is greater
+than a semicircle. And this (segment) is the outer circum-
+ference of the lune.&rsquo;
+<p>[Simplicius observes that Eudemus has omitted the actual
+squaring of the lune, presumably as being obvious. We have
+only to supply the following.
+<p>Since <MATH><I>BD</I><SUP>2</SUP>=3<I>BA</I><SUP>2</SUP>,
+(segment on <I>BD</I>)=3 (segment on <I>BA</I>)
+=(sum of segments on <I>BA, AC, CD</I>)</MATH>.
+<p>Add to each side the area between <I>BA, AC, CD</I>, and the
+circumference of the segment on <I>BD</I>, and we have
+(trapezium <I>ABDC</I>)=(lune bounded by the two circumferences).]
+<FIG>
+<p>&lsquo;A case too where the outer circumference is less than
+a semicircle was solved by Hippocrates,<note>Literally &lsquo;If (the outer circumference) were less than a semicircle,
+Hippocrates solved (<G>kateskeu/asen</G>, constructed) this (case).&rsquo;</note> who gave the follow-
+ing preliminary construction.
+<p>&lsquo;<I>Let there be a circle with diameter AB, and let its centre
+be K.</I>
+<p>&lsquo;<I>Let CD bisect BK at right angles; and let the straight
+line EF be so placed between CD and the circumference that it
+verges towards B</I> [i.e. will, if produced, pass through <I>B</I>], <I>while
+its length is also such that the square on it is</I> 1 1/2 <I>times the square
+on</I> (<I>one of</I>) <I>the radii.</I>
+<pb n=194><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>&lsquo;<I>Let EG be drawn parallel to AB, and let</I> (<I>straight lines</I>)
+<I>be drawn joining K to E and F.</I>
+<p>&lsquo;<I>Let the straight line [KF] joined to F and produced meet
+EG in G, and again let</I> (<I>straight lines</I>) <I>be drawn joining
+B to F, G.</I>
+<p>&lsquo;<I>It is then manifest that BF produced will pass through</I>
+[&ldquo;fall on&rdquo;] <I>E</I> [for by hypothesis <I>EF</I> verges towards <I>B</I>], <I>and
+BG will be equal to EK.</I>&rsquo;
+<p>[Simplicius proves this at length. The proof is easy. The
+triangles <I>FKC, FBC</I> are equal in all respects [Eucl. I. 4].
+Therefore, <I>EG</I> being parallel to <I>KB</I>, the triangles <I>EDF, GDF</I>
+are equal in all respects [Eucl. I. 15, 29, 26]. Hence the
+trapezium is isosceles, and <MATH><I>BG</I>=<I>EK</I></MATH>.
+<p>&lsquo;<I>This being so, I say that the trapezium EKBG can be
+comprehended in a circle.</I>&rsquo;
+<p>[Let the segment <I>EKBG</I> circumscribe it.]
+<p>&lsquo;Next let a segment of a circle be circumscribed about the
+triangle <I>EFG</I> also;
+then manifestly each of the segments [on] <I>EF, FG</I> will be
+similar to each of the segments [on] <I>EK, KB, BG.</I>&rsquo;
+<p>[This is because all the segments contain equal angles,
+namely an angle equal to the supplement of <I>EGK.</I>]
+<p>&lsquo;This being so, the lune so formed, of which <I>EKBG</I> is the
+outer circumference, will be equal to the rectilineal figure made
+up of the three triangles <I>BFG, BFK, EKF.</I>
+<p>&lsquo;For the segments cut off from the rectilineal figure, on the
+inner side of the lune, by the straight lines <I>EF, FG</I>, are
+(together) equal to the segments outside the rectilineal figure
+cut off by the straight lines <I>EK, KB, BG</I>, since each of the
+inner segments is 1 1/2 times each of the outer, because, by
+hypothesis, <MATH><I>EF</I><SUP>2</SUP>(=<I>FG</I><SUP>2</SUP>)=3/2<I>EK</I><SUP>2</SUP></MATH>
+[i.e. <MATH>2<I>EF</I><SUP>2</SUP>=3<I>EK</I><SUP>2</SUP>,
+=<I>EK</I><SUP>2</SUP>+<I>KB</I><SUP>2</SUP>+<I>BG</I><SUP>2</SUP>]</MATH>.
+<p>&lsquo;If then
+<MATH>(lune)=(the three segmts.)+{(rect. fig.)-(the two segmts.)}</MATH>,
+the trapezium including the two segments but not the three,
+while the (sum of the) two segments is equal to the (sum
+of the) three, it follows that
+<MATH>(lune)=(rectilineal figure)</MATH>.
+<pb n=195><head>HIPPOCRATES'S QUADRATURE OF LUNES</head>
+<p>&lsquo;The fact that this lune (is one which) has its outer circum-
+ference less than a semicircle he proves by means of the fact
+that the angle [<I>EKG</I>] in the outer segment is obtuse.
+<p>&lsquo;And the fact that the angle <I>EKG</I> is obtuse he proves as
+follows.&rsquo;
+<p>[This proof is supposed to have been given by Eudemus in
+Hippocrates's own words, but unfortunately the text is con-
+fused. The argument seems to have been substantially as
+follows.
+<p><I>By hypothesis</I>, <MATH><I>EF</I><SUP>2</SUP>=3/2<I>EK</I><SUP>2</SUP></MATH>.
+<p><I>Also</I> <MATH><I>BK</I><SUP>2</SUP>>2<I>BF</I><SUP>2</SUP></MATH> (this is assumed: we shall
+consider the ground later);
+<I>or</I> <MATH><I>EK</I><SUP>2</SUP>>2<I>KF</I><SUP>2</SUP></MATH>.
+<p><I>Therefore</I> <MATH><I>EF</I><SUP>2</SUP>=<I>EK</I><SUP>2</SUP>+1/2<I>EK</I><SUP>2</SUP>
+><I>EK</I><SUP>2</SUP>+<I>KF</I><SUP>2</SUP></MATH>,
+<I>so that the angle EKF is obtuse, and the segment is less than
+a semicircle.</I>
+<p>How did Hippocrates prove that <MATH><I>BK</I><SUP>2</SUP>>2<I>BF</I><SUP>2</SUP></MATH>? The manu-
+scripts have the phrase &lsquo;because the angle at <I>F</I> is greater&rsquo; (where
+presumably we should supply <G>o)rqh=s</G>, &lsquo;than a right angle&rsquo;).
+But, if Hippocrates proved this, he must evidently have proved
+it by means of his hypothesis <MATH><I>EF</I><SUP>2</SUP>=3/2<I>EK</I><SUP>2</SUP></MATH>, and this hypo-
+thesis leads more directly to the consequence that <MATH><I>BK</I><SUP>2</SUP>>2<I>KF</I><SUP>2</SUP></MATH>
+than to the fact that the angle at <I>F</I> is greater than a right
+angle.
+<p>We may supply the proof thus.
+<p>By hypothesis, <MATH><I>EF</I><SUP>2</SUP>=3/2<I>KB</I><SUP>2</SUP></MATH>.
+<p>Also, since <I>A, E, F, C</I> are concyclic,
+<MATH><I>EB.BF</I>=<I>AB.BC</I>
+=<I>KB</I><SUP>2</SUP></MATH>,
+or <MATH><I>EF.FB</I>+<I>BF</I><SUP>2</SUP>=<I>KB</I><SUP>2</SUP>
+=2/3<I>EF</I><SUP>2</SUP></MATH>.
+<p>It follows from the last relations that <I>EF</I>><I>FB</I>, and that
+<MATH><I>KB</I><SUP>2</SUP>>2<I>BF</I><SUP>2</SUP></MATH>.
+<p>The most remarkable feature in the above proof is the
+assumption of the solution of the problem &lsquo;<I>to place a straight</I>
+<pb n=196><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<I>line [EF] of length such that the square on it is</I> 1 1/2 <I>times the
+square on AK between the circumference of the semicircle and
+CD in such a way that it will verge</I> (<G>neu/ein</G>) <I>towards B</I>&rsquo; [i.e. if
+produced, will pass through <I>B</I>]. This is a problem of a type
+which the Greeks called <G>neu/seis</G>, <I>inclinationes</I> or <I>vergings.</I>
+Theoretically it may be regarded as the problem of finding
+a length (<I>x</I>) such that, if <I>F</I> be so taken on <I>CD</I> that <I>BF</I>=<I>x</I>,
+<I>BF</I> produced will intercept between <I>CD</I> and the circumference
+of the semicircle a length <I>EF</I> equal to &radic;3/2.<I>AK.</I>
+<p>If we suppose it done, we have
+<MATH><I>EB.BF</I>=<I>AB.BC</I>=<I>AK</I><SUP>2</SUP></MATH>;
+or <MATH><I>x</I>(<I>x</I>+&radic;(3/2).<I>a</I>)=<I>a</I><SUP>2</SUP> (where <I>AK</I>=<I>a</I>)</MATH>.
+<p>That is, the problem is equivalent to the solution of the
+quadratic equation
+<MATH><I>x</I><SUP>2</SUP>+&radic;3/2.<I>ax</I>=<I>a</I><SUP>2</SUP></MATH>.
+<p>This again is the problem of &lsquo;applying to a straight line
+of length &radic;3/2.<I>a</I> a rectangle exceeding by a square figure and
+equal in area to <I>a</I><SUP>2</SUP>&rsquo;, and would theoretically be solved by the
+Pythagorean method based on the theorem of Eucl. II. 6.
+Undoubtedly Hippocrates could have solved the problem by
+this theoretical method; but he may, on this occasion, have
+used the purely mechanical method of marking on a ruler
+or straight edge a length equal to &radic;3/2.<I>AK</I>, and then moving
+it till the points marked lay on the circumference and on <I>CD</I>
+respectively, while the straight edge also passed through <I>B.</I>
+This method is perhaps indicated by the fact that he first
+<I>places EF</I> (without producing it to <I>B</I>) and afterwards
+<I>joins BF.</I>
+<p>We come now to the last of Hippocrates's quadratures.
+Eudemus proceeds:]
+<p>&lsquo;Thus Hippocrates squared every<note>Tannery brackets <G>pa/nta</G> and <G>ei)/per kai/</G>. Heiberg thinks (<I>l.c</I>, p. 343)
+the <I>wording</I> is that of Simplicius reproducing the <I>content</I> of Eudemus.
+The wording of the sentence is important with reference to the questions
+(1) What was the paralogism with which Aristotle actually charged
+Hippocrates? and (2) What, if any, was the justification for the charge?
+Now the four quadratures as given by Eudemus are clever, and contain in
+themselves no fallacy at all. The supposed fallacy, then, can only have
+consisted in an assumption on the part of Hippocrates that, because he
+had squared one particular lune of each of three types, namely those
+which have for their outer circumferences respectively (1) a semicircle,
+(2) an are greater than a semicircle, (3) an are less than a semicircle, he
+had squared all possible lunes, and therefore also the lune included in his
+last quadrature, the squaring of which (had it been possible) would
+actually have enabled him to square the circle. The question is, did
+<05>ippocrates so delude himself? Heiberg thinks that, in the then
+state of logic, he may have done so. But it seems impossible to believe
+this of so good a mathematician; moreover, if Hippocrates had really
+thought that he had squared the circle, it is inconceivable that he
+would not have said so in express terms at the end of his fourth
+quadrature.
+<p>Another recent view is that of Bj&ouml;rnbo (in Pauly-Wissowa, <I>Real-Ency-
+clop&auml;die</I>, xvi, pp. 1787-99), who holds that Hippocrates realized
+perfectly the limits of what he had been able to do and knew that he had not
+squared the circle, but that he deliberately used language which, without
+being actually untrue, was calculated to mislead any one who read him
+into the belief that he had really solved the problem. This, too, seems
+incredible; for surely Hippocrates must have known that the first expert
+who read his tract would detect the fallacy at once, and that he was
+risking his reputation as a mathematician for no purpose. I prefer to
+think that he was merely trying to put what he had discovered in the
+most favourable light; but it must be admitted that the effect of his
+language was only to bring upon himself a charge which he might easily
+have avoided.</note> (sort of) lune, seeing
+that<note>Tannery brackets <G>pa/nta</G> and <G>ei)/per kai/</G>. Heiberg thinks (<I>l.c</I>, p. 343)
+the <I>wording</I> is that of Simplicius reproducing the <I>content</I> of Eudemus.
+The wording of the sentence is important with reference to the questions
+(1) What was the paralogism with which Aristotle actually charged
+Hippocrates? and (2) What, if any, was the justification for the charge?
+Now the four quadratures as given by Eudemus are clever, and contain in
+themselves no fallacy at all. The supposed fallacy, then, can only have
+consisted in an assumption on the part of Hippocrates that, because he
+had squared one particular lune of each of three types, namely those
+which have for their outer circumferences respectively (1) a semicircle,
+(2) an are greater than a semicircle, (3) an are less than a semicircle, he
+had squared all possible lunes, and therefore also the lune included in his
+last quadrature, the squaring of which (had it been possible) would
+actually have enabled him to square the circle. The question is, did
+<05>ippocrates so delude himself? Heiberg thinks that, in the then
+state of logic, he may have done so. But it seems impossible to believe
+this of so good a mathematician; moreover, if Hippocrates had really
+thought that he had squared the circle, it is inconceivable that he
+would not have said so in express terms at the end of his fourth
+quadrature.
+<p>Another recent view is that of Bj&ouml;rnbo (in Pauly-Wissowa, <I>Real-Ency-
+clop&auml;die</I>, xvi, pp. 1787-99), who holds that Hippocrates realized
+perfectly the limits of what he had been able to do and knew that he had not
+squared the circle, but that he deliberately used language which, without
+being actually untrue, was calculated to mislead any one who read him
+into the belief that he had really solved the problem. This, too, seems
+incredible; for surely Hippocrates must have known that the first expert
+who read his tract would detect the fallacy at once, and that he was
+risking his reputation as a mathematician for no purpose. I prefer to
+think that he was merely trying to put what he had discovered in the
+most favourable light; but it must be admitted that the effect of his
+language was only to bring upon himself a charge which he might easily
+have avoided.</note> (he squared) not only (1) the lune which has for its outer
+<pb n=197><head>HIPPOCRATES'S QUADRATURE OF LUNES</head>
+circumference the arc of a semicircle, but also (2) the lune
+in which the outer circumference is greater, and (3) the lune in
+which it is less, than a semicircle.
+<p>&lsquo;But he also squared the sum of a lune and a circle in the
+following manner.
+<p>&lsquo;<I>Let there be two circles about K as centre, such that the
+square on the diameter of the outer is</I> 6 <I>times the square on
+that of the inner.</I>
+<p>&lsquo;<I>Let a</I> (<I>regular</I>) <I>hexagon ABCDEF be inscribed in the
+inner circle, and let KA, KB, KC be joined from the centre
+and produced as far as the circumference of the outer circle.
+Let GH, HI, GI be joined.</I>&rsquo;
+<p>[Then clearly <I>GH, HI</I> are sides of a hexagon inscribed in
+the outer circle.]
+<p>&lsquo;<I>About GI</I> [i.e. on <I>GI</I>] <I>let a segment be circumscribed
+similar to the segment cut off by GH.</I>
+<p>&lsquo;<I>Then</I> <MATH><I>GI</I><SUP>2</SUP>=3<I>GH</I><SUP>2</SUP></MATH>,
+for <MATH><I>GI</I><SUP>2</SUP>+(side of outer hexagon)<SUP>2</SUP>=(diam. of outer circle)<SUP>2</SUP>
+=4<I>GH</I><SUP>2</SUP></MATH>.
+[The original states this in words without the help of the
+letters of the figure.]
+<p>&lsquo;<I>Also</I> <MATH><I>GH</I><SUP>2</SUP>=6<I>AB</I><SUP>2</SUP></MATH>.
+<pb n=198><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>&lsquo;<I>Therefore
+segment on GI</I> [<MATH>=2(segmt. on <I>GH</I>)+6(segmt. on <I>AB</I>)</MATH>]
+<MATH>=(<I>segmts. on GH, HI</I>)+(<I>all segmts. in
+inner circle</I>)</MATH>.
+<FIG>
+[&lsquo;Add to each side the area bounded by <I>GH, HI</I> and the
+arc <I>GI</I>;]
+<I>therefore</I> <MATH>(&utri;<I>GHI</I>)=(<I>lune GHI</I>)+(<I>all segmts. in inner circle</I>)</MATH>.
+<p>Adding to both sides the hexagon in the inner circle, we have
+<MATH>(&utri; <I>GHI</I>)+(inner hexagon)=(lune <I>GHI</I>)+(inner circle)</MATH>.
+&lsquo;Since, then, the sum of the two rectilineal figures can be
+squared, so can the sum of the circle and the lune in question.&rsquo;
+<p>Simplicius adds the following observations:
+<p>&lsquo;Now, so far as Hippocrates is concerned, we must allow
+that Eudemus was in a better position to know the facts, since
+he was nearer the times, being a pupil of Aristotle. But, as
+regards the &ldquo;squaring of the circle by means of segments&rdquo;
+which Aristotle reflected on as containing a fallacy, there are
+three possibilities, (1) that it indicates the squaring by means
+of lunes (Alexander was quite right in expressing the doubt
+implied by his words, &ldquo;if it is the same as the squaring by
+means of lunes&rdquo;), (2) that it refers, not to the proofs of
+Hippocrates, but some others, one of which Alexander actually
+reproduced, or (3) that it is intended to reflect on the squaring
+by Hippocrates of the circle <I>plus</I> the lune, which Hippocrates
+did in fact prove &ldquo;by means of segments&rdquo;, namely the three
+(in the greater circle) and those in the lesser circle. . . . On
+<pb n=199><head>HIPPOCRATES'S QUADRATURE OF LUNES</head>
+this third hypothesis the fallacy would lie in the fact that
+the sum of the circle and the lune is squared, and not the
+circle alone.&rsquo;
+<p>If, however, the reference of Aristotle was really to Hip-
+pocrates's last quadrature alone, Hippocrates was obviously
+misjudged; there is no fallacy in it, nor is Hippocrates likely
+to have deceived himself as to what his proof actually
+amounted to.
+<p>In the above reproduction of the extract from Eudemus
+I have marked by italics the passages where the writer follows
+the ancient fashion of describing points, lines, angles, &amp;c., with
+reference to the letters in the figure: the ancient practice was
+to write <G>to\ shmei=on e)f) w=(=|</G> (or <G>e)f) ou=(</G>) <I>K</I>, the (point) <I>on which</I> (is)
+the letter <I>K</I>, instead of the shorter form <G>to\</G> <I>K</I> <G>shmei=on</G>, the
+point <I>K</I>, used by Euclid and later geometers; <G>h( e)f) h=(</G> <I>AB</I>
+(<G>eu)qei=a</G>), the straight line <I>on which</I> (are the letters <I>AB</I>, for
+<G>h(</G> <I>AB</I> (<G>eu)qei=a</G>), the straight line <I>AB</I>; <G>to\ tri/gwnon to\ e)f) ou=(</G>
+<I>EZH</I>, the triangle <I>on which</I> (are the letters) <I>EFG</I>, instead of
+<G>to\</G> <I>EZH</I> <G>tri/gwnon</G>, the triangle <I>EFG</I>; and so on. Some have
+assumed that, where the longer archaic form, instead of the
+shorter Euclidean, is used, Eudemus must be quoting Hippocrates
+<I>verbatim</I>; but this is not a safe criterion, because, e.g., Aristotle
+himself uses both forms of expression, and there are, on the
+other hand, some relics of the archaic form even in Archimedes.
+<p>Trigonometry enables us readily to find all the types of
+Hippocratean lunes that can
+be squared by means of the
+straight line and circle. Let
+<I>ACB</I> be the external circum-
+ference, <I>ADB</I> the internal cir-
+cumference of such a lune,
+<I>r, r</I>&prime; the radii, and <I>O, O</I>&prime; the
+centres of the two arcs, <G>q</G>, <G>q</G>&prime;
+the halves of the angles sub-
+tended by the arcs at the centres
+respectively.
+<FIG>
+<p>Now (area of lune)
+<MATH>=(difference of segments <I>ACB, ADB</I>)
+=(sector <I>OACB-&utri;AOB</I>)&prime;-(sector <I>O&prime;ADB-&utri;AO&prime;B</I>)
+=<I>r</I><SUP>2</SUP><G>q</G>-<I>r</I>&prime;<SUP>2</SUP><G>q</G>&prime;+1/2 (<I>r</I>&prime;<SUP>2</SUP> sin2<G>q</G>&prime; - <I>r</I><SUP>2</SUP> sin2<G>q</G>)</MATH>.
+<pb n=200><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>We also have
+<MATH><I>r</I> sin<G>q</G>=1/2<I>AB</I>=<I>r</I>&prime; sin<G>q</G>&prime; . . . . . . (1)</MATH>
+<p>In order that the lune may be squareable, we must have, in
+the first place, <MATH><I>r</I><SUP>2</SUP><G>q</G>=<I>r</I>&prime;<SUP>2</SUP><G>q</G>&prime;</MATH>.
+<p>Suppose that <MATH><G>q</G>=<I>m</I><G>q</G>&prime;</MATH>, and it follows that
+<MATH><I>r</I>&prime;=&radic;<I>m.r.</I></MATH>
+<p>Accordingly the area becomes
+<MATH>1/2<I>r</I><SUP>2</SUP>(<I>m</I> sin2<G>q</G>&prime;-sin2<I>m</I><G>q</G>&prime;)</MATH>;
+and it remains only to solve the equation (1) above, which
+becomes <MATH>sin<I>m</I><G>q</G>&prime;=&radic;<I>m</I>.sin<G>q</G>&prime;</MATH>.
+<p>This reduces to a quadratic equation only when <I>m</I> has one
+of the values 2, 3, 3/2, 5, 5/3.
+<p>The solutions of Hippocrates correspond to the first three
+values of <I>m.</I> But the lune is squareable by &lsquo;plane&rsquo; methods
+in the other two cases also. Clausen (1840) gave the last four
+cases of the problem as new<note>Crelle, xxi, 1840, pp. 375-6.</note> (it was not then known that
+Hippocrates had solved more than the first); but, according
+to M. Simon<note><I>Geschichte der Math. im Altertum</I>, p. 174.</note>, all five cases were given much earlier in
+a dissertation by Martin Johan Wallenius of &Aring;bo (Abveae,
+1766). As early as 1687 Tschirnhausen noted the existence
+of an infinite number of squareable portions of the first of
+Hippocrates's lunes. Vieta<note>Vieta, <I>Variorum de rebus mathematicis responsorum</I> lib. viii, 1593.</note> discussed the case in which <I>m</I>=4,
+which of course leads to a cubic equation.
+<p>(<G>b</G>) <I>Reduction of the problem of doubling the cube to
+the finding of two mean proportionals.</I>
+<p>We have already alluded to Hippocrates's discovery of the
+reduction of the problem of duplicating the cube to that of
+finding two mean proportionals in continued proportion. That
+is, he discovered that, if
+<MATH><I>a</I>:<I>x</I>=<I>x</I>:<I>y</I>=<I>y</I>:<I>b</I></MATH>,
+then <MATH><I>a</I><SUP>3</SUP>:<I>x</I><SUP>3</SUP>=<I>a</I>:<I>b</I></MATH>. This shows that he could work with
+compound ratios, although for him the theory of proportion
+must still have been the incomplete, <I>numerical</I>, theory
+developed by the Pythagoreans. It has been suggested that
+<pb n=201><head>ELEMENTS AS KNOWN TO HIPPOCRATES</head>
+the idea of the reduction of the problem of duplication may
+have occurred to him through analogy. The problem of
+doubling a square is included in that of finding <I>one</I> mean
+proportional between two lines; he might therefore have
+thought of what would be the effect of finding two mean
+proportionals. Alternatively he may have got the idea from
+the theory of numbers. Plato in the <I>Timaeus</I> has the pro-
+positions that between two square numbers there is one mean
+proportional number, but that two cube numbers are connected,
+not by one, but by two mean numbers in continued proportion.<note>Plato, <I>Timaeus</I>, 32 A, B.</note>
+These are the theorems of Eucl. VIII. 11, 12, the latter of
+which is thus enunciated: &lsquo;Between two cube numbers there
+are two mean proportional numbers, and the cube has to the
+cube the ratio triplicate of that which the side has to the side.&rsquo;
+If this proposition was really Pythagorean, as seems prob-
+able enough, Hippocrates had only to give the geometrical
+adaptation of it.
+<p>(<G>g</G>) <I>The Elements as known to Hippocrates.</I>
+<p>We can now take stock of the advances made in the
+Elements up to the time when Hippocrates compiled a work
+under that title. We have seen that the Pythagorean geometry
+already contained the substance of Euclid's Books I and II,
+part of Book IV, and theorems corresponding to a great part
+of Book VI; but there is no evidence that the Pythagoreans
+paid much attention to the geometry of the circle as we find
+it, e.g., in Eucl., Book III. But, by the time of Hippocrates,
+the main propositions of Book III were also known and used,
+as we see from Eudemus's account of the quadratures of
+lunes. Thus it is assumed that &lsquo;similar&rsquo; segments contain
+equal angles, and, as Hippocrates assumes that two segments
+of circles are similar when the obvious thing about the figure
+is that the angles at the circumferences which are the supple-
+ments of the angles in the segments are one and the same,
+we may clearly infer, as above stated, that Hippocrates knew
+the theorems of Eucl. III. 20-2. Further, he assumes the
+construction on a given straight line of a segment similar to
+another given segment (cf. Eucl. III. 33). The theorems of
+Eucl. III. 26-9 would obviously be known to Hippocrates,
+<pb n=202><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+as was that of III. 31 (that the angle in a semicircle is
+a right angle, and that, according as a segment is less or
+greater than a semicircle, the angle in it is obtuse or acute).
+He assumes the solution of the problem of circumscribing
+a circle about a triangle (Eucl. IV. 5), and the theorem that
+the side of a regular hexagon inscribed in a circle is equal
+to the radius (Eucl. IV. 15).
+<p>But the most remarkable fact of all is that, according to
+Eudemus, Hippocrates actually proved the theorem of Eucl.
+XII. 2, that <I>circles are to one another as the squares on their
+diameters</I>, afterwards using this proposition to prove that
+<I>similar segments are to one another as the squares on their
+bases.</I> Euclid of course proves XII. 2 by the <I>method of
+exhaustion</I>, the invention of which is attributed to Eudoxus
+on the ground of notices in Archimedes.<note>Prefaces to <I>On the Sphere and Cylinder</I>, i, and <I>Quadrature of the
+Parabola.</I></note> This method
+depends on the use of a certain lemma known as the Axiom
+of Archimedes, or, alternatively, a lemma similar to it. The
+lemma used by Euclid is his proposition X. 1, which is closely
+related to Archimedes's lemma in that the latter is practically
+used in the proof of it. Unfortunately we have no infor-
+mation as to the nature of Hippocrates's proof; if, however,
+it amounted to a genuine proof, as Eudemus seems to imply,
+it is difficult to see how it could have been effected other-
+wise than by some anticipation in essence of the method of
+exhaustion.
+<p>THEODORUS OF CYRENE, who is mentioned by Proclus along
+with Hippocrates as a celebrated geometer and is claimed by
+Iamblichus as a Pythagorean,<note>Iambl. <I>Vit. Pyth.</I> c. 36.</note> is only known to us from
+Plato's <I>Theaetetus.</I> He is said to have been Plato's teacher
+in mathematics,<note>Diog. L. ii. 103.</note> and it is likely enough that Plato, while on
+his way to or from Egypt, spent some time with Theodorus at
+Cyrene,<note>Cf. Diog. L. iii. 6.</note> though, as we gather from the <I>Theaetetus</I>, Theodorus
+had also been in Athens in the time of Socrates. We learn
+from the same dialogue that he was a pupil of Protagoras, and
+was distinguished not only in geometry but in astronomy,
+arithmetic, music, and all educational subjects.<note>Plato, <I>Theaetetus</I>, 161 B, 162 A; <I>ib.</I> 145 A, C, D.</note> The one notice
+<pb n=203><head>THEODORUS OF CYRENE</head>
+which we have of a particular achievement of his suggests that
+it was he who first carried the theory of irrationals beyond
+the first step, namely the discovery by the Pythagoreans
+of the irrationality of &radic;2. According to the <I>Theaetetus</I>,<note><I>Theaetetus</I>, 147 D sq.</note>
+Theodorus
+<p>&lsquo;was proving<note><G>*peri\ duna/mew/n ti h(mi=n qeo/dwros o(/de e)/grafe, th=s te tri/podos pe/ri kai\
+pente/podos [a)pofai/nwn] o(/ti mh/kei ou) su/mmetroi th= podiai/a|</G>. Certain writers
+(H. Vogt in particular) persist in taking <G>e)/grafe</G> in this sentence to mean
+<I>drew</I> or <I>constructed.</I> The idea is that Theodorus's exposition must have
+included two things, first the <I>construction</I> of straight lines representing
+&radic;3, &radic;5 ... (of course by means of the Pythagorean theorem, Eucl. I. 47),
+in order to show that these straight lines exist, and secondly the <I>proof</I>
+that each of them is incommensurable with 1; therefore, it is argued,
+<G>e)/grafe</G> must indicate the construction and <G>a)pofai/nwn</G> the proof. But in
+the first place it is impossible that <G>e)/grafe/ ti peri/</G>, &lsquo;he wrote <I>something
+about</I>&rsquo; (roots), should mean &lsquo;<I>constructed</I> each of the roots&rsquo;. Moreover, if
+<G>a)pofai/nwn</G> is bracketed (as it is by Burnet), the supposed contrast between
+<G>e)/grafe</G> and <G>a)pofai/nwn</G> disappears, and <G>e)/grafe</G> <I>must</I> mean &lsquo;proved&rsquo;, in
+accordance with the natural meaning of <G>e)grafe/ ti</G>, because there is
+nothing else to govern <G>o(/ti mh/kei, ktl</G>. (&lsquo;that they are not commensurable
+in length ...&rsquo;), which phrase is of course a closer description of <G>ti</G>. There
+are plenty of instances of <G>gra/fein</G> in the sense of &lsquo;prove&rsquo;. Aristotle says
+(<I>Topics</I>, <G>*q</G>. 3, 158 b 29) &lsquo;It would appear that in mathematics too some
+things are difficult to prove (<G>ou) r(a|di/ws gra/fesqai</G>) owing to the want of
+a definition, e.g. that a straight line parallel to the side and cutting a plane
+figure (parallelogram) divides the straight line (side) and the area simi-
+larly&rsquo;. Cf. Archimedes, <I>On the Sphere and Cylinder</I>, ii, Pref., &lsquo;It happens
+that most of them are proved (<G>gra/fesqai</G>) by means of the theorems ...&rsquo;;
+&lsquo;Such of the theorems and problems as are proved (<G>gra/fetai</G>) by means of
+these theorems I have proved (or written out, <G>gra/yas</G>) and send you
+in this book&rsquo;; <I>Quadrature of a Parabola</I>, Pref., &lsquo;I have proved (<G>e)/grafon</G>)
+that every cone is one third of the cylinder with the same base and equal
+height by assuming a lemma similar to that aforesaid.&rsquo;
+<p>I do not deny that Theodorus <I>constructed</I> his &lsquo;roots&rsquo;; I have no doubt
+that he did; but this is not what <G>e)/grafe/ ti</G> means.</note> to us a certain thing about square roots
+(<G>duna/meis</G>), I mean (the square roots, i.e. sides) of three square
+feet and of five square feet, namely that these roots are not
+commensurable in length with the foot-length, and he went on
+in this way, taking all the separate cases up to the root of
+17 square feet, at which point, for some reason, he stopped&rsquo;.
+<p>That is, he proved the irrationality of &radic;3, &radic;5 ... up to
+&radic;17. It does not appear, however, that he had reached any
+definition of a surd in general or proved any general proposi-
+tion about all surds, for Theaetetus goes on to say:
+<p>&lsquo;The idea occurred to the two of us (Theaetetus and the
+younger Socrates), seeing that these square roots appeared
+<pb n=204><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+to be unlimited in multitude, to try to arrive at one collective
+term by which we could designate all these roots . . . We
+divided number in general into two classes. The number
+which can be expressed as equal multiplied by equal (<G>i)/son
+i)sa/kis</G>) we likened to a square in form, and we called it
+square and equilateral (<G>i)so/pleuron</G>) . . . The intermediate
+number, such as three, five, and any number which cannot
+be expressed as equal multiplied by equal, but is either less
+times more or more times less, so that it is always contained
+by a greater and a less side, we likened to an oblong figure
+(<G>promh/kei sxh/mati</G>) and called an oblong number. . . . Such
+straight lines then as square the equilateral and plane number
+we defined as <I>length</I> (<G>mh=kos</G>), and such as square the oblong
+(we called) <I>square roots</I> (<G>duna/meis</G>) as not being commensurable
+with the others in length but only in the plane areas to which
+their squares are equal. And there is another distinction of
+the same sort with regard to solids.&rsquo;
+<p>Plato gives no hint as to how Theodorus proved the proposi-
+tions attributed to him, namely that &radic;3, &radic;5 ... &radic;17 are
+all incommensurable with 1; there is therefore a wide field
+open for speculation, and several conjectures have been put
+forward.
+<p>(1) Hultsch, in a paper on Archimedes's approximations to
+square roots, suggested that Theodorus took the line of seeking
+successive approximations. Just as 7/5, the first approximation
+to &radic;2, was obtained by putting 2=50/25, Theodorus might
+have started from 3=48/16, and found 7/4 or 1 1/2 1/4 as a first
+approximation, and then, seeing that <MATH>1 1/2 1/4 > &radic;3 > 1 1/2</MATH>, might
+(by successive trials, probably) have found that
+<MATH>1 1/2 1/8 1/16 1/32 1/64 > &radic;3 > 1 1/2 1/8 1/16 1/32 1/128</MATH>.
+But the method of finding closer and closer approximations,
+although it might afford a presumption that the true value
+cannot be exactly expressed in fractions, would leave Theodorus
+as far as ever from <I>proving</I> that &radic;3 is incommensurable.
+<p>(2) There is no mention of &radic;2 in our passage, and Theodorus
+probably omitted this case because the incommensurability
+of &radic;2 and the traditional method of proving it were already
+known. The traditional proof was, as we have seen, a <I>reductio
+ad absurdum</I> showing that, if &radic;2 is commensurable with 1,
+it will follow that the same number is both even and odd,
+i.e. both divisible and not divisible by 2. The same method
+<pb n=205><head>THEODORUS OF CYRENE</head>
+of proof can be adapted to the cases of &radic;3, &radic;5, &amp;c., if 3, 5 ...
+are substituted for 2 in the proof; e.g. we can prove that,
+if &radic;3 is commensurable with 1, then the same number will
+be both divisible and not divisible by 3. One suggestion,
+therefore, is that Theodorus may have applied this method
+to all the cases from &radic;3 to &radic;17. We can put the proof
+quite generally thus. Suppose that <I>N</I> is a non-square number
+such as 3, 5 ..., and, if possible, let <MATH>&radic;<I>N</I>=<I>m/n</I></MATH>, where <I>m, n</I>
+are integers prime to one another.
+<p>Therefore <MATH><I>m</I><SUP>2</SUP>=<I>N</I>.<I>n</I><SUP>2</SUP></MATH>;
+therefore <I>m</I><SUP>2</SUP> is divisible by <I>N</I>, so that <I>m</I> also is a multiple
+of <I>N.</I>
+<p>Let <MATH><I>m</I>=<G>m</G>.<I>N</I>, . . . . . . . . (1)</MATH>
+and consequently <MATH><I>n</I><SUP>2</SUP>=<I>N</I>.<G>m</G><SUP>2</SUP></MATH>.
+<p>Then in the same way we can prove that <I>n</I> is a multiple
+of <I>N</I>.
+<p>Let <MATH><I>n</I>=<G>n</G>.<I>N</I> . . . . . . . . (2)</MATH>
+<p>It follows from (1) and (2) that <I>m/n</I>=<G>m</G>/<G>n</G>, where <G>m</G><<I>m</I>
+and <G>n</G><<I>n</I>; therefore <I>m/n</I> is not in its lowest terms, which
+is contrary to the hypothesis.
+<p>The objection to this conjecture as to the nature of
+Theodorus's proof is that it is so easy an adaptation of the
+traditional proof regarding &radic;2 that it would hardly be
+important enough to mention as a new discovery. Also it
+would be quite unnecessary to repeat the proof for every
+case up to &radic;17; for it would be clear, long before &radic;17 was
+reached, that it is generally applicable. The latter objection
+seems to me to have force. The former objection may or may
+not; for I do not feel sure that Plato is necessarily attributing
+any important new discovery to Theodorus. The object of
+the whole context is to show that a definition by mere
+enumeration is no definition; e.g. it is no definition of <G>e)pi-
+sth/mh</G> to enumerate particular <G>e)pisth=mai</G> (as shoemaking,
+carpentering, and the like); this is to put the cart before the
+horse, the general definition of <G>e)pisth/mh</G> being logically prior.
+Hence it was probably Theaetetus's generalization of the
+procedure of Theodorus which impressed Plato as being
+original and important rather than Theodorus's proofs them-
+selves.
+<pb n=206><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>(3) The third hypothesis is that of Zeuthen.<note>Zeuthen, &lsquo;Sur la constitution des livres arithm&eacute;tiques des &Eacute;l&eacute;ments
+d'Euclide et leur rapport &agrave; la question de l'irrationalit&eacute;&rsquo; in <I>Oversigt over
+det kgl. Danske videnskabernes Selskabs Forhandlinger</I>, 1915, pp. 422 sq.</note> He starts
+with the assumptions (<I>a</I>) that the method of proof used by
+Theodorus must have been original enough to call for special
+notice from Plato, and (<I>b</I>) that it must have been of such
+a kind that the application of it to each surd required to be
+set out separately in consequence of the variations in the
+numbers entering into the proofs. Neither of these con-
+ditions is satisfied by the hypothesis of a mere adaptation to
+&radic;3, &radic;5 ... of the traditional proof with regard to &radic;2.
+Zeuthen therefore suggests another hypothesis as satisfying
+both conditions, namely that Theodorus used the criterion
+furnished by the process of finding the greatest common
+measure as stated in the theorem of Eucl. X. 2. &lsquo;If, when
+the lesser of two unequal magnitudes is continually subtracted
+in turn from the greater [this includes the subtraction
+from any term of the highest multiple of another that it
+contains], that which is left never measures the one before
+it, the magnitudes will be incommensurable&rsquo;; that is, if two
+magnitudes are such that the process of finding their G. C. M.
+never comes to an end, the two magnitudes are incommensur-
+able. True, the proposition Eucl. X. 2 depends on the famous
+X. 1 (Given two unequal magnitudes, if from the greater
+there be subtracted more than the half (or the half), from the
+remainder more than the half (or the half), and so on, there
+will be left, ultimately, some magnitude less than the lesser
+of the original magnitudes), which is based on the famous
+postulate of Eudoxus (= Eucl. V, Def. 4), and therefore belongs
+to a later date. Zeuthen gets over this objection by pointing
+out that the necessity of X. 1 for a rigorous demonstration
+of X. 2 may not have been noticed at the time; Theodorus
+may have proceeded by intuition, or he may even have
+postulated the truth proved in X. 1.
+<p>The most obvious case in which incommensurability can be
+proved by using the process of finding the greatest common
+measure is that of the two segments of a straight line divided
+in extreme and mean ratio. For, if <I>AB</I> is divided in this way
+at <I>C</I>, we have only to mark off along <I>CA</I> (the greater segment)
+<pb n=207><head>THEODORUS OF CYRENE</head>
+a length <I>CD</I> equal to <I>CB</I> (the lesser segment), and <I>CA</I> is then
+divided at <I>D</I> in extreme and mean ratio, <I>CD</I> being the
+greater segment. (Eucl. XIII. 5 is the equivalent of this
+<FIG>
+proposition.) Similarly, <I>DC</I> is so divided if we set off <I>DE</I>
+along it equal to <I>DA</I>; and so on. This is precisely the
+process of finding the greatest common measure of <I>AC, CB</I>,
+the quotient being always unity; and the process never comes
+to an end. Therefore <I>AC, CB</I> are incommensurable. What
+is proved in this case is the irrationality of 1/2(&radic;5-1). This
+of course shows incidentally that &radic;5 is incommensurable
+with 1. It has been suggested, in view of the easiness of the
+above proof, that the irrational may first have been discovered
+with reference to the segments of a straight line cut in extreme
+and mean ratio, rather than with reference to the diagonal
+of a square in relation to its side. But this seems, on the
+whole, improbable.
+<p>Theodorus would, of course, give a geometrical form to the
+process of finding the G. C. M., after he had represented in
+a figure the particular surd which he was investigating.
+Zeuthen illustrates by two cases, &radic;5 and &radic;3.
+<p>We will take the former, which is the easier. The process
+of finding the G. C. M. (if any) of &radic;5 and 1 is as follows:
+<table>
+<tr><td>1)</td><td>&radic;5(2</td></tr>
+<tr><td></td><td align=center>2</td></tr>
+<tr><td></td><td>&radic;5-2)</td><td>1</td><td>(4</td></tr>
+<tr><td></td><td></td><td>4(&radic;5-2)</td></tr>
+<tr><td></td><td></td><td>(&radic;5-2)<SUP>2</SUP></td></tr>
+</table>
+<p>[The explanation of the second division is this:
+<MATH>1=(&radic;5-2) (&radic;5+2)=4(&radic;5-2) + (&radic;5-2)<SUP>2</SUP></MATH>.]
+<p>Since, then, the ratio of the last term (&radic;5-2)<SUP>2</SUP> to the pre-
+ceding one, &radic;5-2, is the same as the ratio of &radic;5-2 to 1,
+the process will never end.
+<p>Zeuthen has a geometrical proof which is not difficult; but
+I think the following proof is neater and easier.
+<p>Let <I>ABC</I> be a triangle right-angled at <I>B</I>, such that <I>AB</I>=1,
+<I>BC</I>=2, and therefore <I>AC</I>=&radic;5.
+<pb n=208><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>Cut off <I>CD</I> from <I>CA</I> equal to <I>CB</I>, and draw <I>DE</I> at right
+angles to <I>CA</I>. Then <I>DE</I>=<I>EB</I>.
+<p>Now <MATH><I>AD</I>=&radic;5-2</MATH>, and by similar triangles
+<MATH><I>DE</I>=2<I>AD</I>=2(&radic;5-2)</MATH>.
+<FIG>
+<p>Cut off from <I>EA</I> the portion <I>EF</I> equal to
+<I>ED</I>, and draw <I>FG</I> at right angles to <I>AE.</I>
+<p>Then <MATH><I>AF</I>=<I>AB</I> - <I>BF</I>=<I>AB</I> - 2<I>DE</I>
+= 1-4(&radic;5-2)
+= (&radic;5-2)<SUP>2</SUP></MATH>.
+<p>Therefore <I>ABC, ADE, AFG</I> are diminishing
+similar triangles such that
+<MATH><I>AB</I>:<I>AD</I>:<I>AF</I>=1:(&radic;5-2):(&radic;5-2)<SUP>2</SUP></MATH>,
+and so on.
+<p>Also <I>AB</I> > <I>FB</I>, i.e. 2 <I>DE</I> or 4<I>AD.</I>
+<p>Therefore the side of each triangle in the series is less than
+1/4 of the corresponding side of the preceding triangle.
+<p>In the case of &radic;3 the process of finding the G. C. M. of
+&radic;3 and 1 gives
+<MATH></MATH>
+the ratio of 1/2(&radic;3-1)<SUP>2</SUP> to 1/2(&radic;3-1)<SUP>3</SUP> being the same as that
+of 1 to (&radic;3-1).
+<p>This case is more difficult to show in geometrical form
+because we have to make one more
+<FIG>
+division before recurrence takes place.
+<p>The cases &radic;10 and &radic;17 are exactly
+similar to that of &radic;5.
+<p>The irrationality of &radic;2 can, of course,
+be proved by the same method. If <I>ABCD</I>
+is a square, we mark off along the diagonal
+<I>AC</I> a length <I>AE</I> equal to <I>AB</I> and draw
+<I>EF</I> at right angles to <I>AC.</I> The same
+thing is then done with the triangle <I>CEF</I>
+<pb n=209><head>THEODORUS OF CYRENE</head>
+as with the triangle <I>ABC</I>, and so on. This could not have
+escaped Theodorus if his proof in the cases of &radic;3, &radic;5 ...
+took the form suggested by Zeuthen; but he was presumably
+content to accept the traditional proof with regard to &radic;2.
+<p>The conjecture of Zeuthen is very ingenious, but, as he
+admits, it necessarily remains a hypothesis.
+<p>THEAETETUS<note>On Theaetetus the reader may consult a recent dissertation, <I>De Theae-
+teto Atheniensi mathematico</I>, by Eva Sachs (Berlin, 1914).</note> (about 415-369 B. C.) made important contribu-
+tions to the body of the Elements. These related to two
+subjects in particular, (<I>a</I>) the theory of irrationals, and (<I>b</I>) the
+five regular solids.
+<p>That Theaetetus actually succeeded in generalizing the
+theory of irrationals on the lines indicated in the second part
+of the passage from Plato's dialogue is confirmed by other
+evidence. The commentary on Eucl. X, which has survived
+in Arabic and is attributed to Pappus, says (in the passage
+partly quoted above, p. 155) that the theory of irrationals
+<p>&lsquo;had its origin in the school of Pythagoras. It was con-
+siderably developed by Theaetetus the Athenian, who gave
+proof in this part of mathematics, as in others, of ability
+which has been justly admired. . . . As for the exact dis-
+tinctions of the above-named magnitudes and the rigorous
+demonstrations of the propositions to which this theory gives
+rise, I believe that they were chiefly established by this
+mathematician. For Theaetetus had distinguished square
+roots<note>&lsquo;Square roots&rsquo;. The word in Woepcke's translation is &lsquo;puissances&rsquo;,
+which indicates that the original word was <G>duna/meis</G>. This word is always
+ambiguous; it might mean &lsquo;squares&rsquo;, but I have translated it &lsquo;square
+roots&rsquo; because the <G>du/namis</G> of Theaetetus's definition is undoubtedly the
+square root of a non-square number, a surd. The distinction in that case
+would appear to be between &lsquo;square roots&rsquo; commensurable in length and
+square roots commensurable in square only; thus &radic;3 and &radic;12 are
+commensurable in length, while &radic;3 and &radic;7 are commensurable in
+square only. I do not see how <G>duna/meis</G> could here mean squares; for
+&lsquo;squares commensurable in length&rsquo; is not an intelligible phrase, and it
+does not seem legitimate to expand it into &lsquo;squares <on straight lines>
+commensurable in length&rsquo;.</note> commensurable in length from those which are incom-
+mensurable, and had divided the well-known species of
+irrational lines after the different means, assigning the <I>medial</I>
+to geometry, the <I>binomial</I> to arithmetic, and the <I>apotome</I> to
+harmony, as is stated by Eudemus the Peripatetic.&rsquo;<note>For an explanation of this see <I>The Thirteen Books of Euclid's Elements</I>
+vol. iii, p. 4.</note>
+<pb n=210><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>The irrationals called by the names here italicized are
+described in Eucl. X. 21, 36 and 73 respectively.
+<p>Again, a scholiast<note>X, No. 62 (Heiberg's Euclid, vol. v, p. 450).</note> on Eucl. X. 9 (containing the general
+theorem that squares which have not to one another the ratio
+of a square number to a square number have their sides
+incommensurable in length) definitely attributes the discovery
+of this theorem to Theaetetus. But, in accordance with the
+traditional practice in Greek geometry, it was necessary to
+prove the existence of such incommensurable ratios, and this
+is done in the porism to Eucl. X. 6 by a geometrical con-
+struction; the porism first states that, given a straight line <I>a</I>
+and any two numbers <I>m, n</I>, we can find a straight line <I>x</I> such
+that <MATH><I>a</I>:<I>x</I>=<I>m</I>:<I>n</I></MATH>; next it is shown that, if <I>y</I> be taken a mean
+proportional between <I>a</I> and <I>x</I>, then
+<MATH><I>a</I><SUP>2</SUP>:<I>y</I><SUP>2</SUP>=<I>a</I>:<I>x</I>=<I>m</I>:<I>n</I></MATH>;
+if, therefore, the ratio <I>m</I>:<I>n</I> is not a ratio of a square to
+a square, we have constructed an irrational straight line
+<I>a</I>&radic;(<I>n</I>/<I>m</I>) and therefore shown that such a straight line
+exists.
+<p>The proof of Eucl. X. 9 formally depends on VIII. 11 alone
+(to the effect that between two square numbers there is one
+mean proportional number, and the square has to the square
+the duplicate ratio of that which the side has to the side);
+and VIII. 11 again depends on VII. 17 and 18 (to the effect
+that <MATH><I>ab</I>:<I>ac</I>=<I>b</I>:<I>c</I></MATH>, and <MATH><I>a</I>:<I>b</I>=<I>ac</I>:<I>bc</I></MATH>, propositions which are
+not identical). But Zeuthen points out that these propositions
+are an inseparable part of a whole theory established in
+Book VII and the early part of Book VIII, and that the
+real demonstration of X. 9 is rather contained in propositions
+of these Books which give a rigorous proof of the necessary
+and sufficient conditions for the rationality of the square
+roots of numerical fractions and integral numbers, notably
+VII. 27 and the propositions leading up to it, as well as
+VIII. 2. He therefore suggests that the theory established
+in the early part of Book VII was not due to the Pytha-
+goreans, but was an innovation made by Theaetetus with the
+direct object of laying down a scientific basis for his theory
+of irrationals, and that this, rather than the mere formulation
+<pb n=211><head>THEAETETUS</head>
+of the theorem of Eucl. X. 9, was the achievement which Plato
+intended to hold up to admiration.
+<p>This conjecture is of great interest, but it is, so far as
+I know, without any positive confirmation. On the other
+hand, there are circumstances which suggest doubts. For
+example, Zeuthen himself admits that Hippocrates, who re-
+duced the duplication of the cube to the finding of two mean
+proportionals, must have had a proposition corresponding to
+the very proposition VIII. 11 on which X. 9 formally depends.
+Secondly, in the extract from Simplicius about the squaring
+of lunes by Hippocrates, we have seen that the proportionality
+of similar segments of circles to the circles of which they form
+part is explained by the statement that &lsquo;similar segments are
+those which are <I>the same part</I> of the circles&rsquo;; and if we may
+take this to be a quotation by Eudemus from Hippocrates's
+own argument, the inference is that Hippocrates had a defini-
+tion of numerical proportion which was at all events near
+to that of Eucl. VII, Def. 20. Thirdly, there is the proof
+(presently to be given) by Archytas of the proposition that
+there can be no number which is a (geometric) mean between
+two consecutive integral numbers, in which proof it will
+be seen that several propositions of Eucl., Book VII, are
+pre-supposed; but Archytas lived (say) 430-365 B.C., and
+Theaetetus was some years younger. I am not, therefore,
+prepared to give up the view, which has hitherto found
+general acceptance, that the Pythagoreans already had a
+theory of proportion of a numerical kind on the lines, though
+not necessarily or even probably with anything like the
+fullness and elaboration, of Eucl., Book VII.
+<p>While Pappus, in the commentary quoted, says that Theae-
+tetus distinguished the well-known species of irrationals, and
+in particular the <I>medial</I>, the <I>binomial</I>, and the <I>apotome</I>, he
+proceeds thus:
+<p>&lsquo;As for Euclid, he set himself to give rigorous rules, which
+he established, relative to commensurability and incommen-
+surability in general; he made precise the definitions and
+distinctions between rational and irrational magnitudes, he
+set out a great number of orders of irrational magnitudes,
+and finally he made clear their whole extent.&rsquo;
+<p>As Euclid proves that there are thirteen irrational straight
+<pb n=212><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+lines in all, we may perhaps assume that the subdivision of
+the three species of irrationals distinguished by Theaetetus
+into thirteen was due to Euclid himself, while the last words
+of the quotation seem to refer to Eucl. X. 115, where it is
+proved that from the <I>medial</I> straight line an unlimited number
+of other irrationals can be derived which are all different from
+it and from one another.
+<p>It will be remembered that, at the end of the passage of the
+<I>Theaetetus</I> containing the definition of &lsquo;square roots&rsquo; or surds,
+Theaetetus says that &lsquo;there is a similar distinction in the case
+of solids&rsquo;. We know nothing of any further development
+of a theory of irrationals arising from solids; but Theaetetus
+doubtless had in mind a distinction related to VIII. 12 (the
+theorem that between two cube numbers there are two mean
+proportional numbers) in the same way as the definition of
+a &lsquo;square root&rsquo; or surd is related to VIII. 11; that is to say,
+he referred to the incommensurable cube root of a non-cube
+number which is the product of three factors.
+<p>Besides laying the foundation of the theory of irrationals
+as we find it in Eucl., Book X, Theaetetus contributed no less
+substantially to another portion of the <I>Elements</I>, namely
+Book XIII, which is devoted (after twelve introductory
+propositions) to constructing the five regular solids, circum-
+scribing spheres about them, and finding the relation between
+the dimensions of the respective solids and the circumscribing
+spheres. We have already mentioned (pp. 159, 162) the tradi-
+tions that Theaetetus was the first to &lsquo;construct&rsquo; or &lsquo;write upon&rsquo;
+the five regular solids,<note>Suidas, <I>s. v.</I> <G>*qeai/thtos</G>.</note> and that his name was specially
+associated with the octahedron and the icosahedron.<note>Schol. 1 to Eucl. XIII (Euclid, ed. Heiberg, vol. v, p. 654).</note> There
+can be little doubt that Theaetetus's &lsquo;construction&rsquo; of, or
+treatise upon, the regular solids gave the theoretical con-
+structions much as we find them in Euclid.
+<p>Of the mathematicians of Plato's time, two others are
+mentioned with Theaetetus as having increased the number
+of theorems in geometry and made a further advance towards
+a scientific grouping of them, LEODAMAS OF THASOS and
+ARCHYTAS OF TARAS. With regard to the former we are
+<pb n=213><head>ARCHYTAS</head>
+told that Plato &lsquo;explained (<G>ei)shgh/sato</G>) to Leodamas of Thasos
+the method of inquiry by analysis&rsquo;<note>Diog. L. iii. 24.</note>; Proclus's account is
+fuller, stating that the finest method for discovering lemmas
+in geometry is that &lsquo;which by means of <I>analysis</I> carries the
+thing sought up to an acknowledged principle, a method
+which Plato, as they say, communicated to Leodamas, and
+by which the latter too is said to have discovered many
+things in geometry&rsquo;.<note>Proclus on Eucl. I, p. 211. 19-23.</note> Nothing more than this is known of
+Leodamas, but the passages are noteworthy as having given
+rise to the idea that Plato <I>invented</I> the method of mathe-
+matical analysis, an idea which, as we shall see later on, seems
+nevertheless to be based on a misapprehension.
+<p>ARCHYTAS OF TARAS, a Pythagorean, the friend of Plato,
+flourished in the first half of the fourth century, say 400 to
+365 B.C. Plato made his acquaintance when staying in Magna
+Graecia, and he is said, by means of a letter, to have saved
+Plato from death at the hands of Dionysius. Statesman and
+philosopher, he was famed for every sort of accomplishment.
+He was general of the forces of his city-state for seven years,
+though ordinarily the law forbade any one to hold the post
+for more than a year; and he was never beaten. He is
+said to have been the first to write a systematic treatise on
+<I>mechanics</I> based on mathematical principles.<note>Diog. L. viii. 79-83.</note> Vitruvius men-
+tions that, like Archimedes, Ctesibius, Nymphodorus, and
+Philo of Byzantium, Archytas wrote on machines<note>Vitruvius, <I>De architectura</I>, Praef. vii. 14.</note>; two
+mechanical devices in particular are attributed to him, one
+a mechanical dove made of wood which would fly,<note>Gellius, x. 12. 8, after Favorinus (<I>Vors.</I> i<SUP>3</SUP>, p. 325. 21-9).</note> the
+other a rattle which, according to Aristotle, was found useful
+to &lsquo;give to children to occupy them, and so prevent them
+from breaking things about the house (for the young are
+incapable of keeping still)&rsquo;.<note>Aristotle, <I>Pol&iacute;tics</I>, E (<G>*q</G>). 6, 1340 b 26.</note>
+<p>We have already seen Archytas distinguishing the four
+mathematical sciences, geometry, arithmetic, sphaeric (or
+astronomy), and music, comparing the art of calculation with
+geometry in respect of its relative efficiency and conclusive-
+ness, and defining the three means in music, the arithmetic,
+<pb n=214><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+the geometric, and the harmonic (a name substituted by
+Archytas and Hippasus for the older name &lsquo;sub-contrary&rsquo;).
+<p>From his mention of <I>sphaeric</I> in connexion with his state-
+ment that &lsquo;the mathematicians have given us clear knowledge
+about the speed of the heavenly bodies and their risings and
+settings&rsquo; we gather that in Archytas's time astronomy was
+already treated mathematically, the properties of the sphere
+being studied so far as necessary to explain the movements
+in the celestial sphere. He discussed too the question whether
+the universe is unlimited in extent, using the following
+argument.
+<p>&lsquo;If I were at the outside, say at the heaven of the fixed
+stars, could I stretch my hand or my stick outwards or not?
+To suppose that I could not is absurd; and if I can stretch
+it out, that which is outside must be either body or space (it
+makes no difference which it is, as we shall see). We may
+then in the same way get to the outside of that again, and
+so on, asking on arrival at each new limit the same question;
+and if there is always a new place to which the stick may be
+held out, this clearly involves extension without limit. If
+now what so extends is body, the proposition is proved; but
+even if it is space, then, since space is that in which body
+is or can be, and in the case of eternal things we must treat
+that which potentially is as being, it follows equally that there
+must be body and space (extending) without limit.&rsquo;<note>Simplicius <I>in Phys.</I>, p. 467. 26.</note>
+<p>In <I>geometry</I>, while Archytas doubtless increased the number
+of theorems (as Proclus says), only one fragment of his has
+survived, namely the solution of the problem of finding two
+mean proportionals (equivalent to the duplication of the cube)
+by a remarkable theoretical construction in three dimensions.
+As this, however, belongs to higher geometry and not to the
+Elements, the description of it will come more appropriately
+in another place (pp. 246-9).
+<p>In <I>music</I> he gave the numerical ratios representing the
+intervals of the tetrachord on three scales, the anharmonic,
+the chromatic, and the diatonic.<note>Ptol. <I>harm.</I> i. 13, p. 31 Wall.</note> He held that sound is due
+to impact, and that higher tones correspond to quicker motion
+communicated to the air, and lower tones to slower motion.<note>Porph. <I>in Ptol. harm.</I>, p. 236 (<I>Vors.</I> i<SUP>3</SUP>, p. 232-3); Theon of Smyrna,
+p. 61. 11-17.</note>
+<pb n=215><head>ARCHYTAS</head>
+<p>Of the fragments of Archytas handed down to us the most
+interesting from the point of view of this chapter is a proof
+of the proposition that there can be no number which is
+a (geometric) mean between two numbers in the ratio known
+as <G>e)pimo/rios</G> or <I>superparticularis</I>, that is, (<I>n</I>+1):<I>n.</I> This
+proof is preserved by Boetius<note>Boetius, <I>De inst. mus.</I> iii. 11, pp. 285-6 Friedlein.</note>, and the noteworthy fact about
+it is that it is substantially identical with the proof of the
+same theorem in Prop. 3 of Euclid's tract on the <I>Sectio
+canonis.</I><note><I>Musici scriptores Graeci</I>, ed. Jan, p. 14; Heiberg and Menge's Euclid,
+vol. viii, p. 162.</note> I will quote Archytas's proof in full, in order to
+show the slight differences from the Euclidean form and
+notation.
+<p>Let <I>A, B</I> be the given &lsquo;superparticularis proportio&rsquo; (<G>e)pi-
+mo/rion dia/sthma</G> in Euclid). [Archytas writes the smaller
+number first (instead of second, as Euclid does); we are then
+to suppose that <I>A, B</I> are integral numbers in the ratio of
+<I>n</I> to (<I>n</I>+1).]
+<p>Take <I>C, DE</I> the smallest numbers which are in the ratio
+of <I>A</I> to <I>B.</I> [Here <I>DE</I> means <I>D</I>+<I>E</I>; in this respect the
+notation differs from that of Euclid, who, as usual, takes
+a straight line <I>DF</I> divided into two parts at <I>G</I>, the parts
+<I>DG, GF</I> corresponding to the <I>D</I> and <I>E</I> respectively in
+Archytas's proof. The step of finding <I>C, DE</I> the smallest
+numbers in the same ratio as that of <I>A</I> to <I>B</I> presupposes
+Eucl. VII. 33 applied to two numbers.]
+<p>Then <I>DE</I> exceeds <I>C</I> by an aliquot part of itself and of <I>C</I>
+[cf. the definition of <G>e)pimo/rios a)riqmo/s</G> in Nicomachus, i. 19. 1].
+<p>Let <I>D</I> be the excess [i.e. we suppose <I>E</I> equal to <I>C</I>].
+<p>I say that <I>D</I> is not a number, but a unit.
+<p>For, if <I>D</I> is a number and an aliquot part of <I>DE</I>, it measures
+<I>DE</I>; therefore it measures <I>E</I>, that is, <I>C.</I>
+<p>Thus <I>D</I> measures both <I>C</I> and <I>DE</I>: which is impossible,
+since the smallest numbers which are in the same ratio as
+any numbers are prime to one another. [This presupposes
+Eucl. VII. 22.]
+<p>Therefore <I>D</I> is a unit; that is, <I>DE</I> exceeds <I>C</I> by a unit.
+<p>Hence no number can be found which is a mean between
+the two numbers <I>C, DE</I> [for there is no <I>integer</I> intervening].
+<pb n=216><head>THE ELEMENTS DOWN TO PLATO'S TIME</head>
+<p>Therefore neither can any number be a mean between the
+original numbers <I>A, B</I>, which are in the same ratio as <I>C, DE</I>
+[cf. the more general proposition, Eucl. VIII. 8; the particular
+inference is a consequence of Eucl. VII. 20, to the effect that
+the least numbers of those which have the same ratio with
+them measure the latter the same number of times, the greater
+the greater and the less the less].
+<p>Since this proof cites as known several propositions corre-
+sponding to propositions in Euclid, Book VII, it affords a strong
+presumption that there already existed, at least as early as
+the time of Archytas, a treatise of some sort on the Elements
+of Arithmetic in a form similar to the Euclidean, and con-
+taining many of the propositions afterwards embodied by
+Euclid in his arithmetical books.
+<C>Summary.</C>
+<p>We are now in a position to form an idea of the scope of
+the Elements at the stage which they had reached in Plato's
+time. The substance of Eucl. I-IV was practically complete.
+Book V was of course missing, because the theory of proportion
+elaborated in that book was the creation of Eudoxus. The
+Pythagoreans had a theory of proportion applicable to com-
+mensurable magnitudes only; this was probably a numerical
+theory on lines similar to those of Eucl., Book VII. But the
+theorems of Eucl., Book VI, in general, albeit insufficiently
+established in so far as they depended on the numerical theory
+of proportion, were known and used by the Pythagoreans.
+We have seen reason to suppose that there existed Elements
+of Arithmetic partly (at all events) on the lines of Eucl.,
+Book VII, while some propositions of Book VIII (e.g. Props.
+11 and 12) were also common property. The Pythagoreans,
+too, conceived the idea of perfect numbers (numbers equal to
+the sum of all their divisors) if they had not actually shown
+(as Euclid does in IX. 36) how they are evolved. There can
+also be little doubt that many of the properties of plane and
+solid numbers and of similar numbers of both classes proved in
+Euclid, Books VIII and IX, were known before Plato's time.
+<p>We come next to Book X, and it is plain that the foundation
+of the whole had been well and truly laid by Theaetetus, and
+<pb n=217><head>SUMMARY</head>
+the main varieties of irrationals distinguished, though their
+classification was not carried so far as in Euclid.
+<p>The substance of Book XI. 1-19 must already have been in-
+cluded in the Elements (e.g. Eucl. XI. 19 is assumed in Archytas's
+construction for the two mean proportionals), and the whole
+theory of the section of Book XI in question would be required
+for Theaetetus's work on the five regular solids: XI. 21 must
+have been known to the Pythagoreans: while there is nothing
+in the latter portion of the book about parallelepipedal solids
+which (subject to the want of a rigorous theory of proportion)
+was not within the powers of those who were familiar with
+the theory of plane and solid numbers.
+<p>Book XII employs throughout the <I>method of exhaustion</I>,
+the orthodox form of which is attributed to Eudoxus, who
+grounded it upon a lemma known as Archimedes's Axiom or
+its equivalent (Eucl. X. 1). Yet even XII. 2, to the effect that
+circles are to one another as the square of their diameters, had
+already been anticipated by Hippocrates of Chios, while
+Democritus had discovered the truth of the theorems of
+XII. 7, Por., about the volume of a pyramid, and XII. 10,
+about the volume of a cone.
+<p>As in the case of Book X, it would appear that Euclid was
+indebted to Theaetetus for much of the substance of Book XIII,
+the latter part of which (Props. 12-18) is devoted to the
+construction of the five regular solids, and the inscribing of
+them in spheres.
+<p>There is therefore probably little in the whole compass of
+the <I>Elements</I> of Euclid, except the new theory of proportion due
+to Eudoxus and its consequences, which was not in substance
+included in the recognized content of geometry and arithmetic
+by Plato's time, although the form and arrangement of the
+subject-matter and the methods employed in particular cases
+were different from what we find in Euclid.
+<pb>
+<C>VII
+SPECIAL PROBLEMS</C>
+<p>SIMULTANEOUSLY with the gradual evolution of the Elements,
+the Greeks were occupying themselves with problems in
+higher geometry; three problems in particular, the squaring
+of the circle, the doubling of the cube, and the trisection of
+any given angle, were rallying-points for mathematicians
+during three centuries at least, and the whole course of Greek
+geometry was profoundly influenced by the character of the
+specialized investigations which had their origin in the attempts
+to solve these problems. In illustration we need only refer
+to the subject of conic sections which began with the use
+made of two of the curves for the finding of two mean pro-
+portionals.
+<p>The Greeks classified problems according to the means by
+which they were solved. The ancients, says Pappus, divided
+them into three classes, which they called <I>plane, solid</I>, and
+<I>linear</I> respectively. Problems were <I>plane</I> if they could be
+solved by means of the straight line and circle only, <I>solid</I>
+if they could be solved by means of one or more conic sections,
+and <I>linear</I> if their solution required the use of other curves
+still more complicated and difficult to construct, such as spirals,
+<I>quadratrices</I>, cochloids (conchoids) and cissoids, or again the
+various curves included in the class of &lsquo;loci on surfaces&rsquo; (<G>to/poi
+pro\s e)pifanei/ais</G>), as they were called.<note>Pappus, iii, pp. 54-6, iv, pp. 270-2.</note> There was a corre-
+sponding distinction between loci: <I>plane</I> loci are straight
+lines or circles; <I>solid</I> loci are, according to the most strict
+classification, conics only, which arise from the sections of
+certain solids, namely cones; while <I>linear</I> loci include all
+<pb n=219><head>CLASSIFICATION OF PROBLEMS</head>
+higher curves.<note>Cf. Pappus, vii, p. 662, 10-15.</note> Another classification of loci divides them
+into <I>loci on lines</I> (<G>to/poi pro\s grammai=s</G>) and <I>loci on surfaces</I>
+(<G>to/poi pro\s e)pifanei/ais</G>).<note>Proclus on Eucl. I, p. 394. 19.</note> The former term is found in
+Proclus, and seems to be used in the sense both of loci which
+<I>are</I> lines (including of course curves) and of loci which are
+spaces bounded by lines; e.g. Proclus speaks of &lsquo;the whole
+space between the parallels&rsquo; in Eucl. I. 35 as being the locus
+of the (equal) parallelograms &lsquo;on the same base and in the
+same parallels&rsquo;.<note><I>Ib.</I>, p. 395. 5.</note> Similarly <I>loci on surfaces</I> in Proclus may
+be loci which <I>are</I> surfaces; but Pappus, who gives lemmas
+to the two books of Euclid under that title, seems to imply
+that they were curves drawn on surfaces, e.g. the cylindrical
+helix.<note>Pappus, iv, p. 258 sq.</note>
+<p>It is evident that the Greek geometers came very early
+to the conclusion that the three problems in question were not
+<I>plane</I>, but required for their solution either higher curves
+than circles or constructions more mechanical in character
+than the mere use of the ruler and compasses in the sense of
+Euclid's Postulates 1-3. It was probably about 420 B.C. that
+Hippias of Elis invented the curve known as the <I>quadratrix</I>
+for the purpose of trisecting any angle, and it was in the first
+half of the fourth century that Archytas used for the dupli-
+cation of the cube a solid construction involving the revolution
+of plane figures in space, one of which made a <I>tore</I> or anchor-
+ring with internal diameter <I>nil.</I> There are very few records
+of illusory attempts to do the impossible in these cases. It is
+practically only in the case of the squaring of the circle that
+we read of abortive efforts made by &lsquo;plane&rsquo; methods, and none
+of these (with the possible exception of Bryson's, if the
+accounts of his argument are correct) involved any real
+fallacy. On the other hand, the bold pronouncement of
+Antiphon the Sophist that by inscribing in a circle a series
+of regular polygons each of which has twice as many sides
+as the preceding one, we shall use up or exhaust the area of
+the circle, though it was in advance of his time and was
+condemned as a fallacy on the technical ground that a straight
+line cannot coincide with an arc of a circle however short
+its length, contained an idea destined to be fruitful in the
+<pb n=220><head>SPECIAL PROBLEMS</head>
+hands of later and abler geometers, since it gives a method
+of approximating, with any desired degree of accuracy, to the
+area of a circle, and lies at the root of the <I>method of exhaustion</I>
+as established by Eudoxus. As regards Hippocrates's quadra-
+ture of lunes, we must, notwithstanding the criticism of
+Aristotle charging him with a paralogism, decline to believe
+that he was under any illusion as to the limits of what his
+method could accomplish, or thought that he had actually
+squared the circle.
+<C>The squaring of the circle.</C>
+<p>There is presumably no problem which has exercised such
+a fascination throughout the ages as that of rectifying or
+squaring the circle; and it is a curious fact that its attraction
+has been no less (perhaps even greater) for the non-mathe-
+matician than for the mathematician. It was naturally the
+kind of problem which the Greeks, of all people, would take
+up with zest the moment that its difficulty was realized. The
+first name connected with the problem is Anaxagoras, who
+is said to have occupied himself with it when in prison.<note>Plutarch, <I>De exil.</I> 17, p. 607 F.</note>
+The Pythagoreans claimed that it was solved in their school,
+&lsquo;as is clear from the demonstrations of Sextus the Pythagorean,
+who got his method of demonstration from early tradition&rsquo;<note>Iambl. ap. Simpl. <I>in Categ.</I>, p. 192, 16-19 K., 64 b 11 Brandis.</note>;
+but Sextus, or rather Sextius, lived in the reign of Augustus
+or Tiberius, and, for the usual reasons, no value can be
+attached to the statement.
+<p>The first serious attempts to solve the problem belong to
+the second half of the fifth century B.C. A passage of
+Aristophanes's <I>Birds</I> is quoted as evidence of the popularity
+of the problem at the time (414 B.C.) of its first representation.
+Aristophanes introduces Meton, the astronomer and discoverer
+of the Metonic cycle of 19 years, who brings with him a ruler
+and compasses, and makes a certain construction &lsquo;in order that
+your circle may become square&rsquo;.<note>Aristophanes, <I>Birds</I> 1005.</note> This is a play upon words,
+because what Meton really does is to divide a circle into four
+quadrants by two diameters at right angles to one another;
+the idea is of streets radiating from the agora in the centre
+<pb n=221><head>THE SQUARING OF THE CIRCLE</head>
+of a town; the word <G>tetra/gwnos</G> then really means &lsquo;with four
+(right) angles&rsquo; (at the centre), and not &lsquo;square&rsquo;, but the word
+conveys a laughing allusion to the problem of squaring all
+the same.
+<p>We have already given an account of Hippocrates's quadra-
+tures of lunes. These formed a sort of <I>prolusio</I>, and clearly
+did not purport to be a solution of the problem; Hippocrates
+was aware that &lsquo;plane&rsquo; methods would not solve it, but, as
+a matter of interest, he wished to show that, if circles could
+not be squared by these methods, they could be employed
+to find the area of <I>some</I> figures bounded by arcs of circles,
+namely certain lunes, and even of the sum of a certain circle
+and a certain lune.
+<p>ANTIPHON of Athens, the Sophist and a contemporary of
+Socrates, is the next person to claim attention. We owe
+to Aristotle and his commentators our knowledge of Anti-
+phon's method. Aristotle observes that a geometer is only
+concerned to refute any fallacious arguments that may be
+propounded in his subject if they are based upon the admitted
+principles of geometry; if they are not so based, he is not
+concerned to refute them:
+<p>&lsquo;thus it is the geometer's business to refute the quadrature by
+means of segments, but it is not his business to refute that
+of Antiphon&rsquo;.<note>Arist. <I>Phys.</I> i. 2, 185 a 14-17.</note>
+<FIG>
+<p>As we have seen, the quadrature &lsquo;by means of segments&rsquo; is
+probably Hippocrates's quad-
+rature of lunes. Antiphon's
+method is indicated by Themis-
+tius<note>Them. <I>in Phys.</I>, p. 4. 2 sq., Schenkl.</note> and Simplicius.<note>Simpl. <I>in Phys.</I>, p. 54. 20-55. 24, Diels.</note> Suppose
+there is any regular polygon
+inscribed in a circle, e.g. a square
+or an equilateral triangle. (Ac-
+cording to Themistius, Antiphon
+began with an equilateral triangle,
+and this seems to be the authentic
+version; Simplicius says he in-
+scribed some one of the regular polygons which can be inscribed
+<pb n=222><head>THE SQUARING OF THE CIRCLE</head>
+in a circle, &lsquo;suppose, if it so happen, that the inscribed polygon
+is a square&rsquo;.) On each side of the inscribed triangle or square
+as base describe an isosceles triangle with its vertex on the
+arc of the smaller segment of the circle subtended by the side.
+This gives a regular inscribed polygon with double the number
+of sides. Repeat the construction with the new polygon, and
+we have an inscribed polygon with four times as many sides as
+the original polygon had. Continuing the process,
+<p>&lsquo;Antiphon thought that in this way the area (of the circle)
+would be used up, and we should some time have a polygon
+inscribed in the circle the sides of which would, owing to their
+smallness, coincide with the circumference of the circle. And,
+as we can make a square equal to any polygon ... we shall
+be in a position to make a square equal to a circle.&rsquo;
+<p>Simplicius tells us that, while according to Alexander the
+geometrical principle hereby infringed is the truth that a circle
+touches a straight line in one point (only), Eudemus more
+correctly said it was the principle that magnitudes are divisible
+without limit; for, if the area of the circle is divisible without
+limit, the process described by Antiphon will never result in
+using up the whole area, or in making the sides of the polygon
+take the position of the actual circumference of the circle.
+But the objection to Antiphon's statement is really no more than
+verbal; Euclid uses exactly the same construction in XII. 2,
+only he expresses the conclusion in a different way, saying
+that, if the process be continued far enough, the small seg-
+ments left over will be together less than any assigned area.
+Antiphon in effect said the same thing, which again we express
+by saying that the circle is the <I>limit</I> of such an inscribed
+polygon when the number of its sides is indefinitely increased.
+Antiphon therefore deserves an honourable place in the history
+of geometry as having originated the idea of <I>exhausting</I> an
+area by means of inscribed regular polygons with an ever
+increasing number of sides, an idea upon which, as we said,
+Eudoxus founded his epoch-making <I>method of exhaustion.</I>
+The practical value of Antiphon's construction is illustrated
+by Archimedes's treatise on the <I>Measurement of a Circle</I>,
+where, by constructing inscribed and circumscribed regular
+polygons with 96 sides, Archimedes proves that <MATH>3 1/7 > <G>p</G> > 3 10/71</MATH>,
+the lower limit, <MATH><G>p</G> > 3 10/71</MATH>, being obtained by calculating the
+<pb n=223><head>ANTIPHON AND BRYSON</head>
+perimeter of the <I>inscribed</I> polygon of 96 sides, which is
+constructed in Antiphon's manner from an inscribed equilateral
+triangle. The same construction starting from a square was
+likewise the basis of Vieta's expression for 2/<G>p</G>, namely
+<MATH>2/<G>p</G>=cos<G>p</G>/4.cos<G>p</G>/8.cos<G>p</G>/16 ...
+=&radic;(1/2).&radic;(1/2)(1+&radic;(1/2)).&radic;(1/2)(1+&radic;(1/2)(1+&radic;(1/2))) ... (<I>ad inf.</I>)</MATH>
+<p>BRYSON, who came a generation later than Antiphon, being
+a pupil of Socrates or of Euclid of Megara, was the author
+of another attempted quadrature which is criticized by
+Aristotle as &lsquo;sophistic&rsquo; and &lsquo;eristic&rsquo; on the ground that it
+was based on principles not special to geometry but applicable
+equally to other subjects.<note>Arist. <I>An. Post.</I> i. 9, 75 b 40.</note> The commentators give accounts
+of Bryson's argument which are substantially the same, except
+that Alexander speaks of <I>squares</I> inscribed and circumscribed
+to a circle<note>Alexander on <I>Soph. El.</I>, p. 90. 10-21, Wallies, 306 b 24 sq., Brandis.</note>, while Themistius and Philoponus speak of any
+polygons.<note>Them. on <I>An. Post.</I>, p. 19. 11-20, Wallies, 211 b 19, Brandis; Philop. on <I>An. Post.</I>, p. 111. 20-114. 17 W., 211 b 30, Brandis.</note> According to Alexander, Bryson inscribed a square
+in a circle and circumscribed another about it, while he also
+took a square intermediate between them (Alexander does not
+say how constructed); then he argued that, as the intermediate
+square is less than the outer and greater than the inner, while
+the circle is also less than the outer square and greater than
+the inner, and as <I>things which are greater and less than the
+same things respectively are equal</I>, it follows that the circle is
+equal to the intermediate square: upon which Alexander
+remarks that not only is the thing assumed applicable to
+other things besides geometrical magnitudes, e.g. to numbers,
+times, depths of colour, degrees of heat or cold, &amp;c., but it
+is also false because (for instance) 8 and 9 are both less than
+10 and greater than 7 and yet they are not equal. As regards
+the intermediate square (or polygon), some have assumed that
+it was the arithmetic mean between the inscribed and circum-
+scribed figures, and others that it was the geometric mean.
+Both assumptions seem to be due to misunderstanding<note>Psellus (11th cent. A.D.) says, &lsquo;there are different opinions as to the
+proper method of finding the area of a circle, but that which has found
+the most favour is to take the geometric mean between the inscribed and
+circumscribed squares&rsquo;. I am not aware that he quotes Bryson as the
+authority for this method, and it gives the inaccurate value <MATH><G>p</G>=&radic;8</MATH> or
+2.8284272 .... Isaac Argyrus (14th cent.) adds to his account of Bryson
+the following sentence: &lsquo;For the circumscribed square <I>seems</I> to exceed
+the circle by the same amount as the inscribed square is exceeded by the
+circle.&rsquo;</note>; for
+<pb n=224><head>THE SQUARING OF THE CIRCLE</head>
+the ancient commentators do not attribute to Bryson any such
+statement, and indeed, to judge by their discussions of different
+interpretations, it would seem that tradition was by no means
+clear as to what Bryson actually did say. But it seems
+important to note that Themistius states (1) that Bryson
+declared the circle to be greater than <I>all</I> inscribed, and less
+than <I>all</I> circumscribed, polygons, while he also says (2) that
+the assumed axiom is <I>true</I>, though not peculiar to geometry.
