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CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 GIFT OF 
 
 A, Abraham 
 
 iViatiiematicg 
 
GREEK GEOMETRY, 
 
 FROM 
 
 THALES TO EUCLID. 
 
 BY 
 
 GEORGE JOHNSTON ALLMAN, LL.D., 
 
 PROFESSOR OF MATHEMATICS, AND MEMBER OF THE SENATE, OF THE 
 QUEEN'S UNIVERSITY IN IRELAND. 
 
 DUBLIN: 
 
 PRINTED AT THE UNIVERSITY PRESS, 
 
 BY PONSONBY AND MURPPIY. 
 
 1877. 
 
[From " Hermathena," Vo/. III., No. F.] 
 
i6o J)R. ALLMAN ON GBEEK GEOMETRY 
 
 GREEK GEOMETRY FROM THALES TO 
 EUCLID.' 
 
 IN studying the development of Greek Science, two 
 periods must be carefully distinguished. 
 
 The founders of Greek philosophy — Thales and Pytha- 
 goras — were also the founders of Greek Science, and from 
 the time of Thales to that of Euclid and the foundation of 
 the Museum of Alexandria, the development of science was, 
 for the most part, the work of the Greek philosophers. 
 With the foundation of the School of Alexandria, a second 
 period commences ; and henceforth, until the end of the 
 scientific evolution of Greece, the cultivation of science 
 was separated from that of philosophy, and pursued for 
 its own sake. 
 
 In this Paper I propose to give some account of the 
 progress of geometry during the first of these periods, and 
 
 1 It has been frequently observed,. ut:d die Geometer Vor EukUdes, Leip- 
 
 and is indeed generally admitted, that zig, 1870; Suter, H., Geschichte der 
 
 the present century is characterized by Mathematischen Wissenschaften (ist 
 
 the importance which is attached to Part), Zurich, 1873 ; *HaDkeI, H., ^«f 
 
 historical researches, and by; a widely- Geschichte der Mathematik in Alter- 
 
 difiiised taste for the philosophy of his- thum und MiUel-alter, Leipzig, 1874 
 
 'OT- (a posthumous work); *Hoefer, F., 
 
 In Mathematics, we have evidence of Histoire des Mathematiques, Paris, 
 
 these prevailing views and tastes in two 1874. (This forms the fifth volume by 
 
 distinct ways :— M. Hoefer on the history of the sciences, 
 
 1° The publication of many recent aU being parts of the Histoire Uni. 
 
 works on the history of Mathematics, verselle, published under the direction 
 
 '• S- of M. Duruy.) In studying the subject 
 
 Aiueth, A., Die Geschichte der of this Paper, I have made use of the 
 
 reinen Mathematik, Stuttgart, 1852; works marked thus*. Though the 
 
 • Bretschneider, C. A., Die Geometric work of M. Hoefer is too metaphysical, 
 
FROM THALE8 TO EUCLID. 
 
 i6i 
 
 also to notice briefly the chief organs of its develop • 
 ment. 
 
 For authorities on the early history of geometry we are 
 dependent on scattered notices in ancient writers, many of 
 which have been taken from a work which has unfortu- 
 nately been lost — the History of Geometry by Eudemus of 
 Rhodes, one of the principal pupils of Aristotle. A sum- 
 mary of the history of geometry during the whole period 
 of which I am about to treat has been preserved by Pro- 
 clus, who most probably derived it from the work of 
 Eudemus. I give it here at length, because I shall fre- 
 quently have occasion to refer to it in the following pages. 
 
 After attributing the origin of geometry to the Egyp- 
 tians, who, according to the old story, were obliged to in- 
 
 and is not free from inadvertencies and 
 even errors, yet I have derived advan- 
 tage from the part which concerns Py- 
 thagoras and his ideas. Hankel's book 
 contains some fragments of a great work 
 on the History of Mathematics, which 
 was interrupted by the death of the 
 author. The part treating of the ma- 
 thematics of the Greeks during the first 
 period — from Thales to the foundation 
 of the School of Alexandria — is fortu- 
 nately complete. This is an excellent 
 work, and is in many parts distinguished 
 by its depth and originality. 
 
 The monograph of M. Bretschneider 
 is most valuable, and is greatly in ad- 
 vance of all that preceded it on the 
 origin of geometry amongst the Greeks. 
 He has collected with great care, and 
 has set out in the original, the fragments 
 relating to it, which are scattered in 
 ancient writers ; I have derived much 
 aid from these citations. 
 
 j° New editions of ancient Mathema- 
 tical works, some of which had become 
 extremely scarce, e. g, — 
 VOL. in. 
 
 Theodosii Sphaericorum libri Tres, 
 Nizze, Berlin, 1852; Nicomachi Gera- 
 seni Introduction's Arithmeticae, lib. 
 II., Hoche, Lipsiae, 1866 (Teubner) ; 
 Boetii De Inst. Arithm., &'c., ed. G. 
 Friedlein, Lipsiae, 1867 (Teubner); 
 Procli Diadochi in primum Euclidis 
 Elementorum librum commentarii, ex 
 recog. G. Friedlein, Lipsiae, 1873 (Teu- 
 bner) ; Heronis Alexandrini Geometri- 
 corum et Stereometricorum Reliquiae 
 e librismanuscriptis^ ediditF. Hultsch, 
 Berohni, 1864 ; Pappi Alexandrini 
 Collectiones quae supersunt e libris 
 manuscriptis Latina interpretatione 
 et commentariis instnixit F. Hultsch, 
 vol. t, Berolini, 1876 : vol. 11, ib., 
 
 1877. 
 
 Occasional portions only of the Greek 
 text of Pappus had been published at 
 various times (see De Morgan in Dr. W. 
 Smith's Dictionary of Biography). An 
 Oxford edition, uniform vrith the great 
 editions of Euclid, Apollonius, and 
 Archimedes, published in the last cen- 
 turj', has been long looked for. 
 
 M 
 
1 62 BR. ALLMAN ON GREEK GEOMETRY 
 
 vent it in order to restore the landmarks which had been 
 destroyed by the inundation of the Nile, and observing 
 that it is by no means strange that the invention of the 
 sciences should have originated in practical needs, and that, 
 further, the transition from sensual perception to reflection, 
 and from that to knowledge, is to be expected, Proclus goes 
 on to say that Thales, having visited Egypt, first brought 
 this knowledge into Greece ; that he discovered many things 
 himself, and communicated the beginnings of many to his 
 successors, some of which he attempted in a more abstract 
 manner [KodoXiKwripov), and some in a more intuitional or 
 sensible manner (at(T0rjr£ica)TEjoov). After him, Ameristus [or 
 Mamercus], brother of the poet Stesichorus, is mentioned 
 as celebrated for his zeal in the study of geometry. Then 
 Pythagoras changed it into the form of a liberal science, 
 regarding its principles in a purely abstract manner, and 
 investigated its theorems from the immaterial and intellec- 
 tual point of view(a(5Awc kqi voiftw^); he also discovered the 
 theory of incommensurable quantities (rwv aXoywv vpayfia- 
 Tt'iav), and the construction of the mundane figures [the 
 regular solids]. After him, Anaxagoras of Clazomenae 
 contributed much to geometry, as also did Oenopides of 
 Chios, who was somewhat junior to Anaxagoras. After 
 these, Hippocrates of Chios, who found the quadrature of 
 the lunule, and Theodorus of Cyrene became famous in geo- 
 metry. Of those mentioned above, Hippocrates is the first 
 writer of elements. Plato, who was posterior to these, con- 
 tributed to the progress of geometry, and of the other ma- 
 thematical sciences, through his study of these subjects, and 
 through the mathematical matter introduced in his writ- 
 ings. Amongst his contemporaries were Leodamas of 
 Thasos, Arch)rtas of Tarentum, and Theaetetus of Athens, 
 by all of whom theorems were added or iplaced on a more 
 scientific basis. To Leodamas succeeded Neocleides, and 
 his pupil was Leon, who added much to what had been 
 
FROM THALES TO EUCLID. 163 
 
 done before. Leon also composed elements, which, both in 
 regard to the number and the value of the propositions 
 proved, are put together more carefully ; he also invented 
 that part of the solution of a problem called its determina- 
 tion (SioptiTjuoe) — a test for determining when the problem 
 is possible and when impossible. Eudoxus of Cnidus, a 
 little younger than Leon and a companion of Plato's 
 pupils, in the first place increased the number of general 
 theorems, added three proportions to the three already 
 existing, and also developed further the things begun by 
 Plato concerning the section,'' making use, for the pur- 
 pose, of the analytical method [raig avaXvataiv). Amyclas 
 of Heraclea, one of Plato's companions, and Menaechmus, 
 a pupil of Eudoxus and also an associate of Plato, and his 
 brother, Deinostratus, made the whole of geometry more 
 perfect. Theudius of Magnesia appears to have been dis- 
 tinguished in mathematics, as well as in other branches of 
 philosophy, for he made an excellent arrangement of the 
 elements, and generalized many particular propositions. 
 Athenaeus of Cyzicus [or Cyzicinus of Athens] about the 
 same time became famous in other mathematical studies, 
 but especially in geometry. All these frequented the 
 Academy, and made their researches in common. Her- 
 motimus of Colophon developed further what had been 
 done by Eudoxus and Theaetetus, discovered many ele- 
 mentary theorems, and wrote something on loci. Philip- 
 pus Mendaeus [Medmaeus], a pupil of Plato, and drawn by 
 him to mathematical studies, made researches under Plato's 
 direction, and occupied himself with whatever he thought 
 
 ^ Does this mean the cutting of a and synthesis are first used and de- 
 straight Jine in extreme and mean ratio, fined by him in connection with theo- 
 " sectio aurea" '} or is the reference rems relating to the cutting of a line in 
 to the invention of the conic sections ? extreme and mean ratio. See Bret- 
 Most probably the former. In Euclid'' s Schneider, Die Geometrie vor Euklides, 
 Elements, Lib., xiii., the terms analysis p. l68. 
 
 M 2 
 
1 64 DR. ALLMAN ON GREEK GEOMETRY 
 
 would advance the Platonic philosophy. Thus far those 
 who have written on the history of geometry bring the 
 development of the science.^ 
 
 Proclus goes on to say, Euclid was not much younger 
 than these ; he collected the elements, arranged much 'of 
 what Eudoxus had discovered, and completed much that 
 had been commenced by Theaetetus ; further, he substi- 
 tuted incontrovertible proofs for the lax demonstrations 
 of his predecessors. He lived in the times of the first 
 Ptolemy, by whom, it is said, he was asked whether there 
 was a shorter way to the knowledge of geometry than by 
 his Elements, to which he replied that there was no royal 
 road to geometry. Euclid then was younger than the dis- 
 ciples of Plato, but elder than Eratosthenes and Archimedes 
 — who were contemporaries — the latter of whom mentions 
 him. He was of the Platonic sect, and familiar with its 
 philosophy, whence also he proposed to himself the con- 
 struction of the so-called Platonic bodies [the regular 
 solids] as the final aim of his systematization of the Ele- 
 ments.* 
 
 I. 
 
 The first name, then, which meets us in the history of 
 Greek mathematics is that of Thales of Miletus (640- 
 546 B. c). He lived at the time when his native city, and 
 Ionia in general, were in a flourishing condition, and when 
 an active trade was carried on with Egypt. Thales himself 
 was engaged in trade, and is said to have resided in Egypt, 
 and, on his return to Miletus in his old age, to have brought 
 with him from that country the knowledge of geometry and 
 
 ^ From these words we infer that the pp. 299, 333, 352, and 379. 
 
 History of Geometry by Eudemus is * Vrocli Diadochi in primumSuclidis 
 
 most probably referred to, inasmuch as elsTnentorum librum commentarii. Ex 
 
 he lived at the time here indicated, and recognitione G.Friedlein. Lipsiae, 1873, 
 
 his history is elsewhere mentioned by pp. 64-68. 
 Proclus. — Proclus, ed. G. Friedlein, 
 
FROM THALES TO EUCLID. 165 
 
 astronomy. To the knowledge thus introduced he added the 
 capital creation of the geometry of lines, which was essen- 
 tially abstract in its character. The only geometry known 
 to the Egyptian priests was that of surfaces, together with 
 a sketch of that of solids, a geometry consisting of some 
 simple quadratures and elementary cubatures, which they 
 had obtained empirically ; Thales, on the other hand, intro- 
 duced abstract geometry, the object of which is to establish 
 precise relations between the different parts of a figure, so 
 that some of them could be found by means of others in a 
 manner strictly rigorous. This was a phenomenon quite 
 new in the world, and due, in fact, to the abstract spirit of 
 the Greeks. In connection with the new impulse given to 
 geometry, there arose with Thales, moreover, scientific 
 astronomy, also an abstract science, and undoubtedly a 
 Greek creation. The astronomy of the Greeks differs from ; 
 that of the Orientals in this respect, that the astronomy of 
 the latter, which is altogether concrete and empirical, con- 
 sisted merely in determining the duration of some periods, 
 or in indicating, by means of a mechanical process, the 
 motions of the sun and planets, whilst the astronomy of the 
 Greeks aimed at the discovery of the geometric laws of the 
 motions of the heavenly bodies.' 
 
 s Tlie importance, for the present the Greeks the discovery of truths which 
 research, of bearing in mind this ab- were known to the Egyptians. See, in 
 stract character of Greek science con- relation to the distinction between ab- 
 sists in this, that it furnishes a clue stract and concrete science, and its 
 by means of which we can, in many bearing on the history of Greek Ma- 
 cases, recognise theorems of purely thematics, amongst many passages in 
 Greek growth, and distinguish them the works of Augusta Corate, Systeme 
 from those of eastern extraction. The de Politique Positive, vol. ill., oh. iv., 
 neglect of this consideration has led p. 297, axiiseq., vol.i.,ch. i., pp. 424- 
 some recent writers on the early history 437 ; and see, also, Les Grands Types 
 of geometry greatly to exaggerate the de PHumaniti, par P. Laffitte, vol. 11., 
 obligations of the Greeks to the Orien- Le9on isieme, p. 280, and seq.—Ap- 
 tals ; whilst others have attributed to preciation de la Science Antique. 
 