+This suggests a possible explanation of what otherwise seems
+to be an absurd argument. Bryson may have multiplied the
+number of the sides of both the inscribed and circumscribed
+regular polygons as Antiphon did with inscribed polygons;
+he may then have argued that, if we continue this process
+long enough, we shall have an inscribed and a circumscribed
+polygon differing so little in area that, if we can describe
+a polygon intermediate between them in area, the circle, which
+is also intermediate in area between the inscribed and circum-
+scribed polygons, must be equal to the intermediate polygon.<note>It is true that, according to Philoponus, Proclus had before him an
+explanation of this kind, but rejected it on the ground that it would
+mean that the circle must actually <I>be</I> the intermediate polygon and not
+only be equal to it, in which case Bryson's contention would be tanta-
+mount to Antiphon's, whereas according to Aristotle it was based on
+a quite different principle. But it is sufficient that the circle should
+be taken to be <I>equal</I> to any polygon that can be drawn intermediate
+between the two ultimate polygons, and this gets over Proclus's difficulty.</note>
+If this is the right explanation, Bryson's name by no means
+deserves to be banished from histories of Greek mathematics;
+on the contrary, in so far as he suggested the necessity of
+considering circumscribed as well as inscribed polygons, he
+went a step further than Antiphon; and the importance of
+the idea is attested by the fact that, in the regular method
+of exhaustion as practised by Archimedes, use is made of both
+inscribed and circumscribed figures, and this <I>compression</I>, as it
+were, of a circumscribed and an inscribed figure into one so
+that they ultimately coincide with one another, and with the
+<pb n=225><head>THE SQUARING OF THE CIRCLE</head>
+curvilinear figure to be measured, is particularly characteristic
+of Archimedes.
+<p>We come now to the real rectifications or quadratures of
+circles effected by means of higher curves, the construction
+of which is more &lsquo;mechanical&rsquo; than that of the circle. Some
+of these curves were applied to solve more than one of the
+three classical problems, and it is not always easy to determine
+for which purpose they were originally destined by their
+inventors, because the accounts of the different authorities
+do not quite agree. Iamblichus, speaking of the quadrature
+of the circle, said that
+<p>&lsquo;Archimedes effected it by means of the spiral-shaped curve,
+Nicomedes by means of the curve known by the special name
+<I>quadratrix</I> (<G>tetragwni/zousa</G>), Apollonius by means of a certain
+curve which he himself calls &ldquo;sister of the cochloid&rdquo; but
+which is the same as Nicomedes's curve, and finally Carpus
+by means of a certain curve which he simply calls (the curve
+arising) &ldquo;from a double motion&rdquo;.&rsquo;<note>Iambl. ap. Simpl. <I>in Categ.</I>, p. 192. 19-24 K., 64 b 13-18 Br.</note>
+<p>Pappus says that
+<p>&lsquo;for the squaring of the circle Dinostratus, Nicomedes and
+certain other and later geometers used a certain curve which
+took its name from its property; for those geometers called it
+<I>quadratrix.</I>&rsquo;<note>Pappus, iv, pp. 250. 33-252. 3.</note>
+<p>Lastly, Proclus, speaking of the trisection of any angle,
+says that
+<p>&lsquo;Nicomedes trisected any rectilineal angle by means of the
+conchoidal curves, the construction, order and properties of
+which he handed down, being himself the discoverer of their
+peculiar character. Others have done the same thing by
+means of the <I>quadratrices</I> of Hippias and Nicomedes....
+Others again, starting from the spirals of Archimedes, divided
+any given rectilineal angle in any given ratio.&rsquo;<note>Proclus on Eucl. I, p. 272. 1-12.</note>
+<p>All these passages refer to the <I>quadratrix</I> invented by
+Hippias of Elis. The first two seem to imply that it was not
+used by Hippias himself for squaring the circle, but that it
+was Dinostratus (a brother of Menaechmus) and other later
+geometers who first applied it to that purpose; Iamblichus
+and Pappus do not even mention the name of Hippias. We
+might conclude that Hippias originally intended his curve to
+<pb n=226><head>THE SQUARING OF THE CIRCLE</head>
+be used for trisecting an angle. But this becomes more doubt-
+ful when the passages of Proclus are considered. Pappus's
+authority seems to be Sporus, who was only slightly older
+than Pappus himself (towards the end of the third century A.D.),
+and who was the author of a compilation called <G>*khri/a</G> con-
+taining, among other things, mathematical extracts on the
+quadrature of the circle and the duplication of the cube.
+Proclus's authority, on the other hand, is doubtless Geminus,
+who was much earlier (first century B.C.) Now not only
+does the above passage of Proclus make it possible that the
+name <I>quadratrix</I> may have been used by Hippias himself,
+but in another place Proclus (i.e. Geminus) says that different
+mathematicians have explained the properties of particular
+kinds of curves:
+<p>&lsquo;thus Apollonius shows in the case of each of the conic curves
+what is its property, and similarly Nicomedes with the
+conchoids, <I>Hippias with the quadratrices</I>, and Perseus with
+the spiric curves.&rsquo;<note>Proclus on Eucl. I, p. 356. 6-12.</note>
+<p>This suggests that Geminus had before him a regular treatise
+by Hippias on the properties of the <I>quadratrix</I> (which may
+have disappeared by the time of Sporus), and that Nicomedes
+did not write any such general work on that curve; and,
+if this is so, it seems not impossible that Hippias himself
+discovered that it would serve to rectify, and therefore to
+square, the circle.
+<C>(<G>a</G>) <I>The Quadratrix of Hippias.</I></C>
+<p>The method of constructing the curve is described by
+Pappus.<note>Pappus, iv, pp. 252 sq.</note> Suppose that <I>ABCD</I> is
+a square, and <I>BED</I> a quadrant of a
+circle with centre <I>A.</I>
+<FIG>
+<p>Suppose (1) that a radius of the
+circle moves uniformly about <I>A</I> from
+the position <I>AB</I> to the position <I>AD</I>,
+and (2) that <I>in the same time</I> the
+line <I>BC</I> moves uniformly, always
+parallel to itself and with its ex-
+tremity <I>B</I> moving along <I>BA</I>, from the position <I>BC</I> to the
+position <I>AD.</I>
+<pb n=227><head>THE QUADRATRIX OF HIPPIAS</head>
+<p>Then, in their ultimate positions, the moving straight line
+and the moving radius will both coincide with <I>AD</I>; and at
+any previous instant during the motion the moving line and
+the moving radius will by their intersection determine a point,
+as <I>F</I> or <I>L.</I>
+<p>The locus of these points is the <I>quadratrix.</I>
+<p>The property of the curve is that
+<MATH>&angle;<I>BAD</I>:&angle;<I>EAD</I>=(arc <I>BED</I>):(arc <I>ED</I>)=<I>AB</I>:<I>FH.</I></MATH>
+<p>In other words, if <G>f</G> is the angle <I>FAD</I> made by any radius
+vector <I>AF</I> with <I>AD</I>, <G>r</G> the length of <I>AF</I>, and <G>a</G> the length
+of the side of the square,
+<MATH>(<G>r</G> sin<G>f</G>)/<G>a</G>=<G>f</G>/(1/2)<G>p</G></MATH>.
+<p>Now clearly, when the curve is once constructed, it enables
+us not only to <I>trisect</I> the angle <I>EAD</I> but also to <I>divide it in
+any given ratio.</I>
+<p>For let <I>FH</I> be divided at <I>F</I>&prime; in the given ratio. Draw <I>F</I>&prime;<I>L</I>
+parallel to <I>AD</I> to meet the curve in <I>L</I>: join <I>AL</I>, and produce
+it to meet the circle in <I>N.</I>
+<p>Then the angles <I>EAN, NAD</I> are in the ratio of <I>FF</I>&prime; to <I>F</I>&prime;<I>H</I>,
+as is easily proved.
+<p>Thus the quadratrix lends itself quite readily to the division
+of any angle in a given ratio.
+<p>The application of the <I>quadratrix</I> to the rectification of the
+circle is a more difficult matter, because it requires us to
+know the position of <I>G</I>, the point where the quadratrix
+intersects <I>AD.</I> This difficulty was fully appreciated in ancient
+times, as we shall see.
+<p>Meantime, assuming that the quadratrix intersects <I>AD</I>
+in <I>G</I>, we have to prove the proposition which gives the length
+of the arc of the quadrant <I>BED</I> and therefore of the circum-
+ference of the circle. This proposition is to the effect that
+<MATH>(arc of quadrant <I>BED</I>):<I>AB</I>=<I>AB</I>:<I>AG.</I></MATH>
+<p>This is proved by <I>reductio ad absurdum.</I>
+<p>If the former ratio is not equal to <I>AB</I>:<I>AG</I>, it must be
+equal to <I>AB</I>:<I>AK</I>, where <I>AK</I> is either (1) greater or (2) less
+than <I>AG.</I>
+<p>(1) Let <I>AK</I> be greater than <I>AG</I>; and with <I>A</I> as centre
+<pb n=228><head>THE SQUARING OF THE CIRCLE</head>
+and <I>AK</I> as radius, draw the quadrant <I>KFL</I> cutting the quad-
+ratrix in <I>F</I> and <I>AB</I> in <I>L.</I>
+<p>Join <I>AF</I>, and produce it to meet the circumference <I>BED</I>
+in <I>E</I>; draw <I>FH</I> perpendicular to <I>AD.</I>
+<FIG>
+<p>Now, by hypothesis,
+<MATH>(arc <I>BED</I>):<I>AB</I>=<I>AB</I>:<I>AK</I>
+=(arc <I>BED</I>):(arc <I>LFK</I>)</MATH>;
+therefore <MATH><I>AB</I>=(arc <I>LFK</I>)</MATH>.
+<p>But, by the property of the <I>quadra-
+trix</I>,
+<MATH><I>AB</I>:<I>FH</I>=(arc <I>BED</I>):(arc <I>ED</I>)
+=(arc <I>LFK</I>):(arc <I>FK</I>)</MATH>;
+and it was proved that <MATH><I>AB</I>=(arc <I>LFK</I>)</MATH>;
+therefore <MATH><I>FH</I>=(arc <I>FK</I>)</MATH>:
+which is absurd. Therefore <I>AK</I> is not greater than <I>AG.</I>
+<p>(2) Let <I>AK</I> be less than <I>AG.</I>
+<p>With centre <I>A</I> and radius <I>AK</I> draw the quadrant <I>KML.</I>
+<p>Draw <I>KF</I> at right angles to <I>AD</I> meeting the quadratrix
+in <I>F</I>; join <I>AF</I>, and let it meet the
+quadrants in <I>M, E</I> respectively.
+<FIG>
+<p>Then, as before, we prove that
+<MATH><I>AB</I>=(arc <I>LMK</I>)</MATH>.
+<p>And, by the property of the <I>quad-
+ratrix</I>,
+<MATH><I>AB</I>:<I>FK</I>=(arc <I>BED</I>):(arc <I>DE</I>)
+=(arc <I>LMK</I>):(arc <I>MK</I>)</MATH>.
+<p>Therefore, since <MATH><I>AB</I>=(arc <I>LMK</I>),
+<I>FK</I>=(arc <I>KM</I>)</MATH>:
+which is absurd. Therefore <I>AK</I> is not less than <I>AG.</I>
+<p>Since then <I>AK</I> is neither less nor greater than <I>AG</I>, it is
+equal to it, and
+<MATH>(arc <I>BED</I>):<I>AB</I>=<I>AB</I>:<I>AG.</I></MATH>
+<p>[The above proof is presumably due to Dinostratus (if not
+to Hippias himself), and, as Dinostratus was a brother of
+Menaechmus, a pupil of Eudoxus, and therefore probably
+<pb n=229><head>THE QUADRATRIX OF HIPPIAS</head>
+flourished about 350 B.C., that is to say, some time before
+Euclid, it is worth while to note certain propositions which
+are assumed as known. These are, in addition to the theorem
+of Eucl. VI. 33, the following: (1) the circumferences of
+circles are as their respective radii; (2) any arc of a circle
+is greater than the chord subtending it; (3) any arc of a
+circle less than a quadrant is less than the portion of the
+tangent at one extremity of the arc cut off by the radius
+passing through the other extremity. (2) and (3) are of
+course equivalent to the facts that, if <G>a</G> be the circular measure
+of an angle less than a right angle, sin <MATH><G>a</G> < <G>a</G> < tan <G>a</G></MATH>.]
+<p>Even now we have only rectified the circle. To square it
+we have to use the proposition (1) in Archimedes's <I>Measure-
+ment of a Circle</I>, to the effect 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. This proposition is proved by the method of
+exhaustion and may have been known to Dinostratus, who
+was later than Eudoxus, if not to Hippias.
+<p>The criticisms of Sporus,<note>Pappus, iv, pp. 252. 26-254. 22.</note> in which Pappus concurs, are
+worth quoting:
+<p>(1) &lsquo;The very thing for which the construction is thought
+to serve is actually assumed in the hypothesis. For how is it
+possible, with two points starting from <I>B</I>, to make one of
+them move along a straight line to <I>A</I> and the other along
+a circumference to <I>D</I> in an equal time, unless you first know
+the ratio of the straight line <I>AB</I> to the circumference <I>BED</I>?
+In fact this ratio must also be that of the speeds of motion.
+For, if you employ speeds not definitely adjusted (to this
+ratio), how can you make the motions end at the same
+moment, unless this should sometime happen by pure chance?
+Is not the thing thus shown to be absurd?
+<p>(2) &lsquo;Again, the extremity of the curve which they employ
+for squaring the circle, I mean the point in which the curve
+cuts the straight line <I>AD</I>, is not found at all. For if, in the
+figure, the straight lines <I>CB, BA</I> are made to end their motion
+together, they will then coincide with <I>AD</I> itself and will not
+cut one another any more. In fact they cease to intersect
+before they coincide with <I>AD</I>, and yet it was the intersection
+of these lines which was supposed to give the extremity of the
+<pb n=230><head>THE SQUARING OF THE CIRCLE</head>
+curve, where it met the straight line <I>AD.</I> Unless indeed any
+one should assert that the curve is conceived to be produced
+further, in the same way as we suppose straight lines to be
+produced, as far as <I>AD.</I> But this does not follow from the
+assumptions made; the point <I>G</I> can only be found by first
+assuming (as known) the ratio of the circumference to the
+straight line.&rsquo;
+<p>The second of these objections is undoubtedly sound. The
+point <I>G</I> can in fact only be found by applying the method
+of exhaustion in the orthodox Greek manner; e.g. we may
+first bisect the angle of the quadrant, then the half towards
+<I>AD</I>, then the half of that and so on, drawing each time
+from the points <I>F</I> in which the bisectors cut the quadratrix
+perpendiculars <I>FH</I> on <I>AD</I> and describing circles with <I>AF</I>
+as radius cutting <I>AD</I> in <I>K.</I> Then, if we continue this process
+long enough, <I>HK</I> will get smaller and smaller and, as <I>G</I> lies
+between <I>H</I> and <I>K</I>, we can approximate to the position of <I>G</I> as
+nearly as we please. But this process is the equivalent of
+approximating to <G>p</G>, which is the very object of the whole
+construction.
+<p>As regards objection (1) Hultsch has argued that it is not
+valid because, with our modern facilities for making instru-
+ments of precision, there is no difficulty in making the two
+uniform motions take the same time. Thus an accurate clock
+will show the minute hand describing an exact quadrant in
+a definite time, and it is quite practicable now to contrive a
+uniform rectilinear motion taking exactly the same time.
+I suspect, however, that the rectilinear motion would be the
+result of converting some one or more circular motions into
+rectilinear motions; if so, they would involve the use of an
+approximate value of <G>p</G>, in which case the solution would depend
+on the assumption of the very thing to be found. I am inclined,
+therefore, to think that both Sporus's objections are valid.
+<C>(<G>b</G>) <I>The Spiral of Archimedes.</I></C>
+<p>We are assured that Archimedes actually used the spiral
+for squaring the circle. He does in fact show how to rectify
+a circle by means of a polar subtangent to the spiral. The
+spiral is thus generated: suppose that a straight line with
+one extremity fixed starts from a fixed position (the initial
+<pb n=231><head>THE SPIRAL OF ARCHIMEDES</head>
+line) and revolves uniformly about the fixed extremity, while
+a point also moves uniformly along the moving straight line
+starting from the fixed extremity (the origin) at the com-
+mencement of the straight line's motion; the curve described
+is a spiral.
+<p>The polar equation of the curve is obviously <MATH><G>r</G>=<G>aq</G></MATH>.
+<p>Suppose that the tangent at any point <I>P</I> of the spiral is
+met at <I>T</I> by a straight line drawn from <I>O</I>, the origin or pole,
+perpendicular to the radius vector <I>OP</I>; then <I>OT</I> is the polar
+subtangent.
+<p>Now in the book <I>On Spirals</I> Archimedes proves generally
+the equivalent of the fact that, if <G>r</G> be the radius vector to
+the point <I>P</I>,
+<MATH><I>OT</I>=<G>r</G><SUP>2</SUP>/<G>a</G></MATH>.
+<p>If <I>P</I> is on the <I>n</I>th turn of the spiral, the moving straight
+line will have moved through an angle <MATH>2(<I>n</I>-1)<G>p</G>+<G>q</G></MATH>, say.
+<p>Hence <MATH><G>r</G>=<G>a</G>{2(<I>n</I>-1)<G>p</G>+<G>q</G>}</MATH>,
+and <MATH><I>OT</I>=<G>r</G><SUP>2</SUP>/<G>a</G>=<G>r</G>{2(<I>n</I>-1)<G>p</G>+<G>q</G>}</MATH>.
+<p>Archimedes's way of expressing this is to say (Prop. 20)
+that, if <I>p</I> be the circumference of the circle with radius
+<MATH><I>OP</I>(=<G>r</G>)</MATH>, and if this circle cut the initial line in the point <I>K</I>,
+<MATH><I>OT</I>=(<I>n</I>-1)<I>p</I>+arc<I>KP</I></MATH> measured &lsquo;forward&rsquo; from <I>K</I> to <I>P.</I>
+<p>If <I>P</I> is the end of the <I>n</I>th turn, this reduces to
+<MATH><I>OT</I>=<I>n</I> (circumf. of circle with radius <I>OP</I>)</MATH>,
+and, if <I>P</I> is the end of the first turn in particular,
+<MATH><I>OT</I>=(circumf. of circle with radius <I>OP</I>). (Prop. 19.)</MATH>
+<p>The spiral can thus be used for the rectification of any
+circle. And the quadrature follows directly from <I>Measure-
+ment of a Circle</I>, Prop. 1.
+<C>(<G>g</G>) <I>Solutions by Apollonius and Carpus.</I></C>
+<p>Iamblichus says that Apollonius himself called the curve by
+means of which he squared the circle &lsquo;sister of the cochloid&rsquo;.
+What this curve was is uncertain. As the passage goes on to
+say that it was really &lsquo;the same as the (curve) of Nicomedes&rsquo;,
+and the quadratrix has just been mentioned as the curve used
+<pb n=232><head>THE SQUARING OF THE CIRCLE</head>
+by Nicomedes, some have supposed the &lsquo;sister of the cochloid&rsquo;
+(or conchoid) to be the <I>quadratrix</I>, but this seems highly im-
+probable. There is, however, another possibility. Apollonius
+is known to have written a regular treatise on the <I>Cochlias</I>,
+which was the cylindrical helix.<note>Pappus, viii, p. 1110. 20; Proclus on Eucl. I, p. 105. 5.</note> It is conceivable that he
+might call the <I>cochlias</I> the &lsquo;sister of the <I>cochloid</I>&rsquo; on the
+ground of the similarity of the names, if not of the curves.
+And, as a matter of fact, the drawing of a tangent to the
+helix enables the circular section of the cylinder to be squared.
+For, if a plane be drawn at right angles to the axis of the
+cylinder through the initial position of the moving radius
+which describes the helix, and if we project on this plane
+the portion of the tangent at any point of the helix intercepted
+between the point and the plane, the projection is equal to
+an arc of the circular section of the cylinder subtended by an
+angle at the centre equal to the angle through which the
+plane through the axis and the moving radius has turned
+from its original position. And this squaring by means of
+what we may call the &lsquo;subtangent&rsquo; is sufficiently parallel to
+the use by Archimedes of the polar subtangent to the spiral
+for the same purpose to make the hypothesis attractive.
+<p>Nothing whatever is known of Carpus's curve &lsquo;of double
+motion&rsquo;. Tannery thought it was the cycloid; but there is no
+evidence for this.
+<C>(<G>d</G>) <I>Approximations to the value of</I> <G>p</G>.</C>
+<p>As we have seen, Archimedes, by inscribing and cir-
+cumscribing regular polygons of 96 sides, and calculating
+their perimeters respectively, obtained the approximation
+<MATH>3 1/7 > <G>p</G> > 3 10/71</MATH> (<I>Measurement of a Circle</I>, Prop. 3). But we
+now learn<note>Heron, <I>Metrica</I>, i. 26, p. 66. 13-17.</note> that, in a work on <I>Plinthides and Cylinders</I>, he
+made a nearer approximation still. Unfortunately the figures
+as they stand in the Greek text are incorrect, the lower limit
+being given as the ratio of <G>m<SUP>ka</SUP><SUB>/</SUB>awoe</G> to <G>m<SUP>s</SUP><SUB>/</SUB>zuma</G>, or <MATH>211875:67441
+(=3.141635)</MATH>, and the higher limit as the ratio of <G>m<SUP>iq</SUP><SUB>/</SUB>zwph</G> to
+<G>m<SUP>s</SUP><SUB>/</SUB>btna</G> or <MATH>197888:62351 (=3.17377)</MATH>, so that the lower limit
+<pb n=233><head>APPROXIMATIONS TO THE VALUE OF <I>II</I></head>
+as given is greater than the true value, and the higher limit is
+greater than the earlier upper limit 3 1/7. Slight corrections by
+Tannery (<G>m<SUP>ka</SUP><SUB>/</SUB>awob</G> for <G>m<SUP>ka</SUP><SUB>/</SUB>awoe</G> and <G>m<SUP>iq</SUP><SUB>/</SUB>ewpb</G> for <G>m<SUP>iq</SUP><SUB>/</SUB>zwph</G>) give
+better figures, namely
+<MATH>195882/62351 > <G>p</G> > 211872/67441</MATH>
+or <MATH>3.1416016 > <G>p</G> > 3.1415904 ....</MATH>
+<p>Another suggestion<note>J. L. Heiben in <I>Nordisk Tidsskrift for Filologi</I>, 3<SUP>e</SUP> S&eacute;r. xx. Fasc. 1-2.</note> is to correct <G>m<SUP>s</SUP><SUB>/</SUB>zuma</G> into <G>m<SUP>s</SUP><SUB>/</SUB>zumd</G> and
+<G>m<SUP>iq</SUP><SUB>/</SUB>zwph</G> into <G>m<SUP>iq</SUP><SUB>/</SUB>ewph</G>, giving
+<MATH>195888/62351 > <G>p</G> > 211875/67444</MATH>
+or <MATH>3.141697 ... > <G>p</G> > 3.141495 ....</MATH>
+<p>If either suggestion represents the true reading, the mean
+between the two limits gives the same remarkably close
+approximation 3.141596.
+<p>Ptolemy<note>Ptolemy, <I>Suntaxis</I>, vi. 7, p. 513. 1-5, Heib.</note> gives a value for the ratio of the circumference
+of a circle to its diameter expressed thus in sexagesimal
+fractions, <G>g h l</G>, i.e. <MATH>3+8/60+30/60<SUP>2</SUP></MATH> or 3.1416. He observes
+that this is almost exactly the mean between the Archimedean
+limits 3 1/7 and 3 10/71. It is, however, more exact than this mean,
+and Ptolemy no doubt obtained his value independently. He
+had the basis of the calculation ready to hand in his Table
+of Chords. This Table gives the lengths of the chords of
+a circle subtended by arcs of 1/2&deg;, 1&deg;, 1 1/2&deg;, and so on by half
+degrees. The chords are expressed in terms of 120th parts
+of the length of the diameter. If one such part be denoted
+by 1<SUP><I>p</I></SUP>, the chord subtended by an arc of 1&deg; is given by the
+Table in terms of this unit and sexagesimal fractions of it
+thus, 1<SUP><I>p</I></SUP> 2&prime; 50&Prime;. Since an angle of 1&deg; at the centre subtends
+a side of the regular polygon of 360 sides inscribed in the
+circle, the perimeter of this polygon is 360 times 1<SUP><I>p</I></SUP> 2&prime; 50&Prime;
+or, since <MATH>1<SUP><I>p</I></SUP>=1/120th</MATH> of the diameter, the perimeter of the
+polygon expressed in terms of the diameter is 3 times 1 2&prime; 50&Prime;,
+that is 3 8&prime; 30&Prime;, which is Ptolemy's figure for <G>p</G>.
+<pb n=234><head>THE SQUARING OF THE CIRCLE</head>
+<p>There is evidence of a still closer calculation than Ptolemy's
+due to some Greek whose name we do not know. The Indian
+mathematician Aryabhatta (born A.D. 476) says in his <I>Lessons
+in Calculation</I>:
+<p>&lsquo;To 100 add 4; multiply the sum by 8; add 62000 more
+and thus (we have), for a diameter of 2 myriads, the approxi-
+mate length of the circumference of the circle&rsquo;;
+<p>that is, he gives 62832/20000 or 3.1416 as the value of <G>p</G>. But the
+way in which he expresses it points indubitably to a Greek
+source, &lsquo;for the Greeks alone of all peoples made the myriad
+the unit of the second order&rsquo; (Rodet).
+<p>This brings us to the notice at the end of Eutocius's com-
+mentary on the <I>Measurement of a Circle</I> of Archimedes, which
+records<note>Archimedes, ed. Heib., vol. iii, pp. 258-9.</note> that other mathematicians made similar approxima-
+tions, though it does not give their results.
+<p>&lsquo;It is to be observed that Apollonius of Perga solved the
+same problem in his <G>*w)kuto/kion</G> (&ldquo;means of quick delivery&rdquo;),
+using other numbers and making the approximation closer
+[than that of Archimedes]. While Apollonius's figures seem
+to be more accurate, they do not serve the purpose which
+Archimedes had in view; for, as we said, his object in this
+book was to find an approximate figure suitable for use in
+daily life. Hence we cannot regard as appropriate the censure
+of Sporus of Nicaea, who seems to charge Archimedes with
+having failed to determine with accuracy (the length of) the
+straight line which is equal to the circumference of the circle,
+to judge by the passage in his <I>Keria</I> where Sporus observes
+that his own teacher, meaning Philon of Gadara, reduced (the
+matter) to more exact numerical expression than Archimedes
+did, I mean in his 1/7 and 10/71; in fact people seem, one after the
+other, to have failed to appreciate Archimedes's object. They
+have also used multiplications and divisions of myriads, a
+method not easy to follow for any one who has not gone
+through a course of Magnus's <I>Logistica.</I>&rsquo;
+<p>It is possible that, as Apollonius used myriads, &lsquo;second
+myriads&rsquo;, &lsquo;third myriads&rsquo;, &amp;c., as orders of integral numbers,
+he may have worked with the fractions 1/10000, 1/10000<SUP>2</SUP>, &amp;c.;
+<pb n=235><head>APPROXIMATIONS TO THE VALUE OF <I>II</I></head>
+in any case Magnus (apparently later than Sporus, and therefore
+perhaps belonging to the fourth or fifth century A.D.) would
+seem to have written an exposition of such a method, which,
+as Eutocius indicates, must have been very much more
+troublesome than the method of sexagesimal fractions used
+by Ptolemy.
+<C>The Trisection of any Angle.</C>
+<p>This problem presumably arose from attempts to continue
+the construction of regular polygons after that of the pentagon
+had been discovered. The trisection of an angle would be
+necessary in order to construct a regular polygon the sides
+of which are nine, or any multiple of nine, in number.
+A regular polygon of seven sides, on the other hand, would
+no doubt be constructed with the help of the first discovered
+method of dividing any angle in a given ratio, i.e. by means
+of the <I>quadratrix.</I> This method covered the case of trisection,
+but other more practicable ways of effecting this particular
+construction were in due time evolved.
+<p>We are told that the ancients attempted, and failed, to
+solve the problem by &lsquo;plane&rsquo; methods, i.e. by means of the
+straight line and circle; they failed because the problem is
+not &lsquo;plane&rsquo; but &lsquo;solid&rsquo;. Moreover, they were not yet familiar
+with conic sections, and so were at a loss; afterwards,
+however, they succeeded in trisecting an angle by means of
+conic sections, a method to which they were led by the
+reduction of the problem to another, of the kind known as
+<G>neu/seis</G> (<I>inclinationes</I>, or <I>vergings</I>).<note>Pappus, iv, p. 272. 7-14.</note>
+<C>(<G>a</G>) <I>Reduction to a certain <G>neu=sis</G>, solved by conics.</I></C>
+<p>The reduction is arrived at by the following analysis. It is
+only necessary to deal with the case where the given angle to
+be trisected is acute, since a right angle can be trisected
+by drawing an equilateral triangle.
+<p>Let <I>ABC</I> be the given angle, and let <I>AC</I> be drawn perpen-
+dicular to <I>BC.</I> Complete the parallelogram <I>ACBF</I>, and
+produce the side <I>FA</I> to <I>E.</I>
+<pb n=236><head>THE TRISECTION OF ANY ANGLE</head>
+<p><I>Suppose E to be such a point that, if BE be joined meeting
+AC in D, the intercept DE between AC and AE is equal
+to 2 AB.</I>
+<FIG>
+<p>Bisect <I>DE</I> at <I>G</I>, and join <I>AG.</I>
+<p>Then <MATH><I>DG</I>=<I>GE</I>=<I>AG</I>=<I>AB</I></MATH>.
+<p>Therefore <MATH>&angle;<I>ABG</I>=&angle;<I>AGB</I>=2&angle;<I>AEG</I>
+=2&angle;<I>DBC</I></MATH>, since <I>FE, BC</I> are parallel.
+<p>Hence <MATH>&angle;<I>DBC</I>=1/3&angle;<I>ABC</I></MATH>,
+and the angle <I>ABC</I> is trisected by <I>BE.</I>
+<p>Thus the problem is reduced to <I>drawing BE from B to cut
+AC and AE in such a way that the intercept</I> <MATH><I>DE</I>=2<I>AB</I></MATH>.
+<p>In the phraseology of the problems called <G>neu/seis</G> the
+problem is to insert a straight line <I>ED</I> of given length
+2<I>AB</I> between <I>AE</I> and <I>AC</I> in such a way that <I>ED verges</I>
+towards <I>B.</I>
+<p>Pappus shows how to solve this problem in a more general
+form. Given a parallelogram <I>ABCD</I> (which need not be
+rectangular, as Pappus makes it), to draw <I>AEF</I> to meet <I>CD</I>
+and <I>BC</I> produced in points <I>E</I> and <I>F</I> such that <I>EF</I> has a given
+length.
+<p>Suppose the problem solved, <I>EF</I> being of the given length.
+<FIG>
+<p>Complete the parallelogram
+<I>EDGF.</I>
+<p>Then, <I>EF</I> being given in length,
+<I>DG</I> is given in length.
+<p>Therefore <I>G</I> lies on a circle with
+centre <I>D</I> and radius equal to the
+given length.
+<p>Again, by the help of Eucl. I. 43 relating to the complements
+<pb n=237><head>REDUCTION TO A <G>*n*e*g*s*i*s</G></head>
+of the parallelograms about the diagonal of the complete
+parallelogram, we see that
+<MATH><I>BC.CD</I>=<I>BF.ED</I>
+=<I>BF.FG.</I></MATH>
+<p>Consequently <I>G</I> lies on a hyperbola with <I>BF, BA</I> as
+asymptotes and passing through <I>D.</I>
+<p>Thus, in order to effect the construction, we have only to
+draw this hyperbola as well as the circle with centre <I>D</I> and
+radius equal to the given length. Their intersection gives the
+point <I>G</I>, and <I>E, F</I> are then determined by drawing <I>GF</I> parallel
+to <I>DC</I> to meet <I>BC</I> produced in <I>F</I> and joining <I>AF.</I>
+<C>(<G>b</G>) <I>The <G>neu=sis</G> equivalent to a cubic equation.</I></C>
+<p>It is easily seen that the solution of the <G>neu=sis</G> is equivalent
+to the solution of a cubic equation. For in the first figure on
+p. 236, if <I>FA</I> be the axis of <I>x, FB</I> the axis of <I>y</I>, <MATH><I>FA</I>=<I>a</I>,
+<I>FB</I>=<I>b</I></MATH>, the solution of the problem by means of conics as
+Pappus gives it is the equivalent of finding a certain point
+as the intersection of the conics
+<MATH><I>xy</I>=<I>ab</I>,
+(<I>x</I>-<I>a</I>)<SUP>2</SUP>+(<I>y</I>-<I>b</I>)<SUP>2</SUP>=4(<I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>)</MATH>.
+<p>The second equation gives
+<MATH>(<I>x</I>+<I>a</I>)(<I>x</I>-3<I>a</I>)=(<I>y</I>+<I>b</I>)(3<I>b</I>-<I>y</I>)</MATH>.
+<p>From the first equation it is easily seen that
+<MATH>(<I>x</I>+<I>a</I>):(<I>y</I>+<I>b</I>)=<I>a</I>:<I>y</I></MATH>,
+and that <MATH>(<I>x</I>-3<I>a</I>)<I>y</I>=<I>a</I>(<I>b</I>-3<I>y</I>)</MATH>;
+therefore, eliminating <I>x</I>, we have
+<MATH><I>a</I><SUP>2</SUP>(<I>b</I>-3<I>y</I>)=<I>y</I><SUP>2</SUP>(3<I>b</I>-<I>y</I>)</MATH>,
+or <MATH><I>y</I><SUP>3</SUP>-3<I>by</I><SUP>2</SUP>-3<I>a</I><SUP>2</SUP><I>y</I>+<I>a</I><SUP>2</SUP><I>b</I>=0</MATH>.
+<p>Now suppose that <MATH>&angle;<I>ABC</I>=<G>q</G></MATH>, so that tan <MATH><G>q</G>=<I>b/a</I></MATH>;
+and suppose that <MATH><I>t</I>=tan <I>DBC</I></MATH>,
+so that <MATH><I>y</I>=<I>at.</I></MATH>
+<p>We have then
+<MATH><I>a</I><SUP>3</SUP><I>t</I><SUP>3</SUP>-3<I>ba</I><SUP>2</SUP><I>t</I><SUP>2</SUP>-3<I>a</I><SUP>3</SUP><I>t</I>+<I>a</I><SUP>2</SUP><I>b</I>=0</MATH>,
+<pb n=238><head>THE TRISECTION OF ANY ANGLE</head>
+or <MATH><I>at</I><SUP>3</SUP>-3<I>bt</I><SUP>2</SUP>-3<I>at</I>+<I>b</I>=0</MATH>,
+whence <MATH><I>b</I>(1-3<I>t</I><SUP>2</SUP>)=<I>a</I>(3<I>t</I>-<I>t</I><SUP>3</SUP>)</MATH>,
+or <MATH>tan<G>q</G>=<I>b/a</I>=(3<I>t</I>-<I>t</I><SUP>3</SUP>)/(1-3<I>t</I><SUP>2</SUP>)</MATH>,
+so that, by the well-known trigonometrical formula,
+<MATH><I>t</I>=tan1/3<G>q</G></MATH>;
+that is, <I>BD</I> trisects the angle <I>ABC.</I>
+<C>(<G>g</G>) <I>The Conchoids of Nicomedes.</I></C>
+<p>Nicomedes invented a curve for the specific purpose of
+solving such <G>neu/seis</G> as the above. His date can be fixed with
+sufficient accuracy by the facts (1) that he seems to have
+criticized unfavourably Eratosthenes's solution of the problem
+of the two mean proportionals or the duplication of the cube,
+and (2) that Apollonius called a certain curve the &lsquo;sister of
+the cochloid&rsquo;, evidently out of compliment to Nicomedes.
+Nicomedes must therefore have been about intermediate
+between Eratosthenes (a little younger than Archimedes, and
+therefore born about 280 B.C.) and Apollonius (born probably
+about 264 B.C.).
+<p>The curve is called by Pappus the <I>cochloid</I> (<G>koxloeidh\s
+grammh/</G>), and this was evidently the original name for it;
+later, e.g. by Proclus, it was called the <I>conchoid</I> (<G>kogxoeidh\s
+grammh/</G>). There were varieties of the cochloidal curves;
+Pappus speaks of the &lsquo;first&rsquo;, &lsquo;second&rsquo;, &lsquo;third&rsquo; and &lsquo;fourth&rsquo;,
+observing that the &lsquo;first&rsquo; was used for trisecting an angle and
+duplicating the cube, while the others were useful for other
+investigations.<note>Pappus, iv, p. 244. 18-20.</note> It is the &lsquo;first&rsquo; which concerns us here.
+Nicomedes constructed it by means of a mechanical device
+which may be described thus.<note><I>Ib.</I>, pp. 242-4.</note> <I>AB</I> is a ruler with a slot
+in it parallel to its length, <I>FE</I> a second ruler fixed at right
+angles to the first, with a peg <I>C</I> fixed in it. A third ruler
+<I>PC</I> pointed at <I>P</I> has a slot in it parallel to its length which
+fits the peg <I>C. D</I> is a fixed peg on <I>PC</I> in a straight line
+with the slot, and <I>D</I> can move freely along the slot in <I>AB.</I>
+If then the ruler <I>PC</I> moves so that the peg <I>D</I> describes the
+<pb n=239><head>THE CONCHOIDS OF NICOMEDES</head>
+length of the slot in <I>AB</I> on each side of <I>F</I>, the extremity <I>P</I> of
+the ruler describes the curve which is called a conchoid or
+cochloid. Nicomedes called the straight line <I>AB</I> the <I>ruler</I>
+(<G>kanw/n</G>), the fixed point <I>C</I> the <I>pole</I> (<G>po/los</G>), and the constant
+length <I>PD</I> the <I>distance</I> (<G>dia/sthma</G>).
+<FIG>
+<p>The fundamental property of the curve, which in polar
+coordinates would now be denoted by the equation
+<MATH><I>r</I>=<I>a</I>+<I>b</I>sec<G>q</G></MATH>,
+is that, if any radius vector be drawn from <I>C</I> to the curve, as
+<I>CP</I>, the length intercepted on the radius vector between the
+curve and the straight line <I>AB</I> is constant. Thus any <G>neu=sis</G>
+in which one of the two given lines (between which the
+straight line of given length is to be placed) is a straight line
+can be solved by means of the intersection of the other line
+with a certain conchoid having as its pole the fixed point
+to which the inserted straight line must <I>verge</I> (<G>neu/ein</G>). Pappus
+tells us that in practice the conchoid was not always actually
+drawn but that &lsquo;some&rsquo;, for greater convenience, moved a ruler
+about the fixed point until by trial the intercept was found to
+be equal to the given length.<note>Pappus, iv, p. 246. 15.</note>
+<p>In the figure above (p. 236) showing the reduction of the
+trisection of an angle to a <G>neu=sis</G> the conchoid to be used
+would have <I>B</I> for its <I>pole, AC</I> for the &lsquo;<I>ruler</I>&rsquo; or <I>base</I>, a length
+equal to 2<I>AB</I> for its <I>distance</I>; and <I>E</I> would be found as the
+intersection of the conchoid with <I>FA</I> produced.
+<p>Proclus says that Nicomedes gave the construction, the
+order, and the properties of the conchoidal lines<note>Proclus on Eucl. I, p. 272. 3-7.</note>; but nothing
+<pb n=240><head>THE TRISECTION OF ANY ANGLE</head>
+of his treatise has come down to us except the construction
+of the &lsquo;first&rsquo; conchoid, its fundamental property, and the fact
+that the curve has the <I>ruler</I> or <I>base</I> as an asymptote in
+each direction. The distinction, however, drawn by Pappus
+between the &lsquo;first&rsquo;, &lsquo;second&rsquo;, &lsquo;third&rsquo; and &lsquo;fourth&rsquo; conchoids
+may well have been taken from the original treatise, directly
+or indirectly. We are not told the nature of the conchoids
+other than the &lsquo;first&rsquo;, but it is probable that they were three
+other curves produced by varying the conditions in the figure.
+Let <I>a</I> be the distance or fixed intercept between the curve and
+the base, <I>b</I> the distance of the pole from the base. Then
+clearly, if along each radius vector drawn through the pole
+we measure <I>a</I> backwards from the base towards the pole,
+we get a conchoidal figure on the side of the base towards
+the pole. This curve takes three forms according as <I>a</I> is
+greater than, equal to, or less than <I>b.</I> Each of them has
+the base for asymptote, but in the first of the three cases
+the curve has a loop as shown in the figure, in the second
+case it has a cusp at the pole, in the third it has no double
+point. The most probable hypothesis seems to be that the
+other three cochloidal curves mentioned by Pappus are these
+three varieties.
+<FIG>
+<C>(<G>d</G>) <I>Another reduction to a</I> <G>neu=sis</G> (<I>Archimedes</I>).</C>
+<p>A proposition leading to the reduction of the trisection
+of an angle to another <G>neu=sis</G> is included in the collection of
+Lemmas (<I>Liber Assumptorum</I>) which has come to us under
+<pb n=241><head>ARCHIMEDES'S SOLUTION (BY NE<G>*g*s*i*s</G>)</head>
+the name of Archimedes through the Arabic. Though the
+Lemmas cannot have been written by Archimedes in their
+present form, because his name is quoted in them more than
+once, it is probable that some of them are of Archimedean
+origin, and especially is this the case with Prop. 8, since the
+<G>neu=sis</G> suggested by it is of very much the same kind as those
+the solution of which is assumed in the treatise <I>On Spirals</I>,
+Props. 5-8. The proposition is as follows.
+<p>If <I>AB</I> be any chord of a circle with centre <I>O</I>, and <I>AB</I> be
+produced to <I>C</I> so that <I>BC</I> is
+equal to the radius, and if <I>CO</I>
+meet the circle in <I>D, E</I>, then the
+arc <I>AE</I> will be equal to three
+times the arc <I>BD.</I>
+<FIG>
+<p>Draw the chord <I>EF</I> parallel
+to <I>AB</I>, and join <I>OB, OF.</I>
+<p>Since <MATH><I>BO</I>=<I>BC</I></MATH>,
+<MATH>&angle;<I>BOC</I>=&angle;<I>BCO</I></MATH>.
+<p>Now <MATH>&angle;<I>COF</I>=2&angle;<I>OEF</I>,
+=2&angle;<I>BCO</I></MATH>, by parallels,
+<MATH>=2&angle;<I>BOC</I></MATH>.
+<p>Therefore <MATH>&angle;<I>BOF</I>=3&angle;<I>BOD</I></MATH>,
+and <MATH>(arc <I>BF</I>)=(arc <I>AE</I>)=3(arc <I>BD</I>)</MATH>.
+<p>By means of this proposition we can reduce the trisection of
+the arc <I>AE</I> to a <G>neu=sis</G>. For, in order to find an arc which is
+one-third of the arc <I>AE</I>, we have only to draw through <I>A</I>
+a straight line <I>ABC</I> meeting the circle again in <I>B</I> and <I>EO</I>
+produced in <I>C</I>, and such that <I>BC</I> is equal to the radius of the
+circle.
+<C>(<G>e</G>) <I>Direct solutions by means of conics.</I></C>
+<p>Pappus gives two solutions of the trisection problem in
+which conics are applied directly without any preliminary
+reduction of the problem to a <G>neu=sis</G>.<note>Pappus, iv, pp. 282-4.</note>
+<p>1. The analysis leading to the first method is as follows.
+<p>Let <I>AC</I> be a straight line, and <I>B</I> a point without it such
+that, if <I>BA, BC</I> be joined, the angle <I>BCA</I> is double of the
+angle <I>BAC.</I>
+<pb n=242><head>THE TRISECTION OF ANY ANGLE</head>
+<p>Draw <I>BD</I> perpendicular to <I>AC</I>, and cut off <I>DE</I> along <I>DA</I>
+equal to <I>DC.</I> Join <I>BE.</I>
+<FIG>
+<p>Then, since <MATH><I>BE</I>=<I>BC</I></MATH>,
+<MATH>&angle;<I>BEC</I>=<I>BCE</I></MATH>.
+<p>But <MATH>&angle;<I>BEC</I>=&angle;<I>BAE</I>+&angle;<I>EBA</I></MATH>,
+and, by hypothesis,
+<MATH>&angle;<I>BCA</I>=2&angle;<I>BAE</I></MATH>.
+<p>Therefore <MATH>&angle;<I>BAE</I>+&angle;<I>EBA</I>=2&angle;<I>BAE</I></MATH>;
+therefore <MATH>&angle;<I>BAE</I>=&angle;<I>ABE</I></MATH>,
+or <MATH><I>AE</I>=<I>BE</I></MATH>.
+<p>Divide <I>AC</I> at <I>G</I> so that <MATH><I>AG</I>=2<I>GC</I></MATH>, or <MATH><I>CG</I>=1/3<I>AC</I></MATH>.
+<p>Also let <I>FE</I> be made equal to <I>ED</I>, so that <MATH><I>CD</I>=1/3<I>CF</I></MATH>.
+<p>It follows that <MATH><I>GD</I>=1/3(<I>AC</I>-<I>CF</I>)=1/3<I>AF</I></MATH>.
+<p>Now <MATH><I>BD</I><SUP>2</SUP>=<I>BE</I><SUP>2</SUP>-<I>ED</I><SUP>2</SUP>
+=<I>BE</I><SUP>2</SUP>-<I>EF</I><SUP>2</SUP></MATH>.
+<p>Also <MATH><I>DA.AF</I>=<I>AE</I><SUP>2</SUP>-<I>EF</I><SUP>2</SUP></MATH> (Eucl. II. 6)
+<MATH>=<I>BE</I><SUP>2</SUP>-<I>EF</I><SUP>2</SUP></MATH>.
+<p>Therefore <MATH><I>BD</I><SUP>2</SUP>=<I>DA.AF</I>
+=3<I>AD.DG</I></MATH>, from above,
+so that <MATH><I>BD</I><SUP>2</SUP>:<I>AD.DG</I>=3:1
+=3<I>AG</I><SUP>2</SUP>:<I>AG</I><SUP>2</SUP></MATH>.
+<p>Hence <I>D</I> lies on a hyperbola with <I>AG</I> as transverse axis
+and with conjugate axis equal to &radic;3.<I>AG</I>.
+<FIG>
+<p>Now suppose we are required
+to trisect an arc <I>AB</I> of a circle
+with centre <I>O.</I>
+<p>Draw the chord <I>AB</I>, divide it
+at <I>C</I> so that <MATH><I>AC</I>=2<I>CB</I></MATH>, and
+construct the hyperbola which
+has <I>AC</I> for transverse axis and
+a straight line equal to &radic;3.<I>AC</I> for conjugate axis.
+<p>Let the hyperbola meet the circular arc in <I>P.</I> Join <I>PA,
+PO, PB.</I>
+<pb n=243><head>SOLUTIONS BY MEANS OF CONICS</head>
+<p>Then, by the above proposition,
+<MATH>&angle;<I>PBA</I>=2&angle;<I>PAB</I></MATH>.
+<p>Therefore their doubles are equal,
+or <MATH>&angle;<I>POA</I>=2&angle;<I>POB</I></MATH>,
+and <I>OP</I> accordingly trisects the arc <I>APB</I> and the angle <I>AOB.</I>
+<p>2. &lsquo;Some&rsquo;, says Pappus, set out another solution not in-
+volving recourse to a <G>neu=sis</G>, as follows.
+<p>Let <I>RPS</I> be an arc of a circle which it is required to
+trisect.
+<p>Suppose it done, and let the arc <I>SP</I> be one-third of the
+arc <I>SPR.</I>
+<p>Join <I>RP, SP.</I>
+<p>Then the angle <I>RSP</I> is equal
+to twice the angle <I>SRP.</I>
+<FIG>
+<p>Let <I>SE</I> bisect the angle <I>RSP</I>,
+meeting <I>RP</I> in <I>E</I>, and draw <I>EX, PN</I> perpendicular to <I>RS.</I>
+<p>Then <MATH>&angle;<I>ERS</I>=&angle;<I>ESR</I></MATH>, so that <MATH><I>RE</I>=<I>ES</I></MATH>.
+<p>Therefore <MATH><I>RX</I>=<I>XS</I></MATH>, and <I>X</I> is given.
+<p>Again <MATH><I>RS</I>:<I>SP</I>=<I>RE</I>:<I>EP</I>=<I>RX</I>:<I>XN</I></MATH>;
+therefore <MATH><I>RS</I>:<I>RX</I>=<I>SP</I>:<I>NX</I></MATH>.
+<p>But <MATH><I>RS</I>=2<I>RX</I></MATH>;
+therefore <MATH><I>SP</I>=2<I>NX</I></MATH>.
+<p>It follows that <I>P</I> lies on a hyperbola with <I>S</I> as focus and <I>XE</I>
+as directrix, and with eccentricity 2.
+<p>Hence, in order to trisect the arc, we have only to bisect <I>RS</I>
+at <I>X</I>, draw <I>XE</I> at right angles to <I>RS</I>, and then draw a hyper-
+bola with <I>S</I> as focus, <I>XE</I> as directrix, and 2 as the eccentricity.
+The hyperbola is the same as that used in the first solution.
+<p>The passage of Pappus from which this solution is taken is
+remarkable as being one of three passages in Greek mathe-
+matical works still extant (two being in Pappus and one in
+a fragment of Anthemius on burning mirrors) which refer to
+the focus-and-directrix property of conics. The second passage
+in Pappus comes under the heading of Lemmas to the <I>Surface-
+Loci</I> of Euclid.<note>Pappus, vii, pp. 1004-1114.</note> Pappus there gives a complete proof of the
+<pb n=244><head>THE DUPLICATION OF THE CUBE</head>
+theorem that, <I>if the distance of a point from a fixed point is
+in a given ratio to its distance from <B>a</B> fixed line, the locus of
+the point is a conic section which is an ellipse, a parabola,
+or a hyperbola according as the given ratio is less than, equal
+to, or greater than, unity.</I> The importance of these passages
+lies in the fact that the Lemma was required for the
+understanding of Euclid's treatise. We can hardly avoid
+the conclusion that the property was used by Euclid in his
+<I>Surface-Loci</I>, but was assumed as well known. It was, there-
+fore, probably taken from some treatise current in Euclid's
+time, perhaps from Aristaeus's work on <I>Solid Loci.</I>
+<C><B>The Duplication of the Cube, or the problem
+of the two mean proportionals.</B></C>
+<C>(<G>a</G>) <I>History of the problem.</I></C>
+<p>In his commentary on Archimedes, <I>On the Sphere and
+Cylinder</I>, II. 1, Eutocius has preserved for us a precious
+collection of solutions of this famous problem.<note>Archimedes, ed. Heib., vol. iii, pp. 54. 26-106. 24.</note> One of the
+solutions is that of Eratosthenes, a younger contemporary of
+Archimedes, and it is introduced by what purports to be
+a letter from Eratosthenes to Ptolemy. This was Ptolemy
+Euergetes, who at the beginning of his reign (245 B.C.) per-
+suaded Eratosthenes to come from Athens to Alexandria to be
+tutor to his son (Philopator). The supposed letter gives the
+tradition regarding the origin of the problem and the history of
+its solution up to the time of Eratosthenes. Then, after some
+remarks on its usefulness for practical purposes, the author
+describes the construction by which Eratosthenes himself solved
+it, giving the proof of it at some length and adding directions
+for making the instrument by which the construction could
+be effected in practice. Next he says that the mechanical
+contrivance represented by Eratosthenes was, &lsquo;in the votive
+monument&rsquo;, actually of bronze, and was fastened on with lead
+close under the <G>stefa/nh</G> of the pillar. There was, further,
+on the pillar the proof in a condensed form, with one figure,
+and, at the end, an epigram. The supposed letter of Eratos-
+thenes is a forgery, but the author rendered a real service
+<pb n=245><head>HISTORY OF THE PROBLEM</head>
+by actually quoting the proof and the epigram, which are the
+genuine work of Eratosthenes.
+<p>Our document begins with the story that an ancient tragic
+poet had represented Minos as putting up a tomb to Glaucus
+but being dissatisfied with its being only 100 feet each way;
+Minos was then represented as saying that it must be made
+double the size, by increasing each of the dimensions in that
+ratio. Naturally the poet &lsquo;was thought to have made a mis-
+take&rsquo;. Von Wilamowitz has shown that the verses which
+Minos is made to say cannot have been from any play by
+Aeschylus, Sophocles, or Euripides. They are the work of
+some obscure poet, and the ignorance of mathematics shown
+by him is the only reason why they became notorious and so
+survived. The letter goes on to say that
+<p>&lsquo;Geometers took up the question and sought to find out
+how one could double a given solid while keeping the same
+shape; the problem took the name of &ldquo;the duplication of the
+cube&rdquo; because they started from a cube and sought to double
+it. For a long time all their efforts were vain; then Hippo-
+crates of Chios discovered for the first time that, if we can
+devise a way of finding two mean proportionals in continued
+proportion between two straight lines the greater of which
+is double of the less, the cube will be doubled; that is, one
+puzzle (<G>a)po/rhma</G>) was turned by him into another not less
+difficult. After a time, so goes the story, certain Delians, who
+were commanded by the oracle to double a certain altar, fell
+into the same quandary as before.&rsquo;
+<p>At this point the versions of the story diverge somewhat.
+The pseudo-Eratosthenes continues as follows:
+<p>&lsquo;They therefore sent over to beg the geometers who were
+with Plato in the Academy to find them the solution. The
+latter applying themselves diligently to the problem of finding
+two mean proportionals between two given straight lines,
+Archytas of Taras is said to have found them by means of
+a half cylinder, and Eudoxus by means of the so-called curved
+lines; but, as it turned out, all their solutions were theoretical,
+and no one of them was able to give a practical construction
+for ordinary use, save to a certain small extent Menaechmus,
+and that with difficulty.&rsquo;
+<p>Fortunately we have Eratosthenes's own version in a quota-
+tion by Theon of Smyrna:
+<p>&lsquo;Eratosthenes in his work entitled <I>Platonicus</I> relates that,
+<pb n=246><head>THE DUPLICATION OF THE CUBE</head>
+when the god proclaimed to the Delians by the oracle that, if
+they would get rid of a plague, they should construct an altar
+double of the existing one, their craftsmen fell into great
+perplexity in their efforts to discover how a solid could be made
+double of a (similar) solid; they therefore went to ask Plato
+about it, and he replied that the oracle meant, not that the god
+wanted an altar of double the size, but that he wished, in
+setting them the task, to shame the Greeks for their neglect
+of mathematics and their contempt for geometry.&rsquo;<note>Theon of Smyrna, p. 2. 3-12.</note>
+<p>Eratosthenes's version may well be true; and there is no
+doubt that the question was studied in the Academy, solutions
+being attributed to Eudoxus, Menaechmus, and even (though
+erroneously) to Plato himself. The description by the pseudo-
+Eratosthenes of the three solutions by Archytas, Eudoxus and
+Menaechmus is little more than a paraphrase of the lines about
+them in the genuine epigram of Eratosthenes,
+<p>&lsquo;Do not seek to do the difficult business of the cylinders of
+Archytas, or to cut the cones in the triads of Menaechmus, or
+to draw such a curved form of lines as is described by the
+god-fearing Eudoxus.&rsquo;
+<p>The different versions are reflected in Plutarch, who in one
+place gives Plato's answer to the Delians in almost the same
+words as Eratosthenes,<note>Plutarch, <I>De E apud Delphos</I>, c. 6, 386 E.</note> and in another place tells us that
+Plato referred the Delians to Eudoxus and Helicon of Cyzicus
+for a solution of the problem.<note><I>De genio Socratis</I>, c. 7, 579 C, D.</note>
+<p>After Hippocrates had discovered that the duplication of
+the cube was equivalent to finding two mean proportionals in
+continued proportion between two given straight lines, the
+problem seems to have been attacked in the latter form
+exclusively. The various solutions will now be reproduced
+in chronological order.
+<C>(<G>b</G>) <I>Archytas.</I></C>
+<p>The solution of Archytas is the most remarkable of all,
+especially when his date is considered (first half of fourth cen-
+tury B.C.), because it is not a construction in a plane but a bold
+<pb n=247><head>ARCHYTAS</head>
+construction in three dimensions, determining a certain point
+as the intersection of three surfaces of revolution, (1) a right
+cone, (2) a cylinder, (3) a <I>tore</I> or anchor-ring with inner
+diameter <I>nil.</I> The intersection of the two latter surfaces
+gives (says Archytas) a certain curve (which is in fact a curve
+of double curvature), and the point required is found as the
+point in which the cone meets this curve.
+<FIG>
+<p>Suppose that <I>AC, AB</I> are the two straight lines between
+which two mean proportionals are to be found, and let <I>AC</I> be
+made the diameter of a circle and <I>AB</I> a chord in it.
+<p>Draw a semicircle with <I>AC</I> as diameter, but in a plane at
+right angles to the plane of the circle <I>ABC</I>, and imagine this
+semicircle to revolve about a straight line through <I>A</I> per-
+pendicular to the plane of <I>ABC</I> (thus describing half a <I>tore</I>
+with inner diameter <I>nil</I>).
+<p>Next draw a right half-cylinder on the semicircle <I>ABC</I> as
+base; this will cut the surface of the half-<I>tore</I> in a certain
+curve.
+<p>Lastly let <I>CD</I>, the tangent to the circle <I>ABC</I> at the point <I>C</I>,
+meet <I>AB</I> produced in <I>D</I>; and suppose the triangle <I>ADC</I> to
+revolve about <I>AC</I> as axis. This will generate the surface
+of a right circular cone; the point <I>B</I> will describe a semicircle
+<I>BQE</I> at right angles to the plane of <I>ABC</I> and having its
+diameter <I>BE</I> at right angles to <I>AC</I>; and the surface of the
+cone will meet in some point <I>P</I> the curve which is the inter-
+section of the half-cylinder and the half-<I>tore.</I>
+<pb n=248><head>THE DUPLICATION OF THE CUBE</head>
+<p>Let <I>APC</I>&prime; be the corresponding position of the revolving
+semicircle, and let <I>AC</I>&prime; meet the circumference <I>ABC</I> in <I>M.</I>
+<p>Drawing <I>PM</I> perpendicular to the plane of <I>ABC</I>, we see
+that it must meet the circumference of the circle <I>ABC</I> because
+<I>P</I> is on the cylinder which stands on <I>ABC</I> as base.
+<p>Let <I>AP</I> meet the circumference of the semicircle <I>BQE</I> in <I>Q</I>,
+and let <I>AC</I>&prime; meet its diameter in <I>N.</I> Join <I>PC</I>&prime;, <I>QM, QN.</I>
+<p>Then, since both semicircles are perpendicular to the plane
+<I>ABC</I>, so is their line of intersection <I>QN</I> [Eucl. XI. 19].
+<p>Therefore <I>QN</I> is perpendicular to <I>BE.</I>
+<p>Therefore <MATH><I>QN</I><SUP>2</SUP>=<I>BN.NE</I>=<I>AN.NM</I></MATH>, [Eucl. III. 35]
+so that the angle <I>AQM</I> is a right angle.
+<p>But the angle <I>APC</I>&prime; is also right;
+therefore <I>MQ</I> is parallel to <I>C</I>&prime;<I>P.</I>
+<p>It follows, by similar triangles, that
+<MATH><I>C</I>&prime;<I>A</I>:<I>AP</I>=<I>AP</I>:<I>AM</I>=<I>AM</I>:<I>AQ</I></MATH>;
+that is, <MATH><I>AC</I>:<I>AP</I>=<I>AP</I>:<I>AM</I>=<I>AM</I>:<I>AB</I></MATH>,
+and <I>AB, AM, AP, AC</I> are in continued proportion, so that
+<I>AM, AP</I> are the two mean proportionals required.
+<p>In the language of analytical geometry, if <I>AC</I> is the axis
+of <I>x</I>, a line through <I>A</I> perpendicular to <I>AC</I> in the plane of
+<I>ABC</I> the axis of <I>y</I>, and a line through <I>A</I> parallel to <I>PM</I> the
+axis of <I>z</I>, then <I>P</I> is determined as the intersection of the
+surfaces
+(1) <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>+<I>z</I><SUP>2</SUP>=(<I>a</I><SUP>2</SUP>/<I>b</I><SUP>2</SUP>)<I>x</I><SUP>2</SUP></MATH>, (the cone)
+(2) <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>=<I>ax</I></MATH>, (the cylinder)
+(3) <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>+<I>z</I><SUP>2</SUP>=<I>a</I>&radic;(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>)</MATH>, (the <I>tore</I>)
+where <MATH><I>AC</I>=<I>a, AB</I>=<I>b.</I></MATH>
+<p>From the first two equations we obtain
+<MATH><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>)<SUP>2</SUP>/<I>b</I><SUP>2</SUP></MATH>,
+and from this and (3) we have
+<MATH><I>a</I>/&radic;(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>+<I>z</I><SUP>2</SUP>)=&radic(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>+<I>z</I><SUP>2</SUP>)/&radic(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>)=&radic;(<I>x</I><SUP>2</SUP>+<I>y</I>
+<SUP>2</SUP>)/<I>b</I></MATH>,
+or <MATH><I>AC</I>:<I>AP</I>=<I>AP</I>:<I>AM</I>=<I>AM</I>:<I>AB.</I></MATH>
+<pb n=249><head>ARCHYTAS. EUDOXUS</head>
+<p>Compounding the ratios, we have
+<MATH><I>AC</I>:<I>AB</I>=(<I>AM</I>:<I>AB</I>)<SUP>3</SUP></MATH>;
+therefore the cube of side <I>AM</I> is to the cube of side <I>AB</I> as <I>AC</I>
+is to <I>AB.</I>
+<p>In the particular case where <MATH><I>AC</I>=2<I>AB, AM</I><SUP>3</SUP>=2<I>AB</I><SUP>3</SUP></MATH>,
+and the cube is doubled.
+<C>(<G>g</G>) <I>Eudoxus.</I></C>
+<p>Eutocius had evidently seen some document purporting to
+give Eudoxus's solution, but it is clear that it must have
+been an erroneous version. The epigram of Eratosthenes
+says that Eudoxus solved the problem by means of lines
+of a &lsquo;curved or bent form&rsquo; (<G>kampu/lon ei=)dos e)n grammai=s</G>).
+According to Eutocius, while Eudoxus said in his preface
+that he had discovered a solution by means of &lsquo;curved lines&rsquo;,
+yet, when he came to the proof, he made no use of such
+lines, and further he committed an obvious error in that he
+treated a certain discrete proportion as if it were continuous.<note>Archimedes, ed. Heib., vol. iii, p. 56. 4-8.</note>
+It may be that, while Eudoxus made use of what was really
+a curvilinear locus, he did not actually draw the whole curve
+but only indicated a point or two upon it sufficient for his
+purpose. This may explain the first part of Eutocius's remark,
+but in any case we cannot believe the second part; Eudoxus
+was too accomplished a mathematician to make any confusion
+between a discrete and a continuous proportion. Presumably
+the mistake which Eutocius found was made by some one
+who wrongly transcribed the original; but it cannot be too
+much regretted, because it caused Eutocius to omit the solution
+altogether from his account.
+<p>Tannery<note>Tannery, <I>M&eacute;moires scientifiques</I>, vol. i, pp. 53-61.</note> made an ingenious suggestion to the effect that
+Eudoxus's construction was really adapted from that of
+Archytas by what is practically projection on the plane
+of the circle <I>ABC</I> in Archytas's construction. It is not difficult
+to represent the projection on that plane of the curve of
+intersection between the cone and the <I>tore</I>, and, when this
+curve is drawn in the plane <I>ABC</I>, its intersection with the
+circle <I>ABC</I> itself gives the point <I>M</I> in Archytas's figure.
+<pb n=250><head>THE DUPLICATION OF THE CUBE</head>
+<p>The projection on the plane <I>ABC</I> of the intersection between
+the cone and the <I>tore</I> is seen, by means of their equations
+(1) and (3) above, to be
+<MATH><I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP>/<I>a</I>&radic(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>)</MATH>,
+or, in polar coordinates referred to <I>A</I> as origin and <I>AC</I> as axis,
+<MATH><G>r</G>=<I>b</I><SUP>2</SUP>/(<G>a</G>cos<SUP>2</SUP><G>q</G>)</MATH>
+<p>It is easy to find any number of points on the curve. Take
+the circle <I>ABC</I>, and let <I>AC</I> the diameter and <I>AB</I> a chord
+be the two given straight lines between which two mean
+proportionals have to be found.
+<FIG>
+<p>With the above notation
+<MATH><I>AC</I>=<I>a, AB</I>=<I>b</I></MATH>;
+and, if <I>BF</I> be drawn perpendicular to <I>AC</I>,
+<MATH><I>AB</I><SUP>2</SUP>=<I>AF.AC</I></MATH>,
+or <MATH><I>AF</I>=<I>b</I><SUP>2</SUP>/<I>a</I></MATH>.
+<p>Take any point <I>G</I> on <I>BF</I> and join <I>AG.</I>
+<p>Then, if <MATH>&angle;<I>GAF</I>=<G>q</G>, <I>AG</I>=<I>AF</I>sec<G>q</G></MATH>.
+<p>With <I>A</I> as centre and <I>AG</I> as radius draw a circle meeting
+<I>AC</I> in <I>H</I>, and draw <I>HL</I> at right angles to <I>AC</I>, meeting <I>AG</I>
+produced in <I>L.</I>
+<pb n=251><head>EUDOXUS. MENAECHMUS</head>
+<p>Then <MATH><I>AL</I>=<I>AH</I> sec <G>q</G>=<I>AG</I> sec <G>q</G>=<I>AF</I> sec<SUP>2</SUP> <G>q</G></MATH>.
+<p>That is, if <MATH><G>r</G>=<I>AL,</I> <G>r</G>=<I>b</I><SUP>2</SUP>/<I>a</I> sec<SUP>2</SUP> <G>q</G></MATH>,
+and <I>L</I> is a point on the curve.
+<p>Similarly any number of other points on the curve may be
+found. If the curve meets the circle <I>ABC</I> in <I>M,</I> the length
+<I>AM</I> is the same as that of <I>AM</I> in the figure of Archytas's
+solution.
+<p>And <I>AM</I> is the first of the two mean proportionals between
+<I>AB</I> and <I>AC.</I> The second (= <I>AP</I> in the figure of Archytas's
+solution) is easily found from the relation <MATH><I>AM</I><SUP>2</SUP>=<I>AB.AP</I></MATH>,
+and the problem is solved.
+<p>It must be admitted that Tannery's suggestion as to
+Eudoxus's method is attractive; but of course it is only a con-
+jecture. To my mind the objection to it is that it is too close
+an adaptation of Archytas's ideas. Eudoxus was, it is true,
+a pupil of Archytas, and there is a good deal of similarity
+of character between Archytas's construction of the curve of
+double curvature and Eudoxus's construction of the spherical
+lemniscate by means of revolving concentric spheres; but
+Eudoxus was, I think, too original a mathematician to con-
+tent himself with a mere adaptation of Archytas's method
+of solution.
+<C>(<G>d</G>) <I>Menaechmus.</I></C>
+<p>Two solutions by Menaechmus of the problem of finding
+two mean proportionals are described by Eutocius; both find
+a certain point as the intersection between two conics, in
+the one case two parabolas, in the other a parabola and
+a rectangular hyperbola. The solutions are referred to in
+Eratosthenes's epigram: &lsquo;do not&rsquo;, says Eratosthenes, &lsquo;cut the
+cone in the triads of Menaechmus.&rsquo; From the solutions
+coupled with this remark it is inferred that Menaechmus
+was the discoverer of the conic sections.
+<p>Menaechmus, brother of Dinostratus, who used the <I>quadra-
+trix</I> to square the circle, was a pupil of Eudoxus and flourished
+about the middle of the fourth century B. C. The most attrac-
+tive from of the story about the geometer and the king who
+wanted a short cut to geometry is told of Menaechmus and
+<pb n=252><head>THE DUPLICATION OF THE CUBE</head>
+Alexander: &lsquo;O king,&rsquo; said Menaechmus, &lsquo;for travelling over
+the country there are royal roads and roads for common
+citizens, but in geometry there is one road for all.&rsquo;<note>Stobaeus, <I>Eclogae</I>, ii. 31, 115 (vol. ii, p. 228. 30, Wachsmuth).</note> A similar
+story is indeed told of Euclid and Ptolemy; but there would
+be a temptation to transfer such a story at a later date to
+the more famous mathematician. Menaechmus was evidently
+a considerable mathematician; he is associated by Proclus with
+Amyclas of Heraclea, a friend of Plato, and with Dinostratus
+as having &lsquo;made the whole of geometry more perfect&rsquo;.<note>Proclus on Eucl. I, p. 67. 9.</note>
+Beyond, however, the fact that the discovery of the conic
+sections is attributed to him, we have very few notices relating
+to his work. He is mentioned along with Aristotle and
+Callippus as a supporter of the theory of concentric spheres
+invented by Eudoxus, but as postulating a larger number of
+spheres.<note>Theon of Smyrna, pp. 201. 22-202. 2.</note> We gather from Proclus that he wrote on the
+technology of mathematics; he discussed for instance the
+difference between the broader meaning of the word <I>element</I>
+(in which any proposition leading to another may be said
+to be an element of it) and the stricter meaning of something
+simple and fundamental standing to consequences drawn from
+it in the relation of a <I>principle,</I> which is capable of being
+universally applied and enters into the proof of all manner
+of propositions.<note>Proclus on Eucl. I, pp. 72. 23-73. 14.</note> Again, he did not agree in the distinction
+between theorems and problems, but would have it that they
+were all <I>problems,</I> though directed to two different objects<note><I>Ib.,</I> p. 78. 8-13.</note>;
+he also discussed the important question of the convertibility
+of theorems and the conditions necessary to it.<note><I>Ib.,</I> p. 254. 4-5.</note>
+<p>If <I>x, y</I> are two mean proportionals between straight
+lines <I>a, b,</I>
+that is, if <MATH><I>a</I>:<I>x</I>=<I>x</I>:<I>y</I>=<I>y</I>:<I>b</I></MATH>,
+then clearly <MATH><I>x</I><SUP>2</SUP>=<I>ay</I>, <I>y</I><SUP>2</SUP>=<I>bx</I></MATH>, and <MATH><I>xy</I>=<I>ab</I></MATH>.
+<p>It is easy for us to recognize here the Cartesian equations
+of two parabolas referred to a diameter and the tangent at its
+extremity, and of a hyperbola referred to its asymptotes.
+But Menaechmus appears to have had not only to recognize,
+<pb n=253><head>MENAECHMUS AND CONICS</head>
+but to discover, the existence of curves having the properties
+corresponding to the Cartesian equations. He discovered
+them in plane sections of right circular cones, and it would
+doubtless be the properties of the <I>principal</I> ordinates in
+relation to the abscissae on the axes which he would arrive
+at first. Though only the parabola and the hyperbola are
+wanted for the particular problem, he would certainly not
+fail to find the ellipse and its property as well. But in the
+case of the hyperbola he needed the property of the curve
+with reference to the <I>asymptotes,</I> represented by the equation
+<MATH><I>xy</I>=<I>ab</I></MATH>; he must therefore have discovered the existence of
+the asymptotes, and must have proved the property, at all
+events for the rectangular hyperbola. The original method
+of discovery of the conics will occupy us later. In the mean-
+time it is obvious that the use of any two of the curves
+<MATH><I>x</I><SUP>2</SUP>=<I>ay</I>, <I>y</I><SUP>2</SUP>=<I>bx</I>, <I>xy</I>=<I>ab</I></MATH> gives the solution of our problem,
+and it was in fact the intersection of the second and third
+which Menaechmus used in his first solution, while for his
+second solution he used the first two. Eutocius gives the
+analysis and synthesis of each solution in full. I shall repro-
+duce them as shortly as possible, only suppressing the use of
+four separate lines representing the two given straight lines
+and the two required means in the figure of the first solution.
+<C><I>First solution.</I></C>
+<p>Suppose that <I>AO, OB</I> are two given straight lines of which
+<I>AO</I> > <I>OB,</I> and let them form a right angle at <I>O.</I>
+<p>Suppose the problem solved, and let the two mean propor-
+tionals be <I>OM</I> measured along <I>BO</I> produced and <I>ON</I> measured
+along <I>AO</I> produced. Complete the rectangle <I>OMPN.</I>
+<p>Then, since <MATH><I>AO</I>:<I>OM</I>=<I>OM</I>:<I>ON</I>=<I>ON</I>:<I>OB</I></MATH>,
+we have (1) <MATH><I>OB.OM</I>=<I>ON</I><SUP>2</SUP>=<I>PM</I><SUP>2</SUP></MATH>,
+so that <I>P</I> lies on a parabola which has <I>O</I> for vertex, <I>OM</I> for
+axis, and <I>OB</I> for <I>latus rectum</I>;
+and (2) <MATH><I>AO.OB</I>=<I>OM.ON</I>=<I>PN.PM</I></MATH>,
+so that <I>P</I> lies on a hyperbola with <I>O</I> as centre and <I>OM, ON</I> as
+asymptotes.
+<pb n=254><head>THE DUPLICATION OF THE CUBE</head>
+<p>Accordingly, to find the point <I>P,</I> we have to construct
+(1) a parabola with <I>O</I> as vertex, <I>OM</I> as axis, and <I>latus rectum</I>
+equal to <I>OB,</I>
+<FIG>
+(2) a hyperbola with asymptotes <I>OM, ON</I> and such that
+the rectangle contained by straight lines <I>PM, PN</I> drawn
+from any point <I>P</I> on the curve parallel to one asymptote and
+meeting the other is equal to the rectangle <I>AO.OB.</I>
+<p>The intersection of the parabola and hyperbola gives the
+point <I>P</I> which solves the problem, for
+<MATH><I>AO</I>:<I>PN</I>=<I>PN</I>:<I>PM</I>=<I>PM</I>:<I>OB</I></MATH>.
+<C><I>Second solution.</I></C>
+<p>Supposing the problem solved, as in the first case, we have,
+since <MATH><I>AO</I>:<I>OM</I>=<I>OM</I>:<I>ON</I>=<I>ON</I>:<I>OB</I></MATH>,
+<p>(1) the relation <MATH><I>OB.OM</I>=<I>ON</I><SUP>2</SUP>=<I>PM</I><SUP>2</SUP></MATH>,
+<FIG>
+<pb n=255><head>MENAECHMUS AND CONICS</head>
+so that <I>P</I> lies on a parabola which has <I>O</I> for vertex, <I>OM</I> for
+axis, and <I>OB</I> for <I>latus rectum,</I>
+<p>(2) the similar relation <MATH><I>AO.ON</I>=<I>OM</I><SUP>2</SUP>=<I>PN</I><SUP>2</SUP></MATH>,
+so that <I>P</I> lies on a parabola which has <I>O</I> for vertex, <I>ON</I> for
+axis, and <I>OA</I> for <I>latus rectum.</I>
+<p>In order therefore to find <I>P,</I> we have only to construct the
+two parabolas with <I>OM, ON</I> for axes and <I>OB, OA</I> for <I>latera
+recta</I> respectively; the intersection of the two parabolas gives
+a point <I>P</I> such that
+<MATH><I>AO</I>:<I>PN</I>=<I>PN</I>:<I>PM</I>=<I>PM</I>:<I>OB</I></MATH>,
+and the problem is solved.
+<p>(We shall see later on that Menaechmus did not use the
+names <I>parabola</I> and <I>hyperbola</I> to describe the curves, those
+names being due to Apollonius.)
+<C>(<G>e</G>) <I>The solution attributed to Plato.</I></C>
+<p>This is the first in Eutocius's arrangement of the various
+solutions reproduced by him. But there is almost conclusive
+reason, for thinking that it is wrongly attributed to Plato.
+No one but Eutocius mentions it, and there is no reference to
+it in Eratosthenes's epigram, whereas, if a solution by Plato
+had then been known, it could hardly fail to have been
+mentioned along with those of Archytas, Menaechmus, and
+Eudoxus. Again, Plutarch says that Plato told the Delians
+that the problem of the two mean proportionals was no easy
+one, but that Eudoxus or Helicon of Cyzicus would solve it
+for them; he did not apparently propose to attack it himself.
+And, lastly, the solution attributed to him is mechanical,
+whereas we are twice told that Plato objected to mechanical
+solutions as destroying the good of geometry.<note>Plutarch, <I>Quaest. Conviv.</I> 8. 2. 1, p. 718 E, F; <I>Vita Marcelli,</I> c. 14. 5.</note> Attempts
+have been made to reconcile the contrary traditions. It is
+argued that, while Plato objected to mechanical solutions on
+principle, he wished to show how easy it was to discover
+such solutions and put forward that attributed to him as an
+illustration of the fact. I prefer to treat the silence of
+Eratosthenes as conclusive on the point, and to suppose that
+the solution was invented in the Academy by some one con-
+temporary with or later than Menaechmus.
+<pb n=256><head>THE DUPLICATION OF THE CUBE</head>
+<p>For, if we look at the figure of Menaechmus's second solu-
+tion, we shall see that the given straight lines and the two
+means between them are shown in cyclic order (clockwise)
+as straight lines radiating from <I>O</I> and separated by right
+angles. This is exactly the arrangement of the lines in
+&lsquo;Plato's&rsquo; solution. Hence it seems probable that some one
+who had Menaechmus's second solution before him wished
+to show how the same representation of the four straight
+lines could be got by a mechanical construction as an alterna-
+tive to the use of conics.
+<p>Drawing the two given straight lines with the means, that
+is to say, <I>OA, OM, ON, OB,</I> in cyclic clockwise order, as in
+Menaechmus's second solution, we have
+<MATH><I>AO</I>:<I>OM</I>=<I>OM</I>:<I>ON</I>=<I>ON</I>:<I>OB</I></MATH>,
+and it is clear that, if <I>AM, MN, NB</I> are joined, the angles
+<I>AMN, MNB</I> are both right angles. The problem then is,
+given <I>OA, OB</I> at right angles to one another, to contrive the
+rest of the figure so that the angles at <I>M, N</I> are right.
+<FIG>
+<p>The instrument used is somewhat like that which a shoe-
+maker uses to measure the length of the foot. <I>FGH</I> is a rigid
+right angle made, say, of wood. <I>KL</I> is a strut which, fastened,
+say, to a stick <I>KF</I> which slides along <I>GF,</I> can move while
+remaining always parallel to <I>GH</I> or at right angles to <I>GF.</I>
+<p>Now place the rigid right angle <I>FGH</I> so that the leg <I>GH</I>
+passes through <I>B,</I> and turn it until the angle <I>G</I> lies on <I>AO</I>
+<pb n=257><head>THE SOLUTION ATTRIBUTED TO PLATO</head>
+produced. Then slide the movable strut <I>KL,</I> which remains
+always parallel to <I>GH,</I> until its edge (towards <I>GH</I>) passes
+through <I>A.</I> If now the inner angular point between the
+strut <I>KL</I> and the leg <I>FG</I> does not lie on <I>BO</I> produced,
+the machine has to be turned again and the strut moved
+until the said point does lie on <I>BO</I> produced, as <I>M</I>, care being
+taken that during the whole of the motion the inner edges
+of <I>KL</I> and <I>HG</I> pass through <I>A, B</I> respectively and the inner
+angular point at <I>G</I> moves along <I>AO</I> produced.
+<p>That it is possible for the machine to take up the desired
+position is clear from the figure of Menaechmus, in which
+<I>MO, NO</I> are the means between <I>AO</I> and <I>BO</I> and the angles
+<I>AMN, MNB</I> are right angles, although to get it into the
+required position is perhaps not quite easy.
+<p>The matter may be looked at analytically thus. Let us
+take any other position of the machine in which the strut and
+the leg <I>GH</I> pass through <I>A, B</I> respectively, while <I>G</I> lies on <I>AO</I>
+produced, but <I>P,</I> the angular point between the strut <I>KL</I> and
+<FIG>
+the leg <I>FG,</I> does not lie on <I>OM</I> produced. Take <I>ON, OM</I> as
+the axes of <I>x, y</I> respectively. Draw <I>PR</I> perpendicular to <I>OG,</I>
+and produce <I>GP</I> to meet <I>OM</I> produced in <I>S.</I>
+<p>Let <MATH><I>AO</I>=<I>a</I>, <I>BO</I>=<I>b</I>, <I>OG</I>=<I>r</I></MATH>.
+<pb n=258><head>THE DUPLICATION OF THE CUBE</head>
+<p>Then <MATH><I>AR.RG</I>=<I>PR</I><SUP>2</SUP></MATH>,
+or <MATH>(<I>a</I>+<I>x</I>) (<I>r</I>-<I>x</I>)=<I>y</I><SUP>2</SUP></MATH>. (1)
+<p>Also, by similar triangles,
+<MATH><I>PR</I>:<I>RG</I>=<I>SO</I>:<I>OG</I>
+=<I>OG</I>:<I>OB</I></MATH>;
+or <MATH><I>y</I>/(<I>r</I>-<I>x</I>)=<I>r/b</I></MATH>. (2)
+<p>From the equation (1) we obtain
+<MATH><I>r</I>=(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>+<I>ax</I>)/(<I>a</I>+<I>x</I>)</MATH>,
+and, by multiplying (1) and (2), we have
+<MATH><I>by</I>(<I>a</I>+<I>x</I>)=<I>ry</I><SUP>2</SUP></MATH>,
+whence, substituting the value of <I>r,</I> we obtain, as the locus of
+<I>P,</I> a curve of the third degree,
+<MATH><I>b</I>(<I>a</I>+<I>x</I>)<SUP>2</SUP>=<I>y</I>(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>+<I>ax</I>)</MATH>.
+<p>The intersection (<I>M</I>) of this curve with the axis of <I>y</I> gives
+<MATH><I>OM</I><SUP>3</SUP>=<I>a</I><SUP>2</SUP><I>b</I></MATH>.
+<p>As a theoretical solution, therefore, &lsquo;Plato's&rsquo; solution is
+more difficult than that of Menaechmus.
+<C>(<G>z</G>) <I>Eratosthenes.</I></C>
+<p>This is also a mechanical solution effected by means of
+three plane figures (equal right-angled triangles or rectangles)
+which can move parallel to one another and to their original
+positions between two parallel rulers forming a sort of frame
+and fitted with grooves so arranged that the figures can
+move over one another. Pappus's account makes the figures
+triangles,<note>Pappus, iii, pp. 56-8.</note> Eutocius has parallelograms with diagonals drawn;
+triangles seem preferable. I shall use the lettering of Eutocius
+for the second figure so far as it goes, but I shall use triangles
+instead of rectangles.
+<pb n=259><head>ERATOSTHENES</head>
+<p>Suppose the frame bounded by the parallels <I>AX, EY.</I> The
+<FIG>
+<CAP>FIG. 1.</CAP>
+initial position of the triangles is that shown in the first figure,
+where the triangles are <I>AMF, MNG, NQH.</I>
+<p>In the second figure the straight lines <I>AE, DH</I> which are
+<FIG>
+<CAP>FIG. 2.</CAP>
+parallel to one another are those between which two mean
+proportionals have to be found.
+<p>In the second figure the triangles (except <I>AMF,</I> which
+remains fixed) are moved parallel to their original positions
+towards <I>AMF</I> so that they overlap (as <I>AMF</I>, <I>M</I>&prime;<I>NG</I>, <I>N</I>&prime;<I>QH</I>),
+<I>NQH</I> taking the position <I>N</I>&prime;<I>QH</I> in which <I>QH</I> passes through <I>D,</I>
+and <I>MNG</I> a position <I>M</I>&prime;<I>NG</I> such that the points <I>B, C</I> where
+<I>MF</I>, <I>M</I>&prime;<I>G</I> and <I>NG</I>, <I>N</I>&prime;<I>H</I> respectively intersect are in a straight
+line with <I>A, D.</I>
+<p>Let <I>AD, EH</I> meet in <I>K.</I>
+<p>Then <MATH><I>EK</I>:<I>KF</I>=<I>AK</I>:<I>KB</I>
+=<I>FK</I>:<I>KG</I></MATH>,
+and <MATH><I>EK</I>:<I>KF</I>=<I>AE</I>:<I>BF</I></MATH>, while <MATH><I>FK</I>:<I>KG</I>=<I>BF</I>:<I>CG</I></MATH>;
+therefore <MATH><I>AE</I>:<I>BF</I>=<I>BF</I>:<I>CG</I></MATH>.
+<p>Similarly <MATH><I>BF</I>:<I>CG</I>=<I>CG</I>:<I>DH</I></MATH>,
+so that <I>AE, BF, CG, DH</I> are in continued proportion, and
+<I>BF, CG</I> are the required mean proportionals.
+<p>This is substantially the short proof given in Eratosthenes's
+<pb n=260><head>THE DUPLICATION OF THE CUBE</head>
+inscription on the column; the construction was left to be
+inferred from the single figure which corresponded to the
+second above.
+<p>The epigram added by Eratosthenes was as follows:
+<p>&lsquo;If, good friend, thou mindest to obtain from a small (cube)
+a cube double of it, and duly to change any solid figure into
+another, this is in thy power; thou canst find the measure of
+a fold, a pit, or the broad basin of a hollow well, by this
+method, that is, if thou (thus) catch between two rulers (two)
+means with their extreme ends-converging.<note>Lit. &lsquo;converging with their extreme ends&rsquo; (<G>te/rmasin a(/krois sundro-
+ma/das</G>).</note> Do not thou seek
+to do the difficult business of Archytas's cylinders, or to cut the
+cone in the triads of Menaechmus, or to compass such a curved
+form of lines as is described by the god-fearing Eudoxus.
+Nay thou couldst, on these tablets, easily find a myriad of
+means, beginning from a small base. Happy art thou,
+Ptolemy, in that, as a father the equal of his son in youthful
+vigour, thou hast thyself given him all that is dear to Muses
+and Kings, and may he in the future,<note>Reading with v. Wilamowitz <G>o^ d' e)s u(/steron</G>.</note> O Zeus, god of heaven,
+also receive the sceptre at thy hands. Thus may it be, and
+let any one who sees this offering say &ldquo;This is the gift of
+Eratosthenes of Cyrene&rdquo;.&rsquo;
+<C>(<G>h</G>) <I>Nicomedes.</I></C>
+<p>The solution by Nicomedes was contained in his book on
+conchoids, and, according to Eutocius, he was inordinately
+proud of it, claiming for it much superiority over the method
+of Eratosthenes, which he derided as being impracticable as
+well as ungeometrical.
+<p>Nicomedes reduced the problem to a <G>neu=sis</G> which he solved
+by means of the conchoid. Both Pappus and Eutocius explain
+the method (the former twice over<note>Pappus, iii, pp. 58. 23-62. 13; iv, pp. 246. 20-250. 25.</note>) with little variation.
+<p>Let <I>AB, BC</I> be the two straight lines between which two
+means are to be found. Complete the parallelogram <I>ABCL.</I>
+<p>Bisect <I>AB, BC</I> in <I>D</I> and <I>E.</I>
+<p>Join <I>LD,</I> and produce it to meet <I>CB</I> produced in <I>G.</I>
+<p>Draw <I>EF</I> at right angles to <I>BC</I> and of such length that
+<MATH><I>CF</I>=<I>AD.</I></MATH>
+<p>Join <I>GF,</I> and draw <I>CH</I> parallel to it.
+<pb n=261><head>NICOMEDES</head>
+<p>Then from the point <I>F</I> draw <I>FHK</I> cutting <I>CH</I> and <I>EC</I>
+produced in <I>H</I> and <I>K</I> in such a way that the intercept
+<MATH><I>HK</I>=<I>CF</I>=<I>AD</I></MATH>.
+<p>(This is done by means of a conchoid constructed with <I>F</I> as
+pole, <I>CH</I> as &lsquo;ruler&rsquo;, and &lsquo;distance&rsquo; equal to <I>AD</I> or <I>CF,</I> This
+<FIG>
+conchoid meets <I>EC</I> produced in a point <I>K.</I> We then join <I>FK</I>
+and, by the property of the conchoid, <MATH><I>HK</I> = the &lsquo;distance&rsquo;</MATH>.)
+<p>Join <I>KL,</I> and produce it to meet <I>BA</I> produced in <I>M.</I>
+<p>Then shall <I>CK, MA</I> be the required mean proportionals.
+<p>For, since <I>BC</I> is bisected at <I>E</I> and produced to <I>K,</I>
+<MATH><I>BK.KC</I>+<I>CE</I><SUP>2</SUP>=<I>EK</I><SUP>2</SUP></MATH>.
+<p>Add <I>EF</I><SUP>2</SUP> to each;
+therefore <MATH><I>BK.KC</I>+<I>CF</I><SUP>2</SUP>=<I>KF</I><SUP>2</SUP></MATH>. (1)
+<p>Now, by parallels, <MATH><I>MA</I>:<I>AB</I>=<I>ML</I>:<I>LK</I>
+=<I>BC</I>:<I>CK</I></MATH>.
+<p>But <MATH><I>AB</I>=2<I>AD</I></MATH>, and <MATH><I>BC</I>=1/2<I>GC</I></MATH>;
+therefore <MATH><I>MA</I>:<I>AD</I>=<I>GC</I>:<I>CK</I>
+=<I>FH</I>:<I>HK</I></MATH>,
+and, <I>componendo,</I> <MATH><I>MD</I>:<I>DA</I>=<I>FK</I>:<I>HK</I></MATH>.
+<p>But, by construction, <MATH><I>DA</I>=<I>HK</I></MATH>;
+therefore <MATH><I>MD</I>=<I>FK</I></MATH>, and <MATH><I>MD</I><SUP>2</SUP>=<I>FK</I><SUP>2</SUP></MATH>.
+<pb n=262><head>THE DUPLICATION OF THE CUBE</head>
+<p>Now <MATH><I>MD</I><SUP>2</SUP>=<I>BM.MA</I>+<I>DA</I><SUP>2</SUP></MATH>,
+while, by (1), <MATH><I>FK</I><SUP>2</SUP>=<I>BK.KC</I>+<I>CF</I><SUP>2</SUP></MATH>;
+therefore <MATH><I>BM.MA</I>+<I>DA</I><SUP>2</SUP>=<I>BK.KC</I>+<I>CF</I><SUP>2</SUP></MATH>.
+<p>But <MATH><I>DA</I>=<I>CF</I></MATH>; therefore <MATH><I>BM.MA</I>=<I>BK.KC</I></MATH>.
+<p>Therefore <MATH><I>CK</I>:<I>MA</I>=<I>BM</I>:<I>BK</I>
+=<I>LC</I>:<I>CK</I></MATH>;
+while, at the same time, <MATH><I>BM</I>:<I>BK</I>=<I>MA</I>:<I>AL</I></MATH>.
+<p>Therefore <MATH><I>LC</I>:<I>CK</I>=<I>CK</I>:<I>MA</I>=<I>MA</I>:<I>AL</I></MATH>,
+or <MATH><I>AB</I>:<I>CK</I>=<I>CK</I>:<I>MA</I>=<I>MA</I>:<I>BC</I></MATH>.
+<C>(<G>q</G>) <I>Apollonius, Heron, Philon of Byzantium.</I></C>
+<p>I give these solutions together because they really amount
+to the same thing.<note>Heron's solution is given in his <I>Mechanics</I> (i. 11) and <I>Belopoeica</I>, and is
+reproduced by Pappus (iii, pp. 62-4) as well as by Eutocius (loc. cit.).</note>
+<p>Let <I>AB, AC,</I> placed at right angles, be the two given straight
+<FIG>
+lines. Complete the rectangle <I>ABDC,</I> and let <I>E</I> be the point
+at which the diagonals bisect one another.
+<p>Then a circle with centre <I>E</I> and radius <I>EB</I> will circumscribe
+the rectangle <I>ABDC.</I>
+<p>Now (Apollonius) draw with centre <I>E</I> a circle cutting
+<I>AB, AC</I> produced in <I>F, G</I> but such that <I>F, D, G</I> are in one
+straight line.
+<p>Or (Heron) place a ruler so that its edge passes through <I>D,</I>
+<pb n=263><head>APOLLONIUS, HERON, PHILON OF BYZANTIUM</head>
+and move it about <I>D</I> until the edge intersects <I>AB, AC</I> pro-
+duced in points (<I>F, G</I>) which are equidistant from <I>E.</I>
+<p>Or (Philon) place a ruler so that it passes through <I>D</I> and
+turn it round <I>D</I> until it cuts <I>AB, AC</I> produced and the circle
+about <I>ABDC</I> in points <I>F, G, H</I> such that the intercepts <I>FD,
+HG</I> are equal.
+<p>Clearly all three constructions give the same points <I>F, G.</I>
+For in Philon's construction, since <MATH><I>FD</I>=<I>HG</I></MATH>, the perpendicular
+from <I>E</I> on <I>DH,</I> which bisects <I>DH,</I> must also bisect <I>FG,</I> so
+that <MATH><I>EF</I>=<I>EG</I></MATH>.
+<p>We have first to prove that <MATH><I>AF.FB</I>=<I>AG.GC</I></MATH>.
+<p>(<I>a</I>) With Apollonius's and Heron's constructions we have, if
+<I>K</I> be the middle point of <I>AB,</I>
+<MATH><I>AF.FB</I>+<I>BK</I><SUP>2</SUP>=<I>FK</I><SUP>2</SUP></MATH>.
+<p>Add <I>KE</I><SUP>2</SUP> to both sides;
+therefore <MATH><I>AF.FB</I>+<I>BE</I><SUP>2</SUP>=<I>EF</I><SUP>2</SUP></MATH>.
+<p>Similarly <MATH><I>AG.GC</I>+<I>CE</I><SUP>2</SUP>=<I>EG</I><SUP>2</SUP></MATH>.
+<p>But <MATH><I>BE</I>=<I>CE</I></MATH>, and <MATH><I>EF</I>=<I>EG</I></MATH>;
+therefore <MATH><I>AF.FB</I>=<I>AG.GC</I></MATH>.
+<p>(<I>b</I>) With Philon's construction, since <MATH><I>GH</I>=<I>FD</I></MATH>,
+<MATH><I>HF.FD</I>=<I>DG.GH</I></MATH>.
+<p>But, since the circle <I>BDHC</I> passes through <I>A,</I>
+<MATH><I>HF.FD</I>=<I>AF.FB</I></MATH>, and <MATH><I>DG.GH</I>=<I>AG.GC</I></MATH>;
+therefore <MATH><I>AF.FB</I>=<I>AG.GC</I></MATH>.
+<p>Therefore <MATH><I>FA</I>:<I>AG</I>=<I>CG</I>:<I>FB</I></MATH>.
+<p>But, by similar triangles,
+<MATH><I>FA</I>:<I>AG</I>=<I>DC</I>:<I>CG</I></MATH>, and also <MATH>=<I>FB</I>:<I>BD</I></MATH>;
+therefore <MATH><I>DC</I>:<I>CG</I>=<I>CG</I>:<I>FB</I>=<I>FB</I>:<I>BD</I></MATH>,
+or <MATH><I>AB</I>:<I>CG</I>=<I>CG</I>:<I>FB</I>=<I>FB</I>:<I>AC</I></MATH>.
+<p>The connexion between this solution and that of Menaech-
+mus can be seen thus. We saw that, if <MATH><I>a</I>:<I>x</I>=<I>x</I>:<I>y</I>=<I>y</I>:<I>b</I></MATH>,
+<MATH><I>x</I><SUP>2</SUP>=<I>ay</I>, <I>y</I><SUP>2</SUP>=<I>bx</I>, <I>xy</I>=<I>ab</I></MATH>,
+which equations represent, in Cartesian coordinates, two
+parabolas and a hyperbola. Menaechmus in effect solved the
+<pb n=264><head>THE DUPLICATION OF THE CUBE</head>
+problem of the two mean proportionals by means of the points
+of intersection of any two of these conics.
+<p>But, if we add the first two equations, we have
+<MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>-<I>bx</I>-<I>ay</I>=0</MATH>,
+which is a circle passing through the points common to the
+two parabolas <MATH><I>x</I><SUP>2</SUP>=<I>ay</I>, <I>y</I><SUP>2</SUP>=<I>bx</I></MATH>.
+<p>Therefore we can equally obtain a solution by means of
+the intersections of the circle <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>-<I>bx</I>-<I>ay</I>=0</MATH> and the
+rectangular hyperbola <MATH><I>xy</I>=<I>ab</I></MATH>.
+<p>This is in effect what Philon does, for, if <I>AF, AG</I> are the
+coordinate axes, the circle <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>-<I>bx</I>-<I>ay</I>=0</MATH> is the circle
+<I>BDHC,</I> and <MATH><I>xy</I>=<I>ab</I></MATH> is the rectangular hyperbola with
+<I>AF, AG</I> as asymptotes and passing through <I>D,</I> which
+hyperbola intersects the circle again in <I>H,</I> a point such
+that <MATH><I>FD</I>=<I>HG</I></MATH>.
+<C>(<G>i</G>) <I>Diocles and the cissoid.</I></C>
+<p>We gather from allusions to the cissoid in Proclus's com-
+mentary on Eucl. I that the curve which Geminus called by
+that name was none other than the curve invented by Diocles
+and used by him for doubling the cube or finding two mean
+proportionals. Hence Diocles must have preceded Geminus
+(fl. 70 B.C.). Again, we conclude from the two fragments
+preserved by Eutocius of a work by him, <G>pepi\ purei/wn</G>, <I>On
+burning-mirrors,</I> that he was later than Archimedes and
+Apollonius. He may therefore have flourished towards the
+end of the second century or at the beginning of the first
+century B.C. Of the two fragments given by Eutocius one
+contains a solution by means of conics of the problem of
+dividing a sphere by a plane in such a way that the volumes
+of the resulting segments shall be in a given ratio&mdash;a problem
+equivalent to the solution of a certain cubic equation&mdash;while
+the other gives the solution of the problem of the two mean
+proportionals by means of the cissoid.
+<p>Suppose that <I>AB, DC</I> are diameters of a circle at right
+angles to one another. Let <I>E, F</I> be points on the quadrants
+<I>BD, BC</I> respectively such that the arcs <I>BE, BF</I> are equal.
+<p>Draw <I>EG, FH</I> perpendicular to <I>DC.</I> Join <I>CE,</I> and let <I>P</I> be
+the point in which <I>CE, FH</I> intersect.
+<pb n=265><head>DIOCLES AND THE CISSOID</head>
+<p>The cissoid is the locus of all the points <I>P</I> corresponding to
+different positions of <I>E</I> on the quadrant <I>BD</I> and of <I>F</I> at an
+equal distance from <I>B</I> on the quadrant <I>BC.</I>
+<p>If <I>P</I> is any point found by the above construction, it is
+<FIG>
+required to prove that <I>FH, HC</I> are two mean proportionals in
+continued proportion between <I>DH</I> and <I>HP,</I> or that
+<MATH><I>DH</I>:<I>HF</I>=<I>HF</I>:<I>HC</I>=<I>HC</I>:<I>HP</I></MATH>.
+<p>Now it is clear from the construction that <MATH><I>EG</I>=<I>FH</I>,
+<I>DG</I>=<I>HC</I></MATH>, so that <MATH><I>CG</I>:<I>GE</I>&equals3;<I>DH</I>:<I>HF</I></MATH>.
+<p>And, since <I>FH</I> is a mean proportional between <I>DH, HC,</I>
+<MATH><I>DH</I>:<I>HF</I>=<I>HF</I>:<I>CH</I></MATH>.
+<p>But, by similar triangles,
+<MATH><I>CG</I>:<I>GE</I>=<I>CH</I>:<I>HP</I></MATH>.
+<p>It follows that
+<MATH><I>DH</I>:<I>HF</I>=<I>HF</I>:<I>CH</I>=<I>CH</I>:<I>HP</I></MATH>,
+or <I>FH, HC</I> are the two mean proportionals between <I>DH, HP.</I>
+<p>[Since <MATH><I>DH.HP</I>=<I>HF.CH</I></MATH>, we have, if <I>a</I> is the radius of
+the circle and if <MATH><I>OH</I>=<I>x</I>, <I>HP</I>=<I>y</I></MATH>, or (in other words) if we
+use <I>OC, OB</I> as axes of coordinates,
+<MATH>(<I>a</I>+<I>x</I>)<I>y</I>=&radic;(<I>a</I><SUP>2</SUP>-<I>x</I><SUP>2</SUP>).(<I>a</I>-<I>x</I>)</MATH>
+or <MATH><I>y</I><SUP>2</SUP>(<I>a</I>+<I>x</I>)=(<I>a</I>-<I>x</I>)<SUP>3</SUP></MATH>,
+which is the Cartesian equation of the curve. It has a cusp
+at <I>C,</I> and the tangent to the circle at <I>D</I> is an asymptote to it.]
+<pb n=266><head>THE DUPLICATION OF THE CUBE</head>
+<p>Suppose now that the cissoid has been drawn as shown by
+the dotted line in the figure, and that we are required to find
+two mean proportionals between two straight lines <I>a, b.</I>
+<p>Take the point <I>K</I> on <I>OB</I> such that <MATH><I>DO</I>:<I>OK</I>=<I>a</I>:<I>b</I></MATH>.
+<p>Join <I>DK,</I> and produce it to meet the cissoid in <I>Q.</I>
+<p>Through <I>Q</I> draw the ordinate <I>LM</I> perpendicular to <I>DC.</I>
+<p>Then, by the property of the cissoid, <I>LM, MC</I> are the two
+mean proportionals between <I>DM, MQ.</I> And
+<MATH><I>DM</I>:<I>MQ</I>=<I>DO</I>:<I>OK</I>=<I>a</I>:<I>b</I></MATH>.
+<p>In order, then, to obtain the two mean proportionals between
+<I>a</I> and <I>b,</I> we have only to take straight lines which bear respec-
+tively the same ratio to <I>DM, LM, MC, MQ</I> as <I>a</I> bears to <I>DM.</I>
+The extremes are then <I>a, b,</I> and the two mean proportionals
+are found.
+<C>(<G>k</G>) <I>Sporus and Pappus.</I></C>
+<p>The solutions of Sporus and Pappus are really the same as
+that of Diocles, the only difference being that, instead of using
+the cissoid, they use a ruler which they turn about a certain
+point until certain intercepts which it cuts off between two
+pairs of lines are equal.
+<p>In order to show the identity of the solutions, I shall draw
+Sporus's figure with the same lettering as above for corre-
+sponding points, and I shall add dotted lines to show the
+additional auxiliary lines used by Pappus.<note>Pappus, iii, pp. 64-8; viii, pp. 1070-2.</note> (Compared with
+my figure, Sporus's is the other way up, and so is Pappus's
+where it occurs in his own <I>Synagoge,</I> though not in Eutocius.)
+<p>Sporus was known to Pappus, as we have gathered from
+Pappus's reference to his criticisms on the <I>quadratrix,</I> and
+it is not unlikely that Sporus was either Pappus's master or
+a fellow-student of his. But when Pappus gives (though in
+better form, if we may judge by Eutocius's reproduction of
+Sporus) the same solution as that of Sporus, and calls it
+a solution <G>kaq) h(ma=s</G>, he clearly means &lsquo;according to my
+method&rsquo;, not &lsquo;<I>our</I> method&rsquo;, and it appears therefore that he
+claimed the credit of it for himself.
+<p>Sporus makes <I>DO, OK</I> (at right angles to one another) the
+actual given straight lines; Pappus, like Diocles, only takes
+<pb n=267><head>SPORUS AND PAPPUS</head>
+them in the same proportion as the given straight lines.
+Otherwise the construction is the same.
+<p>A circle being drawn with centre <I>O</I> and radius <I>DO,</I> we join
+<I>DK</I> and produce it to meet the circle in <I>I.</I>
+<p>Now conceive a ruler to pass through <I>C</I> and to be turned
+about <I>C</I> until it cuts <I>DI, OB</I> and the circumference of the
+<FIG>
+circle in points <I>Q, T, R</I> such that <MATH><I>QT</I>=<I>TR</I></MATH>. Draw <I>QM, RN</I>
+perpendicular to <I>DC.</I>
+<p>Then, since <MATH><I>QT</I>=<I>TR</I>, <I>MO</I>=<I>ON</I></MATH>, and <I>MQ, NR</I> are equi-
+distant from <I>OB.</I> Therefore in reality <I>Q</I> lies on the cissoid of
+Diocles, and, as in the first part of Diocles's proof, we prove
+(since <I>RN</I> is equal to the ordinate through <I>Q,</I> the foot of
+which is <I>M</I>) that
+<MATH><I>DM</I>:<I>RN</I>=<I>RN</I>:<I>MC</I>=<I>MC</I>:<I>MQ</I></MATH>,
+and we have the two means between <I>DM, MQ,</I> so that we can
+easily construct the two means between <I>DO, OK.</I>
+<p>But Sporus actually proves that the first of the two means
+between <I>DO</I> and <I>OK</I> is <I>OT.</I> This is obvious from the above
+relations, because
+<MATH><I>RN</I>:<I>OT</I>=<I>CN</I>:<I>CO</I>=<I>DM</I>:<I>DO</I>=<I>MQ</I>:<I>OK</I></MATH>.
+<p>Sporus has an <I>ab initio</I> proof of the fact, but it is rather
+confused, and Pappus's proof is better worth giving, especially
+as it includes the actual duplication of the cube.
+<p>It is required to prove that <MATH><I>DO</I>:<I>OK</I>=<I>DO</I><SUP>3</SUP>:<I>OT</I><SUP>3</SUP></MATH>.
+<pb n=268><head>THE DUPLICATION OF THE CUBE</head>
+<p>Join <I>RO,</I> and produce it to meet the circle at <I>S.</I> Join
+<I>DS, SC.</I>
+<p>Then, since <MATH><I>RO</I>=<I>OS</I></MATH> and <MATH><I>RT</I>=<I>TQ</I></MATH>, <I>SQ</I> is parallel to <I>AB</I>
+and meets <I>OC</I> in <I>M.</I>
+<p>Now
+<MATH><I>DM</I>:<I>MC</I>=<I>SM</I><SUP>2</SUP>:<I>MC</I><SUP>2</SUP>=<I>CM</I><SUP>2</SUP>:<I>MQ</I><SUP>2</SUP></MATH> (since &angle;<I>RCS</I> is right).
+<p>Multiply by the ratio <I>CM</I>:<I>MQ</I>;
+therefore <MATH>(<I>DM</I>:<I>MC</I>).(<I>CM</I>:<I>MQ</I>)=(<I>CM</I><SUP>2</SUP>:<I>MQ</I><SUP>2</SUP>).(<I>CM</I>:<I>MQ</I>)</MATH>
+or <MATH><I>DM</I>:<I>MQ</I>=<I>CM</I><SUP>3</SUP>:<I>MQ</I><SUP>3</SUP></MATH>.
+<p>But <MATH><I>DM</I>:<I>MQ</I>=<I>DO</I>:<I>OK</I></MATH>,
+and <MATH><I>CM</I>:<I>MQ</I>=<I>CO</I>:<I>OT</I></MATH>.
+<p>Therefore <MATH><I>DO</I>:<I>OK</I>=<I>CO</I><SUP>3</SUP>:<I>OT</I><SUP>3</SUP>=<I>DO</I><SUP>3</SUP>:<I>OT</I><SUP>3</SUP></MATH>.
+<p>Therefore <I>OT</I> is the first of the two mean proportionals to
+<I>DO, OK</I>; the second is found by taking a third proportional
+to <I>DO, OT.</I>
+<p>And a cube has been increased in any given ratio.
+<C>(<G>l</G>) <I>Approximation to a solution by plane methods only.</I></C>
+<p>There remains the procedure described by Pappus and
+criticized by him at length at the beginning of Book III of
+his <I>Collection.</I><note>Pappus, iii, pp. 30-48.</note> It was suggested by some one &lsquo;who was
+thought to be a great geometer&rsquo;, but whose name is not given.
+Pappus maintains that the author did not understand what
+he was about, &lsquo;for he claimed that he was in possession of
+a method of finding two mean proportionals between two
+straight lines by means of plane considerations only&rsquo;; he
+gave his construction to Pappus to examine and pronounce
+upon, while Hierius the philosopher and other friends of his
+supported his request for Pappus's opinion. The construction
+is as follows.
+<p>Let the given straight lines be <I>AB, AD</I> placed at right
+angles to one another, <I>AB</I> being the greater.
+<p>Draw <I>BC</I> parallel to <I>AD</I> and equal to <I>AB.</I> Join <I>CD</I> meeting
+<I>BA</I> produced in <I>E.</I> Produce <I>BC</I> to <I>L,</I> and draw <I>EL</I>&prime; through
+<I>E</I> parallel to <I>BL.</I> Along <I>CL</I> cut off lengths <I>CF, FG, GK, KL,</I>
+<pb n=269><head>APPROXIMATION BY PLANE METHODS</head>
+each of which is equal to <I>BC.</I> Draw <I>CC</I>&prime;, <I>FF</I>&prime;, <I>GG</I>&prime;, <I>KK</I>&prime;, <I>LL</I>&prime;
+parallel to <I>BA.</I>
+<p>On <I>LL</I>&prime;, <I>KK</I>&prime; take <I>LM, KR</I> equal to <I>BA,</I> and bisect <I>LM</I>
+in <I>N.</I>
+<p>Take <I>P, Q</I> on <I>LL</I>&prime; such that <I>L</I>&prime;<I>L</I>, <I>L</I>&prime;<I>N</I>, <I>L</I>&prime;<I>P</I>, <I>L</I>&prime;<I>Q</I> are in con-
+<FIG>
+tinued proportion; join <I>QR, RL,</I> and through <I>N</I> draw <I>NS</I>
+parallel to <I>QR</I> meeting <I>RL</I> in <I>S.</I>
+<p>Draw <I>ST</I> parallel to <I>BL</I> meeting <I>GG</I>&prime; in <I>T.</I>
+<p>To <I>G</I>&prime;<I>G</I>, <I>G</I>&prime;<I>T</I> take continued proportionals <I>G</I>&prime;<I>O</I>, <I>G</I>&prime;<I>U,</I> as before.
+Take <I>W</I> on <I>FF</I>&prime; such that <MATH><I>FW</I>=<I>BA</I></MATH>, join <I>UW, WG,</I> and
+through <I>T</I> draw <I>TI</I> parallel to <I>UW</I> meeting <I>WG</I> in <I>I.</I>
+<p>Through <I>I</I> draw <I>IV</I> parallel to <I>BC</I> meeting <I>CC</I>&prime; in <I>V.</I>
+<p>Take continued proportionals <I>C</I>&prime;<I>C</I>, <I>C</I>&prime;<I>V</I>, <I>C</I>&prime;<I>X</I>, <I>C</I>&prime;<I>Y,</I> and draw
+<I>XZ, VZ</I>&prime; parallel to <I>YD</I> meeting <I>EC</I> in <I>Z, Z</I>&prime;. Lastly draw
+<I>ZX</I>&prime;, <I>Z</I>&prime;<I>Y</I>&prime; parallel to <I>BC.</I>
+<p>Then, says the author, it is required to prove that <I>ZX</I>&prime;, <I>Z</I>&prime;<I>Y</I>&prime;
+are two mean proportionals in continued proportion between
+<I>AD, BC.</I>
+<p>Now, as Pappus noticed, the supposed conclusion is clearly
+not true unless <I>DY</I> is parallel to <I>BC,</I> which in general it is not.
+But what Pappus failed to observe is that, if the operation of
+taking the continued proportionals as described is repeated,
+not three times, but an infinite number of times, the length of
+the line <I>C</I>&prime;<I>Y</I> tends continually towards equality with <I>EA.</I>
+Although, therefore, by continuing the construction we can
+never exactly determine the required means, the method gives
+an endless series of approximations tending towards the true
+lengths of the means.
+<pb n=270><head>THE DUPLICATION OF THE CUBE</head>
+<p>Let <MATH><I>LL</I>&prime;=<I>BE</I>=<I>a</I>, <I>AB</I>=<I>b</I>, <I>L</I>&prime;<I>N</I>=<G>a</G></MATH> (for there is no
+necessity to take <I>N</I> at the middle point of <I>LM</I>).
+<p>Then <MATH><I>L</I>&prime;<I>Q</I>=<G>a</G><SUP>3</SUP>/<I>a</I><SUP>2</SUP></MATH>,
+therefore <MATH><I>LQ</I>=(<I>a</I><SUP>3</SUP>-<G>a</G><SUP>3</SUP>)/<I>a</I><SUP>2</SUP></MATH>.
+<p>And <MATH><I>TG/RK</I>=<I>SL/RL</I>=<I>NL/QL</I>=((<I>a</I>-<G>a</G>)<I>a</I><SUP>2</SUP>)/(<I>a</I><SUP>3</SUP>-<G>a</G><SUP>3</SUP>)</MATH>;
+therefore <MATH><I>TG</I>=((<I>a</I>-<G>a</G>)<I>a</I><SUP>2</SUP><I>b</I>)/(<I>a</I><SUP>3</SUP>-<G>a</G><SUP>3</SUP>)</MATH>,
+and accordingly <MATH><I>G</I>&prime;<I>T</I>=<I>a</I>-((<I>a</I>-<G>a</G>)<I>a</I><SUP>2</SUP><I>b</I>)/(<I>a</I><SUP>3</SUP>-<G>a</G><SUP>3</SUP>)</MATH>.
+<p>Now let <G>a</G><SUB><I>n</I></SUB> be the length corresponding to <I>G</I>&prime;<I>T</I> after <I>n</I>
+operations; then it is clear that
+<MATH><I>a</I>-<G>a</G><SUB><I>n</I>+1</SUB>=(<I>a</I>-<G>a</G><SUB><I>n</I></SUB>)<I>a</I><SUP>2</SUP><I>b</I>/(<I>a</I><SUP>3</SUP>-<G>a</G><SUB><I>n</I></SUB><SUP>3</SUP>)</MATH>.
+<p><G>a</G><SUB><I>n</I></SUB> must approach some finite limit when <MATH><I>n</I>=<*></MATH>. Taking <G>x</G>
+as this limit, we have
+<MATH><I>a</I>-<G>x</G>=(<I>a</I>-<G>x</G>)<I>a</I><SUP>2</SUP><I>b</I>/(<I>a</I><SUP>3</SUP>-<G>x</G><SUP>3</SUP>)</MATH>,
+and, <MATH><G>x</G>=<I>a</I></MATH> not being a root of this equation, we get at once
+<MATH><G>x</G><SUP>3</SUP>=<I>a</I><SUP>3</SUP>-<I>a</I><SUP>2</SUP><I>b</I>=<I>a</I><SUP>2</SUP>(<I>a</I>-<I>b</I>)</MATH>.
+Therefore, ultimately <I>C</I>&prime;<I>V</I> is one of the mean proportionals
+between <I>EA</I> and <I>EB,</I> whence <I>Y</I>&prime;<I>Z</I>&prime; will be one of the mean
+proportionals between <I>AD, BC,</I> that is, between <I>AD</I> and <I>AB.</I>
+<p>The above was pointed out for the first time by R. Pendle-
+bury,<note><I>Messenger of Mathematics,</I> ser. 2, vol. ii (1873), pp. 166-8.</note> and I have followed his way of stating the matter.
+<pb>
+<C>VIII
+ZENO OF ELEA</C>
+<p>WE have already seen how the consideration of the subject
+of infinitesimals was forced upon the Greek mathematicians so
+soon as they came to close grips with the problem of the
+quadrature of the circle. Antiphon the Sophist was the first
+to indicate the correct road upon which the solution was to
+be found, though he expressed his idea in a crude form which
+was bound to provoke immediate and strong criticism from
+logical minds. Antiphon had inscribed a series of successive
+regular polygons in a circle, each of which had double as
+many sides as the preceding, and he asserted that, by con-
+tinuing this process, we should at length exhaust the circle:
+&lsquo;he thought that in this way the area of the circle would
+sometime be used up and a polygon would be inscribed in the
+circle the sides of which on account of their smallness would
+coincide with the circumference.&rsquo;<note>Simpl. <I>in Arist. Phys.</I>, p. 55. 6 Diels.</note> Aristotle roundly said that
+this was a fallacy which it was not even necessary for a
+geometer to trouble to refute, since an expert in any science
+is not called upon to refute <I>all</I> fallacies, but only those which
+are false deductions from the admitted principles of the
+science; if the fallacy is based on anything which is in con-
+tradiction to any of those principles, it may at once be ignored.<note>Arist. <I>Phys.</I> i. 2, 185 a 14-17.</note>
+Evidently therefore, in Aristotle's view, Antiphon's argument
+violated some &lsquo;geometrical principle&rsquo;, whether this was the
+truth that a straight line, however short, can never coincide
+with an arc of a circle, or the principle assumed by geometers
+that geometrical magnitudes can be divided <I>ad infinitum.</I>
+<p>But Aristotle is only a representative of the criticisms
+directed against the ideas implied in Antiphon's argument;
+those ideas had already, as early as the time of Antiphon
+<pb n=272><head>ZENO OF ELEA</head>
+himself (a contemporary of Socrates), been subjected to a
+destructive criticism expressed with unsurpassable piquancy
+and force. No wonder that the subsequent course of Greek
+geometry was profoundly affected by the arguments of Zeno
+on motion. Aristotle indeed called them &lsquo;fallacies&rsquo;, without
+being able to refute them. The mathematicians, however, knew
+better, and, realizing that Zeno's arguments were fatal to
+infinitesimals, they saw that they could only avoid the diffi-
+culties connected with them by once for all banishing the idea
+of the infinite, even the potentially infinite, altogether from
+their science; thenceforth, therefore, they made no use of
+magnitudes increasing or diminishing <I>ad infinitum</I>, but con-
+tented themselves with finite magnitudes that can be made as
+great or as small <I>as we please.</I><note>Cf. Arist. <I>Phys.</I> iii. 7, 207 b 31.</note> If they used infinitesimals
+at all, it was only as a tentative means of <I>discovering</I> proposi-
+tions; they <I>proved</I> them afterwards by rigorous geometrical
+methods. An illustration of this is furnished by the <I>Method</I> of
+Archimedes. In that treatise Archimedes finds (<I>a</I>) the areas
+of curves, and (<I>b</I>) the volumes of solids, by treating them
+respectively as the sums of an infinite number (<I>a</I>) of parallel
+<I>lines</I>, i.e. infinitely narrow strips, and (<I>b</I>) of parallel <I>planes</I>,
+i.e. infinitely thin laminae; but he plainly declares that this
+method is only useful for discovering results and does not
+furnish a proof of them, but that to establish them scientific-
+ally a geometrical proof by the method of exhaustion, with
+its double <I>reductio ad absurdum</I>, is still necessary.
+<p>Notwithstanding that the criticisms of Zeno had so impor-
+tant an influence upon the lines of development of Greek
+geometry, it does not appear that Zeno himself was really
+a mathematician or even a physicist. Plato mentions a work
+of his (<G>ta\ tou= *zh/nwnos gra/mmata</G>, or <G>to\ su/ggramma</G>) in terms
+which imply that it was his only known work.<note>Plato, <I>Parmenides</I>, 127 c sq.</note> Simplicius
+too knows only one work of his, and this the same as that
+mentioned by Plato<note>Simpl. <I>in Phys.</I>, pp. 139. 5, 140. 27 Diels.</note>; when Suidas mentions four, a <I>Commen-
+tary on</I> or <I>Exposition of Empedocles, Controversies, Against
+the philosophers</I> and <I>On Nature</I>, it may be that the last three
+titles are only different designations for the one work, while
+the book on Empedocles may have been wrongly attributed
+<pb n=273><head>ZENO OF ELEA</head>
+to Zeno.<note>Zeller, i<SUP>5</SUP>, p. 587 note.</note> Plato puts into the mouth of Zeno himself an
+explanation of the character and object of his book.<note>Plato, <I>Parmenides</I> 128 C-E.</note> It was
+a youthful effort, and it was stolen by some one, so that the
+author had no opportunity of considering whether to publish
+it or not. Its object was to defend the system of Parmenides
+by attacking the common conceptions of things. Parmenides
+held that only the One exists; whereupon common sense
+pointed out that many contradictions and absurdities will
+follow if this be admitted. Zeno replied that, if the popular
+view that Many exist be accepted, still more absurd results
+will follow. The work was divided into several parts (<G>lo/goi</G>
+according to Plato) and each of these again into sections
+(&lsquo;hypotheses&rsquo; in Plato, &lsquo;contentions&rsquo;, <G>e)pixeirh/mata</G>, in Sim-
+plicius): each of the latter (which according to Proclus
+numbered forty in all<note>Proclus <I>in Parm.</I>, p. 694. 23 seq.</note>) seems to have taken one of the
+assumptions made on the ordinary view of life and to have
+shown that it leads to an absurdity. It is doubtless on
+account of this systematic use of indirect proof by the <I>reductio
+ad absurdum</I> of particular hypotheses that Zeno is said to
+have been called by Aristotle the discoverer of Dialectic<note>Diog. L. viii. 57, ix. 25; Sext. Emp. <I>Math.</I> vii. 6.</note>;
+Plato, too, says of him that he understood how to make one
+and the same thing appear like and unlike, one and many, at
+rest and in motion.<note>Plato, <I>Phaedrus</I> 261 D.</note>
+<C>Zeno's arguments about motion.</C>
+<p>It does not appear that the full significance and value of
+Zeno's paradoxes have ever been realized until these latter
+days. The most modern view of them shall be expressed in
+the writer's own words:
+<p>&lsquo;In this capricious world nothing is more capricious than
+posthumous fame. One of the most notable victims of pos-
+terity's lack of judgement is the Eleatic Zeno. Having
+invented four arguments all immeasurably subtle and pro-
+found, the grossness of subsequent philosophers pronounced
+him to be a mere ingenious juggler, and his arguments to be
+<pb n=274><head>ZENO OF ELEA</head>
+one and all sophisms. After two thousand years of continual
+refutation, these sophisms were reinstated, and made the
+foundation of a mathematical renaissance, by a German
+professor who probably never dreamed of any connexion
+between himself and Zeno. Weierstrass, by strictly banishing
+all infinitesimals, has at last shown that we live in an
+unchanging world, and that the arrow, at every moment of its
+flight, is truly at rest. The only point where Zeno probably
+erred was in inferring (if he did infer) that, because there
+is no change, the world must be in the same state at one time
+as at another. This consequence by no means follows, and in
+this point the German professor is more constructive than the
+ingenious Greek. Weierstrass, being able to embody his
+opinions in mathematics, where familiarity with truth elimi-
+nates the vulgar prejudices of common sense, has been able to
+give to his propositions the respectable air of platitudes; and
+if the result is less delightful to the lover of reason than Zeno's
+bold defiance, it is at any rate more calculated to appease the
+mass of academic mankind.&rsquo;<note>Bertrand Russell, <I>The Principles of Mathematics</I>, vol. i, 1903, pp.
+347, 348.</note>
+<p>Thus, while in the past the arguments of Zeno have been
+treated with more or less disrespect as mere sophisms, we have
+now come to the other extreme. It appears to be implied that
+Zeno anticipated Weierstrass. This, I think, a calmer judge-
+ment must pronounce to be incredible. If the arguments of
+Zeno are found to be &lsquo;immeasurably subtle and profound&rsquo;
+because they contain ideas which Weierstrass used to create
+a great mathematical theory, it does not follow that for Zeno
+they meant at all the same thing as for Weierstrass. On the
+contrary, it is probable that Zeno happened upon these ideas
+without realizing any of the significance which Weierstrass
+was destined to give them; nor shall we give Zeno any less
+credit on this account.
+<p>It is time to come to the arguments themselves. It is the
+four arguments on the subject of motion which are most
+important from the point of view of the mathematician; but
+they have points of contact with the arguments which Zeno
+used to prove the non-existence of Many, in refutation of
+those who attacked Parmenides's doctrine of the One. Accord-
+ing to Simplicius, he showed that, if Many exist, they must
+<pb n=275><head>ZENO'S ARGUMENTS ABOUT MOTION</head>
+be both great and small, so great on the one hand as to be
+infinite in size and so small on the other as to have no size.<note>Simpl. <I>in Phys.</I>, p. 139. 5, Diels.</note>
+To prove the latter of these contentions, Zeno relied on the
+infinite divisibility of bodies as evident; assuming this, he
+easily proved that division will continually give smaller and
+smaller parts, there will be no limit to the diminution, and, if
+there is a final element, it must be absolutely <I>nothing.</I> Conse-
+quently to add any number of these <I>nil</I>-elements to anything
+will not increase its size, nor will the subtraction of them
+diminish it; and of course to add them to one another, even
+in infinite number, will give <I>nothing</I> as the total. (The
+second horn of the dilemma, not apparently stated by Zeno
+in this form, would be this. A critic might argue that infinite
+division would only lead to parts having <I>some</I> size, so that the
+last element would itself have some size; to this the answer
+would be that, as there would, by hypothesis, be an infinite
+number of such parts, the original magnitude which was
+divided would be infinite in size.) The connexion between
+the arguments against the Many and those against motion
+lies in the fact that the former rest on the assumption of
+the divisibility of matter <I>ad infinitum</I>, and that this is the
+hypothesis assumed in the first two arguments against motion.
+We shall see that, while the first two arguments proceed on
+this hypothesis, the last two appear to proceed on the opposite
+hypothesis that space and time are not infinitely divisible, but
+that they are composed of <I>indivisible</I> elements; so that the
+four arguments form a complete dilemma.
+<p>The four arguments against motion shall be stated in the
+words of Aristotle.
+<p>I. The <I>Dichotomy.</I>
+<p>&lsquo;There is no motion because that which is moved must
+arrive at the middle (of its course) before it arrives at the
+end.&rsquo;<note>Aristotle, <I>Phys.</I> vi. 9, 239 b 11.</note> (And of course it must traverse the half of the half
+before it reaches the middle, and so on <I>ad infinitum.</I>)
+<p>II. The <I>Achilles.</I>
+<p>&lsquo;This asserts that the slower when running will never be
+<pb n=276><head>ZENO OF ELEA</head>
+overtaken by the quicker; for that which is pursuing must
+first reach the point from which that which is fleeing started,
+so that the slower must necessarily always be some distance
+ahead.&rsquo;<note>Aristotle, <I>Phys.</I> vi. 9, 239 b 14.</note>
+<p>III. The <I>Arrow.</I>
+<p>&lsquo;If, says Zeno, everything is either at rest or moving when
+it occupies a space equal (to itself), while the object moved is
+always in the instant (<G>e)/sti d) a)ei\ to\ fero/menon e)n tw=| nu=n</G>, in
+the <I>now</I>), the moving arrow is unmoved.&rsquo;<note><I>Ib.</I> 239 b 5-7.</note>
+<p>I agree in Brochard's interpretation of this passage,<note>V. &Bdot;rochard, <I>&Eacute;tudes de Philosophie ancienne et de Philosophie moderne</I>,
+Paris 1912, p. 6.</note> from
+which Zeller<note>Zeller, i<SUP>5</SUP>, p. 599.</note> would banish <G>h)\ kinei=tai</G>, &lsquo;or is moved&rsquo;. The
+argument is this. It is strictly impossible that the arrow can
+move in the <I>instant</I>, supposed indivisible, for, if it changed its
+position, the instant would be at once divided. Now the
+moving object is, in the instant, either at rest or in motion;
+but, as it is not in motion, it is at rest, and as, by hypothesis,
+time is composed of nothing but instants, the moving object is
+always at rest. This interpretation has the advantage of
+agreeing with that of Simplicius,<note>Simpl. <I>in Phys.</I>, pp. 1011-12, Diels.</note> which seems preferable
+to that of Themistius<note>Them. (<I>ad loc.</I>, p. 392 Sp., p. 199 Sch.)</note> on which Zeller relies.
+<p>IV. The <I>Stadium.</I> I translate the first two sentences of
+Aristotle's account<note><I>Phys.</I> vi, 9, 239 b 33-240 a 18.</note>:
+<p>&lsquo;The fourth is the argument concerning the two rows of
+bodies each composed of an equal number of bodies of equal
+size, which pass one another on a race-course as they proceed
+with equal velocity in opposite directions, one row starting
+from the end of the course and the other from the middle.
+This, he thinks, involves the conclusion that half a given time
+is equal to its double. The fallacy of the reasoning lies in
+the assumption that an equal magnitude occupies an equal
+time in passing with equal velocity a magnitude that is in
+motion and a magnitude that is at rest, an assumption which
+is false.&rsquo;
+<p>Then follows a description of the process by means of
+<pb n=277><head>ZENO'S ARGUMENTS ABOUT MOTION</head>
+letters <I>A, B, C</I> the <I>exact</I> interpretation of which is a matter
+of some doubt<note>The interpretation of the passage 240 a 4-18 is elaborately discussed
+by R. K. Gaye in the <I>Journal of Philology</I>, xxxi, 1910, pp. 95-116. It is
+a question whether in the above quotation Aristotle means that Zeno
+argued that half the given time would be equal to double the half, i. e.
+the whole time simply, or to double the whole, i.e. <I>four</I> times the half.
+Gaye contends (unconvincingly, I think) for the latter.</note>; the essence of it, however, is clear. The first
+diagram below shows the original positions of the rows of
+<FIG>
+bodies (say eight in number). The <I>A</I>'s represent a row which
+is stationary, the <I>B</I>'s and <I>C</I>'s are rows which move with equal
+velocity alongside the <I>A</I>'s and one another, in the directions
+shown by the arrows. Then clearly there will be (1) a moment
+<FIG>
+when the <I>B</I>'s and <I>C</I>'s will be exactly under the respective <I>A</I>'s,
+as in the second diagram, and after that (2) a moment when
+the <I>B</I>'s and <I>C</I>'s will have exactly reversed their positions
+relatively to the <I>A</I>'s, as in the third figure.
+<FIG>
+<p>The observation has been made<note>Brochard, <I>loc. cit.</I>, pp. 4, 5.</note> that the four arguments
+form a system curiously symmetrical. The first and fourth
+consider the continuous and movement within given limits,
+the second and third the continuous and movement over
+<pb n=278><head>ZENO OF ELEA</head>
+lengths which are indeterminate. In the first and third there
+is only one moving object, and it is shown that it cannot even
+begin to move. The second and fourth, comparing the motions
+of two objects, make the absurdity of the hypothesis even
+more palpable, so to speak, for they prove that the movement,
+even if it has once begun, cannot continue, and that relative
+motion is no less impossible than absolute motion. The first
+two establish the impossibility of movement by the nature of
+space, supposed continuous, without any implication that time
+is otherwise than continuous in the same way as space; in the
+last two it is the nature of time (considered as made up of
+indivisible elements or instants) which serves to prove the
+impossibility of movement, and without any implication that
+space is not likewise made up of indivisible elements or points.
+The second argument is only another form of the first, and
+the fourth rests on the same principle as the third. Lastly, the
+first pair proceed on the hypothesis that continuous magni-
+tudes are divisible <I>ad infinitum</I>; the second pair give the
+other horn of the dilemma, being directed against the assump-
+tion that continuous magnitudes are made up of <I>indivisible</I>
+elements, an assumption which would scarcely suggest itself
+to the imagination until the difficulties connected with the
+other were fully realized. Thus the logical order of the argu-
+ments corresponds exactly to the historical order in which
+Aristotle has handed them down and which was certainly the
+order adopted by Zeno.
+<p>Whether or not the paradoxes had for Zeno the profound
+meaning now claimed for them, it is clear that they have
+been very generally misunderstood, with the result that the
+criticisms directed against them have been wide of the mark.
+Aristotle, it is true, saw that the first two arguments, the
+<I>Dichotomy</I> and the <I>Achilles</I>, come to the same thing, the latter
+differing from the former only in the fact that the ratio of
+each space traversed by Achilles to the preceding space is not
+that of 1 : 2 but a ratio of 1 : <I>n</I>, where <I>n</I> may be any number,
+however large; but, he says, both proofs rest on the fact that
+a certain moving object &lsquo;cannot reach the end of the course if
+the magnitude is divided in a certain way&rsquo;.<note>Arist. <I>Phys.</I> vi. 9, 239 b 18-24.</note> But another
+passage shows that he mistook the character of the argument
+<pb n=279><head>ZENO'S ARGUMENTS ABOUT MOTION</head>
+in the <I>Dichotomy.</I> He observes that time is divisible in
+exactly the same way as a length; if therefore a length is
+infinitely divisible, so is the corresponding time; he adds
+&lsquo;<I>this is why</I> (<G>dio/</G>) Zeno's argument falsely assumes that it is
+not possible to traverse or touch each of an infinite number of
+points in a finite time&rsquo;,<note><I>Ib.</I> vi. 2, 233 a 16-23.</note> thereby implying that Zeno did not
+regard time as divisible <I>ad infinitum</I> like space. Similarly,
+when Leibniz declares that a space divisible <I>ad infinitum</I>
+is traversed in a time divisible <I>ad infinitum</I>, he, like Aristotle,
+is entirely beside the question. Zeno was perfectly aware that,
+in respect of divisibility, time and space have the same
+property, and that they are alike, always, and concomitantly,
+divisible <I>ad infinitum.</I> The question is how, in the one as
+in the other, this series of divisions, by definition inexhaustible,
+can be exhausted; and it must be exhausted if motion is to
+be possible. It is not an answer to say that the two series
+are exhausted simultaneously.
+<p>The usual mode of refutation given by mathematicians
+from Descartes to Tannery, correct in a sense, has an analogous
+defect. To show that the sum of the infinite series <MATH>1 + 1/2 + 1/4 + ...</MATH>
+is equal to 2, or to calculate (in the <I>Achilles</I>) the exact moment
+when Achilles will overtake the tortoise, is to answer the
+question <I>when</I>? whereas the question actually asked is <I>how</I>?
+On the hypothesis of divisibility <I>ad infinitum</I> you will, in the
+<I>Dichotomy</I>, never reach the limit, and, in the <I>Achilles</I>, the
+distance separating Achilles from the tortoise, though it con-
+tinually decreases, will never vanish. And if you introduce
+the limit, or, with a numerical calculation, the discontinuous,
+Zeno is quite aware that his arguments are no longer valid.
+We are then in presence of another hypothesis as to the com-
+position of the continuum; and this hypothesis is dealt with
+in the third and fourth arguments.<note>Brochard, <I>loc. cit.</I>, p. 9.</note>
+<p>It appears then that the first and second arguments, in their
+full significance, were not really met before G. Cantor formu-
+lated his new theory of continuity and infinity. On this I
+can only refer to Chapters xlii and xliii of Mr. Bertrand
+Russell's <I>Principles of Mathematics</I>, vol. i. Zeno's argument
+in the <I>Dichotomy</I> is that, whatever motion we assume to have
+taken place, this presupposes another motion; this in turn
+<pb n=280><head>ZENO OF ELEA</head>
+another, and so on <I>ad infinitum.</I> Hence there is an endless
+regress in the mere idea of any assigned motion. Zeno's
+argument has then to be met by proving that the &lsquo;infinite
+regress&rsquo; in this case is &lsquo;harmless&rsquo;.
+<p>As regards the <I>Achilles</I>, Mr. G. H. Hardy remarks that &lsquo;the
+kernel of it lies in the perfectly valid proof which it affords
+that the tortoise passes through as many points as Achilles,
+a view which embodies an accepted doctrine of modern mathe-
+matics&rsquo;.<note><I>Encyclopaedia Britannica</I>, art. Zeno.</note>
+<p>The argument in the <I>Arrow</I> is based on the assumption that
+time is made up of <I>indivisible</I> elements or instants. Aristotle
+meets it by denying the assumption. &lsquo;For time is not made
+up of indivisible instants (<I>nows</I>), any more than any other
+magnitude is made up of indivisible elements.&rsquo; &lsquo;(Zeno's result)
+follows through assuming that time is made up of (indivisible)
+instants (<I>nows</I>); if this is not admitted, his conclusion does
+not follow.&rsquo;<note>Arist. <I>Phys.</I> vi. 9, 239 b 8, 31.</note> On the other hand, the modern view is that
+Zeno's contention is <I>true</I>: &lsquo;If&rsquo; (said Zeno) &lsquo;everything is at
+rest or in motion when it occupies a space equal to itself, and
+if what moves is always in the instant, it follows that the
+moving arrow is unmoved.&rsquo; Mr. Russell<note>Russell, <I>Principles of Mathematics</I>, i, pp. 350, 351.</note> holds that this is
+&lsquo;a very plain statement of an elementary fact&rsquo;;
+<p>&lsquo;it is a very important and very widely applicable platitude,
+namely &ldquo;Every possible value of a variable is a constant&rdquo;.
+If <I>x</I> be a variable which can take all values from 0 to 1,
+all the values it can take are definite numbers such as 1/2 or 1/3,
+which are all absolute constants ... Though a variable is
+always connected with some class, it is not the class, nor
+a particular member of the class, nor yet the whole class, but
+<I>any</I> member of the class.&rsquo; The usual <I>x</I> in algebra &lsquo;denotes
+the disjunction formed by the various members&rsquo; ... &lsquo;The
+values of <I>x</I> are then the terms of the disjunction; and each
+of these is a constant. This simple logical fact seems to
+constitute the essence of Zeno's contention that the arrow
+is always at rest.&rsquo; &lsquo;But Zeno's argument contains an element
+which is specially applicable to continua. In the case of
+motion it denies that there is such a thing as a <I>state</I> of motion.
+In the general case of a continuous variable, it may be taken
+as denying actual infinitesimals. For infinitesimals are an
+<pb n=281><head>ZENO'S ARGUMENTS ABOUT MOTION</head>
+attempt to extend to the <I>values</I> of a variable the variability
+which belongs to it alone. When once it is firmly realized
+that all the values of a variable are constants, it becomes easy
+to see, by taking <I>any</I> two such values, that their difference is
+always finite, and hence that there are no infinitesimal differ-
+ences. If <I>x</I> be a variable which may take all real values
+from 0 to 1, then, taking any two of these values, we see that
+their difference is finite, although <I>x</I> is a continuous variable.
+It is true the difference might have been less than the one we
+chose; but if it had been, it would still have been finite. The
+lower limit to possible differences is zero, but all possible
+differences are finite; and in this there is no shadow of
+contradiction. This static theory of the variable is due to the
+mathematicians, and its absence in Zeno's day led him to
+suppose that continuous change was impossible without a state
+of change, which involves infinitesimals and the contradiction
+of a body's being where it is not.&rsquo;
+<p>In his later chapter on Motion Mr. Russell concludes as
+follows:<note><I>Op. cit.</I>, p. 473.</note>
+<p>&lsquo;It is to be observed that, in consequence of the denial
+of the infinitesimal and in consequence of the allied purely
+technical view of the derivative of a function, we must
+entirely reject the notion of a <I>state</I> of motion. Motion consists
+<I>merely</I> in the occupation of different places at different times,
+subject to continuity as explained in Part V. There is no
+transition from place to place, no consecutive moment or
+consecutive position, no such thing as velocity except in the
+sense of a real number which is the limit of a certain set
+of quotients. The rejection of velocity and acceleration as
+physical facts (i. e. as properties belonging <I>at each instant</I> to
+a moving point, and not merely real numbers expressing limits
+of certain ratios) involves, as we shall see, some difficulties
+in the statement of the laws of motion; but the reform
+introduced by Weierstrass in the infinitesimal calculus has
+rendered this rejection imperative.&rsquo;
+<p>We come lastly to the fourth argument (the <I>Stadium</I>).
+Aristotle's representation of it is obscure through its extreme
+brevity of expression, and the matter is further perplexed by
+an uncertainty of reading. But the meaning intended to be
+conveyed is fairly clear. The eight <I>A</I>'s, <I>B</I>'s and <I>C</I>'s being
+<pb n=282><head>ZENO OF ELEA</head>
+initially in the position shown in Figure 1, suppose, e.g., that
+the <I>B</I>'s move to the right and the <I>C</I>'s to the left with equal
+<FIG>
+velocity until the rows are vertically under one another as in
+Figure 2. Then <I>C</I><SUB>1</SUB> has passed alongside all the eight <I>B</I>'s (and <I>B</I><SUB>1</SUB>
+<FIG>
+alongside all the eight <I>C</I>'s), while <I>B</I><SUB>1</SUB> has passed alongside only
+half the <I>A</I>'s (and similarly for <I>C</I><SUB>1</SUB>). But (Aristotle makes Zeno
+say) <I>C</I><SUB>1</SUB> <I>is the same time in passing each of the B's as it is in
+passing each of the A's.</I> It follows that the time occupied by <I>C</I><SUB>1</SUB>
+in passing all the <I>A</I>'s is the same as the time occupied by
+<I>C</I><SUB>1</SUB> in passing half the <I>A</I>'s, or a given time is equal to its half.
+Aristotle's criticism on this is practically that Zeno did not
+understand the difference between absolute and relative motion.
+This is, however, incredible, and another explanation must be
+found. The real explanation seems to be that given by
+<FIG>
+Brochard, No&euml;l and Russell. Zeno's object is to prove that
+time is not made up of indivisible elements or instants.
+Suppose the <I>B</I>'s have moved one place to the right and the <I>C</I>'s
+one place to the left, so that <I>B</I><SUB>1</SUB>, which was under <I>A</I><SUB>4</SUB>, is now
+under <I>A</I><SUB>5</SUB>, and <I>C</I><SUB>1</SUB>, which was under <I>A</I><SUB>5</SUB>, is now under <I>A</I><SUB>4</SUB>. We
+must suppose that <I>B</I><SUB>1</SUB> and <I>C</I><SUB>1</SUB> are absolute indivisible elements
+of space, and that they move to their new positions in an
+<pb n=283><head>ZENO'S ARGUMENTS ABOUT MOTION</head>
+instant, the absolute indivisible element of time; this is Zeno's
+hypothesis. But, in order that <I>B</I><SUB>1</SUB>, <I>C</I><SUB>1</SUB> may have taken up
+their new positions, there must have been a moment at which
+they crossed or <I>B</I><SUB>1</SUB> was vertically over <I>C</I><SUB>1</SUB>. Yet the motion
+has, by hypothesis, taken place in an indivisible instant.
+Therefore, either they have <I>not</I> crossed (in which case there
+is no movement), or in the particular indivisible instant two
+positions have been occupied by the two moving objects, that
+is to say, the instant is no longer indivisible. And, if the
+instant is divided into two equal parts, this, on the hypothesis
+of indivisibles, is equivalent to saying that an instant is double
+of itself.
+<p>Two remarks may be added. Though the first two argu-
+ments are directed against those who assert the divisibility <I>ad
+infinitum</I> of magnitudes and times, there is no sufficient
+justification for Tannery's contention that they were specially
+directed against a view, assumed by him to be Pythagorean,
+that bodies, surfaces and lines are made up of <I>mathematical</I>
+points. There is indeed no evidence that the Pythagoreans
+held this view at all; it does not follow from their definition
+of a point as a &lsquo;unit having position&rsquo; (<G>mona\s qe/sin e)/xousa</G>);
+and, as we have seen, Aristotle says that the Pythagoreans
+maintained that units and numbers have magnitude.<note>Arist. <I>Metaph.</I> M. 6, 1080 b 19, 32.</note>
+<p>It would appear that, after more than 2,300 years, con-
+troversy on Zeno's arguments is yet by no means at an end.
+But the subject cannot here be pursued further.<note>It is a pleasure to be able to refer the reader to a most valuable and
+comprehensive series of papers by Professor Florian Cajori, under the
+title &lsquo;The History of Zeno's arguments on Motion&rsquo;, published in the
+<I>American Mathematical Monthly</I> of 1915, and also available in a reprint.
+This work carries the history of the various views and criticisms of
+Zeno's arguments down to 1914. I may also refer to the portions of
+Bertrand Russell's work, <I>Our Knowledge of the External World as a Field
+for Scientific Method in Philosophy</I>, 1914, which deal with Zeno, and to
+Philip E. B. Jourdain's article, &lsquo;The Flying Arrow; an Anachronism&rsquo;, in
+<I>Mind</I>, January 1916, pp. 42-55.</note>
+<pb><C>IX
+PLATO</C>
+<p>IT is in the Seventh Book of the <I>Republic</I> that we find
+the most general statement of the attitude of Plato towards
+mathematics. Plato regarded mathematics in its four branches,
+arithmetic, geometry, stereometry and astronomy, as the first
+essential in the training of philosophers and of those who
+should rule his ideal State; &lsquo;let no one destitute of geometry
+enter my doors&rsquo;, said the inscription over the door of his
+school. There could be no better evidence of the supreme
+importance which he attached to the mathematical sciences.
+<p>What Plato emphasizes throughout when speaking of mathe-
+matics is its value for the training of the mind; its practical
+utility is of no account in comparison. Thus arithmetic must
+be pursued for the sake of knowledge, not for any practical
+ends such as its use in trade<note><I>Rep.</I> vii. 525 C, D.</note>; the real science of arithmetic
+has nothing to do with actions, its object is knowledge.<note><I>Politicus</I> 258 D.</note>
+A very little geometry and arithmetical calculation suffices
+for the commander of an army; it is the higher and more
+advanced portions which tend to lift the mind on high and
+to enable it ultimately to see the final aim of philosophy,
+the idea of the Good<note><I>Rep.</I> 526 D, E.</note>; the value of the two sciences consists
+in the fact that they draw the soul towards truth and create
+the philosophic attitude of mind, lifting on high the things
+which our ordinary habit would keep down.<note><I>Ib.</I> 527 B.</note>
+<p>The extent to which Plato insisted on the purely theoretical
+character of the mathematical sciences is illustrated by his
+peculiar views about the two subjects which the ordinary
+person would regard as having, at least, an important practical
+side, namely astronomy and music. According to Plato, true
+astronomy is not concerned with the movements of the visible
+<pb n=285><head>PLATO</head>
+heavenly bodies. The arrangement of the stars in the heaven
+and their apparent movements are indeed wonderful and
+beautiful, but the observation of and the accounting for them
+falls far short of true astronomy. Before we can attain to
+this we must get beyond mere observational astronomy, &lsquo;we
+must leave the heavens alone&rsquo;. The true science of astronomy
+is in fact a kind of ideal kinematics, dealing with the laws
+of motion of true stars in a sort of mathematical heaven of
+which the visible heaven is an imperfect expression in time
+and space. The visible heavenly bodies and their apparent
+motions we are to regard merely as illustrations, comparable
+to the diagrams which the geometer draws to illustrate the
+true straight lines, circles, &amp;c., about which his science reasons;
+they are to be used as &lsquo;problems&rsquo; only, with the object of
+ultimately getting rid of the apparent irregularities and
+arriving at &lsquo;the true motions with which essential speed
+and essential slowness move in relation to one another in the
+true numbers and the true forms, and carry their contents
+with them&rsquo; (to use Burnet's translation of <G>ta\ e)no/nta</G>).<note><I>Rep.</I> vii. 529 C-530 C.</note>
+&lsquo;Numbers&rsquo; in this passage correspond to the periods of the
+apparent motions; the &lsquo;true forms&rsquo; are the true orbits con-
+trasted with the apparent. It is right to add that according
+to one view (that of Burnet) Plato means, not that true
+astronomy deals with an &lsquo;ideal heaven&rsquo; different from the
+apparent, but that it deals with the true motions of the visible
+bodies as distinct from their apparent motions. This would
+no doubt agree with Plato's attitude in the <I>Laws,</I> and at the
+time when he set to his pupils as a problem for solution
+the question by what combinations of uniform circular revolu-
+tions the apparent movements of the heavenly bodies can be
+accounted for. But, except on the assumption that an ideal
+heaven is meant, it is difficult to see what Plato can mean
+by the contrast which he draws between the visible broideries
+of heaven (the visible stars and their arrangement), which
+are indeed beautiful, and the true broideries which they
+only imitate and which are infinitely more beautiful and
+marvellous.
+<p>This was not a view of astronomy that would appeal to
+the ordinary person. Plato himself admits the difficulty.
+<pb n=286><head>PLATO</head>
+When Socrates's interlocutor speaks of the use of astronomy
+for distinguishing months and seasons, for agriculture and
+navigation, and even for military purposes, Socrates rallies
+him on his anxiety that his curriculum should not consist
+of subjects which the mass of people would regard as useless:
+&lsquo;it is by no means an easy thing, nay it is difficult, to believe
+that in studying these subjects a certain organ in the mind
+of every one is purified and rekindled which is destroyed and
+blinded by other pursuits, an organ which is more worthy
+of preservation than ten thousand eyes; for by it alone is
+truth discerned.&rsquo;<note><I>Rep.</I> 527 D, E.</note>
+<p>As with astronomy, so with harmonics.<note><I>Ib.</I> 531 A-C.</note> The true science of
+harmonics differs from that science as commonly understood.
+Even the Pythagoreans, who discovered the correspondence
+of certain intervals to certain numerical ratios, still made
+their theory take too much account of audible sounds. The
+true science of harmonics should be altogether independent
+of observation and experiment. Plato agreed with the Pytha-
+goreans as to the nature of sound. Sound is due to concussion of
+air, and when there is rapid motion in the air the tone is high-
+pitched, when the motion is slow the tone is low; when the
+speeds are in certain arithmetical proportions, consonances or
+harmonies result. But audible movements produced, say, by
+different lengths of strings are only useful as illustrations;
+they are imperfect representations of those mathematical
+movements which produce mathematical consonances, and
+it is these true consonances which the true <G>a(rmoniko/s</G> should
+study.
+<p>We get on to easier ground when Plato discusses geometry.
+The importance of geometry lies, not in its practical use, but
+in the fact that it is a study of objects eternal and unchange-
+able, and tends to lift the soul towards truth. The essence
+of geometry is therefore directly opposed even to the language
+which, for want of better terms, geometers are obliged to use;
+thus they speak of &lsquo;squaring&rsquo;, &lsquo;applying (a rectangle)&rsquo;,
+&lsquo;adding&rsquo;, &amp;c., as if the object were to <I>do</I> something, whereas
+the true purpose of geometry is knowledge.<note><I>Ib.</I> vii. 526 D-527 B.</note> Geometry is
+concerned, not with material things, but with mathematical
+<pb n=287><head>PLATO</head>
+points, lines, triangles, squares, &amp;c., as objects of pure thought.
+If we use a diagram in geometry, it is only as an illustration;
+the triangle which we draw is an imperfect representation
+of the real triangle of which we think. <I>Constructions,</I> then,
+or the <I>processes</I> of squaring, adding, and so on, are not of the
+essence of geometry, but are actually antagonistic to it. With
+these views before us, we can without hesitation accept as
+well founded the story of Plutarch that Plato blamed Eudoxus,
+Archytas and Menaechmus for trying to reduce the dupli-
+cation of the cube to mechanical constructions by means of
+instruments, on the ground that &lsquo;the good of geometry is
+thereby lost and destroyed, as it is brought back to things
+of sense instead of being directed upward and grasping at
+eternal and incorporeal images&rsquo;.<note>Plutarch, <I>Quaest. Conviv.</I> viii. 2. 1, p. 718 F.</note> It follows almost inevitably
+that we must reject the tradition attributing to Plato himself
+the elegant mechanical solution of the problem of the two
+mean proportionals which we have given in the chapter on
+Special Problems (pp. 256-7). Indeed, as we said, it is certain
+on other grounds that the so-called Platonic solution was later
+than that of Eratosthenes; otherwise Eratosthenes would
+hardly have failed to mention it in his epigram, along
+with the solutions by Archytas and Menaechmus. Tannery,
+indeed, regards Plutarch's story as an invention based on
+nothing more than the general character of Plato's philosophy,
+since it took no account of the real nature of the solutions
+of Archytas and Menaechmus; these solutions are in fact
+purely theoretical and would have been difficult or impossible
+to carry out in practice, and there is no reason to doubt that
+the solution by Eudoxus was of a similar kind.<note>Tannery, <I>La g&eacute;om&eacute;trie grecque</I>, pp. 79, 80.</note> This is true,
+but it is evident that it was the practical difficulty quite as
+much as the theoretical elegance of the constructions which
+impressed the Greeks. Thus the author of the letter, wrongly
+attributed to Eratosthenes, which gives the history of the
+problem, says that the earlier solvers had all solved the
+problem in a theoretical manner but had not been able to
+reduce their solutions to practice, except to a certain small
+extent Menaechmus, and that with difficulty; and the epigram
+of Eratosthenes himself says, &lsquo;do not attempt the impracticable
+<pb n=288><head>PLATO</head>
+business of the cylinders of Archytas or the cutting of the
+cone in the three curves of Menaechmus&rsquo;. It would therefore
+be quite possible for Plato to regard Archytas and Menaechmus
+as having given constructions that were ultra-mechanical, since
+they were more mechanical than the ordinary constructions by
+means of the straight line and circle; and even the latter, which
+alone are required for the processes of &lsquo;squaring&rsquo;, &lsquo;applying
+(a rectangle)&rsquo; and &lsquo;adding&rsquo;, are according to Plato no part of
+theoretic geometry. This banning even of simple constructions
+from true geometry seems, incidentally, to make it impossible
+to accept the conjecture of Hankel that we owe to Plato the
+limitation, so important in its effect on the later development
+of geometry, of the instruments allowable in constructions to
+the ruler and compasses.<note>Hankel, <I>op. cit.,</I> p. 156.</note> Indeed, there are signs that the
+limitation began before Plato's time (e.g. this may be the
+explanation of the two constructions attributed to Oenopides),
+although no doubt Plato's influence would help to keep the
+restriction in force; for other instruments, and the use of
+curves of higher order than circles in constructions, were
+expressly barred in any case where the ruler and compasses
+could be made to serve (cf. Pappus's animadversion on a solu-
+tion of a &lsquo;plane&rsquo; problem by means of conics in Apollonius's
+<I>Conics,</I> Book V).
+<C>Contributions to the philosophy of mathematics.</C>
+<p>We find in Plato's dialogues what appears to be the first
+serious attempt at a philosophy of mathematics. Aristotle
+says that between sensible objects and the ideas Plato placed
+&lsquo;things mathematical&rsquo; (<G>ta\ maqhmatika/</G>), which differed from
+sensibles in being eternal and unmoved, but differed again
+from the ideas in that there can be many mathematical
+objects of the same kind, while the idea is one only; e.g. the
+idea of triangle is one, but there may be any number of
+mathematical triangles as of visible triangles, namely the
+perfect triangles of which the visible triangles are imper-
+fect copies. A passage in one of the <I>Letters</I> (No. 7, to the
+friends of Dion) is interesting in this connexion.<note>Plato, <I>Letters,</I> 342 B, C, 343 A, B.</note> Speaking
+of a circle by way of example, Plato says there is (1) some-
+<pb n=289><head>THE PHILOSOPHY OF MATHEMATICS</head>
+thing called a circle and known by that name; next there
+is (2) its definition as that in which the distances from its
+extremities in all directions to the centre are always equal,
+for this may be said to be the definition of that to which the
+names &lsquo;round&rsquo; and &lsquo;circle&rsquo; are applied; again (3) we have
+the circle which is drawn or turned: this circle is perishable
+and perishes; not so, however, with (4) <G>au)to\s o( ku/klos</G>, the
+essential circle, or the idea of circle: it is by reference to
+this that the other circles exist, and it is different from each
+of them. The same distinction applies to anything else, e. g.
+the straight, colour, the good, the beautiful, or any natural
+or artificial object, fire, water, &amp;c. Dealing separately with
+the four things above distinguished, Plato observes that there
+is nothing essential in (1) the name: it is merely conventional;
+there is nothing to prevent our assigning the name &lsquo;straight&rsquo;
+to what we now call &lsquo;round&rsquo; and vice versa; nor is there any
+real definiteness about (2) the definition, seeing that it too
+is made up of parts of speech, nouns and verbs. The circle
+(3), the particular circle drawn or turned, is not free from
+admixture of other things: it is even full of what is opposite
+to the true nature of a circle, for it will anywhere touch
+a straight line&rsquo;, the meaning of which is presumably that we
+cannot in practice draw a circle and a tangent with only <I>one</I>
+point common (although a mathematical circle and a mathe-
+matical straight line touching it meet in one point only). It
+will be observed that in the above classification there is no
+place given to the many particular mathematical circles which
+correspond to those which we draw, and are intermediate
+between these imperfect circles and the idea of circle which
+is one only.
+<C>(<G>a</G>) <I>The hypotheses of mathematics.</I></C>
+<p>The <I>hypotheses</I> of mathematics are discussed by Plato in
+the <I>Republic.</I>
+<p>&lsquo;I think you know that those who occupy themselves with
+geometries and calculations and the like take for granted the
+odd and the even, figures, three kinds of angles, and other
+things cognate to these in each subject; assuming these things
+as known, they take them as hypotheses and thenceforward
+they do not feel called upon to give any explanation with
+<pb n=290><head>PLATO</head>
+regard to them either to themselves or any one else, but treat
+them as manifest to every one; basing themselves on these
+hypotheses, they proceed at once to go through the rest of
+the argument till they arrive, with general assent, at the
+particular conclusion to which their inquiry was directed.
+Further you know that they make use of visible figures and
+argue about them, but in doing so they are not thinking of
+these figures but of the things which they represent; thus
+it is the absolute square and the absolute diameter which is
+the object of their argument, not the diameter which they
+draw; and similarly, in other cases, the things which they
+actually model or draw, and which may also have their images
+in shadows or in water, are themselves in turn used as
+images, the object of the inquirer being to see their abso-
+lute counterparts which cannot be seen otherwise than by
+thought.&rsquo;<note><I>Republic,</I> vi. 510 C-E.</note>
+<C>(<G>b</G>) <I>The two intellectual methods.</I></C>
+<p>Plato distinguishes two processes: both begin from hypo-
+theses. The one method cannot get above these hypotheses,
+but, treating them as if they were first principles, builds upon
+them and, with the aid of diagrams or images, arrives at
+conclusions: this is the method of geometry and mathematics
+in general. The other method treats the hypotheses as being
+really hypotheses and nothing more, but uses them as stepping-
+stones for mounting higher and higher until the principle
+of all things is reached, a principle about which there is
+nothing hypothetical; when this is reached, it is possible to
+descend again, by steps each connected with the preceding
+step, to the conclusion, a process which has no need of any
+sensible images but deals in ideas only and ends in them<note><I>Ib.</I> vi. 510 B 511 A-C.</note>;
+this method, which rises above and puts an end to hypotheses,
+and reaches the first principle in this way, is the dialectical
+method. For want of this, geometry and the other sciences
+which in some sort lay hold of truth are comparable to one
+dreaming about truth, nor can they have a waking sight of
+it so long as they treat their hypotheses as immovable
+truths, and are unable to give any account or explanation
+of them.<note><I>Ib.</I> vii. 533 B-E.</note>
+<pb n=291><head>THE TWO INTELLECTUAL METHODS</head>
+<p>With the above quotations we should read a passage of
+Proclus.
+<p>&lsquo;Nevertheless certain methods have been handed down. The
+finest is the method which by means of <I>analysis</I> carries
+the thing sought up to an acknowledged principle; a method
+which Plato, as they say, communicated to Leodamas, and by
+which the latter too is said to have discovered many things
+in geometry. The second is the method of <I>division,</I> which
+divides into its parts the genus proposed for consideration,
+and gives a starting-point for the demonstration by means of
+the elimination of the other elements in the construction
+of what is proposed, which method also Plato extolled as
+being of assistance to all sciences.&rsquo;<note>Proclus, <I>Comm. on Eucl.</I> I, pp. 211. 18-212. 1.</note>
+<p>The first part of this passage, with a like dictum in Diogenes
+Laertius that Plato &lsquo;explained to Leodamas of Thasos the
+method of inquiry by analysis&rsquo;,<note>Diog. L. iii. 24, p. 74, Cobet.</note> has commonly been under-
+stood as attributing to Plato the <I>invention</I> of the method
+of mathematical analysis. But, analysis being according to
+the ancient view nothing more than a series of successive
+reductions of a theorem or problem till it is finally reduced
+to a theorem or problem already known, it is difficult to
+see in what Plato's supposed discovery could have consisted;
+for analysis in this sense must have been frequently used
+in earlier investigations. Not only did Hippocrates of Chios
+reduce the problem of duplicating the cube to that of finding
+two mean proportionals, but it is clear that the method of
+analysis in the sense of reduction must have been in use by
+the Pythagoreans. On the other hand, Proclus's language
+suggests that what he had in mind was the philosophical
+method described in the passage of the <I>Republic,</I> which of
+course does not refer to mathematical analysis at all; it may
+therefore well be that the idea that Plato discovered the
+method of analysis is due to a misapprehension. But analysis
+and synthesis following each other are related in the same
+way as the upward and downward progressions in the dialec-
+tician's intellectual method. It has been suggested, therefore,
+that Plato's achievement was to observe the importance
+from the point of view of logical rigour, of the confirma-
+tory synthesis following analysis. The method of <I>division</I>
+<pb n=292><head>PLATO</head>
+mentioned by Proclus is the method of successive bipartitions
+of genera into species such as we find in the <I>Sophist</I> and
+the <I>Politicus,</I> and has little to say to geometry; but the
+mention of it side by side with analysis itself suggests that
+Proclus confused the latter with the philosophical method
+referred to.