1 66 BR. ALLMAN ON GREEK GEOMETRY 
 
 The following notices of the geometrical work of Thales 
 have been preserved : — 
 
 [a). He is reported to have first demonstrated that the 
 circle was bisected by its diameter ; ^ 
 
 {b). He is said first to have stated the theorem that the 
 angles at the base of every isosceles triangle are equal, " or, 
 as in archaic fashion he phrased it, like {b/iolai) ; " ' 
 
 (c). Eudemus attributes to him the theorem that when 
 two straight lines cut each other, the vertically opposite 
 angles are equal ; * 
 
 {d). Pamphila' relates that he, having learned geometry 
 from the Egyptians, was the first person to describe a right- 
 angled triangle in a circle ; others, however, of whom 
 ApoUodorus (6 XoyiariKo^) is one, say the same of Pythago- 
 ras ; 1" 
 
 [e). He* never had any teacher except during the time 
 when he went to Egypt and associated with the priests. 
 Hieronymus also says that he measured the pyramids, 
 making an observation on our shadows when they are of 
 the same length as ourselves, and applying it to the pyra- 
 mids." To the same effect Pliny — " Mensuram altitudi- 
 nis earum omniumque similium deprehendere invenit 
 Thales Milesius, umbram metiendo, qua hora par esse cor- 
 pori solet ; " " 
 
 (This is told in a different manner by Plutarch. Niloxe- 
 nus is introduced as conversing with Thales concerning 
 Amasis, King of Egypt. — " Although he [Amasis] admired 
 you [Thales] for other things, yet he particularly liked the 
 
 6 Proclus, ed. Friedlein, p. 157. ed. C. G. Cobet, p. 6. 
 
 "^ Ididj p. 250. 11 6 Se^lfpi^vvfios KaX iKfieTpTJffal tpTiffiv 
 
 Joia, p. 299- avrhv tos Trupa/ilSas iK ttjs ffKtas xapo- 
 
 " Pamphila was a female historian ttj/j^itoi'to Ste tiixTv iVo/ieye'fleis eiVf. 
 
 who lived at the time of Nero ; an Epi- Diog. Laert., I,, c. i, n. 6., ed. Cobet, 
 
 daurian according to Suidas, an Egyp- p. 6. 
 
 tian according to Photius. " Plin. Stsi. Nat., xxxvi. 17. 
 
 '"Diogenes Laertius, I,, c. I, n. 3, 
 
FROM TRALE8 TO EUCLID. 167 
 
 manner by which you measured thfe height of the p3rramid 
 without any trouble or instrument ; for, by merely placing 
 a staff at the extremity of the shadow which the pyramid 
 casts, you formed two triangles by the contact of the sun- 
 beams, and showed that the height of the p)rramid was to 
 the length of the staff in the same ratio as their respective 
 shadows").'' 
 
 {/). Proclus tells us that Thales measured the distance 
 of vessels from the shore by a geometrical process, and that 
 Eudemus, in his history of geometry, refers the theorem 
 Etccl. i. 26 to Thales, for he says that it is necessary to use 
 this theorem in determining, the distance of ships at sea 
 according to the method employed by Thales in this inves- 
 tigation ; " 
 
 [g). Proclus, or rather Eudemus, tells us in the passage 
 quoted above in extenso that Thales brought the knowledge 
 of geometry to Greece, and added many things, attempt- 
 ing some in a more abstract manner, and some in a more 
 intuitional or sensible manner." 
 
 Let us now examine what inferences as to the geometri- 
 cal knowledge of Thales can be drawn from the preceding 
 notices. 
 
 First inference. — Thales must have known the theorem 
 that the sum of the three angles of a triangle is equal to 
 two right angles. 
 
 Pamphila, in [d], refers to the discovery of the property 
 of a circle that all triangles described on a diameter as base 
 with their vertices on the circumference have their vertical 
 angles right." 
 
 " Plut. Sept. Sap. Conviv. 2.voLiii., which it has been stated by Diogenes 
 
 p. 174, ed. Didot. Laertius shows that he did not distin- 
 
 1* Proclus, ed. Friedlein, p. 352. guish between a problem and a theo- 
 
 •s Ibid, p. 65. rem ; and further, that he was ignorant 
 
 16 This is unquestionably the dis- of geometry. To this effect Proclus — 
 
 covery referred to. The manner in " When, therefore, anyone proposes to 
 
i68 DR. ALLMAN ON GREEK GEOMETRY 
 
 Assuming, then, that this theorem was known to Thales, 
 he must have known that the sum of the three angles of 
 any right-angled triangle is equal to two right angles, for, 
 if the vertex of any of these right-angled triangles be con- 
 nected with the centre of the circle, the right-angled tri- 
 angle will be resolved into two isosceles triangles, and 
 since the angles at the base of an isosceles triangle are 
 equal — a theorem attributed to Thales (b) — it follows that 
 the sum of the angles at the base of the right-angled tri- 
 angle is equal to the vertical angle, and that therefore the 
 sura of the three angles of the right-angled triangle is equal 
 to two right angles. Further, since any triangle can be 
 resolved into two right-angled triangles, it follows imme- 
 diately that the sum of the three angles of any triangle is 
 equal to two right angles. If, then, we accept the evidence 
 of Pamphila as satisfactory, we are forced to the conclusion 
 that Thales must have known this theorem. No doubt the 
 knowledge of this theorem [Euclid i., 32) is required in the 
 proof given in the elements of Euclid of the property of the 
 circle (iii., 31), the discovery of which is attributed to 
 Thales by Pamphila, and some writers have inferred hence 
 that Thales must have known the theorem (i., 32)." Al- 
 though I agree with this conclusion, for the reasons given 
 
 nscribe an equilateral triangle in a. every angle in a semicirde is necessa- 
 
 circle, he proposes a problem : for it is rily a right one." — Taylor's Proclus, 
 
 possible to inscribe one that is not Vol. I., p. no. Procl. ed. Friedlein, 
 
 equilateral. But when anyone asserts pp. 79, 80. 
 
 that the angles at the base of an isosce- Sir G. C. Lewis has subjected himself 
 
 les triangle are equal, he must affirm to the same criticism when he says — 
 
 that he proposes a theorem : for it is ' According to Pamphila, he first solved 
 
 not possible that the angles at the base the problem of inscribing a right-angled 
 
 of an isosceles triangle should be un- triangle in a circle." — G. Comewall 
 
 equal to each other. On which account "Lev/is, Historical Survey of the Astro- 
 
 if anyone, stating it as a problem, should nomy of the A ncients, p . 83 . 
 say that he wishes to inscribe a right " So F. A. Finger, De Primordiis 
 
 angle in ix semicircle, he must be con- Geometriae apud Graecos, p. 20, Heidel- 
 
 sidered as ignorant of geometry, since bergae, 1831. 
 
FROM THALE8 TO EUCLID. 169 
 
 above, yet I consider the inference founded on the demon- 
 stration given by Euclid to be inadmissible, for we are in- 
 formed by Proclus, on the authority of Eudemus, that the 
 theorem [Euclid i., 32) was first proved in a general way by 
 the Pythagoreans, and their proof, which does not differ 
 substantially from that given by Euclid, has been preserved 
 by Proclus.'* Further, Geminus states that the ancient 
 geometers observed the equality to two right angles in 
 each species of triangle separately, first in equilateral, then 
 in isosceles, and lastly in scalene triangles,^' and it is plain 
 that the geometers older than the Pythagoreans can be no 
 other than Thales and his successors in the Ionic school. 
 
 If I may be permitted to offer a conjecture, in confor- 
 mity with the notice of Geminus, as to the manner in which 
 the theorem was arrived at in the different species of tri- 
 angles, I would suggest that Thales had been led by the 
 concrete geometry of the Egyptians to contemplate floors 
 covered with tiles in the form of equilateral triangles or 
 regular hexagons,^" and had observed that six equilateral 
 triangles could be placed round a common vertex, from 
 which he saw that six such angles made up four right 
 angles, and that consequently the sum of the three angles 
 of an equilateral triangle is equal to two right angles {c). 
 The observation of a floor covered with square tiles 
 would lead to a similar conclusion with respect to the 
 isosceles right-angled tfiangle." Further, if a perpen- 
 
 '8 Proclus, ed. Friedlein, p. 379. so as to fill a space," is attributed by 
 
 '9 ApoUonii Conica, ed. Hallejus ,p. Proclus to Pythagoras or his school 
 
 9, Oxon. I710. (^(TTi t!) Bedprifia tovto Xlv8ay6f>eioi'. 
 
 2° Floors or walls covered with tiles of Proclus, ed. Friedlein, p. 305), yet it 
 
 various colours were common in Egypt. is difficult to conceive that the 'Egfpt- 
 
 Se&Vfii^nsoTi's " Ancient Egyptians " jans — who erected the pyramids — had 
 
 vol. ii., pp. 287 and 292. not a practical knowledge of the fact 
 
 21 Although the theorem that " only jjjaj (jjes of the forms above mentioned 
 three kinds of regular polygons— the could be placed so as to form a con- 
 equilateral triangle, the square and the tinuous plane surface, 
 hexagon — can be placed about a point 
 
170 DR. ALLMAN ON GREEK GEOMETRY 
 
 dicular be drawn from a vertex of an equilateral triangle 
 on the opposite side,'"' the triangle is divided into two 
 right-angled triangles, which are in every respect equal 
 to each other, hence the sum of the three angles of each of 
 these right-angled triangles is easily seen to be two right 
 angles. If now we suppose that Thales was led to examine 
 whether the property, which he had observed in two dis- 
 tinct kinds of right-angled triangles, held generally for 
 all right-angled triangles, it seems to me that, by com- 
 pleting the rectangle and drawing the second diagonal, he 
 could easily see that the diagonals are equal, that they 
 bisect each other, and that the vertical angle of the right- 
 angled triangle is equal to the sum of the base angles. 
 Further, if he constructed several right-angled triangles 
 on the same hypotenuse he could see that their vertices 
 are all equally distant from the middle point of their com- 
 mon hypotenuse, and therefore lie on the circumference 
 of a circle described on that line as diameter, which is the 
 theorem in question. It may be noticed that this remark- 
 able property of the circle, with which, in fact, abstract 
 geometry was inaugurated, struck the imagination of 
 Dante : — 
 
 " O se del mezzo cerchio far si puote 
 Triangol si, ch'un retto non avesse." 
 
 Par. c. xiii. loi. 
 
 Second inference. — The conc^tion of geometrical loci 
 is due to Thales. 
 
 We are informed by Eudemus (/) that Thales knew 
 that a triangle is determined if its base and base angles 
 are given; further, we have seen that Thales knew that, 
 
 22 Though we are infonned by Pro- the square, could not be ignorant of its 
 
 clus (ed. Friedlein, p. 283), that Oeno- mechanical solution. Observe that we 
 
 pides of Chios first solved (^f^TTjirei') are expressly told by Proclus that Thales 
 
 this problem, yet Thales, and indeed attempted some things in an intuitional 
 
 the Egyptians, who were furnished with or sensible manner. 
 
FROM THALES TO EUCLID. 171 
 
 if the base is given, and the base angles not given sepa- 
 rately, but their sum known to be a right angle, then there 
 could be described an unlimited number of triangles 
 satisfying the conditions of the question, and that their 
 vertices all lie on the circumference of a circle described 
 on the base as diameter. Hence it is manifest that the 
 important conception oi geometrical loci, which is attributed 
 by Montucla, and after him by Chasles and other writers 
 on the History of Mathematics, to the School of Plato," 
 had been formed by Thales. 
 
 Third inference. — Thales discovered the theorem that 
 the sides of equiangular triangles are proportional. 
 
 The knowledge of this theorem is distinctly attributed 
 to Thales by Plutarch in a passage quoted above [e). On 
 the other hand, Hieronymus of Rhodes, a pupil of Aris- 
 totle, according to the testimony of Diogenes Laertius," 
 says that Thales measured the height of the pyramids by 
 watching when bodies cast shadows of their own length, 
 and to the same effect Pliny in the passage quoted above [e). 
 Bretschneider thinks that Plutarch has spun out the story 
 told by Hieronymus, attributing to Thales the knowledge 
 of his own times, denies to Thales the knowledge of the 
 theorem in question, and says that there is no trace of any 
 theorems concerning similarity before Pythagoras." He 
 says further, that the Egyptians were altogether ignorant 
 of the doctrine of the similarity of figures, that we do not 
 find amongst them any trace of the doctrine of proportion, 
 and that Greek writers say that this part of their mathe- 
 
 ^ Montucla, Histoire des Mathhna- " ce chef du Lycie." 
 
 tiques. Tome i., p. 183, Paris, 1758. 24 But we have seen that the account 
 
 Chasles, Aperfu Historique des Mitho- given by Diogenes Laertius of the dis- 
 
 des en Giometrie, p. 5, BraxeUes, 1837. covery of Thales mentioned by Pam- 
 
 Chasles in the history of geometry be- phila is unintelligible and evinces 
 
 fore Euclid copies Montucla, and we ignorance of geometry on his part, 
 
 have a remarkable instance of this here, ^ Bretsch. Die Geometrie und Geo- 
 
 for Chasles, after Montucla, calls Plato meter vor Euklides, pp. 45, 46.; 
 
172 DR. ALLMAN ON GREEK GEOMETRY 
 
 matical knowledge was derived from the Babylonians or 
 Chaldaeans.^* Bretschneider also endeavours to show that 
 Thales could have obtained the solution of the second 
 practical problem — the determination of the distance of a 
 ship from the shore — by geometrical construction, a method 
 long before known to the Egyptians.*' Now, as Bretschnei- 
 der denies to the Egyptians and to Thales any knowledge 
 of the doctrine of proportion, it was plainly necessary, on 
 this supposition, that Thales should find a sufficient extent 
 of free and level ground on which to construct a triangle 
 of the same dimensions as that he wished to measure ; and 
 even if he could have found such ground, the great length 
 of the sides would have rendered the operations very diffi- 
 cult."* It is much simpler to accept the testimony of 
 Plutarch, and suppose that the method of superseding such 
 operations by using similar triangles is due to Thales. 
 
 If Thales had employed a right-angled triangle,'" he 
 could have solved this problem by the same principle which, 
 we are told by Plutarch, he used in measuring the height 
 of the pyramid, the only difference being that the right- 
 
 ^^ /bid, p. 1 8. abaisser une perpendiciilaire sur une- 
 
 " Ibid, pp. 43, 44. ligne du point qui en est eloigne seule- 
 
 28 In reference to this I may quote ment de 500 toises, ce seroit un ouvrage 
 
 the following passage from Clairaut, extremement penible, et peut-etre im- 
 
 Elemens de Geomitrie, pp. 34-35. practicable. II importe done d'avoir 
 
 Paris, 1 74 1. un moyen qui supplee a ces grandes 
 
 " La methode qu'ou vient de don- operations. Ce moyen s' ofire comme 
 
 ner pour mesurer les terrains, dans de lui-meme. II vient, &c." 
 
 lesquels on ne s9auroit tirer de lignes, ,9 Observe that the inventions of the 
 
 fait souvent nattre de grandes difficultes sq^^re and level are attributed by Pliny 
 
 dans la pratique. On trouve rarement (^jsfat. Hist., vii., 57) to Theodoras of 
 
 unespaceunietlibre.assez grand pour gamos, who was a contemporary of 
 
 fah-e des triangles egaux S ceux du ter- Thales. They were, however, known 
 
 rain dont on cherche la mesure. Et long before this period to the Egyptians; 
 
 raeme quand on en trouveroit, la grande ^q that to Theodorus is due at most the 
 
 longueur des c6t6s des triangles pour- honour of having introduced tKem into 
 
 roit rendre les operations tres-difliciles : Greece. 
 
FROM T HALE 8 TO EUCLID. 173 
 
 angled triangle is in one case in a vertical, and in the 
 other in a horizontal plane. 
 
 From what has been said it is plain that there is a 
 natural connection between the several theorems attributed 
 to Thales, and that the two practical applications which 
 he made of his geometrical knowledge are also connected 
 with each other. 
 
 Let us now proceed to consider the importance of the 
 work of Thales : — 
 
 I. In a scientific point of view : — 
 
 {a). We see, in the first place, that by his two theorems 
 he founded the geometry of lines, which has ever since 
 remained the principal part of geometry.^" 
 
 Vainly do some recent writers refer these geometrical 
 discoveries of Thales to the Egyptians ; in doing so they 
 ignore the distinction between the geometry of lines, which 
 we owe to the genius of the Greeks, and that of areas and 
 volumes — the only geometry known, and that empirically, 
 to the ancient priesthoods. This view is confirmed by an 
 ancient papyrus, that of Rhind," which is now in the 
 British Museum. It contains a complete applied mathe- 
 matics, in which the measurement of figures and solids 
 plays the principal part ; there are no -theorems properly so 
 called ; everything is stated in the form of problems, not 
 in general terms but in distinct numbers, e.g. — to measure 
 a rectangle the sides of which contain two and ten units of 
 length ; to find the surface of a circular area whose diame- 
 ter is six units ; to mark out in a field a right-angled triangle 
 
 30 Auguste Comte, Syst^me de Poli- thematiques, p. 69. Since this Paper 
 
 tique Positive, vol. iii., p. 297. was sent to the press, Dr. August 
 
 " Birch, in Lepsius' Zeitschrift fiir Eisenlohr, of Heidelberg, has published 
 
 Aegyptische Sprache und Alterthums- this papyrus with a translation and 
 
 ]iunde (year 1868, p. 108). Bret- commentary under the title " £zn il/a- 
 
 schneider, Geometrie vor Euilides, thematisches Handbuch der alten 
 
 p. 16. F. Hoefer, Ilistoire des Ma- ^gypter." 
 
174 I>R- ALLMAN ON GREEK GEOMETRY 
 
 whose sides measure ten and four units ; to describe a 
 trapezium whose parallel sides are six and four units, and 
 each of the other sides twenty units. We find also in it 
 indications for the measurement of solids, particularly of 
 pyramids, whole and truncated. 
 
 It appears from the above that the Egyptians had 
 made great progress in practical geometry. Of their pro- 
 ficiency and skill in geometrical constructions we have 
 also the direct testimony of the ancients ; for example, 
 Democritus says : " No one has ever excelled me in the 
 construction of lines according to cei-tain indications — not 
 even the so-called Egyptian Harpedonaptae." ^* 
 
 [b). Thales may, in the second place, be fairly con- 
 sidered to have laid the foundation of Algebra, for his first 
 theorem establishes an equation in the true sense of the 
 word, while the second institutes a proportion.'' 
 
 II. In a philosophic point of view : — 
 
 We see that in these two theorems of Thales the first 
 type of a natural law — i. e., the expression of a fixed de- 
 pendence between difl^erent quantities, or, in another form, 
 the disentanglement of constancy in the midst of variety — 
 has decisively arisen." 
 
 III. Lastly, in a practical point of view : — 
 
 Thales furnished the first example of an application of 
 theoretical geometry to practice,'* and laid the foundation 
 of an important branch of the same — the measurement of 
 heights and distances. 
 
 I have now pointed out the importance of the geome- 
 trical discoveries of Thales, and attempted to appreciate 
 his work. His successors of the Ionic School followed 
 
 ^ Mullach, Fragmenta Philosopho- Pos. vol. iii., p. 300). 
 
 »-K«(?racc£i?-aOT, p. 37l,Democritusap. 34 p_ Laffitte, Z^j Grands Types de 
 
 Clem. Alex. Strom. I. p. 357, ed. Pot- ^' Humanity, vol. ii., p. 292. 
 
 ter. '5 ]l)id, p. 294. 
 
 33 Auguste Comte {Systhme de Pol. 
 
FROM THALES TO EUCLID. 175 
 
 him in other lines of thought, and were, for the most part, 
 occupied with physical theories on the nature of the 
 universe — speculations which have their representatives at 
 the present time — and added little or nothing to the de- 
 velopment of science, except in astronomy. The further 
 progress of geometry was certainly not due to them. 
 
 "Without_ doubt Anaxagoras of Clazomenae, one of the 
 latest representatives of this School, is said to have been 
 occupied during his exile with the problem of the qua- 
 drature of the circle, but this was in his old age, and after 
 the works of another School — to which the early progress 
 of geometry was really due — had become the common 
 property of the Hellenic race. I refer to the immortal 
 School of Pythagoras. 
 
 II. 
 
 About the middle of the sixth century before the Chris- 
 tian era, a great change had taken place : Ionia, no longer 
 free and prosperous, had fallen under the yoke, first of Lydia, 
 then of Persia, and the very name Ionian — the name by 
 which the Greeks were known in the whole East — had 
 become a reproach, and was shunned by their kinsmen on 
 the other side of the Aegean.'^ On the other hand, Athens 
 and Sparta had not become pre-eminent ; the days of Ma- 
 rathon and Salamis were yet to come. Meanwhile the 
 glory of the Hellenic name was maintained chiefly by the 
 Italic Greeks, who were then in the height of their pros- 
 perity, and had recently obtained for their territory the 
 well-earned appellation of 11 juEyaXij 'EXXac-" It should be 
 noted, too, that at this period there was great commercial 
 intercourse between the Hellenic cities of Italy and Asia ; 
 and further, that some of them, as Sybaris and Miletus on 
 the one hand, and Tarentum and Cnidus on the other, were 
 
 36 Herodotus, i. 143. i-> P- 14'- '844- 
 
 3' Polybius, ii., 39 ; ed. Bekker, vol. 
 
176 DR. ALLMAN ON GREEK GEOMETRY 
 
 bound by ties of the most intimate character.^' It is not 
 surprising, then, that after the Persian conquest of Ionia, 
 Pythagoras, Xenophanes, and others, left their native 
 country, and, following the current of civilization, removed 
 to Magna Graecia. 
 
 As the introduction of geometry into Greece is by com- 
 mon consent attributed to Thales, so alP' are agreed that 
 to Pythagoras of Samos, the second of the great philoso- 
 phers of Greece, and founder of the Italic School, is due 
 the honour of having raised mathematics to the rank of a 
 science. 
 
 The statements of ancient writers concerning this great 
 man are most conflicting, and all that relates to him per- 
 sonally is involved in obscurity; for example, the dates 
 given for his birth vary within the limits of eighty-four 
 years — 43rd to 64th Olympiad." It seems desirable, how- 
 ever, if for no other reason than to fix our ideas, that we 
 should adopt some definite date for the birth of Pythagoras ; 
 and there is an additional reason for doing so, inasmuch as 
 some writers, by neglecting this, have become confused, 
 and fallen into inconsistencies in the notices which they 
 have given of his life. Of the various dates which have 
 been assigned for the birth of Pythagoras, the one which 
 seems to me to harmonise best with the records of the most 
 trustworthy writers is that given by Ritter, and adopted by 
 Grote, Brandis, Ueberweg, and Hankel, namely, about 
 580 B. c. (49th Olymp.) This date would accord with the 
 following statements : — 
 
 That Pythagoras had personal relations with Thales, 
 then old, of whom he was regarded by all antiquity as the 
 
 '8 Herod., vi. 21, and iii. 138. History of Philosophy, Book ii., c. ii., 
 
 '' Aristotle, Diogenes Laertius, Pro- where the various dates given by 
 
 clus, amongst others. scholars are cited. 
 *" See G. H. Lewes, Biographical 
 
FROM THALES TO EUCLID. 177 
 
 successor, and by whom he was incited to visit Egypt," — 
 mother of all the civilization of the West ; 
 
 That he left his country being still a young man, and, 
 on this supposition as to the date of his birth, in the early 
 years of the reign of Croesus (560-546 B. c), when Ionia 
 was still free ; 
 
 That he resided in Egypt many years, so that he learned 
 the Egyptian language, and became imbued with the philo- 
 sophy of the priests of the country ;" 
 
 That he probably visited Crete and Tyre, and may have 
 even extended his journeys to Babylon, at that time Chal- 
 daean and free ; 
 
 That on his return to Samos, finding his country under 
 the tyranny of Polycrates," and Ionia under the dominion 
 of the Persians, he migrated to Italy in the early years of 
 Tarquinius Superbus ; ** 
 
 And that he founded his Brotherhood at Crotona, where 
 for the space of twenty years or more he lived and taught, 
 being held in the highest estimation, and even looked on 
 almost as divine by the population — native as well as Hel- 
 lenic; and then, soon after the destruction of Sybaris 
 (510 B. c), being banished by a democratic party under 
 Cylon, he removed to Metapontum, where he died soon 
 afterwards. 
 
 All who have treated of Pythagoras and the P)rthago- 
 reans have experienced great difficulties. These difficulties 
 are due partly to the circumstance that the reports of the 
 earlier and most reliable authorities have for the most part 
 been lost, while those which have come down to us are not 
 always consistent with each other. On the other hand, we 
 have pretty full accounts fi-om later writers, especially those 
 
 "" lamUichus, de Vita Fyth., c. ii., 12. ap. Porphyr., de Vita Pytk., 9. 
 *- Isocrates is the oldest authority for " Cicero, de Rep. 11., 15 ; Tusc. Disp. , 
 
 this, Busiris, c. li. I., xvi., 38. 
 
 " Diog. Laert., viii. 3 ; Aristoxenus, 
 VOL. III. N 
 
1 78 DR. ALLMAN ON GREEK GEOMETRY 
 
 of the Neo-P)^hagorean School ; but these notices, which 
 are mixed up with fables, were written with a particular 
 object in view, and are in general highly coloured; they 
 are particularly to be suspected, as Zeller has remarked, 
 because the notices are fuller and more circumstantial the 
 greater the interval from Pythagoras. Some recent authors, 
 therefore, even go to the length of omitting from their ac- 
 count of the Pythagoreans everything which depends solely 
 on the evidence of the Neo-Pythagoreans. In doing so, 
 these authors no doubt effect a simplification, but it seems 
 to me that they are not justified in this proceeding, as the 
 Neo-P)rthagdreans had access to ancient and reliable au- 
 thorities which have unfortunately been lost since.*^ 
 
 Though the difB.culties to which I refer have been felt 
 chiefly by those who have treated of the Pythagorean phi- 
 losophy, yet we cannot, in the present inquiry, altogether 
 escape from them ; for, in the first place, there was, in the 
 whole period of which we treat, an intimate connection 
 between the growth of philosophy and that of science, each 
 re-acting on the other; and, further, this was particularly 
 the case in the School of Pythagoras, owing to the fact, 
 that whilst on the one hand he united the study of geo- 
 metry with that of arithmetic, on the other he made num- 
 bers the base of his philosophical system, as well physical 
 as metaphysical. 
 
 It is to be observed, too, that the early Pythagoreans 
 published nothing, and that, moreover, with a noble self- 
 denial, they referred back to their master all their discover- 
 ies. Hence, it is not possible to separate what was done 
 by him from what was done by his early disciples, and we 
 
 *^ For example, the History of Geo- of whom lived in the reign of Justinian. 
 
 metry, by Eudemus of Rhodes, one of Eudemus also wrote a History of Astro- 
 
 the principal pupils of Aristotle, is nomy. Theophrastus, too, Aristotle's 
 
 quoted by Theon of Smyrna, Proclus, successor, wrote Histories of Arithme- 
 
 Simplicius, and Eutocius, the last two tic, Geometry, and Astronomy. 
 
FROM TEALE8 TO EUCLID. 179 
 
 are under the necessity, therefore, of treating the work of 
 the early Pythagorean School as a whole." 
 
 All ag^ee, as was stated above, that Pythagoras first 
 raised mathematics to the rank of a science, and that we 
 ■owe to him two new branches — arithmetic and music. 
 
 We have the following statements on the subject : — 
 
 [a). In the age of these philosophers [the Eleats and 
 Atomists], and even before them, lived those called Pytha- 
 goreans, who first applied themselves to mathematics, a 
 science they improved : and, penetrated with it, they fancied 
 that the principles of mathematics were the principles of all 
 things ; " 
 
 [b.) Eudemus informs us, in the passage quoted above ift 
 cxtenso, that P5rthagoras changed geometry into the form 
 -of a liberal science, regarding its principles in a purely 
 abstract manner, and investigated his theorems from the 
 immaterial and intellectual point of view ; and that he also 
 discovered the theory of irrational qualities, and the con- 
 struction of the mundane figures [the five regular solids] ; " 
 
 (c.) It was Pythagoras, also, who carried geometry to 
 perfection, after Moeris " had first found out the principles 
 of the elements of that science, as Anticlides tells us in 
 the second book of his History of Alexander ; and the part 
 
 46 " Pythagoras and his earliest sue- laus." — Smith's Dictionary, in v. Phi- 
 
 cessors do not appear to have commit- lolaus. Philolaus was bom at Cro- 
 
 ted any of their doctrines to writing. tona, or Tarentum, and was a contem- 
 
 According to Porphyrins {de Vita. Pyth. porary of Socrates and Democritus. 
 
 p. 40), Lysis and Archippus collectedin See Diog. Laert. in Vita Pythag., viii., 
 
 ,a written form some of the principal i., 15 ; in Vita Empedodis, viii., ii., 2 ; 
 
 Pythagorean doctrines, which were and in Vita Democriti, ix., vii., 6. 
 
 handed down as heirlooms in their See also lamblichus, de Vita Pythag., 
 
 families, under strict injunctions that c. 18, s. 88. 
 
 they should not be made public. But " Aristot. Met., i., 5, 985, N. 23, 
 
 amid the different and inconsistent ed. Bekker. 
 
 accounts of the matter, the first publi- « Procl. Comm.,^A. Friedlein, p. 65. 
 
 nation of the Pythagorean doctrines is *" An ancient King of Egypt, who 
 
 pretty uniformly attributed to Philo- reigned 900 years before Herodotus. 
 