+<C>(<G>g</G>) <I>Definitions.</I></C>
+<p>Among the fundamentals of mathematics Plato paid a good
+deal of attention to definitions. In some cases his definitions
+connect themselves with Pythagorean tradition; in others he
+seems to have struck out a new line for himself. The division
+of numbers into odd and even is one of the most common of
+his illustrations; number, he says, is divided equally, i. e.
+there are as many odd numbers as even, and this is the true
+division of number; to divide number (e. g.) into myriads and
+what are not myriads is not a proper division.<note><I>Politicus,</I> 262 D, E.</note> An even
+number is defined as a number divisible into two equal parts<note><I>Laws,</I> 895 E.</note>;
+in another place it is explained as that which is not scalene
+but isosceles<note><I>Euthyphro,</I> 12 D.</note>: a curious and apparently unique application
+of these terms to number, and in any case a defective state-
+ment unless the term &lsquo;scalene&rsquo; is restricted to the case in which
+one part of the number is odd and the other even; for of
+course an even number can be divided into two unequal odd
+numbers or two unequal even numbers (except 2 in the first
+case and 2 and 4 in the second). The further distinction
+between even-times-even, odd-times-even, even-times-odd and
+odd-times-odd occurs in Plato<note><I>Parmenides,</I> 143 E-144 A.</note>: but, as thrice two is called
+odd-times-even and twice three is even-times-odd, the number
+in both cases being the same, it is clear that, like Euclid,
+Plato regarded even-times-odd and odd-times-even as con-
+vertible terms, and did not restrict their meaning in the way
+that Nicomachus and the neo-Pythagoreans did.
+<p>Coming to geometry we find an interesting view of the
+term &lsquo;figure&rsquo;. What is it, asks Socrates, that is true of the
+round, the straight, and the other things that you call figures,
+and is the same for all? As a suggestion for a definition
+of &lsquo;figure&rsquo;, Socrates says, &lsquo;let us regard as <I>figure</I> that which
+alone of existing things is associated with colour&rsquo;. Meno
+<pb n=293><head>DEFINITIONS</head>
+asks what is to be done if the interlocutor says he does not
+know what colour is; what alternative definition is there?
+Socrates replies that it will be admitted that in geometry
+there are such things as what we call a surface or a solid,
+and so on; from these examples we may learn what we mean
+by figure; figure is that in which a solid ends, or figure is
+the limit (or extremity, <G>pe/ras</G>) of a solid.<note><I>Meno,</I> 75 A-76 A.</note> Apart from
+&lsquo;figure&rsquo; as form or shape, e.g. the round or straight, this
+passage makes &lsquo;figure&rsquo; practically equivalent to surface, and
+we are reminded of the Pythagorean term for surface, <G>xroia/</G>,
+colour or skin, which Aristotle similarly explains as <G>xrw=ma</G>,
+colour, something inseparable from <G>pe/ras</G>, extremity.<note>Arist. <I>De sensu,</I> 439 a 31, &amp;c.</note> In
+Euclid of course <G>o(/ros</G>, limit or boundary, is defined as the
+extremity (<G>pe/ras</G>) of a thing, while &lsquo;figure&rsquo; is that which is
+contained by one or more boundaries.
+<p>There is reason to believe, though we are not specifically
+told, that the definition of a line as &lsquo;breadthless length&rsquo;
+originated in the Platonic School, and Plato himself gives
+a definition of a straight line as &lsquo;that of which the middle
+covers the ends&rsquo;<note><I>Parmenides,</I> 137 E.</note> (i. e. to an eye placed at either end and
+looking along the straight line); this seems to me to be the
+origin of the Euclidean definition &lsquo;a line which lies evenly
+with the points on it&rsquo;, which, I think, can only be an attempt
+to express the sense of Plato's definition in terms to which
+a geometer could not take exception as travelling outside the
+subject matter of geometry, i. e. in terms excluding any appeal
+to vision. A <I>point</I> had been defined by the Pythagoreans as
+a &lsquo;monad having position&rsquo;; Plato apparently objected to this
+definition and substituted no other; for, according to Aristotle,
+he regarded the genus of points as being a &lsquo;geometrical
+fiction&rsquo;, calling a point the beginning of a line, and often using
+the term &lsquo;indivisible lines&rsquo; in the same sense.<note>Arist. <I>Metaph.</I> A. 9, 992 a 20.</note> Aristotle
+points out that even indivisible lines must have extremities,
+and therefore they do not help, while the definition of a point
+as &lsquo;the extremity of a line&rsquo; is unscientific.<note>Arist. <I>Topics,</I> vi. 4, 141 b 21.</note>
+<p>The &lsquo;round&rsquo; (<G>stroggu/lon</G>) or the circle is of course defined
+as &lsquo;that in which the furthest points (<G>ta\ e)/sxata</G>) in all
+<pb n=294><head>PLATO</head>
+directions are at the same distance from the middle (centre)&rsquo;.<note><I>Parmenides,</I> 137 E.</note>
+The &lsquo;sphere&rsquo; is similarly defined as &lsquo;that which has the
+distances from its centre to its terminations or ends in every
+direction equal&rsquo;, or simply as that which is &lsquo;equal (equidistant)
+from the centre in all directions&rsquo;.<note><I>Timaeus,</I> 33 B, 34 B.</note>
+<p>The <I>Parmenides</I> contains certain phrases corresponding to
+what we find in Euclid's preliminary matter. Thus Plato
+speaks of something which is &lsquo;a part&rsquo; but not &lsquo;parts&rsquo; of the
+One,<note><I>Parmenides,</I> 153 D.</note> reminding us of Euclid's distinction between a fraction
+which is &lsquo;a part&rsquo;, i. e. an aliquot part or submultiple, and one
+which is &lsquo;parts&rsquo;, i. e. some number more than one of such
+parts, e. g. 3/7. If equals be added to unequals, the sums differ
+by the same amount as the original unequals did:<note><I>Ib.</I> 154 B.</note> an axiom
+in a rather more complete form than that subsequently inter-
+polated in Euclid.
+<C>Summary of the mathematics in Plato.</C>
+<p>The actual arithmetical and geometrical propositions referred
+to or presupposed in Plato's writings are not such as to suggest
+that he was in advance of his time in mathematics; his
+knowledge does not appear to have been more than up to
+date. In the following paragraphs I have attempted to give
+a summary, as complete as possible, of the mathematics con-
+tained in the dialogues.
+<p>A proposition in proportion is quoted in the <I>Parmenides,</I><note><I>Ib.</I> 154 D.</note>
+namely that, if <I>a</I> > <I>b</I>, then <MATH>(<I>a</I>+<I>c</I>):(<I>b</I>+<I>c</I>)<<I>a</I>:<I>b</I></MATH>.
+<p>In the <I>Laws</I> a certain number, 5,040, is selected as a most
+convenient number of citizens to form a state; its advantages
+are that it is the product of 12, 21 and 20, that a twelfth
+part of it is again divisible by 12, and that it has as many as
+59 different divisors in all, including all the natural numbers
+from 1 to 12 with the exception of 11, while it is nearly
+divisible by 11 (5038 being a multiple of 11).<note><I>Laws,</I> 537 E-538 A.</note>
+<C>(<G>a</G>) <I>Regular and semi-regular solids.</I></C>
+<p>The &lsquo;so-called Platonic figures&rsquo;, by which are meant the
+five regular solids, are of course not Plato's discovery, for they
+had been partly investigated by the Pythagoreans, and very
+<pb n=295><head>REGULAR AND SEMI-REGULAR SOLIDS</head>
+fully by Theaetetus; they were evidently only called Platonic
+because of the use made of them in the <I>Timaeus,</I> where the
+particles of the four elements are given the shapes of the first
+four of the solids, the pyramid or tetrahedron being appro-
+priated to fire, the octahedron to air, the icosahedron to water,
+and the cube to earth, while the Creator used the fifth solid,
+the dodecahedron, for the universe itself.<note><I>Timaeus,</I> 55 D-56 B, 55 C.</note>
+<p>According to Heron, however, Archimedes, who discovered
+thirteen semi-regular solids inscribable in a sphere, said that
+<p>&lsquo;Plato also knew one of them, the figure with fourteen faces,
+of which there are two sorts, one made up of eight triangles
+and six squares, of earth and air, and already known to some
+of the ancients, the other again made up of eight squares and
+six triangles, which seems to be more difficult.&rsquo;<note>Heron, <I>Definitions,</I> 104, p. 66, Heib.</note>
+<p>The first of these is easily obtained; if we take each square
+face of a cube and make in it a smaller square by joining
+the middle points of each pair of consecutive sides, we get six
+squares (one in each face); taking the three out of the twenty-
+four sides of these squares which are about any one angular
+point of the cube, we have an equilateral triangle; there are
+eight of these equilateral triangles, and if we cut off-from the
+corners of the cube the pyramids on these triangles as bases,
+<FIG>
+we have a semi-regular polyhedron
+inscribable in a sphere and having
+as faces eight equilateral triangles
+and six squares. The description of
+the second semi-regular figure with
+fourteen faces is wrong: there are
+only two more such figures, (1) the
+figure obtained by cutting off from
+the corners of the cube smaller
+pyramids on equilateral triangular bases such that regular
+<I>octagons,</I> and not squares, are left in the six square faces,
+the figure, that is, contained by eight triangles and six
+octagons, and (2) the figure obtained by cutting off from the
+corners of an <I>octahedron</I> equal pyramids with square bases
+such as to leave eight regular hexagons in the eight faces,
+that is, the figure contained by six squares and eight hexagons.
+<pb n=296><head>PLATO</head>
+<C>(<G>b</G>) <I>The construction of the regular solids.</I></C>
+<p>Plato, of course, constructs the regular solids by simply
+putting together the plane faces. These faces are, he observes,
+made up of triangles; and all triangles are decomposable into
+two right-angled triangles. Right-angled triangles are either
+(1) isosceles or (2) not isosceles, having the two acute angles
+unequal. Of the latter class, which is unlimited in number,
+one triangle is the most beautiful, that in which the square on
+the perpendicular is triple of the square on the base (i. e. the
+triangle which is the half of an equilateral triangle obtained
+by drawing a perpendicular from a vertex on the opposite
+side). (Plato is here Pythagorizing.<note>Cf. Speusippus in <I>Theol. Ar.,</I> p. 61, Ast.</note>) One of the regular
+solids, the cube, has its faces (squares) made up of the first
+<FIG>
+kind of right-angled triangle, the isosceles, four of
+them being put together to form the square; three
+others with equilateral triangles for faces, the tetra-
+hedron, octahedron and icosahedron, depend upon
+the other species of right-angled triangle only,
+each face being made up of six (not two) of those right-angled
+triangles, as shown in the figure; the fifth solid, the dodeca-
+<FIG>
+hedron, with twelve regular pentagons for
+faces, is merely alluded to, not described, in
+the passage before us, and Plato is aware that
+its faces cannot be constructed out of the two
+elementary right-angled triangles on which the
+four other solids depend. That an attempt was made to divide
+the pentagon into a number of triangular elements is clear
+<FIG>
+from three passages, two in Plutarch<note>Plutarch, <I>Quaest. Plat.</I> 5. 1, 1003 D; <I>De defectu Oraculorum,</I> c. 33, 428 A.</note>
+and one in Alcinous.<note>Alcinous, <I>De Doctrina Platonis,</I> c. 11.</note> Plutarch says
+that each of the twelve faces of a
+dodecahedron is made up of thirty
+elementary scalene triangles which are
+different from the elementary triangle
+of the solids with triangular faces.
+Alcinous speaks of the 360 elements
+which are produced when each pen-
+tagon is divided into five isosceles triangles and each of the
+<pb n=297><head>THE REGULAR SOLIDS</head>
+latter into six scalene triangles. If we draw lines in a pen-
+tagon as shown in the accompanying figure, we obtain such
+a set of triangles in a way which also shows the Pythagorean
+pentagram (cf. p. 161, above).
+<C>(<G>g</G>) <I>Geometric means between two square numbers
+or two cubes.</I></C>
+<p>In the <I>Timaeus</I> Plato, speaking of numbers &lsquo;whether solid
+or square&rsquo; with a (geometric) mean or means between them,
+observes that between <I>planes</I> one mean suffices, but to connect
+two <I>solids</I> two means are necessary.<note><I>Timaeus,</I> 31 C-32 B.</note> By <I>planes</I> and <I>solids</I>
+Plato probably meant <I>square</I> and <I>cube numbers</I> respectively,
+so that the theorems quoted are probably those of Eucl. VIII.
+11, 12, to the effect that between two square numbers there is
+one mean proportional number, and between two cube numbers
+two mean proportional numbers. Nicomachus quotes these
+very propositions as constituting &lsquo;a certain Platonic theorem&rsquo;.<note>Nicom. ii. 24. 6.</note>
+Here, too, it may be that the theorem is called &lsquo;Platonic&rsquo; for
+the sole reason that it is quoted by Plato in the <I>Timaeus</I>;
+it may well be older, for the idea of two mean proportionals
+between two straight lines had already appeared in Hippo-
+crates's reduction of the problem of doubling the cube. Plato's
+allusion does not appear to be to the duplication of the cube
+in this passage any more than in the expression <G>ku/bwn au)/xh</G>,
+&lsquo;cubic increase&rsquo;, in the <I>Republic,</I><note><I>Republic,</I> 528 B.</note> which appears to be nothing
+but the addition of the third dimension to a square, making
+a cube (cf. <G>tri/th au)/xh</G>, &lsquo;third increase&rsquo;,<note><I>Ib.</I> 587 D.</note> meaning a cube
+number as compared with <G>du/namis</G>, a square number, terms
+which are applied, e. g. to the numbers 729 and 81 respec-
+tively).
+<C>(<G>d</G>) <I>The two geometrical passages in the</I> MENO.</C>
+<p>We come now to the two geometrical passages in the <I>Meno.</I>
+In the first<note><I>Meno,</I> 82 B-85 B.</note> Socrates is trying to show that teaching is only
+reawaking in the mind of the learner the memory of some-
+thing. He illustrates by putting to the slave a carefully
+prepared series of questions, each requiring little more than
+<pb n=298><head>PLATO</head>
+&lsquo;yes&rsquo; or &lsquo;no&rsquo; for an answer, but leading up to the geometrical
+construction of &radic;2. Starting with a straight line <I>AB</I> 2 feet
+long, Socrates describes a square <I>ABCD</I> upon it and easily
+shows that the area is 4 square feet. Producing the sides
+<I>AB, AD</I> to <I>G, K</I> so that <I>BG, DK</I> are equal to <I>AB, AD,</I> and
+completing the figure, we have a square of side 4 feet, and this
+square is equal to four times the original square and therefore
+has an area of 16 square feet. Now, says Socrates, a square
+8 feet in area must have its side
+<FIG>
+greater than 2 and less than 4 feet.
+The slave suggests that it is 3 feet
+in length. By taking <I>N</I> the
+middle point of <I>DK</I> (so that <I>AN</I>
+is 3 feet) and completing the square
+on <I>AN,</I> Socrates easily shows that
+the square on <I>AN</I> is not 8 but 9
+square feet in area. If <I>L, M</I> be
+the middle points of <I>GH, HK</I> and
+<I>CL, CM</I> be joined, we have four
+squares in the figure, one of which is <I>ABCD,</I> while each of the
+others is equal to it. If now we draw the diagonals <I>BL, LM,
+MD, DB</I> of the four squares, each diagonal bisects its square,
+and the four make a square <I>BLMD,</I> the area of which is half
+that of the square <I>AGHK,</I> and is therefore 8 square feet;
+<I>BL</I> is a side of this square. Socrates concludes with the
+words:
+<p>&lsquo;The Sophists call this straight line (<I>BD</I>) the <I>diameter</I>
+(diagonal); this being its name, it follows that the square
+which is double (of the original square) has to be described on
+the diameter.&rsquo;
+<p>The other geometrical passage in the <I>Meno</I> is much more
+difficult,<note><I>Meno,</I> 86 E-87 C.</note> and it has gathered round it a literature almost
+comparable in extent to the volumes that have been written
+to explain the Geometrical Number of the <I>Republic.</I> C. Blass,
+writing in 1861, knew thirty different interpretations; and
+since then many more have appeared. Of recent years
+Benecke's interpretation<note>Dr. Adolph Benecke, <I>Ueber die geometrische Hypothesis in Platon's
+Menon</I> (Elbing, 1867). See also below, pp. 302-3.</note> seems to have enjoyed the most
+<pb n=299><head>TWO GEOMETRICAL PASSAGES IN THE <I>MENO</I></head>
+acceptance; nevertheless, I think that it is not the right one,
+but that the essentials of the correct interpretation were given
+by S. H. Butcher<note><I>Journal of Philology,</I> vol. xvii, pp. 219-25; cf. E. S. Thompson's edition
+of the <I>Meno.</I></note> (who, however, seems to have been com-
+pletely anticipated by E. F. August, the editor of Euclid, in
+1829). It is necessary to begin with a literal translation of
+the passage. Socrates is explaining a procedure &lsquo;by way
+of hypothesis&rsquo;, a procedure which, he observes, is illustrated
+by the practice of geometers
+<p>&lsquo;when they are asked, for example, as regards a given area,
+whether it is possible for this area to be inscribed in the form
+of a triangle in a given circle. The answer might be, &ldquo;I do
+not yet know whether this area is such as can be so inscribed,
+but I think I can suggest a hypothesis which will be useful for
+the purpose; I mean the following. If the given area is such
+as, when one has applied it (as a rectangle) to the given
+straight line in the circle [<G>th\n doqei=san au)tou= grammh/n</G>, the
+given straight line <I>in it,</I> cannot, I think, mean anything
+but the <I>diameter</I> of the circle<note>The obvious &lsquo;line&rsquo; of a circle is its diameter, just as, in the first
+geometrical passage about the squares, the <G>grammh/</G>, the &lsquo;line&rsquo;, of a square
+is its <I>side.</I></note>], it is deficient by a figure
+(rectangle) similar to the very figure which is applied, then
+one alternative seems to me to result, while again another
+results if it is impossible for what I said to be done with it.
+Accordingly, by using a hypothesis, I am ready to tell you what
+results with regard to the inscribing of the figure in the circle,
+namely, whether the problem is possible or impossible.&rdquo;&rsquo;
+<p>Let <I>AEB</I> be a circle on <I>AB</I> as diameter, and let <I>AC</I> be the
+tangent at <I>A.</I> Take <I>E</I> any point on the circle and draw
+<I>ED</I> perpendicular to <I>AB.</I> Complete the rectangles <I>ACED,
+EDBF.</I>
+<p>Then it is clear that the rectangle <I>CEDA</I> is &lsquo;applied&rsquo; to
+the diameter <I>AB,</I> and also that it &lsquo;falls short&rsquo; by a figure, the
+rectangle <I>EDBF,</I> similar to the &lsquo;applied&rsquo; rectangle, for
+<MATH><I>AD</I>:<I>DE</I> = <I>ED</I>:<I>DB</I></MATH>.
+<p>Also, if <I>ED</I> be produced to meet the circle again in <I>G,
+AEG</I> is an isosceles triangle bisected by the diameter <I>AB,</I>
+and therefore equal in area to the rectangle <I>ACED.</I>
+<p>If then the latter rectangle, &lsquo;applied&rsquo; to <I>AB</I> in the manner
+<pb n=300><head>PLATO</head>
+described, is equal to the given area, that area is inscribed in
+the form of a triangle in the given circle.<note>Butcher, after giving the essentials of the interpretation of the
+passage quite correctly, finds a difficulty. &lsquo;If&rsquo;, he says, &lsquo;the condition&rsquo;
+(as interpreted by him) &lsquo;holds good, the given <G>xwri/on</G> can be inscribed in
+a circle. But the converse proposition is not true. The <G>xwri/on</G> can still
+be inscribed, as required, even if the condition laid down is not fulfilled;
+the true and necessary condition being that the given area is not greater
+than that of the equilateral triangle, i. e. the <I>maximum</I> triangle, which
+can be inscribed in the given circle.&rsquo; The difficulty arises in this way.
+Assuming (quite fairly) that the given area is given in the form of a rect-
+angle (for any given rectilineal figure can be transformed into a rectangle
+of equal area), Butcher seems to suppose that it is identically the given
+rectangle that is applied to <I>AB.</I> But this is not necessary. The termi-
+nology of mathematics was not quite fixed in Plato's time, and he allows
+himself some latitude of expression, so that we need not be surprised to
+find him using the phrase &lsquo;to apply the area (<G>xwri/on</G>) to a given straight
+line&rsquo; as short for &lsquo;to apply to a given straight line a <I>rectangle equal</I> (but not
+similar) to the given area&rsquo; (cf. Pappus vi, p. 544. 8-10 <G>mh\ pa=n to\ doqe\n
+para\ th\n doqei=san paraba/llesqai e)llei=pon tetragw/nw|</G>, &lsquo;that it is not every
+given (area) that can be applied (in the form of a rectangle) falling short
+by a square figure&rsquo;). If we interpret the expression in this way, the
+converse <I>is</I> true; if we cannot apply, in the way described, a rectangle
+<I>equal</I> to the given rectangle, it is because the given rectangle is greater
+than the equilateral, i. e. the maximum, triangle that can be inscribed in
+the circle, and the problem is therefore impossible of solution. (It was
+not till long after the above was written that my attention was drawn to
+the article on the same subject in the <I>Journal of Philology,</I> xxviii, 1903,
+pp. 222-40, by Professor Cook Wilson. I am gratified to find that my
+interpretation of the passage agrees with his.)</note>
+<p>In order, therefore, to inscribe in the circle an isosceles
+triangle equal to the given area (<I>X</I>), we have to find a point <I>E</I>
+on the circle such that, if <I>ED</I> be drawn perpendicular to <I>AB,</I>
+<FIG>
+the rectangle <I>AD. DE</I> is equal to the given area <I>X</I> (&lsquo;applying&rsquo;
+to <I>AB</I> a rectangle equal to <I>X</I> and falling short by a figure
+similar to the &lsquo;applied&rsquo; figure is only another way of ex-
+pressing it). Evidently <I>E</I> lies on the rectangular hyperbola
+<pb n=301><head>TWO GEOMETRICAL PASSAGES IN THE <I>MENO</I></head>
+the equation of which referred to <I>AB, AC</I> as axes of <I>x, y</I> is
+<MATH><I>xy</I> = <I>b</I><SUP>2</SUP></MATH>, where <I>b</I><SUP>2</SUP> is equal to the given area. For a real
+solution it is necessary that <I>b</I><SUP>2</SUP> should be not greater than the
+equilateral triangle inscribed in the circle, i. e. not greater than
+<MATH>3 &radic;3.<I>a</I><SUP>2</SUP>/4</MATH>, where <I>a</I> is the radius of the circle. If <I>b</I><SUP>2</SUP> is equal
+to this area, there is only one solution (the hyperbola in that
+case touching the circle); if <I>b</I><SUP>2</SUP> is less than this area, there are
+two solutions corresponding to two points <I>E, E</I>&prime; in which the
+hyperbola cuts the circle. If <MATH><I>AD</I> = <I>x</I></MATH>, we have <MATH><I>OD</I> = <I>x-a</I></MATH>,
+<MATH><I>DE</I> = &radic;(2 <I>ax</I>-<I>x</I><SUP>2</SUP>)</MATH>, and the problem is the equivalent of
+solving the equation
+<MATH><I>x</I>&radic;(2 <I>ax</I>-<I>x</I><SUP>2</SUP>) = <I>b</I><SUP>2</SUP></MATH>,
+or
+<MATH><I>x</I><SUP>2</SUP> (2 <I>ax</I>-<I>x</I><SUP>2</SUP>) = <I>b</I><SUP>4</SUP></MATH>.
+<p>This is an equation of the fourth degree which can be solved
+by means of conics but not by means of the straight line
+and circle. The solution is given by the points of intersec-
+tion of the hyperbola <MATH><I>xy</I> = <I>b</I><SUP>2</SUP></MATH> and the circle <MATH><I>y</I><SUP>2</SUP> = 2 <I>ax</I>-<I>x</I><SUP>2</SUP></MATH> or
+<MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP> = 2 <I>ax</I></MATH>. In this respect therefore the problem is like
+that of finding the two mean proportionals, which was likewise
+solved, though not till later, by means of conics (Menaechmus).
+I am tempted to believe that we have here an allusion to
+another actual problem, requiring more than the straight
+line and circle for its solution,
+<FIG>
+which had exercised the minds
+of geometers by the time of
+Plato, the problem, namely, of
+inscribing in a circle a triangle
+equal to a given area, a problem
+which was still awaiting a
+solution, although it had been
+reduced to the problem of
+applying a rectangle satisfying the condition described by
+Plato, just as the duplication of the cube had been reduced
+to the problem of finding two mean proportionals. Our
+problem can, like the latter problem, easily be solved by the
+&lsquo;mechanical&rsquo; use of a ruler. Suppose that the given rectangle
+is placed so that the side <I>AD</I> lies along the diameter <I>AB</I> of
+the circle. Let <I>E</I> be the angle of the rectangle <I>ADEC</I> opposite
+to <I>A.</I> Place a ruler so that it passes through <I>E</I> and turn
+<pb n=302><head>PLATO</head>
+it about <I>E</I> until it passes through a point <I>P</I> of the circle such
+that, if <I>EP</I> meets <I>AB</I> and <I>AC</I> produced in <I>T, R, PT</I> shall be
+equal to <I>ER.</I> Then, since <MATH><I>RE</I> = <I>PT, AD</I> = <I>MT</I></MATH>, where <I>M</I> is
+the foot of the ordinate <I>PM.</I>
+<p>Therefore <MATH><I>DT</I> = <I>AM</I></MATH>, and
+<MATH><I>AM</I>:<I>AD</I> = <I>DT</I>:<I>MT</I>
+= <I>ED</I>:<I>PM</I></MATH>,
+whence <MATH><I>PM.MA</I> = <I>ED.DA</I></MATH>,
+and <I>APM</I> is the half of the required (isosceles) triangle.
+<p>Benecke criticizes at length the similar interpretation of the
+passage given by E. F. August. So far, however, as his objec-
+tions relate to the translation of particular words in the
+Greek text, they are, in my opinion, not well founded.<note>The main point of Benecke's criticisms under this head has reference
+to <G>toiou/tw| xwri/w| oi=(on</G> in the phrase <G>e)llei/pein toiou/tw| xwri/w| oi=(on a)\n au)to\ to\
+parat tame/non h=)|</G>. He will have it that <G>toiou/tw| oi=(on</G> cannot mean &lsquo;similar to&rsquo;,
+and he maintains that, if Plato had meant it in this sense, he should
+have added that the &lsquo;defect&rsquo;, although &lsquo;similar&rsquo;, is not similarly situated.
+I see no force in this argument in view of the want of fixity in mathe-
+matical terminology in Plato's time, and of his own habit of varying his
+phrases for literary effect. Benecke makes the words mean &lsquo;of the same
+<I>kind</I>&rsquo;, e. g. a square with a square or a rectangle with a rectangle. But
+this would have no point unless the figures are <I>squares,</I> which begs the
+whole question.</note> For
+the rest, Benecke holds that, in view of the difficulty of the
+problem which emerges, Plato is unlikely to have introduced
+it in such an abrupt and casual way into the conversation
+between Socrates and Meno. But the problem is only one
+of the same nature as that of the finding of two mean
+proportionals which was already a famous problem, and, as
+regards the form of the allusion, it is to be noted that Plato
+was fond of dark hints in things mathematical.
+<p>If the above interpretation is too difficult (which I, for one,
+do not admit), Benecke's is certainly too easy. He connects
+his interpretation of the passage with the earlier passage
+about the square of side 2 feet; according to him the problem
+<FIG>
+is, can an isosceles <I>right-angled</I> tri-
+angle equal to the said square be
+inscribed in the given circle? This
+is of course only possible if the
+radius of the circle is 2 feet in length.
+If <I>AB, DE</I> be two diameters at right
+angles, the inscribed triangle is <I>ADE</I>;
+the square <I>ACDO</I> formed by the radii
+<I>AO, OD</I> and the tangents at <I>D, A</I>
+is then the &lsquo;applied&rsquo; rectangle, and
+the rectangle by which it falls short is also a square and equal
+<pb n=303><head>TWO GEOMETRICAL PASSAGES IN THE <I>MENO</I></head>
+to the other square. If this were the correct interpretation,
+Plato is using much too general language about the applied
+rectangle and that by which it is deficient; it would be
+extraordinary that he should express the condition in this
+elaborate way when he need only have said that the radius
+of the circle must be equal to the side of the square and
+therefore 2 feet in length. The explanation seems to me
+incredible. The criterion sought by Socrates is evidently
+intended to be a real <G>diorismo/s</G>, or determination of the
+conditions or limits of the possibility of a solution of the pro-
+blem whether in its original form or in the form to which
+it is reduced; but it is no real <G>diorismo/s</G> to say what is
+equivalent to saying that the problem is possible of solution
+if the circle is of a particular size, but impossible if the circle
+is greater or less than that size.
+<p>The passage incidentally shows that the idea of a formal
+<G>diorismo/s</G> defining the limits of possibility of solution was
+familiar even before Plato's time, and therefore that Proclus
+must be in error when he says that Leon, the pupil of
+Neoclides, &lsquo;<I>invented</I> <G>diorismoi/</G> (determining) when the problem
+which is the subject of investigation is possible and when
+impossible&rsquo;,<note>Proclus on Eucl. I, p. 66. 20-2.</note> although Leon may have been the first to intro-
+duce the term or to recognize formally the essential part
+played by <G>diorismoi/</G> in geometry.
+<C>(<G>e</G>) <I>Plato and the doubling of the cube.</I></C>
+<p>The story of Plato's relation to the problem of doubling
+the cube has already been told (pp. 245-6, 255). Although the
+solution attributed to him is not his, it may have been with
+this problem in view that he complained that the study of
+solid geometry had been unduly neglected up to his time.<note><I>Republic</I>, vii. 528 A-C.</note>
+<pb n=304><head>PLATO</head>
+<C>(<G>z</G>) <I>Solution of <MATH><I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>=<I>z</I><SUP>2</SUP></MATH> in integers</I></C>.
+<p>We have already seen (p. 81) that Plato is credited with
+a rule (complementary to the similar rule attributed to Pytha-
+goras) for finding a whole series of square numbers the sum
+of which is also a square; the formula is
+<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>.
+<C>(<G>h</G>) <I>Incommensurables</I>.</C>
+<p>On the subject of incommensurables or irrationals we have
+first the passage of the <I>Theaetetus</I> recordin that Theodorus
+proved the incommensurability of &radic;3, &radic;5 ... &radic;17, after
+which Theaetetus generalized the theory of such &lsquo;roots&rsquo;.
+This passage has already been fully discussed (pp. 203-9).
+The subject of incommensurables comes up again in the <I>Laws,</I>
+where Plato inveighs against the ignorance prevailing among
+the Greeks of his time of the fact that lengths, breadths and
+depths may be incommensurable as well as commensurable
+with one another, and appears to imply that he himself had
+not learnt the fact till late (<G>a)kou/sas o)ye/ pote</G>), so that he
+was ashamed for himself as well as for his countrymen in
+general.<note><I>Laws,</I> 819 D-820 C.</note> But the irrationals known to Plato included more
+than mere &lsquo;surds&rsquo; or the sides of non-squares; in one place
+he says that, just as an even number may be the sum of
+either two odd or two even numbers, the sum of two irra-
+tionals may be either rational or irrational.<note><I>Hippias Maior,</I> 303 B, C.</note> An obvious
+illustration of the former case is afforded by a rational straight
+line divided &lsquo;in extreme and mean ratio&rsquo;. Euclid (XIII. 6)
+proves that each of the segments is a particular kind of
+irrational straight line called by him in Book X an <I>apotome</I>;
+and to suppose that the irrationality of the two segments was
+already known to Plato is natural enough if we are correct in
+supposing that &lsquo;the theorems which&rsquo; (in the words of Proclus)
+&lsquo;Plato originated regarding <I>the section</I>&rsquo;<note>Proclus on Eucl. I, p. 67. 6.</note> were theorems about
+what came to be called the &lsquo;golden section&rsquo;, namely the
+division of a straight line in extreme and mean ratio as in
+Eucl. II. 11 and VI. 30. The appearance of the latter problem
+in Book II, the content of which is probably all Pythagorean,
+suggests that the incommensurability of the segments with
+<pb n=305><head>INCOMMENSURABLES</head>
+the whole line was discovered before Plato's time, if not as
+early as the irrationality of &radic;2.
+<C>(<G>q</G>) <I>The Geometrical Number</I>.</C>
+<p>This is not the place to discuss at length the famous passage
+about the Geometrical Number in the <I>Republic.</I><note><I>Republic,</I> viii. 546 B-D. The number of interpretations of this passage
+is legion. For an exhaustive discussion of the language as well as for
+one of the best interpretations that has been put forward, see Dr. Adam's
+edition of the <I>Republic,</I> vol. ii, pp. 204-8, 264-312.</note> Nor is its
+mathematical content of importance; the whole thing is
+mystic rather than mathematical, and is expressed in
+rhapsodical language, veiling by fanciful phraseology a few
+simple mathematical conceptions. The numbers mentioned
+are supposed to be two. Hultsch and Adam arrive at the
+same two numbers, though by different routes. The first
+of these numbers is 216, which according to Adam is the sum
+of three cubes 3<SUP>3</SUP>+4<SUP>3</SUP>+5<SUP>3</SUP>; 2<SUP>3</SUP>.3<SUP>3</SUP> is the form in which
+Hultsch obtains it.<note>The Greek is <G>e)n w=( prw/tw| au)xh/seis duna/menai/ te kai\ dunasteuo/menai, trei=s
+a)posta/seis, te/ttaras de\ o(/rous labou=sai o(moiou/ntwn te kai\ a)nomoiou/ntwn kai\
+au)xo/ntwn kai\ fqino/ntwn, pa/nta prosh/gora kai\ r(hta\ pro\s a)/llhla a)pe/fhnan</G>,
+which Adam translates by &lsquo;the first number in which root and
+square increases, comprehending three distances and four limits, of
+elements that make like and unlike and wax and wane, render all
+things conversable and rational with one another&rsquo;. <G>au)xh/seis</G> are
+clearly multiplications. <G>duna/menai/ te kai\ dunasteuo/menai</G> are explained in
+this way. A straight line is said <G>du/nasqai</G> (&lsquo;to be capable of&rsquo;) an area,
+e.g. a rectangle, when the square on it is equal to the rectangle; hence
+<G>duname/nh</G> should mean a side of a square. <G>dunasteuome/nh</G> represents a sort
+of passive of <G>duname/nh</G>, meaning that of which the <G>duname/nh</G> is &lsquo;capable&rsquo;;
+hence Adam takes it here to be the square of which the <G>duname/nh</G> is the
+side, and the whole expression to mean the product of a square and its
+side, i.e. simply the cube of the side. The cubes 3<SUP>3</SUP>, 4<SUP>3</SUP>, 5<SUP>3</SUP> are supposed
+to be meant because the words in the description of the second number
+&lsquo;of which the ratio in its lowest terms 4:3 when joined to 5&rsquo; clearly
+refer to the right-angled triangle 3, 4, 5, and because at least three
+authors, Plutarch (<I>De Is. et Os.</I> 373 F), Proclus (on Eucl. I, p. 428. 1) and
+Aristides Quintilianus (<I>De mus.,</I> p. 152 Meibom. = p. 90 Jahn) say that
+<FIG>
+Plato used the Pythagorean or &lsquo;cosmic&rsquo; triangle in
+his Number. The &lsquo;three distances&rsquo; are regarded
+as &lsquo;dimensions&rsquo;, and the &lsquo;three distances and
+four limits&rsquo; are held to confirm the interpretation
+&lsquo;cube&rsquo;, because a solid (parallelepiped) was said to
+have &lsquo;three dimensions and four limits&rsquo; (<I>Theol. Ar.,</I>
+p. 16 Ast, and Iambl. <I>in Nicom.,</I> p. 93. 10), the limits
+being bounding points as <I>A, B, C, D</I> in the accom-
+panying figure. &lsquo;Making like and unlike&rsquo; is sup-
+posed to refer to the square and oblong forms in which the second
+number is stated.
+<p>Another view of the whole passage has recently appeared (A. G. Laird,
+<I>Plato's Geometrical Number and the comment of Proclus,</I> Madison, Wiscon-
+sin, 1918). Like all other solutions, it is open to criticism in some
+details, but it is attractive in so far as it makes greater use of Proclus
+(<I>in Platonis remp.,</I> vol. ii, p. 36 seq. Kroll) and especially of the passage
+(p. 40) in which he illustrates the formation of the &lsquo;harmonies&rsquo; by means
+of geometrical figures. According to Mr. Laird there are not <I>two</I> separ-
+ate numbers, and the description from which Hultsch and Adam derive
+the number 216 is not a description of a number but a statement of a
+general method of formation of &lsquo;harmonies&rsquo;, which is then applied to
+the triangle 3, 4, 5 as a particular case, in order to produce the one
+Geometrical Number. The basis of the whole thing is the use of figures
+like that of Eucl. VI. 8 (a right-angled triangle divided by a perpendicular
+from the right angle on the opposite side into two right-angled triangles
+similar to one another and to the original triangle). Let <I>ABC</I> be a
+right-angled triangle in which the sides <I>CB, BA</I> containing the right
+<FIG>
+angle are rational numbers <I>a, b</I> respectively.
+Draw <I>AF</I> at right angles to <I>AC</I> meeting <I>CB</I>
+produced in <I>F.</I> Then the figure <I>AFC</I> is that of
+Eucl. VI. 8, and of course <MATH><I>AB</I><SUP>2</SUP>=<I>CB.BF</I></MATH>.
+Complete the rectangle <I>ABFL,</I> and produce
+<I>FL, CA</I> to meet at <I>K.</I> Then, by similar tri-
+angles, <I>CB, BA, FB</I> (=<I>AL</I>) and <I>KL</I> are four
+straight lines in continued proportion, and their
+lengths are <I>a, b, b</I><SUP>2</SUP>/<I>a, b</I><SUP>3</SUP>/<I>a</I><SUP>2</SUP> respectively. Mul-
+tiplying throughout by <I>a</I><SUP>2</SUP> in order to get rid of
+fractions, we may take the lengths to be <I>a</I><SUP>3</SUP>,
+<I>a</I><SUP>2</SUP><I>b, ab</I><SUP>2</SUP>, <I>b</I><SUP>3</SUP> respectively. Now, on Mr. Laird's
+view, <G>au)xh/seis duna/menai</G> are <I>squares,</I> as <I>AB</I><SUP>2</SUP>, and
+<G>au)xh/seis dunasteuo/menai</G> <I>rectangles</I>, as <I>FB, BC, to
+which the squares are equal.</I> &lsquo;Making like and
+unlike&rsquo; refers to the equal factors of <I>a</I><SUP>3</SUP>, <I>b</I><SUP>3</SUP> and the unequal factors of
+<I>a</I><SUP>2</SUP><I>b, ab</I><SUP>2</SUP>; the terms <I>a</I><SUP>3</SUP>, <I>a</I><SUP>2</SUP><I>b, ab</I><SUP>2</SUP>, <I>b</I><SUP>3</SUP> are four <I>terms</I> (<G>o(/roi</G>) of a continued
+proportion with three <I>intervals</I> (<G>a)posta/seis</G>), and of course are all &lsquo;con-
+versable and rational with one another&rsquo;. (Incidentally, out of such
+terms we can even obtain the number 216, for if we put <I>a</I>=2, <I>b</I>=3, we
+have 8, 12, 18, 27, and the product of the extremes 8.27=the product
+of the means 12.18=216). Applying the method to the triangle 3, 4, 5
+(as Proclus does) we have the terms 27, 36, 48, 64, and the first three
+numbers, multiplied respectively by 100, give the elements of the
+Geometrical Number 3600<SUP>2</SUP>=2700.4800. On this interpretation <G>tri\s
+au)xhqei/s</G> simply means raised to the third dimension or &lsquo;made solid&rsquo; (as
+Aristotle says, <I>Politics</I> *q (E). 12, 1316 a 8), the factors being of course
+3.3.3=27, 3.3.4=36, and 3.4.4=48; and &lsquo;the ratio 4:3 joined
+to 5&rsquo; does not mean either the product or the sum of 3, 4, 5, but simply
+the triangle 3, 4, 5.</note>
+<pb n=306><head>PLATO</head>
+<p>The second number is described thus:
+<p>&lsquo;The ratio 4:3 in its lowest terms (&lsquo;the base&rsquo;, <G>puqmh/n</G>, of
+the ratio <G>e)pi/tritos</G>) joined or wedded to 5 yields two harmonies
+when thrice increased (<G>tri\s au)xhqei/s</G>), the one equal an equal
+number of times, so many times 100, the other of equal length
+one way, but oblong, consisting on the one hand of 100 squares
+of rational diameters of 5 diminished by one each or, if of
+<pb n=307><head>THE GEOMETRICAL NUMBER</head>
+irrational diameters, by two, and on the other hand of 100
+cubes of 3.&rsquo;
+<p>The ratio 4:3 must be taken in the sense of &lsquo;the numbers
+4 and 3&rsquo;, and Adam takes &lsquo;joined with 5&rsquo; to mean that 4, 3
+and 5 are multiplied together, making 60; 60 &lsquo;thrice increased&rsquo;
+he interprets as &lsquo;60 thrice multiplied by 60&rsquo;, that is to say,
+60x60x60x60 or 3600<SUP>2</SUP>; &lsquo;so many times 100&rsquo; must then
+be the &lsquo;equal&rsquo; side of this, or 36 times 100; this 3600<SUP>2</SUP>, or
+12960000, is one of the &lsquo;harmonies&rsquo;. The other is the same
+number expressed as the product of two unequal factors, an
+&lsquo;oblong&rsquo; number; the first factor is 100 times a number
+which can be described either as 1 less than the square of the
+&lsquo;rational diameter of 5&rsquo;, or as 2 less than the square of
+the &lsquo;irrational diameter&rsquo; of 5, where the irrational diameter
+of 5 is the diameter of a square of side 5, i. e. &radic;50, and the
+rational diameter is the nearest whole number to this, namely
+7, so that the number which is multiplied by 100 is 49-1, or
+50-2, i. e. 48, and the first factor is therefore 4800; the
+second factor is 100 cubes of 3, or 2700; and of course
+<MATH>4800x2700=3600<SUP>2</SUP></MATH> or 12960000. Hultsch obtains the side,
+3600, of the first &lsquo;harmony&rsquo; in another way; he takes 4 and 3
+joined to 5 to be the <I>sum</I> of 4, 3 and 5, i. e. 12, and <G>tri\s au)xhqei/s</G>,
+&lsquo;thrice increased&rsquo;, to mean that the 12 is &lsquo;multiplied by three&rsquo;
+<note>Adam maintains that <G>tri\s au)xhqei/s</G> cannot mean &lsquo;multiplied by 3&rsquo;. He
+observes (p. 278, note) that the Greek for &lsquo;multiplied by 3&rsquo;, if we
+use <G>au)xa/nw</G>, would be <G>tria/di au)xhqei/s</G>, this being the construction used by
+Nicomachus (ii. 15. 2 <G>i(/na o( q tri\s g w)\n pa/lin tria/di e)p) a)/llo dia/sthma
+au)xhqh= kai\ ge/nhtai o( kz</G>) and in <I>Theol. Ar.</I> (p. 39, Ast <G>e(xa/di au)xhqei/s</G>). Never-
+theless I think that <G>tri\s au)xhqei/s</G> would not be an unnatural expression for
+a mathematician to use for &lsquo;multiplied by 3&rsquo;, let alone Plato in a passage
+like this. It is to be noted that <G>pollaplasia/zw</G> and <G>pollapla/sios</G> are
+likewise commonly used with the dative of the multiplier; yet <G>i)sa/kis
+pollapla/sios</G> is the regular expression for &lsquo;equimultiple&rsquo;. And <G>au)xa/nw</G> is
+actually found with <G>tosauta/kis</G>: see Pappus ii, p. 28. 15, 22, where <G>tosau-
+ta/kis au)xh/somen</G> means &lsquo;we have to multiply by such a power&rsquo; of 10000 or
+of 10 (although it is true that the chapter in which the expression occurs
+may be a late addition to Pappus's original text). On the whole, I prefer
+Hultsch's interpretation to Adam's. <G>tri\s au)xhqei/s</G> can hardly mean that
+60 is raised to the <I>fourth</I> power, 60<SUP>4</SUP>; and if it did, &lsquo;so many times 100&rsquo;,
+immediately following the expression for 3600<SUP>2</SUP>, would be pointless and
+awkward. On the other hand, &lsquo;so many times 100&rsquo; following the ex-
+pression for 36 would naturally indicate 3600.</note>
+making 36; &lsquo;so many times 100&rsquo; is then 36 times 100, or 3600.
+<p>But the main interest of the passage from the historical
+<pb n=308><head>PLATO</head>
+point of view lies in the terms &lsquo;rational&rsquo; and &lsquo;irrational
+diameter of 5&rsquo;. A fair approximation to &radic;2 was obtained
+by selecting a square number such that, if 2 be multiplied by
+it, the product is nearly a square; 25 is such a square number,
+since 25 times 2, or 50, only differs by 1 from 7<SUP>2</SUP>; conse-
+quently 7/5 is an approximation to &radic;2. It may have been
+arrived at in the tentative way here indicated; we cannot
+doubt that it was current in Plato's time; nay, we know that
+the general solution of the equations
+<MATH><I>x</I><SUP>2</SUP>-2<I>y</I><SUP>2</SUP>=&plusmn;1</MATH>
+by means of successive &lsquo;side-&rsquo; and &lsquo;diameter-&rsquo; numbers was
+Pythagorean, and Plato was therefore, here as in so many
+other places, &lsquo;Pythagorizing&rsquo;.
+<p>The diameter is again mentioned in the <I>Politicus,</I> where
+Plato speaks of &lsquo;the diameter which is in square (<G>duna/mei</G>)
+two feet&rsquo;, meaning the diagonal of the square with side
+1 foot, and again of the diameter of the square on this
+diameter, i.e. the diagonal of a square 2 square feet in area,
+in other words, the side of a square 4 square feet in area,
+or a straight line 2 feet in length.<note><I>Politicus,</I> 266 B.</note>
+<p>Enough has been said to show that Plato was abreast of
+the mathematics of his day, and we can understand the
+remark of Proclus on the influence which he exerted upon
+students and workers in that field:
+<p>&lsquo;he caused mathematics in general and geometry in particular
+to make a very great advance by reason of his enthusiasm
+for them, which of course is obvious from the way in which
+he filled his books with mathematical illustrations and every-
+where tries to kindle admiration for these subjects in those
+who make a pursuit of philosophy.&rsquo;<note>Proclus on Eucl. I, p. 66. 8-14.</note>
+<C>Mathematical &lsquo;arts&rsquo;</C>.
+<p>Besides the purely theoretical subjects, Plato recognizes the
+practical or applied mathematical &lsquo;arts&rsquo;; along with arith-
+metic, he mentions the art of measurement (for purposes of
+trade or craftsmanship) and that of weighing<note><I>Philebus,</I> 55 E-56 E.</note>; in the former
+connexion he speaks of the instruments of the craftsman,
+the circle-drawer (<G>to/rnos</G>), the compasses (<G>diabh/ths</G>), the rule
+<pb n=309><head>MATHEMATICAL &lsquo;ARTS&rsquo;</head>
+(<G>sta/qmh</G>) and &lsquo;a certain elaborate <G>prosagw/gion</G>&rsquo; (? approxi-
+mator). The art of weighing, he says,<note><I>Charmides,</I> 166 B.</note> &lsquo;is concerned with
+the heavier and lighter weight&rsquo;, as &lsquo;logistic&rsquo; deals with odd
+and even in their relation to one another, and geometry with
+magnitudes greater and less or equal; in the <I>Protagoras</I> he
+speaks of the man skilled in weighing
+<p>&lsquo;who puts together first the pleasant, and second the painful
+things, and adjusts the near and the far on the balance&rsquo;<note><I>Protagoras,</I> 356 B.</note>;
+<p>the principle of the lever was therefore known to Plato, who
+was doubtless acquainted with the work of Archytas, the
+reputed founder of the science of mechanics.<note>Diog. L. viii. 83.</note>
+<C>(<I>a</I>) <I>Optics.</I></C>
+<p>In the physical portion of the <I>Timaeus</I> Plato gives his
+explanation of the working of the sense organs. The account
+of the process of vision and the relation of vision to the
+light of day is interesting,<note><I>Timaeus,</I> 45 B-46 C.</note> and at the end of it is a reference
+to the properties of mirrors, which is perhaps the first indica-
+tion of a science of optics. When, says Plato, we see a thing
+in a mirror, the fire belonging to the face combines about the
+bright surface of the mirror with the fire in the visual current;
+the right portion of the face appears as the left in the image
+seen, and vice versa, because it is the mutually opposite parts
+of the visual current and of the object seen which come into
+contact, contrary to the usual mode of impact. (That is, if you
+imagine your reflection in the mirror to be another person
+looking at you, <I>his</I> left eye is the image of your right, and the
+left side of <I>his</I> left eye is the image of the right side of your
+right.) But, on the other hand, the right side really becomes
+the right side and the left the left when the light in com-
+bination with that with which it combines is transferred from
+one side to the other; this happens when the smooth part
+of the mirror is higher at the sides than in the middle (i. e. the
+mirror is a hollow cylindrical mirror held with its axis
+vertical), and so diverts the right portion of the visual current
+to the left and vice versa. And if you turn the mirror so that
+its axis is horizontal, everything appears upside down.
+<pb n=310><head>PLATO</head>
+<C>(<G>b</G>) <I>Music.</I></C>
+<p>In music Plato had the advantage of the researches of
+Archytas and the Pythagorean school into the numerical
+relations of tones. In the <I>Timaeus</I> we find an elaborate
+filling up of intervals by the interposition of arithmetic and
+harmonic means<note><I>Timaeus</I>, 35 C-36 B.</note>; Plato is also clear that higher and lower
+pitch are due to the more or less rapid motion of the air.<note><I>Ib.</I> 67 B.</note>
+In like manner the different notes in the &lsquo;harmony of the
+spheres&rsquo;, poetically turned into Sirens sitting on each of the
+eight whorls of the Spindle and each uttering a single sound,
+a single musical note, correspond to the different speeds of
+the eight circles, that of the fixed stars and those of the sun,
+the moon, and the five planets respectively.<note><I>Republic,</I> 617 B.</note>
+<C>(<G>g</G>) <I>Astronomy.</I></C>
+<p>This brings us to Plato's astronomy. His views are stated
+in their most complete and final form in the <I>Timaeus,</I> though
+account has to be taken of other dialogues, the <I>Phaedo,</I> the
+<I>Republic,</I> and the <I>Laws.</I> He based himself upon the early
+Pythagorean system (that of Pythagoras, as distinct from
+that of his successors, who were the first to abandon the
+geocentric system and made the earth, with the sun, the
+moon and the other planets, revolve in circles about the &lsquo;cen-
+tral fire&rsquo;); while of course he would take account of the
+results of the more and more exact observations made up
+to his own time. According to Plato, the universe has the
+most perfect of all shapes, that of a sphere. In the centre
+of this sphere rests the earth, immovable and kept there by
+the equilibrium of symmetry as it were (&lsquo;for a thing in
+equilibrium in the middle of any uniform substance will not
+have cause to incline more or less in any direction&rsquo;<note><I>Phaedo,</I> 109 A.</note>). The
+axis of the sphere of the universe passes through the centre of
+the earth, which is also spherical, and the sphere revolves
+uniformly about the axis in the direction from east to west.
+The fixed stars are therefore carried round in small circles
+of the sphere. The sun, the moon and the five planets are
+also carried round in the motion of the outer sphere, but they
+have independent circular movements of their own in addition.
+<pb n=311><head>ASTRONOMY</head>
+These latter movements take place in a plane which cuts
+at an angle the equator of the heavenly sphere; the several
+orbits are parts of what Plato calls the &lsquo;circle of the Other&rsquo;,
+as distinguished from the &lsquo;circle of the Same&rsquo;, which is the
+daily revolution of the heavenly sphere as a whole and which,
+carrying the circle of the Other and the seven movements
+therein along with it, has the mastery over them. The result
+of the combination of the two movements in the case of any
+one planet is to twist its actual path in space into a spiral<note><I>Timaeus,</I> 38 E-39 B.</note>;
+the spiral is of course included between two planes parallel to
+that of the equator at a distance equal to the maximum
+deviation of the planet in its course from the equator on
+either side. The speeds with which the sun, the moon and
+the five planets describe their own orbits (independently
+of the daily rotation) are in the following order; the moon is
+the quickest; the sun is the next quickest and Venus and
+Mercury travel in company with it, each of the three taking
+about a year to describe its orbit; the next in speed is Mars,
+the next Jupiter, and the last and slowest is Saturn; the
+speeds are of course angular speeds, not linear. The order
+of distances from the earth is, beginning with the nearest,
+as follows: moon, sun, Venus, Mercury, Mars, Jupiter, Saturn.
+In the <I>Republic</I> all these heavenly bodies describe their own
+orbits in a sense opposite to that of the daily rotation, i. e. in
+the direction from west to east; this is what we should
+expect; but in the <I>Timaeus</I> we are distinctly told, in one
+place, that the seven circles move &lsquo;in opposite senses to one
+another&rsquo;,<note><I>Ib.</I> 36 D.</note> and, in another place, that Venus and Mercury
+have &lsquo;the contrary tendency&rsquo; to the sun.<note><I>Ib.</I> 38 D.</note> This peculiar
+phrase has not been satisfactorily interpreted. The two state-
+ments taken together in their literal sense appear to imply
+that Plato actually regarded Venus and Mercury as describing
+their orbits the contrary way to the sun, incredible as this
+may appear (for on this hypothesis the angles of divergence
+between the two planets and the sun would be capable of any
+value up to 180&deg;, whereas observation shows that they are
+never far from the sun). Proclus and others refer to attempts
+to explain the passages by means of the theory of epicycles;
+Chalcidius in particular indicates that the sun's motion on its
+<pb n=312><head>PLATO</head>
+epicycle (which is from east to west) is in the contrary sense
+to the motion of Venus and Mercury on their epicycles
+respectively (which is from west to east)<note>Chalcidius on <I>Timaeus,</I> cc. 81, 109, 112.</note>; and this would
+be a satisfactory explanation if Plato could be supposed to
+have been acquainted with the theory of epicycles. But the
+probabilities are entirely against the latter supposition. All,
+therefore, that can be said seems to be this. Heraclides of
+Pontus, Plato's famous pupil, is known on clear evidence to
+have discovered that Venus and Mercury revolve round the
+sun like satellites. He may have come to the same conclusion
+about the superior planets, but this is not certain; and in any
+case he must have made the discovery with reference to
+Mercury and Venus first. Heraclides's discovery meant that
+Venus and Mercury, while accompanying the sun in its annual
+motion, described what are really epicycles about it. Now
+discoveries of this sort are not made without some preliminary
+seeking, and it may have been some vague inkling of the
+truth that prompted the remark of Plato, whatever the precise
+meaning of the words.
+<p>The differences between the angular speeds of the planets
+account for the overtakings of one planet by another, and
+the combination of their independent motions with that of the
+daily rotation causes one planet to <I>appear</I> to be overtaking
+another when it is really being overtaken by it and vice
+versa.<note><I>Timaeus,</I> 39 A.</note> The sun, moon and planets are instruments for
+measuring time.<note><I>Ib.</I> 41 E, 42 D.</note> Even the earth is an instrument for making
+night and day by virtue of its <I>not</I> rotating about its axis,
+while the rotation of the fixed stars carrying the sun with
+it is completed once in twenty-four hours; a month has passed
+when the moon after completing her own orbit overtakes the
+sun (the &lsquo;month&rsquo; being therefore the <I>synodic</I> month), and
+a year when the sun has completed its own circle. According
+to Plato the time of revolution of the other planets (except
+Venus and Mercury, which have the same speed as the sun)
+had not been exactly calculated; nevertheless the Perfect
+Year is completed &lsquo;when the relative speeds of all the eight
+revolutions [the seven independent revolutions and the daily
+rotation] accomplish their course together and reach their
+<pb n=313><head>ASTRONOMY</head>
+starting-point&rsquo;.<note><I>Timaeus,</I> 39 B-D.</note> There was apparently a tradition that the
+Great Year of Plato was 36000 years: this corresponds to
+the minimum estimate of the precession of the equinoxes
+quoted by Ptolemy from Hipparchus's treatise on the length
+of the year, namely at least one-hundredth of a degree in
+a year, or 1&deg; in 100 years,<note>Ptolemy, <I>Syntaxis,</I> vii. 2, vol. ii, p. 15. 9-17, Heib.</note> that is to say, 360&deg; in 36000 years.
+The period is connected by Adam with the Geometrical Num-
+ber 12960000 because this number of days, at the rate of 360
+days in the year, makes 36000 years. The coincidence may,
+it is true, have struck Ptolemy and made him describe the
+Great Year arrived at on the basis of 1&deg; per 100 years
+as the &lsquo;Platonic&rsquo; year; but there is nothing to show that
+Plato himself calculated a Great Year with reference to pre-
+cession: on the contrary, precession was first discovered by
+Hipparchus.
+<p>As regards the distances of the sun, moon and planets
+Plato has nothing more definite than that the seven circles
+are &lsquo;in the proportion of the double intervals, three of each&rsquo;<note><I>Timaeus,</I> 36 D.</note>:
+the reference is to the Pythagorean <G>tetraktu/s</G> represented in
+<FIG>
+the annexed figure, the numbers after 1 being
+on the one side successive powers of 2, and on
+the other side successive powers of 3. This
+gives 1, 2, 3, 4, 8, 9, 27 in ascending order.
+What precise estimate of relative distances
+Plato based upon these figures is uncertain.
+It is generally supposed (1) that the radii of the successive
+orbits are in the ratio of the numbers; but (2) Chalcidius
+considered that 2, 3, 4 ... are the successive differences
+between these radii,<note>Chalcidius on <I>Timaeus,</I> c. 96, p. 167, Wrobel</note> so that the radii themselves are in
+the ratios of 1, <MATH>1+2=3, 1+2+3=6</MATH>, &amp;c.; and again (3),
+according to Macrobius,<note>Macrobius, <I>In somn. Scip.</I> ii. 3. 14.</note> the Platonists held that the successive
+radii are as 1, 1.2=2, 1.2.3=6, 6.4=24, 24.9=216,
+216.8=1728 and 1728.27=46656. In any case the
+figures have no basis in observation.
+<p>We have said that Plato made the earth occupy the centre
+of the universe and gave it no movement of any kind. Other
+<pb n=314><head>PLATO</head>
+views, however, have been attributed to Plato by later writers.
+In the <I>Timacus</I> Plato had used of the earth the expression
+which has usually been translated &lsquo;our nurse, globed (<G>i)llo-
+me/nhn</G>) round the axis stretched from pole to pole through
+the universe&rsquo;.<note><I>Timaeus,</I> 40 B.</note> It is well known that Aristotle refers to the
+passage in these terms:
+<p>&lsquo;Some say that the earth, actually lying at the centre (<G>kai\
+keime/nhn e)pi\ tou= ke/ntrou</G>), is yet wound <I>and moves</I> (<G>i)/llesqai
+kai\ kinei=sqai</G>) about the axis stretched through the universe
+from pole to pole.&rsquo;<note>Arist. <I>De caelo,</I> ii. 13, 293 b 20; cf. ii. 14, 296 a 25.</note>
+<p>This naturally implies that Aristotle attributed to Plato
+the view that the earth rotates about its axis. Such a view
+is, however, entirely inconsistent with the whole system
+described in the <I>Timaeus</I> (and also in the <I>Laws,</I> which Plato
+did not live to finish), where it is the sphere of the fixed
+stars which by its revolution about the earth in 24 hours
+makes night and day; moreover, there is no reason to doubt
+the evidence that it was Heraclides of Pontus who was the
+first to affirm the rotation of the earth about its own axis
+in 24 hours. The natural inference seems to be that Aristotle
+either misunderstood or misrepresented Plato, the ambiguity
+of the word <G>i)llome/nhn</G> being the contributing cause or the
+pretext as the case may be. There are, however, those who
+maintain that Aristotle <I>must</I> have known what Plato meant
+and was incapable of misrepresenting him on a subject like
+this. Among these is Professor Burnet,<note><I>Greek Philosophy,</I> Part I, Thales to Plato, pp. 347-8.</note> who, being satisfied
+that Aristotle understood <G>i)llome/nhn</G> to mean motion of some
+sort, and on the strength of a new reading which he has
+adopted from two MSS. of the first class, has essayed a new
+interpretation of Plato's phrase. The new reading differs
+from the former texts in having the article <G>th\n</G> after
+<G>i)llome/nhn</G>, which makes the phrase run thus, <G>gh=n de\ trofo\n
+me\n h(mete/ran, i)llome/nhn de\ th\n peri\ to\n dia\ panto\s po/lon
+tetame/non</G>. Burnet, holding that we can only supply with
+<G>th\n</G> some word like <G>o(do/n</G>, understands <G>peri/odon</G> or <G>perifora/n</G>,
+and translates &lsquo;earth our nurse going to and fro on its path
+round the axis which stretches right through the universe&rsquo;.
+<pb n=315><head>ASTRONOMY</head>
+In confirmation of this Burnet cites the &lsquo;unimpeachable
+testimony&rsquo; of Theophrastus, who said that
+&lsquo;Plato in his old age repented of having given the earth
+the central place in the universe, to which it had no right&rsquo;<note>Plutarch, <I>Quaest. Plat.</I> 8. 1, 1006 c; cf. <I>Life of Numa,</I> c. 11.</note>;
+and he concludes that, according to Plato in the <I>Timaeus,</I>
+the earth is not the centre of the universe. But the sentences
+in which Aristotle paraphrases the <G>i)llome/nhn</G> in the <I>Timaeus</I>
+by the words <G>i)/llesqai kai\ kinei=sqai</G> both make it clear that
+the persons who held the view in question also declared
+that the earth <I>lies</I> or <I>is placed at the centre</I> (<G>keime/nhn e)pi\
+tou= ke/ntrou</G>), or &lsquo;placed the earth at the centre&rsquo; (<G>e)pi\ tou= me/sou
+qe/ntes</G>). Burnet's explanation is therefore in contradiction to
+part of Aristotle's statement, if not to the rest; so that he
+does not appear to have brought the question much nearer
+to a solution. Perhaps some one will suggest that the rotation
+or oscillation about the axis of the universe is <I>small,</I> so small
+as to be fairly consistent with the statement that the earth
+remains at the centre. Better, I think, admit that, on our
+present information, the puzzle is insoluble.
+<p>The dictum of Theophrastus that Plato in his old age
+repented of having placed the earth in the centre is incon-
+sistent with the theory of the <I>Timaeus,</I> as we have said.
+Boeckh explained it as a misapprehension. There appear
+to have been among Plato's immediate successors some who
+altered Plato's system in a Pythagorean sense and who may
+be alluded to in another passage of the <I>De caelo</I><note>Arist. <I>De caelo,</I> ii. 13, 293 a 27-b 1.</note>; Boeckh
+suggested, therefore, that the views of these Pythagorizing
+Platonists may have been put down to Plato himself. But
+the tendency now seems to be to accept the testimony of
+Theophrastus literally. Heiberg does so, and so does Burnet,
+who thinks it probable that Theophrastus heard the statement
+which he attributes to Plato from Plato himself. But I would
+point out that, if the <I>Timaeus,</I> as Burnet contends, contained
+Plato's explicit recantation of his former view that the earth
+was at the centre, there was no need to supplement it by an
+oral communication to Theophrastus. In any case the question
+has no particular importance in comparison with the develop-
+ments which have next to be described.
+<pb><C>X
+FROM PLATO TO EUCLID</C>
+<p>WHATEVER original work Plato himself did in mathematics
+(and it may not have been much), there is no doubt that his
+enthusiasm for the subject in all branches and the pre-eminent
+place which he gave it in his system had enormous influence
+upon its development in his lifetime and the period following.
+In astronomy we are told that Plato set it as a problem to
+all earnest students to find &lsquo;what are the uniform and ordered
+movements by the assumption of which the apparent move-
+ments of the planets can be accounted for&rsquo;; our authority for
+this is Sosigenes, who had it from Eudemus.<note>Simpl. on <I>De caelo</I>, ii. 12 (292 b 10), p. 488. 20-34, Heib.</note> One answer
+to this, representing an advance second to none in the history
+of astronomy, was given by Heraclides of Pontus, one of
+Plato's pupils (<I>circa</I> 388-310 B.C.); the other, which was
+by Eudoxus and on purely mathematical lines, constitutes
+one of the most remarkable achievements in pure geometry
+that the whole of the history of mathematics can show.
+Both were philosophers of extraordinary range. Heraclides
+wrote works of the highest class both in matter and style:
+the catalogue of them covers subjects ethical, grammatical,
+musical and poetical, rhetorical, historical; and there were
+geometrical and dialectical treatises as well. Similarly
+Eudoxus, celebrated as philosopher, geometer, astronomer,
+geographer, physician and legislator, commanded and enriched
+almost the whole field of learning.
+<C>Heraclides of Pontus: astronomical discoveries.</C>
+<p>Heraclides held that the apparent daily revolution of the
+heavenly bodies round the earth was accounted for, not by
+<pb n=317><head>HERACLIDES. ASTRONOMICAL DISCOVERIES</head>
+the circular motion of the stars round the earth, but by the
+rotation of the earth about its own axis; several passages
+attest this, e.g.
+<p>&lsquo;Heraclides of Pontus supposed that the earth is in the
+centre and rotates (lit. &lsquo;moves in a circle&rsquo;) while the heaven
+is at rest, and he thought by this supposition to save the
+phenomena.&rsquo;<note>Simpl. on <I>De caelo</I>, p. 519. 9-11, Heib.; cf.pp.441. 31-445. 5, pp. 541.
+27-542. 2; Proclus <I>in Tim.</I> 281 E.</note>
+<p>True, Heraclides may not have been alone in holding this
+view, for we are told that Ecphantus of Syracuse, a Pytha-
+gorean, also asserted that &lsquo;the earth, being in the centre
+of the universe, moves about its own centre in an eastward
+direction&rsquo;<note>Hippolytus, <I>Refut.</I> i. 15 (<I>Vors.</I> i<SUP>3</SUP>, p. 340. 31), cf. A&euml;tius, iii. 13. 3
+(<I>Vors.</I> i<SUP>3</SUP>, p. 341. 8-10).</note>; when Cicero<note>Cic. <I>Acad. Pr.</I> ii. 39, 123.</note> says the same thing of Hicetas, also
+of Syracuse, this is probably due to a confusion. But there
+is no doubt of the originality of the other capital discovery
+made by Heraclides, namely that Venus and Mercury revolve,
+like satellites, round the sun as centre. If, as Schiaparelli
+argued, Heraclides also came to the same conclusion about
+Mars, Jupiter and Saturn, he anticipated the hypothesis of
+Tycho Brahe (or rather improved on it), but the evidence is
+insufficient to establish this, and I think the probabilities are
+against it; there is some reason for thinking that it was
+Apollonius of Perga who thus completed what Heraclides had
+begun and put forward the full Tychonic hypothesis.<note><I>Aristarchus of Samos, the ancient Copernicus</I>, ch. xviii.</note> But
+there is nothing to detract from the merit of Heraclides in
+having pointed the way to it.
+<p>Eudoxus's theory of concentric spheres is even more re-
+markable as a mathematical achievement; it is worthy of the
+man who invented the great theory of proportion set out
+in Euclid, Book V, and the powerful <I>method of exhaustion</I>
+which not only enabled the areas of circles and the volumes
+of pyramids, cones, spheres, &amp;c., to be obtained, but is at the
+root of all Archimedes's further developments in the mensura-
+tion of plane and solid figures. But, before we come to
+Eudoxus, there are certain other names to be mentioned.
+<pb n=318><head>FROM PLATO TO EUCLID</head>
+<C>Theory of numbers (Speusippus, Xenocrates).</C>
+<p>To begin with arithmetic or the theory of numbers. SPEU-
+SIPPUS, nephew of Plato, who succeeded him as head of the
+school, is said to have made a particular study of Pythagorean
+doctrines, especially of the works of Philolaus, and to have
+written a small treatise <I>On the Pythagorean Numbers</I> of
+which a fragment, mentioned above (pp. 72, 75, 76) is pre-
+served in the <I>Theologumena Arithmetices.</I><note><I>Theol. Ar.</I>, Ast, p. 61.</note> To judge by the
+fragment, the work was not one of importance. The arith-
+metic in it was evidently of the geometrical type (polygonal
+numbers, for example, being represented by dots making up
+the particular figures). The portion of the book dealing with
+&lsquo;the five figures (the regular solids) which are assigned to the
+cosmic elements, their particularity and their community
+with one another&rsquo;, can hardly have gone beyond the putting
+together of the figures by faces, as we find it in the <I>Timaeus.</I>
+To Plato's distinction of the fundamental triangles, the equi-
+lateral, the isosceles right-angled, and the half of an equilateral
+triangle cut off by a perpendicular from a vertex on the
+opposite side, he adds a distinction (&lsquo;passablement futile&rsquo;,
+as is the whole fragment in Tannery's opinion) of four
+pyramids (1) the regular pyramid, with an equilateral triangle
+for base and all the edges equal, (2) the pyramid on a square
+base, and (evidently) having its four edges terminating at the
+corners of the base equal, (3) the pyramid which is the half of
+the preceding one obtained by drawing a plane through the
+vertex so as to cut the base perpendicularly in a diagonal
+of the base, (4) a pyramid constructed on the half of an
+equilateral triangle as base; the object was, by calling these
+pyramids a monad, a dyad, a triad and a tetrad respectively,
+to make up the number 10, the special properties and virtues
+of which as set forth by the Pythagoreans were the subject of
+the second half of the work. Proclus quotes a few opinions
+of Speusippus; e.g., in the matter of theorems and problems,
+he differed from Menaechmus, since he regarded both alike
+as being more properly <I>theorems</I>, while Menaechmus would
+call both alike <I>problems.</I><note>Proclus on Eucl. I, pp. 77. 16; 78. 14.</note>
+<pb n=319><head>THEORY OF NUMBERS</head>
+<p>XENOCRATES of Chalcedon (396-314 B.C.), who succeeded
+Speusippus as head of the school, having been elected by
+a majority of only a few votes over Heraclides, is also said
+to have written a book <I>On Numbers</I> and a <I>Theory of Numbers</I>,
+besides books on geometry.<note>Diog. L. iv. 13, 14.</note> These books have not survived,
+but we learn that Xenocrates upheld the Platonic tradition in
+requiring of those who would enter the school a knowledge of
+music, geometry and astronomy; to one who was not pro-
+ficient in these things he said &lsquo;Go thy way, for thou hast not
+the means of getting a grip of philosophy&rsquo;. Plutarch says
+that he put at 1,002,000,000,000 the number of syllables which
+could be formed out of the letters of the alphabet.<note>Plutarch, <I>Quaest. Conviv.</I> viii. 9. 13, 733 A.</note> If the
+story is true, it represents the first attempt on record to solve
+a difficult problem in permutations and combinations. Xeno-
+crates was a supporter of &lsquo;indivisible lines&rsquo;(and magnitudes)
+by which he thought to get over the paradoxical arguments
+of Zeno.<note>Simpl. <I>in Phys.</I>, p. 138. 3, &amp;c.</note>
+<C>The Elements. Proclus's summary (<I>continued</I>).</C>
+<p>In geometry we have more names mentioned in the sum-
+mary of Proclus.<note>Proclus on Eucl. I, p. 66. 18-67. 1.</note>
+<p>&lsquo;Younger than Leodamas were Neoclides and his pupil Leon,
+who added many things to what was known before their
+time, so that Leon was actually able to make a collection
+of the elements more carefully designed in respect both of
+the number of propositions proved and of their utility, besides
+which he invented <I>diorismi</I> (the object of which is to deter-
+mine) when the problem under investigation is possible of
+solution and when impossible.&rsquo;
+<p>Of Neoclides and Leon we know nothing more than what
+is here stated; but the definite recognition of the <G>diorismo/s</G>,
+that is, of the necessity of finding, as a preliminary to the
+solution of a problem, the conditions for the possibility of
+a solution, represents an advance in the philosophy and
+technology of mathematics. Not that the thing itself had
+not been met with before: there is, as we have seen, a
+<pb n=320><head>FROM PLATO TO EUCLID</head>
+<G>diorismo/s</G> indicated in the famous geometrical passage of the
+<I>Meno</I><note>Plato, <I>Meno</I>, 87 A.</note>; no doubt, too, the geometrical solution by the Pytha-
+goreans of the quadratic equation would incidentally make
+clear to them the limits of possibility corresponding to the
+<G>diorismo/s</G> in the solution of the most general form of quad-
+ratic in Eucl. VI. 27-9, where, in the case of the &lsquo;deficient&rsquo;
+parallelogram (Prop. 28), the enunciation states that &lsquo;the
+given rectilineal figure must not be greater than the parallelo-
+gram described on half of the straight line and similar to the
+defect&rsquo;. Again, the condition of the possibility of constructing
+a triangle out of three given straight lines (Eucl. I. 22),
+namely that any two of them must be together greater than
+the third, must have been perfectly familiar long before Leon
+or Plato.
+<p>Proclus continues:<note>Proclus on Eucl. I., p. 67. 2-68. 4.</note>
+<p>&lsquo;Eudoxus of Cnidos, a little younger than Leon, who had
+been associated with the school of Plato, was the first to
+increase the number of the so-called general theorems; he
+also added three other proportions to the three already known,
+and multiplied the theorems which originated with Plato
+about the section, applying to them the method of analysis.
+Amyclas [more correctly Amyntas] of Heraclea, one of the
+friends of Plato, Menaechmus, a pupil of Eudoxus who had
+also studied with Plato, and Dinostratus, his brother, made
+the whole of geometry still more perfect. Theudius of
+Magnesia had the reputation of excelling in mathematics as
+well as in the other branches of philosophy; for he put
+together the elements admirably and made many partial (or
+limited) theorems more general. Again, Athenaeus of Cyzicus,
+who lived about the same time, became famous in other
+branches of mathematics and most of all in geometry. These
+men consorted together in the Academy and conducted their
+investigations in common. Hermotimus of Colophon carried
+further the investigations already opened up by Eudoxus and
+Theaetetus, discovered many propositions of the Elements
+and compiled some portion of the theory of Loci. Philippus
+of Medma, who was a pupil of Plato and took up mathematics
+at his instance, not only carried out his investigations in
+accordance with Plato's instructions but also set himself to
+do whatever in his view contributed to the philosophy of
+Plato.&rsquo;
+<pb n=321><head>THE ELEMENTS</head>
+<p>It will be well to dispose of the smaller names in this
+list before taking up Eudoxus, the principal subject of
+this chapter. The name of Amyclas should apparently be
+Amyntas,<note>See <I>Ind. Hercul.</I>, ed. B cheler, <I>Ind. Schol. Gryphisw.</I>, 1869/70, col.
+6 in.</note> although Diogenes Laertius mentions Amyclos of
+Heraclea in Pontus as a pupil of Plato<note>Diog. L. iii. 46.</note> and has elsewhere an
+improbable story of one Amyclas, a Pythagorean, who with
+Clinias is supposed to have dissuaded Plato from burning the
+works of Democritus in view of the fact that there were
+many other copies in circulation.<note><I>Ib.</I> ix. 40.</note> Nothing more is known
+of Amyntas, Theudius, Athenaeus and Hermotimus than what
+is stated in the above passage of Proclus. It is probable,
+however, that the propositions, &amp;c., in elementary geometry
+which are quoted by Aristotle were taken from the Elements
+of Theudius, which would no doubt be the text-book of the
+time just preceding Euclid. Of Menaechmus and Dinostratus
+we have already learnt that the former discovered conic
+sections, and used them for finding two mean proportionals,
+and that the latter applied the quadratrix to the squaring
+of the circle. Philippus of Medma (vulg. Mende) is doubtless
+the same person as Philippus of Opus, who is said to have
+revised and published the <I>Laws</I> of Plato which had been left
+unfinished, and to have been the author of the <I>Epinomis.</I>
+He wrote upon astronomy chiefly; the astronomy in the
+<I>Epinomis</I> follows that of the <I>Laws</I> and the <I>Timaeus</I>; but
+Suidas records the titles of other works by him as follows:
+<I>On the distance of the sun and moon, On the eclipse of the
+moon, On the size of the sun, the moon and the earth, On
+the planets.</I> A passage of A&euml;tius<note><I>Dox. Gr.</I>, p. 360.</note> and another of Plutarch<note><I>Non posse suaviter vivi secundum Epicurum</I>, c. 11, 1093 E.</note>
+alluding to his <I>proofs</I> about the shape of the moon may
+indicate that Philippus was the first to establish the complete
+theory of the phases of the moon. In mathematics, accord-
+ing to the same notice by Suidas, he wrote <I>Arithmetica,
+Means, On polygonal numbers, Cyclica, Optics, Enoptrica</I>
+(On mirrors); but nothing is known of the contents of these
+works.
+<pb n=322><head>FROM PLATO TO EUCLID</head>
+<p>According to Apollodorus, EUDOXUS flourished in Ol. 103 =
+368-365 B.C., from which we infer that he was born about 408
+B.C., and (since he lived 53 years) died about 355 B.C. In his
+23rd year he went to Athens with the physician Theomedon,
+and there for two months he attended lectures on philosophy
+and oratory, and in particular the lectures of Plato; so poor
+was he that he took up his abode at the Piraeus and trudged
+to Athens and back on foot each day. It would appear that
+his journey to Italy and Sicily to study geometry with
+Archytas, and medicine with Philistion, must have been
+earlier than the first visit to Athens, for from Athens he
+returned to Cnidos, after which he went to Egypt with
+a letter of introduction to King Nectanebus, given him by
+Agesilaus; the date of this journey was probably 381-380 B.C.
+or a little later, and he stayed in Egypt sixteen months.
+After that he went to Cyzicus, where he collected round him
+a large school which he took with him to Athens in 368 B.C.
+or a little later. There is apparently no foundation for the
+story mentioned by Diogenes Laertius that he took up a hostile
+attitude to Plato,<note>Diog. L. viii. 87.</note> nor on the other side for the statements
+that he went with Plato to Egypt and spent thirteen years
+in the company of the Egyptian priests, or that he visited
+Plato when Plato was with the younger Dionysius on his
+third visit to Sicily in 361 B.C. Returning later to his native
+place, Eudoxus was by a popular vote entrusted with legisla-
+tive office.
+<p>When in Egypt Eudoxus assimilated the astronomical
+knowledge of the priests of Heliopolis and himself made
+observations. The observatory between Heliopolis and Cerce-
+sura used by him was still pointed out in Augustus's time;
+he also had one built at Cnidos, and from there he observed
+the star Canopus which was not then visible in higher
+latitudes. It was doubtless to record the observations thus
+made that he wrote the two books attributed to him by
+Hipparchus, the <I>Mirror</I> and the <I>Phaenomena</I><note>Hipparchus, <I>in Arati et Eudoxi phaenomena commentarii</I>, i. 2. 2, p. 8.
+15-20 Manitius.</note>; it seems, how-
+ever, unlikely that there could have been two independent
+works dealing with the same subject, and the latter, from which
+<pb n=323><head>EUDOXUS</head>
+the poem of Aratus was drawn, so far as verses 19-732 are
+concerned, may have been a revision of the former work and
+even, perhaps, posthumous.
+<p>But it is the theoretical side of Eudoxus's astronomy rather
+than the observational that has importance for us; and,
+indeed, no more ingenious and attractive hypothesis than
+that of Eudoxus's system of concentric spheres has ever been
+put forward to account for the apparent motions of the sun,
+moon and planets. It was the first attempt at a purely
+mathematical theory of astronomy, and, with the great and
+immortal contributions which he made to geometry, puts him
+in the very first rank of mathematicians of all time. He
+was a <I>man of science</I> if there ever was one. No occult or
+superstitious lore appealed to him; Cicero says that Eudoxus,
+&lsquo;in astrologia iudicio doctissimorum hominum facile princeps&rsquo;,
+expressed the opinion and left it on record that no sort of
+credence should be given to the Chaldaeans in their predic-
+tions and their foretelling of the life of individuals from the
+day of their birth.<note>Cic., <I>De div.</I> ii. 42.</note> Nor would he indulge in vain physical
+speculations on things which were inaccessible to observation
+and experience in his time; thus, instead of guessing at
+the nature of the sun, he said that he would gladly be
+burnt up like Phaethon if at that price he could get to the
+sun and so ascertain its form, size, and nature.<note>Plutarch, <I>Non posse suaviter vivi secundum Epicurum</I>, c. 11, 1094 B.</note> Another
+story (this time presumably apocryphal) is to the effect
+that he grew old at the top of a very high mountain in
+the attempt to discover the movements of the stars and the
+heavens.<note>Petronius Arbiter, <I>Satyricon</I>, 88.</note>
+<p>In our account of his work we will begin with the sentence
+about him in Proclus's summary. First, he is said to have
+increased &lsquo;the number of the <I>so-called general</I> theorems&rsquo;.
+&lsquo;So-called general theorems&rsquo; is an odd phrase; it occurred to
+me whether this could mean theorems which were true of
+everything falling under the conception of magnitude, as are
+the definitions and theorems forming part of Eudoxus's own
+theory of proportion, which applies to numbers, geometrical
+magnitudes of all sorts, times, &amp;c. A number of propositions
+<pb n=324><head>FROM PLATO TO EUCLID</head>
+at the beginning of Euclid's Book X similarly refer to magni-
+tudes in general, and the proposition X. 1 or its equivalent
+was actually used by Eudoxus in his <I>method of exhaustion</I>,
+as it is by Euclid in his application of the same method to the
+theorem (among others) of XII. 2 that circles are to one
+another as the squares on their diameters.
+<p>The three &lsquo;proportions&rsquo; or means added to the three pre-
+viously known (the arithmetic, geometric and harmonic) have
+already been mentioned (p. 86), and, as they are alterna-
+tively attributed to others, they need not detain us here.
+<p>Thirdly, we are told that Eudoxus &lsquo;extended&rsquo; or &lsquo;increased
+the number of the (propositions) about <I>the section</I> (<G>ta\ peri\
+th\n tomh/n</G>) which originated with Plato, applying to them
+the method of analysis&rsquo;. What is <I>the section</I>? The sugges-
+tion which has been received with most favour is that of
+Bretschneider,<note>Bretschneider, <I>Die Geometrie und die Geometer vor Eukleides</I>, pp.
+167-9.</note> who pointed out that up to Plato's time there
+was only one &lsquo;section&rsquo; that had any real significance in
+geometry, namely the section of a straight line in extreme
+and mean ratio which is obtained in Eucl. II. 11 and is used
+again in Eucl. IV. 10-14 for the construction of a pentagon.
+These theorems were, as we have seen, pretty certainly Pytha-
+gorean, like the whole of the substance of Euclid, Book II.
+Plato may therefore, says Bretschneider, have directed atten-
+tion afresh to this subject and investigated the metrical rela-
+tions between the segments of a straight line so cut, while
+Eudoxus may have continued the investigation where Plato
+left off. Now the passage of Proclus says that, in extending
+the theorems about &lsquo;the section&rsquo;, Eudoxus applied the method
+of analysis; and we actually find in Eucl. XIII. 1-5 five
+propositions about straight lines cut in extreme and mean
+ratio followed, in the MSS., by definitions of analysis and
+synthesis, and alternative proofs of the same propositions
+in the form of analysis followed by synthesis. Here, then,
+Bretschneider thought he had found a fragment of some actual
+work by Eudoxus corresponding to Proclus's description.
+But it is certain that the definitions and the alternative proofs
+were interpolated by some scholiast, and, judging by the
+figures (which are merely straight lines) and by comparison
+<pb n=325><head>EUDOXUS</head>
+with the remarks on analysis and synthesis quoted from
+Heron by An-Nair&imacr;z&imacr; at the beginning of his commentary on
+Eucl. Book II, it seems most likely that the interpolated defini-
+tions and proofs were taken from Heron. Bretschneider's
+argument based on Eucl. XIII. 1-5 accordingly breaks down,
+and all that can be said further is that, if Eudoxus investi-
+gated the relation between the segments of the straight line,
+he would find in it a case of incommensurability which would
+further enforce the necessity for a theory of proportion which
+should be applicable to incommensurable as well as to com-
+mensurable magnitudes. Proclus actually observes that
+&lsquo;theorems about sections like those in Euclid's Second Book
+are common to both [arithmetic and geometry] <I>except that in
+which the straight line is cut in extreme and mean ratio</I>&rsquo;<note>Proclus on Eucl. I, p. 60. 16-19.</note>
+(cf. Eucl. XIII. 6 for the actual proof of the irrationality
+in this case). Opinion, however, has not even in recent years
+been unanimous in favour of Bretschneider's interpretation;
+Tannery<note>Tann&egrave;ry, <I>La g&eacute;om&eacute;trie grecque</I>, p. 76.</note> in particular preferred the old view, which pre-
+vailed before Bretschneider, that &lsquo;section&rsquo; meant section <I>of
+solids</I>, e.g. by planes, a line of investigation which would
+naturally precede the discovery of conics; he pointed out that
+the use of the singular, <G>th\n tomh/n</G>, which might no doubt
+be taken as &lsquo;section&rsquo; in the abstract, is no real objection, that
+there is no other passage which speaks of a certain section
+<I>par excellence</I>, and that Proclus in the words just quoted
+expresses himself quite differently, speaking of &lsquo;sections&rsquo; of
+which the particular section in extreme and mean ratio is
+only one. Presumably the question will never be more defi-
+nitely settled unless by the discovery of new documents.
+<C>(<G>a</G>) <I>Theory of proportion.</I></C>
+<p>The anonymous author of a scholium to Euclid's Book V,
+who is perhaps Proclus, tells us that &lsquo;some say&rsquo; that this
+Book, containing the general theory of proportion which is
+equally applicable to geometry, arithmetic, music and all
+mathematical science, &lsquo;is the discovery of Eudoxus, the teacher
+of Plato&rsquo;.<note>Euclid, ed. Heib., vol. v, p. 280.</note> There is no reason to doubt the truth of this
+<pb n=326><head>FROM PLATO TO EUCLID</head>
+statement. The new theory appears to have been already
+familiar to Aristotle. Moreover, the fundamental principles
+show clear points of contact with those used in the <I>method
+of exhaustion</I>, also due to Eudoxus. I refer to the definition
+(Eucl. V, Def. 4) of magnitudes having a ratio to one another,
+which are said to be &lsquo;such as are capable, when (sufficiently)
+multiplied, of exceeding one another&rsquo;; compare with this
+Archimedes's &lsquo;lemma&rsquo; by means of which he says that the
+theorems about the volume of a pyramid and about circles
+being to one another as the squares on their diameters were
+proved, namely that &lsquo;of unequal lines, unequal surfaces, or
+unequal solids, the greater exceeds the less by such a
+magnitude as is capable, if added (continually) to itself, of
+exceeding any magnitude of those which are comparable to
+one another&rsquo;, i.e. of magnitudes of the same kind as the
+original magnitudes.
+<p>The essence of the new theory was that it was applicable
+to incommensurable as well as commensurable quantities;
+and its importance cannot be overrated, for it enabled
+geometry to go forward again, after it had received the blow
+which paralysed it for the time. This was the discovery of
+the irrational, at a time when geometry still depended on the
+Pythagorean theory of proportion, that is, the numerical
+theory which was of course applicable only to commensurables.
+The discovery of incommensurables must have caused what
+Tannery described as &lsquo;un v&eacute;ritable scandale logique&rsquo; in
+geometry, inasmuch as it made inconclusive all the proofs
+which had depended on the old theory of proportion. One
+effect would naturally be to make geometers avoid the use
+of proportions as much as possible; they would have to use
+other methods wherever they could. Euclid's Books I-IV no
+doubt largely represent the result of the consequent remodel-
+ling of fundamental propositions; and the ingenuity of the
+substitutes devised is nowhere better illustrated than in I. 44,
+45, where the equality of the complements about the diagonal
+of a parallelogram is used (instead of the construction, as
+in Book VI, of a fourth proportional) for the purpose of
+applying to a given straight line a parallelogram in a given
+angle and equal to a given triangle or rectilineal area.
+<p>The greatness of the new theory itself needs no further
+<pb n=327><head>EUDOXUS'S THEORY OF PROPORTION</head>
+argument when it is remembered that the definition of equal
+ratios in Eucl. V, Def. 5 corresponds exactly to the modern
+theory of irrationals due to Dedekind, and that it is word for
+word the same as Weierstrass's definition of equal numbers.
+<C>(<G>b</G>) <I>The method of exhaustion.</I></C>
+<p>In the preface to Book I of his treatise <I>On the Sphere and
+Cylinder</I> Archimedes attributes to Eudoxus the proof of the
+theorems that the volume of a pyramid is one-third of
+the volume of the prism which has the same base and equal
+height, and that the volume of a cone is one-third of the
+cylinder with the same base and height. In the <I>Method</I> he
+says that these facts were discovered, though not proved
+(i. e. in Archimedes's sense of the word), by Democritus,
+who accordingly deserved a great part of the credit for the
+theorems, but that Eudoxus was the first to supply the
+scientific proof. In the preface to the <I>Quadrature of the Para-
+bola</I> Archimedes gives further details. He says that for the
+proof of the theorem that the area of a segment of a parabola
+cut off by a chord is (4/3)rds of the triangle on the same base and
+of equal height with the segment he himself used the &lsquo;lemma&rsquo;
+quoted above (now known as the Axiom of Archimedes), and
+he goes on:
+<p>&lsquo;The earlier geometers have also used this lemma; for it is
+by the use of this lemma that they have proved the proposi-
+tions (1) that circles are to one another in the duplicate ratio
+of their diameters, (2) that spheres are to one another in the
+triplicate ratio of their diameters, and further (3) that every
+pyramid is one third part of the prism which has the same
+base with the pyramid and equal height; also (4) that every
+cone is one third part of the cylinder having the same base
+with the cone and equal height they proved by assuming
+a certain lemma similar to that aforesaid.&rsquo;
+<p>As, according to the other passage, it was Eudoxus who
+first proved the last two of these theorems, it is a safe
+inference that he used for this purpose the &lsquo;lemma&rsquo; in ques-
+tion or its equivalent. But was he the first to use the lemma?
+This has been questioned on the ground that one of the
+theorems mentioned as having been proved by &lsquo;the earlier
+geometers&rsquo; in this way is the theorem that circles are to one
+<pb n=328><head>FROM PLATO TO EUCLID</head>
+another as the squares on their diameters, which proposition,
+as we are told on the authority of Eudemus, was proved
+(<G>dei=xai</G>) by Hippocrates of Chios. This suggested to Hankel
+that the lemma in question must have been formulated by
+Hippocrates and used in his proof.<note>Hankel, <I>Zur Geschichte der Mathematik in Alterthum und Mittelalter</I>,
+p. 122.</note> But seeing that, accord-
+ing to Archimedes, &lsquo;the earlier geometers&rsquo; proved by means
+of the same lemma <I>both</I> Hippocrates's proposition, (1) above,
+and the theorem (3) about the volume of a pyramid, while
+the first proof of the latter was certainly given by Eudoxus,
+it is simplest to suppose that it was Eudoxus who first formu-
+lated the &lsquo;lemma&rsquo; and used it to prove both propositions, and
+that Hippocrates's &lsquo;proof&rsquo; did not amount to a rigorous
+demonstration such as would have satisfied Eudoxus or
+Archimedes. Hippocrates may, for instance, have proceeded
+on the lines of Antiphon's &lsquo;quadrature&rsquo;, gradually exhausting
+the circles and <I>taking the limit</I>, without clinching the proof
+by the formal <I>reductio ad absurdum</I> used in the method of
+exhaustion as practised later. Without therefore detracting
+from the merit of Hippocrates, whose argument may have
+contained the germ of the method of exhaustion, we do not
+seem to have any sufficient reason to doubt that it was
+Eudoxus who established this method as part of the regular
+machinery of geometry.
+<p>The &lsquo;lemma&rsquo; itself, we may observe, is not found in Euclid
+in precisely the form that Archimedes gives it, though it
+is equivalent to Eucl. V, Def. 4 (Magnitudes are said to have
+a ratio to one another which are capable, when multiplied,
+of exceeding one another). When Euclid comes to prove the
+propositions about the content of circles, pyramids and cones
+(XII. 2, 4-7 Por., and 10), he does not use the actual lemma of
+Archimedes, but another which forms Prop. 1 of Book X, to
+the effect that, if there are two unequal magnitudes and from
+the greater there be subtracted more than its half (or the
+half itself), from the remainder more than its half (or the half),
+and if this be done continually, there will be left some magni-
+tude which will be less than the lesser of the given magnitudes.
+This last lemma is frequently used by Archimedes himself
+(notably in the second proof of the proposition about the area
+<pb n=329><head>EUDOXUS. METHOD OF EXHAUSTION</head>
+of a parabolic segment), and it may be the &lsquo;lemma similar
+to the aforesaid&rsquo; which he says was used in the case of the
+cone. But the existence of the two lemmas constitutes no
+real difficulty, because Archimedes's lemma (under the form
+of Eucl. V, Def. 4) is in effect used by Euclid to prove X. 1.
+<p>We are not told whether Eudoxus proved the theorem that
+spheres are to one another in the triplicate ratio of their
+diameters. As the proof of this in Eucl. XII. 16-18 is likewise
+based on X. 1 (which is used in XII. 16), it is probable enough
+that this proposition, mentioned along with the others by
+Archimedes, was also first proved by Eudoxus.
+<p>Eudoxus, as we have seen, is said to have solved the problem
+of the two mean proportionals by means of &lsquo;curved lines&rsquo;.
+This solution has been dealt with above (pp. 249-51).
+<p>We pass on to the
+<C>(<G>g</G>) <I>Theory of concentric spheres.</I></C>
+<p>This was the first attempt to account by purely geometrical
+hypotheses for the apparent irregularities of the motions of
+the planets; it included similar explanations of the apparently
+simpler movements of the sun and moon. The ancient
+evidence of the details of the system of concentric spheres
+(which Eudoxus set out in a book entitled <I>On speeds</I>, <G>*peri\
+taxw=n</G>, now lost) is contained in two passages. The first is in
+Aristotle's <I>Metaphysics</I>, where a short notice is given of the
+numbers and relative positions of the spheres postulated by
+Eudoxus for the sun, moon and planets respectively, the
+additions which Callippus thought it necessary to make to
+the numbers of those spheres, and lastly the modification
+of the system which Aristotle himself considers necessary
+&lsquo;if the phenomena are to be produced by all the spheres
+acting in combination&rsquo;.<note>Aristotle, <I>Metaph.</I> A. 8. 1073 b 17-1074 a 14.</note> A more elaborate and detailed
+account of the system is contained in Simplicius's commentary
+on the <I>De caelo</I> of Aristotle<note>Simpl. on <I>De caelo</I>, p. 488. 18-24, pp. 493. 4-506. 18 Heib.; p. 498
+a 45-b 3, pp. 498 b 27-503 a 33.</note>; Simplicius quotes largely from
+Sosigenes the Peripatetic (second century A. D.), observing that
+Sosigenes drew from Eudemus, who dealt with the subject
+in the second book of his <I>History of Astronomy.</I> Ideler was
+<pb n=330><head>FROM PLATO TO EUCLID</head>
+the first to appreciate the elegance of the theory and to
+attempt to explain its working (1828, 1830); E. F. Apelt, too,
+gave a fairly full exposition of it in a paper of 1849. But it
+was reserved for Schiaparelli to work out a complete restora-
+tion of the theory and to investigate in detail the extent
+to which it could be made to account for the phenomena; his
+paper has become a classic,<note>Schiaparelli, <I>Le sfere omocentriche di Eudosso, di Callippo e di Aristotele</I>,
+Milano 1875; Germ. trans. by W. Horn in <I>Abh. zur Gesch. d. Math.</I>, i.
+Heft, 1877, pp. 101-98.</note> and all accounts must necessarily
+follow his.
+<p>I shall here only describe the system so far as to show its
+mathematical interest. I have given fuller details elsewhere.<note><I>Aristarchus of Samos, the ancient Copernicus</I>, pp. 193-224.</note>
+Eudoxus adopted the view which prevailed from the earliest
+times to the time of Kepler, that circular motion was sufficient
+to account for the movements of all the heavenly bodies.
+With Eudoxus this circular motion took the form of the
+revolution of different spheres, each of which moves about
+a diameter as axis. All the spheres were concentric, the
+common centre being the centre of the earth; hence the name
+of &lsquo;homocentric&rsquo; spheres used in later times to describe the
+system. The spheres were of different sizes, one inside the
+other. Each planet was fixed at a point in the equator of
+the sphere which carried it, the sphere revolving at uniform
+speed about the diameter joining the corresponding poles;
+that is, the planet revolved uniformly in a great circle of the
+sphere perpendicular to the axis of rotation. But one such
+circular motion was not enough; in order to explain the
+changes in the apparent speed of the planets' motion, their
+stations and retrogradations, Eudoxus had to assume a number
+of such circular motions working on each planet and producing
+by their combination that single apparently irregular motion
+which observation shows us. He accordingly held that the
+poles of the sphere carrying the planet are not fixed, but
+themselves move on a greater sphere concentric with the
+carrying sphere and moving about two different poles with
+uniform speed. The poles of the second sphere were simi-
+larly placed on a third sphere concentric with and larger
+than the first and second, and moving about separate poles
+<pb n=331><head>THEORY OF CONCENTRIC SPHERES</head>
+of its own with a speed peculiar to itself. For the planets
+yet a fourth sphere was required, similarly related to the
+others; for the sun and moon Eudoxus found that, by a
+suitable choice of the positions of the poles and of speeds
+of rotation, he could make three spheres suffice. Aristotle
+and Simplicius describe the spheres in the reverse order, the
+sphere carrying the planet being the last; this makes the
+description easier, because we begin with the sphere represent-
+ing the daily rotaton of the heavens. The spheres which
+move each planet Eudoxus made quite separate from those
+which move the others; but one sphere sufficed to produce
+the daily rotation of the heavens. The hypothesis was purely
+mathematical; Eudoxus did not trouble himself about the
+material of the spheres or their mechanical connexion.
+<p>The moon has a motion produced by three spheres; the
+first or outermost moves in the same sense as the fixed stars
+from east to west in 24 hours; the second moves about an
+axis perpendicular to the plane of the zodiac circle or the
+ecliptic, and in the sense of the daily rotation, i.e. from
+east to west; the third again moves about an axis inclined
+to the axis of the second at an angle equal to the highest
+latitude attained by the moon, and from west to east;
+the moon is fixed on the equator of this third sphere. The
+speed of the revolution of the second sphere was very slow
+(a revolution was completed in a period of 223 lunations);
+the third sphere produced the revolution of the moon from
+west to east in the draconitic or nodal month (of 27 days,
+5 hours, 5 minutes, 36 seconds) round a circle inclined to
+the ecliptic at an angle equal to the greatest latitude of the
+moon.<note>Simplicius (and presumably Aristotle also) confused the motions of
+the second and third spheres. The above account represents what
+Eudoxus evidently intended.</note> The moon described the latter circle, while the
+circle itself was carried round by the second sphere in
+a retrograde sense along the ecliptic in a period of 223
+lunations; and both the inner spheres were bodily carried
+round by the first sphere in 24 hours in the sense of the daily
+rotation. The three spheres thus produced the motion of the
+moon in an orbit inclined to the ecliptic, and the retrogression
+of the nodes, completed in a period of about 181/2 years.
+<pb n=332><head>FROM PLATO TO EUCLID</head>
+<p>The system of three spheres for the sun was similar, except
+that the orbit was less inclined to the ecliptic than that of the
+moon, and the second sphere moved from west to east instead
+of from east to west, so that the nodes moved slowly forward
+in the direct order of the signs instead of backward.
+<p>But the case to which the greatest mathematical interest
+attaches is that of the planets, the motion of which is pro-
+duced by sets of four spheres for each. Of each set the first
+and outermost produced the daily rotation in 24 hours; the
+second, the motion round the zodiac in periods which in the
+case of superior planets are equal to the sidereal periods of
+revolution, and for Mercury and Venus (on a geocentric
+system) one year. The third sphere had its poles fixed at two
+opposite points on the zodiac circle, the poles being carried
+round in the motion of the second sphere; the revolution
+of the third sphere about its poles was again uniform and
+was completed in the synodic period of the planet or the time
+which elapsed between two successive oppositions or conjunc-
+tions with the sun. The poles of the third sphere were the
+same for Mercury and Venus but different for all the other
+planets. On the surface of the third sphere the poles of the
+fourth sphere were fixed, the axis of the latter being inclined
+to that of the former at an angle which was constant for each
+planet but different for the different planets. The rotation of
+the fourth sphere about its axis took place in the same time
+as the rotation of the third about its axis but in the opposite
+sense. On the equator of the fourth sphere the planet was
+fixed. Consider now the actual path of a planet subject to
+the rotations of the third and fourth spheres only, leaving out
+of account for the moment the first two spheres the motion of
+which produces the daily rotation and the motion along the
+zodiac respectively. The problem is the following. A sphere
+rotates uniformly about the fixed diameter <I>AB. P, P</I>&prime; are
+two opposite poles on this sphere, and a second sphere con-
+centric with the first rotates uniformly about the diameter
+<I>PP</I>&prime; in the same time as the former sphere rotates about <I>AB,</I>
+but in the opposite direction. <I>M</I> is a point on the second
+sphere equidistant from <I>P, P</I>&prime;, i. e. a point on the equator
+of the second sphere. Required to find the path of the
+point <I>M.</I> This is not difficult nowadays for any one familiar
+<pb n=333><head>THEORY OF CONCENTRIC SPHERES</head>
+with spherical trigonometry and analytical geometry; but
+Schiaparelli showed, by means of a series of seven propositions
+or problems involving only elementary geometry, that it was
+well within the powers of such a geometer as Eudoxus. The
+path of <I>M</I> in space turns out in fact to be a curve like
+a lemniscate or figure-of-eight described on the surface of a
+sphere, namely the fixed sphere about <I>AB</I> as diameter. This
+<FIG>
+&lsquo;spherical lemniscate&rsquo; is roughly shown in the second figure
+above. The curve is actually the intersection of the sphere
+with a certain cylinder touching it internally at the double
+point <I>O,</I> namely a cylinder with diameter equal to <I>AS</I> the
+<I>sagitta</I> (shown in the other figure) of the diameter of the
+small circle on which <I>P</I> revolves. But the curve is also
+the intersection of <I>either</I> the sphere <I>or</I> the cylinder with
+a certain cone with vertex <I>O,</I> axis parallel to the axis of the
+cylinder (i. e. touching the circle <I>AOB</I> at <I>O</I>) and vertical angle
+equal to the &lsquo;inclination&rsquo; (the angle <I>AO</I>&prime;<I>P</I> in the first figure).
+That this represents the actual result obtained by Eudoxus
+himself is conclusively proved by the facts that Eudoxus
+called the curve described by the planet about the zodiac
+circle the <I>hippopede</I> or <I>horse-fetter,</I> and that the same term
+<I>hippopede</I> is used by Proclus to describe the plane curve of
+similar shape formed by a plane section of an anchor-ring or
+<I>tore</I> touching the tore internally and parallel to its axis.<note>Proclus on Eucl. I, p. 112. 5.</note>
+<p>So far account has only been taken of the motion due to
+the combination of the rotations of the third and fourth
+<pb n=334><head>FROM PLATO TO EUCLID</head>
+spheres. But <I>A, B,</I> the poles of the third sphere, are carried
+round the zodiac or ecliptic by the motion of the second
+sphere in a time equal to the &lsquo;zodiacal&rsquo; period of the planet.
+Now the axis of symmetry of the &lsquo;spherical lemniscate&rsquo; (the
+arc of the great circle bisecting it longitudinally) always lies
+on the ecliptic. We may therefore substitute for the third
+and fourth spheres the &lsquo;lemniscate&rsquo; moving bodily round
+the ecliptic. The combination of the two motions (that of the
+&lsquo;lemniscate&rsquo; and that of the planet on it) gives the motion of
+the planet through the constellations. The motion of the
+planet round the curve is an oscillatory motion, now forward in
+acceleration of the motion round the ecliptic due to the motion
+of the second sphere, now backward in retardation of the same
+motion; the period of the oscillation is the period of the syno-
+dic revolution, and the acceleration and retardation occupy
+half the period respectively. When the retardation in the
+sense of longitude due to the backward oscillation is greater
+than the speed of the forward motion of the lemniscate itself,
+the planet will for a time have a retrograde motion, at the
+beginning and end of which it will appear stationary for a little
+while, when the two opposite motions balance each other.
+<p>It will be admitted that to produce the retrogradations
+in this theoretical way by superimposed axial rotations of
+spheres was a remarkable stroke of genius. It was no slight
+geometrical achievement, for those days, to demonstrate the
+<I>effect</I> of the hypotheses; but this is nothing in comparison
+with the speculative power which enabled the man to invent
+the hypothesis which would produce the effect. It was, of
+course, a much greater achievement than that of Eudoxus's
+teacher Archytas in finding the two mean proportionals by
+means of the intersection of three surfaces in space, a <I>tore</I>
+with internal diameter <I>nil,</I> a cylinder and a cone; the problem
+solved by Eudoxus was much more difficult, and yet there
+is the curious resemblance between the two solutions that
+Eudoxus's <I>hippopede</I> is actually the section of a sphere with
+a cylinder touching it internally and also with a certain
+cone; the two cases together show the freedom with which
+master and pupil were accustomed to work with figures in
+three dimensions, and in particular with surfaces of revolution,
+their intersections, &amp;c.
+<pb n=335><head>THEORY OF CONCENTRIC SPHERES</head>
+<p>Callippus (about 370-300 B.C.) tried to make the system of
+concentric spheres suit the phenomena more exactly by adding
+other spheres; he left the number of the spheres at four in
+the case of Jupiter and Saturn, but added one each to the
+other planets and two each in the case of the sun and moon
+(making five in all). This would substitute for the hippopede
+a still more complicated elongated figure, and the matter is
+not one to be followed out here. Aristotle modified the system
+in a mechanical sense by introducing between each planet
+and the one below it reacting spheres one less in number than
+those acting on the former planet, and with motions equal
+and opposite to each of them, except the outermost, respec-
+tively; by neutralizing the motions of all except the outermost
+sphere acting on any planet he wished to enable that outer-
+most to be the outermost acting on the planet below, so that
+the spheres became one connected system, each being in actual
+contact with the one below and acting on it, whereas with
+Eudoxus and Callippus the spheres acting on each planet
+formed a separate set independent of the others. Aristotle's
+modification was not an improvement, and has no mathe-
+matical interest.
+<p>The works of ARISTOTLE are of the greatest importance to
+the history of mathematics and particularly of the Elements.
+His date (384-322/1) comes just before that of Euclid, so
+that from the differences between his statement of things
+corresponding to what we find in Euclid and Euclid's own we
+can draw a fair inference as to the innovations which were
+due to Euclid himself. Aristotle was no doubt a competent
+mathematician, though he does not seem to have specialized
+in mathematics, and fortunately for us he was fond of mathe-
+matical illustrations. His allusions to particular definitions,
+propositions, &amp;c., in geometry are in such a form as to suggest
+that his pupils must have had at hand some text-book where
+they could find the things he mentions. The particular text-
+book then in use would presumably be that which was the
+immediate predecessor of Euclid's, namely the Elements of
+Theudius; for Theudius is the latest of pre-Euclidean
+geometers whom the summary of Proclus mentions as a com-
+piler of Elements.<note>Proclus on Eucl. I, p. 67. 12-16.</note>
+<pb n=336><head>FROM PLATO TO EUCLID</head>
+<C>The mathematics in Aristotle comes under the
+following heads.</C>
+<C>(<G>a</G>) <I>First principles.</I></C>
+<p>On no part of the subject does Aristotle throw more light
+than on the first principles as then accepted. The most
+important passages dealing with this subject are in the
+<I>Posterior Analytics.</I><note><I>Anal. Post.</I> i. 6. 74 b 5, i. 10. 76 a 31-77 a 4.</note> While he speaks generally of &lsquo;demon-
+strative sciences&rsquo;, his illustrations are mainly mathematical,
+doubtless because they were readiest to his hand. He gives
+the clearest distinctions between axioms (which are common
+to all sciences), definitions, hypotheses and postulates (which
+are different for different sciences since they relate to the
+subject-matter of the particular science). If we exclude from
+Euclid's axioms (1) the assumption that two straight lines
+cannot enclose a space, which is interpolated, and (2) the
+so-called &lsquo;Parallel-Axiom&rsquo; which is the 5th Postulate, Aris-
+totle's explanation of these terms fits the classification of
+Euclid quite well. Aristotle calls the axioms by various
+terms, &lsquo;<I>common</I> (things)&rsquo;, &lsquo;common axioms&rsquo;, &lsquo;common opinions&rsquo;,
+and this seems to be the origin of &lsquo;common notions&rsquo; (<G>koinai\
+e)/nnoiai</G>), the term by which they are described in the text
+of Euclid; the particular axiom which Aristotle is most fond
+of quoting is No. 3, stating that, if equals be subtracted from
+equals, the remainders are equal. Aristotle does not give any
+instance of a geometrical postulate. From this we may fairly
+make the important inference that Euclid's Postulates are all
+his own, the momentous Postulate 5 as well as Nos. 1, 2, 3
+relating to constructions of lines and circles, and No. 4 that
+all right angles are equal. These postulates as well as those
+which Archimedes lays down at the beginning of his book
+<I>On Plane Equilibriums</I> (e.g. that &lsquo;equal weights balance at
+equal lengths, but equal weights at unequal lengths do not
+balance but incline in the direction of the weight which is
+at the greater length&rsquo;) correspond exactly enough to Aristotle's
+idea of a postulate. This is something which, e.g., the
+geometer assumes (for reasons known to himself) without
+demonstration (though properly a subject for demonstration)
+<pb n=337><head>ARISTOTLE</head>
+and without any assent on the part of the learner, or even
+against his opinion rather than otherwise. As regards defini-
+tions, Aristotle is clear that they do not assert existence or
+non-existence; they only require to be understood. The only
+exception he makes is in the case of the <I>unit</I> or <I>monad</I> and
+<I>magnitude,</I> the existence of which has to be assumed, while
+the existence of everything else has to be proved; the things
+actually necessary to be assumed in geometry are points and
+lines only; everything constructed out of them, e.g. triangles,
+squares, tangents, and their properties, e.g. incommensura-
+bility, has to be <I>proved</I> to exist. This again agrees sub-
+stantially with Euclid's procedure. Actual construction is
+with him the proof of existence. If triangles other than the
+equilateral triangle constructed in I. 1 are assumed in I. 4-21,
+it is only provisionally, pending the construction of a triangle
+out of three straight lines in I. 22; the drawing and producing
+of straight lines and the describing of circles is postulated
+(Postulates 1-3). Another interesting statement on the
+philosophical side of geometry has reference to the geometer's
+hypotheses. It is untrue, says Aristotle, to assert that a
+geometer's hypotheses are false because he assumes that a line
+which he has drawn is a foot long when it is not, or straight
+when it is not straight. The geometer bases no conclusion on
+the particular line being that which he has assumed it to be;
+he argues about what it <I>represents,</I> the figure itself being
+a mere illustration.<note>Arist. <I>Anal. Post.</I> i. 10. 76 b 39-77 a 2; cf. <I>Anal. Prior.</I> i. 41. 49 b 34 sq.;
+<I>Metaph.</I> N. 2. 1089 a 20-5.</note>
+<p>Coming now to the first definitions of Euclid, Book I, we
+find that Aristotle has the equivalents of Defs. 1-3 and 5, 6.
+But for a straight line he gives Plato's definition only:
+whence we may fairly conclude that Euclid's definition
+was his own, as also was his definition of a plane which
+he adapted from that of a straight line. Some terms seem
+to have been defined in Aristotle's time which Euclid leaves
+undefined, e.g. <G>kekla/sqai</G>, &lsquo;to be inflected&rsquo;, <G>neu/ein</G>, to &lsquo;verge&rsquo;.<note><I>Anal. Post.</I> i. 10. 76 b 9.</note>
+Aristotle seems to have known Eudoxus's new theory of pro-
+portion, and he uses to a considerable extent the usual
+<pb n=338><head>FROM PLATO TO EUCLID</head>
+terminology of proportions; he defines similar figures as
+Euclid does.
+<C>(<G>b</G>) <I>Indications of proofs differing from Euclid's.</I></C>
+<p>Coming to theorems, we find in Aristotle indications of
+proofs differing entirely from those of Euclid. The most
+remarkable case is that of the theorem of I. 5. For the
+purpose of illustrating the statement that in any syllogism
+one of the propositions must be affirmative and universal
+he gives a proof of the proposition as follows.<note><I>Anal. Prior.</I> i. 24. 41 b 13-22.</note>
+<p>&lsquo;For let <I>A, B</I> be drawn [i. e. joined] to the centre.
+<p>&lsquo;If then we assumed (1) that the angle <I>AC</I> [i. e. <I>A</I>+<I>C</I>]
+is equal to the angle <I>BD</I> [i. e. <I>B</I>+<I>D</I>] without asserting
+generally that <I>the angles of semicircles are equal,</I> and again
+<FIG>
+(2) that the angle <I>C</I> is equal to the
+angle <I>D</I> without making the further
+assumption that <I>the two angles of all
+segments are equal,</I> and if we then
+inferred, lastly, that since the whole
+angles are equal, and equal angles are
+subtracted from them, the angles which
+remain, namely <I>E, F,</I> are equal, without
+assuming generally that, if equals be
+subtracted from equals, the remainders are equal, we should
+commit a <I>petitio principii.</I>&rsquo;
+<p>There are obvious peculiarities of notation in this extract;
+the angles are indicated by single letters, and sums of two
+angles by two letters in juxtaposition (cf. <I>DE</I> for <I>D</I>+<I>E</I> in
+the proof cited from Archytas above, p. 215). The angles
+<I>A, B</I> are the angles at <I>A, B</I> of the <I>isosceles triangle OAB,</I> the
+same angles as are afterwards spoken of as <I>E, F.</I> But the
+differences of substance between this and Euclid's proof are
+much more striking. First, it is clear that &lsquo;mixed&rsquo; angles
+(&lsquo;angles&rsquo; formed by straight lines with circular arcs) played
+a much larger part in earlier text-books than they do in
+Euclid, where indeed they only appear once or twice as a
+survival. Secondly, it is remarkable that the equality of
+the two &lsquo;angles&rsquo; of a semicircle and of the two &lsquo;angles&rsquo; of any
+segment is assumed as a means of proving a proposition so
+<pb n=339><head>ARISTOTLE</head>
+elementary as I. 5, although one would say that the assump-
+tions are no more obvious than the proposition to be proved;
+indeed some kind of proof, e.g. by superposition, would
+doubtless be considered necessary to justify the assumptions.
+It is a natural inference that Euclid's proof of I. 5 was his
+own, and it would appear that his innovations as regards
+order of propositions and methods of proof began at the very
+threshold of the subject.
+<p>There are two passages<note><I>Anal. Post.</I> i. 5. 74 a 13-16; <I>Anal. Prior.</I> ii. 17. 66 a 11-15.</note> in Aristotle bearing on the theory
+of parallels which seem to show that the theorems of Eucl.
+I. 27, 28 are pre-Euclidean; but another passage<note><I>Anal. Prior.</I> ii. 16. 65 a 4.</note> appears to
+indicate that there was some vicious circle in the theory of
+parallels then current, for Aristotle alludes to a <I>petitio prin-
+cipii</I> committed by &lsquo;those who think that they draw parallels&rsquo;
+(or &lsquo;establish the theory of parallels&rsquo;, <G>ta\s parallh/lous
+gra/fein</G>), and, as I have tried to show elsewhere,<note>See <I>The Thirteen Books of Euclid's Elements,</I> vol. i, pp. 191-2 (cf.
+pp. 308-9).</note> a note of
+Philoponus makes it possible that Aristotle is criticizing a
+<I>direction</I>-theory of parallels such as has been adopted so
+often in modern text-books. It would seem, therefore, to have
+been Euclid who first got rid of the <I>petitio principii</I> in earlier
+text-books by formulating the famous Postulate 5 and basing
+I. 29 upon it.
+<p>A difference of method is again indicated in regard to the
+theorem of Eucl. III. 31 that the angle in a semicircle is right.
+Two passages of Aristotle taken together<note><I>Anal. Post.</I> ii. 11. 94 a 28; <I>Metaph.</I> <G>*q</G>. 9. 1051 a 26.</note> show that before
+Euclid the proposition was proved by means of the radius
+drawn to the middle point of the
+<FIG>
+arc of the semicircle. Joining the
+extremity of this radius to the ex-
+tremities of the diameter respec-
+tively, we have two isosceles right-
+angled triangles, and the two angles,
+one in each triangle, which are at the middle point of the arc,
+being both of them halves of right angles, make the angle in
+the semicircle <I>at that point</I> a right angle. The proof of the
+theorem must have been completed by means of the theorem
+<pb n=340><head>FROM PLATO TO EUCLID</head>
+of III. 21 that angles in the same segment are equal, a proposi-
+tion which Euclid's more general proof does not need to use.
+<p>These instances are sufficient to show that Euclid was far
+from taking four complete Books out of an earlier text-book
+without change; his changes began at the very beginning,
+and there are probably few, if any, groups of propositions in
+which he did not introduce some improvements of arrange-
+ment or method.
+<p>It is unnecessary to go into further detail regarding
+Euclidean theorems found in Aristotle except to note the
+interesting fact that Aristotle already has the principle of
+the method of exhaustion used by Eudoxus: &lsquo;If I continually
+add to a finite magnitude, I shall exceed every assigned
+(&lsquo;defined&rsquo;, <G>w(risme/nou</G>) magnitude, and similarly, if I subtract,
+I shall fall short (of any assigned magnitude).&rsquo;<note>Arist. <I>Phys.</I> viii. 10. 266 b 2.</note>
+<C>(<G>g</G>) <I>Propositions not found in Euclid.</I></C>
+<p>Some propositions found in Aristotle but not in Euclid
+should be mentioned. (1) The exterior angles of any polygon
+are together equal to four right angles<note><I>Anal. Post.</I> i. 24. 85 b 38; ii. 17. 99 a 19.</note>; although omitted
+in Euclid and supplied by Proclus, this is evidently a Pytha-
+gorean proposition. (2) The locus of a point such that its
+distances from two given points are in a given ratio (not
+being a ratio of equality) is a circle<note><I>Meteorologica,</I> iii. 5. 376 a 3 sq.</note>; this is a proposition
+quoted by Eutocius from Apollonius's <I>Plane Loci,</I> but the
+proof given by Aristotle differs very little from that of
+Apollonius as reproduced by Eutocius, which shows that the
+proposition was fully known and a standard proof of it was in
+existence before Euclid's time. (3) Of all closed lines starting
+from a point, returning to it again, and including a given
+area, the circumference of a circle is the shortest<note><I>De caelo,</I> ii. 4. 287 a 27.</note>; this shows
+that the study of isoperimetry (comparison of the perimeters
+of different figures having the same area) began long before
+the date of Zenodorus's treatise quoted by Pappus and Theon
+of Alexandria. (4) Only two solids can fill up space, namely
+the pyramid and the cube<note><I>Ib.</I> iii. 8. 306 b 7.</note>; this is the complement of the
+Pythagorean statement that the only three figures which can
+<pb n=341><head>ARISTOTLE</head>
+by being put together fill up space in a plane are the equi-
+lateral triangle, the square and the regular hexagon.
+<C>(<G>d</G>) <I>Curves and solids known to Aristotle.</I></C>
+<p>There is little beyond elementary plane geometry in Aris-
+totle. He has the distinction between straight and &lsquo;curved&rsquo;
+lines (<G>kampu/lai grammai/</G>), but the only curve mentioned
+specifically, besides circles, seems to be the spiral<note><I>Phys.</I> v. 4. 228 b 24.</note>; this
+term may have no more than the vague sense which it has
+in the expression &lsquo;the spirals of the heaven&rsquo;<note><I>Metaph.</I> B. 2. 998 a 5.</note>; if it really
+means the cylindrical helix, Aristotle does not seem to have
+realized its property, for he includes it among things which
+are not such that &lsquo;any part will coincide with any other
+part&rsquo;, whereas Apollonius later proved that the cylindrical
+helix has precisely this property.
+<p>In solid geometry he distinguishes clearly the three dimen-
+sions belonging to &lsquo;body&rsquo;, and, in addition to parallelepipedal
+solids, such as cubes, he is familiar with spheres, cones and
+cylinders. A sphere he defines as the figure which has all its
+radii (&lsquo;lines from the centre&rsquo;) equal,<note><I>Phys.</I> ii. 4. 287 a 19.</note> from which we may infer
+that Euclid's definition of it as the solid generated by the revo-
+lution of a semicircle about its diameter is his own (Eucl. XI,
+Def. 14). Referring to a cone, he says<note><I>Meteorologica,</I> iii. 5. 375 b 21.</note> &lsquo;the straight lines
+thrown out from <I>K</I> in the form of a cone make <I>GK as a sort
+of axis</I> (<G>w(/sper a)/xona</G>)&rsquo;, showing that the use of the word
+&lsquo;axis&rsquo; was not yet quite technical; of conic sections he does
+not seem to have had any knowledge, although he must have
+been contemporary with Menaechmus. When he alludes to
+&lsquo;two cubes being a cube&rsquo; he is not speaking, as one might
+suppose, of the duplication of the cube, for he is saying that
+no science is concerned to prove anything outside its own
+subject-matter; thus geometry is not required to prove &lsquo;that
+two cubes are a cube&rsquo;<note><I>Anal. Post.</I> i. 7. 75 b 12.</note>; hence the sense of this expression
+must be not geometrical but arithmetical, meaning that the
+product of two cube numbers is also a cube number. In the
+Aristotelian <I>Problems</I> there is a question which, although not
+mathematical in intention, is perhaps the first suggestion of
+<pb n=342><head>FROM PLATO TO EUCLID</head>
+a certain class of investigation. If a book in the form of a
+cylindrical roll is cut by a plane and then unrolled, why is it
+that the cut edge appears as a straight line if the section
+is parallel to the base (i. e. is a right section), but as a crooked
+line if the section is obliquely inclined (to the axis).<note><I>Probl.</I> xvi. 6. 914 a 25.</note> The
+<I>Problems</I> are not by Aristotle; but, whether this one goes
+back to Aristotle or not, it is unlikely that he would think of
+investigating the form of the curve mathematically.
+<C>(<G>e</G>) <I>The continuous and the infinite.</I></C>
+<p>Much light was thrown by Aristotle on certain general
+conceptions entering into mathematics such as the &lsquo;continuous&rsquo;
+and the &lsquo;infinite&rsquo;. The continuous, he held, could not be
+made up of indivisible parts; the continuous is that in which
+the boundary or limit between two consecutive parts, where
+they touch, is one and the same, and which, as the name
+itself implies, is <I>kept together,</I> which is not possible if the
+extremities are two and not one.<note><I>Phys.</I> v. 3. 227 a 11; vii. 1. 231 a 24.</note> The &lsquo;infinite&rsquo; or &lsquo;un-
+limited&rsquo; only exists potentially, not in actuality. The infinite
+is so in virtue of its endlessly changing into something else,
+like day or the Olympic games, and is manifested in different
+forms, e.g. in time, in Man, and in the division of magnitudes.
+For, in general, the infinite consists in something new being
+continually taken, that something being itself always finite
+but always different. There is this distinction between the
+forms above mentioned that, whereas in the case of magnitudes
+what is once taken remains, in the case of time and Man it
+passes or is destroyed, but the succession is unbroken. The
+case of addition is in a sense the same as that of division;
+in the finite magnitude the former takes place in the converse
+way to the latter; for, as we see the finite magnitude divided
+<I>ad infinitum,</I> so we shall find that addition gives a sum
+tending to a definite limit. Thus, in the case of a finite
+magnitude, you may take a definite fraction of it and add to
+it continually in the same ratio; if now the successive added
+terms do not include one and the same magnitude, whatever
+it is [i. e. if the successive terms diminish in geometrical
+progression], you will not come to the end of the finite
+magnitude, but, if the ratio is increased so that each term
+<pb n=343><head>ARISTOTLE ON THE INFINITE</head>
+does include one and the same magnitude, whatever it is, you
+will come to the end of the finite magnitude, for every finite
+magnitude is exhausted by continually taking from it any
+definite fraction whatever. In no other sense does the infinite
+exist but only in the sense just mentioned, that is, potentially
+and by way of diminution.<note><I>Phys.</I> iii. 6. 206 a 15-6 13.</note> And in this sense you may have
+potentially infinite addition, the process being, as we say, in
+a manner the same as with division <I>ad infinitum</I>; for in the
+case of addition you will always be able to find something
+outside the total for the time being, but the total will never
+exceed every definite (or assigned) magnitude in the way that,
+in the direction of division, the result will pass every definite
+magnitude, that is, by becoming smaller than it. The infinite
+therefore cannot exist, even potentially, in the sense of exceed-
+ing every finite magnitude as the result of successive addition.
+It follows that the correct view of the infinite is the opposite
+of that commonly held; it is not that which has nothing
+outside it, but that which always has something outside it.<note><I>Ib.</I> iii. 6. 206 b 16-207 a 1.</note>
+Aristotle is aware that it is essentially of physical magnitudes
+that he is speaking: it is, he says, perhaps a more general
+inquiry that would be necessary to determine whether the
+infinite is possible in mathematics and in the domain of
+thought and of things which have no magnitude.<note><I>Ib.</I> iii. 5. 204 a 34.</note>
+<p>&lsquo;But&rsquo;, he says, &lsquo;my argument does not anyhow rob
+mathematicians of their study, although it denies the existence
+of the infinite in the sense of actual existence as something
+increased to such an extent that it cannot be gone through
+(<G>a)diexi/thton</G>); for, as it is, they do not even need the infinite
+or use it, but only require that the finite (straight line) shall
+be as long <I>as they please.</I> . . . Hence it will make no difference
+to them for the purpose of demonstration.&rsquo;<note><I>Ib.</I> iii. 7. 207 b 27.</note>
+<p>The above disquisition about the infinite should, I think,
+be interesting to mathematicians for the distinct expression
+of Aristotle's view that the existence of an infinite series the
+terms of which are <I>magnitudes</I> is impossible unless it is
+convergent and (with reference to Riemann's developments)
+that it does not matter to geometry if the straight line is not
+infinite in length provided that it is as long as we please.
+<pb n=344><head>FROM PLATO TO EUCLID</head>
+Aristotle's denial of even the potential existence of a sum
+of magnitudes which shall exceed every definite magnitude
+was, as he himself implies, inconsistent with the lemma or
+assumption used by Eudoxus in his method of exhaustion.
+We can, therefore, well understand why, a century later,
+Archimedes felt it necessary to justify his own use of the
+lemma:
+<p>&lsquo;the earlier geometers too have used this lemma: for it is by
+its help that they have proved that circles have to one another
+the duplicate ratio of their diameters, that spheres have to
+one another the triplicate ratio of their diameters, and so on.
+And, in the result, each of the said theorems has been accepted
+no less than those proved without the aid of this lemma.&rsquo;<note>Archimedes, <I>Quadrature of a Parabola,</I> Preface.</note>
+<C>(<G>z</G>) <I>Mechanics.</I></C>
+<p>An account of the mathematics in Aristotle would be incom-
+plete without a reference to his ideas in mechanics, where he
+laid down principles which, even though partly erroneous,
+held their ground till the time of Benedetti (1530-90) and
+Galilei (1564-1642). The <I>Mechanica</I> included in the Aris-
+totelian writings is not indeed Aristotle's own work, but it is
+very close in date, as we may conclude from its terminology;
+this shows more general agreement with the terminology of
+Euclid than is found in Aristotle's own writings, but certain
+divergences from Euclid's terms are common to the latter and
+to the <I>Mechanica</I>; the conclusion from which is that the
+<I>Mechanica</I> was written before Euclid had made the termino-
+logy of mathematics more uniform and convenient, or, in the
+alternative, that it was composed after Euclid's time by persons
+who, though they had partly assimilated Euclid's terminology,
+were close enough to Aristotle's date to be still influenced
+by his usage. But the Aristotelian origin of many of the
+ideas in the <I>Mechanica</I> is proved by their occurrence in
+Aristotle's genuine writings. Take, for example, the principle
+of the lever. In the <I>Mechanica</I> we are told that,
+<p>&lsquo;as the weight moved is to the moving weight, so is the
+length (or distance) to the length inversely. In fact the mov-
+ing weight will more easily move (the system) the farther it
+is away from the fulcrum. The reason is that aforesaid,
+<pb n=345><head>ARISTOTELIAN MECHANICS</head>
+namely that the line which is farther from the centre describes
+the greater circle, so that, if the power applied is the same,
+that which moves (the system) will change its position the
+more, the farther it is away from the fulcrum.&rsquo;<note><I>Mechanica,</I> 3. 850 b 1.</note>
+<p>The idea then is that the greater power exerted by the
+weight at the greater distance corresponds to its greater
+velocity. Compare with this the passage in the <I>De caelo</I>
+where Aristotle is speaking of the speeds of the circles of
+the stars:
+<p>&lsquo;it is not at all strange, nay it is inevitable, that the speeds of
+circles should be in the proportion of their sizes.&rsquo;<note><I>De caelo,</I> ii. 8. 289 b 15.</note> . . . &lsquo;Since
+in two concentric circles the segment (sector) of the outer cut
+off between two radii common to both circles is greater than
+that cut off on the inner, it is reasonable that the greater circle
+should be carried round in the same time.&rsquo;<note><I>Ib.</I> 290 a 2.</note>
+<p>Compare again the passage of the <I>Mechanica</I>:
+<p>&lsquo;what happens with the balance is reduced to (the case of the)
+circle, the case of the lever to that of the balance, and
+practically everything concerning mechanical movements to
+the case of the lever. Further it is the fact that, given
+a radius of a circle, no two points of it move at the same
+speed (as the radius itself revolves), but the point more distant
+from the centre always moves more quickly, and this is the
+reason of many remarkable facts about the movements of
+circles which will appear in the sequel.&rsquo;<note><I>Mechanica,</I> 848 a 11.</note>
+<p>The axiom which is regarded as containing the germ of the
+principle of virtual velocities is enunciated, in slightly different
+forms, in the <I>De caelo</I> and the <I>Physics</I>:
+<p>&lsquo;A smaller and lighter weight will be given more movement
+if the force acting on it is the same. . . . The speed of the
+lesser body will be to that of the greater as the greater body
+is to the lesser.&rsquo;<note><I>De caelo,</I> iii. 2. 301 b 4, 11.</note>
+<p>&lsquo;If <I>A</I> be the movent, <I>B</I> the thing moved, <I>C</I> the length
+through which it is moved, <I>D</I> the time taken, then
+<p><I>A</I> will move 1/2<I>B</I> over the distance 2 <I>C</I> in the time <I>D,</I>
+and <I>A</I> &rdquo; 1/2<I>B</I> &rdquo; &rdquo; <I>C</I> &rdquo; &rdquo; 1/2<I>D</I>;
+thus proportion is maintained.&rsquo;<note><I>Phys.</I> vii. 5. 249 b 30-250 a 4.</note>
+<pb n=346><head>FROM PLATO TO EUCLID</head>
+<p>Again, says Aristotle,
+<p><I>A</I> will move <I>B</I> over the distance 1/2<I>C</I> in the time 1/2<I>D,</I>
+and 1/2<I>A</I> &rdquo; 1/2<I>B</I> a distance <I>C</I> &rdquo; &rdquo; <I>D</I>;<note><I>Phys.</I> vii. 5. 250 a 4-7.</note>
+and so on.
+<p>Lastly, we have in the <I>Mechanica</I> the parallelogram of
+velocities:
+<p>&lsquo;When a body is moved in a certain ratio (i. e. has two linear
+movements in a constant ratio to one another), the body must
+move in a straight line, and this straight line is the diameter
+of the figure (parallelogram) formed from the straight lines
+which have the given ratio.&rsquo;<note><I>Mechanica,</I> 2. 848 b 10.</note>
+<p>The author goes on to say<note><I>Ib.</I> 848 b 26 sq.</note> that, if the ratio of the two
+movements does not remain the same from one instant to the
+next, the motion will not be in a straight line but in a curve.
+He instances a circle in a vertical plane with a point moving
+along it downwards from the topmost point; the point has
+two simultaneous movements; one is in a vertical line, the
+other displaces this vertical line parallel to itself away from
+the position in which it passes through the centre till it
+reaches the position of a tangent to the circle; if during this
+time the ratio of the two movements were constant, say one of
+equality, the point would not move along the circumference
+at all but along the diagonal of a rectangle.
+<p>The parallelogram of <I>forces</I> is easily deduced from the
+parallelogram of velocities combined with Aristotle's axiom
+that the force which moves a given weight is directed along
+the line of the weight's motion and is proportional to the
+distance described by the weight in a given time.
+<p>Nor should we omit to mention the Aristotelian tract <I>On
+indivisible lines.</I> We have seen (p. 293) that, according to
+Aristotle, Plato objected to the genus &lsquo;point&rsquo; as a geometrical
+fiction, calling a point the beginning of a line, and often
+positing &lsquo;indivisible lines&rsquo; in the same sense.<note><I>Metaph.</I> A. 9. 992 a 20.</note> The idea of
+indivisible lines appears to have been only vaguely conceived
+by Plato, but it took shape in his school, and with Xenocrates
+<pb n=347><head>THE TRACT ON INDIVISIBLE LINES</head>
+became a definite doctrine. There is plenty of evidence for
+this<note>Cf. Zeller, ii. 1<SUP>4</SUP>, p. 1017.</note>; Proclus, for instance, tells us of &lsquo;a discourse or argu-
+ment by Xenocrates introducing indivisible lines&rsquo;.<note>Proclus on Eucl. I, p. 279. 5.</note> The tract
+<I>On indivisible lines</I> was no doubt intended as a counterblast
+to Xenocrates. It can hardly have been written by Aristotle
+himself; it contains, for instance, some expressions without
+parallel in Aristotle. But it is certainly the work of some
+one belonging to the school; and we can imagine that, having
+on some occasion to mention &lsquo;indivisible lines&rsquo;, Aristotle may
+well have set to some pupil, as an exercise, the task of refuting
+Xenocrates. According to Simplicius and Philoponus, the
+tract was attributed by some to Theophrastus<note>See Zeller, ii. 2<SUP>3</SUP>, p. 90, note.</note>; and this
+seems the most likely supposition, especially as Diogenes
+Laertius mentions, in a list of works by Theophrastus, &lsquo;<I>On
+indivisible lines,</I> one Book&rsquo;. The text is in many places
+corrupt, so that it is often difficult or impossible to restore the
+argument. In reading the book we feel that the writer is
+for the most part chopping logic rather than contributing
+seriously to the philosophy of mathematics. The interest
+of the work to the historian of mathematics is of the slightest.
+It does indeed cite the equivalent of certain definitions and
+propositions in Euclid, especially Book X (on irrationals), and
+in particular it mentions the irrationals called &lsquo;binomial&rsquo; or
+&lsquo;apotome&rsquo;, though, as far as irrationals are concerned, the
+writer may have drawn on Theaetetus rather than Euclid.
+The mathematical phraseology is in many places similar to
+that of Euclid, but the writer shows a tendency to hark back
+to older and less fixed terminology such as is usual in
+Aristotle. The tract begins with a section stating the argu-
+ments for indivisible lines, which we may take to represent
+Xenocrates's own arguments. The next section purports to
+refute these arguments one by one, after which other con-
+siderations are urged against indivisible lines. It is sought to
+show that the hypothesis of indivisible lines is not reconcilable
+with the principles assumed, or the conclusions proved, in
+mathematics; next, it is argued that, if a line is made up
+of indivisible lines (whether an odd or even number of such
+lines), or if the indivisible line has any point in it, or points
+<pb n=348><head>FROM PLATO TO EUCLID</head>
+terminating it, the indivisible line must be divisible; and,
+lastly, various arguments are put forward to show that a line
+can no more be made up of points than of indivisible lines,
+with more about the relation of points to lines, &amp;c.<note>A revised text of the work is included in Aristotle, <I>De plantis,</I> edited
+by O. Apelt, who also gave a German translation of it in <I>Beitr&auml;ge zur
+Geschichte der griechischen Philosophie</I> (1891), pp. 271-86. A translation
+by H. H. Joachim has since appeared (1908) in the series of Oxford
+Translations of Aristotle's works.</note>
+<C>Sphaeric.</C>
+<p>AUTOLYCUS of Pitane was the teacher of Arcesilaus (about
+315-241/40 B.C.), also of Pitane, the founder of the so-called
+Middle Academy. He may be taken to have flourished about
+310 B.C. or a little earlier, so that he was an elder con-
+temporary of Euclid. We hear of him in connexion with
+Eudoxus's theory of concentric spheres, to which he adhered.
+The great difficulty in the way of this theory was early seen,
+namely the impossibility of reconciling the assumption of the
+invariability of the distance of each planet with the observed
+differences in the brightness, especially of Mars and Venus,
+at different times, and the apparent differences in the relative
+sizes of the sun and moon. We are told that no one before
+Autolycus had even attempted to deal with this difficulty
+&lsquo;by means of hypotheses&rsquo;, i. e. (presumably) in a theoretical
+manner, and even he was not successful, as clearly appeared
+from his controversy with Aristotherus<note>Simplicius on <I>De caelo,</I> p. 504. 22-5 Heib.</note> (who was the teacher
+of Aratus); this implies that Autolycus's argument was in
+a written treatise.
+<p>Two works by Autolycus have come down to us. They
+both deal with the geometry of the sphere in its application
+to astronomy. The definite place which they held among
+Greek astronomical text-books is attested by the fact that, as
+we gather from Pappus, one of them, the treatise <I>On the
+moving Sphere,</I> was included in the list of works forming
+the &lsquo;Little Astronomy&rsquo;, as it was called afterwards, to distin-
+guish it from the &lsquo;Great Collection&rsquo; (<G>mega/lh su/ntaxis</G>) of
+Ptolemy; and we may doubtless assume that the other work
+<I>On Risings and Settings</I> was similarly included.
+<pb n=349><head>AUTOLYCUS OF PITANE</head>
+<p>Both works have been well edited by Hultsch with Latin
+translation.<note><I>Autolyci De sphaera quae movetur liber, De ortibus et occasibus libri duo</I>
+edidit F. Hultsch (Teubner 1885).</note> They are of great interest for several reasons.
+First, Autolycus is the earliest Greek mathematician from
+whom original treatises have come down to us entire, the next
+being Euclid, Aristarchus and Archimedes. That he wrote
+earlier than Euclid is clear from the fact that Euclid, in his
+similar work, the <I>Phaenomena,</I> makes use of propositions
+appearing in Autolycus, though, as usual in such cases, giving
+no indication of their source. The form of Autolycus's proposi-
+tions is exactly the same as that with which we are familiar
+in Euclid; we have first the enunciation of the proposition in
+general terms, then the particular enunciation with reference
+to a figure with letters marking the various points in it, then
+the demonstration, and lastly, in some cases but not in all, the
+conclusion in terms similar to those of the enunciation. This
+shows that Greek geometrical propositions had already taken
+the form which we recognize as classical, and that Euclid did
+not invent this form or introduce any material changes.
+<C>A lost text-book on Sphaeric.</C>
+<p>More important still is the fact that Autolycus, as well as
+Euclid, makes use of a number of propositions relating to the
+sphere without giving any proof of them or quoting any
+authority. This indicates that there was already in existence
+in his time a text-book of the elementary geometry of the
+sphere, the propositions of which were generally known to
+mathematicians. As many of these propositions are proved
+in the <I>Sphaerica</I> of Theodosius, a work compiled two or three
+centuries later, we may assume that the lost text-book proceeded
+on much the same lines as that of Theodosius, with much the
+same order of propositions. Like Theodosius's <I>Sphaerica</I>
+it treated of the stationary sphere, its sections (great and
+small circles) and their properties. The geometry of the
+sphere at rest is of course prior to the consideration of the
+sphere in motion, i. e. the sphere rotating about its axis, which
+is the subject of Autolycus's works. Who was the author of
+the lost pre-Euclidean text-book it is impossible to say;
+<pb n=350><head>FROM PLATO TO EUCLID</head>
+Tannery thought that we could hardly help attributing it to
+Eudoxus. The suggestion is natural, seeing that Eudoxus
+showed, in his theory of concentric spheres, an extraordinary
+mastery of the geometry of the sphere; on the other hand,
+as Loria observes, it is, speaking generally, dangerous to
+assume that a work of an unknown author appearing in
+a certain country at a certain time must have been written
+by a particular man of science simply because he is the only
+man of the time of whom we can certainly say that he was
+capable of writing it.<note>Loria, <I>Le scienze esatte nell' antica Grecia,</I> 1914, p. 496-7.</note> The works of Autolycus also serve to
+confirm the pre-Euclidean origin of a number of propositions
+in the <I>Elements.</I> Hultsch<note><I>Berichte der Kgl. S&auml;chs. Gesellschaft der Wissenschaften zu Leipzig,</I>
+Phil.-hist. Classe, 1886, pp. 128-55.</note> examined this question in detail
+in a paper of 1886. There are (1) the propositions pre-
+supposed in one or other of Autolycus's theorems. We have
+also to take account of (2) the propositions which would be
+required to establish the propositions in sphaeric assumed by
+Autolycus as known. The best clue to the propositions under
+(2) is the actual course of the proofs of the corresponding
+propositions in the <I>Sphaerica</I> of Theodosius; for Theodosius
+was only a compiler, and we may with great probability
+assume that, where Theodosius uses propositions from Euclid's
+<I>Elements,</I> propositions corresponding to them were used to
+prove the analogous propositions in the fourth-century
+<I>Sphaeric.</I> The propositions which, following this criterion,
+we may suppose to have been directly used for this purpose
+are, roughly, those represented by Eucl. I. 4, 8, 17, 19, 26, 29,
+47; III. 1-3, 7, 10, 16 Cor., 26, 28, 29; IV. 6; XI. 3, 4, 10, 11,
+12, 14, 16, 19, and the interpolated 38. It is, naturally, the
+subject-matter of Books I, III, and XI that is drawn upon,
+but, of course, the propositions mentioned by no means
+exhaust the number of pre-Euclidean propositions even in
+those Books. When, however, Hultsch increased the list of
+propositions by adding the whole chain of propositions (in-
+cluding Postulate 5) leading up to them in Euclid's arrange-
+ment, he took an unsafe course, because it is clear that many
+of Euclid's proofs were on different lines from those used
+by his predecessors.
+<pb n=351><head>AUTOLYCUS AND EUCLID</head>
+<p>The work <I>On the moving Sphere</I> assumes abstractly a
+sphere moving about the axis stretching from pole to pole,
+and different series of circular sections, the first series being
+great circles passing through the poles, the second small
+circles (as well as the equator) which are sections of the
+sphere by planes at right angles to the axis and are called
+the &lsquo;parallel circles&rsquo;, while the third kind are great circles
+inclined obliquely to the axis of the sphere; the motion of
+points on these circles is then considered in relation to the
+section by a fixed plane through the centre of the sphere.
+It is easy to recognize in the oblique great circle in the sphere
+the ecliptic or zodiac circle, and in the section made by the
+fixed plane the horizon, which is described as the circle
+in the sphere &lsquo;which defines (<G>o(ri/zwn</G>) the visible and the
+invisible portions of the sphere&rsquo;. To give an idea of the
+content of the work, I will quote a few enunciations from
+Autolycus and along with two of them, for the sake of
+comparison with Euclid, the corresponding enunciations from
+the <I>Phaenomena.</I>
+<table>
+<tr><th>Autolycus.</th><th>Euclid.</th></tr>
+<tr><td>1. If a sphere revolve uni-</td><td></td></tr>
+<tr><td>formly about its own axis, all</td><td></td></tr>
+<tr><td>the points on the surface of the</td><td></td></tr>
+<tr><td>sphere which are not on the</td><td></td></tr>
+<tr><td>axis will describe parallel</td><td></td></tr>
+<tr><td>circles which have the same</td><td></td></tr>
+<tr><td>poles as the sphere and are</td><td></td></tr>
+<tr><td>also at right angles to the axis.</td><td></td></tr>
+<tr><td>7. If the circle in the sphere</td><td>3. The circles which are at</td></tr>
+<tr><td>defining the visible and the</td><td>right angles to the axis and</td></tr>
+<tr><td>invisible portions of the sphere</td><td>cut the horizon make both</td></tr>
+<tr><td>be obliquely inclined to the</td><td>their risings and settings at</td></tr>
+<tr><td>axis, the circles which are at</td><td>the same points of the horizon.</td></tr>
+<tr><td>right angles to the axis and cut</td><td></td></tr>
+<tr><td>the defining circle [horizon]</td><td></td></tr>
+<tr><td>always make both their risings</td><td></td></tr>
+<tr><td>and settings at the same points</td><td></td></tr>
+<tr><td>of the defining circle [horizon]</td><td></td></tr>
+<tr><td>and further will also be simi-</td><td></td></tr>
+<tr><td>larly inclined to that circle.</td><td></td></tr>
+</table>
+<pb n=352><head>FROM PLATO TO EUCLID</head>
+<table>
+<tr><th>Autolycus.</th><th>Euclid.</th></tr>
+<tr><td>9. If in a sphere a great</td><td></td></tr>
+<tr><td>circle which is obliquely in-</td><td></td></tr>
+<tr><td>clined to the axis define the</td><td></td></tr>
+<tr><td>visible and the invisible por-</td><td></td></tr>
+<tr><td>tions of the sphere, then, of</td><td></td></tr>
+<tr><td>the points which rise at the</td><td></td></tr>
+<tr><td>same time, those towards the</td><td></td></tr>
+<tr><td>visible pole set later and, of</td><td></td></tr>
+<tr><td>those which set at the same</td><td></td></tr>
+<tr><td>time, those towards the visible</td><td></td></tr>
+<tr><td>pole rise earlier.</td><td></td></tr>
+<tr><td>11. If in a sphere a great</td><td>7. That the circle of the</td></tr>
+<tr><td>circle which is obliquely in-</td><td>zodiac rises and sets over the</td></tr>
+<tr><td>clined to the axis define the</td><td>whole extent of the horizon</td></tr>
+<tr><td>visible and the invisible por-</td><td>between the tropics is mani-</td></tr>
+<tr><td>tions of the sphere, and any</td><td>fest, forasmuch as it touches</td></tr>
+<tr><td>other oblique great circle</td><td>circles greater than those</td></tr>
+<tr><td>touch greater (parallel) circles</td><td>which the horizon touches.</td></tr>
+<tr><td>than those which the defin-</td><td></td></tr>
+<tr><td>ing circle (horizon) touches,</td><td></td></tr>
+<tr><td>the said other oblique circle</td><td></td></tr>
+<tr><td>makes its risings and settings</td><td></td></tr>
+<tr><td>over the whole extent of the</td><td></td></tr>
+<tr><td>circumference (arc) of the de-</td><td></td></tr>
+<tr><td>fining circle included between</td><td></td></tr>
+<tr><td>the parallel circles which it</td><td></td></tr>
+<tr><td>touches.</td><td></td></tr>
+</table>
+<p>It will be noticed that Autolycus's propositions are more
+abstract in so far as the &lsquo;other oblique circle&rsquo; in Autolycus
+is any other oblique circle, whereas in Euclid it definitely
+becomes the zodiac circle. In Euclid &lsquo;the great circle defining
+the visible and the invisible portions of the sphere&rsquo; is already
+shortened into the technical term &lsquo;horizon&rsquo; (<G>o(ri/zwn</G>), which is
+defined as if for the first time; &lsquo;Let the name <I>horizon</I> be
+given to the plane through us (as observers) passing through
+the universe and separating off the hemisphere which is visible
+above the earth.&rsquo;
+<p>The book <I>On Risings and Settings</I> is of astronomical interest
+only, and belongs to the region of <I>Phaenomena</I> as understood
+by Eudoxus and Aratus, that is, observational astronomy.
+It begins with definitions distinguishing between &lsquo;true&rsquo; and
+<pb n=353><head>AUTOLYCUS ON RISINGS AND SETTINGS</head>
+&lsquo;apparent&rsquo; morning- and evening-risings and settings of fixed
+stars. The &lsquo;true&rsquo; morning-rising (setting) is when the star
+rises (sets) at the moment of the sun's rising; the &lsquo;true&rsquo;
+morning-rising (setting) is, therefore invisible to us, and so is
+the &lsquo;true&rsquo; evening-rising (setting) which takes place at the
+moment when the sun is setting. The &lsquo;apparent&rsquo; morning-
+rising (setting) takes place when the star is first seen rising
+(setting) before the sun rises, and the &lsquo;apparent&rsquo; evening-
+rising (setting) when the star is last seen rising (setting) after
+the sun has set. The following are the enunciations of a few
+of the propositions in the treatise.
+<p>I. 1. In the case of each of the fixed stars the apparent
+morning-risings and settings are later than the true, and
+the apparent evening-risings and settings are earlier than
+the true.
+<p>I. 2. Each of the fixed stars is seen rising each night from
+the (time of its) apparent morning-rising to the time of its
+apparent evening-rising but at no other period, and the time
+during which the star is seen rising is less than half a year.
+<p>I. 5. In the case of those of the fixed stars which are on the
+zodiac circle, the interval from the time of their apparent
+evening-rising to the time of their apparent evening-setting is
+half a year, in the case of those north of the zodiac circle
+more than half a year, and in the case of those south of the
+zodiac circle less than half a year.
+<p>II. 1. The twelfth part of the zodiac circle in which the
+sun is, is neither seen rising nor setting, but is hidden; and
+similarly the twelfth part which is opposite to it is neither
+seen setting nor rising but is visible above the earth the whole
+of the nights.
+<p>II. 4. Of the fixed stars those which are cut off by the
+zodiac circle in the northerly or the southerly direction will
+reach their evening-setting at an interval of five months from
+their morning-rising.
+<p>II. 9. Of the stars which are carried on the same (parallel-)
+circle those which are cut off by the zodiac circle in the
+northerly direction will be hidden a shorter time than those
+on the southern side of the zodiac.
+<pb><C>XI
+EUCLID
+Date and traditions.</C>
+<p>WE have very few particulars of the lives of the great
+mathematicians of Greece. Even Euclid is no exception.
+Practically all that is known about him is contained in a few
+sentences of Proclus's summary:
+<p>&lsquo;Not much younger than these (sc. Hermotimus of Colophon
+and Philippus of Mende or Medma) is Euclid, who put to-
+gether the Elements, collecting many of Eudoxus's theorems,
+perfecting many of Theaetetus's, and also bringing to irre-
+fragable demonstration the things which were only somewhat
+loosely proved by his predecessors. This man lived in the
+time of the first Ptolemy. For Archimedes, who came
+immediately after the first (Ptolemy), makes mention of
+Euclid; and further they say that Ptolemy once asked him if
+there was in geometry any shorter way than that of the
+Elements, and he replied that there was no royal road to
+geometry. He is then younger than the pupils of Plato, but
+older than Eratosthenes and Archimedes, the latter having
+been contemporaries, as Eratosthenes somewhere says.&rsquo;<note>Proclus on Eucl. I, p. 68. 6-20.</note>
+<p>This passage shows that even Proclus had no direct know-
+ledge of Euclid's birthplace, or of the dates of his birth and
+death; he can only infer generally at what period he flourished.
+All that is certain is that Euclid was later than the first
+pupils of Plato and earlier than Archimedes. As Plato died
+in 347 B.C. and Archimedes lived from 287 to 212 B.C., Euclid
+must have flourished about 300 B.C., a date which agrees well
+with the statement that he lived under the first Ptolemy, who
+reigned from 306 to 283 B.C.
+<pb n=355><head>DATE AND TRADITIONS</head>
+<p>More particulars are, it is true, furnished by Arabian
+authors. We are told that
+<p>&lsquo;Euclid, son of Naucrates, and grandson of Zenarchus [the
+<I>Fihrist</I> has &lsquo;son of Naucrates, the son of Berenice (?)&rsquo;], called
+the author of geometry, a philosopher of somewhat ancient
+date, a Greek by nationality, domiciled at Damascus, born at
+Tyre, most learned in the science of geometry, published
+a most excellent and most useful work entitled the foundation
+or elements of geometry, a subject in which no more general
+treatise existed before among the Greeks: nay, there was no
+one even of later date who did not walk in his footsteps and
+frankly profess his doctrine. Hence also Greek, Roman,
+and Arabian geometers not a few, who undertook the task of
+illustrating this work, published commentaries, scholia, and
+notes upon it, and made an abridgement of the work itself.
+For this reason the Greek philosophers used to post up on the
+doors of their schools the well-known notice, &ldquo;Let no one
+come to our school, who has not first learnt the elements
+of Euclid&rdquo;.&rsquo;<note>Casiri, <I>Bibliotheca Arabico-Hispana Escurialensis</I>, i, p. 339 (Casiri's
+source is the <I>Ta)r&imacr;kh al-&Hdot;ukam&amacr;</I> of al-Qif&tdot;&imacr; (d. 1248).</note>
+<p>This shows the usual tendency of the Arabs to romance.
+They were in the habit of recording the names of grand-
+fathers, while the Greeks were not; Damascus and Tyre were
+no doubt brought in to gratify the desire which the Arabians
+always showed to connect famous Greeks in some way or other
+with the east (thus they described Pythagoras as a pupil of the
+wise Salomo, and Hipparchus as &lsquo;the Chaldaean&rsquo;). We recog-
+nize the inscription over the doors of the schools of the Greek
+philosophers as a variation of Plato's <G>mhdei\s a)gewme/trhtos
+ei)si/tw</G>; the philosopher has become Greek philosophers in
+general, the school their schools, while geometry has become
+the <I>Elements</I> of Euclid. The Arabs even explained that the
+name of Euclid, which they pronounced variously as <I>Uclides</I> or
+<I>Icludes</I>, was compounded of <I>Ucli</I>, a key, and <I>Dis</I>, a measure, or,
+as some say, geometry, so that Uclides is equivalent to the
+<I>key of geometry</I>!
+<p>In the Middle Ages most translators and editors spoke of
+Euclid as Euclid <I>of Megara</I>, confusing our Euclid with Euclid
+the philosopher, and the contemporary of Plato, who lived about
+400 B.C. The first trace of the confusion appears in Valerius
+<pb n=356><head>EUCLID</head>
+Maximus (in the time of Tiberius) who says<note>viii. 12, ext. 1.</note> that Plato,
+on being appealed to for a solution of the problem of doubling
+the cube, sent the inquirers to &lsquo;Euclid the geometer&rsquo;. The
+mistake was seen by one Constantinus Lascaris (d. about
+1493), and the first translator to point it out clearly was
+Commandinus (in his translation of Euclid published in 1572).
+<p>Euclid may have been a Platonist, as Proclus says, though
+this is not certain. In any case, he probably received his
+mathematical training in Athens from the pupils of Plato;
+most of the geometers who could have taught him were of
+that school. But he himself taught and founded a school
+at Alexandria, as we learn from Pappus's statement that
+Apollonius &lsquo;spent a very long time with the pupils of Euclid
+at Alexandria&rsquo;.<note>Pappus, vii, p. 678. 10-12.</note> Here again come in our picturesque
+Arabians,<note>The authorities are al-Kind&imacr;, <I>De instituto libri Euclidis</I> and a commen-
+tary by Q&amacr;&ddot;&imacr;z&amacr;de on the <I>Ashkal at-ta)s&imacr;s</I> of Ashraf Shamsadd&imacr;n as-Samar-
+qand&imacr; (quoted by Casiri and &Hdot;&amacr;j&imacr; Khalfa).</note> who made out that the <I>Elements</I> were originally
+written by a man whose name was Apollonius, a carpenter,
+who wrote the work in fifteen books or sections (this idea
+seems to be based on some misunderstanding of Hypsicles's
+preface to the so-called Book XIV of Euclid), and that, as
+some of the work was lost in course of time and the rest
+disarranged, one of the kings at Alexandria who desired to
+study geometry and to master this treatise in particular first
+questioned about it certain learned men who visited him, and
+then sent for Euclid, who was at that time famous as a
+geometer, and asked him to revise and complete the work
+and reduce it to order, upon which Euclid rewrote the work
+in thirteen books, thereafter known by his name.
+<p>On the character of Euclid Pappus has a remark which,
+however, was probably influenced by his obvious animus
+against Apollonius, whose preface to the <I>Conics</I> seemed to him
+to give too little credit to Euclid for his earlier work in the same
+subject. Pappus contrasts Euclid's attitude to his predecessors.
+Euclid, he says, was no such boaster or controversialist: thus
+he regarded Aristaeus as deserving credit for the discoveries
+he had made in conics, and made no attempt to anticipate
+him or to construct afresh the same system, such was his
+scrupulous fairness and his exemplary kindliness to all who
+<pb n=357><head>DATE AND TRADITIONS</head>
+could advance mathematical science to however small an
+extent.<note>Pappus, vii, pp. 676. 25-678. 6.</note> Although, as I have indicated, Pappus's motive was
+rather to represent Apollonius in a relatively unfavourable
+light than to state a historical fact about Euclid, the state-
+ment accords well with what we should gather from Euclid's
+own works. These show no sign of any claim to be original;
+in the <I>Elements</I>, for instance, although it is clear that he
+made great changes, altering the arrangement of whole Books,
+redistributing propositions between them, and inventing new
+proofs where the new order made the earlier proofs inappli-
+cable, it is safe to say that he made no more alterations than
+his own acumen and the latest special investigations (such as
+Eudoxus's theory of proportion) showed to be imperative in
+order to make the exposition of the whole subject more
+scientific than the earlier efforts of writers of elements. His
+respect for tradition is seen in his retention of some things
+which were out of date and useless, e. g. certain definitions
+never afterwards used, the solitary references to the angle
+of a semicircle or the angle of a segment, and the like; he
+wrote no sort of preface to his work (would that he had!)
+such as those in which Archimedes and Apollonius introduced
+their treatises and distinguished what they claimed as new in
+them from what was already known: he plunges at once into
+his subject, &lsquo;<I>A point is that which has no part</I>&rsquo;!
+<p>And what a teacher he must have been! One story enables
+us to picture him in that capacity. According to Stobaeus,
+<p>&lsquo;some one who had begun to read geometry with Euclid,
+when he had learnt the first theorem, asked Euclid, &ldquo;what
+shall I get by learning these things?&rdquo; Euclid called his slave
+and said, &ldquo;Give him threepence, since he must make gain out
+of what he learns&rdquo;.&rsquo;<note>Stobaeus, <I>Floril.</I> iv. p. 205.</note>
+<p>Ancient commentaries, criticisms, and references.