 N 2 
 
i8o BE. ALLMAN ON GREEK GEOMETRY 
 
 of the science to which Pythagoras applied himself above 
 all others was arithmetic ; *° 
 
 {d.) P)rthagoras seems to have esteemed arithmetic above 
 everything, and to have advanced it by diverting it from the 
 service of commerce, and likening all things to numbers ; ^' 
 
 (c.) He was the first person who introduced measures 
 and weights among the Greeks, as Aristoxenus the musi- 
 cian informs us ; ''^ 
 
 (/.) He discovered the numerical relations of the musical 
 scale ; ^ 
 
 [g.) The word mathematics originated with the Pytha- 
 goreans ; ^' 
 
 [h.) The Pythagoreans made a four-fold division of 
 mathematical science, attributing one of its parts to the 
 how many, to iruaov, and the other to the how much, t» 
 TrrjXtKov ; and they assigned to each of these parts a two- 
 fold division. Discrete quantity, or the how many, either 
 subsists by itself, or must be considered with relation to 
 some other ; and continued quantity, or the how much, is 
 either stable or in motion. Hence arithmetic contem- 
 plates that discrete quantity which subsists by itself, but 
 music that which is related to another; and geometry con- 
 siders continued quantity so far as it is immovable ; but 
 astronomy (ttjv a^aipiKnv) contemplates continued quantity 
 so far as it is of a self-motive nature ; " 
 
 {t.) Favorinus says that he employed definitions on 
 
 50 Diog. Laert., viii. ii, ed. Cobet, eipetv. Diog. Laert., viii., ii, ed. 
 
 p. 207. Cobet, p. 207. 
 
 *' Aristoxenus, Fragm. ap. Stob. ** Procli Comm., Friedlein, p. 45. 
 
 Eclog. Phys., I., ii., 6 ; ed. Heeren, ^ Procli Comm., ed. Friedlein, p. 
 
 vol. I., p. 17. 35. As to the distinction between rh 
 
 5s Diog. Laert., viii., 13, ed. Cobet, mfXiKov, continuous, and ri iiaaiv, 
 
 p. 208. discrete, quantity, see Iambi., in Nic. 
 
 '' -7611 Te Kav6va. rhv ix /iias xop^V^ ^- Arithm. introd. ed. Ten., p. 148. 
 
FROM THALES TO EUCLID. i8i 
 
 account of the mathematical subjects to which he applied 
 himself (opoic Xp{)aatjBat Sia Trjg jua&qjuariKqc wXiJc)-'" 
 
 As to the particular work done by this school in geo- 
 metry, the following statements have been handed down 
 to us : — 
 
 (a.). The Pythagoreans define a point as unity having 
 position (fjiovaSa wpoaXaliovaav diatv) ; " 
 
 (3.) They considered a point as analogous to the monad, 
 a line to the duad, a superficies to the triad, and a body to 
 the tetrad ; " 
 
 (c.) The plane around a point is completely filled by six 
 ■equilateral triangles, four squares, or three regular hexa- 
 gons : this is a Pythagorean theorem ; " 
 
 (d.) The peripatetic Eudemus ascribes to the Pythago- 
 reans the discovery of the theorem that the interior angles 
 of a triangle are equal to two right angles [Eucl. i. 32), and 
 states their method of proving it, which was substantially 
 the same as that of Euclid ; °° 
 
 (£.) Proclus informs us in his commentary on Euclid, 
 i., 44, that Eudemus says that the problems concerning the 
 application of areas — in which the term application is not 
 to be taken in its restricted sense (ira/oa/3oXjj) in which it 
 is used in this proposition, but also in its wider significa- 
 tion, embracing uirtpjSoXij and thXio\it.g, in which it is used in 
 the 28th and 29th propositions of the Sixth Book, — are old, 
 and inventions of the Pythagoreans ; *' 
 
 5* Diog. Laert., viii., 25, ed. Cobet, and defect of areas are ancient, and are 
 
 p. 215. due to the Pythagoreans. Modems bor- 
 
 " Procli Comtn. ed. Friedlein, p. 95. rowing these names transferred them to 
 
 S8 Ibid., p. 97. the so-called conic lines — the parabola, 
 
 *' Ibid., p. 305. the hyperbola, the ellipse ; as the older 
 
 ™ Ibid., p. 379. school in their nomenclature concerning 
 
 *' Ibid., p. 419. The words of Pro- the description of areas in piano on a 
 
 <:lus are interesting : — finite right line regarded the terms 
 
 "According to Eudemus, the inven- thus : — 
 Hons respecting the application, excess, " An area is said to be applied (xopa 
 
i82 BR. ALLMAN ON GREEK GEOMETRY 
 
 {/.) This is to some extent confirmed by Plutarch, who 
 says that Pythagoras sacrificed an ox on account of the 
 geometrical diagram, as Apollodotus [-rus] says : — 
 
 'HviKa TlvOayopr)': to ^cpiKXees cvpero ypdix/JLO., 
 Keiv' icj) OTO) XaiJLTrpyjv ^ytro PovOv<TLriv, 
 
 either the one relating to the hypotenuse— namely, that the- 
 square on it is equal to the sum of the squares on the 
 sides — or that relating to the problem concerning the ap- 
 plication of areas (tiVe irpo^Xrifia trtgl tov \(i)qiov t^c Trapu- 
 
 {g.) One of the most elegant {yiwutrpiKtoTaroic;) theorems,, 
 or rather problems, is to construct a figure equal to one 
 and similar to another given figure, for the solution of 
 which also they say that Pythagoras offered a sacrifice : 
 and indeed it is finer and more elegant than the theorem 
 which shows that the square on the hypotenuse is equal 
 to the sum of the squares on the sides ; *' 
 
 [k.) Eudemus, in the passage already quoted from Pro- 
 clus, says Pythagoras discovered the construction of the 
 regular solids ; " 
 
 0d\\eiy) to a given right line when an rendering irep! tov xi'piov Tijs itapaPoKrjs: 
 
 area equal in content to some given one " concerning the area of the parabola," 
 
 is described thereon; but when the base have ascribed to Pythagoras the qua- 
 
 of the area is greater than the given drature of the parabola — which was in 
 
 line, then the area is said to be in ex- fact o^^ of the great discoveries of Ar- 
 
 cess {intpPd^Xuv) ; but when the base chimedes ; and this, though Archimedes 
 
 is less, so that some part of the given himself tells us that no one before him 
 
 line lies without the described area, had considered thequestion; and though 
 
 then the area is said to be in defect further he gives in his letter to Dosi- 
 
 {iWflvfiv). Euclid uses in this way, in *eus the history of his discovery, 
 
 his Sixth Book, the terms excess and which, as is well known, was first ob- 
 
 defect. . . . The term application tained from mechanical considerations,. 
 
 {trafafiiKKioi), which we owe to the and then by geometrical reasonings. 
 
 Pythagoreans, has this signification." " Plutarch, Symp., viii., Quaestio 2, 
 
 '^ Plutarch, non posse suaviter vivi c. 4. Plut. Opera, ed. Didot, vol. iv.,. 
 
 sec. Epicurum, c. xi. ; Pint., Opera, ed. p. 877. 
 
 Didot, vol. iv., p. 1338. Some authors, " Procl. Comm., ed. Friedlein, p. 65. 
 
FROM THALE8 TO EUCLID. i8 
 
 o 
 
 [t.) But particularly as to Hippasus, who was a Pytha- 
 gorean, they say that he perished in the sea on account 
 of his impiety, inasmuch as he boasted that he first 
 divulged the knowledge of the sphere with the twelve 
 pentagons [the ordinate dodecahedron inscribed in the 
 sphere] : Hippasus assumed the glory of the discovery to 
 himself, whereas everything belonged to Him — for thus 
 they designate Pythagoras, and do not call him by 
 name ;" 
 
 (/.) The triple interwoven triangle or Pentagram — star- 
 shaped regular pentagon — ^was used as a symbol or sign of 
 recognition by the Pythagoreans, and was called by them 
 Health (iyuia) ; " 
 
 [k.) The discovery of the law of the three squares [Eucl. 
 I., 47), commonly called the Theorem, of Pythagoras, is 
 attributed to him by — amongst others — Vitruvius, " Dio- 
 genes Laertius,^' Proclus,*' and Plutarch [/}. Plutarch, 
 however, attributes to the Egyptians the knowledge of this 
 theorem in the particular case where the sides are 3, 4, 
 and 5 T" 
 
 (/.) One of the methods of finding right-angled tri- 
 angles whose sides can be expressed in numbers — that 
 
 ** Iambi., de Vit. Pyth., c. i8, a. 88. racelsus it was regarded by him as the 
 •^ Scholiast on Aristophanes, Nub. symbol of health. See Chasles, Histoire 
 6i I ; also Lucian, pro Lapsu in Sa- de Geometrie, pp. 477 et seqq. 
 Int., s. 5. That the Pythagoreans used •" De Arch., vs.., Praef. 5, 6, and 7. 
 such symbols we learn from lamblichus *' Where the same couplet from 
 (de Vit. Pyth., c. 33, ss. 237 and 238). Apollodorus as that in (/) is found, 
 This figure is referred to Pythagoras except that kKuv^v ^yoyt occurs in 
 himself, and in the middle ages was place of \aju.irf>^v jj^cTo. Diog. Laeit., 
 called Pythagorae figura. It is said to viii., 11, p.,'207, ed. Cobet. 
 have obtained its special name from his "Procli Ciwim., p. 426, ed. Fried- 
 having written the letters v, 7, 1, 9 {= et), lein. 
 
 o, at its prominent vertices. We learn '" De Is. et Osir., u. 56. Plut. Op., 
 
 from Kepler (Opera Omnia, ed. Frisch, vol. iii.,' p. 457, Didot. 
 vol. v., p. 122) that even so late as Pa- 
 
1 84 DR. ALLMAN ON GREEK GEOMETRY 
 
 setting out from the odd numbers — is attributed to Pytha- 
 goras ;" 
 
 {m.) The discovery of irrational quantities is ascribed 
 to Pythagoras by Eudemus in the passage quoted above 
 from Proclus ;" 
 
 [n.) The three proportions — arithmetical, geometrical, 
 and harmonical, were known to Pythagoras ;" 
 
 [o.) Formerly, in the time of Pythagoras and the mathe- 
 maticians under him, there were three means only — the 
 arithmetical, the geometrical, and the third in order which 
 was known by the name v-irivavTia, but which Archytas and 
 Hippasus designated the harmonical, since it appeared 
 to include the ratios concerning harmony and melody 
 {fitTaKXr\Qtiaa on tovq Kara to apfioafiivov koI ififjiiXlg tj)aivtTO 
 Aoyoue TTf/otlxouffa) ;'* 
 
 (p.) With reference to the means corresponding to these 
 proportions, lamblichus says : " — We must now speak of 
 the most perfect proportion, consisting of four terms, and 
 properly called the musical, for it clearly contains the 
 musical ratios of harmonical symphonies. It is said to be 
 an invention of the Babylonians, and to have been brought 
 first into Greece by Pythagoras ;" 
 
 "1 Procli Comm., ed. Friedlein, p. with the numbers themselves. (Nicom. 
 428 ; Heronis Alex., Geom. et Ster. Instit. Arithm. ed. Ast. p. 153, and 
 Rel., ed. F. Hultsch, pp. 56, 146. Animad., p. 329 ; see, also, Iambi., in 
 ■"2 Procli CoTO«., ed. Friedlein, p. 65. Nicom. Arithm. ed. Ten., pp. 172 et 
 '' Nicom. G. Introd. Ar. c. xxii., ed. seq.) 
 R. Hoche, p. 122. Hankel, commenting on this pas- 
 " lamblichus in Nicomachi Arith- sage of lamblichus, says : " What we 
 meticam a S. Tennulio, p. 141. are to do with the report, that this 
 " Ibid., p. 168. proportion was known to the Baby- 
 's Hid., p. 168. As an example of lonians, and only brought into Greece 
 this proportion, Nicomachus gives the by Pythagoras, must be left to the 
 numbers 6,8, 9, 12, the harmonical and judgment of the reader." — Geschichte 
 arithmetical means between two num- der Matkematik, p. 105. In another 
 bers forming a geometrical proportion part of his book,, however, after refer- 
 
FROM THALES TO EUCLID. 185 
 
 (q.) The doctrine of arithmetical progressions is attribu- 
 ted to Pythagoras ;" 
 
 [r.) It would appear that he had considered the special 
 case oi triangular numbers. Thus Lucian :— IlYe. Eit' l-aX 
 Towr£ot(Ttv apidftUtv. AT. OiSa icai vvv apiO/jiiiv. IIYG. Ilfcic 
 apiOfiisig ; AT. "Ev, Suo, rpia, rirrapa. IlYe. 'Opag ; a av Bo- 
 Kieig TiTTapa, ravra SIko iari koi rpiyuvov kvrtXig Koi riptTipov 
 
 " 7ft 
 
 ■OpKlOV. 
 
 {s.) Another of his doctrines was, that of all solid 
 figures the sphere was the most beautiful ; and of all plane 
 figiures, the circle.'* 
 
 {t.) Also lamblichus, in his commentary on the Catego- 
 ries of Aristotle, says that Aristotle may perhaps not have 
 squared the circle ; but that the Pythagoreans had done so, 
 as is evident, he adds, from the demonstrations of the Py- 
 thagorean Sextos who had got by tradition the manner of 
 proof.*" 
 
 On examining the purely geometrical work of Pythago- 
 ras and his early disciples, we observe that it is much 
 concerned with the geometry of areas, and we are indeed 
 struck with its Egyptian character. This appears in the 
 theorem [c] concerning the filling up a plane by regular 
 polygons, as already noted; in the construction of the 
 regular solids {/i) — for some of them are found in the Egyp- 
 tian architecture ; in the problems concerning the applica- 
 tion of areas [e) ; and lastly, in the law of the three 
 
 ling to two authentic documents of the '■■ Theologumena Arithmetica,p. 153, 
 
 Babylonians which have come down to ed. F. Ast, Lipsiae, 1817. 
 
 us, he says: "We cannot, therefore, '* Lucian, Biaiv rpaais, 4, voL i., 
 
 •doubt that the Babylonians occupied p. 3 1 7, ed. C. Jacobitz. 
 
 themselves with such progressions " Kal r&v (rxwcfToii' rh koAAio'toi' 
 
 [arithmetical and geometrical] ; and a ff(pa7pav ^ivat rav trreptav kvk\ov, 
 
 Greek notice that they knew propor- Diog. Laert., in Vita Pyth., viii., 19. 
 
 tions, nay, even invented, the so-called so Simplicius, Comment., See, ap. 
 
 perfect or musical proportion, gains Bretsch., ZHe Geometrie vor Suilides, 
 
 thereby in value." — Ibid., p. 67. p, 108. 
 
i86 DR. ALLMAN ON GREEK GEOMETRY 
 
 squares [k), coupled with the rule given by Pythagoras 
 for the construction of right-angled triangles in num- 
 bers (/). 
 
 According to Plutarch, the Egyptians knew that a tri- 
 angle whose sides consist of 3, 4, and 5 parts, must be 
 right-angled. "The Egyptians may perhaps have ima- 
 gined the nature of the universe like the most beautiful 
 triangle, as also Plato appears to have made use of it in 
 his work on the State, where he sketches the picture of 
 matrimony. That triangle contains one of the perpendicu- 
 lars of 3, the base of 4, and the hypotenuse of 5 parts, the 
 square of which is equal to those of the contairiing sides. 
 The perpendicular may be regarded as the male, the base 
 as the female, the hypotenuse as the offspring of both, and 
 thus Osiris as the originating principle {apxH), Isis as the 
 receptive principle [viroZoxv), and Horus as the product 
 {aitOTiXi<Jna):' ^"^ 
 
 This passage is remarkable, and seems to indicate the 
 way in which the knowledge of the useful geometrical 
 fact enunciated in it may have been arrived at by the 
 Egyptians. The contemplation of a draught-board, or of 
 a floor covered with square tiles, or of a wall ruled with 
 squares," would at once show that the square constructed 
 on the diagonal of a square is equal to the sum of the 
 squares constructed on the sides — each containing four ot 
 the right-angled isosceles triangles into which one of the 
 squares is divided by its diagonal. 
 
 Although this observation would not serve them for 
 practical uses, on account of the impossibility of presenting 
 it arithmetically, yet it must have shown the possibility of 
 
 80 a Plutarch, De Is. et Osir. t. 56, rately with squares before the figures, 
 
 vol. iii., p. 4i7, ed. Didot were introduced. See Wilkinson's 
 
 8' It was the custom of the Egyp- Ancient Egyptians, vol. ii., pp. 265, 
 
 tians, where a subject was to be drawn, 267. 
 to rule the walls of the building accu- 
 
FROM TEALES TO ETTCLIB. 
 
 187 
 
 constructing a square which would be the sum of two 
 squares, and encouraged them to attempt the solution of the 
 problem numerically. Now, the Egyptians, with whom 
 speculations concerning generation were in vogue, could 
 scarcely fail to have perceived, from the observation of a 
 chequered board, that the element in the successive forma- 
 tion of squares is the gnomon (yvw/xeuv),*'' or common car- 
 penter's square, which was known to them.'' It remained 
 then for them only to examine whether some particular 
 gnomon might not be metamorphosed into a square, and, 
 therefore, vice versa. The solution would then be easy, 
 being furnished at once from the contemplation of a floor 
 or board composed of squares. 
 
 Each gnomon consists of an odd number of squares, 
 and the successive gnomons correspond to the successive 
 
 82 Tv^iuav means that by which any- 
 thing is known, or criterion ; its oldest 
 concrete signification seems to be the 
 carpenter's square (norma), by which 
 a right angle is known. Hence, it 
 came to denote a perpendicular, of 
 which, indeed, it was the archaic name, 
 as we leam from Proclus on Euclid, i., 
 12 : — TovTO ri irp6fi\Ti/j.a irpwrov Olvo' 
 ■jriSiji i^iinifffv ^(pf)aifio» avrh irphs 
 affrpoKoyiav oiSfievos' ovofid^ei 8^ t^v 
 KiStrov apx"^"''^ Kwrh yvdfiova, Si(!ti 
 Kal & yv^fjuap vpbs opdiis 4ffTi t^ bpl^ovri 
 (Procli Comm., ed. Friedlein, p. 283). 
 Gnomon is also an instrument for mea- 
 suring altitudes, by means of which the 
 Meridian can be found ; it denotes, 
 further, the index or style of a sundial, 
 the shadow of which points out the 
 hours. 
 
 In geometry it means the square or 
 rectangle about the diagonal of a square 
 or rectangle, together with the two 
 
 complements, on account of the resem- 
 blance of the figure to a carpenter's 
 square ; and then, more generally, the 
 similar figure with regard to any paral- 
 lelogram, as defined by Euclid, ii., 
 Def. 2. Again, in a still more general 
 signification, it means the figure which, 
 being added to any figure, preserves 
 the original form. See Hero, Defini- 
 tiones (59). 
 
 When gnomons are added succes- 
 sively in this manner to a square 
 monad, the first gnomon may be re- 
 garded as that consisting of three 
 square monads, and is indeed the con- 
 stituent of a simple Greek fret ; the 
 second, of [five square monads, &c. ; 
 hence we have the gnomonic numbers, 
 which were also looked on as male, or 
 generating. 
 
 83 Wilkinson's Ancient Egyptians, 
 vol. ii., p. III. 
 
i88 DR. ALLMAN ON GREEK GEOMETRY 
 
 odd numbers," and include, therefore, all odd squares. 
 Suppose, now, two squares are given, one consisting of i6 
 and the other of 9 unit squares, and that it is proposed to 
 form another square out of them. It is plain that the square 
 consisting of 9 unit squares can take the form of the fourth 
 gnomon, which, being placed round the former square, will 
 generate a new square containing 25 unit squares. Simi- 
 larly, it may have been observed that the 12th gnomon, 
 consisting of 25 unit squares, could be transformed into a 
 square, each of whose sides contain 5 units, and thus it 
 may have been seen conversely that the latter square, by 
 taking the gnomonic, or generating, form with respect to 
 the square on 1 2 units as base, would produce the square of 
 13 units, and so on. 
 
 This, then, is my attempt to interpret what Plutarch 
 has told us concerning Isis, Osiris, and Horus, bearing in 
 mind that the odd, or gnomonic, numbers were regarded 
 by Pythagoras as male, or generating.^" 
 
 w It may be observed here that we in the table of principles attributed by 
 first count with counters, as is indicated Aristotle to certain Pythagoreans {JHe- 
 by the Greek r^ipl^eiv and the Latin taph., i., 5, 986 a, ed. Bekker). 
 calculare. The counters might be The odd — or gnomonic — numbers are 
 equal squares, as well as any other like finite ; the even, infinite. Odd num- 
 objects. There is an indication that bers were regarded also as male, 
 the odd numbers were first regarded in or generating. Further, by the ad- 
 this maimer in the name gnomonic dition of successive gnomons — con- 
 numbers, which the Pythagoreans ap- sisting, as we have seen, each of an 
 plied to them, and that term was used odd number of units — to the original 
 in the same signification by Aristotle, unit square or monad, the square form 
 and by subsequent writers, even up to is preserved. On the other hand, if we 
 Kepler. See Arist. Phys., lib. iii., ed. start from the simplest oblong (Irepo.* 
 Bekker, vol. i. p. 203 ; Stob., Bclog., niixes), consisting of two unit squares, 
 ab Heeren, vol. i., p. 24, and note ; or monads, in juxtaposition, and place 
 Kepleri Opera Omnia, ed. Ch. Frisch, about it, after the manner of a gnomon 
 vol. viii., Matkematica, pp. 164 et seq. — and gnomon, as we have seen, was 
 
 '* This seems to me to throw light on used in this more extended sense also at 
 
 some of the oppositions which are found a later period — 4 unit squares, and 
 
FROM TRALES TO EUCLID. 
 
 i8(^ 
 
 It is another matter to see that the triangle formed by 
 4, and 5 units is right-angled, and this I think the 
 
 then in succession in like manner 6, 8, 
 . . . unit squares, the oblong form 
 fTepo-fi'liKfS wiU be preserved. The 
 elements, then, which generate a square 
 are odd, while those of which the ob- 
 long is made up are even. The limited, 
 the odd, the male, and the square, 
 occur on one side of the table : while 
 the unlimited, the even, the female, and 
 the oblong, are met with on the other 
 side. 
 
 The correctness of this view is con- 
 firmed by the following passage pre- 
 served by Stobaeus :— 'En Si Tij /lovdSi 
 Tuv i(pe^ris -aepttnTtcv yva)fi6vav trepiri- 
 Bcfievtayy & yivSfievos ael T€Tpd.'yuv6s itrn, 
 ruv a aprtofv dfioivs ireptnOe/Mevav, ere- 
 po/i'fiKets Kcd Hytffoi vdyres aTofiaivovfftv. 
 Xffov 5t iffoKis ovBeis. 
 
 " Explicanda haec sunt ex antiqua 
 Pythagoricorum terminologia. rvd/iovts 
 nempe de quibus hie loquitur auctor, 
 vocabantur apud eos omnes numeri im- 
 pares, yoh. Philof. ad Aristot. Phys., 
 L iii., p. 131 : Kal ol apiBiirjTiKo) Sc 
 yviofiovas Ka\ov<ri iravras Toiis Trcptrrobs 
 apiS/iois. Causam adjicit Simplicius 
 ad eundem locum, ry<i/iovas Si iKoAovv 
 Tovs trcpiTToiis 01 Tlv6ay6peiQt Si6rt irpoff' 
 rtdffievoi Tois r^rpaydivoiSy rb ainh 
 ffXVf"^ 4>i'A(£TT0tKri, &<rvep Kal ol iv yew- 
 lifTplif yyiifioves. Quae nostro loco le- 
 guntur jam satis clara erunt. Vult 
 nempe auctor, monade addita ad pri- 
 mum gnomonem, ad sequentes autem 
 summam, quam proxime antecedentes 
 numeri efficiunt, semper prodlre nume- 
 ros quadratos, v. c. positis gnomonibus 
 3, S, 7, 9 primum i + 3 = 2', tunc porro 
 1 + 3 {i. ^. 4) + 5 = 3', 9 + 7 = 4'. 
 16 + 9 = S", et sic porro, cf. Tiedem. 
 
 Geist der Speculat. Philos., pp. 107, 
 108. Reliqua expedita sunt." Stob. 
 Eclog. ab Heeren, lib. I., p. 24 and 
 note. 
 
 The passage of Aristotle referred to 
 IS — irTf/icioi' 8' eXyai Toirov ri ffvftfiatyoy 
 iv\ rwy apiB/ivy. ireptTtBefievotv yitp raiv 
 yv(0fi6vup V€pl rh ey Kal x^P^^ ^^ M^ 
 iX\o ad ylyveaSai rh ttSos. Phys., 
 iii., 4, p. 203", 14. 
 
 Compare, aA.\* ctrrt riva av^avifteya 
 & oiiK aWoiovyTai, oloy rb Terpdyavoy 
 yydfwvos vfpiTtBfyros riH^rrrat /le'c, oA.- 
 \oi6Tepoy Si oi/Siy yeyevTirat. Cat, 14, 
 IS«, 30, Arist., ed. Bekker. 
 
 Hankel gives a different explanation 
 of the opposition between the square 
 and oblong — 
 
 ' ' When the Pythagoreans discovered 
 the theory of the Irrational, and recog- 
 nised its importance, it must, as wiU be 
 at once admitted, appear most striking 
 that the oppositions, which present 
 themselves so naturally, of Rational 
 and Irrational have no place in their 
 table. Should they not be contained 
 imder the imageof square and rectangle, 
 which, in the extraction of the square 
 root, have led precisely to those ideas ? " 
 Geschichte der Mathematik, p. no, 
 note. 
 
 Hankel also says — " Upon what the 
 comparison of the odd with the hmited 
 may have been based, and whether 
 upon the theory of the gnomons, can 
 scarcely be made out now." Ibid. 
 p. 109, note. 
 
 May not the gnomon be looked on 
 as framing, as it were, or hmiting the 
 squares ? 
 
I90 DR. ALLMAN ON GREEK GEOMETRY 
 
 Egyptians may have first arrived at by an induction 
 founded on direct measvirement, the opportunity for which 
 was furnished to them by their pavements, or chequered 
 plane surfaces. 
 
 The method given above for the formation of the square 
 constructed on 5 units as the sum of those constructed on 
 4 units and on 3 units, and of that constructed on 1 3 units 
 as the sum of those constructed on 1 2 units and 5 units, 
 required only to be generalized in order to enable Pytha- 
 goras to arrive at his rule for finding right-angled triangles, 
 which we are told sets out from the odd numbers. 
 
 The two rules of Pjrthagoras and of Plato are given by 
 Proclus: — "But there are delivered certain methods of 
 finding triangles of this kind [sc, right-angled triangles 
 whose sides can be expressed by numbers], one of which 
 they refer to Plato, but the other to Pythagoras, as origi- 
 nating from odd numbers. For Pythagoras places a given 
 odd number as the lesser of the sides about the right angle, 
 and when he has taken the square constructed on it, and 
 diminished it by unity, he places half the remainder as 
 the greater of the sides about the right angle ; and when 
 he has added unity to this, he gets the hypotenuse. Thus, 
 for example, when he has taken 3, and has formed from it 
 a square number, and fi-om this number 9 has taken unity, 
 he takes the half of 8, that is 4, and to this again he adds 
 unity, and makes 5 ; and thus obtains a right-angled tri- 
 angle, having one of its sides of 3, the other of 4, and the 
 hypotenuse of 5 units. But the Platonic method originates 
 from even numbers. For when he has taken a given even 
 number, he places it as one of the sides about the right 
 angle, and when he has divided this into half, and squared 
 the half, by adding unity to this square he gets the hypo- 
 tenuse, but by subtracting unity from the square he forms 
 the remaining side about the right angle. Thus, for ex- 
 ample, taking 4, and squaring its half, 2, and thus getting 
 
FROM TEALES TO EUCLID. 191 
 
 4, then subtracting i he gets 3, and by adding i he gets 
 5 ; and he obtains the same triangle as by the former 
 method." *^ It should be observed, however, that this is not 
 necessarily the case ; for example, we may obtain by the 
 method of Plato a triangle whose sides are 8, 15, and 17 
 units, which cannot be got by the Pythagorean method. 
 
 The «*'' square together with the «** gnomon is the 
 [n + i)"' square; if the «'* gnomon contains m^ unit squares 
 
 m being an odd number, we have zn+ i = m^, .•. n = ; 
 
 hence the rule of Pythagoras. Similarly the sum of two 
 successive gnomons contains an even number of unit 
 squares, and may therefore consist of m^ unit squares, 
 where m is an even number; we have then (2 « - i) + (2 « 
 
 + i) = m^, or « = ( — ) : hence the rule ascribed to Plato by 
 
 Proclus." This passage of Proclus, which is correctly in- 
 terpreted by Hoefer,*^ was understood by Kepler,*" who, 
 indeed, was familiar with this work of Proclus, and often 
 quotes it in his Harmonia Mundi. 
 