+<p>Euclid has, of course, always been known almost exclusively
+as the author of the <I>Elements.</I> From Archimedes onwards
+the Greeks commonly spoke of him as <G>o( stoixeiw/ths</G>, the
+writer of the <I>Elements</I>, instead of using his name. This
+wonderful book, with all its imperfections, which indeed are
+slight enough when account is taken of the date at which
+<pb n=358><head>EUCLID</head>
+it appeared, is and will doubtless remain the greatest mathe-
+matical text-book of all time. Scarcely any other book
+except the Bible can have circulated more widely the world
+over, or been more edited and studied. Even in Greek times
+the most accomplished mathematicians occupied themselves
+with it; Heron, Pappus, Porphyry, Proclus and Simplicius
+wrote commentaries; Theon of Alexandria re-edited it, alter-
+ing the language here and there, mostly with a view to
+greater clearness and consistency, and interpolating inter-
+mediate steps, alternative proofs, separate &lsquo;cases&rsquo;, porisms
+(corollaries) and lemmas (the most important addition being
+the second part of VI. 33 relating to <I>sectors</I>). Even the great
+Apollonius was moved by Euclid's work to discuss the first
+principles of geometry; his treatise on the subject was in
+fact a criticism of Euclid, and none too successful at that;
+some alternative definitions given by him have point, but his
+alternative solutions of some of the easy problems in Book I
+do not constitute any improvement, and his attempt to prove
+the axioms (if one may judge by the case quoted by Proclus,
+that of Axiom 1) was thoroughly misconceived.
+<p>Apart from systematic commentaries on the whole work or
+substantial parts of it, there were already in ancient times
+discussions and controversies on special subjects dealt with by
+Euclid, and particularly his theory of parallels. The fifth
+Postulate was a great stumbling-block. We know from
+Aristotle that up to his time the theory of parallels had not
+been put on a scientific basis<note><I>Anal. Prior.</I> ii. 16. 65 a 4.</note>: there was apparently some
+<I>petitio principii</I> lurking in it. It seems therefore clear that
+Euclid was the first to apply the bold remedy of laying down
+the indispensable principle of the theory in the form of an
+indemonstrable Postulate. But geometers were not satisfied
+with this solution. Posidonius and Geminus tried to get
+over the difficulty by substituting an <I>equidistance</I> theory of
+parallels. Ptolemy actually tried to prove Euclid's postulate,
+as also did Proclus, and (according to Simplicius) one Diodorus,
+as well as &lsquo;Aganis&rsquo;; the attempt of Ptolemy is given by
+Proclus along with his own, while that of &lsquo;Aganis&rsquo; is repro-
+duced from Simplicius by the Arabian commentator an-
+Nair&imacr;z&imacr;.
+<pb n=359><head>COMMENTARIES, CRITICISMS &amp; REFERENCES</head>
+<p>Other very early criticisms there were, directed against the
+very first steps in Euclid's work. Thus Zeno of Sidon, an
+Epicurean, attacked the proposition I. 1 on the ground that it
+is not conclusive unless it be first assumed that neither two
+straight lines nor two circumferences can have a common
+segment; and this was so far regarded as a serious criticism
+that Posidonius wrote a whole book to controvert Zeno.<note>Proclus on Eucl. I, p. 200. 2.</note>
+Again, there is the criticism of the Epicureans that I. 20,
+proving that any two sides in a triangle are together greater
+than the third, is evident even to an ass and requires no
+proof. I mention these isolated criticisms to show that the
+<I>Elements</I>, although they superseded all other Elements and
+never in ancient times had any rival, were not even at the
+first accepted without question.
+<p>The first Latin author to mention Euclid is Cicero; but
+it is not likely that the <I>Elements</I> had then been translated
+into Latin. Theoretical geometry did not appeal to the
+Romans, who only cared for so much of it as was useful for
+measurements and calculations. Philosophers studied Euclid,
+but probably in the original Greek; Martianus Capella speaks
+of the effect of the mention of the proposition &lsquo;how to con-
+struct an equilateral triangle on a given straight line&rsquo; among
+a company of philosophers, who, recognizing the first pro-
+position of the <I>Elements</I>, straightway break out into encomiums
+on Euclid.<note>Mart. Capella, vi. 724.</note> Beyond a fragment in a Verona palimpsest of
+a free rendering or rearrangement of some propositions from
+Books XII and XIII dating apparently from the fourth century,
+we have no trace of any Latin version before Bo&euml;tius (born
+about A.D. 480), to whom Magnus Aurelius Cassiodorus and
+Theodoric attribute a translation of Euclid. The so-called
+geometry of Bo&euml;tius which has come down to us is by no
+means a translation of Euclid; but even the redaction of this
+in two Books which was edited by Friedlein is not genuine,
+having apparently been put together in the eleventh century
+from various sources; it contains the definitions of Book I,
+the Postulates (five in number), the Axioms (three only), then
+some definitions from Eucl. II, III, IV, followed by the
+<I>enunciations</I> only (without proofs) of Eucl. I, ten propositions
+<pb n=360><head>EUCLID</head>
+of Book II, and a few of Books III and IV, and lastly a
+passage indicating that the editor will now give something of
+his own, which turns out to be a literal translation of the
+proofs of Eucl. I. 1-3. This proves that the Pseudo-Bo&euml;tius
+had a Latin translation of Euclid from which he extracted
+these proofs; moreover, the text of the definitions from
+Book I shows traces of perfectly correct readings which are
+not found even in the Greek manuscripts of the tenth century,
+but which appear in Proclus and other ancient sources.
+Fragments of such a Latin translation are also found in
+the <I>Gromatici veteres.</I><note>Ed. Lachmann, pp. 377 sqq.</note>
+<C>The text of the Elements.</C>
+<p>All our Greek texts of the <I>Elements</I> up to a century ago
+depended upon manuscripts containing Theon's recension of the
+work; these manuscripts purport, in their titles, to be either
+&lsquo;from the edition of Theon&rsquo; (<G>e)k th=s *qe/wnos e)kdo/sews</G>) or
+&lsquo;from the lectures of Theon&rsquo; (<G>a)po\ sunousiw=n tou= *qe/wnos</G>).
+Sir Henry Savile in his <I>Praelectiones</I> had drawn attention
+to the passage in Theon's Commentary on Ptolemy<note>I, p. 201, ed. Halma.</note> quoting
+the second part of VI. 33 about sectors as having been proved
+by <I>himself</I> in his edition of the <I>Elements</I>; but it was not
+till Peyrard discovered in the Vatican the great MS.
+gr. 190, containing neither the words from the titles of the
+other manuscripts quoted above nor the addition to VI. 33,
+that scholars could get back from Theon's text to what thus
+represents, on the face of it, a more ancient edition than
+Theon's. It is also clear that the copyist of P (as the manu-
+script is called after Peyrard), or rather of its archetype,
+had before him the two recensions and systematically gave
+the preference to the earlier one; for at XIII. 6 in P the first
+hand has a marginal note, &lsquo;This theorem is not given in most
+copies of the <I>new edition</I>, but is found in those of the old&rsquo;.
+The <I>editio princeps</I> (Basel, 1533) edited by Simon Grynaeus
+was based on two manuscripts (Venetus Marcianus 301 and
+Paris. gr. 2343) of the sixteenth century, which are among
+the worst. The Basel edition was again the foundation
+of the text of Gregory (Oxford, 1703), who only consulted the
+<pb n=361><head>THE TEXT OF THE ELEMENTS</head>
+manuscripts bequeathed by Savile to the University in
+places where the Basel text differed from the Latin version
+of Commandinus which he followed in the main. It was
+a pity that even Peyrard in his edition (1814-18) only
+corrected the Basel text by means of P, instead of rejecting
+it altogether and starting afresh; but he adopted many of the
+readings of P and gave a conspectus of them in an appendix.
+E. F. August's edition (1826-9) followed P more closely, and
+he consulted the Viennese MS. gr. 103 also; but it was
+left for Heiberg to bring out a new and definitive Greek text
+(1883-8) based on P and the best of the Theonine manuscripts,
+and taking account of external sources such as Heron and
+Proclus. Except in a few passages, Proclus's manuscript does
+not seem to have been of the best, but authors earlier than
+Theon, e.g. Heron, generally agree with our best manuscripts.
+Heiberg concludes that the <I>Elements</I> were most spoiled by
+interpolations about the third century, since Sextus Empiricus
+had a correct text, while Iamblicus had an interpolated one.
+<p>The differences between the inferior Theonine manuscripts
+and the best sources are perhaps best illustrated by the arrange-
+ment of postulates and axioms in Book I. Our ordinary
+editions based on Simson have three postulates and twelve
+axioms. Of these twelve axioms the eleventh (stating that
+all right angles are equal) is, in the genuine text, the fourth
+Postulate, and the twelfth Axiom (the Parallel-Postulate) is
+the fifth Postulate; the Postulates were thus originally five
+in number. Of the ten remaining Axioms or Common
+Notions Heron only recognized the first three, and Proclus
+only these and two others (that things which coincide are
+equal, and that the whole is greater than the part); it is fairly
+certain, therefore, that the rest are interpolated, including the
+assumption that two straight lines cannot enclose a space
+(Euclid himself regarded this last fact as involved in Postu-
+late 1, which implies that a straight line joining one point
+to another is <I>unique</I>).
+<C>Latin and Arabic translations.</C>
+<p>The first Latin translations which we possess in a complete
+form were made not from the Greek but from the Arabic.
+It was as early as the eighth century that the <I>Elements</I> found
+<pb n=362><head>EUCLID</head>
+their way to Arabia. The Caliph al-Man&sdot;&umacr;r (754-75), as the
+result of a mission to the Byzantine Emperor, obtained a copy
+of Euclid among other Greek books, and the Caliph al-Ma'm&umacr;n
+(813-33) similarly obtained manuscripts of Euclid, among
+others, from the Byzantines. Al-Hajj&amacr;j b. Y&umacr;suf b. Ma&tdot;ar made
+two versions of the <I>Elements</I>, the first in the reign of H&amacr;r&umacr;n
+ar-Rash&imacr;d (786-809), the second for al-Ma)m&umacr;n; six Books of
+the second of these versions survive in a Leyden manuscript
+(Cod. Leidensis 399. 1) which is being edited along with
+an-Nair&imacr;z&imacr;'s commentary by Besthorn and Heiberg<note>Parts I, i. 1893, I, ii. 1897, II, i. 1900, II, ii. 1905, III, i. 1910 (Copen-
+hagen).</note>; this
+edition was abridged, with corrections and explanations, but
+without change of substance, from the earlier version, which
+appears to be lost. The work was next translated by Ab&umacr;
+Ya`q&umacr;b Is&hdot;&amacr;q b. &Hdot;unain b. Is&hdot;&amacr;q al-(Ib&amacr;d&imacr; (died 910), evidently
+direct from the Greek; this translation seems itself to have
+perished, but we have it as revised by Th&amacr;bit b. Qurra (died
+901) in two manuscripts (No. 279 of the year 1238 and No. 280
+written in 1260-1) in the Bodleian Library; Books I-XIII in
+these manuscripts are in the Is&hdot;&amacr;q-Th&amacr;bit version, while the
+non-Euclidean Books XIV, XV are in the translation of Qus&tdot;&amacr;
+b. L&umacr;q&amacr; al-Ba`labakk&imacr; (died about 912). Is&hdot;&amacr;q's version seems
+to be a model of good translation; the technical terms are
+simply and consistently rendered, the definitions and enun-
+ciations differ only in isolated cases from the Greek, and the
+translator's object seems to have been only to get rid of
+difficulties and unevennesses in the Greek text while at the
+same time giving a faithful reproduction of it. The third
+Arabic version still accessible to us is that of Na&sdot;&imacr;radd&imacr;n
+a&tdot;-&Tdot;&umacr;s&imacr; (born in 1201 at &Tdot;&umacr;s in Khur&amacr;s&amacr;n); this, however,
+is not a translation of Euclid but a rewritten version based
+upon the older Arabic translations. On the whole, it appears
+probable that the Arabic tradition (in spite of its omission
+of lemmas and porisms, and, except in a very few cases, of
+the interpolated alternative proofs) is not to be preferred
+to that of the Greek manuscripts, but must be regarded as
+inferior in authority.
+<p>The known Latin translations begin with that of Athelhard,
+an Englishman, of Bath; the date of it is about 1120. That
+<pb n=363><head>LATIN AND ARABIC TRANSLATIONS</head>
+it was made from the Arabic is clear from the occurrence
+of Arabic words in it; but Athelhard must also have had
+before him a translation of (at least) the enunciations of
+Euclid based ultimately upon the Greek text, a translation
+going back to the old Latin version which was the common
+source of the passage in the <I>Gromatici</I> and &lsquo;Bo&euml;tius&rsquo;. But
+it would appear that even before Athelhard's time some sort
+of translation, or at least fragments of one, were available
+even in England if one may judge by the Old English verses:
+&lsquo;The clerk Euclide on this wyse hit fonde<lb>
+Thys craft of gemetry yn Egypte londe<lb>
+Yn Egypte he tawghte hyt ful wyde,<lb>
+In dyvers londe on every syde.<lb>
+Mony erys afterwarde y understonde<lb>
+Yer that the craft com ynto thys londe.<lb>
+Thys craft com into England, as y yow say,<lb>
+Yn tyme of good Kyng Adelstone's day&rsquo;,<lb>
+which would put the introduction of Euclid into England
+as far back as A.D. 924-40.
+<p>Next, Gherard of Cremona (1114&mdash;87) is said to have
+translated the &lsquo;15 Books of Euclid&rsquo; from the Arabic as he
+undoubtedly translated an-Nair&imacr;z&imacr;'s commentary on Books
+I-X; this translation of the <I>Elements</I> was till recently
+supposed to have been lost, but in 1904 A. A. Bj&ouml;rnbo dis-
+covered in manuscripts at Paris, Boulogne-sur-Mer and Bruges
+the whole, and at Rome Books X-XV, of a translation which
+he gives good ground for identifying with Gherard's. This
+translation has certain Greek words such as <I>rombus, romboides</I>,
+where Athelhard keeps the Arabic terms; it was thus clearly
+independent of Athelhard's, though Gherard appears to have
+had before him, in addition, an old translation of Euclid from
+the Greek which Athelhard also used. Gherard's translation
+is much clearer than Athelhard's; it is neither abbreviated
+nor &lsquo;edited&rsquo; in the same way as Athelhard's, but it is a word
+for word translation of an Arabic manuscript containing a
+revised and critical edition of Th&amacr;bit's version.
+<p>A third translation from the Arabic was that of Johannes
+Campanus, which came some 150 years after that of Athelhard,
+That Campanus's translation was not independent of Athel-
+hard's is proved by the fact that, in all manuscripts and
+<pb n=364><head>EUCLID</head>
+editions, the definitions, postulates and axioms, and the 364
+enunciations are word for word identical in Athelhard and
+Campanus. The exact relation between the two seems even
+yet not to have been sufficiently elucidated. Campanus may
+have used Athelhard's translation and only developed the
+proofs by means of another redaction of the Arabian Euclid.
+Campanus's translation is the clearer and more complete,
+following the Greek text more closely but still at some
+distance; the arrangement of the two is different; in Athel-
+hard the proofs regularly precede the enunciations, while
+Campanus follows the usual order. How far the differences
+in the proofs and the additions in each are due to the
+translators themselves or go back to Arabic originals is a
+moot question; but it seems most probable that Campanus
+stood to Athelhard somewhat in the relation of a commen-
+tator, altering and improving his translation by means of
+other Arabic originals.
+<C>The first printed editions.</C>
+<p>Campanus's translation had the luck to be the first to be
+put into print. It was published at Venice by Erhard Ratdolt
+in 1482. This beautiful and very rare book was not only
+the first printed edition of Euclid, but also the first printed
+mathematical book of any importance. It has margins of
+2 1/2 inches and in them are placed the figures of the proposi-
+tions. Ratdolt says in his dedication that, at that time,
+although books by ancient and modern authors were being
+printed every day in Venice, little or nothing mathematical
+had appeared; this fact he puts down to the difficulty involved
+by the figures, which no one had up to that time succeeded in
+printing; he adds that after much labour he had discovered
+a method by which figures could be produced as easily as
+letters. Experts do not seem even yet to be agreed as to the
+actual way in which the figures were made, whether they
+were woodcuts or whether they were made by putting together
+lines and circular arcs as letters are put together to make
+words. How eagerly the opportunity of spreading geometrical
+knowledge was seized upon is proved by the number of
+editions which followed in the next few years. Even the
+<pb n=365><head>THE FIRST PRINTED EDITIONS</head>
+year 1482 saw two forms of the book, though they only differ
+in the first sheet. Another edition came out at Ulm in 1486,
+and another at Vicenza in 1491.
+<p>In 1501 G. Valla gave in his encyclopaedic work <I>De ex-
+petendis et fugiendis rebus</I> a number of propositions with
+proofs and scholia translated from a Greek manuscript which
+was once in his possession; but Bartolomeo Zamberti (Zam-
+bertus) was the first to bring out a translation from the
+Greek text of the whole of the <I>Elements</I>, which appeared
+at Venice in 1505. The most important Latin translation
+is, however, that of Commandinus (1509-75), who not only
+followed the Greek text more closely than his predecessors,
+but added to his translation some ancient scholia as well
+as good notes of his own; this translation, which appeared
+in 1572, was the foundation of most translations up to the
+time of Peyrard, including that of Simson, and therefore of
+all those editions, numerous in England, which gave Euclid
+&lsquo;chiefly after the text of Dr. Simson&rsquo;.
+<C>The study of Euclid in the Middle Ages.</C>
+<p>A word or two about the general position of geometry in
+education during the Middle Ages will not be out of place in
+a book for English readers, in view of the unique place which
+Euclid has till recently held as a text-book in this country.
+From the seventh to the tenth century the study of geometry
+languished: &lsquo;We find in the whole literature of that time
+hardly the slightest sign that any one had gone farther
+in this department of the Quadrivium than the definitions
+of a triangle, a square, a circle, or of a pyramid or cone, as
+Martianus Capella and Isidorus (Hispalensis, died as Bishop
+of Seville in 636) left them.&rsquo;<note>Hankel, <I>op. cit.</I>, pp. 311-12.</note> (Isidorus had disposed of the
+four subjects of Arithmetic, Geometry, Music and Astronomy
+in <I>four pages</I> of his encyclopaedic work <I>Origines</I> or <I>Ety-
+mologiae</I>). In the tenth century appeared a &lsquo;reparator
+studiorum&rsquo; in the person of the great Gerbert, who was born
+at Aurillac, in Auvergne, in the first half of the tenth century,
+and after a very varied life ultimately (in 999) became Pope
+Sylvester II; he died in 1003. About 967 he went on
+<pb n=366><head>EUCLID</head>
+a journey to Spain, where he studied mathematics. In 970 he
+went to Rome with Bishop Hatto of Vich (in the province of
+Barcelona), and was there introduced by Pope John XIII
+to the German king Otto I. To Otto, who wished to find
+him a post as a teacher, he could say that &lsquo;he knew enough of
+mathematics for this, but wished to improve his knowledge
+of logic&rsquo;. With Otto's consent he went to Reims, where he
+became Scholasticus or teacher at the Cathedral School,
+remaining there for about ten years, 972 to 982. As the result
+of a mathematico-philosophic argument in public at Ravenna
+in 980, he was appointed by Otto II to the famous monastery
+at Bobbio in Lombardy, which, fortunately for him, was rich
+in valuable manuscripts of all sorts. Here he found the
+famous &lsquo;Codex Arcerianus&rsquo; containing fragments of the
+works of the <I>Gromatici</I>, Frontinus, Hyginus, Balbus, Nipsus,
+Epaphroditus and Vitruvius Rufus. Although these frag-
+ments are not in themselves of great merit, there are things
+in them which show that the authors drew upon Heron of
+Alexandria, and Gerbert made the most of them. They
+formed the basis of his own &lsquo;Geometry&rsquo;, which may have
+been written between the years 981 and 983. In writing this
+book Gerbert evidently had before him Bo&euml;tius's <I>Arithmetic</I>,
+and in the course of it he mentions Pythagoras, Plato's
+<I>Timaeus</I>, with Chalcidius's commentary thereon, and Eratos-
+thenes. The geometry in the book is mostly practical; the
+theoretical part is confined to necessary preliminary matter,
+definitions, &amp;c., and a few proofs; the fact that the sum of the
+angles of a triangle is equal to two right angles is proved in
+Euclid's manner. A great part is taken up with the solution
+of triangles, and with heights and distances. The Archimedean
+value of <G>p</G> (22/7) is used in stating the area of a circle; the
+surface of a sphere is given as 11/21 <I>D</I><SUP>3</SUP>. The plan of the book
+is quite different from that of Euclid, showing that Gerbert
+could neither have had Euclid's <I>Elements</I> before him, nor,
+probably, Bo&euml;tius's <I>Geometry</I>, if that work in its genuine
+form was a version of Euclid. When in a letter written
+probably from Bobbio in 983 to Adalbero, Archbishop of
+Reims, he speaks of his expectation of finding &lsquo;eight volumes
+of Bo&euml;tius on astronomy, also the most famous of figures
+(presumably propositions) in geometry and other things not
+<pb n=367><head>STUDY OF EUCLID IN THE MIDDLE AGES</head>
+less admirable&rsquo;, it is not clear that he actually found these
+things, and it is still less certain that the geometrical matter
+referred to was Bo&euml;tius's <I>Geometry.</I>
+<p>From Gerbert's time, again, no further progress was made
+until translations from the Arabic began with Athelhard and
+the rest. Gherard of Cremona (died 1187), who translated
+the <I>Elements</I> and an-Nair&imacr;z&imacr;'s commentary thereon, is credited
+with a whole series of translations from the Arabic of Greek
+authors; they included the <I>Data</I> of Euclid, the <I>Sphaerica</I> of
+Theodosius, the <I>Sphaerica</I> of Menelaus, the <I>Syntaxis</I> of Ptolemy;
+besides which he translated Arabian geometrical works such
+as the <I>Liber trium fratrum</I>, and also the algebra of Mu&hdot;ammad
+b. M&umacr;s&amacr;. One of the first results of the interest thus aroused
+in Greek and Arabian mathematics was seen in the very
+remarkable works of Leonardo of Pisa (Fibonacci). Leonardo
+first published in 1202, and then brought out later (1228) an
+improved edition of, his <I>Liber abaci</I> in which he gave the
+whole of arithmetic and algebra as known to the Arabs, but
+in a free and independent style of his own; in like manner in
+his <I>Practica geometriae</I> of 1220 he collected (1) all that the
+<I>Elements</I> of Euclid and Archimedes's books on the <I>Measure-
+ment of a Circle</I> and <I>On the Sphere and Cylinder</I> had taught
+him about the measurement of plane figures bounded by
+straight lines, solid figures bounded by planes, the circle and
+the sphere respectively, (2) divisions of figures in different
+proportions, wherein he based himself on Euclid's book <I>On the
+divisions of figures</I>, but carried the subject further, (3) some
+trigonometry, which he got from Ptolemy and Arabic sources
+(he uses the terms <I>sinus rectus</I> and <I>sinus versus</I>); in the
+treatment of these varied subjects he showed the same mastery
+and, in places, distinct originality. We should have expected
+a great general advance in the next centuries after such a
+beginning, but, as Hankel says, when we look at the work of
+Luca Paciuolo nearly three centuries later, we find that the
+talent which Leonardo had left to the Latin world had lain
+hidden in a napkin and earned no interest. As regards the
+place of geometry in education during this period we have
+the evidence of Roger Bacon (1214-94), though he, it
+is true, seems to have taken an exaggerated view of the
+incompetence of the mathematicians and teachers of his
+<pb n=368><head>EUCLID</head>
+time; the philosophers of his day, he says, despised geo-
+metry, languages, &amp;c., declaring that they were useless;
+people in general, not finding utility in any science such as
+geometry, at once recoiled, unless they were boys forced to
+it by the rod, from the idea of studying it, so that they
+would hardly learn as much as three or four propositions;
+the fifth proposition of Euclid was called <I>Elefuga</I> or <I>fuga
+miserorum.</I><note>Roger Bacon, <I>Opus Tertium</I>, cc. iv, vi.</note>
+<p>As regards Euclid at the Universities, it may be noted that
+the study of geometry seems to have been neglected at the
+University of Paris. At the reformation of the University in
+1336 it was only provided that no one should take a Licentiate
+who had not attended lectures on some mathematical books;
+the same requirement reappears in 1452 and 1600. From the
+preface to a commentary on Euclid which appeared in 1536
+we learn that a candidate for the degree of M.A. had to take
+a solemn oath that he had attended lectures on the first six
+Books; but it is doubtful whether for the examinations more
+than Book I was necessary, seeing that the proposition I. 47
+was known as <I>Magister matheseos.</I> At the University of
+Prague (founded in 1348) mathematics were more regarded.
+Candidates for the Baccalaureate had to attend lectures on
+the <I>Tractatus de Sphaera materiali</I>, a treatise on the funda-
+mental ideas of spherical astronomy, mathematical geography
+and the ordinary astronomical phenomena, but without the
+help of mathematical propositions, written by Johannes de
+Sacrobosco (i. e. of Holywood, in Yorkshire) in 1250, a book
+which was read at all Universities for four centuries and
+many times commented upon; for the Master's degree lectures
+on the first six Books of Euclid were compulsory. Euclid
+was lectured upon at the Universities of Vienna (founded 1365),
+Heidelberg (1386), Cologne (1388); at Heidelberg an oath was
+required from the candidate for the Licentiate corresponding
+to M.A. that he had attended lectures on some whole books and
+not merely parts of several books (not necessarily, it appears,
+of Euclid); at Vienna, the first five Books of Euclid were
+required; at Cologne, no mathematics were required for the
+Baccalaureate, but the candidate for M.A. must have attended
+<pb n=369><head>STUDY OF EUCLID IN THE MIDDLE AGES</head>
+lectures on the <I>Sphaera mundi</I>, planetary theory, three Books
+of Euclid, optics and arithmetic. At Leipzig (founded 1409),
+as at Vienna and Prague, there were lectures on Euclid for
+some time at all events, though Hankel says that he found no
+mention of Euclid in a list of lectures given in the consecutive
+years 1437-8, and Regiomontanus, when he went to Leipzig,
+found no fellow-students in geometry. At Oxford, in the
+middle of the fifteenth century, the first two Books of Euclid
+were read, and doubtless the Cambridge course was similar.
+<C>The first English editions.</C>
+<p>After the issue of the first printed editions of Euclid,
+beginning with the translation of Campano, published by
+Ratdolt, and of the <I>editio princeps</I> of the Greek text (1533),
+the study of Euclid received a great impetus, as is shown
+by the number of separate editions and commentaries which
+appeared in the sixteenth century. The first complete English
+translation by Sir Henry Billingsley (1570) was a monumental
+work of 928 pages of folio size, with a preface by John Dee,
+and notes extracted from all the most important commentaries
+from Proclus down to Dee himself, a magnificent tribute to
+the immortal Euclid. About the same time Sir Henry Savile
+began to give <I>unpaid</I> lectures on the Greek geometers; those
+on Euclid do not indeed extend beyond I. 8, but they are
+valuable because they deal with the difficulties connected with
+the preliminary matter, the definitions, &amp;c., and the tacit
+assumptions contained in the first propositions. But it was
+in the period from about 1660 to 1730, during which Wallis
+and Halley were Professors at Oxford, and Barrow and
+Newton at Cambridge, that the study of Greek mathematics
+was at its height in England. As regards Euclid in particular
+Barrow's influence was doubtless very great. His Latin
+version (<I>Euclidis Elementorum Libri XV breviter demon-
+strati</I>) came out in 1655, and there were several more editions
+of the same published up to 1732; his first English edition
+appeared in 1660, and was followed by others in 1705, 1722,
+1732, 1751. This brings us to Simson's edition, first published
+both in Latin and English in 1756. It is presumably from
+this time onwards that Euclid acquired the unique status as
+<pb n=370><head>EUCLID</head>
+a text-book which it maintained till recently. I cannot help
+thinking that it was Barrow's influence which contributed
+most powerfully to this. We are told that Newton, when
+he first bought a Euclid in 1662 or 1663, thought it &lsquo;a trifling
+book&rsquo;, as the propositions seemed to him obvious; after-
+wards, however, on Barrow's advice, he studied the <I>Elements</I>
+carefully and derived, as he himself stated, much benefit
+therefrom.
+<C>Technical terms connected with the classical form
+of a proposition.</C>
+<p>As the classical form of a proposition in geometry is that
+which we find in Euclid, though it did not originate with
+him, it is desirable, before we proceed to an analysis of the
+<I>Elements</I>, to give some account of the technical terms used by
+the Greeks in connexion with such propositions and their
+proofs. We will take first the terms employed to describe the
+formal divisions of a proposition.
+<C>(<G>a</G>) <I>Terms for the formal divisions of a proposition.</I></C>
+<p>In its completest form a proposition contained six parts,
+(1) the <G>pro/tasis</G>, or <I>enunciation</I> in general terms, (2) the
+<G>e)/kqesis</G>, or <I>setting-out</I>, which states the particular <I>data</I>, e.g.
+a given straight line <I>AB</I>, two given triangles <I>ABC, DEF</I>, and
+the like, generally shown in a figure and constituting that
+upon which the proposition is to operate, (3) the <G>diorismo/s</G>,
+<I>definition</I> or <I>specification</I>, which means the restatement of
+what it is required to do or to prove in terms of the particular
+data, the object being to fix our ideas, (4) the <G>kataskeuh/</G>, the
+<I>construction</I> or <I>machinery</I> used, which includes any additions
+to the original figure by way of construction that are necessary
+to enable the proof to proceed, (5) the <G>a)po/deixis</G>, or the <I>proof</I>
+itself, and (6) the <G>sumpe/rasma</G>, or <I>conclusion</I>, which reverts to
+the enunciation, and states what has been proved or done;
+the conclusion can, of course, be stated in as general terms
+as the enunciation, since it does not depend on the particular
+figure drawn; that figure is only an illustration, a type of the
+<I>class</I> of figure, and it is legitimate therefore, in stating
+the conclusion, to pass from the particular to the general.
+<pb n=371><head>FORMAL DIVISIONS OF A PROPOSITION</head>
+In particular cases some of these formal divisions may be
+absent, but three are always found, the <I>enunciation, proof</I>
+and <I>conclusion.</I> Thus in many propositions no construction
+is needed, the given figure itself sufficing for the proof;
+again, in IV. 10 (to construct an isosceles triangle with each
+of the base angles double of the vertical angle) we may, in
+a sense, say with Proclus<note>Proclus on Eucl. I, p. 203. 23 sq.</note> that there is neither <I>setting-out</I> nor
+<I>definition</I>, for there is nothing <I>given</I> in the enunciation, and
+we set out, not a given straight line, but any straight line <I>AB</I>,
+while the proposition does not state (what might be said by
+way of <I>definition</I>) that the required triangle is to have <I>AB</I> for
+one of its equal sides.
+<C>(<G>b</G>) <I>The</I> <G>diorismo/s</G> <I>or statement of conditions of possibility.</I></C>
+<p>Sometimes to the statement of a problem there has to be
+added a <G>diorismo/s</G> in the more important and familiar sense of
+a criterion of the conditions of possibility or, in its most
+complete form, a criterion as to &lsquo;whether what is sought
+is impossible or possible and how far it is practicable and in
+how many ways&rsquo;.<note><I>Ib.</I>, p. 202. 3.</note> Both kinds of <G>diorismo/s</G> begin with the
+words <G>dei= dh/</G>, which should be translated, in the case of the
+<I>definition</I>, &lsquo;thus it is required (to prove or do so and so)&rsquo; and,
+in the case of the criterion of possibility, &lsquo;thus it is necessary
+that ...&rsquo; (not &lsquo;<I>but</I> it is necessary ...&rsquo;). Cf. I. 22, &lsquo;Out of
+three straight lines which are equal to three given straight
+lines to construct a triangle: thus it is necessary that two
+of the straight lines taken together in any manner should be
+greater than the remaining straight line&rsquo;.
+<C>(<G>g</G>) <I>Analysis, synthesis, reduction, reductio ad absurdum.</I></C>
+<p>The <I>Elements</I> is a synthetic treatise in that it goes directly
+forward the whole way, always proceeding from the known
+to the unknown, from the simple and particular to the more
+complex and general; hence <I>analysis</I>, which reduces the
+unknown or the more complex to the known, has no place
+in the exposition, though it would play an important part in
+the discovery of the proofs. A full account of the Greek
+<I>analysis</I> and <I>synthesis</I> will come more conveniently elsewhere.
+<pb n=372><head>EUCLID</head>
+In the meantime we may observe that, where a proposition
+is worked out by analysis followed by synthesis, the analysis
+comes between the <I>definition</I> and the <I>construction</I> of the
+proposition; and it should not be forgotten that <I>reductio ad
+absurdum</I> (called in Greek <G>h( ei)s to\ a)du/naton a)pagwgh/</G>,
+&lsquo;reduction to the impossible&rsquo;, or <G>h( dia\ to|u= a)duna/tou dei=xis</G>
+or <G>a)po/deixis</G>, &lsquo;proof <I>per impossibile</I>&rsquo;), a method of proof
+common in Euclid as elsewhere, is a variety of analysis.
+For analysis begins with <I>reduction</I> (<G>a)pagwgh/</G>) of the original
+proposition, which we hypothetically assume to be true, to
+something simpler which we can recognize as being either
+true or false; the case where it leads to a conclusion known
+to be false is the <I>reductio ad absurdum.</I>
+<C>(<G>d</G>) <I>Case, objection, porism, lemma.</I></C>
+<p>Other terms connected with propositions are the following.
+A proposition may have several <I>cases</I> according to the different
+arrangements of points, lines, &amp;c., in the figure that may
+result from variations in the positions of the elements given;
+the word for <I>case</I> is <G>ptw=sis</G>. The practice of the great
+geometers was, as a rule, to give only one case, leaving the
+others for commentators or pupils to supply for themselves.
+But they were fully alive to the existence of such other
+cases; sometimes, if we may believe Proclus, they would even
+give a proposition solely with a view to its use for the purpose
+of proving a case of a later proposition which is actually
+omitted. Thus, according to Proclus,<note>Proclus on Eucl. I, pp. 248. 8-11; 263. 4-8.</note> the second part of I. 5
+(about the angles beyond the base) was intended to enable
+the reader to meet an <I>objection</I> (<G>e)/nstasis</G>) that might be
+raised to I. 7 as given by Euclid on the ground that it was
+incomplete, since it took no account of what was given by
+Proclus himself, and is now generally given in our text-books,
+as the second case.
+<p>What we call a <I>corollary</I> was for the Greeks a <I>porism</I>
+(<G>po/risma</G>), i. e. something provided or ready-made, by which
+was meant some result incidentally revealed in the course
+of the demonstration of the main proposition under discussion,
+a sort of incidental gain' arising out of the demonstration,
+<pb n=373><head>TECHNICAL TERMS</head>
+as Proclus says.<note><I>Ib.</I>, p. 212. 16.</note> The name <I>porism</I> was also applied to a
+special kind of substantive proposition, as in Euclid's separate
+work in three Books entitled <I>Porisms</I> (see below, pp. 431-8).
+<p>The word <I>lemma</I> (<G>lh=mma</G>) simply means something <I>assumed.</I>
+Archimedes uses it of what is now known as the Axiom of
+Archimedes, the principle assumed by Eudoxus and others in
+the method of exhaustion; but it is more commonly used
+of a subsidiary proposition requiring proof, which, however,
+it is convenient to assume in the place where it is wanted
+in order that the argument may not be interrupted or unduly
+lengthened. Such a lemma might be proved in advance, but
+the proof was often postponed till the end, the assumption
+being marked as something to be afterwards proved by some
+such words as <G>w(s e(xh=s deixqh/setai</G>, &lsquo;as will be proved in due
+course&rsquo;.
+<C>Analysis of the <I>Elements.</I></C>
+<p>Book I of the <I>Elements</I> necessarily begins with the essential
+preliminary matter classified under the headings <I>Definitions</I>
+(<G>o(/roi</G>), <I>Postulates</I> (<G>ai)th/mata</G>) and <I>Common Notions</I> (<G>koinai\
+e)/nnoiai</G>). In calling the axioms <I>Common Notions</I> Euclid
+followed the lead of Aristotle, who uses as alternatives for
+&lsquo;axioms&rsquo; the terms &lsquo;common (things)&rsquo;, &lsquo;common opinions&rsquo;.
+<p>Many of the <I>Definitions</I> are open to criticism on one ground
+or another. Two of them at least seem to be original, namely,
+the definitions of a straight line (4) and of a plane surface (7);
+unsatisfactory as these are, they seem to be capable of a
+simple explanation. The definition of a straight line is
+apparently an attempt to express, without any appeal to
+sight, the sense of Plato's definition &lsquo;that of which the middle
+covers the ends&rsquo; (<I>sc.</I> to an eye placed at one end and looking
+along it); and the definition of a plane surface is an adaptation
+of the same definition. But most of the definitions were
+probably adopted from earlier text-books; some appear to be
+inserted merely out of respect for tradition, e.g. the defini-
+tions of <I>oblong, rhombus, rhomboid</I>, which are never used
+in the <I>Elements.</I> The definitions of various figures assume
+the existence of the thing defined, e.g. the square, and the
+<pb n=374><head>EUCLID</head>
+different kinds of triangle under their twofold classification
+(<I>a</I>) with reference to their sides (equilateral, isosceles and
+scalene), and (<I>b</I>) with reference to their angles (right-angled,
+obtuse-angled and acute-angled); such definitions are pro-
+visional pending the proof of existence by means of actual con-
+struction. A <I>parallelogram</I> is not defined; its existence is
+first proved in I. 33, and in the next proposition it is called a
+&lsquo;parallelogrammic area&rsquo;, meaning an area contained by parallel
+lines, in preparation for the use of the simple word &lsquo;parallelo-
+gram&rsquo; from I. 35 onwards. The definition of a diameter
+of a circle (17) includes a theorem; for Euclid adds that &lsquo;such
+a straight line also bisects the circle&rsquo;, which is one of the
+theorems attributed to Thales; but this addition was really
+necessary in view of the next definition (18), for, without
+this explanation, Euclid would not have been justified in
+describing a <I>semi</I>-circle as a portion of a circle cut off by
+a diameter.
+<p>More important by far are the five Postulates, for it is in
+them that Euclid lays down the real principles of Euclidean
+geometry; and nothing shows more clearly his determination
+to reduce his original assumptions to the very minimum.
+The first three Postulates are commonly regarded as the
+postulates of <I>construction</I>, since they assert the possibility
+(1) of drawing the straight line joining two points, (2) of
+producing a straight line in either direction, and (3) of describ-
+ing a circle with a given centre and &lsquo;distance&rsquo;. But they
+imply much more than this. In Postulates 1 and 3 Euclid
+postulates the existence of straight lines and. circles, and
+implicitly answers the objections of those who might say that,
+as a matter of fact, the straight lines and circles which we
+can draw are not mathematical straight lines and circles;
+Euclid may be supposed to assert that we can nevertheless
+assume our straight lines and circles to be such for the purpose
+of our proofs, since they are only illustrations enabling us to
+<I>imagine</I> the real things which they imperfectly represent.
+But, again, Postulates 1 and 2 further imply that the straight
+line drawn in the first case and the produced portion of the
+straight line in the second case are <I>unique</I>; in other words,
+Postulate 1 implies that two straight lines cannot enclose a
+space, and so renders unnecessary the &lsquo;axiom&rsquo; to that effect
+<pb n=375><head>THE <I>ELEMENTS.</I> BOOK I</head>
+interpolated in Proposition 4, while Postulate 2 similarly im-
+plies the theorem that two straight lines cannot have a
+common segment, which Simson gave as a corollary to I. 11.
+<p>At first sight the Postulates 4 (that all right angles are
+equal) and 5 (the Parallel-Postulate) might seem to be of
+an altogether different character, since they are rather of the
+nature of theorems unproved. But Postulate 5 is easily seen
+to be connected with constructions, because so many con-
+structions depend on the existence and use of points in which
+straight lines intersect; it is therefore absolutely necessary to
+lay down some criterion by which we can judge whether two
+straight lines in a figure will or will not meet if produced.
+Postulate 5 serves this purpose as well as that of providing
+a basis for the theory of parallel lines. Strictly speaking,
+Euclid ought to have gone further and given criteria for
+judging whether other pairs of lines, e.g. a straight line and
+a circle, or two circles, in a particular figure will or will not
+intersect one another. But this would have necessitated a
+considerable series of propositions, which it would have been
+difficult to frame at so early a stage, and Euclid preferred
+to assume such intersections provisionally in certain cases,
+e.g. in I. 1.
+<p>Postulate 4 is often classed as a theorem. But it had in any
+case to be placed before Postulate 5 for the simple reason that
+Postulate 5 would be no criterion at all unless right angles
+were determinate magnitudes; Postulate 4 then declares them
+to be such. But this is not all. If Postulate 4 were to be
+proved as a theorem, it could only be proved by applying one
+pair of &lsquo;adjacent&rsquo; right angles to another pair. This method
+would not be valid unless on the assumption of the <I>invaria-
+bility of figures</I>, which would therefore have to be asserted as
+an antecedent postulate. Euclid preferred to assert as a
+postulate, directly, the fact that all right angles are equal;
+hence his postulate may be taken as equivalent to the prin-
+ciple of the <I>invariability of figures</I>, or, what is the same thing,
+the <I>homogeneity of space.</I>
+<p>For reasons which I have given above (pp. 339, 358), I think
+that the great Postulate 5 is due to Euclid himself; and it
+seems probable that Postulate 4 is also his, if not Postulates
+1-3 as well.
+<pb n=376><head>EUCLID</head>
+<p>Of the <I>Common Notions</I> there is good reason to believe
+that only five (at the most) are genuine, the first three and
+two others, namely &lsquo;Things which coincide when applied to
+one another are equal to one another&rsquo; (4), and &lsquo;The whole
+is greater than the part&rsquo; (5). The objection to (4) is that
+it is incontestably geometrical, and therefore, on Aristotle's
+principles, should not be classed as an &lsquo;axiom&rsquo;; it is a more
+or less sufficient definition of geometrical equality, but not
+a real axiom. Euclid evidently disliked the method of super-
+position for proving equality, no doubt because it assumes the
+possibility of motion <I>without deformation.</I> But he could not
+dispense with it altogether. Thus in I. 4 he practically had
+to choose between using the method and assuming the whole
+proposition as a postulate. But he does not there quote
+<I>Common Notion</I> 4; he says &lsquo;the base <I>BC</I> will coincide with
+the base <I>EF</I> and will be equal to it&rsquo;. Similarly in I. 6 he
+does not quote <I>Common Notion</I> 5, but says &lsquo;the triangle
+<I>DBC</I> will be equal to the triangle <I>ACB</I>, the less to the greater,
+which is absurd&rsquo;. It seems probable, therefore, that even
+these two <I>Common Notions</I>, though apparently recognized
+by Proclus, were generalizations from particular inferences
+found in Euclid and were inserted after his time.
+<p>The propositions of Book I fall into three distinct groups.
+The first group consists of Propositions 1-26, dealing mainly
+with triangles (without the use of parallels) but also with
+perpendiculars (11, 12), two intersecting straight lines (15),
+and one straight line standing on another but not cutting it,
+and making &lsquo;adjacent&rsquo; or supplementary angles (13, 14).
+Proposition 1 gives the construction of an equilateral triangle
+on a given straight line as base; this is placed here not so
+much on its own account as because it is at once required for
+constructions (in 2, 9, 10, 11). The construction in 2 is a
+direct continuation of the minimum constructions assumed
+in Postulates 1-3, and enables us (as the Postulates do not) to
+transfer a given length of straight line from one place to
+another; it leads in 3 to the operation so often required of
+cutting off from one given straight line a length equal to
+another. 9 and 10 are the problems of bisecting a given angle
+and a given straight line respectively, and 11 shows how
+to erect a perpendicular to a given straight line from a given
+<pb n=377><head>THE <I>ELEMENTS.</I> BOOK I</head>
+point on it. Construction as a means of proving existence is
+in evidence in the Book, not only in 1 (the equilateral triangle)
+but in 11, 12 (perpendiculars erected and let fall), and in
+22 (construction of a triangle in the general case where the
+lengths of the sides are given); 23 constructs, by means of 22,
+an angle equal to a given rectilineal angle. The propositions
+about triangles include the congruence-theorems (4, 8, 26)&mdash;
+omitting the &lsquo;ambiguous case&rsquo; which is only taken into
+account in the analogous proposition (7) of Book VI&mdash;and the
+theorems (allied to 4) about two triangles in which two sides
+of the one are respectively equal to two sides of the other, but
+of the included angles (24) or of the bases (25) one is greater
+than the other, and it is proved that the triangle in which the
+included angle is greater has the greater base and vice versa.
+Proposition 7, used to prove Proposition 8, is also important as
+being the Book I equivalent of III. 10 (that two circles cannot
+intersect in more points than two). Then we have theorems
+about single triangles in 5, 6 (isosceles triangles have the
+angles opposite to the equal sides equal&mdash;Thales's theorem&mdash;
+and the converse), the important propositions 16 (the exterior
+angle of a triangle is greater than either of the interior and
+opposite angles) and its derivative 17 (any two angles of
+a triangle are together less than two right angles), 18, 19
+(greater angle subtended by greater side and vice versa),
+20 (any two sides together greater than the third). This last
+furnishes the necessary <G>diorismo/s</G>, or criterion of possibility, of
+the problem in 22 of constructing a triangle out of three
+straight lines of given length, which problem had therefore
+to come after and not before 20. 21 (proving that the two
+sides of a triangle other than the base are together greater,
+but include a lesser angle, than the two sides of any other
+triangle on the same base but with vertex within the original
+triangle) is useful for the proof of the proposition (not stated
+in Euclid) that of all straight lines drawn from an external
+point to a given straight line the perpendicular is the
+shortest, and the nearer to the perpendicular is less than the
+more remote.
+<p>The second group (27-32) includes the theory of parallels
+(27-31, ending with the construction through a given point
+of a parallel to a given straight line); and then, in 32, Euclid
+<pb n=378><head>EUCLID</head>
+proves that the sum of the three angles of a triangle is equal
+to two right angles by means of a parallel to one side drawn
+from the opposite vertex (cf. the slightly different Pytha-
+gorean proof, p. 143).
+<p>The third group of propositions (33-48) deals generally
+with parallelograms, triangles and squares with reference to
+their areas. 33, 34 amount to the proof of the existence and
+the property of a parallelogram, and then we are introduced
+to a new conception, that of <I>equivalent</I> figures, or figures
+equal in area though not equal in the sense of congruent:
+parallelograms on the same base or on equal bases and between
+the same parallels are equal in area (35, 36); the same is true
+of triangles (37, 38), and a parallelogram on the same (or an
+equal) base with a triangle and between the same parallels is
+double of the triangle (41). 39 and the interpolated 40 are
+partial converses of 37 and 38. The theorem 41 enables us
+&lsquo;to construct in a given rectilineal angle a parallelogram
+equal to a given triangle&rsquo; (42). Propositions 44, 45 are of
+the greatest importance, being the first cases of the Pytha-
+gorean method of &lsquo;application of areas&rsquo;, &lsquo;to apply to a given
+straight line, in a given rectilineal angle, a parallelogram
+equal to a given triangle (or rectilineal figure)&rsquo;. The con-
+struction in 44 is remarkably ingenious, being based on that
+of 42 combined with the proposition (43) proving that the
+&lsquo;complements of the parallelograms about the diameter&rsquo; in any
+parallelogram are equal. We are thus enabled to transform
+a parallelogram of any shape into another with the same
+angle and of equal area but with one side of any given length,
+say a <I>unit</I> length; this is the geometrical equivalent of the
+algebraic operation of dividing the product of two quantities
+by a third. Proposition 46 constructs a square on any given
+straight line as side, and is followed by the great Pythagorean
+theorem of the square on the hypotenuse of a right-angled
+triangle (47) and its converse (48). The remarkably clever
+proof of 47 by means of the well-known &lsquo;windmill&rsquo; figure
+and the application to it of I. 41 combined with I. 4 seems to
+be due to Euclid himself; it is really equivalent to a proof by
+the methods of Book VI (Propositions 8, 17), and Euclid's
+achievement was that of avoiding the use of proportions and
+making the proof dependent upon Book I only.
+<pb n=379><head>THE <I>ELEMENTS.</I> BOOKS I-II</head>
+<p>I make no apology for having dealt at some length with
+Book I and, in particular, with the preliminary matter, in
+view of the unique position and authority of the <I>Elements</I>
+as an exposition of the fundamental principles of Greek
+geometry, and the necessity for the historian of mathematics
+of a clear understanding of their nature and full import.
+It will now be possible to deal more summarily with the
+other Books.
+<p>Book II is a continuation of the third section of Book I,
+relating to the transformation of areas, but is specialized in
+that it deals, not with parallelograms in general, but with
+<I>rectangles</I> and squares, and makes great use of the figure
+called the <I>gnomon.</I> The <I>rectangle</I> is introduced (Def. 1) as
+a &lsquo;rectangular parallelogram&rsquo;, which is said to be &lsquo;contained
+by the two straight lines containing the right angle&rsquo;. The
+<I>gnomon</I> is defined (Def. 2) with reference to any parallelo-
+gram, but the only gnomon actually used is naturally that
+which belongs to a square. The whole Book constitutes an
+essential part of the <I>geometrical algebra</I> which really, in
+Greek geometry, took the place of our algebra. The first ten
+propositions give the equivalent of the following algebraical
+identities.
+<p>1. <MATH><I>a</I>(<I>b</I>+<I>c</I>+<I>d</I>+...)=<I>ab</I>+<I>ac</I>+<I>ad</I>+...</MATH>,
+<p>2. <MATH>(<I>a</I>+<I>b</I>)<I>a</I>+(<I>a</I>+<I>b</I>)<I>b</I>=(<I>a</I>+<I>b</I>)<SUP>2</SUP></MATH>,
+<p>3. <MATH>(<I>a</I>+<I>b</I>)<I>a</I>=<I>ab</I>+<I>a</I><SUP>2</SUP></MATH>,
+<p>4. <MATH>(<I>a</I>+<I>b</I>)<SUP>2</SUP>=<I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>+2<I>ab</I></MATH>,
+<p>5. <MATH><I>ab</I>+{1/2(<I>a</I>+<I>b</I>)-<I>b</I>}<SUP>2</SUP>={1/2(<I>a</I>+<I>b</I>)}<SUP>2</SUP></MATH>,
+or <MATH>(<G>a</G>+<G>b</G>)(<G>a</G>-<G>b</G>)+<G>b</G><SUP>2</SUP>=<G>a</G><SUP>2</SUP></MATH>,
+<p>6. <MATH>(2<I>a</I>+<I>b</I>)<I>b</I>+<I>a</I><SUP>2</SUP>=(<I>a</I>+<I>b</I>)<SUP>2</SUP></MATH>,
+or <MATH>(<G>a</G>+<G>b</G>)(<G>b</G>-<G>a</G>)+<G>a</G><SUP>2</SUP>=<G>b</G><SUP>2</SUP></MATH>,
+<p>7. <MATH>(<I>a</I>+<I>b</I>)<SUP>2</SUP>+<I>a</I><SUP>2</SUP>=2(<I>a</I>+<I>b</I>)<I>a</I>+<I>b</I><SUP>2</SUP></MATH>,
+or <MATH><G>a</G><SUP>2</SUP>+<G>b</G><SUP>2</SUP>=2<G>ab</G>+(<G>a</G>-<G>b</G>)<SUP>2</SUP></MATH>,
+<p>8. <MATH>4(<I>a</I>+<I>b</I>)<I>a</I>+<I>b</I><SUP>2</SUP>={(<I>a</I>+<I>b</I>)+<I>a</I>}<SUP>2</SUP></MATH>,
+or <MATH>4<G>ab</G>+(<G>a</G>-<G>b</G>)<SUP>2</SUP>=(<G>a</G>+<G>b</G>)<SUP>2</SUP></MATH>,
+<pb n=380><head>EUCLID</head>
+<p>9. <MATH><I>a</I><SUP>2</SUP>+<I>b</I><SUP>2</SUP>=2[{1/2(<I>a</I>+<I>b</I>)}<SUP>2</SUP>+{1/2(<I>a</I>+<I>b</I>)-<I>b</I>}<SUP>2</SUP>]</MATH>,
+or <MATH>(<G>a</G>+<G>b</G>)<SUP>2</SUP>+(<G>a</G>-<G>b</G>)<SUP>2</SUP>=2(<G>a</G><SUP>2</SUP>+<G>b</G><SUP>2</SUP>)</MATH>,
+<p>10. <MATH>(2<I>a</I>+<I>b</I>)<SUP>2</SUP>+<I>b</I><SUP>2</SUP>=2{<I>a</I><SUP>2</SUP>+(<I>a</I>+<I>b</I>)<SUP>2</SUP>}</MATH>,
+or <MATH>(<G>a</G>+<G>b</G>)<SUP>2</SUP>+(<G>b</G>-<G>a</G>)<SUP>2</SUP>=2(<G>a</G><SUP>2</SUP>+<G>b</G><SUP>2</SUP>)</MATH>.
+As we have seen (pp. 151-3), Propositions 5 and 6 enable us
+to solve the quadratic equations
+<MATH>
+(1) <I>ax</I>-<I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP> or
+<BRACE><I>x</I>+<I>y</I>=<I>a</I>
+<I>xy</I>=<I>b</I><SUP>2</SUP>
+</BRACE>,
+and (2) <I>ax</I>+<I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP> or
+<BRACE>
+<I>y</I>-<I>x</I>=<I>a</I>
+<I>xy</I>=<I>b</I><SUP>2</SUP>
+</BRACE>.
+</MATH>
+The procedure is <I>geometrical</I> throughout; the areas in all
+the Propositions 1-8 are actually shown in the figures.
+Propositions 9 and 10 were really intended to solve a problem
+in <I>numbers</I>, that of finding any number of successive pairs
+of integral numbers (&lsquo;side-&rsquo; and &lsquo;diameter-&rsquo; numbers) satisfy-
+ing the equations
+<MATH>2<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>=&plusmn;1</MATH>
+(see p. 93, above).
+<p>Of the remaining propositions, II. 11 and II. 14 give the
+geometrical equivalent of solving the quadratic equations
+<MATH><I>x</I><SUP>2</SUP>+<I>ax</I>=<I>a</I><SUP>2</SUP></MATH>
+and <MATH><I>x</I><SUP>2</SUP>=<I>ab</I></MATH>,
+while the intervening propositions 12 and 13 prove, for any
+triangle with sides <I>a, b, c</I>, the equivalent of the formula
+<MATH><I>a</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP>+<I>c</I><SUP>2</SUP>-2<I>bc</I> cos <I>A.</I></MATH>
+<p>It is worth noting that, while I. 47 and its converse con-
+clude Book I as if that Book was designed to lead up to the
+great proposition of Pythagoras, the last propositions but one
+of Book II give the generalization of the same proposition
+with <I>any</I> triangle substituted for a right-angled triangle.
+<p>The subject of Book III is the geometry of the circle,
+including the relations between circles cutting or touching
+each other. It begins with some definitions, which are
+<pb n=381><head>THE <I>ELEMENTS.</I> BOOKS II-III</head>
+generally of the same sort as those of Book I. Definition 1,
+stating that <I>equal circles</I> are those which have their diameters
+or their radii equal, might alternatively be regarded as a
+postulate or a theorem; if stated as a theorem, it could only
+be proved by superposition and the congruence-axiom. It is
+curious that the Greeks had no single word for <I>radius</I>, which
+was with them &lsquo;the (straight line) from the centre&rsquo;, <G>h( e)k tou=
+ke/ntrou</G>. A tangent to a circle is defined (Def. 2) as a straight
+line which meets the circle but, if produced, does not cut it;
+this is provisional pending the proof in III. 16 that such lines
+exist. The definitions (4, 5) of straight lines (in a circle),
+i. e. chords, equally distant or more or less distant from the
+centre (the test being the length of the perpendicular from
+the centre on the chord) might have referred, more generally,
+to the distance of any straight line from any point. The
+definition (7) of the &lsquo;angle <I>of</I> a segment&rsquo; (the &lsquo;mixed&rsquo; angle
+made by the circumference with the base at either end) is
+a survival from earlier text-books (cf. Props. 16, 31). The
+definitions of the &lsquo;angle <I>in</I> a segment&rsquo; (8) and of &lsquo;similar
+segments&rsquo; (11) assume (provisionally pending III. 21) that the
+angle in a segment is one and the same at whatever point of
+the circumference it is formed. A <I>sector</I> (<G>tomeu/s</G>, explained by
+a scholiast as <G>skutotomiko\s tomeu/s</G>, a shoemaker's knife) is
+defined (10), but there is nothing about &lsquo;similar sectors&rsquo; and
+no statement that similar segments belong to similar sectors.
+<p>Of the propositions of Book III we may distinguish certain
+groups. Central properties account for four propositions,
+namely 1 (to find the centre of a circle), 3 (any straight line
+through the centre which bisects any chord not passing
+through the centre cuts it at right angles, and vice versa),
+4 (two chords not passing through the centre cannot bisect
+one another) and 9 (the centre is the only point from which
+more than two equal straight lines can be drawn to the
+circumference). Besides 3, which shows that any diameter
+bisects the whole series of chords at right angles to it, three
+other propositions throw light on the <I>form</I> of the circum-
+ference of a circle, 2 (showing that it is everywhere concave
+towards the centre), 7 and 8 (dealing with the varying lengths
+of straight lines drawn from any point, internal or external,
+to the concave or convex circumference, as the case may be,
+<pb n=382><head>EUCLID</head>
+and proving that they are of maximum or minimum length
+when they pass through the centre, and that they diminish or
+increase as they diverge more and more from the maximum
+or minimum straight lines on either side, while the lengths of
+any two which are equally inclined to them, one on each side,
+are equal).
+<p>Two circles which cut or touch one another are dealt with
+in 5, 6 (the two circles cannot have the same centre), 10, 13
+(they cannot cut in more points than two, or touch at more
+points than one), 11 and the interpolated 12 (when they touch,
+the line of centres passes through the point of contact).
+<p>14, 15 deal with chords (which are equal if equally distant
+from the centre and vice versa, while chords more distant
+from the centre are less, and chords less distant greater, and
+vice versa).
+<p>16-19 are concerned with tangent properties including the
+drawing of a tangent (17); it is in 16 that we have the
+survival of the &lsquo;angle <I>of</I> a semicircle&rsquo;, which is proved greater
+than any acute rectilineal angle, while the &lsquo;remaining&rsquo; angle
+(the &lsquo;angle&rsquo;, afterwards called <G>keratoeidh/s</G>, or &lsquo;hornlike&rsquo;,
+between the curve and the tangent at the point of contact)
+is less than any rectilineal angle. These &lsquo;mixed&rsquo; angles,
+occurring in 16 and 31, appear no more in serious Greek
+geometry, though controversy about their nature went on
+in the works of commentators down to Clavius, Peletarius
+(Pel&eacute;tier), Vieta, Galilei and Wallis.
+<p>We now come to propositions about segments. 20 proves
+that the angle at the centre is double of the angle at the
+circumference, and 21 that the angles in the same segment are
+all equal, which leads to the property of the quadrilateral
+in a circle (22). After propositions (23, 24) on &lsquo;similar
+segments&rsquo;, it is proved that in equal circles equal arcs subtend
+and are subtended by equal angles at the centre or circum-
+ference, and equal arcs subtend and are subtended by equal
+chords (26-9). 30 is the problem of bisecting a given arc,
+and 31 proves that the angle in a segment is right, acute or
+obtuse according as the segment is a semicircle, greater than
+a semicircle or less than a semicircle. 32 proves that the
+angle made by a tangent with a chord through the point
+of contact is equal to the angle in the alternate segment;
+<pb n=383><head>THE <I>ELEMENTS.</I> BOOKS III-IV</head>
+33, 34 are problems of constructing or cutting off a segment
+containing a given angle, and 25 constructs the complete circle
+when a segment of it is given.
+<p>The Book ends with three important propositions. Given
+a circle and any point <I>O</I>, internal (35) or external (36), then,
+if any straight line through <I>O</I> meets the circle in <I>P, Q</I>, the
+rectangle <I>PO.OQ</I> is constant and, in the case where <I>O</I> is
+an external point, is equal to the square on the tangent from
+<I>O</I> to the circle. Proposition 37 is the converse of 36.
+<p>Book IV, consisting entirely of problems, again deals with
+circles, but in relation to rectilineal figures inscribed or circum-
+scribed to them. After definitions of these terms, Euclid
+shows, in the preliminary Proposition 1, how to fit into a circle
+a chord of given length, being less than the diameter. The
+remaining problems are problems of inscribing or circum-
+scribing rectilineal figures. The case of the triangle comes
+first, and we learn how to inscribe in or circumscribe about
+a circle a triangle equiangular with a given triangle (2, 3) and
+to inscribe a circle in or circumscribe a circle about a given
+triangle (4, 5). 6-9 are the same problems for a square, 11-
+14 for a regular pentagon, and 15 (with porism) for a regular
+hexagon. The porism to 15 also states that the side of the
+inscribed regular hexagon is manifestly equal to the radius
+of the circle. 16 shows how to inscribe in a circle a regular
+polygon with fifteen angles, a problem suggested by astronomy,
+since the obliquity of the ecliptic was taken to be about 24&deg;,
+or one-fifteenth of 360&deg;. IV. 10 is the important proposition,
+required for the construction of a regular pentagon, &lsquo;to
+construct an isosceles triangle such that each of the base
+angles is double of the vertical angle&rsquo;, which is effected by
+dividing one of the equal sides in extreme and mean ratio
+(II. 11) and fitting into the circle with this side as radius
+a chord equal to the greater segment; the proof of the con-
+struction depends on III. 32 and 37.
+<p>We are not surprised to learn from a scholiast that the
+whole Book is &lsquo;the discovery of the Pythagoreans&rsquo;.<note>Euclid, ed. Heib., vol. v, pp. 272-3.</note> The
+same scholium says that &lsquo;it is proved in this Book that
+the perimeter of a circle is not triple of its diameter, as many
+<pb n=384><head>EUCLID</head>
+suppose, but greater than that (the reference is clearly to
+IV. 15 Por.), and likewise that neither is the circle three-
+fourths of the triangle circumscribed about it&rsquo;. Were these
+fallacies perhaps exposed in the lost <I>Pseudaria</I> of Euclid?
+<p>Book V is devoted to the new theory of proportion,
+applicable to incommensurable as well as commensurable
+magnitudes, and to magnitudes of every kind (straight lines,
+areas, volumes, numbers, times, &amp;c.), which was due to
+Eudoxus. Greek mathematics can boast no finer discovery
+than this theory, which first put on a sound footing so much
+of geometry as depended on the use of proportions. How far
+Eudoxus himself worked out his theory in detail is unknown;
+the scholiast who attributes the discovery of it to him says
+that &lsquo;it is recognized by all&rsquo; that Book V is, as regards its
+arrangement and sequence in the <I>Elements</I>, due to Euclid
+himself.<note>Euclid, ed. Heib., vol. v, p. 282.</note> The ordering of the propositions and the develop-
+ment of the proofs are indeed masterly and worthy of Euclid;
+as Barrow said, &lsquo;There is nothing in the whole body of the
+elements of a more subtile invention, nothing more solidly
+established, and more accurately handled, than the doctrine of
+proportionals&rsquo;. It is a pity that, notwithstanding the pre-
+eminent place which Euclid has occupied in English mathe-
+matical teaching, Book V itself is little known in detail; if it
+were, there would, I think, be less tendency to seek for
+substitutes; indeed, after reading some of the substitutes,
+it is with relief that one turns to the original. For this
+reason, I shall make my account of Book V somewhat full,
+with the object of indicating not only the whole content but
+also the course of the proofs.
+<p>Of the Definitions the following are those which need
+separate mention. The definition (3) of <I>ratio</I> as &lsquo;a sort of
+relation (<G>poia\ sxe/sis</G>) in respect of size (<G>phliko/ths</G>) between
+two magnitudes of the same kind&rsquo; is as vague and of as
+little practical use as that of a straight line; it was probably
+inserted for completeness' sake, and in order merely to aid the
+conception of a ratio. Definition 4 (&lsquo;Magnitudes are said to
+have a ratio to one another which are capable, when multi-
+plied, of exceeding one another&rsquo;) is important not only because
+<pb n=385><head>THE <I>ELEMENTS.</I> BOOK V</head>
+it shows that the magnitudes must be of the same kind,
+but because, while it includes incommensurable as well as
+commensurable magnitudes, it excludes the relation of a finite
+magnitude to a magnitude of the same kind which is either
+infinitely great or infinitely small; it is also practically equiva-
+lent to the principle which underlies the method of exhaustion
+now known as the Axiom of Archimedes. Most important
+of all is the fundamental definition (5) of magnitudes which
+are in the same ratio: &lsquo;Magnitudes are said to be in the same
+ratio, the first to the second and the third to the fourth, when,
+if any equimultiples whatever be taken of the first and third,
+and any equimultiples whatever of the second and fourth, the
+former equimultiples alike exceed, are alike equal to, or alike
+fall short of, the latter equimultiples taken in corresponding
+order.&rsquo; Perhaps the greatest tribute to this marvellous defini-
+tion is its adoption by Weierstrass as a definition of equal
+numbers. For a most attractive explanation of its exact
+significance and its absolute sufficiency the reader should turn
+to De Morgan's articles on Ratio and Proportion in the <I>Penny
+Cyclopaedia.</I><note>Vol. xix (1841). I have largely reproduced the articles in <I>The
+Thirteen Books of Euclid's Elements</I>, vol. ii, pp. 116-24.</note> The definition (7) of <I>greater ratio</I> is an adden-
+dum to Definition 5: &lsquo;When, of the equimultiples, the multiple
+of the first exceeds the multiple of the second, but the
+multiple of the third does not exceed the multiple of the
+fourth, then the first is said to have a <I>greater ratio</I> to
+the second than the third has to the fourth&rsquo;; this (possibly
+for brevity's sake) states only one criterion, the other possible
+criterion being that, while the multiple of the first is <I>equal</I>
+to that of the second, the multiple of the third is <I>tess</I> than
+that of the fourth. A proportion may consist of three or
+four terms (Defs. 8, 9, 10); &lsquo;corresponding&rsquo; or &lsquo;homologous&rsquo;
+terms are antecedents in relation to antecedents and conse-
+quents in relation to consequents (11). Euclid proceeds to
+define the various transformations of ratios. <I>Alternation</I>
+(<G>e)nalla/x</G>, <I>alternando</I>) means taking the alternate terms in
+the proportion <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, i.e. transforming it into <MATH><I>a</I>:<I>c</I>=<I>b</I>:<I>d</I></MATH>
+(12). <I>Inversion</I> (<G>a)na/palin</G>, inversely) means turning the ratio
+<I>a:b</I> into <I>b:a</I> (13). <I>Composition</I> of a ratio, <G>su/nqesis lo/gou</G>
+(<I>componendo</I> is in Greek <G>sunqe/nti</G>, &lsquo;to one who has compounded
+<pb n=386><head>EUCLID</head>
+or added&rsquo;, i. e. if one compounds or adds) is the turning of
+<I>a:b</I> into <MATH>(<I>a</I>+<I>b</I>):<I>b</I></MATH> (14). <I>Separation</I>, <G>diai/resis</G> (<G>dielo/nti</G>=
+<I>separando</I>) turns <I>a:b</I> into <MATH>(<I>a</I>-<I>b</I>):<I>b</I></MATH> (15). <I>Conversion</I>, <G>a)na-
+strofh/</G> (<G>a)nastre/yanti</G>=<I>convertendo</I>) turns <I>a</I>:<I>b</I> into <MATH><I>a</I>:<I>a</I>-<I>b</I></MATH>
+(16). Lastly, <I>ex aequali</I> (sc. <I>distantia</I>), <G>di) i)/sou</G>, and <I>ex aequali
+in disturbed proportion</I> (<G>e)n tetaragme/nh a)nalogi/a|</G>) are defined
+(17, 18). If <MATH><I>a</I>:<I>b</I>=<I>A</I>:<I>B</I>, <I>b</I>:<I>c</I>=<I>B</I>:<I>C</I> ... <I>k</I>:<I>l</I>=<I>K</I>:<I>L</I></MATH>, then
+the inference <I>ex aequali</I> is that <MATH><I>a</I>:<I>l</I>=<I>A</I>:<I>L</I></MATH> (proved in V. 22).
+If again <MATH><I>a</I>:<I>b</I>=<I>B</I>:<I>C</I></MATH> and <MATH><I>b</I>:<I>c</I>=<I>A</I>:<I>B</I></MATH>, the inference <I>ex aequali
+in disturbed proportion</I> is <MATH><I>a</I>:<I>c</I>=<I>A</I>:<I>C</I></MATH> (proved in V. 23).
+<p>In reproducing the content of the Book I shall express
+magnitudes in general (which Euclid represents by straight
+lines) by the letters <I>a, b, c</I> ... and I shall use the letters
+<I>m, n, p</I> ... to express integral numbers: thus <I>ma, mb</I> are
+equimultiples of <I>a, b.</I>
+<p>The first six propositions are simple theorems in concrete
+arithmetic, and they are practically all proved by separating
+into their units the multiples used.
+<MATH>
+<BRACE>
+1. <I>ma</I>+<I>mb</I>+<I>mc</I>+...=<I>m</I>(<I>a</I>+<I>b</I>+<I>c</I>+...).
+5. <I>ma</I>-<I>mb</I>=<I>m</I>(<I>a</I>-<I>b</I>).
+</BRACE>
+</MATH>
+5 is proved by means of 1. As a matter of fact, Euclid
+assumes the construction of a straight line equal to 1/<I>m</I>th of
+<MATH><I>ma</I>-<I>mb</I></MATH>. This is an anticipation of VI. 9, but can be avoided;
+for we can draw a straight line equal to <MATH><I>m</I>(<I>a</I>-<I>b</I>)</MATH>; then,
+by 1, <MATH><I>m</I>(<I>a</I>-<I>b</I>)+<I>mb</I>=<I>ma</I></MATH>, or <MATH><I>ma</I>-<I>mb</I>=<I>m</I>(<I>a</I>-<I>b</I>)</MATH>.
+<MATH>
+<BRACE>
+2. <I>ma</I>+<I>na</I>+<I>pa</I>+...=(<I>m</I>+<I>n</I>+<I>p</I>+...)<I>a</I>.
+6. <I>ma</I>-<I>na</I>=(<I>m</I>-<I>n</I>)<I>a</I>.
+</BRACE>
+</MATH>
+Euclid actually expresses 2 and 6 by saying that <I>ma</I>&plusmn;<I>na</I> is
+the same multiple of <I>a</I> that <I>mb</I>&plusmn;<I>nb</I> is of <I>b.</I> By separation
+of <I>m, n</I> into units he in fact shows (in 2) that
+<MATH><I>ma</I>+<I>na</I>=(<I>m</I>+<I>n</I>)<I>a</I></MATH>, and <MATH><I>mb</I>+<I>nb</I>=(<I>m</I>+<I>n</I>)<I>b</I></MATH>.
+6 is proved by means of 2, as 5 by means of 1.
+<p>3. If <I>m.na, m.nb</I> are equimultiples of <I>na, nb</I>, which are
+themselves equimultiples of <I>a, b</I>, then <I>m.na, m.nb</I> are also
+equimultiples of <I>a, b.</I>
+<p>By separating <I>m, n</I> into their units Euclid practically proves
+that <MATH><I>m.na</I>=<I>mn.a</I></MATH> and <MATH><I>m.nb</I>=<I>mn.b</I></MATH>.
+<pb n=387><head>THE <I>ELEMENTS.</I> BOOK V</head>
+<p>4. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, then <MATH><I>ma</I>:<I>nb</I>=<I>mc</I>:<I>nd</I></MATH>.
+<p>Take any equimultiples <I>p.ma, p.mc</I> of <I>ma, mc</I>, and any
+equimultiples <I>q.nb, q.nd</I> of <I>nb, nd.</I> Then, by 3, these equi-
+multiples are also equimultiples of <I>a, c</I> and <I>b, d</I> respectively,
+so that by Def. 5, since <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+<MATH><I>p.ma</I>>=<<I>q.nb</I></MATH> according as <MATH><I>p.mc</I>>=<<I>q.nd</I></MATH>,
+whence, again by Def. 5, since <I>p, q</I> are any integers,
+<MATH><I>ma</I>:<I>nb</I>=<I>mc</I>:<I>nd</I></MATH>.
+<MATH>
+<BRACE><note>; and conversely.</note>
+7, 9. If <I>a</I>=<I>b</I>, then <I>a</I>:<I>c</I>=<I>b</I>:<I>c</I>
+and <I>c</I>:<I>a</I>=<I>c</I>:<I>b</I>
+</BRACE>
+</MATH>
+<MATH>
+<BRACE><note>; and conversely.</note>
+8, 10. If <I>a</I>><I>b</I>, then <I>a</I>:<I>c</I>><I>b</I>:<I>c</I>
+and <I>c</I>:<I>b</I>><I>c</I>:<I>a</I>
+</BRACE>
+</MATH>
+<p>7 is proved by means of Def. 5. Take <I>ma, mb</I> equi-
+multiples of <I>a, b</I>, and <I>nc</I> a multiple of <I>c.</I> Then, since <I>a</I>=<I>b</I>,
+<MATH><I>ma</I>>=<<I>nc</I></MATH> according as <MATH><I>mb</I>>=<<I>nc</I></MATH>,
+and <MATH><I>nc</I>>=<<I>ma</I></MATH> according as <MATH><I>nc</I>>=<<I>mb</I></MATH>,
+whence the results follow.
+<p>8 is divided into two cases according to which of the two
+magnitudes <I>a</I>-<I>b</I>, <I>b</I> is the less. Take <I>m</I> such that
+<MATH><I>m</I>(<I>a</I>-<I>b</I>)><I>c</I></MATH> or <MATH><I>mb</I>><I>c</I></MATH>
+in the two cases respectively. Next let <I>nc</I> be the first
+multiple of <I>c</I> which is greater than <I>mb</I> or <MATH><I>m</I>(<I>a</I>-<I>b</I>)</MATH> respec-
+tively, so that
+<MATH><I>nc</I>><I>mb</I> or <I>m</I>(<I>a</I>-<I>b</I>)&ge;(<I>n</I>-1)<I>c</I></MATH>.
+Then, (i) since <MATH><I>m</I>(<I>a</I>-<I>b</I>)><I>c</I></MATH>, we have, by addition, <MATH><I>ma</I>><I>nc</I></MATH>.
+(ii) since <MATH><I>mb</I>><I>c</I></MATH>, we have similarly <MATH><I>ma</I>><I>nc</I></MATH>.
+In either case <MATH><I>mb</I><<I>nc</I></MATH>, since in case (ii) <MATH><I>m</I>(<I>a</I>-<I>b</I>)><I>mb</I></MATH>.
+Thus in either case, by the definition (7) of greater ratio,
+<MATH><I>a</I>:<I>c</I>><I>b</I>:<I>c</I></MATH>,
+and <MATH><I>c</I>:<I>b</I>><I>c</I>:<I>a</I></MATH>.
+<p>The converses 9, 10 are proved from 7, 8 by <I>reductio ad
+absurdum.</I>
+<pb n=388><head>EUCLID</head>
+<p>11. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+and <MATH><I>c</I>:<I>d</I>=<I>e</I>:<I>f</I></MATH>,
+then <MATH><I>a</I>:<I>b</I>=<I>e</I>:<I>f</I></MATH>.
+<p>Proved by taking any equimultiples of <I>a, c, e</I> and any other
+equimultiples of <I>b, d, f</I>, and using Def. 5.
+<p>12. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I>=<I>e</I>:<I>f</I>=...</MATH>
+then <MATH><I>a</I>:<I>b</I>=(<I>a</I>+<I>c</I>+<I>e</I>+...):(<I>b</I>+<I>d</I>+<I>f</I>+...)</MATH>.
+<p>Proved by means of V. 1 and Def. 5, after taking equi-
+multiples of <I>a, c, e</I> ... and other equimultiples of <I>b, d, f</I> ....
+<p>13. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+and <MATH><I>c</I>:<I>d</I>><I>e</I>:<I>f</I></MATH>,
+then <MATH><I>a</I>:<I>b</I>><I>e</I>:<I>f</I></MATH>.
+<p>Equimultiples <I>mc, me</I> of <I>c, e</I> are taken and equimultiples
+<I>nd, nf</I> of <I>d, f</I> such that, while <MATH><I>mc</I>><I>nd</I></MATH>, <I>me</I> is not greater
+than <I>nf</I> (Def. 7). Then the same equimultiples <I>ma, mc</I> of
+<I>a, c</I> and the same equimultiples <I>nb, nd</I> of <I>b, d</I> are taken, and
+Defs. 5 and 7 are used in succession.
+<p>14. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, then, according as <MATH><I>a</I>>=<<I>c</I>, <I>b</I>>=<<I>d</I></MATH>.
+<p>The first case only is proved; the others are dismissed with
+&lsquo;Similarly&rsquo;.
+<p>If <MATH><I>a</I>><I>c</I>, <I>a</I>:<I>b</I>><I>c</I>:<I>b</I></MATH>. (8)
+<p>But <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, whence (13) <MATH><I>c</I>:<I>d</I>><I>c</I>:<I>b</I></MATH>, and therefore (10)
+<I>b</I>><I>d</I>.
+<p>15. <MATH><I>a</I>:<I>b</I>=<I>ma</I>:<I>mb</I></MATH>.
+<p>Dividing the multiples into their units, we have <I>m</I> equal
+ratios <I>a</I>:<I>b</I>; the result follows by 12.
+<p>Propositions 16-19 prove certain cases of the transformation
+of proportions in the sense of Defs. 12-16. The case of
+<I>inverting</I> the ratios is omitted, probably as being obvious.
+For, if <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, the application of Def. 5 proves simul-
+taneously that <MATH><I>b</I>:<I>a</I>=<I>d</I>:<I>c</I></MATH>.
+<p>16. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+then, <I>alternando</I>, <MATH><I>a</I>:<I>c</I>=<I>b</I>:<I>d</I></MATH>.
+<p>Since <MATH><I>a</I>:<I>b</I>=<I>ma</I>:<I>mb</I></MATH>, and <MATH><I>c</I>:<I>d</I>=<I>nc</I>:<I>nd</I></MATH>, (15)
+<pb n=389><head>THE <I>ELEMENTS.</I> BOOK V</head>
+we have <MATH><I>ma</I>:<I>mb</I>=<I>nc</I>:<I>nd</I></MATH>, (11)
+whence (14), according as <MATH><I>ma</I>>=<<I>nc</I>, <I>mb</I>>=<<I>nd</I></MATH>;
+therefore (Def. 5) <MATH><I>a</I>:<I>c</I>=<I>b</I>:<I>d</I></MATH>.
+<p>17. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+then, <I>separando</I>, <MATH>(<I>a</I>-<I>b</I>):<I>b</I>=(<I>c</I>-<I>d</I>):<I>d</I></MATH>.
+<p>Take <I>ma, mb, mc, md</I> equimultiples of all four magnitudes,
+and <I>nb, nd</I> other equimultiples of <I>b, d.</I> It follows (2) that
+<MATH>(<I>m</I>+<I>n</I>)<I>b</I>, (<I>m</I>+<I>n</I>)<I>d</I></MATH> are also equimultiples of <I>b, d.</I>
+<p>Therefore, since <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+<MATH><I>ma</I>>=<(<I>m</I>+<I>n</I>)<I>b</I></MATH> according as <MATH><I>mc</I>>=<(<I>m</I>+<I>n</I>)<I>d</I></MATH>. (Def. 5)
+<p>Subtracting <I>mb</I> from both sides of the former relation and
+<I>md</I> from both sides of the latter, we have (5)
+<MATH><I>m</I>(<I>a</I>-<I>b</I>)>=<<I>nb</I></MATH> according as <MATH><I>m</I>(<I>c</I>-<I>d</I>)>=<<I>nd</I></MATH>.