 Let us now examine how Pythagoras proved the the- 
 orem of the three squares. Though he could have disco- 
 vered it as a consequence of the theorem concerning the 
 proportionality of the sides of equiangfular triangles, attri- 
 buted above to Thales, yet there is no indication whatever of 
 bis having arrived at it in that deductive manner. On the 
 
 *« Prodi Comm., ed. Friedlein, p. capable of further extension, «. ^. : the 
 428. Hero, Geom., ed. Hultsch, pp. sum of 9 (an odd square number) Suc- 
 re, 57. cessive gnomons may contain an odd 
 
 87 This rule is ascribed to Architas number (say 49 x 9) of square units ; 
 
 [no doubt, Archytas of Tarentum] by hence we obtain a right-angled triangle 
 
 Boetius, Geom., ed. Friedlein, p. 408. in numbers, whose hypotenuse exceeds 
 
 eslloe{eT,HisiotredesMatk.,p. 112. one side by 9 units— the three sides 
 
 •' Kepleri Opera Omnia, ed. Frisch, being 20, 21, and 29. Plato's method 
 
 vol. viii., pp. 163 et seq. It may may be extended in like manner, 
 be observed that this method is 
 
192 BE. ALLMAN ON GREEK GEOMETRY 
 
 other hand the proof given in the Elements of Euclid clearly 
 points to such an origin, for it depends on the theorem that 
 the square on a side of a right-angled triangle is equal to- 
 the rectangle under the hypotenuse and its adjacent seg- 
 ment made by the perpendicular on it from the right an- 
 gle — a theorem which follows at once from the similarity 
 of each of the partial triangles, into which the original 
 right-angled triangle is broken up by the perpendicular,, 
 with the whole. That the proof in the Elements is not the 
 way in which the theorem was discovered is indeed stated 
 directly by Proclus, who says : — 
 
 " If we attend to those who wish to investigate anti- 
 quity, we shall find them referring the present theorem to 
 P)^hagoras, . . . For my own part, I admire those who first 
 investigated the truth of this theorem : but I admire still 
 more the author of the Elements, because he has not only 
 secured it by evident demonstration, but because he re- 
 duced it into a more general theorem in his sixth book by 
 strict reasoning [Euclid, vi., 31]."'° 
 
 The simplest and most natural way of arriving at the 
 theorem is the following, as suggested by Bretschneid- 
 er " :— 
 
 A square can be dissected into the sum of two squares 
 and two equal rectangles, as in Euclid, ii., 4 ; these two rect- 
 angles can, by drawing their diagonals, be decomposed 
 into four equal right-angled triangles, the sum of the sides 
 of each being the side of the square : again, these four 
 right-angled triangles can be placed so that a vertex of 
 each shall be in one of the corners of the square in such a 
 way that a greater and less side are in continuation. The 
 original square is thus dissected into the four triangles as 
 
 »» Prodi Comm. ed. Friedlein, p. 426. Camerer, Euclidis Element, vol. i. , p. 
 9' Bretsch., Die Geometrie vor Eu- 444, and references given there. 
 Hides, p. 82. This proof is old : see 
 
FROM TEALES TO EUCLID. 19:, 
 
 before and the figure within, which is the square on the hy- 
 potenuse. This square then must be equal to the sum of the 
 squares on the sides of the right-angled triangle. Hankel, 
 in quoting this proof from Bretschneider, says that it may 
 be objected that it bears by no means a specifically Greek 
 colouring, but reminds us of the Indian method. This hy- 
 pothesis as to the oriental origin of the theorem seems 
 to me to be well founded. I would, however, attribute the 
 discovery to the Egyptians, inasmuch as the theorem con- 
 cerns the geometry' of areas, and as the method used is 
 that of the dissection of figures, for which the Egyptians 
 were famous, as we have already seen. Moreover, the theo- 
 rem concerning the areas connected with two lines and their 
 sum (Euclid, ii., 4), which admits also of arithmetical in- 
 terpretation, was certainly within their reach. The gnomon 
 by which any square exceeds another breaks up naturally 
 into a square and two equal rectangles. 
 
 I think also that the Egyptians knew that the difference 
 between the squares on two lines is equal to the rectangle 
 under their sum and difference — though they would not have 
 stated it in that abstract manner. The two squares maybe 
 placed with a common vertex and adjacent sides coinciding 
 in direction, so that their difference is a gnomon. This 
 gnomon can, on account of the equality of the two com- 
 plements,"' be transformed into a rectangle which can be 
 constructed by producing the side of the greater square so 
 that it shall be equal to itself, and then we have the figure 
 of Euclid, ii., 5, or to the side of the lesser square, in which 
 case we have the figure of Euclid, ii., 6. Indeed I have 
 little hesitation in attributing to the Egyptians the contents 
 
 «i This theorem (Euclid, i. 43) Bret- gnomon was not used in it either as de- 
 schneider says was called the " theorem fined by Euclid (ii., Def. 2), or in the 
 of the gnomon." I do not know of more general signification in Hero 
 any authority for this statement. If {Def. 58). 
 the theorem were so called, the word 
 VOL. III. O 
 
194 DR- ALLMAN ON GREEK GEOMETRY 
 
 of the first ten propositions of the second book of Euclid. 
 In the demonstrations of propositions 5, 6, 7, and 8, use is 
 made of the gnomon, and propositions 9 and lo also can 
 be proved similarly without the aid of Euclid, i., 47. 
 
 It is well known that the Pythagoreans were much oc- 
 cupied with the construction of regular polygons and solids, 
 which in their cosmology played an essential part as the 
 fundamental forms of the elements of the universe." 
 
 We can trace the origin of these mathematical specula- 
 tions in the theorem [c) that " the plane around a point 
 is completely filled by six equilateral triangles or four 
 squares, or three regular hexagons," a theorem attributed 
 to the Pythagoreans, but which must have been known as 
 a fact to the Egyptians. Plato also makes the Pythago- 
 rean Timaeus explain — " Each straight-lined figure consists 
 of triangles, but all triangles can be dissected into rectan- 
 gular ones which are either isosceles or scalene. Among 
 the latter the most beautiful is that out of the doubling of 
 which an equilateral arises, or in which the square of the 
 greater perpendicular is three times that of the smaller, or 
 in which the smaller perpendicular is half the hypotenuse. 
 But two or four right-angled isosceles triangles, properly 
 put together, form the square; two or six of the most 
 beautiful scalene right-angled triangles form the equilateral 
 triangle ; and out of these two figures arise the solids which 
 correspond with the four elements of the real world, the 
 tetrahedron, octahedron, icosahedron, and the cube." *' 
 
 This dissection of figures into right-angled triangles 
 may be fairly referred to Pythagoras, and indeed may have 
 been derived by him from the Egyptians. 
 
 «« Hankel says it cannot be ascer- nate dodecahedron was known to them, 
 
 lained with precision how far the Py- Hankel, Geschichtt der Mathematik, 
 
 thagoreans had penetrated into this p. 95, note, 
 theory, namely, whether the construe- «9 Plato, Tim., c. 20, s. 107. 
 
 tion of the regular pentagon and ordi- 
 
FROM THALES TO EUCLID. 195 
 
 The construction of the regular solids is distinctly 
 ascribed to Pythagoras himself by Eudemus, in the passage 
 in which he briefly states the principal services of Pytha- 
 goras to geometry. Of the five regular solids, three — the 
 tetrahedron, the cube, and the octahedron — ^were certainly 
 known to the Egyptians, and are to be found in their archi- 
 tecture. There remain, then, the icosahedron and the 
 dodecahedron. Let us now examine what is required for 
 the construction of these two solids. 
 
 In the formation of the tetrahedron, three, and in that 
 of the octahedron, four, equal equilateral triangles had 
 been placed with a common vertex and adjacent sides co- 
 incident, and it was knovim too that if six such triangles 
 were placed round a common vertex with their adjacent 
 sides coincident, they would lie in a plane, and that, there- 
 fore, no solid could be formed in that manner from them. 
 It remained then to try whether five such equilateral tri- 
 angles could be placed at a common vertex in like man- 
 ner: on trial it would be found that they could be so 
 placed, and that their bases would form a regular penta- 
 gon. The existence of a regular pentagon would thus be 
 known. It was also known from the formation of the cube 
 that three squares could be placed in a similar way with a 
 common vertex, and that, further, if three equal and regu- 
 lar hexagons were placed round a point as common vertex 
 with adjacent sides coincident, they would form a plane. 
 It remained then only to try whether three equal regular 
 pentagons could be placed with a common vertex, and in 
 a similar way ; this on trial would be found possible, and 
 would lead to the construction of the regular dodecahedron, 
 which was the regular solid last arrived at.'° 
 
 We see then that the construction of the regular penta- 
 gon is required for the formation of each of these two 
 
 «) The four elements had been repre- the dodecahedron was then taken sym- 
 sented by the four other regular solids ; bolically for the universe. 
 
 O 2 
 
196 DR. ALLMAN ON GREEK GEOMETRY 
 
 regular solids, and that therefore it must have been a dis- 
 covery of Pythagoras. We have now to examine what 
 knowledge of geometry was required for the solution of 
 this problem. 
 
 If any vertex of a regular pentagon be connected with 
 the two remote ones, an isosceles triangle will be formed 
 having each of the base angles double the vertical angle. 
 The construction of the regular pentagon depends, there- 
 fore, on the description of such a triangle (Euclid, iv., lo). 
 Now, if either base angle of such a triangle be bisected, 
 the isosceles triangle will be decomposed into two trian- 
 gles, which are evidently also both isosceles. It is also 
 evident that the one of which the base of the proposed is a 
 side is equiangular with it. From a comparison of the 
 sides of these two triangles it will appear at once by the 
 second theorem, attributed above to Thales,, that the pro- 
 blem is reduced to cutting~a straight line so that one seg- 
 ment shall be a mean proportional between the whole line 
 and the other segment (Euclid, vi., 30), or so that the rect- 
 angle under the whole line and one part shall be equal to 
 the square on the other part (Euclid, ii., 11). To effect this, 
 let us suppose the square on the greater segment to be 
 constructed on one side of the line, and the rectangle under 
 the whole line and the lesser segment on the other side. 
 It is evident that by adding to both the rectangle under 
 the whole line and the greater segment, the problem is 
 reduced to the following: — To produce a given straight 
 line so that the rectangle under the whole line thus pro- 
 duced and the part produced shall be equal to the square 
 on the given line, or, in the language of the ancients, to 
 apply to a given straight line a rectangle which shall be 
 equal to a given area — in this case the square on the given 
 line — and which shall be excessive by a square. Now it is 
 to be observed that the problem is solved in this manner 
 by Euclid (vi., 30, ist method), and that we learn from 
 
FROM THALES TO EUCLID. 
 
 197 
 
 Eudemus that the problems concerning- the application of 
 areas and their excess and defect are old, and inventions of 
 the Pythagoreans (e)." 
 
 The statements, then, of lamblichus concerning Hip- 
 pasus (?■) — that he divulged the sphere with the twelve 
 pentagons; and of Lucian and the scholiast on Aristo- 
 phanes {j) — that the pentagram was used as a symbol of 
 recognition amongst the Pythagoreans, become of greater 
 importance. We learn too from lamblichus that the Py- 
 thagoreans made use of signs for that purpose."* 
 
 Further, the discovery of irrational magnitudes is 
 ascribed to Pythagoras in the same passage of Eude- 
 
 " It may be objected that this reason- 
 ing presupposes a loiowledge, on the 
 part of Pythagoras, of the method of 
 geometrical analysis, which was in- 
 vented by Plato more than a century 
 later. 
 
 While admitting that it contains the 
 germ of that method, I reply in the 
 first place, that this manner of reason- 
 ing was not only natural and sponta- 
 neous, but that in fact in the solution 
 of problems there was no other way of 
 proceeding. And, to anticipate a little, 
 we shall see, secondly, that the oldest 
 fragment of Greek geometry extant — 
 that namely by Hippocrates of Chios — 
 contains traces of an analytical method, 
 and that, moreover, Proclus ascribes 
 to Hippocrates, who, it wUi appear, 
 was taught by the Pythagoreans the 
 method of reduction (dirayay^), a sys- 
 tematization, as it seems to me, of the 
 manner of reasoning that was sponta- 
 neous with Pythagoras. Proclus de- 
 fines dira-ywT^ to be " a transition from 
 one problem or theorem to another, 
 which being known or determined, the 
 thing proposed is also plain. For ex- 
 ample : when the duplication of the 
 cube is investigated, geometers reduce 
 
 the question to another to which this 
 is consequent, i. e. the finding of two 
 mean proportionals, and afterwards 
 they inquire how between two given 
 straight lines two mean proportionals 
 may be found. But Hippocrates of 
 Chios is reported to have been the first 
 inventor of geometrical reduction {aira.- 
 yuyi]) : who also squared the lunula, 
 and made many other discoveries in 
 geometry, and who was excelled by no 
 geometer in his powers of construc- 
 tion." — Proclus, ed. Friedlein, p. 212. 
 Lastly, we shall find that the passages 
 in Diogenes Laertius and Proclus, 
 which are relied on in support of the 
 statement that Plato invented this me- 
 thod, prove nothing more than that 
 Plato communicated it to Leodamas 
 of Thasos. For my part, I am con- 
 vinced that the gradual elaboration 
 of this famous method — by which ma- 
 thematics rose above the elements — is 
 due to the Pythagorean philosophers 
 from the founder to Theodorus of 
 Cyrene and Archytas of Tarentum, 
 who were Plato's masters in mathema- 
 tics. 
 
 ^^ Iambi, de Pyth. Vita, cxxxiii., 
 p. 77, ed. Didot. 
 
198 I)R. ALLMAN ON GREEK GEOMETRY 
 
 mus [m], and this discovery has been ever regarded as one 
 of the greatest of antiquity. It is commonly assumed that 
 Pythagoras was led to this theory from the consideration 
 of the isosceles right-angled triangle. It seems to me, 
 however, more probable that the discovery of incommen- 
 surable magnitudes was rather owing to the problem — To 
 cut a line in extreme and mean ratio. From the solution 
 of this problem it follows at once that, if on the greater 
 segment of a line so cut a part be taken equal to the less, 
 the greater segment, regarded as a new line, will be cut in 
 a similar manner ; and this process can be continued with- 
 out end. On the other hand, if a similar method be adopted 
 in the case of any two lines which are capable of numerical 
 representation, the process would end. Hence would arise 
 the distinction between commensurable and incommensur- 
 able quantities. 
 
 A reference to Euclid, x., 2, will show that the method 
 above is the one used to prove that two magnitudes are in- 
 commensurable. And in Euclid, x., 3, it will be seen that 
 the greatest common measure of two commensurable mag- 
 nitudes is found by this process of continued subtraction. 
 
 It seems probable that Pythagoras, to whom is attribu- 
 ted one of the rules for representing the sides of right- 
 angled triangles in numbers, tried to find the sides of an 
 isosceles right-angled triangle numerically, and that, fail- 
 ing in the attempt, he suspected that the hypotenuse and a 
 side had no common measure. He may have demonstrated 
 the incommensurability of the side of a square and its dia- 
 gonal. The nature of the old proof — which consisted of a 
 redudio ad absurdum, showing that if the diagonal be com- 
 mensurable with the side, it would follow that the same 
 number would be odd and even" — makes it more probable, 
 however, that this was accomplished by his successors. 
 
 "3 Aristoteles,^«a^<. Prior., i., c. 23, for its historical interest only, since the 
 
 41, u, 2fa, and c. 44, 50, a, 37, ed. Bek- irrationality follows self-evidently from 
 
 ker. X., 9; and x., 117, is merely an ap- 
 
 Euclid has preserved this proof, x., pendix. — 'H3.nksl,GeschichtederMath.. 
 
 117. Hankel thinks he did so probably p. 102, note. 
 
FROM THALES TO EUCLID. 199 
 
 The existence of the irrational, as well as that of the 
 regfular dodecahedron, appears to have been regarded also 
 by the school as one of their chief discoveries, and to have 
 been preserved as a secret ; it is remarkable, too, that a 
 story similar to that told by lamblichus of Hippasus is 
 narrated of the person who first published the idea of the 
 irrational, namely, that he suffered shipwreck, &c." 
 
 Eudemus ascribes the problems concerning the appli- 
 cation of figures to the Pythagoreans. The simplest cases 
 of the problems (Euclid, vi., 28, 29) — those, namely, in 
 which the given parallelogram is a square — correspond to 
 the problem : To cut a straight line internally, or externally, 
 so that the rectangle under the segments shall be equal to 
 a given rectilineal figure. On examination it will be found 
 that the solution of these problems depends on the problem 
 Euclid, ii., 14, and the theorems Euclid, ii., 5 and 6, which 
 we have seen were probably known to the Egyptians, to- 
 gether with the law of the three squares (Euclid, i., 47). 
 
 The finding of a mean proportional between two given 
 lines, or the construction of a square which shall be equal 
 to a given rectangle, must be referred, I have no doubt, to 
 Pythagoras. The rectangle can be easily thrown into the 
 form of a gnomon, and then exhibited as the difference 
 between two squares, and therefore as a square by means 
 of the law of the three squares. 
 
 Lastly, the solution of the problem to construct a 
 rectilineal figure which shall be equal to one and similar 
 to another given rectilineal figure is attributed by Plutarch 
 to P3^hagoras. The solution of this problem depends on 
 the application of areas, and requires a knowledge of the 
 theorems : — that similar rectilineal figures are to each other 
 as the squares on their homologous sides; that if three 
 
 91 Untersuchungenuberdieneuauf- Dr. Joachim Heinrich Knoche, Her- 
 irefundenen Scholien des Proklus Dia- ford, 1865, pp. 20 and 23. 
 dochus zu Euclid's Elementen, von 
 
2 00 DM. ALLMAN ON GREEK GEOMETRY 
 
 lines be in geometrical proportion, the first is to the third 
 as the square on the first is to the square on the second ; 
 and also on the solution of the problem, to find a mean 
 proportional between two given straight lines. Now, we 
 shall see later that Hippocrates of Chios — ^who was in- 
 structed in geometry by the Pythagoreans — must have 
 known these theorems and the solution of this problem. 
 We are justified, therefore, in ascribing this theorem also, 
 if not with Plutarch to Pythagoras, at least to his early 
 successors. 
 
 The theorem that similar polygons are to each other in 
 the duplicate ratio of their homologous sides involves a 
 first^sketch, at least, of the doctrine of proportion. 
 
 That we owe the foundation and development of the 
 doctrine of proportion to Pythagoras and his disciples is 
 confirmed by the testimony of Nicomachus [n] and lanabli- 
 chus [o and/). 
 
 From these passages it appears that the early Pythago- 
 reans were acquainted not only with the arithmetical and 
 geometrical means between two magnitudes, but also with 
 their harmonical mean, which was then called virivavTia. 
 
 When two quantities are compared, it may be con- 
 sidered how much the one is greater than the other, what is 
 their difference ; or it may be considered how many times 
 the one is contained in the other, what is their quotient. 
 The former relation of the two quantities is called their 
 arithmetical ratio ; the latter their geometrical ratio. 
 
 Let now three magnitudes, lines or numbers, a, b, c, be 
 taken, li a-b = b-c, the three magnitudes are in arithmeti- 
 cal proportion ; but \i a : b : : b : c, they are in geometrical 
 proportion."* In the latter case, it follows at once, from the 
 
 95 In lines we may have c = a —i,or then the sum of the other two lines, 
 
 a : h : a - b. This particular case, in and is said to be cut in extreme and 
 
 which the geometrical and arithmetical mean ratio. This section, as we have 
 
 ratios both occur in the same proper- seen, has arisen out of the construction 
 
 tion, is worth noticing. The line a is of the regular pentagon, and we learn 
 
FROM THALES TO EUCLID. 201 
 
 second theorem of Thales (Euclid, vi., 4), that a- b : b - c 
 :: a : b, whereas in the former case we have plainly a - b 
 : b - c : : a : a. This might have suggested the considera- 
 tion of three magnitudes, so taken that a - b : b - c :: a : c; 
 three such magnitudes are in harmonical proportion. 
 
 The probability of the correctness of this view is indica- 
 ted by the consideration of the three later proportions — 
 a : c :: b - c : a - b ... the contrary of the harmonical ; 
 
 a • b • • b - c ■ a - b\ ' ' ' *^^ contrary of the geometrical. 
 The discovery of these proportions is attributed" to Hip- 
 pasus, Archytas, and Eudoxus. 
 
 We have seen also (J>) that a knowledge of the so-called 
 most perfect or musical proportion, which comprehends in 
 it all the former ratios, is attributed by lamblichus to Py- 
 thagoras — 
 
 a + b zab , 
 
 a 
 
 a + b 
 
 We have also seen [q] that a knowledge of the doctrine , 
 of arithmetical progressions is attributed to Pythagoras. 
 This much at least seems certain, that he was acquainted 
 with the summation of the natural numbers, the odd num- 
 bers, and the even numbers, all of which are capable of 
 geometrical representation. 
 
 Montucla says that Pythagoras laid the foundation of 
 the doctrine of Isoperimetry by proving that of all figures 
 having the same perimeter the circle is the greatest, and 
 
 from Kepler that it was called by the vol. v., pp. 90 and 187 (Hannonia 
 
 modems, on account of its many won- Mundi) ; also vol. i. p. 377 (Literae de 
 
 derful properties, sectio divina, et pro- Rebus Astrologicis). The [pentagram 
 
 fortio dimna. He sees in it a fine might be taken as the image of all this, 
 
 image of generation, since the addition as each of its sides and part of a side 
 
 to the line of its greater part produces are cut in this divine proportion, 
 
 a new line cut similarly, and so on. ^^ Y'ncAA.Tn.'&iQ.. Arith., pp. 142, 159, 
 
 See Kepleri Opera Omnia, ed. Frisch, 163. See above, p. 163. 
 
202 DR. ALLMAN ON GREEK GEOMETRY 
 
 that of all solids having the same surface the sphere is the 
 greatest.^' 
 
 There is no evidence to support this assertion, though it 
 is repeated by Chasles, Arneth, and others ; it rests merely 
 on an erroneous interpretation of the passage [s] in Dioge- 
 nes Laertius, which says only that " of all solid figures the 
 sphere is the most beautiful ; and of all plane figures, the 
 circle." Pythagoras attributes perfection and beauty to 
 the sphere and circle on account of their regularity and 
 uniformity. That this is the true signification of the pas- 
 sage is confirmed by Plato in the Timaeus,'* when speaking 
 of the Pythagorean cosmogony.'' 
 
 We must also deny to Pythagoras and his school a 
 knowledge of the conic sections, and, in particular, of the 
 quadrature of the parabola, attributed to him by some 
 authors, and we have already noticed the misconception 
 which gave rise to this erroneous conclusion.^™ 
 
 Let us now see what conclusions can be drawn from the 
 foregoing examination of the mathematical work of Pytha- 
 goras and his school, and thus form an estimate of the state 
 of geometry about 480 B. C. : — 
 
 First, then, as to matter: — 
 
 It forms the bulk of the first two books of Euclid, and 
 includes, further, a sketch of the doctrine of proportion — 
 which was probably limited to commensurable magni- 
 tudes — together with some of the contents of the sixth 
 book. It contains, too, the discovery of the irrational 
 {aXoyov), and the construction of the regular solids ; the 
 
 S' " Suivant Diogene, dont le texte Histoire des Mathhnatiques , torn. I., 
 
 est ici fort corrompu, et probabk- p. 113. 
 
 ment transpose, il ebaucha aussi la '8 Timaeus, 33, B., vol. vii., ed. 
 
 doctrine des Isoperimetres, en dSmon- Stallbaum, p. 129. 
 
 trant que de toutes les figures de meme ** See Bretschneider, Die Geometric 
 
 contour, parmi les figures planes, c'est vor Euklides, pp. 89, 90. 
 
 le cercle qui est la plus grande, et par- ""' See above, p. 182, note, 
 mi les solides, la sphere." — Montucla, 
 
FROM THALES TO EUCLID. 203 
 
 latter requiring the description of certain regular polygons 
 — the foundation, in fact, of the fourth book of Euclid. 
 
 The properties of the circle were not much known at 
 this period, as may be inferred from the fact that not one 
 remarkable theorem on this subject is mentioned ; and we 
 shall see later that Hippocrates of Chios did not know 
 the theorem — that the angles in the same segment of a 
 circle are equal to each other. Though this be so, there is, 
 as we have seen, a tradition {t) that the problem of the 
 quadrature of the circle also engaged the attention of the 
 Pythagorean school — a problem which they probably de- 
 rived from the Egyptians."" 
 
 Second, as to form : — 
 
 The Pythagoreans first severed geometry from the needs 
 of practical life, and treated it as a liberal science, giving 
 definitions, and introducing the manner of proof which 
 has ever since been in use. Further, they distinguished be- 
 tween discrete and continuous quantities, and regarded geo- 
 metry as a branch of mathematics, of which they made the 
 fourfold division that lasted to the Middle Ages — the quad- 
 rivium (fourfold way to knowledge) of Boetius and the 
 scholastic philosophy. And it may be observed, too, that 
 the name of mathematics, as well as that of philosophy, is 
 ascribed to them. 
 
 Third, as to method : — 
 
 One chief characteristic of the mathematical work of 
 Pythagoras was the combination of arithmetic with geo- 
 
 '<" This problem is considered in the diameter the side of a square whose 
 
 Papyrus Rhind, pp. 97, 98, 117. The area should be equal to that of the 
 
 point ofview from which it was regarded circle. Their approximation was as 
 
 by the Egyptians was diiFerent from that follows : — The diameter being divided 
 
 of Archimedes. Whilst he made it to into nine equal parts, the side of the 
 
 depend on the determination of the equivalent square was taken by them to 
 
 ratio of the circumference to the dia- consist of eight of those parts, 
 meter, they sought to find from the 
 
204 DR- ALLMAN ON GREEK GEOMETRY 
 
 metry. The notions of an equation and a proportion — which 
 are common to both, and contain the first germ of algebra 
 — ^were, as we have seen, introduced amongst the Greeks 
 by Thales. These notions, especially the latter, were ela- 
 borated by Pythagoras and his school, so that they reached 
 the rank of a true scientific method in their Theory of Pro- 
 portion. To Pythagoras, then, is due the honour of having 
 supplied a method which is common to all branches of 
 mathematics, and in this respect he is fully comparable to 
 Descartes, to whom we owe the decisive combination of 
 algebra with geometry. 
 
 It is necessary to dwell on this at some length, as mo- 
 dem writers are in the habit of looking on proportion as a 
 branch of arithmetic""— no doubt on account of the arith- 
 metical point of view having finally prevailed in it — 
 whereas for a long period it bore much more the marks of 
 its geometrical origin.'"' 
 
 That proportion was not thus regarded by the ancients, 
 merely as a branch of arithmetic, is perfectly plain. We 
 learn from Proclus that " Eratosthenes looked on propor- 
 tion as the bond [avvisaiiov) of mathematics." "* 
 
 We are told, too, in an anonymous scholium on the Ele- 
 ments of Euclid, which Knoche attributes to Proclus, that 
 the Jifth book, which treats of proportion, is common to 
 geometry, arithmetic, music, and, in a word, to all mathe- 
 matical science.'" 
 
 And Kepler, who lived near enough to the ancients to 
 reflect the spirit of their methods, says that one part of 
 
 "" Bretschneider (Die Geometrie vor '"'' Procl. Comm.,^A.. Freidlein, p. 43. 
 
 Euklides, p. 74) and Hankel (Ce- ""^EucUdis Elem. Graece ed. ab 
 
 schichte der Mathematik, p. 104) do so, E. F. August, pars ii., p. 328, Berolini, 
 
 although they are treating of the history 1829. Untersuchungen uber die neu 
 
 of Greek geometry, which is clearly a aufgefundenen Scholien des Proklus zu 
 
 mistake. Euclid's Elementen, von Dr. J. H. 
 
 '"' On this see A. Comte, Politique Knoche, p. 10, Herford, 1865. 
 Positive, vol. iii., ch. iv., p. 300. 
 
FROM THALES TO EUCLID. 205 
 
 geometry is concerned with the comparison of figures and 
 quantities, whence proportion arises [" unde proportio ex- 
 istit "). He also adds that arithmetic and geometry afford 
 mutual aid to each other, and that they cannot be separa- 
 ted.'"" 
 
 And since Pythagoras they have never been separated. 
 On the contrary, the union between them, and indeed be- 
 tween the various branches of mathematics, first instituted 
 by Pythagoras and his school, has ever since become more 
 intimate and profound. We are plainly in presence of not 
 merely a great mathematician, but of a great philosopher. 
 It has been ever so — the greatest steps in the deve- 
 lopment of mathematics have been made by philoso- 
 phers. 
 
 Modern writers are surprised that Thales, and indeed 
 all the principal Greek philosophers prior to Pythagoras, 
 are named as his masters. They are surprised, too, at the 
 extent of the travels attributed to him. Yet there is no 
 cause to wonder that he was believed by the ancients to 
 have had these philosophers as his teachers, and to have 
 extended his travels so widely in Greece, Egypt, and the 
 East, in search of knowledge, for — like the geometrical 
 figures on whose properties he loved to meditate — his phi- 
 losophy was many-sided, and had points of contact with 
 all these : — 
 
 He introduced the knowledge of arithmetic from the 
 Phoenicians, and the doctrine of proportion from the 
 Babylonians ; 
 
 Like Moses, he was learned in all the wisdom of the 
 
 106 "Et quidem geometriae theoreti- geometria speculativa, rautuas tradunt 
 
 cae initio hujus tractatus duas fecimus operas nee ab invicem separari possunt, 
 
 partes, unam de magnitudinibus, qua- quamvis et arithmetica sit principium 
 
 tenus finnt figurae, alteram de compara- cognitioms." — ^Kepleii Opera Omnia, 
 
 tione figurarum et quantitatum, unde ed. Dr. Ch. Frisch, vol. viii., p. i6o, 
 
 proportio existit. Francofurti, 1870. 
 