+<p>Therefore (Def. 5) <MATH><I>a</I>-<I>b</I>:<I>b</I>=<I>c</I>-<I>d</I>:<I>d</I></MATH>.
+(I have here abbreviated Euclid a little, without altering the
+substance.)
+<p>18. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+then, <I>componendo</I>, <MATH>(<I>a</I>+<I>b</I>):<I>b</I>=(<I>c</I>+<I>d</I>):<I>d</I></MATH>.
+<p>Proved by <I>reductio ad absurdum.</I> Euclid assumes that
+<MATH><I>a</I>+<I>b</I>:<I>b</I>=(<I>c</I>+<I>d</I>):(<I>d</I>&plusmn;<I>x</I>)</MATH>, if that is possible. (This .implies
+that to any three given magnitudes, two of which at least
+are of the same kind, there exists a fourth proportional, an
+assumption which is not strictly legitimate until the fact has
+been proved by construction.)
+<p>Therefore, <I>separando</I> (17), <MATH><I>a</I>:<I>b</I>=(<I>c</I>&mnplus;<I>x</I>):(<I>d</I>&plusmn;<I>x</I>)</MATH>,.
+whence (11), <MATH>(<I>c</I>&mnplus;<I>x</I>):(<I>d</I>&plusmn;<I>x</I>)=<I>c</I>:<I>d</I></MATH>, which relations are im-
+possible, by 14.
+<p>19. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+then <MATH>(<I>a</I>-<I>c</I>):(<I>b</I>-<I>d</I>)=<I>a</I>:<I>b</I></MATH>.
+<p>Alternately (16),
+<MATH><I>a</I>:<I>c</I>=<I>b</I>:<I>d</I></MATH>, whence <MATH>(<I>a</I>-<I>c</I>):<I>c</I>=(<I>b</I>-<I>d</I>):<I>d</I></MATH> (17).
+<p>Alternately again, <MATH>(<I>a</I>-<I>c</I>):(<I>b</I>-<I>d</I>)=<I>c</I>:<I>d</I></MATH> (16);
+whence (11) <MATH>(<I>a</I>-<I>c</I>):(<I>b</I>-<I>d</I>)=<I>a</I>:<I>b</I></MATH>.
+<pb n=390><head>EUCLID</head>
+<p>The transformation <I>convertendo</I> is only given in an inter-
+polated Porism to 19. But it is easily obtained by using 17
+(<I>separando</I>) combined with <I>alternando</I> (16). Euclid himself
+proves it in X. 14 by using successively <I>separando</I> (17), <I>inver-
+sion</I> and <I>ex aequali</I> (22).
+<p>The <I>composition</I> of ratios <I>ex aequali</I> and <I>ex aequali in
+disturbed proportion</I> is dealt with in 22, 23, each of which
+depends on a preliminary proposition.
+<p>20. If <MATH><I>a</I>:<I>b</I>=<I>d</I>:<I>e</I></MATH>,
+and <MATH><I>b</I>:<I>c</I>=<I>e</I>:<I>f</I></MATH>,
+then, <I>ex aequali</I>, according as <MATH><I>a</I>>=<<I>c</I>, <I>d</I>>=<<I>f</I></MATH>.
+<p>For, according as <MATH><I>a</I>>=<<I>c</I>, <I>a</I>:<I>b</I>>=<<I>c</I>:<I>b</I></MATH> (7, 8),
+and therefore, by means of the above relations and 13, 11,
+<MATH><I>d</I>:<I>e</I>>=<<I>f</I>:<I>e</I></MATH>,
+and therefore again (9, 10)
+<MATH><I>d</I>>=<<I>f</I></MATH>.
+<p>21. If <MATH><I>a</I>:<I>b</I>=<I>e</I>:<I>f</I></MATH>,
+and <MATH><I>b</I>:<I>c</I>=<I>d</I>:<I>e</I></MATH>,
+then, <I>ex aequali in disturbed proportion</I>,
+according as <MATH><I>a</I>>=<<I>c</I>, <I>d</I>>=<<I>f</I></MATH>.
+<p>For, according as <MATH><I>a</I>>=<<I>c</I>, <I>a</I>:<I>b</I>>=<<I>c</I>:<I>b</I></MATH> (7, 8),
+or <MATH><I>e</I>:<I>f</I>>=<<I>e</I>:<I>d</I></MATH> (13, 11),
+and therefore <MATH><I>d</I>>=<<I>f</I></MATH> (9, 10).
+<p>22. If <MATH><I>a</I>:<I>b</I>=<I>d</I>:<I>e</I></MATH>,
+and <MATH><I>b</I>:<I>c</I>=<I>e</I>:<I>f</I></MATH>,
+then, <I>ex aequali</I>, <MATH><I>a</I>:<I>c</I>=<I>d</I>:<I>f</I></MATH>.
+<p>Take equimultiples <I>ma, md; nb, ne; pc, pf</I>, and it follows
+<MATH>
+<BRACE><note>(4)</note>
+that <I>ma</I>:<I>nb</I>=<I>md</I>:<I>ne</I>,
+and <I>nb</I>:<I>pc</I>=<I>ne</I>:<I>pf</I>
+</BRACE>
+</MATH>
+<p>Therefore (20), according as <MATH><I>ma</I>>=<<I>pc</I>, <I>md</I>>=<<I>pf</I></MATH>,
+whence (Def. 5) <MATH><I>a</I>:<I>c</I>=<I>d</I>:<I>f</I></MATH>.
+<pb n=391><head>THE <I>ELEMENTS.</I> BOOK V</head>
+<p>23. If <MATH><I>a</I>:<I>b</I>=<I>e</I>:<I>f</I></MATH>,
+and <MATH><I>b</I>:<I>c</I>=<I>d</I>:<I>e</I></MATH>,
+then, <I>ex aequali in disturbed proportion</I>, <MATH><I>a</I>:<I>c</I>=<I>d</I>:<I>f</I></MATH>.
+<p>Equimultiples <I>ma, mb, md</I> and <I>nc, ne, nf</I> are taken, and
+it is proved, by means of 11, 15, 16, that
+<MATH><I>ma</I>:<I>mb</I>=<I>ne</I>:<I>nf</I></MATH>,
+and <MATH><I>mb</I>:<I>nc</I>=<I>md</I>:<I>ne</I></MATH>,
+whence (21) <MATH><I>ma</I>>=<<I>nc</I></MATH> according as <MATH><I>md</I>>=<<I>nf</I></MATH>
+and (Def. 5) <MATH><I>a</I>:<I>c</I>=<I>d</I>:<I>f</I></MATH>.
+<p>24. If <MATH><I>a</I>:<I>c</I>=<I>d</I>:<I>f</I></MATH>,
+and also <MATH><I>b</I>:<I>c</I>=<I>e</I>:<I>f</I></MATH>,
+then <MATH>(<I>a</I>+<I>b</I>):<I>c</I>=(<I>d</I>+<I>e</I>):<I>f</I></MATH>.
+<p>Invert the second proportion to <MATH><I>c</I>:<I>b</I>=<I>f</I>:<I>e</I></MATH>, and compound
+the first proportion with this (22);
+therefore <MATH><I>a</I>:<I>b</I>=<I>d</I>:<I>e</I></MATH>.
+<p><I>Componendo,</I> <MATH>(<I>a</I>+<I>b</I>):<I>b</I>=(<I>d</I>+<I>e</I>):<I>e</I></MATH>, which compounded (22)
+with the second proportion gives <MATH>(<I>a</I>+<I>b</I>):<I>c</I>=(<I>d</I>+<I>e</I>):<I>f</I></MATH>.
+<p>25. If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, and of the four terms <I>a</I> is the greatest
+(so that <I>d</I> is also the least), <MATH><I>a</I>+<I>d</I>><I>b</I>+<I>c</I></MATH>.
+<p>Since <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>,
+<MATH><I>a-c</I>:<I>b-d</I>=<I>a</I>:<I>b</I></MATH>; (19)
+and, since <MATH><I>a</I>><I>b</I>, (<I>a-c</I>)>(<I>b-d</I>)</MATH>. (16, 14)
+<p>Add <I>c</I>+<I>d</I> to each;
+therefore <MATH><I>a</I>+<I>d</I>><I>b</I>+<I>c</I></MATH>.
+<p>Such slight defects as are found in the text of this great
+Book as it has reached us, like other slight imperfections of
+form in the <I>Elements</I>, point to the probability that the work
+never received its final touches from Euclid's hand; but they
+can all be corrected without much difficulty, as Simson showed
+in his excellent edition.
+<p>Book VI contains the application to plane geometry of the
+general theory of proportion established in Book V. It begins
+with definitions of &lsquo;similar rectilineal figures&rsquo; and of what is
+<pb n=392><head>EUCLID</head>
+meant by cutting a straight line &lsquo;in extreme and mean ratio&rsquo;.
+The first and last propositions are analogous; 1 proves that
+triangles and parallelograms of the same height are to one
+another as their bases, and 33 that in equal circles angles
+at the centre or circumference are as the arcs on which they
+stand; both use the method of equimultiples and apply
+V, Def. 5 as the test of proportion. Equally fundamental
+are 2 (that two sides of a triangle cut by any parallel to
+the third side are divided proportionally, and the converse),
+and 3 (that the internal bisector of an angle of a triangle cuts
+the opposite side into parts which have the same ratio as the
+sides containing the angle, and the converse); 2 depends
+directly on 1 and 3 on 2. Then come the alternative con-
+ditions for the similarity of two triangles: equality of all the
+angles respectively (4), proportionality of pairs of sides in
+order (5), equality of one angle in each with proportionality
+of sides containing the equal angles (6), and the &lsquo;ambiguous
+case&rsquo; (7), in which one angle is equal to one angle and the
+sides about other angles are proportional. After the important
+proposition (8) that the perpendicular from the right angle
+in a right-angled triangle to the opposite side divides the
+triangle into two triangles similar to the original triangle and
+to one another, we pass to the proportional division of
+straight lines (9, 10) and the problems of finding a third
+proportional to two straight lines (11), a fourth proportional
+to three (12), and a mean proportional to two straight lines
+(13, the Book VI version of II. 14). In 14, 15 Euclid proves
+the reciprocal proportionality of the sides about the equal
+angles in parallelograms or triangles of equal area which have
+one angle equal to one angle and the converse; by placing the
+equal angles vertically opposite to one another so that the sides
+about them lie along two straight lines, and completing the
+figure, Euclid is able to apply VI. 1. From 14 are directly
+deduced 16, 17 (that, if four or three straight lines be propor-
+tionals, the rectangle contained by the extremes is equal to
+the rectangle contained by the two means or the square on the
+one mean, and the converse). 18-22 deal with similar recti-
+lineal figures; 19 (with Porism) and 20 are specially important,
+proving that similar triangles, and similar polygons generally,
+are to one another in the duplicate ratio of corresponding
+<pb n=393><head>THE <I>ELEMENTS.</I> BOOK VI</head>
+sides, and that, if three straight lines are proportional, then,
+as the first is to the third, so is the figure described on the first
+to the similar figure similarly described on the second. The
+fundamental case of the two similar triangles is prettily proved
+thus. The triangles being <I>ABC, DEF</I>, in which <I>B, E</I> are equal
+angles and <I>BC, EF</I> corresponding sides, find a third propor-
+tional to <I>BC, EF</I> and measure it off along <I>BC</I> as <I>BG</I>; join <I>AG.</I>
+Then the triangles <I>ABG, DEF</I> have their sides about the equal
+angles <I>B, E</I> reciprocally proportional and are therefore equal
+(VI. 15); the rest follows from VI. 1 and the definition of
+duplicate ratio (V, Def. 9).
+<p>Proposition 23 (equiangular parallelograms have to one
+another the ratio compounded of the ratios of their sides) is
+important in itself, and also because it introduces us to the
+practical use of the method of compounding, i.e. multiplying,
+ratios which is of such extraordinarily wide application in
+Greek geometry. Euclid has never defined &lsquo;compound ratio&rsquo;
+or the &lsquo;compounding&rsquo; of ratios; but the meaning of the terms
+<FIG>
+and the way to compound ratios are made clear in this pro-
+position. The equiangular parallelograms are placed so that
+two equal angles as <I>BCD, GCE</I> are vertically opposite at <I>C.</I>
+Complete the parallelogram <I>DCGH.</I> Take any straight line <I>K</I>,
+and (12) find another, <I>L</I>, such that
+<MATH><I>BC</I>:<I>CG</I>=<I>K</I>:<I>L</I></MATH>,
+and again another straight line <I>M</I>, such that
+<MATH><I>DC</I>:<I>CE</I>=<I>L</I>:<I>M</I></MATH>.
+Now the ratio compounded of <I>K</I>:<I>L</I> and <I>L</I>:<I>M</I> is <I>K</I>:<I>M</I>; there-
+fore <I>K</I>:<I>M</I> is the &lsquo;ratio compounded of the ratios of the sides&rsquo;.
+<p>And <MATH>(<I>ABCD</I>):(<I>DCGH</I>)=<I>BC</I>:<I>CG</I>, (1)
+=<I>K</I>:<I>L</I>;
+(<I>DCGH</I>):(<I>CEFG</I>)=<I>DC</I>:<I>CE</I> (1)
+=<I>L</I>:<I>M</I></MATH>,
+<pb n=394><head>EUCLID</head>
+<p>Therefore, <I>ex aequali</I> (V. 22),
+<MATH>(<I>ABCD</I>):(<I>CEFG</I>)=<I>K</I>:<I>M</I></MATH>.
+<p>The important Proposition 25 (to construct a rectilineal figure
+similar to one, and equal to another, given rectilineal figure) is
+one of the famous problems alternatively associated with the
+story of Pythagoras's sacrifice<note>Plutarch, <I>Non posse suaviter vivi secundum Epicurum</I>, c. 11.</note>; it is doubtless Pythagorean.
+The given figure (<I>P,</I> say) to which the required figure is to be
+similar is transformed (I. 44) into a parallelogram on the same
+base <I>BC.</I> Then the other figure (<I>Q,</I> say) to which the required
+figure is to be <I>equal</I> is (I. 45) transformed into a parallelo-
+gram on the base <I>CF</I> (in a straight line with <I>BC</I>) and of equal
+height with the other parallelogram. Then <MATH>(<I>P</I>):(<I>Q</I>)=<I>BC</I>:<I>CF</I></MATH>
+(1). It is then only necessary to take a straight line <I>GH</I>
+a mean proportional between <I>BC</I> and <I>CF</I>, and to describe on
+<I>GH</I> as base a rectilineal figure similar to <I>P</I> which has <I>BC</I> as
+base (VI. 18). The proof of the correctness of the construction
+follows from VI. 19 Por.
+<p>In 27, 28, 29 we reach the final problems in the Pythagorean
+<I>application of areas</I>, which are the geometrical equivalent of
+the algebraical solution of the most general form of quadratic
+equation where that equation has a real and positive root.
+Detailed notice of these propositions is necessary because of
+their exceptional historic importance, which arises from the
+fact that the method of these propositions was constantly used
+<FIG>
+by the Greeks in the solution of problems. They constitute,
+for example, the foundation of Book X of the <I>Elements</I> and of
+<pb n=395><head>THE <I>ELEMENTS.</I> BOOK VI</head>
+the whole treatment of conic sections by Apollonius. The
+problems themselves are enunciated in 28, 29: &lsquo;To a given
+straight line to apply a parallelogram equal to a given recti-
+lineal figure and <I>deficient</I> (or <I>exceeding</I>) by a parallelogrammic
+figure similar to a given parallelogram&rsquo;; and 27 supplies the
+<G>diorismo/s</G>, or determination of the condition of possibility,
+which is necessary in the case of <I>deficiency</I> (28): &lsquo;The given
+rectilineal figure must (in that case) not be greater than the
+parallelogram described on the half of the straight line and
+similar to the defect.&rsquo; We will take the problem of 28 for
+examination.
+<p>We are already familiar with the notion of applying a
+parallelogram to a straight line <I>AB</I> so that it <I>falls short</I> or
+<I>exceeds</I> by a certain other parallelogram. Suppose that <I>D</I> is
+the given parallelogram to which the <I>defect</I> in this case has to
+be similar. Bisect <I>AB</I> in <I>E</I>, and on the half <I>EB</I> describe the
+parallelogram <I>GEBF</I> similar and similarly situated to <I>D.</I>
+Draw the diagonal <I>GB</I> and complete the parallelogram
+<I>HABF.</I> Now, if we draw through any point <I>T</I> on <I>HA</I> a
+straight line <I>TR</I> parallel to <I>AB</I> meeting the diagonal <I>GB</I> in
+<I>Q</I>, and then draw <I>PQS</I> parallel to <I>TA</I>, the parallelogram <I>TASQ</I>
+is a parallelogram applied to <I>AB</I> but falling short by a
+parallelogram similar and similarly situated to <I>D</I>, since the
+deficient parallelogram is <I>QSBR</I> which is similar to <I>EF</I> (24).
+(In the same way, if <I>T</I> had been on <I>HA produced</I> and <I>TR</I> had
+met <I>GB produced</I> in <I>R</I>, we should have had a parallelogram
+applied to <I>AB</I> but <I>exceeding</I> by a parallelogram similar and
+similarly situated to <I>D.</I>)
+<p>Now consider the parallelogram <I>AQ</I> falling short by <I>SR</I>
+similar and similarly situated to <I>D.</I> Since (<I>AO</I>) = (<I>ER</I>), and
+(<I>OS</I>) = (<I>QF</I>), it follows that the parallelogram <I>AQ</I> is equal to
+the gnomon <I>UWV</I>, and the problem is therefore that of
+constructing the gnomon <I>UWV</I> such that its area is equal to
+that of the given rectilineal figure <I>C.</I> The gnomon obviously
+cannot be greater than the parallelogram <I>EF</I>, and hence the
+given rectilineal figure <I>C</I> must not be greater than that
+parallelogram. This is the <G>diorismo/s</G> proved in 27.
+<p>Since the gnomon is equal to <I>C</I>, it follows that the parallelo-
+gram <I>GOQP</I> which with it makes up the parallelogram <I>EF</I> is
+equal to the difference between (<I>EF</I>) and <I>C.</I> Therefore, in
+<pb n=396><head>EUCLID</head>
+order to construct the required gnomon, we have only to draw
+in the angle <I>FGE</I> the parallelogram <I>GOQP</I> equal to (<I>EF</I>)-<I>C</I>
+and similar and similarly situated to <I>D.</I> This is what Euclid
+in fact does; he constructs the parallelogram <I>LKNM</I> equal to
+(<I>EF</I>) &mdash; <I>C</I> and similar and similarly situated to <I>D</I> (by means of
+25), and then draws <I>GOQP</I> equal to it. The problem is thus
+solved, <I>TASQ</I> being the required parallelogram.
+<p>To show the correspondence to the solution of a quadratic
+equation, let <MATH><I>AB</I>=<I>a, QS</I>=<I>x</I></MATH>, and let <I>b</I>:<I>c</I> be the ratio of the
+sides of <I>D</I>; therefore <MATH><I>SB</I>=<I>(b/c)x.</I></MATH> Then, if <I>m</I> is a certain con-
+stant (in fact the sine of an angle of one of the parallelograms),
+<MATH>(<I>AQ</I>)=<I>m</I>(<I>ax - (b/c)x</I><SUP>2</SUP>)</MATH>, so that the equation solved is
+<MATH><I>m</I>(<I>ax-(b/c)x</I><SUP>2</SUP>)=<I>C.</I></MATH>
+The algebraical solution is <MATH><I>x</I>=<I>c/b.a</I>/2&plusmn;&radic;{<I>a/b</I>(<I>c/b.a</I><SUP>2</SUP>/4-<I>C/m</I>)}</MATH>.
+Euclid gives only one solution (that corresponding to the
+<I>negative</I> sign), but he was of course aware that there are two,
+and how he could exhibit the second in the figure.
+<p>For a real solution we must have <I>C</I> not greater than
+<MATH><I>m(c/b).a</I><SUP>2</SUP>/4</MATH>, which is the area of <I>EF.</I> This corresponds to Pro-
+position 27.
+<p>We observe that what Euclid in fact does is to find the
+parallelogram <I>GOQP</I> which is of given shape (namely such
+that its area <MATH><I>m.GO.OQ</I>=<I>m.GO</I><SUP>2</SUP>(<I>b/c</I>)</MATH>) and is equal to (<I>EF</I>)-<I>C</I>;
+that is, he finds <I>GO</I> such that <MATH><I>GO</I><SUP>2</SUP>=<I>c/b</I>((<I>c/b</I>).(<I>a</I><SUP>2</SUP>/4)-<I>C/m</I>)</MATH>. In other
+words, he finds the straight line equal to <MATH>&radic;{<I>c/b</I>((<I>c/b</I>).(<I>a</I><SUP>2</SUP>/4)-<I>C/m</I>)}</MATH>;
+and <I>x</I> is thus known, since <MATH><I>x</I>=<I>GE - GO</I>=(<I>c/b</I>).(<I>a</I>/2)-<I>GO</I></MATH>.
+Euclid's procedure, therefore, corresponds closely to the alge-
+braic solution.
+<p>The solution of 29 is exactly similar, <I>mutatis mutandis.</I>
+A solution is always possible, so that no <G>diorismo/s</G> is required.
+<pb n=397><head>THE <I>ELEMENTS.</I> BOOKS VI-VII</head>
+<p>VI. 31 gives the extension of the Pythagorean proposition
+I. 47 showing that for squares in the latter proposition we
+may substitute similar plane figures of any shape whatever.
+30 uses 29 to divide a straight line in extreme and mean
+ratio (the same problem as II. 11).
+<p>Except in the respect that it is based on the new theory of
+proportion, Book VI does not appear to contain any matter
+that was not known before Euclid's time. Nor is the generali-
+zation of I. 47 in VI. 31, for which Proclus professes such
+admiration, original on Euclid's part, for, as we have already
+seen (p. 191), Hippocrates of Chios assumes its truth for semi-
+circles described on the three sides of a right-angled triangle.
+<p>We pass to the arithmetical Books, VII, VIII, IX. Book VII
+begins with a set of definitions applicable in all the three
+Books. They include definitions of a <I>unit</I>, a <I>number</I>, and the
+following varieties of numbers, <I>even, odd, even-times-even, even-
+times-odd, odd-times-odd, prime, prime to one another, com-
+posite, composite to one another, plane, solid, square, cube,
+similar plane</I> and <I>solid</I> numbers, and a <I>perfect</I> number,
+definitions of terms applicable in the numerical theory of pro-
+portion, namely <I>a part</I> (= a submultiple or aliquot part),
+<I>parts</I> (=a proper fraction), <I>multiply</I>, and finally the defini-
+tion of (four) proportional numbers, which states that &lsquo;num-
+bers are proportional when the first is the same multiple, the
+same part, or the same parts, of the second that the third is of
+the fourth&rsquo;, i.e. numbers <I>a, b, c, d</I> are proportional if, when
+<MATH><I>a</I>=<I>(m/n)b</I>, <I>c</I>=<I>(m/n)d</I></MATH>, where <I>m, n</I> are any integers (although the
+definition does not in terms cover the case where <I>m>n</I>).
+<p>The propositions of Book VII fall into four main groups.
+1-3 give the method of finding the greatest common mea-
+sure of two or three unequal numbers in essentially the same
+form in which it appears in our text-books, Proposition 1
+giving the test for two numbers being prime to one another,
+namely that no remainder measures the preceding quotient
+till 1 is reached. The second group, 4-19, sets out the
+numerical theory of proportion. 4-10 are preliminary, deal-
+ing with numbers which are &lsquo;a part&rsquo; or &lsquo;parts&rsquo; of other num-
+bers, and numbers which are the same &lsquo;part&rsquo; or &lsquo;parts&rsquo; of
+other numbers, just as the preliminary propositions of Book V
+<pb n=398><head>EUCLID</head>
+deal with multiples and equimultiples. 11-14 are transforma-
+tions of proportions corresponding to similar transformations
+(<I>separando</I>, alternately, &c.) in Book V. The following are
+the results, expressed with the aid of letters which here repre-
+sent integral numbers exclusively.
+<p>If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I> (<I>a</I>><I>c, b</I>><I>d</I>)</MATH>, then
+<MATH>(<I>a-c</I>):(<I>b-d</I>)=<I>a</I>:<I>b</I></MATH>. (11)
+<p>If <MATH><I>a</I>:<I>a</I>&prime;=<I>b</I>:<I>b</I>&prime;=<I>c</I>:<I>c</I>&prime;...</MATH>, then each of the ratios is equal to
+<MATH>(<I>a</I>+<I>b</I>+<I>c</I>+...):(<I>a</I>&prime;+<I>b</I>&prime;+<I>c</I>&prime;+...)</MATH>. (12)
+<p>If <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, then <MATH><I>a</I>:<I>c</I>=<I>b</I>:<I>d</I></MATH>. (13)
+<p>If <MATH><I>a</I>:<I>b</I>=<I>d</I>:<I>e</I></MATH> and <MATH><I>b</I>:<I>c</I>=<I>e</I>:<I>f</I></MATH>, then, <I>ex aequali,</I>
+<MATH><I>a</I>:<I>c</I>=<I>d</I>:<I>f</I></MATH>. (14)
+<p>If 1:<I>m</I>=<I>a</I>:<I>ma</I> (expressed by saying that the third
+number measures the fourth the same number of times that
+the unit measures the second), then alternately
+<MATH>1:<I>a</I>=<I>m</I>:<I>ma.</I></MATH> (15)
+<p>The last result is used to prove that <I>ab</I>=<I>ba</I>; in other
+words, that the order of multiplication is indifferent (16), and
+this is followed by the propositions that <I>b</I>:<I>c</I>=<I>ab</I>:<I>ac</I> (17)
+and that <MATH><I>a</I>:<I>b</I>=<I>ac</I>:<I>bc</I></MATH> (18), which are again used to prove
+the important proposition (19) that, if <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, then
+<MATH><I>ad</I>=<I>bc</I></MATH>, a theorem which corresponds to VI. 16 for straight
+lines.
+<p>Zeuthen observes that, while it was necessary to use the
+numerical definition of proportion to carry the numerical
+theory up to this point, Proposition 19 establishes the necessary
+point of contact between the two theories, since it is now
+shown that the definition of proportion in V, Def. 5, has,
+when applied to numbers, the same import as that in VII,
+Def. 20, and we can henceforth without hesitation borrow any
+of the propositions established in Book V.<note>Zeuthen, &lsquo;Sur la constitution des livres arithm&eacute;tiques des &Eacute;l&eacute;ments
+d'Euclide&rsquo; (<I>Oversigt over det kgl. Danske Videnskabernes Selskabs Forhand-
+linger</I>, 1910, pp. 412, 413).</note>
+<p>Propositions 20, 21 about &lsquo;the least numbers of those which
+have the same ratio with them&rsquo; prove that, if <I>m, n</I> are such
+numbers and <I>a, b</I> any other numbers in the same ratio, <I>m</I>
+<pb n=399><head>THE <I>ELEMENTS.</I> BOOKS VII-VIII</head>
+measures <I>a</I> the same number of times that <I>n</I> measures <I>b</I>, and
+that numbers prime to one another are the least of those which
+have the same ratio with them. These propositions lead up to
+Propositions 22-32 about numbers prime to one another, prime
+numbers, and composite numbers. This group includes funda-
+mental theorems such as the following. If two numbers be
+prime to any number, their product will be prime to the same
+(24). If two numbers be prime to one another, so will their
+squares, their cubes, and so on generally (27). If two numbers
+be prime to one another, their sum will be prime to each
+of them; and, if the sum be prime to either, the original
+numbers will be prime to one another (28). Any prime number
+is prime to any number which it does not measure (29). If two
+numbers are multiplied, and any prime number measures the
+product, it will measure one of the original numbers (30).
+Any composite number is measured by some prime number
+(31). Any number either is prime or is measured by some
+prime number (32).
+<p>Propositions 33 to the end (39) are directed to the problem
+of finding the least common multiple of two or three numbers;
+33 is preliminary, using the G. C. M. for the purpose of solving
+the problem, &lsquo;Given as many numbers as we please, to find the
+least of those which have the same ratio with them.&rsquo;
+<p>It seems clear that in Book VII Euclid was following
+earlier models, while no doubt making improvements in the
+exposition. This is, as we have seen (pp. 215-16), partly con-
+firmed by the fact that in the proof by Archytas of the
+proposition that &lsquo;no number can be a mean between two
+consecutive numbers&rsquo; propositions are presupposed correspond-
+ing to VII. 20, 22, 33.
+<p>Book VIII deals largely with series of numbers &lsquo;in con-
+tinued proportion&rsquo;, i.e. in geometrical progression (Propositions
+1-3, 6-7, 13). If the series in G.P. be
+<MATH><I>a<SUP>n</SUP>, a<SUP>n-1</SUP>b, a<SUP>n-2</SUP>b<SUP>2</SUP>,... a<SUP>2</SUP>b<SUP>n-2</SUP>, ab<SUP>n-1</SUP>, b<SUP>n</SUP></I></MATH>,
+Propositions 1-3 deal with the case where the terms are the
+smallest that are in the ratio <I>a</I>:<I>b</I>, in which case <I>a</I><SUP><I>n</I></SUP>, <I>b<SUP>n</SUP></I> are
+prime to one another. 6-7 prove that, if <I>a<SUP>n</SUP></I> does not measure
+<I>a</I><SUP><I>n</I>-1</SUP><I>b</I>, no term measures any other, but if <I>a<SUP>n</SUP></I> measures <I>b<SUP>n</SUP></I>,
+it measures <I>a</I><SUP><I>n</I>-1</SUP><I>b.</I> Connected with these are Propositions 14-17
+<pb n=400><head>EUCLID</head>
+proving that, according as <I>a</I><SUP>2</SUP> does or does not measure <I>b</I><SUP>2</SUP>,
+<I>a</I> does or does not measure <I>b</I> and vice versa; and similarly,
+according as <I>a</I><SUP>3</SUP> does or does not measure <I>b</I><SUP>3</SUP>, <I>a</I> does or does not
+measure <I>b</I> and vice versa. 13 proves that, if <I>a, b, c</I> ... are in
+G. P., so are <I>a</I><SUP>2</SUP>, <I>b</I><SUP>2</SUP>, <I>c</I><SUP>2</SUP> ... and <I>a</I><SUP>3</SUP>, <I>b</I><SUP>3</SUP>, <I>c</I><SUP>3</SUP> ... respectively.
+<p>Proposition 4 is the problem, Given as many ratios as we
+please, <I>a</I>:<I>b, c</I>:<I>d</I> ... to find a series <I>p, q, r,</I> ... in the least
+possible terms such that <MATH><I>p</I>:<I>q</I>=<I>a</I>:<I>b, q</I>:<I>r</I>=<I>c</I>:<I>d,</I></MATH> .... This is
+done by finding the L. C. M., first of <I>b, c</I>, and then of other
+pairs of numbers as required. The proposition gives the
+means of compounding two or more ratios between numbers
+in the same way that ratios between pairs of straight lines
+are compounded in VI. 23; the corresponding proposition to
+VI. 23 then follows (5), namely, that plane numbers have
+to one another the ratio compounded of the ratios of their
+sides.
+<p>Propositions 8-10 deal with the interpolation of geometric
+means between numbers. If <MATH><I>a</I>:<I>b</I>=<I>e</I>:<I>f</I></MATH>, and there are <I>n</I>
+geometric means between <I>a</I> and <I>b</I>, there are <I>n</I> geometric
+means between <I>e</I> and <I>f</I> also (8). If <I>a</I><SUP><I>n</I></SUP>, <I>a</I><SUP><I>n</I>-1</SUP><I>b</I> ... <I>ab<SUP>n</I>-1</SUP>, <I>b<SUP>n</SUP></I> is a
+G. P. of <I>n</I>+1 terms, so that there are (<I>n</I>-1) means between
+<I>a<SUP>n</SUP>, b<SUP>n</SUP></I>, there are the same number of geometric means between
+1 and <I>a<SUP>n</SUP></I> and between 1 and <I>b</I><SUP><I>n</I></SUP> respectively (9); and con-
+versely, if 1, <I>a, a</I><SUP>2</SUP> ... <I>a<SUP>n</SUP></I> and 1, <I>b, b</I><SUP>2</SUP> ... <I>b<SUP>n</SUP></I> are terms in G. P.,
+there are the same number (<I>n</I>-1) of means between <I>a<SUP>n</SUP>, b<SUP>n</SUP></I> (10).
+In particular, there is one mean proportional number between
+square numbers (11) and between similar plane numbers (18),
+and conversely, if there is one mean between two numbers, the
+numbers are similar plane numbers (20); there are two means
+between cube numbers (12) and between similar solid numbers
+(19), and conversely, if there are two means between two num-
+bers, the numbers are similar solid numbers (21). So far as
+squares and cubes are concerned, these propositions are stated by
+Plato in the <I>Timaeus,</I> and Nicomachus, doubtless for this reason,
+calls them &lsquo;Platonic&rsquo;. Connected with them are the proposi-
+tions that similar plane numbers have the same ratio as a square
+has to a square (26), and similar solid numbers have the same
+ratio as a cube has to a cube (27). A few other subsidiary
+propositions need no particular mention.
+<p>Book IX begins with seven simple propositions such as that
+<pb n=401><head>THE <I>ELEMENTS.</I> BOOK IX</head>
+the product of two similar plane numbers is a square (1) and,
+if the product of two numbers is a square number, the num-
+bers are similar plane numbers (2); if a cube multiplies itself
+or another cube, the product is a cube (3, 4); if <I>a</I><SUP>3</SUP><I>B</I> is a
+cube, <I>B</I> is a cube (5); if <I>A</I><SUP>2</SUP> is a cube, <I>A</I> is a cube (6). Then
+follow six propositions (8-13) about a series of terms in geo-
+metrical progression beginning with 1. If 1, <I>a, b, c ... k</I> are
+<I>n</I> terms in geometrical progression, then (9), if <I>a</I> is a square
+(or a cube), all the other terms <I>b, c, ... k</I> are squares (or
+cubes); if <I>a</I> is not a square, then the only squares in the series
+are the term after <I>a,</I> i.e. <I>b,</I> and all alternate terms after <I>b;</I> if
+<I>a</I> is not a cube, the only cubes in the series are the fourth
+term (<I>c</I>), the seventh, tenth, &c., terms, being terms separated
+by two throughout; the seventh, thirteenth, &c., terms (leaving
+out five in each case) will be both square and cube (8, 10).
+These propositions are followed by the interesting theorem
+that, if 1, <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB> ... <I>a<SUB>n</SUB></I> ... are terms in geometrical progression,
+and if <I>a<SUB>r</SUB>, a<SUB>n</SUB></I> are any two terms where <I>r<n, a<SUB>r</SUB></I> measures <I>a<SUB>n</SUB>,</I>
+and <MATH><I>a</I><SUB><I>n</I></SUB>=<I>a</I><SUB><I>r</I></SUB>.<I>a</I><SUB><I>n-r</I></SUB></MATH> (11 and Por.); this is, of course, equivalent
+to the formula <MATH><I>a</I><SUP><I>m</I>+<I>n</I></SUP>=<I>a</I><SUP><I>m</I></SUP>.<I>a</I><SUP><I>n</I></SUP></MATH>. Next it is proved that, if the
+last term <I>k</I> in a series 1, <I>a, b, c ... k</I> in geometrical progression
+is measured by any primes, <I>a</I> is measured by the same (12);
+and, if <I>a</I> is prime, <I>k</I> will not be measured by any numbers
+except those which have a place in the series (13). Proposi-
+tion 14 is the equivalent of the important theorem that <I>a
+number can only be resolved into prime factors in one way.</I>
+Propositions follow to the effect that, if <I>a, b</I> be prime to one
+another, there can be no integral third proportional to them
+(16) and, if <I>a, b, c ... k</I> be in G.P. and <I>a, k</I> are prime to one
+another, then there is no integral fourth proportional to <I>a, b, k</I>
+(17). The conditions for the possibility of an integral third
+proportional to two numbers and of an integral fourth propor-
+tional to three are then investigated (18, 19). Proposition 20
+is the important proposition that <I>the number of prime numbers
+is infinite</I>, and the proof is the same as that usually given in
+our algebraical text-books. After a number of easy proposi-
+tions about odd, even, &lsquo;even-times-odd&rsquo;, &lsquo;even-times-even&rsquo;
+numbers respectively (Propositions 21-34), we have two im-
+portant propositions which conclude the Book. Proposition 35
+gives the summation of a G.P. of <I>n</I> terms, and a very elegant
+<pb n=402><head>EUCLID</head>
+solution it is. Suppose that <I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB>, <I>a</I><SUB>3</SUB>, ... <I>a<SUB>n</I>+1</SUB> are <I>n</I>+1 terms
+in G. P.; Euclid proceeds thus:
+<p>We have <MATH><I>a</I><SUB><I>n</I>+1</SUB>/<I>a<SUB>n</SUB></I>=<I>a<SUB>n</SUB></I>/<I>a</I><SUB><I>n</I>-1</SUB>=...=<I>a</I><SUB>2</SUB>/<I>a</I><SUB>1</SUB></MATH>,
+and, <I>separando</I>, <MATH><I>a</I><SUB><I>n</I>+1</SUB>-<I>a<SUB>n</SUB></I>/<I>a<SUB>n</SUB></I>=<I>a<SUB>n</SUB></I>-<I>a</I><SUB><I>n</I>-1</SUB>/<I>a</I><SUB><I>n</I>-1</SUB>=...=<I>a</I><SUB>2</SUB>-<I>a</I><SUB>1</SUB>/<I>a</I><SUB>1</SUB></MATH>
+<p>Adding antecedents and consequents, we have (VII. 12)
+<MATH><I>a</I><SUB><I>n</I>+1</SUB>-<I>a</I><SUB>1</SUB>/<I>a<SUB>n</SUB></I>+<I>a</I><SUB><I>n</I>-1</SUB>+...+<I>a</I><SUB>1</SUB>=<I>a</I><SUB>2</SUB>-<I>a</I><SUB>1</SUB>/<I>a</I><SUB>1</SUB></MATH>,
+which gives <MATH><I>a</I><SUB><I>n</I></SUB>+<I>a</I><SUB><I>n</I>-1</SUB>+... +<I>a</I><SUB>1</SUB></MATH> or <I>S</I><SUB><I>n</I></SUB>.
+<p>The last proposition (36) gives the criterion for <I>perfect
+numbers</I>, namely that, if, the sum of any number of terms of
+the series 1, 2, 2<SUP>2</SUP> ... 2<SUP><I>n</I></SUP> is prime, the product of the said sum
+and of the last term, viz. (1+2+2<SUP>2</SUP>+...+2<SUP><I>n</I></SUP>) 2<SUP><I>n</I></SUP>, is a perfect
+number, i.e. is equal to the sum of all its factors.
+<p>It should be added, as regards all the arithmetical Books,
+that all numbers are represented in the diagrams as simple
+straight lines, whether they are linear, plane, solid, or any
+other kinds of numbers; thus a product of two or more factors
+is represented as a new straight line, not as a rectangle or a
+solid.
+<p>Book X is perhaps the most remarkable, as it is the most
+perfect in form, of all the Books of the <I>Elements.</I> It deals
+with irrationals, that is to say, irrational straight lines in rela-
+tion to any particular straight line assumed as rational, and
+it investigates every possible variety of straight lines which
+can be represented by &radic;(&radic;<I>a</I>&plusmn;&radic;<I>b</I>), where <I>a, b</I> are two com-
+mensurable lines. The theory was, of course, not invented by
+Euclid himself. On the contrary, we know that not only the
+fundamental proposition X. 9 (in which it is proved that
+squares which have not to one another the ratio of a square
+number to a square number have their sides incommen-
+surable in length, and conversely), but also a large part of
+the further development of the subject, was due to Theae-
+tetus. Our authorities for this are a scholium to X. 9 and a
+passage from Pappus's commentary on Book X preserved
+in the Arabic (see pp. 154-5, 209-10, above). The passage
+<pb n=403><head>THE <I>ELEMENTS.</I> BOOKS IX-X</head>
+of Pappus goes on to speak of the share of Euclid in the
+investigation:
+<p>&lsquo;As for Euclid, he set himself to give rigorous rules, which he
+established, relative to commensurability and incommensura-
+bility in general; he made precise the definitions and the
+distinctions between rational and irrational magnitudes, he set
+out a great number of orders of irrational magnitudes, and
+finally he made clear their whole extent.&rsquo;
+<p>As usual, Euclid begins with definitions. &lsquo;Commensurable&rsquo;
+magnitudes can be measured by one and the same measure;
+&lsquo;incommensurable&rsquo; magnitudes cannot have any common
+measure (1). Straight lines are &lsquo;commensurable in square&rsquo;
+when the squares on them can be measured by the same area,
+but &lsquo;incommensurable in square&rsquo; when the squares on them
+have no common measure (2). Given an assigned straight
+line, which we agree to call &lsquo;rational&rsquo;, any straight line which
+is commensurable with it either in length or in square only is
+also called rational; but any straight line which is incommen-
+surable with it (i.e. not commensurable with it either in length
+or in square) is &lsquo;irrational&rsquo; (3). The square on the assigned
+straight line is &lsquo;rational&rsquo;, and any area commensurable with
+it is &lsquo;rational&rsquo;, but any area incommensurable with it is
+&lsquo;irrational&rsquo;, as also is the side of the square equal to that
+area (4). As regards straight lines, then, Euclid here takes
+a wider view of &lsquo;rational&rsquo; than we have met before. If a
+straight line <G>r</G> is assumed as rational, not only is (<I>m/n</I>)<G>r</G> also
+&lsquo;rational&rsquo; where <I>m, n</I> are integers and <I>m/n</I> in its lowest terms
+is not square, but any straight line is rational which is either
+commensurable in length or commensurable <I>in square only</I>
+with <G>r</G>; that is, <MATH>&radic;(<I>m/n</I>).<G>r</G></MATH> is rational according to Euclid. In
+the case of squares, <G>r</G><SUP>2</SUP> is of course rational, and so is <MATH>(<I>m/n</I>)<G>r</G><SUP>2</SUP></MATH>; but
+<MATH>&radic;(<I>m/n</I>).<G>r</G><SUP>2</SUP></MATH> is not rational, and of course the side of the latter
+square <MATH>&radic;<SUP>4</SUP>(<I>m/n</I>).<G>r</G></MATH> is irrational, as are all straight lines commen-
+surable neither in length nor in square with <G>r</G>, e. g. <MATH>&radic;<I>a</I>&plusmn;&radic;<I>b</I></MATH>
+or (<MATH>&radic;<I>k</I>&plusmn;&radic;<G>l</G>).<G>r</G></MATH>.
+<pb n=404>
+<head>EUCLID</head>
+<p>The Book begins with the famous proposition, on which the
+&lsquo;method of exhaustion&rsquo; as used in Book XII depends, to the
+effect that, if from any magnitude there be subtracted more
+than its half (or its half simply), from the remainder more than
+its half (or its half), and so on continually, there will at length
+remain a magnitude less than any assigned magnitude of the
+same kind. Proposition 2 uses the process for finding the
+G. C. M. of two magnitudes as a test of their commensurability
+or incommensurability: they are incommensurable if the process
+never comes to an end, i.e. if no remainder ever measures the
+preceding divisor; and Propositions 3, 4 apply to commen-
+surable magnitudes the method of finding the G. C. M. of two
+or three <I>numbers</I> as employed in VII. 2, 3. Propositions 5
+to 8 show that two magnitudes are commensurable or incom-
+mensurable according as they have or have not to one another
+the ratio of one number to another, and lead up to the funda-
+mental proposition (9) of Theaetetus already quoted, namely
+that the sides of squares are commensurable or incommen-
+surable in length according as the squares have or have not to
+one another the ratio of a square number to a square number,
+and conversely. Propositions 11-16 are easy inferences as to
+the commensurability or incommensurability of magnitudes
+from the known relations of others connected with them;
+e.g. Proposition 14 proves that, if <MATH><I>a</I>:<I>b</I>=<I>c</I>:<I>d</I></MATH>, then, according
+as <MATH>&radic;(<I>a</I><SUP>2</SUP>-<I>b</I><SUP>2</SUP>)</MATH> is commensurable or incommensurable with <I>a</I>,
+<MATH>&radic;(<I>c</I><SUP>2</SUP>-<I>d</I><SUP>2</SUP></MATH> is commensurable or incommensurable with <I>c.</I>
+Following on this, Propositions 17, 18 prove that the roots of
+the quadratic equation <MATH><I>ax</I>-<I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP>/4</MATH> are commensurable or
+incommensurable with <I>a</I> according as <MATH>&radic;(<I>a</I><SUP>2</SUP>-<I>b</I><SUP>2</SUP>)</MATH> is commen-
+surable or incommensurable with <I>a.</I> Propositions 19-21 deal
+with rational and irrational <I>rectangles</I>, the former being
+contained by straight lines commensurable in length, whereas
+rectangles contained by straight lines commensurable in square
+only are irrational. The side of a square equal to a rectangle
+of the latter kind is called <I>medial</I>; this is the first in Euclid's
+classification of irrationals. As the sides of the rectangle may
+be expressed as <G>r</G>, <G>r</G>&radic;<I>k</I>, where <G>r</G> is a rational straight line,
+the <I>medial</I> is <I>k</I><SUP>1/4</SUP><G>r</G>. Propositions 23-8 relate to medial straight
+lines and rectangles; two medial straight lines may be either
+commensurable in length or commensurable in square only:
+<pb n=405>
+<head>THE <I>ELEMENTS.</I> BOOK X</head>
+thus <I>k</I><SUP>1/4</SUP><G>r</G> and <G>l</G><I>k</I><SUP>1/4</SUP><G>r</G> are commensurable in length, while <I>k</I><SUP>1/4</SUP><G>r</G>
+and &radic;<G>l</G>.<I>k</I><SUP>1/4</SUP><G>r</G> are commensurable in square only: the rectangles
+formed by such pairs are in general <I>medial</I>, as <G>l</G><I>k</I><SUP>1/2</SUP><G>r</G><SUP>2</SUP> and
+&radic;<G>l</G>.<I>k</I><SUP>1/2</SUP><G>r</G><SUP>2</SUP>; but if <MATH>&radic;<G>l</G>=<I>k</I>&prime;&radic;<I>k</I></MATH> in the second case, the rectangle
+(<I>k&prime;k</I><G>r</G><SUP>2</SUP>) is rational (Propositions 24, 25). Proposition 26 proves
+that the difference between two medial areas cannot be
+rational; as any two medial areas can be expressed in the
+form &radic;<I>k</I>.<G>r</G><SUP>2</SUP>, &radic;<G>l</G>.<G>r</G><SUP>2</SUP>, this is equivalent to proving, as we do in
+algebra, that (&radic;<I>k</I>-&radic;<G>l</G>) cannot be equal to <I>k</I>&prime;. Finally,
+Propositions 27, 28 find medial straight lines commensurable
+in square only (1) which contain a rational rectangle, viz. <I>k</I><SUP>1/4</SUP><G>r</G>,
+<I>k</I><SUP>3/4</SUP><G>r</G>, and (2) which contain a medial rectangle, viz.<I>k</I><SUP>1/4</SUP><G>r,l</G><SUP>1/2</SUP><G>r</G>/<I>k</I><SUP>1/4</SUP>. It
+should be observed that, as <G>r</G> may take either of the forms <I>a</I>
+or &radic;<I>A</I>, a medial straight line may take the alternative forms
+&radic;(<I>a</I>&radic;<I>B</I>) or &radic;<SUP>4</SUP>(<I>AB</I>), and the pairs of medial straight lines just
+mentioned may take respectively the forms
+(1) <MATH>&radic;(<I>a</I>&radic;<I>B</I>), &radic;(<I>B</I>&radic;<I>B</I>/<I>a</I>)</MATH> or <MATH>&radic;<SUP>4</SUP>(<I>AB</I>), &radic;(<I>B</I>(&radic;<I>B</I>/&radic;<I>A</I>))</MATH>
+and (2) <MATH>&radic;(<I>a</I>&radic;<I>B</I>), &radic;(<I>aC</I>/&radic;<I>B</I>)</MATH> or <MATH>&radic;<SUP>4</SUP>(<I>AB</I>), &radic;(<I>C</I>&radic;<I>A</I>/&radic;<I>B</I>)</MATH>
+I shall henceforth omit reference to these obvious alternative
+forms. Next follow two lemmas the object of which is to find
+(1) two square numbers the sum of which is a square, Euclid's
+solution being
+<MATH><I>mnp</I><SUP>2</SUP>.<I>mnq</I><SUP>2</SUP>+(<I>mnp</I><SUP>2</SUP>-<I>mnq</I><SUP>2</SUP>/2)<SUP>2</SUP>=(<I>mnp</I><SUP>2</SUP>+<I>mnq</I><SUP>2</SUP>/2)<SUP>2</SUP></MATH>,
+where <I>mnp</I><SUP>2</SUP>, <I>mnq</I><SUP>2</SUP> are either both odd or both even, and
+(2) two square numbers the sum of which is not square,
+Euclid's solution being
+<MATH><I>mp</I><SUP>2</SUP>.<I>mq</I><SUP>2</SUP>, ((<I>mp</I><SUP>2</SUP>-<I>mq</I><SUP>2</SUP>/2)-1)<SUP>2</SUP></MATH>.
+Propositions 29-35 are problems the object of which is to find
+(<I>a</I>) two rational straight lines commensurable in square only,
+(<I>b</I>) two medial straight lines commensurable in square only,
+(<I>c</I>) two straight lines incommensurable in square, such that
+the difference or sum of their squares and the rectangle
+<pb n=406>
+<head>EUCLID</head>
+contained by them respectively have certain characteristics.
+The solutions are
+<p>(<I>a</I>) <I>x, y</I> rational and commensurable in square only.
+<p>Prop. 29: <MATH><G>r, r</G>&radic;(1-<I>k</I><SUP>2</SUP>) [&radic;(<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>)</MATH> commensurable with <I>x</I>].
+&rdquo; 30: <MATH><G>r, r</G>/&radic;(1+<I>k</I><SUP>2</SUP>) [&radic;(<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>)</MATH> incommensurable with <I>x</I>].
+<p>(<I>b</I>) <I>x, y</I> medial and commensurable in square only.
+<p>Prop. 31: <MATH><G>r</G>(1-<I>k</I><SUP>2</SUP>)<SUP>1/4</SUP>, <G>r</G>(1-<I>k</I><SUP>2</SUP>)<SUP>3/4</SUP> [<I>xy</I> rational, &radic;(<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>)</MATH> commen-
+surable with <I>x</I>];
+<MATH><G>r</G>/(1+<I>k</I><SUP>2</SUP>)<SUP>1/4</SUP>, <G>r</G>/(1+<I>k</I><SUP>2</SUP>)<SUP>3/4</SUP> [<I>xy</I> rational, &radic;(<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>)</MATH> incom-
+mensurable with <I>x</I>].
+&rdquo; 32: <MATH><G>rl</G><SUP>1/4</SUP>, <G>rl</G><SUP>1/4</SUP>&radic;(1-<I>k</I><SUP>2</SUP>) [<I>xy</I> medial, &radic;(<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>)</MATH> commensur-
+able with <I>x</I>];
+<MATH><G>rl</G><SUP>1/4</SUP>, <G>rl</G><SUP>1/4</SUP>/&radic;(1+<I>k</I><SUP>2</SUP>) [<I>xy</I> medial, &radic;(<I>x</I><SUP>2</SUP>-<I>y</I><SUP>2</SUP>)</MATH> incommen-
+surable with <I>x</I>].
+<p>(<I>c</I>) <I>x, y</I> incommensurable in square.
+<p>Prop. 33: <MATH><G>r</G>/&radic;2&radic;(1+<I>k</I>/&radic;1+<I>k</I><SUP>2</SUP>), <G>r</G>/&radic;2&radic;(1 - <I>k</I>/&radic;1+<I>k</I><SUP>2</SUP>)
+[(<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP>)</MATH> rational, <I>xy</I> medial].
+&rdquo; 34: <MATH><G>r</G>/&radic;{2(1+<I>k</I><SUP>2</SUP>)}.&radic;{&radic;(1+<I>k</I><SUP>2</SUP>)+<I>k</I>},
+<G>r</G>/&radic;{2(1+<I>k</I><SUP>2</SUP>)}.&radic;{&radic;(1+<I>k</I><SUP>2</SUP>)-<I>k</I>}
+[<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP></MATH> medial, <I>xy</I> rational].
+&rdquo; 35: <MATH><G>rl</G><SUP>1/4</SUP>/&radic;2&radic;{1 + <I>k</I>/&radic;(1+<I>k</I><SUP>2</SUP>)}, <G>rl</G><SUP>1/4</SUP>/&radic;2&radic;{1 - <I>k</I>/&radic;(1+<I>k</I><SUP>2</SUP>)}
+[<I>x</I><SUP>2</SUP>+<I>y</I><SUP>2</SUP></MATH> and <I>xy</I> both medial and
+incommensurable with one another].
+With Proposition 36 begins Euclid's exposition of the several
+compound irrationals, twelve in number Those which only
+differ in the sign separating the two component parts can be
+<pb n=407>
+<head>THE <I>ELEMENTS.</I> BOOK X</head>
+taken together. The twelve compound irrationals, with their
+names, are as follows:
+<MATH>
+<BRACE>
+<note>(<I>A</I><SUB>1</SUB>) (<I>A</I><SUB>2</SUB>)</note>
+Binomial, <G>r</G> + &radic;<I>k</I>.<G>r</G> (Prop. 36)
+Apotome, <G>r</G> - &radic;<I>k</I>.<G>r</G> (Prop. 73)
+</BRACE>
+<BRACE>
+<note>(<I>B</I><SUB>1</SUB>) (<I>B</I><SUB>2</SUB>)</note>
+<note><I>k</I><SUP>1/4</SUP><G>r</G> &plusmn; <I>k</I><SUP>3/4</SUP><G>r</G> (Props. 37, 74)</note>
+First bimedial
+First apotome of a medial
+</BRACE>
+<BRACE>
+<note>(<I>C</I><SUB>1</SUB>) (<I>C</I><SUB>2</SUB>)</note>
+<note><I>k</I><SUP>1/4</SUP><G>r</G> &plusmn; <G>l</G><SUP>1/2</SUP><G>r</G>/<I>k</I><SUP>1/4</SUP> (Props. 38, 75)</note>
+Second bimedial
+Second apotome of a medial
+</BRACE>
+<BRACE>
+<note>(<I>D</I><SUB>1</SUB>) (<I>D</I><SUB>2</SUB>)</note>
+<note><G>r</G>/&radic;2&radic;(1 + <I>k</I>/&radic;(1+<I>k</I><SUP>2</SUP>)) &plusmn; <G>r</G>/&radic;2&radic;(1 - <I>k</I>/&radic;(1 + <I>k</I><SUP>2</SUP>)) (Props. 39, 76)</note>
+Major
+Minor
+</BRACE>
+<BRACE>
+<note>(<I>E</I><SUB>1</SUB>) (<I>E</I><SUB>2</SUB>)</note>
+<note><G>r</G>/&radic;2(1+<I>k</I><SUP>2</SUP>)&radic;(&radic;(1+<I>k</I><SUP>2</SUP>) + <I>k</I>) &plusmn; <G>r</G>/&radic;2(1+<I>k</I><SUP>2</SUP>)&radic;(&radic;(1+<I>k</I><SUP>2</SUP>)-<I>k</I>) (Props. 40, 77)</note>
+Side of a rational plus a medial area
+That which &lsquo;produces&rsquo; with a rational area a medial whole
+</BRACE>
+<BRACE>
+<note>(<I>F</I><SUB>1</SUB>) (<I>F</I><SUB>2</SUB>)</note>
+<note><G>rl</G><SUP>1/4</SUP>/&radic;2&radic;(1 + <I>k</I>/&radic;(1+<I>k</I><SUP>2</SUP>)) &plusmn; <G>rl</G><SUP>1/4</SUP>/&radic;2&radic;(1 - <I>k</I>/&radic;(1+<I>k</I><SUP>2</SUP>)) (Props. 41, 78).</note>
+Side of the sum of two medial areas
+That which &lsquo;produces&rsquo; with a medial area a medial whole
+</BRACE>
+</MATH>
+<p>As regards the above twelve compound irrationals, it is
+to be noted that
+<p><I>A</I><SUB>1</SUB>, <I>A</I><SUB>2</SUB> are the positive roots of the equation
+<MATH><I>x</I><SUP>4</SUP>-2(1+<I>k</I>)<G>r</G><SUP>2</SUP>.<I>x</I><SUP>2</SUP>+(1-<I>k</I>)<SUP>2</SUP><G>r</G><SUP>4</SUP>=0</MATH>;
+<p><I>B</I><SUB>1</SUB>, <I>B</I><SUB>2</SUB> are the positive roots of the equation
+<MATH><I>x</I><SUP>4</SUP>-2&radic;<I>k</I>(1+<I>k</I>)<G>r</G><SUP>2</SUP>.<I>x</I><SUP>2</SUP>+<I>k</I>(1-<I>k</I>)<SUP>2</SUP><G>r</G><SUP>4</SUP>=0</MATH>;
+<p><I>C</I><SUB>1</SUB>, <I>C</I><SUB>2</SUB> are the positive roots of the equation
+<MATH><I>x</I><SUP>4</SUP>-2<I>k</I>+<G>l</G>/&radic;<I>k</I><G>r</G><SUP>2</SUP>.<I>x</I><SUP>2</SUP>+(<I>k</I>-<G>l</G>)<SUP>2</SUP>/<I>k</I><G>r</G><SUP>4</SUP>=0</MATH>;
+<pb n=408>
+<head>EUCLID</head>
+<p><I>D</I><SUB>1</SUB>, <I>D</I><SUB>2</SUB> are the positive roots of the equation
+<MATH><I>x</I><SUP>4</SUP>-2<G>r</G><SUP>2</SUP>.<I>x</I><SUP>2</SUP>+<I>k</I><SUP>2</SUP>/1+<I>k</I><SUP>2</SUP><G>r</G><SUP>4</SUP>=0</MATH>;
+<p><I>E</I><SUB>1</SUB>, <I>E</I><SUB>2</SUB> are the positive roots of the equation
+<MATH><I>x</I><SUP>4</SUP>-2/&radic;(1+<I>k</I><SUP>2</SUP>)<G>r</G><SUP>2</SUP>.<I>x</I><SUP>2</SUP>+<I>k</I><SUP>2</SUP>/(1+<I>k</I><SUP>2</SUP>)<SUP>2</SUP><G>r</G><SUP>4</SUP>=0</MATH>;
+<p><I>F</I><SUB>1</SUB>, <I>F</I><SUB>2</SUB> are the positive roots of the equation
+<MATH><I>x</I><SUP>4</SUP>-2&radic;<G>l</G>.<I>x</I><SUP>2</SUP><G>r</G><SUP>2</SUP>+<G>l</G><I>k</I><SUP>2</SUP>/1+<I>k</I><SUP>2</SUP><G>r</G><SUP>4</SUP>=0</MATH>.
+<p>Propositions 42-7 prove that each of the above straight lines,
+made up of the <I>sum</I> of two terms, is divisible into its terms
+in only one way. In particular, Proposition 42 proves the
+equivalent of the well-known theorem in algebra that,
+if <MATH><I>a</I>+&radic;<I>b</I>=<I>x</I>+&radic;<I>y</I></MATH>, then <MATH><I>a</I>=<I>x</I>, <I>b</I>=<I>y</I></MATH>;
+and if <MATH>&radic;<I>a</I>+&radic;<I>b</I>=&radic;<I>x</I>+&radic;<I>y</I></MATH>,
+then <MATH><I>a</I>=<I>x</I>, <I>b</I>=<I>y</I> (or <I>a</I>=<I>y</I>, <I>b</I>=<I>x</I>)</MATH>.
+<p>Propositions 79-84 prove corresponding facts in regard to
+the corresponding irrationals with the negative sign between
+the terms: in particular Proposition 79 shows that,
+if <MATH><I>a</I>-&radic;<I>b</I>=<I>x</I>-&radic;<I>y</I></MATH>, then <MATH><I>a</I>=<I>x</I>, <I>b</I>=<I>y</I></MATH>;
+and if <MATH>&radic;<I>a</I>-&radic;<I>b</I>=&radic;<I>x</I>-&radic;<I>y</I></MATH>, then <MATH><I>a</I>=<I>x</I>, <I>b</I>=<I>y</I></MATH>.
+<p>The next sections of the Book deal with binomials and
+apotomes classified according to the relation of their terms to
+another given rational straight line. There are six kinds,
+which are first defined and then constructed, as follows:
+<MATH>
+<BRACE>
+<note>(<G>a</G><SUB>1</SUB>) (<G>a</G><SUB>2</SUB>)</note>
+<note><I>k</I><G>r</G>&plusmn;<I>k</I><G>r</G>&radic;(1-<G>l</G><SUP>2</SUP>); (Props. 48, 85)</note>
+First binomial
+First apotome
+</BRACE>
+<BRACE>
+<note>(<G>b</G><SUB>1</SUB>) (<G>b</G><SUB>2</SUB>)</note>
+<note><I>k</I><G>r</G>/&radic;(1-<G>l</G><SUP>2</SUP>) &plusmn; <I>k</I><G>r</G>; (Props. 49, 86)</note>
+Second binomial
+Second apotome
+</BRACE>
+<BRACE>
+<note>(<G>g</G><SUB>1</SUB>) (<G>g</G><SUB>2</SUB>)</note>
+<note><I>m</I>&radic;<I>k</I>.<G>r</G> &plusmn; <I>m</I>&radic;<I>k</I>.<G>r</G>&radic;(1-<G>l</G><SUP>2</SUP>); (Props. 50, 87)</note>
+Third binomial
+Third apotome
+</BRACE>
+</MATH>
+<pb n=409>
+<head>THE <I>ELEMENTS.</I> BOOK X</head>
+<MATH>
+<BRACE>
+<note>(<G>d</G><SUB>1</SUB>) (<G>d</G><SUB>2</SUB>)</note>
+<note><I>k</I><G>r</G> &plusmn; <I>k</I><G>r</G>/&radic;(1+<G>l</G>); (Props. 51, 88)</note>
+Fourth binomial
+Fourth apotome
+</BRACE>
+<BRACE>
+<note>(<G>e</G><SUB>1</SUB>) (<G>e</G><SUB>2</SUB>)</note>
+<note><I>k</I><G>r</G>&radic;(1+<G>l</G>) &plusmn; <I>k</I><G>r</G>; (Props. 52, 89)</note>
+Fifth binomial
+Fifth apotome
+</BRACE>
+<BRACE>
+<note>(<G>z</G><SUB>1</SUB>) (<G>z</G><SUB>2</SUB>)</note>
+<note>&radic;<I>k</I>.<G>r</G> &plusmn; &radic;<G>l.r</G> (Prop. 53, 90)</note>
+Sixth binomial
+Sixth apotome
+</BRACE>
+</MATH>
+<p>Here again it is to be observed that these binomials and
+apotomes are the greater and lesser roots respectively of
+certain quadratic equations,
+<p><G>a</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB> being the roots of <MATH><I>x</I><SUP>2</SUP>-2<I>k</I><G>r</G>.<I>x</I>+<G>l</G><SUP>2</SUP><I>k</I><SUP>2</SUP><G>r</G><SUP>2</SUP>=0</MATH>,
+<p><G>b</G><SUB>1</SUB>, <G>b</G><SUB>2</SUB> &rdquo; &rdquo; <MATH><I>x</I><SUP>2</SUP>- 2<I>k</I><G>r</G>/&radic;(1-<G>l</G><SUP>2</SUP>).<I>x</I> + <G>l</G><SUP>2</SUP>/1-<G>l</G><SUP>2</SUP><I>k</I><SUP>2</SUP><G>r</G><SUP>2</SUP>=0</MATH>,
+<p><G>g</G><SUB>1</SUB>, <G>g</G><SUB>2</SUB> &rdquo; &rdquo; <MATH><I>x</I><SUP>2</SUP>-2<I>m</I>&radic;<I>k</I>.<G>r</G><I>x</I>+<G>l</G><SUP>2</SUP><I>m</I><SUP>2</SUP><I>k</I><G>r</G><SUP>2</SUP>=0</MATH>,
+<p><G>d</G><SUB>1</SUB>, <G>d</G><SUB>2</SUB> &rdquo; &rdquo; <MATH><I>x</I><SUP>2</SUP>-2<I>k</I><G>r</G>.<I>x</I> + <G>l</G>/1+<G>l</G><I>k</I><SUP>2</SUP><G>r</G><SUP>2</SUP>=0</MATH>,
+<p><G>e</G><SUB>1</SUB>, <G>e</G><SUB>2</SUB> &rdquo; &rdquo; <MATH><I>x</I><SUP>2</SUP>-2<I>k</I><G>r</G>&radic;(1+<G>l</G>).<I>x</I>+<G>l</G><I>k</I><SUP>2</SUP><G>r</G><SUP>2</SUP>=0</MATH>,
+<p><G>z</G><SUB>1</SUB>, <G>z</G><SUB>2</SUB> &rdquo; &rdquo; <MATH><I>x</I><SUP>2</SUP>-2&radic;<I>k</I>.<G>r</G><I>x</I>+(<I>k</I>-<G>l</G>)<G>r</G><SUP>2</SUP>=0</MATH>.
+<p>The next sets of propositions (54-65 and 91-102) prove the
+connexion between the first set of irrationals (<I>A</I><SUB>1</SUB>, <I>A</I><SUB>2</SUB> ... <I>F</I><SUB>1</SUB>, <I>F</I><SUB>2</SUB>)
+and the second set (<G>a</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB> ... <G>z</G><SUB>1</SUB>, <G>z</G><SUB>2</SUB>) respectively. It is shown
+e.g., in Proposition 54, that the side of a square equal to the
+rectangle contained by <G>r</G> and the first binomial <G>a</G><SUB>1</SUB> is a binomial
+of the type <I>A</I><SUB>1</SUB>, and the same thing is proved in Proposition 91
+for the first apotome. In fact
+<MATH>&radic;{<G>r</G>(<I>k</I><G>r</G> &plusmn; <I>k</I><G>r</G>&radic;1-<G>l</G><SUP>2</SUP>)}=<G>r</G>&radic;{1/2<I>k</I>(1+<G>l</G>)} &plusmn; <G>r</G>&radic;{1/2<I>k</I>(1-<G>l</G>)}</MATH>.
+Similarly &radic;(<G>rb</G><SUB>1</SUB>), &radic;(<G>rb</G><SUB>2</SUB>) are irrationals of the type <I>B</I><SUB>1</SUB>, <I>B</I><SUB>2</SUB>
+respectively, and so on.
+<p>Conversely, the square on <I>A</I><SUB>1</SUB> or <I>A</I><SUB>2</SUB>, if applied as a rectangle
+to a rational straight line (<G>s</G>, say), has for its breadth a binomial
+or apotome of the types <G>a</G><SUB>1</SUB>, <G>a</G><SUB>2</SUB> respectively (60, 97).
+<p>In fact <MATH>(<G>r</G>&plusmn;&radic;<I>k</I>.<G>r</G>)<SUP>2</SUP>/<G>s</G>=<G>r</G><SUP>2</SUP>/<G>s</G> {(1+<I>k</I>) &plusmn; 2&radic;<I>k</I>}</MATH>,
+and <I>B</I><SUB>1</SUB><SUP>2</SUP>, <I>B</I><SUB>2</SUB><SUP>2</SUP> are similarly related to irrationals of the type
+<G>b</G><SUB>1</SUB>, <G>b</G><SUB>2</SUB>, and so on.
+<pb n=410>
+<head>EUCLID</head>
+<p>Propositions 66-70 and Propositions 103-7 prove that
+straight lines commensurable in length with <I>A</I><SUB>1</SUB>, <I>A</I><SUB>2</SUB> ... <I>F</I><SUB>1</SUB>, <I>F</I><SUB>2</SUB>
+respectively are irrationals of the same type and order.
+<p>Propositions 71, 72, 108-10 show that the irrationals
+<I>A</I><SUB>1</SUB>, <I>A</I><SUB>2</SUB> ... <I>F</I><SUB>1</SUB>, <I>F</I><SUB>2</SUB> arise severally as the sides of squares equal
+to the sum or difference of a rational and a medial area, or the
+sum or difference of two medial areas incommensurable with
+one another. Thus <I>k</I><G>r</G><SUP>2</SUP> &plusmn; &radic;<G>l.r</G><SUP>2</SUP> is the sum or difference of a
+rational and a medial area, &radic;<I>k</I>.<G>r</G><SUP>2</SUP> &plusmn; &radic;<G>l</G>.<G>r</G><SUP>2</SUP> is the sum or
+difference of two medial areas incommensurable with one
+another provided that &radic;<I>k</I> and &radic;<G>l</G> are incommensurable, and
+the propositions prove that
+<MATH>&radic;(<I>k</I><G>r</G><SUP>2</SUP> &plusmn; &radic;<G>l</G>.<G>r</G><SUP>2</SUP>) and &radic;(&radic;<I>k</I>.<G>r</G><SUP>2</SUP> &plusmn; &radic;<G>l</G>.<G>r</G><SUP>2</SUP>)</MATH>
+take one or other of the forms <I>A</I><SUB>1</SUB>, <I>A</I><SUB>2</SUB> ... <I>F</I><SUB>1</SUB>, <I>F</I><SUB>2</SUB> according to
+the different possible relations between <I>k</I>, <G>l</G> and the sign
+separating the two terms, but no other forms.
+<p>Finally, it is proved at the end of Proposition 72, in Proposi-
+tion 111 and the explanation following it that the thirteen
+irrational straight lines, the medial and the twelve other
+irrationals <I>A</I><SUB>1</SUB>, <I>A</I><SUB>2</SUB> ... <I>F</I><SUB>1</SUB>, <I>F</I><SUB>2</SUB>, are all different from one another.
+E.g. (Proposition 111) a binomial straight line cannot also be
+an apotome; in other words, &radic;<I>x</I>+&radic;<I>y</I> cannot be equal to
+&radic;<I>x</I>&prime; - &radic;.<I>y</I>&prime;, and <I>x</I>+&radic;<I>y</I> cannot be equal to <I>x</I>&prime; - &radic;<I>y</I>&prime;. We
+prove the latter proposition by squaring, and Euclid's proce-
+dure corresponds exactly to this. Propositions 112-14 prove
+that, if a rectangle equal to the square on a rational straight
+line be applied to a binomial, the other side containing it is an
+apotome of the same order, with terms commensurable with
+those of the binomial and in the same ratio, and vice versa;
+also that a binomial and apotome of the same order and with
+terms commensurable respectively contain a rational rectangle.
+Here we have the equivalent of rationalizing the denominators
+of the fractions <MATH><I>c</I><SUP>2</SUP>/&radic;<I>A</I> &plusmn; &radic;<I>B</I></MATH> or <MATH><I>c</I><SUP>2</SUP>/<I>a</I> &plusmn; &radic;<I>B</I></MATH> by multiplying the
+numerator and denominator by <MATH>&radic;<I>A</I> &mnplus; &radic;<I>B</I></MATH> or <MATH><I>a</I> &mnplus; &radic;<I>B</I></MATH> respec-
+tively. Euclid in fact proves that
+<MATH><G>s</G><SUP>2</SUP>/(<G>r</G>+&radic;<I>k</I>.<G>r</G>)=<G>lr</G> - &radic;<I>k</I>.<G>lr</G> (<I>k</I>&angle;1)</MATH>,
+and his method enables us to see that <MATH><G>l</G>=<G>s</G><SUP>2</SUP>/(<G>r</G><SUP>2</SUP>-<I>k</I><G>r</G><SUP>2</SUP>)</MATH>.
+Proposition 115 proves that from a medial straight line an
+<pb n=411>
+<head>THE <I>ELEMENTS.</I> BOOK X</head>
+infinite number of other irrational straight lines arise each
+of which is different from the preceding. <I>k</I><SUP>1/4</SUP><G>r</G> being medial,
+we take another rational straight line <G>s</G> and find the mean
+proportional &radic;(<I>k</I><SUP>1/4</SUP><G>rs</G>); this is a new irrational. Take the
+mean between this and <G>s</G>&prime;, and so on.
+<p>I have described the contents of Book X at length because
+it is probably not well known to mathematicians, while it is
+geometrically very remarkable and very finished. As regards
+its object Zeuthen has a remark which, I think, must come
+very near the truth. &lsquo;Since such roots of equations of the
+second degree as are incommensurable with the given magni-
+tudes cannot be expressed by means of the latter and of num-
+bers, it is conceivable that the Greeks, in exact investigations,
+introduced no approximate values, but worked on with the
+magnitudes they had found, which were represented by
+straight lines obtained by the construction corresponding to
+the solution of the equation. That is exactly the same thing
+which happens when we do not evaluate roots but content
+ourselves with expressing them by radical signs and other
+algebraical symbols. But, inasmuch as one straight line looks
+like another, the Greeks did not get the same clear view of
+what they denoted (i.e. by simple inspection) as our system
+of symbols assures to us. For this reason then it was neces-
+sary to undertake a classification of the irrational magnitudes
+which had been arrived at by successive solutions of equations
+of the second degree.&rsquo; That is, Book X formed a repository
+of results to which could be referred problems depending on
+the solution of certain types of equations, quadratic and
+biquadratic but reducible to quadratics, namely the equations
+<MATH><I>x</I><SUP>2</SUP> &plusmn; 2<G>m</G><I>x</I>.<G>r</G> &plusmn; <G>n.r</G><SUP>2</SUP>=0</MATH>,
+and <MATH><I>x</I><SUP>4</SUP> &plusmn; 2<G>m</G><I>x</I><SUP>2</SUP>.<G>r</G><SUP>2</SUP> &plusmn; <G>n</G>.<G>r</G><SUP>4</SUP>=0</MATH>,
+where <G>r</G> is a rational straight line and <G>m, n</G> are coefficients.
+According to the values of <G>m, n</G> in relation to one another and
+their character (<G>m</G>, but not <G>n</G>, may contain a surd such as
+&radic;<I>m</I> or &radic;(<I>m</I>/<I>n</I>)) the two positive roots of the first equations are
+the binomial and apotome respectively of some one of the
+orders &lsquo;first&rsquo;, &lsquo;second&rsquo;, . . . &lsquo;sixth&rsquo;, while the two positive
+roots of the latter equation are of some one of the other forms
+of irrationals (<I>A</I><SUB>1</SUB>, <I>A</I><SUB>2</SUB>), (<I>B</I><SUB>1</SUB>, <I>B</I><SUB>2</SUB>) ... (<I>F</I><SUB>1</SUB>, <I>F</I><SUB>2</SUB>).
+<pb n=412>
+<head>EUCLID</head>
+<p>Euclid himself, in Book XIII, makes considerable use of the
+second part of Book X dealing with <I>apotomes</I>; he regards a
+straight line as sufficiently defined in character if he can say
+that it is, e.g., an <I>apotome</I> (XIII. 17), a <I>first apotome</I> (XIII. 6),
+a <I>minor</I> straight line (XIII. 11). So does Pappus.<note>Cf. Pappus, iv, pp. 178, 182.</note>
+<p>Our description of Books XI-XIII can be shorter. They
+deal with geometry in three dimensions. The definitions,
+belonging to all three Books, come at the beginning of Book XI.
+They include those of a straight line, or a plane, at right angles
+to a plane, the inclination of a plane to a plane (dihedral angle),
+parallel planes, equal and similar solid figures, solid angle,
+pyramid, prism, sphere, cone, cylinder and parts of them, cube,
+octahedron, icosahedron and dodecahedron. Only the defini-
+tion of the sphere needs special mention. Whereas it had
+previously been defined as the figure which has all points of
+its surface equidistant from its centre, Euclid, with an eye to
+his use of it in Book XIII to &lsquo;comprehend&rsquo; the regular solids
+in a sphere, defines it as the figure comprehended by the revo-
+lution of a semicircle about its diameter.
+<p>The propositions of Book XI are in their order fairly
+parallel to those of Books I and VI on plane geometry. First
+we have propositions that a straight line is wholly in a plane
+if a portion of it is in the plane (1), and that two intersecting
+straight lines, and a triangle, are in one plane (2). Two
+intersecting planes cut in a straight line (3). Straight lines
+perpendicular to planes are next dealt with (4-6, 8, 11-14),
+then parallel straight lines not all in the same plane (9, 10, 15),
+parallel planes (14, 16), planes at right angles to one another
+(18, 19), solid angles contained by three angles (20, 22, 23, 26)
+or by more angles (21). The rest of the Book deals mainly
+with parallelepipedal solids. It is only necessary to mention
+the more important propositions. Parallelepipedal solids on the
+same base or equal bases and between the same parallel planes
+(i.e. having the same height) are equal (29-31). Parallele-
+pipedal solids of the same height are to one another as their
+bases (32). Similar parallelepipedal solids are in the tripli-
+cate ratio of corresponding sides (33). In equal parallele-
+pipedal solids the bases are reciprocally proportional to their
+heights and conversely (34). If four straight lines be propor-
+<pb n=413>
+<head>THE <I>ELEMENTS.</I> BOOKS XI-XII</head>
+tional, so are parallelepipedal solids similar and similarly
+described upon them, and conversely (37). A few other
+propositions are only inserted because they are required as
+lemmas in later books, e.g. that, if a cube is bisected by two
+planes each of which is parallel to a pair of opposite faces, the
+common section of the two planes and the diameter of the
+cube bisect one another (38).
+<p>The main feature of Book XII is the application of the
+<I>method of exhaustion</I>, which is used to prove successively that
+circles are to one another as the squares on their diameters
+(Propositions 1, 2), that pyramids of the same height and with
+triangular bases are to one another as the bases (3-5), that
+any cone is, in content, one third part of the cylinder which
+has the same base with it and equal height (10), that cones
+and cylinders of the same height are to one another as their
+bases (11), that similar cones and cylinders are to one another
+in the triplicate ratio of the diameters of their bases (12), and
+finally that spheres are to one another in the triplicate ratio
+of their respective diameters (16-18). Propositions 1, 3-4 and
+16-17 are of course preliminary to the main propositions 2, 5
+and 18 respectively. Proposition 5 is extended to pyramids
+with polygonal bases in Proposition 6. Proposition 7 proves
+that any prism with triangular bases is divided into three
+pyramids with triangular bases and equal in content, whence
+any pyramid with triangular base (and therefore also any
+pyramid with polygonal base) is equal to one third part of
+the prism having the same base and equal height. The rest
+of the Book consists of propositions about pyramids, cones,
+and cylinders similar to those in Book XI about parallele-
+pipeds and in Book VI about parallelograms: similar pyra-
+mids with triangular bases, and therefore also similar pyramids
+with polygonal bases, are in the triplicate ratio of correspond-
+ing sides (8); in equal pyramids, cones and cylinders the bases
+are reciprocally proportional to the heights, and conversely
+(9, 15).
+<p>The method of exhaustion, as applied in Euclid, rests upon
+X. 1 as lemma, and no doubt it will be desirable to insert here
+an example of its use. An interesting case is that relating to
+the pyramid. Pyramids with triangular bases and of the same
+height, says Euclid, are to one another as their bases (Prop. 5).
+<pb n=414>
+<head>EUCLID</head>
+It is first proved (Proposition 3) that, given any pyramid, as
+<I>ABCD</I>, on the base <I>BCD</I>, if we bisect the six edges at the
+<FIG>
+points <I>E, F, G, H, K, L</I>, and draw the straight lines shown in
+the figure, we divide the pyramid <I>ABCD</I> into two equal
+prisms and two equal pyramids <I>AFGE, FBHK</I> similar to the
+original pyramid (the equality of the prisms is proved in
+XI. 39), and that the sum of the two prisms is greater than
+half the original pyramid. Proposition 4 proves that, if each
+of two given pyramids of the same height be so divided, and
+if the small pyramids in each are similarly divided, then the
+smaller pyramids left over from that division are similarly
+divided, and so on to any extent, the sums of all the pairs of
+prisms in the two given pyramids respectively will be to one
+another as the respective bases. Let the two pyramids and
+their volumes be denoted by <I>P</I>, <I>P</I>&prime; respectively, and their bases
+by <I>B</I>, <I>B</I>&prime; respectively. Then, if <I>B</I>:<I>B</I>&prime; is not equal to <I>P</I>:<I>P</I>&prime;, it
+must be equal to <I>P</I>:<I>W</I>, where <I>W</I> is some volume either less or
+greater than <I>P</I>&prime;.
+<p>I. Suppose <I>W</I> < <I>P</I>&prime;.
+<p>By X. 1 we can divide <I>P</I>&prime; and the successive pyramids in
+it into prisms and pyramids until the sum of the small
+pyramids left over in it is less that <I>P</I>&prime; - <I>W</I>, so that
+<MATH><I>P</I>&prime; > (prisms in <I>P</I>&prime;) > <I>W</I></MATH>.
+<p>Suppose this done, and <I>P</I> divided similarly.
+<p>Then (XII. 4)
+<MATH>(sum of prisms in <I>P</I>):(sum of prisms in <I>P</I>&prime;)=<I>B</I>:<I>B</I>&prime;
+=<I>P</I>:<I>W</I></MATH>, by hypothesis.
+<p>But <MATH><I>P</I> > (sum of prisms in <I>P</I>)</MATH>:
+therefore <MATH><I>W</I> > (sum of prisms in <I>P</I>&prime;)</MATH>.
+<pb n=415>
+<head>THE <I>ELEMENTS.</I> BOOKS XII-XIII</head>
+<p>But <I>W</I> is also less than the sum of the prisms in <I>P</I>&prime;: which
+is impossible.
+<p>Therefore <I>W</I> is <I>not</I> less than <I>P</I>&prime;.
+<p>II. Suppose <I>W</I> > <I>P</I>&prime;.
+<p>We have, inversely,
+<MATH><I>B</I>&prime;:<I>B</I>=<I>W</I>:<I>P</I>
+= <I>P</I>&prime;:<I>V</I></MATH>, where <I>V</I> is some solid less than <I>P.</I>
+<p>But this can be proved impossible, exactly as in Part I.
+Therefore <I>W</I> is neither greater nor less than <I>P</I>&prime;, so that
+<MATH><I>B</I>:<I>B</I>&prime;=<I>P</I>:<I>P</I>&prime;</MATH>.
+<p>We shall see, when we come to Archimedes, that he extended
+this method of exhaustion. Instead of merely taking the one
+approximation, from underneath as it were, by constructing
+successive figures <I>within</I> the figure to be measured and so
+exhausting it, he combines with this an approximation from
+<I>outside.</I> He takes sets both of inscribed and circumscribed
+figures, approaching from both sides the figure to be measured,
+and, as it were, <I>compresses</I> them into one, so that they coincide
+as nearly as we please with one another and with the curvi-
+linear figure itself. The two parts of the proof are accordingly
+separate in Archimedes, and the second is not merely a reduction
+to the first.
+<p>The object of Book XIII is to construct, and to &lsquo;comprehend
+in a sphere&rsquo;, each of the five regular solids, the pyramid
+(Prop. 13), the octahedron (Prop. 14), the cube (Prop. 15),
+the icosahedron (Prop. 16) and the dodecahedron (Prop. 17);
+&lsquo;comprehending in a sphere&rsquo; means the construction of the
+circumscribing sphere, which involves the determination of
+the relation of a &lsquo;side&rsquo; (i.e. edge) of the solid to the radius
+of the sphere; in the case of the first three solids the relation
+is actually determined, while in the case of the icosahedron
+the side of the figure is shown to be the irrational straight
+line called &lsquo;minor&rsquo;, and in the case of the dodecahedron an
+&lsquo;apotome&rsquo;. The propositions at the beginning of the Book
+are preliminary. Propositions 1-6 are theorems about straight
+lines cut in extreme and mean ratio, Propositions 7, 8 relate
+to pentagons, and Proposition 8 proves that, if, in a regular
+pentagon, two diagonals (straight lines joining angular points
+<pb n=416>
+<head>EUCLID</head>
+next but one to each other) are drawn intersecting at a point,
+each of them is divided at the point in extreme and mean
+ratio, the greater segment being equal to the side of the pen-
+tagon. Propositions 9 and 10 relate to the sides of a pentagon,
+a decagon and a hexagon all inscribed in the same circle,
+and are preliminary to proving (in Prop. 11) that the side of
+the inscribed pentagon is, in relation to the diameter of the
+circle, regarded as rational, the irrational straight line called
+&lsquo;minor&rsquo;. If <I>p, d, h</I> be the sides of the regular pentagon,
+decagon, and hexagon inscribed in the same circle, Proposition 9
+proves that <I>h</I> + <I>d</I> is cut in extreme and mean ratio, <I>h</I> being the
+greater segment; this is equivalent to saying that <MATH>(<I>r</I> + <I>d</I>)<I>d</I>=<I>r</I><SUP>2</SUP></MATH>,
+where <I>r</I> is the radius of the circle, or, in other words, that
+<MATH><I>d</I>=1/2<I>r</I>(&radic;5-1)</MATH>. Proposition 10 proves that <MATH><I>p</I><SUP>2</SUP> = <I>h</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP></MATH> or
+<I>r</I><SUP>2</SUP> + <I>d</I><SUP>2</SUP>, whence we obtain <MATH><I>p</I>=1/2<I>r</I>&radic;(10-2&radic;5)</MATH>. Expressed as
+a &lsquo;minor&rsquo; irrational straight line, which Proposition 11 shows
+it to be, <MATH><I>p</I>=1/2<I>r</I>&radic;(5+2&radic;5)-1/2<I>r</I>&radic;(5-2&radic;5)</MATH>.
+<p>The constructions for the several solids, which have to be
+inscribed in a given sphere, may be briefly indicated, thus:
+<p>1. The regular pyramid or <I>tetrahedron</I>.
+<p>Given <I>D</I>, the diameter of the sphere which is to circum-
+scribe the tetrahedron, Euclid draws a circle with radius <I>r</I>
+such that <MATH><I>r</I><SUP>2</SUP>=1/3<I>D</I>.2/3<I>D</I></MATH>, or <MATH><I>r</I>=1/3&radic;2.<I>D</I></MATH>, inscribes an equi-
+lateral triangle in the circle, and then erects from the centre
+of it a straight line perpendicular to its plane and of length
+2/3<I>D</I>. The lines joining the extremity of the perpendicular to
+the angular points of the equilateral triangle determine the
+tetrahedron. Each of the upstanding edges (<I>x</I>, say) is such
+that <MATH><I>x</I><SUP>2</SUP>=<I>r</I><SUP>2</SUP>+4/9<I>D</I><SUP>2</SUP>=3<I>r</I><SUP>2</SUP></MATH>, and it has been proved (in XIII. 12)
+that the square on the side of the triangle inscribed in the
+circle is also 3<I>r</I><SUP>2</SUP>. Therefore the edge <I>a</I> of the tetrahedron
+=&radic;3.<I>r</I> = 1/3&radic;6.<I>D.</I>
+<p>2. The <I>octahedron.</I>
+<p>If <I>D</I> be the diameter of the circumscribing sphere, a square
+is inscribed in a circle of diameter <I>D</I>, and from its centre
+straight lines are drawn in both directions perpendicular to
+its plane and of length equal to the radius of the circle or half
+the diagonal of the square. Each of the edges which stand up
+from the square=&radic;2.1/2<I>D</I>, which is equal to the side of the
+<pb n=417>
+<head>THE <I>ELEMENTS</I>. BOOK XIII</head>
+square. Each of the edges <I>a</I> of the octahedron is therefore
+equal to &radic;2.1/2<I>D.</I>
+<p>3. The <I>cube.</I>
+<p><I>D</I> being the diameter of the circumscribing sphere, draw
+a square with side <I>a</I> such that <MATH><I>a</I><SUP>2</SUP>=<I>D</I>.1/3<I>D</I></MATH>, and describe a cube
+on this square as base. The edge <MATH><I>a</I>=1/3&radic;3.<I>D</I></MATH>.
+<p>4. The <I>icosahedron.</I>
+<p>Given <I>D</I>, the diameter of the sphere, construct a circle with
+radius <I>r</I> such that <MATH><I>r</I><SUP>2</SUP>=<I>D</I>.1/5<I>D</I></MATH>. Inscribe in it a regular
+decagon. Draw from its angular points straight lines perpen-
+dicular to the plane of the circle and equal in length to its
+radius <I>r</I>; this determines the angular points of a regular
+decagon inscribed in an equal parallel circle. By joining
+alternate angular points of one of the decagons, describe a
+regular pentagon in the circle circumscribing it, and then do
+the same in the other circle but so that the angular points are
+not opposite those of the other pentagon. Join the angular
+points of one pentagon to the nearest angular points of the
+other; this gives ten triangles. Then, if <I>p</I> be the side of each
+pentagon, <I>d</I> the side of each decagon, the upstanding sides
+of the triangles (=<I>x</I>, say) are given by <MATH><I>x</I><SUP>2</SUP>=<I>d</I><SUP>2</SUP>+<I>r</I><SUP>2</SUP>=<I>p</I><SUP>2</SUP></MATH>
+(Prop. 10); therefore the ten triangles are equilateral. We
+have lastly to find the common vertices of the five equilateral
+triangles standing on the pentagons and completing the icosa-
+hedron. If <I>C</I>, <I>C</I>&prime; be the centres of the parallel circles, <I>CC</I>&prime; is
+produced in both directions to <I>X, Z</I> respectively so that
+<I>CX</I>=<I>C&prime;Z</I>=<I>d</I> (the side of the decagon). Then again the
+upstanding edges connecting to <I>X, Z</I> the angular points of the
+two pentagons respectively (=<I>x</I>, say) are given by
+<MATH><I>x</I><SUP>2</SUP>=<I>r</I><SUP>2</SUP>+<I>d</I><SUP>2</SUP>=<I>p</I><SUP>2</SUP></MATH>.
+<p>Hence each of the edges
+<MATH><I>a</I>=<I>p</I>=1/2<I>r</I>&radic;(10-2&radic;5)=<I>D</I>/2&radic;5&radic;(10-2&radic;5)
+=(1/10)<I>D</I>&radic;{10(5-&radic;5)}</MATH>.
+It is finally shown that the sphere described on <I>XZ</I> as
+diameter circumscribes the icosahedron, and
+<MATH><I>XZ</I>=<I>r</I>+2<I>d</I>=<I>r</I>+<I>r</I>(&radic;5-1)=<I>r</I>.&radic;5=<I>D</I></MATH>.
+<pb n=418>
+<head>EUCLID</head>
+<p>5. The <I>dodecahedron</I>.
+<p>We start with the cube inscribed in the given sphere with
+diameter <I>D</I>. We then draw pentagons which have the edges
+of the cube as diagonals in the manner shown in the figure.
+If <I>H, N, M, O</I> be the middle points of the sides of the face
+<I>BF</I>, and <I>H, G, L, K</I> the middle points of the sides of the
+face <I>BD</I>, join <I>NO, GK</I> which are then parallel to <I>BC</I>, and
+draw <I>MH, HL</I> bisecting them at right angles at <I>P, Q</I>.
+<p>Divide <I>PN, PO, QH</I> in extreme and mean ratio at <I>R, S, T</I>,
+and let <I>PR, PS, QT</I> be the greater segments. Draw <I>RU, PX,
+SV</I> at right angles to the plane <I>BF</I>, and <I>TW</I> at right angles to
+<FIG>
+the plane <I>BD</I>, such that each of these perpendiculars =<I>PR</I>
+or <I>PS</I>. Join <I>UV, VC, CW, WB, BU</I>. These determine one
+of the pentagonal faces, and the others are drawn similarly.
+<p>It is then proved that each of the pentagons, as <I>UVCWB</I>,
+is (1) equilateral, (2) in the same plane, (3) equiangular.
+<p>As regards the sides we see, e. g., that
+<MATH><I>BU</I><SUP>2</SUP>=<I>BR</I><SUP>2</SUP>+<I>RU</I><SUP>2</SUP>=<I>BN</I><SUP>2</SUP>+<I>NR</I><SUP>2</SUP>+<I>RP</I><SUP>2</SUP>
+=<I>PN</I><SUP>2</SUP>+<I>NR</I><SUP>2</SUP>+<I>RP</I><SUP>2</SUP>=4<I>RP</I><SUP>2</SUP> (by means of XIII. 4) = <I>UV</I><SUP>2</SUP></MATH>,
+and so on.
+<pb n=419><head>THE <I>ELEMENTS.</I> BOOK XIII</head>
+<p>Lastly, it is proved that the same sphere of diameter <I>D</I>
+which circumscribes the cube also circumscribes the dodeca-
+hedron. For example, if <I>Z</I> is the centre of the sphere,
+<MATH><I>ZU</I><SUP>2</SUP>=<I>ZX</I><SUP>2</SUP>+<I>XU</I><SUP>2</SUP>=<I>NS</I><SUP>2</SUP>+<I>PS</I><SUP>2</SUP>=3<I>PN</I><SUP>2</SUP></MATH>, (XIII. 4)
+while <MATH><I>ZB</I><SUP>2</SUP>=3<I>ZP</I><SUP>2</SUP>=3<I>PN</I><SUP>2</SUP></MATH>.
+<p>If <I>a</I> be the edge of the dodecahedron, <I>c</I> the edge of the cube,
+<MATH><I>a</I>=2<I>RP</I>=2.(&radic;5-1)/4 <I>c</I>
+=(2&radic;3)/3.(&radic;5-1)/4 <I>D</I>
+=1/6 <I>D</I> (&radic;15-&radic;3)</MATH>.
+<p>Book XIII ends with Proposition 18, which arranges the
+edges of the five regular solids inscribed in one and the same
+sphere in order of magnitude, while an addendum proves that
+no other regular solid figures except the five exist.
+<C>The so-called Books XIV, XV.</C>
+<p>This is no doubt the place to speak of the continuations
+of Book XIII which used to be known as Books XIV, XV of
+the <I>Elements</I>, though they are not by Euclid. The former
+is the work of Hypsicles, who probably lived in the second
+half of the second century B.C., and who is otherwise known
+as the reputed author of an astronomical tract <G>)*anaforiko/s</G>
+(<I>De ascensionibus</I>) still extant (the earliest extant Greek book
+in which the division of the circle into 360 degrees appears),
+besides other works, which have not survived, on the harmony
+of the spheres and on polygonal numbers. The preface to
+&lsquo;Book XIV&rsquo; is interesting historically. It appears from
+it that Apollonius wrote a tract on the comparison of the
+dodecahedron and icosahedron inscribed in one and the same
+sphere, i.e. on the ratio between them, and that there were two
+editions of this work, the first of which was in some way
+incorrect, while the second gave a correct proof of the pro-
+position that, as the surface of the dodecahedron is to
+the surface of the icosahedron, so is the solid content of the
+<pb n=420><head>EUCLID</head>
+dodecahedron to that of the icosahedron, &lsquo;because the per-
+pendicular from the centre of the sphere to the pentagon of
+the dodecahedron and to the triangle of the icosahedron is the
+same&rsquo;. Hypsicles says also that Aristaeus, in a work entitled
+<I>Comparison of the five figures</I>, proved that &lsquo;the same circle
+circumscribes both the pentagon of the dodecahedron and the
+triangle of the icosahedron inscribed in the same sphere&rsquo;;
+whether this Aristaeus is the same as the Aristaeus of the
+<I>Solid Loci</I>, the elder contemporary of Euclid, we do not
+know. The proposition of Aristaeus is proved by Hypsicles
+as Proposition 2 of his book. The following is a summary
+of the results obtained by Hypsicles. In a lemma at the end
+he proves that, if two straight lines be cut in extreme and
+mean ratio, the segments of both are in one and the same
+ratio; the ratio is in fact <MATH>2:(&radic;5-1)</MATH>. If then <I>any</I> straight
+line <I>AB</I> be divided at <I>C</I> in extreme and mean ratio, <I>AC</I> being
+the greater segment, Hypsicles proves that, if we have a cube,
+a dodecahedron and an icosahedron all inscribed in the same
+sphere, then:
+<MATH>(Prop. 7) (side of cube):(side of icosahedron)
+=&radic;(<I>AB</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>):&radic;(<I>AB</I><SUP>2</SUP>+<I>BC</I><SUP>2</SUP>);
+(Prop. 6) (surface of dodecahedron):(surface of icosahedron)
+=(side of cube):(side of icosahedron);
+(Prop. 8) (content of dodecahedron):(content of icosahedron)
+=(surface of dodecahedron):(surface of icosahedron);
+and consequently
+(content of dodecahedron):(content of icosahedron)
+=&radic;(<I>AB</I><SUP>2</SUP>+<I>AC</I><SUP>2</SUP>):&radic;(<I>AB</I><SUP>2</SUP>+<I>BC</I><SUP>2</SUP>)</MATH>.
+<p>The second of the two supplementary Books (&lsquo;Book XV&rsquo;) is
+also concerned with the regular solids, but is much inferior to
+the first. The exposition leaves much to be desired, being
+in some places obscure, in others actually inaccurate. The
+Book is in three parts unequal in length. The first<note>Heiberg's Euclid, vol. v, pp. 40-8.</note> shows
+how to inscribe certain of the regular solids in certain others,
+<pb n=421><head>THE SO-CALLED BOOKS XIV, XV</head>
+(<I>a</I>) a tetrahedron in a cube, (<I>b</I>) an octahedron in a tetrahedron,
+(<I>c</I>) an octahedron in a cube, (<I>d</I>) a cube in an octahedron,
+(<I>e</I>) a dodecahedron in an icosahedron. The second portion<note>Heiberg's Euclid, vol. v. pp. 48-50.</note>
+explains how to calculate the number of edges and the number
+of solid angles in the five solids respectively. The third
+portion<note><I>Ib.</I>, pp. 50-66.</note> shows how to determine the dihedral angles between
+the faces meeting in any edge of any one of the solids. The
+method is to construct an isosceles triangle with vertical angle
+equal to the said angle; from the middle point of any edge
+two perpendiculars are drawn to it, one in each of the two
+faces intersecting in that edge; these perpendiculars (forming
+the dihedral angle) are used to determine the two equal sides
+of an isosceles triangle, and the base of the triangle is easily
+found from the known properties of the particular solid. The
+rules for drawing the respective isosceles triangles are first
+given all together in general terms; and the special interest
+of the passage consists in the fact that the rules are attributed
+to &lsquo;Isidorus our great teacher&rsquo;. This Isidorus is doubtless
+Isidorus of Miletus, the architect of the church of Saint Sophia
+at Constantinople (about A.D. 532). Hence the third portion
+of the Book at all events was written by a pupil of Isidorus
+in the sixth century.
+<C>The <I>Data.</I></C>
+<p>Coming now to the other works of Euclid, we will begin
+with those which have actually survived. Most closely con-
+nected with the <I>Elements</I> as dealing with plane geometry, the
+subject-matter of Books I-VI, is the <I>Data</I>, which is accessible
+in the Heiberg-Menge edition of the Greek text, and also
+in the translation annexed by Simson to his edition of the
+<I>Elements</I> (although this translation is based on an inferior
+text). The book was regarded as important enough to be
+included in the <I>Treasury of Analysis</I> (<G>to/pos a)naluo/menos</G>) as
+known to Pappus, and Pappus gives a description of it; the
+description shows that there were differences between Pappus's
+text and ours, for, though Propositions 1-62 correspond to the
+description, as also do Propositions 87-94 relating to circles
+at the end of the book, the intervening propositions do not
+<pb n=422><head>EUCLID</head>
+exactly agree, the differences, however, affecting the distribu-
+tion and numbering of the propositions rather than their
+substance. The book begins with definitions of the senses
+in which things are said to be <I>given.</I> Things such as areas,
+straight lines, angles and ratios are said to be &lsquo;given in
+<I>magnitude</I> when we can make others equal to them&rsquo; (Defs.
+1-2). Rectilineal figures are &lsquo;given <I>in species</I>&rsquo; when their
+angles are severally given as well as the ratios of the sides to
+one another (Def. 3). Points, lines and angles are &lsquo;given
+<I>in position</I>&rsquo; &lsquo;when they always occupy the same place&rsquo;: a not
+very illuminating definition (4). A circle is given <I>in position
+and in magnitude</I> when the centre is given <I>in position</I> and
+the radius <I>in magnitude</I> (6); and so on. The object of the
+proposition called a Datum is to prove that, if in a given figure
+certain parts or relations are given, other parts or relations are
+also given, in one or other of these senses.
+<p>It is clear that a systematic collection of <I>Data</I> such as
+Euclid's would very much facilitate and shorten the procedure
+in <I>analysis</I>; this no doubt accounts for its inclusion in the
+<I>Treasury of Analysis.</I> It is to be observed that this form of
+proposition does not actually determine the thing or relation
+which is shown to be given, but merely proves that it can be
+determined when once the facts stated in the hypothesis
+are known; if the proposition stated that a certain thing <I>is</I>
+so and so, e.g. that a certain straight line in the figure is of
+a certain length, it would be a theorem; if it directed us to
+<I>find</I> the thing instead of proving that it is &lsquo;given&rsquo;, it would
+be a problem; hence many propositions of the form of the
+<I>Data</I> could alternatively be stated in the form of theorems or
+problems.
+<p>We should naturally expect much of the subject-matter of
+the <I>Elements</I> to appear again in the <I>Data</I> under the different
+aspect proper to that book; and this proves to be the case.
+We have already mentioned the connexion of Eucl. II. 5, 6
+with the solution of the mixed quadratic equations <MATH><I>ax</I>&plusmn;<I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP></MATH>.
+The solution of these equations is equivalent to the solution of
+the simultaneous equations
+<MATH>
+<BRACE><I>y</I>&plusmn;<I>x</I>=<I>a</I>
+<I>xy</I>=<I>b</I><SUP>2</SUP></BRACE>
+</MATH>
+and Euclid shows how to solve these equations in Propositions
+<pb n=423><head>THE <I>DATA</I></head>
+84, 85 of the <I>Data</I>, which state that &lsquo;If two straight lines
+contain a given area in a given angle, and if the difference
+(sum) of them be given, then shall each of them be given.&rsquo;
+The proofs depend directly upon those of Propositions 58, 59,
+&lsquo;If a given area be applied to a given straight line, falling
+short (exceeding) by a figure given in species, the breadths
+of the deficiency (excess) are given.&rsquo; All the &lsquo;areas&rsquo; are
+parallelograms.
+<p>We will give the proof of Proposition 59 (the case of
+&lsquo;excess&rsquo;). Let the given area <I>AB</I>
+<FIG>
+be applied to <I>AC</I>, exceeding by the
+figure <I>CB</I> given in species. I say
+that each of the sides <I>HC, CE</I> is
+given.
+<p>Bisect <I>DE</I> in <I>F</I>, and construct
+on <I>EF</I> the figure <I>FG</I> similar and
+similarly situated to <I>CB</I> (VI. 18).
+Therefore <I>FG, CB</I> are about the same diagonal (VI. 26).
+Complete the figure.
+<p>Then <I>FG</I>, being similar to <I>CB</I>, is given in species, and,
+since <I>FE</I> is given, <I>FG</I> is given in magnitude (Prop. 52).
+<p>But <I>AB</I> is given; therefore <MATH><I>AB</I>+<I>FG</I></MATH>, that is to say, <I>KL</I>, is
+given in magnitude. But it is also given in species, being
+similar to <I>CB</I>; therefore the sides of <I>KL</I> are given (Prop. 55).
+<p>Therefore <I>KH</I> is given, and, since <MATH><I>KC</I>=<I>EF</I></MATH> is also given,
+the difference <I>CH</I> is given. And <I>CH</I> has a given ratio to <I>HB</I>;
+therefore <I>HB</I> is also given (Prop. 2).
+<p>Eucl. III. 35, 36 about the &lsquo;power&rsquo; of a point with reference
+to a circle have their equivalent in <I>Data</I> 91, 92 to the effect
+that, given a circle and a point in the same plane, the rectangle
+contained by the intercepts between this point and the points
+in which respectively the circumference is cut by any straight
+line passing through the point and meeting the circle is
+also given.
+<p>A few more enunciations may be quoted. Proposition 8
+(compound ratio): Magnitudes which have given ratios to the
+same magnitude have a given ratio to one another also.
+Propositions 45, 46 (similar triangles): If a triangle have one
+angle given, and the ratio of the sum of the sides containing
+that angle, or another angle, to the third side (in each case) be
+<pb n=424><head>EUCLID</head>
+given, the triangle is given in species. Proposition 52: If a
+(rectilineal) figure given in species be described on a straight
+line given in magnitude, the figure is given in magnitude.
+Proposition 66: If a triangle have one angle given, the rect-
+angle contained by the sides including the angle has to the
+(area of the) triangle a given ratio. Proposition 80: If a
+triangle have one angle given, and if the rectangle contained
+by the sides including the given angle have to the square on
+the third side a given ratio, the triangle is given in species.
+<p>Proposition 93 is interesting: If in a circle given in magni-
+tude a straight line be drawn cutting off a segment containing
+a given angle, and if this angle be bisected (by a straight line
+cutting the base of the segment and the circumference beyond
+it), the sum of the sides including the given angle will have a
+given ratio to the chord bisecting the angle, and the rectangle
+contained by the sum of the said sides and the portion of the
+bisector cut off (outside the segment) towards the circum-
+ference will also be given.
+<p>Euclid's proof is as follows. In the circle <I>ABC</I> let the
+chord <I>BC</I> cut off a segment containing a given angle <I>BAC</I>,
+and let the angle be bisected by <I>AE</I> meeting <I>BC</I> in <I>D.</I>
+<p>Join <I>BE.</I> Then, since the circle is given in magnitude, and
+<FIG>
+<I>BC</I> cuts off a segment containing a given
+angle, <I>BC</I> is given (Prop. 87).
+<p>Similarly <I>BE</I> is given; therefore the
+ratio <I>BC</I>:<I>BE</I> is given. (It is easy to
+see that the ratio <I>BC</I>:<I>BE</I> is equal to
+2 cos 1/2 <I>A.</I>)
+<p>Now, since the angle <I>BAC</I> is bisected,
+<MATH><I>BA</I>:<I>AC</I>=<I>BD</I>:<I>DC</I></MATH>.
+<p>It follows that <MATH>(<I>BA</I>+<I>AC</I>):(<I>BD</I>+<I>DC</I>)=<I>AC</I>:<I>DC</I></MATH>.
+<p>But the triangles <I>ABE, ADC</I> are similar;
+therefore <MATH><I>AE</I>:<I>BE</I>=<I>AC</I>:<I>DC</I>
+=(<I>BA</I>+<I>AC</I>):<I>BC</I></MATH>, from above.
+<p>Therefore <MATH>(<I>BA</I>+<I>AC</I>):<I>AE</I>=<I>BC</I>:<I>BE</I></MATH>, which is a given
+ratio.
+<pb n=425><head>THE <I>DATA</I></head>
+<p>Again, since the triangles <I>ADC, BDE</I> are similar,
+<MATH><I>BE</I>:<I>ED</I>=<I>AC</I>:<I>CD</I>=(<I>BA</I>+<I>AC</I>):<I>BC</I></MATH>.
+<p>Therefore <MATH>(<I>BA</I>+<I>AC</I>).<I>ED</I>=<I>BC.BE</I></MATH>, which is given.
+<C>On divisions (of figures).</C>
+<p>The only other work of Euclid in pure geometry which has
+survived (but not in Greek) is the book <I>On divisions</I> (<I>of
+figures</I>), <G>peri\ diaire/sewn bibli/on</G>. It is mentioned by Proclus,
+who gives some hints as to its content<note>Proclus on Eucl. I, p. 144. 22-6.</note>; he speaks of the
+business of the author being divisions of figures, circles or
+rectilineal figures, and remarks that the parts may be like
+in definition or notion, or unlike; thus to divide a triangle
+into triangles is to divide it into like figures, whereas to
+divide it into a triangle and a quadrilateral is to divide it into
+unlike figures. These hints enable us to check to some extent
+the genuineness of the books dealing with divisions of figures
+which have come down through the Arabic. It was John Dee
+who first brought to light a treatise <I>De divisionibus</I> by one
+Muhammad Bagdadinus (died 1141) and handed over a copy
+of it (in Latin) to Commandinus in 1563; it was published by
+the latter in Dee's name and his own in 1570. Dee appears
+not to have translated the book from the Arabic himself, but
+to have made a copy for Commandinus from a manuscript of
+a Latin translation which he himself possessed at one time but
+which was apparently stolen and probably destroyed some
+twenty years after the copy was made. The copy does not
+seem to have been made from the Cotton MS. which passed to
+the British Museum after it had been almost destroyed by
+a fire in 1731.<note>The question is fully discussed by R. C. Archibald, <I>Euclid's Book on
+Divisions of Figures with a restoration based on Woepcke's text and on the
+Practica Geometriae of Leonardo Pisano</I> (Cambridge 1915).</note> The Latin translation may have been that
+made by Gherard of Cremona (1114-87), since in the list of
+his numerous translations a &lsquo;liber divisionum&rsquo; occurs. But
+the Arabic original cannot have been a direct translation from
+Euclid, and probably was not even a direct adaptation of it,
+since it contains mistakes and unmathematical expressions;
+moreover, as it does not contain the propositions about the
+<pb n=426><head>EUCLID</head>
+division of a circle alluded to by Proclus, it can scarcely have
+contained more than a fragment of Euclid's original work.
+But Woepcke found in a manuscript at Paris a treatise in
+Arabic on the division of figures, which he translated and
+published in 1851. It is expressly attributed to Euclid in the
+manuscript and corresponds to the indications of the content
+given by Proclus. Here we find divisions of different recti-
+linear figures into figures of the same kind, e.g. of triangles
+into triangles or trapezia into trapezia, and also divisions into
+&lsquo;unlike&rsquo; figures, e.g. that of a triangle by a straight line parallel
+to the base. The missing propositions about the division of
+a circle are also here: &lsquo;to divide into two equal parts a given
+figure bounded by an arc of a circle and two straight lines
+including a given angle&rsquo; (28), and &lsquo;to draw in a given circle
+two parallel straight lines cutting off a certain fraction from
+the circle&rsquo; (29). Unfortunately the proofs are given of only
+four propositions out of 36, namely Propositions 19, 20, 28, 29,
+the Arabic translator having found the rest too easy and
+omitted them. But the genuineness of the treatise edited by
+Woepcke is attested by the facts that the four proofs which
+remain are elegant and depend on propositions in the
+<I>Elements</I>, and that there is a lemma with a true Greek ring,
+&lsquo;to apply to a straight line a rectangle equal to the rectangle
+contained by <I>AB, AC</I> and deficient by a square&rsquo; (18). Moreover,
+the treatise is no fragment, but ends with the words, &lsquo;end of
+the treatise&rsquo;, and is (but for the missing proofs) a well-ordered
+and compact whole. Hence we may safely conclude that
+Woepcke's tract represents not only Euclid's work but the
+whole of it. The portion of the <I>Practica geometriae</I> of
+Leonardo of Pisa which deals with the division of figures
+seems to be a restoration and extension of Euclid's work;
+Leonardo must presumably have come across a version of it
+from the Arabic.
+<p>The type of problem which Euclid's treatise was designed
+to solve may be stated in general terms as that of dividing a
+given figure by one or more straight lines into parts having
+prescribed ratios to one another or to other given areas. The
+figures divided are the triangle, the parallelogram, the trape-
+zium, the quadrilateral, a figure bounded by an arc of a circle
+and two straight lines, and the circle. The figures are divided
+<pb n=427><head>ON DIVISIONS OF FIGURES</head>
+into two equal parts, or two parts in a given ratio; or again,
+a given fraction of the figure is to be cut off, or the figure is
+to be divided into several parts in given ratios. The dividing
+straight lines may be transversals drawn through a point
+situated at a vertex of the figure, or a point on any side, on one
+of two parallel sides, in the interior of the figure, outside the
+figure, and so on; or again, they may be merely parallel lines,
+or lines parallel to a base. The treatise also includes auxiliary
+propositions, (1) &lsquo;to apply to a given straight line a rectangle
+equal to a given area and deficient by a square&rsquo;, the proposi-
+tion already mentioned, which is equivalent to the algebraical
+solution of the equation <MATH><I>ax</I>-<I>x</I><SUP>2</SUP>=<I>b</I><SUP>2</SUP></MATH> and depends on Eucl. II. 5
+(cf. p. 152 above); (2) propositions in proportion involving
+unequal instead of equal ratios:
+<MATH>If <I>a.d</I>>or<<I>b.c</I>, then <I>a</I>:<I>b</I>>or<<I>c</I>:<I>d</I> respectively.
+If <I>a</I>:<I>b</I>><I>c</I>:<I>d</I>, then (<I>a</I>&mnplus;<I>b</I>:<I>b</I>>(<I>c</I>&mnplus;<I>d</I>):<I>d</I>.
+If <I>a</I>:<I>b</I><<I>c</I>:<I>d</I>, then (<I>a</I>-<I>b</I>):<I>b</I><(<I>c</I>-<I>d</I>):<I>d</I></MATH>.
+<p>By way of illustration I will set out shortly three proposi-
+tions from the Woepcke text.
+<p>(1) Propositions 19, 20 (slightly generalized): To cut off
+a certain fraction (<I>m</I>/<I>n</I>) from a given triangle by a straight
+<FIG>
+line drawn through a given point within the triangle (Euclid
+gives two cases corresponding to <MATH><I>m</I>/<I>n</I>=1/2</MATH> and <MATH><I>m</I>/<I>n</I>=1/3</MATH>).
+<p>The construction will be best understood if we work out
+the analysis of the problem (not given by Euclid).
+<p>Suppose that <I>ABC</I> is the given triangle, <I>D</I> the given
+<pb n=428><head>EUCLID</head>
+internal point; and suppose the problem solved, i.e. <I>GH</I>
+drawn through <I>D</I> in such a way that <MATH>&utri;<I>GBH</I>=<I>m</I>/<I>n</I>.&utri;<I>ABC</I></MATH>.
+<p>Therefore <MATH><I>GB.BH</I>=<I>m</I>/<I>n.AB.BC</I></MATH>. (This is assumed by
+Euclid.)
+<p>Now suppose that the unknown quantity is <MATH><I>GB</I>=<I>x</I></MATH>, say.
+<p>Draw <I>DE</I> parallel to <I>BC</I>; then <I>DE, EB</I> are given.
+<p>Now <MATH><I>BH</I>:<I>DE</I>=<I>GB</I>:<I>GE</I>=<I>x</I>:(<I>x</I>-<I>BE</I>)</MATH>,
+or <MATH><I>BH</I>=(<I>x.DE</I>)/(<I>x</I>-<I>BE</I>)</MATH>;
+therefore <MATH><I>GB.BH</I>=<I>x</I><SUP>2</SUP>.<I>DE</I>/(<I>x</I>-<I>BE</I>)</MATH>.
+<p>And, by hypothesis, <MATH><I>GB.BH</I>=<I>m</I>/<I>n.AB.BC</I></MATH>;
+therefore <MATH><I>x</I><SUP>2</SUP>=<I>m/n.(AB.BC)/DE</I> (<I>x</I>-<I>BE</I>)</MATH>,
+or, if <MATH><I>k</I>=<I>m</I>/<I>n.(AB.BC)/DE</I></MATH>, we have to solve the equation
+<MATH><I>x</I><SUP>2</SUP>=<I>k</I>(<I>x</I>-<I>BE</I>)</MATH>,
+or <MATH><I>kx</I>-<I>x</I><SUP>2</SUP>=<I>k.BE</I></MATH>.
+<p>This is exactly what Euclid does; he first finds <I>F</I> on <I>BA</I>
+such that <MATH><I>BF.DE</I>=<I>m</I>/<I>n.AB.BC</I></MATH> (the length of <I>BF</I> is deter-
+mined by applying to <I>DE</I> a rectangle equal to <MATH><I>m</I>/<I>n.AB.BC</I></MATH>,
+Eucl. I. 45), that is, he finds <I>BF</I> equal to <I>k.</I> Then he gives
+the geometrical solution of the equation <MATH><I>kx</I>-<I>x</I><SUP>2</SUP>=<I>k.BE</I></MATH> in the
+form &lsquo;apply to the straight line <I>BF</I> a rectangle equal to
+<I>BF.BE</I> and deficient by a square&rsquo;; that is to say, he deter-
+mines <I>G</I> so that <MATH><I>BG.GF</I>=<I>BF.BE</I></MATH>. We have then only
+to join <I>GD</I> and produce it to <I>H</I>; and <I>GH</I> cuts off the required
+triangle.
+<p>(The problem is subject to a <G>diorismo/s</G> which Euclid does
+not give, but which is easily supplied.)
+<p>(2) Proposition 28: To divide into two equal parts a given
+<pb n=429><head>ON DIVISIONS OF FIGURES</head>
+figure bounded by an arc of a circle and by two straight lines
+which form a given angle.
+<p>Let <I>ABEC</I> be the given figure, <I>D</I> the middle point of <I>BC</I>,
+and <I>DE</I> perpendicular to <I>BC.</I> Join <I>AD.</I>
+<p>Then the broken line <I>ADE</I> clearly divides the figure into
+two equal parts. Join <I>AE</I>, and draw
+<FIG>
+<I>DF</I> parallel to it meeting <I>BA</I> in <I>F.</I>
+Join <I>FE.</I>
+<p>The triangles <I>AFE, ADE</I> are then
+equal, being in the same parallels.
+Add to each the area <I>AEC.</I>
+<p>Therefore the area <I>AFEC</I> is equal to the area <I>ADEC</I>, and
+therefore to half the area of the given figure.
+<p>(3) Proposition 29: To draw in a given circle two parallel
+chords cutting off a certain fraction (<I>m</I>/<I>n</I>) of the circle.
+<p>(The fraction <I>m</I>/<I>n</I> must be
+<FIG>
+such that we can, by plane
+methods, draw a chord cutting off
+<I>m</I>/<I>n</I> of the circumference of
+the circle; Euclid takes the case
+where <MATH><I>m</I>/<I>n</I>=1/3</MATH>.)
+<p>Suppose that the arc <I>ADB</I> is
+<I>m</I>/<I>n</I> of the circumference of the
+circle. Join <I>A, B</I> to the centre <I>O.</I>
+Draw <I>OC</I> parallel to <I>AB</I> and join
+<I>AC, BC.</I> From <I>D</I>, the middle point
+of the arc <I>AB</I>, draw the chord <I>DE</I> parallel to <I>BC.</I> Then shall
+<I>BC, DE</I> cut off <I>m</I>/<I>n</I> of the area of the circle.
+<p>Since <I>AB, OC</I> are parallel,
+<MATH>&utri;<I>AOB</I>=&utri;<I>ACB</I></MATH>.
+<p>Add to each the segment <I>ADB</I>;
+therefore
+<MATH>(sector <I>ADBO</I>)=figure bounded by <I>AC, CB</I> and arc <I>ADB</I>
+=(segmt. <I>ABC</I>)-(segmt. <I>BFC</I>)</MATH>.
+<p>Since <I>BC, DE</I> are parallel, <MATH>(arc <I>DB</I>)=(arc <I>CE</I>)</MATH>;
+<pb n=430><head>EUCLID</head>
+therefore
+<MATH>(arc <I>ABC</I>)=(arc <I>DCE</I>), and (segmt. <I>ABC</I>)=(segmt. <I>DCE</I>);
+therefore (sector <I>ADBO</I>), or <I>m</I>/<I>n</I> (circle <I>ABC</I>)
+=(segmt. <I>DCE</I>)-(segmt. <I>BFC</I>)</MATH>.
+<p>That is <I>BC, DE</I> cut off an area equal to <MATH><I>m</I>/<I>n</I> (circle <I>ABC</I>)</MATH>.
+<C>Lost geometrical works.</C>
+<C>(<I>a</I>) The <I>Pseudaria.</I></C>
+<p>The other purely geometrical works of Euclid are lost so far
+as is known at present. One of these again belongs to the
+domain of elementary geometry. This is the <I>Pseudaria</I>, or
+&lsquo;Book of Fallacies&rsquo;, as it is called by Proclus, which is clearly
+the same work as the &lsquo;Pseudographemata&rsquo; of Euclid men-
+tioned by a commentator on Aristotle in terms which agree
+with Proclus's description.<note>Michael Ephesius, <I>Comm. on Arist. Soph. El.</I>, fol. 25<SUP>v</SUP>, p. 76. 23 Wallies.</note> Proclus says of Euclid that,
+<p>&lsquo;Inasmuch as many things, while appearing to rest on truth
+and to follow from scientific principles, really tend to lead one
+astray from the principles and deceive the more superficial
+minds, he has handed down methods for the discriminative
+understanding of these things as well, by the use of which
+methods we shall be able to give beginners in this study
+practice in the discovery of paralogisms, and to avoid being
+ourselves misled. The treatise by which he puts this machinery
+in our hands he entitled (the book) of Pseudaria, enumerating
+in order their various kinds, exercising our intelligence in each
+case by theorems of all sorts, setting the true side by side
+with the false, and combining the refutation of error with
+practical illustration. This book then is by way of cathartic
+and exercise, while the Elements contain the irrefragable and
+complete guide to the actual scientific investigation of the
+subjects of geometry.&rsquo;<note>Proclus on Eucl. I, p. 70. 1-18. Cf. a scholium to Plato's <I>Theaetetus</I>
+191 B, which says that the fallacies did not arise through any importation
+of sense-perception into the domain of non-sensibles.</note>
+<p>The connexion of the book with the <I>Elements</I> and the refer-
+ence to its usefulness for beginners show that it did not go
+beyond the limits of elementary geometry.
+<pb n=431><head>LOST GEOMETRICAL WORKS</head>
+<p>We now come to the lost works belonging to higher
+geometry. The most important was evidently
+<C>(<G>b</G>) The <I>Porisms.</I></C>
+<p>Our only source of information about the nature and con-
+tents of the <I>Porisms</I> is Pappus. In his general preface about
+the books composing the <I>Treasury of Analysis</I> Pappus writes
+as follows<note>Pappus, vii, pp. 648-60.</note> (I put in square brackets the words bracketed by
+Hultsch).
+<p>&lsquo;After the Tangencies (of Apollonius) come, in three Books,
+the Porisms of Euclid, a collection [in the view of many] most
+ingeniously devised for the analysis of the more weighty
+problems, [and] although nature presents an unlimited num-
+ber of such porisms, [they have added nothing to what was
+originally written by Euclid, except that some before my time
+have shown their want of taste by adding to a few (of the
+propositions) second proofs, each (proposition) admitting of
+a definite number of demonstrations, as we have shown, and
+Euclid having given one for each, namely that which is the
+most lucid. These porisms embody a theory subtle, natural,
+necessary, and of considerable generality, which is fascinating
+to those who can see and produce results].
+<p>&lsquo;Now all the varieties of porisms belong, neither to theorems
+nor problems, but to a species occupying a sort of intermediate
+position [so that their enunciations can be formed like those of
+either theorems or problems], the result being that, of the great
+number of geometers, some regarded them as of the class of
+theorems, and others of problems, looking only to the form of
+the proposition. But that the ancients knew better the differ-
+ence between these three things is clear from the definitions.
+For they said that a theorem is that which is proposed with a
+view to the demonstration of the very thing proposed, a pro-
+blem that which is thrown out with a view to the construction
+of the very thing proposed, and a porism that which is pro-
+posed with a view to the producing of the very thing proposed.
+[But this definition of the porism was changed by the more
+recent writers who could not produce everything, but used
+these elements and proved only the fact that that which is
+sought really exists, but did not produce it, and were accord-
+ingly confuted by the definition and the whole doctrine. They
+based their definition on an incidental characteristic, thus:
+A porism is that which falls short of a locus-theorem in
+<pb n=432><head>EUCLID</head>
+respect of its hypothesis. Of this kind of porisms loci are
+a species, and they abound in the Treasury of Analysis; but
+this species has been collected, named, and handed down
+separately from the porisms, because it is more widely diffused
+than the other species] . . . But it has further become charac-
+teristic of porisms that, owing to their complication, the enun-
+ciations are put in a contracted form, much being by usage
+left to be understood; so that many geometers understand
+them only in a partial way and are ignorant of the more
+essential features of their content.
+<p>&lsquo;[Now to comprehend a number of propositions in one
+enunciation is by no means easy in these porisms, because
+Euclid himself has not in fact given many of each species, but
+chosen, for examples, one or a few out of a great multitude.
+But at the beginning of the first book he has given some pro-
+positions, to the number of ten, of one species, namely that
+more fruitful species consisting of loci.] Consequently, finding
+that these admitted of being comprehended in our enunciation,
+we have set it out thus:
+<p>If, in a system of four straight lines which cut one
+another two and two, three points on one straight line
+be given, while the rest except one lie on different straight
+lines given in position, the remaining point also will lie
+on a straight line given in position.
+<p>&lsquo;This has only been enunciated of four straight lines, of
+which not more than two pass through the same point, but it
+is not known (to most people) that it is true of any assigned
+number of straight lines if enunciated thus:
+<p>If any number of straight lines cut one another, not
+more than two (passing) through the same point, and all
+the points (of intersection situated) on one of them be
+given, and if each of those which are on another (of
+them) lie on a straight line given in position&mdash;
+<p>or still more generally thus:
+<p>if any number of straight lines cut one another, not more
+than two (passing) through the same point, and all the
+points (of intersection situated) on one of them be given,
+while of the other points of intersection in multitude
+equal to a triangular number a number corresponding
+to the side of this triangular number lie respectively on
+straight lines given in position, provided that of these
+latter points no three are at the angular points of a
+triangle (sc. having for sides three of the given straight
+<pb n=433><head>THE <I>PORISMS</I></head>
+lines)&mdash;each of the remaining points will lie on a straight
+line given in position.<note>Loria (<I>Le scienze esatte nell'antica Grecia,</I> pp. 256-7) gives the mean-
+ing of this as follows, pointing out that Simson first discovered it: &lsquo;If
+a complete <I>n</I>-lateral be deformed so that its sides respectively turn about
+<I>n</I> points on a straight line, and (<I>n</I> - 1) of its 1/2 <I>n</I> (<I>n</I> - 1) vertices move on
+as many straight lines, the other 1/2 (<I>n</I> - 1) (<I>n</I> - 2) of its vertices likewise
+move on as many straight lines: but it is necessary that it should be
+impossible to form with the (<I>n</I> - 1) vertices any triangle having for sides
+the sides of the polygon.&rsquo;</note>
+<p>&lsquo;It is probable that the writer of the Elements was not
+unaware of this, but that he only set out the principle; and
+he seems, in the case of all the porisms, to have laid down the
+principles and the seed only [of many important things],
+the kinds of which should be distinguished according to the
+differences, not of their hypotheses, but of the results and
+the things sought. [All the hypotheses are different from one
+another because they are entirely special, but each of the
+results and things sought, being one and the same, follow from
+many different hypotheses.]
+<p>&lsquo;We must then in the first book distinguish the following
+kinds of things sought:
+<p>&lsquo;At the beginning of the book is this proposition:
+<p>I. <I>If from two given points straight lines be drawn
+meeting on a straight line given in position, and one cut
+off from a straight line given in position</I> (<I>a segment
+measured</I>) <I>to a given point on it, the other will also cut
+off from another</I> (<I>straight line a segment</I>) <I>having to the
+first a given ratio.</I>
+<p>&lsquo;Following on this (we have to prove)
+<p>II. that such and such a point lies on a straight line
+given in position;
+<p>III. that the ratio of such and such a pair of straight
+lines is given&rsquo;;
+<p>&amp;c. &amp;c. (up to XXIX).
+<p>&lsquo;The three books of the porisms contain 38 lemmas; of the
+theorems themselves there are 171.&rsquo;
+<p>Pappus further gives lemmas to the <I>Porisms.</I><note>Pappus, vii, pp. 866-918; Euclid, ed. Heiberg-Menge, vol. viii,
+pp. 243-74.</note>
+<p>With Pappus's account of Porisms must be compared the
+passages of Proclus on the same subject. Proclus distinguishes
+<pb n=434><head>EUCLID</head>
+the two senses of the word <G>po/risma</G>. The first is that of
+a <I>corollary,</I> where something appears as an incidental result
+of a proposition, obtained without trouble or special seeking,
+a sort of bonus which the investigation has presented us
+with.<note>Proclus on Eucl. I, pp. 212. 14; 301. 22.</note> The other sense is that of Euclid's <I>Porisms.</I> In
+this sense
+<p>&lsquo;<I>porism</I> is the name given to things which are sought, but
+need some finding and are neither pure bringing into existence
+nor simple theoretic argument. For (to prove) that the angles
+at the base of isosceles triangles are equal is matter of theoretic
+argument, and it is with reference to things existing that such
+knowledge is (obtained). But to bisect an angle, to construct
+a triangle, to cut off, or to place&mdash;all these things demand the
+making of something; and to find the centre of a given circle,
+or to find the greatest common measure of two given commen-
+surable magnitudes, or the like, is in some sort intermediate
+between theorems and problems. For in these cases there is
+no bringing into existence of the things sought, but finding
+of them; nor is the procedure purely theoretic. For it is
+necessary to bring what is sought into view and exhibit it
+to the eye. Such are the porisms which Euclid wrote and
+arranged in three books of Porisms.&rsquo;<note><I>Ib.,</I> p. 301. 25 sq.</note>
+<p>Proclus's definition thus agrees well enough with the first,
+the &lsquo;older&rsquo;, definition of Pappus. A porism occupies a place
+between a theorem and a problem; it deals with something
+already existing, as a theorem does, but has to <I>find</I> it (e.g. the
+centre of a circle), and, as a certain operation is therefore
+necessary, it partakes to that extent of the nature of a problem,
+which requires us to construct or produce something not
+previously existing. Thus, besides III. 1 and X. 3, 4 of the
+<I>Elements</I> mentioned by Proclus, the following propositions are
+real porisms: III. 25, VI. 11-13, VII. 33, 34, 36, 39, VIII. 2, 4,
+X. 10, XIII. 18. Similarly, in Archimedes's <I>On the Sphere and
+Cylinder,</I> I. 2-6 might be called porisms.
+<p>The enunciation given by Pappus as comprehending ten of
+Euclid's propositions may not reproduce the <I>form</I> of Euclid's
+enunciations; but, comparing the result to be proved, that
+certain points lie on straight lines given in position, with the
+<I>class</I> indicated by II above, where the question is of such and
+such a point lying on a straight line given in position, and
+<pb n=435><head>THE <I>PORISMS</I></head>
+with other classes, e.g. (V) that such and such a line is given
+in position, (VI) that such and such a line verges to a given point,
+(XXVII) that there exists a given point such that straight
+lines drawn from it to such and such (circles) will contain
+a triangle given in species, we may conclude that a usual form
+of a porism was &lsquo;to prove that it is possible to find a point
+with such and such a property&rsquo; or &lsquo;a straight line on which
+lie all the points satisfying given conditions&rsquo;, and so on.
+<p>The above exhausts all the positive information which we
+have about the nature of a porism and the contents of Euclid's
+<I>Porisms.</I> It is obscure and leaves great scope for speculation
+and controversy; naturally, therefore, the problem of restoring
+the <I>Porisms</I> has had a great fascination for distinguished
+mathematicians ever since the revival of learning. But it has
+proved beyond them all. Some contributions to a solution have,
+it is true, been made, mainly by Simson and Chasles. The first
+claim to have restored the <I>Porisms</I> seems to be that of Albert
+Girard (about 1590-1633), who spoke (1626) of an early pub-
+lication of his results, which, however, never saw the light.
+The great Fermat (1601-65) gave his idea of a &lsquo;porism&rsquo;,
+illustrating it by five examples which are very interesting in
+themselves<note><I>&OElig;uvres de Fermat,</I> ed. Tannery and Henry, I, p. 76-84.</note>; but he did not succeed in connecting them with
+the description of Euclid's <I>Porisms</I> by Pappus, and, though he
+expressed a hope of being able to produce a complete restoration
+of the latter, his hope was not realized. It was left for Robert
+Simson (1687-1768) to make the first decisive step towards the
+solution of the problem.<note>Roberti Simson <I>Opera quaedam reliqua,</I> 1776, pp. 315-594.</note> He succeeded in explaining the mean-
+ing of the actual porisms enunciated in such general terms by
+Pappus. In his tract on Porisms he proves the first porism
+given by Pappus in its ten different cases, which, according to
+Pappus, Euclid distinguished (these propositions are of the
+class connected with <I>loci</I>); after this he gives a number of
+other propositions from Pappus, some auxiliary proposi-
+tions, and some 29 &lsquo;porisms&rsquo;, some of which are meant to
+illustrate the classes I, VI, XV, XXVII-XXIX distin-
+guished by Pappus. Simson was able to evolve a definition
+of a porism which is perhaps more easily understood in
+Chasles's translation: &lsquo;Le porisme est une proposition dans
+<pb n=436><head>EUCLID</head>
+laquelle on demande de d&eacute;montrer qu'une chose ou plusieurs
+choses sont <I>donn&eacute;es,</I> qui, ainsi que l'une quelconque d'une
+infinit&eacute; d'autres choses non donn&eacute;es, mais dont chacune est
+avec des choses donn&eacute;es dans une m&ecirc;me relation, ont une
+propri&eacute;t&eacute; commune, d&eacute;crite dans la proposition.&rsquo; We need
+not follow Simson's English or Scottish successors, Lawson
+(1777), Playfair (1794), W. Wallace (1798), Lord Brougham
+(1798), in their further speculations, nor the controversies
+between the Frenchmen, A. J. H. Vincent and P. Breton (de
+Champ), nor the latter's claim to priority as against Chasles;
+the work of Chasles himself (<I>Les trois livres des Porismes
+d'Euclide r&eacute;tablis . . .</I> Paris, 1860) alone needs to be men-
+tioned. Chasles adopted the definition of a porism given by
+Simson, but showed how it could be expressed in a different
+form. &lsquo;Porisms are incomplete theorems which express
+certain relations existing between things variable in accord-
+ance with a common law, relations which are indicated in
+the enunciation of the porism, but which need to be completed
+by determining the magnitude or position of certain things
+which are the consequences of the hypotheses and which
+would be determined in the enunciation of a theorem properly
+so called or a complete theorem.&rsquo; Chasles succeeded in eluci-
+dating the connexion between a porism and a locus as de-
+scribed by Pappus, though he gave an inexact translation of
+the actual words of Pappus: &lsquo;<I>Ce qui constitue le porisme est
+ce qui manque &agrave; l'hypoth&egrave;se d'un th&eacute;or&egrave;me local</I> (en d'autres
+termes, le porisme est inf&eacute;rieur, par l'hypoth&egrave;se, au th&eacute;or&egrave;me
+local; c'est &agrave; dire que quand quelques parties d'une proposi-
+tion locale n'ont pas dans l'&eacute;nonc&eacute; la d&eacute;termination qui leur
+est propre, cette proposition cesse d'&ecirc;tre regard&eacute;e comme un
+th&eacute;or&egrave;me et devient un porisme)&rsquo;; here the words italicized
+are not quite what Pappus said, viz. that &lsquo;a porism is that
+which falls short of a locus-theorem in respect of its hypo-
+thesis&rsquo;, but the explanation in brackets is correct enough if
+we substitute &lsquo;in respect of&rsquo; for &lsquo;par&rsquo; (&lsquo;by&rsquo;). The work of
+Chasles is historically important because it was in the course
+of his researches on this subject that he was led to the idea of
+anharmonic ratios; and he was probably right in thinking
+that the <I>Porisms</I> were propositions belonging to the modern
+theory of transversals and to projective geometry. But, as a
+<pb n=437><head>THE <I>PORISMS</I></head>
+restoration of Euclid's work, Chasles's Porisms cannot be re-
+garded as satisfactory. One consideration alone is, to my
+mind, conclusive on this point. Chasles made &lsquo;porisms&rsquo; out
+of Pappus's various <I>lemmas</I> to Euclid's porisms and com-
+paratively easy deductions from those lemmas. Now we
+have experience of Pappus's lemmas to books which still
+survive, e.g. the <I>Conics</I> of Apollonius; and, to judge by these
+instances, his lemmas stood in a most ancillary relation to
+the propositions to which they relate, and do not in the
+least compare with them in difficulty and importance. Hence
+it is all but impossible to believe that the lemmas to the
+porisms were themselves porisms such as were Euclid's own
+porisms; on the contrary, the analogy of Pappus's other sets
+of lemmas makes it all but necessary to regard the lemmas in
+question as merely supplying proofs of simple propositions
+assumed by Euclid without proof in the course of the demon-
+stration of the actual porisms. This being so, it appears that
+the problem of the complete restoration of Euclid's three
+Books still awaits a solution, or rather that it will never be
+solved unless in the event of discovery of fresh documents.
+<p>At the same time the lemmas of Pappus to the <I>Porisms</I>
+are by no means insignificant propositions in themselves,
+and, if the usual relation of lemmas to substantive proposi-
+tions holds, it follows that the <I>Porisms</I> was a distinctly
+advanced work, perhaps the most important that Euclid ever
+wrote; its loss is therefore much to be deplored. Zeuthen
+has an interesting remark &agrave; propos of the proposition which
+Pappus quotes as the first proposition of Book I, &lsquo;If from two
+given points straight lines be drawn meeting on a straight
+line given in position, and one of them cut off from a straight
+line given in position (a segment measured) towards a given
+point on it, the other will also cut off from another (straight
+line a segment) bearing to the first a given ratio.&rsquo; This pro-
+position is also true if there be substituted for the first given
+straight line a conic regarded as the &lsquo;locus with respect to
+four lines&rsquo;, and the proposition so extended can be used for
+completing Apollonius's exposition of that locus. Zeuthen
+suggests, on this ground, that the <I>Porisms</I> were in part by-
+products of the theory of conics and in part auxiliary means
+for the study of conics, and that Euclid called them by the
+<pb n=438><head>EUCLID</head>
+same name as that applied to corollaries because they were
+corollaries with respect to conics.<note>Zeuthen, <I>Die Lehre von den Kegelschnitten im Altertum,</I> 1886, pp. 168,
+173-4.</note> This, however, is a pure
+conjecture.
+<C>(<G>g</G>) The <I>Conics.</I></C>
+<p>Pappus says of this lost work: &lsquo;The four books of Euclid's
+Conics were completed by Apollonius, who added four more
+and gave us eight books of Conics.&rsquo;<note>Pappus, vii, p. 672. 18.</note> It is probable that
+Euclid's work was already lost by Pappus's time, for he goes
+on to speak of &lsquo;Aristaeus who wrote the <I>still extant</I> five books
+of Solid Loci <G>sunexh= toi=s kwnikoi=s</G>, connected with, or supple-
+mentary to, the conics&rsquo;.<note>Cf. Pappus, vii, p. 636. 23.</note> This latter work seems to have
+been a treatise on conics regarded as loci; for &lsquo;solid loci&rsquo; was
+a term appropriated to conics, as distinct from &lsquo;plane loci&rsquo;,
+which were straight lines and circles. In another passage
+Pappus (or an interpolator) speaks of the &lsquo;conics&rsquo; of Aristaeus
+the &lsquo;elder&rsquo;,<note><I>Ib.</I> vii, p. 672. 12.</note> evidently referring to the same book. Euclid no
+doubt wrote on the general theory of conics, as Apollonius did,
+but only covered the ground of Apollonius's first three books,
+since Apollonius says that no one before him had touched the
+subject of Book IV (which, however, is not important). As in
+the case of the <I>Elements,</I> Euclid would naturally collect and
+rearrange, in a systematic exposition, all that had been dis-
+covered up to date in the theory of conics. That Euclid's
+treatise covered most of the essentials up to the last part of
+Apollonius's Book III seems clear from the fact that Apol-
+lonius only claims originality for some propositions connected
+with the &lsquo;three- and four-line locus&rsquo;, observing that Euclid
+had not completely worked out the synthesis of the said locus,
+which, indeed, was not possible without the propositions
+referred to. Pappus (or an interpolator)<note><I>Ib.</I> vii, pp. 676. 25-678. 6.</note> excuses Euclid on
+the ground that he made no claim to go beyond the discoveries
+of Aristaeus, but only wrote so much about the locus as was
+possible with the aid of Aristaeus's conics. We may conclude
+that Aristaeus's book preceded Euclid's, and that it was, at
+least in point of originality, more important. When Archi-
+medes refers to propositions in conics as having been proved
+<pb n=439><head>THE <I>CONICS</I> AND <I>SURFACE-LOCI</I></head>
+in the &lsquo;elements of conics&rsquo;, he clearly refers to these two
+treatises, and the other propositions to which he refers as well
+known and not needing proof were doubtless taken from the
+same sources. Euclid still used the old names for the conic
+sections (sections of a right-angled, acute-angled, and obtuse-
+angled cone respectively), but he was aware that an ellipse
+could be obtained by cutting (through) a cone in any manner
+by a plane not parallel to the base, and also by cutting a
+cylinder; this is clear from a sentence in his <I>Phaenomena</I> to
+the effect that, &lsquo;If a cone or a cylinder be cut by a plane not
+parallel to the base, this section is a section of an acute-angled
+cone, which is like a shield (<G>qureo/s</G>).&rsquo;
+<C>(<G>d</G>) The <I>Surface-Loci</I> (<G>to/poi pro\s e)pifanei/a|</G>).</C>
+<p>Like the <I>Data</I> and the <I>Porisms,</I> this treatise in two Books
+is mentioned by Pappus as belonging to the <I>Treasury of
+Analysis.</I> What is meant by surface-loci, literally &lsquo;loci on a
+surface&rsquo; is not entirely clear, but we are able to form a con-
+jecture on the subject by means of remarks in Proclus and
+Pappus. The former says (1) that a locus is &lsquo;a position of a
+line or of a surface which has (throughout it) one and the
+same property&rsquo;,<note>Proclus on Eucl. I, p. 394. 17.</note> and (2) that &lsquo;of locus-theorems some are
+constructed on lines and others on surfaces&rsquo;<note><I>Ib.,</I> p. 394. 19.</note>; the effect of
+these statements together seems to be that &lsquo;loci on lines&rsquo; are
+loci which <I>are</I> lines, and &lsquo;loci on surfaces&rsquo; loci which <I>are</I>
+surfaces. On the other hand, the possibility does not seem to
+be excluded that loci on surfaces may be loci <I>traced</I> on sur-
+faces; for Pappus says in one place that the equivalent of the
+<I>quadratrix</I> can be got geometrically &lsquo;by means of loci on
+surfaces as follows&rsquo;<note>Pappus, iv, p. 258. 20-25.</note> and then proceeds to use a spiral de-
+scribed on a cylinder (the cylindrical helix), and it is consis-
+tent with this that in another passage<note><I>Ib.</I> vii. 662. 9.</note> (bracketed, however, by
+Hultsch) &lsquo;linear&rsquo; loci are said to be exhibited (<G>dei/knuntai</G>) or
+realized from loci on surfaces, for the quadratrix is a &lsquo;linear&rsquo;
+locus, i.e. a locus of an order higher than a plane locus
+(a straight line or circle) and a &lsquo;solid&rsquo; locus (a conic). How-
+ever this may be, Euclid's <I>Surface-Loci</I> probably included
+<pb n=440><head>EUCLID</head>
+such loci as were cones, cylinders and spheres. The two
+lemmas given by Pappus lend some colour to this view. The
+first of these<note>Pappus, vii, p. 1004. 17; Euclid, ed. Heiberg-Menge, vol. viii, p. 274.</note> and the figure attached to it are unsatisfactory
+as they stand, but Tannery indicated a possible restoration.<note>Tannery in <I>Bulletin des sciences math&eacute;matiques,</I> 2&deg; s&eacute;rie, VI, p. 149.</note>
+If this is right, it suggests that one of the loci contained all
+the points on the elliptical parallel sections of a cylinder, and
+was therefore an oblique circular cylinder. Other assump-
+tions with regard to the conditions to which the lines in the
+figure may be subject would suggest that other loci dealt with
+were cones regarded as containing all points on particular
+parallel elliptical sections of the cones. In the second lemma
+Pappus states and gives a complete proof of the focus-and-
+directrix property of a conic, viz. that <I>the locus of a point
+the distance of which from a given point is in a given ratio
+to its distance from a fixed straight line is a conic section,
+which is an ellipse, a parabola or a hyperbola according as the
+given ratio is less than, equal to, or greater than unity.</I><note>Pappus, vii, pp. 1004. 23-1014; Euclid, vol. viii, pp. 275-81.</note> Two
+conjectures are possible as to the application of this theorem in
+Euclid's <I>Surface-Loci.</I> (<I>a</I>) It may have been used to prove that
+the locus of a point the distance of which from a given straight
+line is in a given ratio to its distance from a given plane
+is a certain cone. Or (<I>b</I>) it may have been used to prove
+that the locus of a point the distance of which from a given
+point is in a given ratio to its distance from a given plane is
+the surface formed by the revolution of a conic about its major
+or conjugate axis.<note>For further details, see <I>The Works of Archimedes,</I> pp. lxii-lxv.</note>
+<p>We come now to Euclid's works under the head of
+<C>Applied mathematics.</C>
+<C>(<G>a</G>) The <I>Phaenomena.</I></C>
+<p>The book on <I>sphaeric</I> intended for use in astronomy and
+entitled <I>Phaenomena</I> has already been noticed (pp. 349, 351-2).
+It is extant in Greek and was included in Gregory's edition of
+Euclid. The text of Gregory, however, represents the later
+of two recensions which differ considerably (especially in
+Propositions 9 to 16). The best manuscript of this later
+recension (b) is the famous Vat. gr. 204 of the tenth century,
+<pb n=441><head>THE <I>PHAENOMENA</I> AND <I>OPTICS</I></head>
+while the best manuscript of the older and better version (a)
+is the Viennese MS.Vind. gr. XXXI. 13 of the twelfth century.
+A new text edited by Menge and taking account of both
+recensions is now available in the last volume of the Heiberg-
+Menge edition of Euclid.<note><I>Euclidis Phaenomena et scripta Musica</I> edidit Henricus Menge.
+<I>Fragmenta</I> collegit et disposuit J. L. Heiberg, Teubner, 1916.</note>
+<C>(<G>b</G>) <I>Optics</I> and <I>Catoptrica.</I></C>
+<p>The <I>Optics,</I> a treatise included by Pappus in the collection of
+works known as the Little Astronomy, survives in two forms.
+One is the recension of Theon translated by Zambertus in
+1505; the Greek text was first edited by Johannes Pena
+(de la P&egrave;ne) in 1557, and this form of the treatise was alone
+included in the editions up to Gregory's. But Heiberg dis-
+covered the earlier form in two manuscripts, one at Vienna
+(Vind. gr. XXXI. 13) and one at Florence (Laurent. XXVIII. 3),
+and both recensions are contained in vol. vii of the Heiberg-
+Menge text of Euclid (Teubner, 1895). There is no reason to
+doubt that the earlier recension is Euclid's own work; the
+style is much more like that of the <I>Elements,</I> and the proofs of
+the propositions are more complete and clear. The later recen-
+sion is further differentiated by a preface of some length, which
+is said by a scholiast to be taken from the commentary or
+elucidation by Theon. It would appear that the text of this
+recension is Theon's, and that the preface was a reproduction
+by a pupil of what was explained by Theon in lectures. It
+cannot have been written much, if anything, later than Theon's
+time, for it is quoted by Nemesius about A.D. 400. Only the
+earlier and genuine version need concern us here. It is
+a kind of elementary treatise on perspective, and it may have
+been intended to forearm students of astronomy against
+paradoxical theories such as those of the Epicureans, who
+maintained that the heavenly bodies <I>are</I> of the size that they
+<I>look.</I> It begins in the orthodox fashion with Definitions, the
+first of which embodies the same idea of the process of vision
+as we find in Plato, namely that it is due to rays proceeding
+from our eyes and impinging upon the object, instead of
+the other way about: &lsquo;the straight lines (rays) which issue
+from the eye traverse the distances (or dimensions) of great
+<pb n=442><head>EUCLID</head>
+magnitudes&rsquo;; Def. 2: &lsquo;The figure contained by the visual rays
+is a cone which has its vertex in the eye, and its base at the
+extremities of the objects seen&rsquo;; Def. 3: &lsquo;And those things
+are seen on which the visual rays impinge, while those are
+not seen on which they do not&rsquo;; Def. 4: &lsquo;Things seen under
+a greater angle appear greater, and those under a lesser angle
+less, while things seen under equal angles appear equal&rsquo;;
+Def. 7: &lsquo;Things seen under more angles appear more distinctly.&rsquo;
+Euclid assumed that the visual rays are not &lsquo;continuous&rsquo;,
+i.e. not absolutely close together, but are separated by a
+certain distance, and hence he concluded, in Proposition 1,
+that we can never really see the whole of any object, though
+we seem to do so. Apart, however, from such inferences as
+these from false hypotheses, there is much in the treatise that
+is sound. Euclid has the essential truth that the rays are
+straight; and it makes no difference geometrically whether
+they proceed from the eye or the object. Then, after pro-
+positions explaining the differences in the apparent size of an
+object according to its position relatively to the eye, he proves
+that the apparent sizes of two equal and parallel objects are
+not proportional to their distances from the eye (Prop. 8); in
+this proposition he proves the equivalent of the fact that, if <G>a</G>,
+<G>b</G> are two angles and <MATH><G>a</G> < <G>b</G> < (1/2)<G>p</G></MATH>, then
+<MATH>(tan <G>a</G>)/(tan <G>b</G>) < <G>a</G>/<G>b</G></MATH>,
+the equivalent of which, as well as of the corresponding
+formula with sines, is assumed without proof by Aristarchus
+a little later. From Proposition 6 can easily be deduced the
+fundamental proposition in perspective that parallel lines
+(regarded as equidistant throughout) appear to meet. There
+are four simple propositions in heights and distances, e.g. to
+find the height of an object (1) when the sun is shining
+(Prop. 18), (2) when it is not (Prop. 19): similar triangles are,
+of course, used and the horizontal mirror appears in the second
+case in the orthodox manner, with the assumption that the
+angles of incidence and reflection of a ray are equal, &lsquo;as
+is explained in the Catoptrica (or theory of mirrors)&rsquo;. Pro-
+positions 23-7 prove that, if an eye sees a sphere, it sees
+less than half of the sphere, and the contour of what is seen
+<pb n=443><head><I>OPTICS</I></head>
+appears to be a circle; if the eye approaches nearer to
+the sphere the portion seen becomes less, though it appears
+greater; if we see the sphere with two eyes, we see a hemi-
+sphere, or more than a hemisphere, or less than a hemisphere
+according as the distance between the eyes is equal to, greater
+than, or less than the diameter of the sphere; these pro-
+positions are comparable with Aristarchus's Proposition 2
+stating that, if a sphere be illuminated by a larger sphere,
+the illuminated portion of the former will be greater
+than a hemisphere. Similar propositions with regard to the
+cylinder and cone follow (Props. 28-33). Next Euclid con-
+siders the conditions for the apparent equality of different
+diameters of a circle as seen from an eye occupying various
+positions outside the plane of the circle (Props. 34-7); he
+shows that all diameters will appear equal, or the circle will
+really look like a circle, if the line joining the eye to the
+centre is perpendicular to the plane of the circle, <I>or,</I> not being
+perpendicular to that plane, is equal to the length of the
+radius, but this will not otherwise be the case (35), so that (36)
+a chariot wheel will sometimes appear circular, sometimes
+awry, according to the position of the eye. Propositions
+37 and 38 prove, the one that there is a locus such that, if the
+eye remains at one point of it, while a straight line moves so
+that its extremities always lie on it, the line will always
+<I>appear</I> of the same length in whatever position it is placed
+(not being one in which either of the extremities coincides
+with, or the extremities are on opposite sides of, the point
+at which the eye is placed), the locus being, of course, a circle
+in which the straight line is placed as a chord, when it
+necessarily subtends the same angle at the circumference or at
+the centre, and therefore at the eye, if placed at a point of the
+circumference or at the centre; the other proves the same thing
+for the case where the line is fixed with its extremities on the
+locus, while the eye moves upon it. The same idea underlies
+several other propositions, e.g. Proposition 45, which proves
+that a common point can be found from which unequal
+magnitudes will appear equal. The unequal magnitudes are
+straight lines <I>BC, CD</I> so placed that <I>BCD</I> is a straight line.
+A segment greater than a semicircle is described on <I>BC,</I> and
+a similar segment on <I>CD.</I> The segments will then intersect
+<pb n=444><head>EUCLID</head>
+at <I>F,</I> and the angles subtended by <I>BC</I> and <I>CD</I> at <I>F</I> are
+equal. The rest of the treatise is of the same character, and
+it need not be further described.
+<p>The <I>Catoptrica</I> published by Heiberg in the same volume is
+not by Euclid, but is a compilation made at a much later date,
+possibly by Theon of Alexandria, from ancient works on the
+subject and mainly no doubt from those of Archimedes and
+Heron. Theon<note>Theon, <I>Comm. on Ptolemy's Syntaxis,</I> i, p. 10.</note> himself quotes a <I>Catoptrica</I> by Archimedes,
+and Olympiodorus<note><I>Comment. on Arist. Meteorolog.</I> ii, p. 94, Ideler, p. 211. 18 Busse.</note> quotes Archimedes as having proved the
+fact which appears as an axiom in the <I>Catoptrica</I> now in
+question, namely that, if an object be placed just out of sight
+at the bottom of a vessel, it will become visible over the edge
+when water is poured in. It is not even certain that Euclid
+wrote <I>Catoptrica</I> at all, since, if the treatise was Theon's,
+Proclus may have assigned it to Euclid through inadvertence.
+<C>(<G>g</G> <I>Music.</I></C>
+<p>Proclus attributes to Euclid a work on the <I>Elements of
+Music</I> (<G>ai( kata\ mousikh\n stoixeiw/seis</G><note>Proclus on Eucl. I, p. 69. 3.</note>; so does Marinus.<note>Marinus, <I>Comm. on the Data</I> (Euclid, vol. vi, p. 254. 19).</note>
+As a matter of fact, two musical treatises attributed to Euclid
+are still extant, the <I>Sectio Canonis</I> (<G>*katatomh\ kano/nos</G>) and the
+<I>Introductio harmonica</I> (<G>*ei)sagwgh\ a(rmonikh/</G>). The latter,
+however, is certainly not by Euclid, but by Cleonides, a pupil
+of Aristoxenus. The question remains, in what relation does
+the <I>Sectio Canonis</I> stand to the &lsquo;Elements&rsquo; mentioned by
+Proclus and Marinus? The <I>Sectio</I> gives the Pythagorean
+theory of music, but is altogether too partial and slight to
+deserve the title &lsquo;Elements of Music&rsquo;. Jan, the editor of the
+<I>Musici Graeci,</I> thought that the <I>Sectio</I> was a sort of summary
+account extracted from the &lsquo;Elements&rsquo; by Euclid himself,
+which hardly seems likely; he maintained that it is the
+genuine work of Euclid on the grounds (1) that the style and
+diction and the form of the propositions agree well with what
+we find in Euclid's <I>Elements,</I> and (2) that Porphyry in his
+commentary on Ptolemy's <I>Harmonica</I> thrice quotes Euclid as
+the author of a <I>Sectio Canonis.</I><note>See Wallis, <I>Opera mathematica,</I> vol. iii, 1699, pp. 267, 269, 272.</note> The latest editor, Menge,
+<pb n=445><head>ON MUSIC</head>
+points out that the extract given by Porphyry shows some
+differences from our text and contains some things quite
+unworthy of Euclid; hence he is inclined to think that the
+work as we have it is not actually by Euclid, but was ex-
+tracted by some other author of less ability from the genuine
+&lsquo;Elements of Music&rsquo; by Euclid.
+<C>(<G>d</G>) Works on mechanics attributed to Euclid.</C>
+<p>The Arabian list of Euclid's works further includes among
+those held to be genuine &lsquo;the book of the Heavy and Light&rsquo;.
+This is apparently the tract <I>De levi et ponderoso</I> included by.
+Hervagius in the Basel Latin translation of 1537 and by
+Gregory in his edition. That it comes from the Greek is
+made clear by the lettering of the figures; and this is con-
+firmed by the fact that another, very slightly different, version
+exists at Dresden (Cod. Dresdensis Db. 86), which is evidently
+a version of an Arabic translation from the Greek, since the
+lettering of the figures follows the order characteristic of such
+Arabic translations, <I>a, b, g, d, e, z, h, t.</I> The tract consists of
+nine definitions or axioms and five propositions. Among the
+definitions are these: Bodies are equal, different, or greater in
+size according as they occupy equal, different, or greater spaces
+(1-3). Bodies are equal in <I>power</I> or in <I>virtue</I> which move
+over equal distances in the same medium of air or water in
+equal times (4), while the <I>power</I> or <I>virtue</I> is greater if the
+motion takes less time, and less if it takes more (6). Bodies
+are <I>of the same kind</I> if, being equal in size, they are also equal
+in <I>power</I> when the medium is the same; they are different in
+kind when, being equal in size, they are not equal in <I>power</I> or
+<I>virtue</I> (7, 8). Of bodies different in kind, that has more <I>power</I>
+which is more dense (<I>solidius</I>) (9). With these hypotheses, the
+author attempts to prove (Props. 1, 3, 5) that, of bodies which
+traverse unequal spaces in equal times, that which traverses
+the greater space has the greater <I>power</I> and that, of bodies of
+the same kind, the <I>power</I> is proportional to the size, and con-
+versely, if the <I>power</I> is proportional to the size, the bodies are
+of the same kind. We recognize in the <I>potentia</I> or <I>virtus</I>
+the same thing as the <G>du/namis</G> and <G>i)sxu/s</G> of Aristotle.<note>Aristotle, <I>Physics,</I> Z. 5.</note> The
+<pb n=446><head>EUCLID</head>
+property assigned by the author to bodies <I>of the same kind</I> is
+quite different from what we attribute to bodies of the same
+specific gravity; he purports to prove that bodies of the
+same kind have <I>power</I> proportional to their size, and the effect
+of this, combined with the definitions, is that they move at
+speeds proportional to their volumes. Thus the tract is the
+most precise statement that we possess of the principle of
+Aristotle's dynamics, a principle which persisted until Bene-
+detti (1530-90) and Galilei (1564-1642) proved its falsity.
+<p>There are yet other fragments on mechanics associated with
+the name of Euclid. One is a tract translated by Woepcke
+from the Arabic in 1851 under the title &lsquo;Le livre d'Euclide
+sur la balance&rsquo;, a work which, although spoiled by some com-
+mentator, seems to go back to a Greek original and to have
+been an attempt to establish a theory of the lever, not from a
+general principle of dynamics like that of Aristotle, but from
+a few simple axioms such as the experience of daily life might
+suggest. The original work may have been earlier than
+Archimedes and may have been written by a contemporary of
+Euclid. A third fragment, unearthed by Duhem from manu-
+scripts in the Biblioth&egrave;que Nationale in Paris, contains four
+propositions purporting to be &lsquo;liber Euclidis de ponderibus
+secundum terminorum circumferentiam&rsquo;. The first of the
+propositions, connecting the law of the lever with the size of
+the circles described by its ends, recalls the similar demon-
+stration in the Aristotelian <I>Mechanica</I>; the others attempt to
+give a theory of the balance, taking account of the weight of
+the lever itself, and assuming that a portion of it (regarded as
+cylindrical) may be supposed to be detached and replaced by
+an equal weight suspended from its middle point. The three
+fragments supplement each other in a curious way, and it is a
+question whether they belonged to one treatise or were due to
+different authors. In any case there seems to be no indepen-
+dent evidence that Euclid was the author of any of the
+fragments, or that he wrote on mechanics at all.<note>For further details about these mechanical fragments see P. Duhem,
+<I>Les origines de la statique,</I> 1905, esp. vol. i, pp. 61-97.</note>

Mercurial > hg > mpdl-xml-content

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