 " Hae duae scientiae, arithnaetica et 
 
2o6 BR. ALLMAN ON GREEK GEOMETRY 
 
 Egyptians, and carried their geometry and philosophy into 
 Greece. 
 
 He continued the work commenced by Thales in ab- 
 stract science, and invested geometry with the form which 
 it has preserved to the present day. 
 
 In establishing the existence of the regular solids he 
 showed his deductive power ; in investigating the elemen- 
 tary laws of sound he proved his capacity for induction ; 
 and in combining arithmetic with geometry, and thereby 
 instituting the theory of proportion, he gave an instance of 
 his philosophic power. 
 
 These services, though great, do not form, however, the 
 chief title of this Sage to the gratitude of mankind. He 
 resolved that the knowledge which he had acquired with 
 so great labour, and the doctrine which he had taken such 
 pains to elaborate, should not be lost ; and, as a husband- 
 man selects good ground, and is careful to prepare it for 
 the reception of the seed, which he trusts will produce firuit 
 in due season, so Pythagoras devoted himself to the forma- 
 tion of a society of eliie, which would be fit for the reception 
 and transmission of his science and philosophy, and thus 
 became one of the chief benefactors of humanity, and 
 earned the gratitude of countless generationis. 
 
 His disciples proved themselves worthy of their high mis- 
 sion. We have had already occasion to notice their noble 
 self-renunciation, which they inherited from their master. 
 
 The moral dignity of these men is, further, shown by 
 their admirable maxim — a maxim conceived in the spirit 
 of true social philosophers — a figure and a step ; hut not a 
 figure and three oboli {ayafia koi /3a/ua, oAA' oi ayafxa Kai rpiio- 
 
 '»' Prodi Comm.,ed. Friedlein, p. 84. which are extant, so that it is probably 
 
 Taylor's Commentaries of Proclus, nowhere mentioned but in the present 
 
 vol. i., p. 113. Taylor, in a note on work." 
 
 this passage, says— " I do not find this Taylor is not correct in this state- 
 
 aenigmaamongthePythagoric symbols ment. This symbol occurs in lambli- 
 
FROM THALES TO EUCLID. 207 
 
 Such, then, were the men by whom the first steps in 
 mathematics — the first steps ever the most difiScult — ^were 
 made. 
 
 In the continuation of the present paper we shall 
 notice the events which led to the publication, through 
 Hellas, of the results arrived at by this immortal School. 
 
 chus. See Iambi., Adhortaiio ad p.374. Ti B^5r/)0T(/iaTi)<rxw'''"'''8^M'' 
 Philosophiam, ed. Kiessling, Symb. to 5 ayjwi-o. koX Tpiii0a\ov. 
 xxxvi., cap. xxi., p. 317; also ExJ>l. 
 
 GEORGE J. ALLMAN. 
 
GREEK GEOMETRY, 
 
 FROM 
 
 THALES TO EUCLID. 
 
 PART II. 
 
 BY 
 
 GEORGE JOHNSTON ALLMAN, LL.D., 
 
 OF TRINITY COLLEGE, DUBLIN ; 
 
 PROFESSOR OF MATHEMATICS, AND MEMBER OF THE SENATE, OF THE 
 
 queen's UNIVERSITY IN IRELAND ; 
 
 MEMBER OF THE SENATE OF THE ROYAL UNIVERSITY OF IRELAND. 
 
 DUBLIN : 
 
 PRINTED AT THE UNIVERSITY PRESS, 
 
 BY PONSONBY AND WELDRICK. 
 
 1881. 
 
IFrom " Hermathena," Vol. IV., No. VII.'] 
 
i8o BR. ALLMAN ON GREEK GEOMETRY 
 
 GREEK GEOMETRY FROM THALES TO 
 EUCLID* 
 
 \Continued from Vol. III., No. V7\ 
 
 III. 
 
 THE first twenty years of the fifth century before 
 the Christian era was a period of deep gloom and 
 despondency throughout the Hellenic world. The lonians 
 had revolted and were conquered, for the third time ; this 
 time, however, the conquest was complete and final : they 
 were overcome by sea as well as by land. Miletus, till 
 then the chief city of Hellas, and rival of Tyre and Car- 
 thage, was taken and destroyed ; the Phoenician fleet ruled 
 the sea, and the islands of the ^gean became subject 
 to Persia. The fall of Ionia, and the maritime supremacy 
 of the Phoenicians, involving the interruption of Greek 
 commerce, must have exercised a disastrous influence on 
 
 • In the former part of this Paper mathematical works given in the note 
 
 (Hermathena, vol. iii. p. i6o, note) referred to above, I have to add : 
 
 I acknowledged my obligations to the Theonis Smymaei Expositio rerum 
 
 works of Bretschneider and Hankel : I Mathematicarum ad legendum Pla- 
 
 have again made use of them in the tonem utilium. Recensuit Eduardus 
 
 preparation of this part. Since it was Hiller, Lipsiae, 1878 (Teubner) ; Pappi 
 
 written, I have received from Dr. Alexandrini CoUectionis quae super- 
 
 Moritz Cantor, of Heidelberg, the sunt, &c., instruxit F. Hultsch, vol. 
 
 ^ar\.\on oi \a% History of Mathematics iii., Berolini, 1878; (to the latter the 
 
 which treats of the Greeks (Vorlesun- editor has appended xa Index Graeci- 
 
 gen Hber Geschichte der Mathematik, tatis, a valuable addition ; for as he 
 
 von Moritz Cantor, Erster Band. Von remarks, ' Mathematicam Graecorum 
 
 den altesten Zeiten bis zum Jahre 1200 dictionem nemo adhuc in lexici formam 
 
 n. Chr. Leipzig, 1880 (Teubner)). redegit." Praef., vol. iii., torn, ii.) ; 
 
 To the list of new editions of ancient Archimedis Opera omnia cum com- 
 
FROM THALES TO EUCLID. i8i 
 
 the cities of Magna Graecia.' The events which occurred 
 there after the destruction of Sybaris are involved in great 
 obscurity. We are told that some years after this event 
 there was an uprising of the democracy — which had been 
 repressed under the influence of the Pythagoreans — not 
 only in Crotona, but also in the other cities of Magna 
 Graecia. The Pythagoreans were attacked, and the house 
 in which they were assembled was burned ; the whole 
 country was thrown into a state of confusion and anarchy ; 
 the Pythagorean Brotherhood was suppressed, and the 
 chief men in each city perished. 
 
 The Italic Greeks, as well as the lonians, ceased to 
 prosper. 
 
 Towards the end of this period Athens was in the 
 hands of the Persians, and Sicily was threatened by the 
 Carthaginians. Then followed the glorious struggle ; the 
 gloom was dispelled, the war which had been at first 
 defensive became offensive, and the ^gean Sea was 
 cleared of Phoenicians and pirates. A solid basis was thus 
 laid for the development of Greek commerce and for the 
 interchange of Greek thought, and a brilliant period fol- 
 lowed — one of the most memorable in the history of the 
 world. 
 
 mentariis Eutocii. E codice Florentino is historical. 
 
 recensuit, Latine vertit notisque illus- * The names Ionian Sea, and Ionian 
 
 travit J. L. Heiberg, Dr. Phil. Vol.i., Isles, still bear testimony to the inter- 
 
 Lipsiae, 1880 (Teubner). Since the course between these cities and Ionia, 
 
 above was in type, the following work The writer of the article iji Smith's 
 
 has been published : An Introduction Dictionary of Geography thinks that 
 
 to the Ancient and Modem Geometry the name Ionian Sea was derived from 
 
 of Conies : being a geometrical treatise lonians residing, in very early times, 
 
 on the Conic Sections, with a collection on the west coast of the Peloponnesus. 
 
 of Problems and Historical Notes, and Is it not more probable that it was so 
 
 Prolegomena. ByCharles Taylor, M. A., called from being the highway of the 
 
 Fellow of St. John's College, Cam- Ionian ships, just as, now-a-days, in a 
 
 bridge. Cambridge, 1881. The matter provincial town we have the London 
 
 of the Prolegomena, pp. xvii.-lxxxviii., road.'' 
 
1 82 BE. ALLMAN ON GREEK GEOMETRT 
 
 Athens now exercised a powerful attraction on all that 
 was eminent in Hellas, and became the centre of the intel- 
 lectual movement. Anaxagoras settled there, and brought 
 with him the Ionic philosophy, numbering Pericles and 
 Euripides amongst his pupils ; many of the dispersed Py- 
 thagoreans no doubt found a refuge in that city, always 
 hospitable to strangers ; subsequently the Eleatic philoso- 
 phy was taught there by Parmenides and Zeno. Eminent 
 teachers flocked from all parts of Hellas to the Athens of 
 Pericles. All were welcome ; but the spirit of Athenian 
 life required that there should be no secrets, whether con- 
 fined to priestly families^ or to philosophic sects : every- 
 thing should be made public. 
 
 In this city, then, geometry was first published ; and 
 with that publication, as we have seen, the name of Hip- 
 pocrates of Chios is connected. 
 
 Before proceeding, however, to give an account of the 
 work of Hippocrates of Chios, and the geometers of the fifth 
 century before the Christian era, we must take a cursory 
 glance at the contemporaneous philosophical movement. 
 Proclus makes no mention of any of the philosophers of 
 the Eleatic School in the summary of the history of geome- 
 try which he has handed down — they seem, indeed, not 
 to have made any addition to geometry or astronomy, but 
 rather to have affected a contempt for both these sciences — 
 and most writers' on the history of mathematics either take 
 no notice whatever of that School, or merely refer to it as 
 outside their province. Yet the visit of Parmenides and 
 Zeno to Athens [circ. 450 B.C.), the invention of dialectics 
 by Zeno, and his famous polemic against multiplicity and 
 
 " E.g. the Asclepiadae. SeeCurtius, I have adopted. See a fine chapter of 
 
 History of Greece,'En^. transl., vol. ii. his Gesch. der Math., pp. 115 et seq., 
 
 P- S'O- bom which much of what follows is 
 
 3 Not so Hankel, whose views as to taken. 
 the influence of the Eleatic philosophy 
 
FROM TEALE8 TO EUCLID. 183 
 
 motion, not only exercised an important influence on "the 
 development of geometry at that time, but, further, had a 
 lasting effect on its subsequent progress in respect of 
 method.^ 
 
 Zeno argued that neither multiplicity nor motion is 
 possible, because these notions lead to contradictory con- 
 sequences. In order to prove a contradiction in the idea of 
 motion, Zeno argues : 'Before a moving body can arrive at 
 its destination, it must have arrived at the middle of its 
 path ; before getting there it must have accomplished the 
 half of that distance, and so on ad infinitum : in short, 
 every body, in order to move from one place to another, 
 must pass through an infinite number of spaces, which is 
 impossible.' Similarly he argued that 'Achilles cannot 
 overtake the tortoise, if the latter has got any start, 
 because in order to overtake it he would be obliged first 
 to reach every one of the infinitely many places which the 
 tortoise had previously occupied.' In like manner, ' The 
 flying arrow is always at rest; for it is at each moment 
 only in one place.' 
 
 Zeno applied a similar argument to show that the 
 notion of multiplicity involves a contradiction. ' If the 
 manifold exists, it must be at the same time infinitely 
 small and infinitely great — the. former, because its last 
 divisions are without magnitude ; the latter, on account of 
 the infinite number of these divisions.' Zeno seems to 
 have been unable to see that if xy = a, x and y may both 
 
 * This influence is noticed by Clairaut, il est renferme; on n'en sera pas sur- 
 
 ElemensdeGeometrie,7tet.-p-^;^3.ns, pris. Ce Ggometre avoit a convaincre 
 
 1 741 : 'Qu' Euclide se donne la peine des Sophistes obstines, qui se faisoient 
 
 de demontrer, que deux cercles qui se gloire de se refuser aux verites les plus 
 
 coupent n'ont pas le merae centre, Svidentes : il falloit done qu'alors la 
 
 qu'un triangle renferme dans un autre Geometrie eut, comme la Logique, le 
 
 a la somme de ses cotes plus petite que secours des raisonnemens en forme, 
 
 celle des cotes du triangle dans lequel pour fermer la bouche a la chicanne.' 
 
1 84 BR. ALLMAN ON GREEK GEOMETRY 
 
 vary, and that the number of parts taken may make up for 
 their minuteness. 
 
 Subsequently the Atomists endeavoured to reconcile 
 the notions of unity and multiplicity; stability and mo- 
 tion ; permanence and change ; being and becoming — in 
 short, the Eleatic and Ionic philosophy. The atomic 
 philosophy was founded by Leucippus and Democritus ; 
 and we are told by Diogenes Laertius that Leucippus was 
 a pupil of Zeno : the filiation of this philosophy to the 
 Eleatic can, however, be seen independently of this state- 
 ment. In accordance with the atomic philosophy, mag- 
 nitudes were considered to be composed of indivisible 
 elements (oro/uoi) in finite numbers : and indeed Aristotle — 
 who, a century later, wrote a treatise on Indivisible Lines 
 (TTEpl aTOfiwv ypafi/iwv), in order to show their mathematical 
 and logical impossibility — tells us that Zepo's disputation 
 was taken as compelling such a view.^ We shall see, too, 
 that in Antiphon's attempt to square the circle, it is 
 assumed that straight and curved lines are ultimately 
 reducible to the same indivisible elements.' 
 
 Insuperable difficulties were found, however, in this 
 conception; for no matter how far we proceed with the 
 divisipn, the distinction between the straight and curved 
 still exists. A like difficulty had been already met with 
 in the case of straight lines themselves, for the incommen- 
 surability of certain lines had been established by the 
 Pythagoreans. The diagonal of a square, for example, 
 cannot be made up of submultiples of the side, no matter 
 how minute these submultiples may be. It is possible 
 that Democritus may have attempted to get over this diffi- 
 culty, and reconcile incommensurability with his atomic 
 theory ; for we are told by Diogenes Laertius that he 
 
 ^ Arist. De insecah. lineis, p. 968, a, ^ Vid. Bretsch., Geom. vor Eukl., 
 
 ed. Bek. p. loi, et infra, p. 194. 
 
FROM THALE8 TO EUCLID. 185 
 
 wrote on incommensurable lines and solids (n-Epl aXo^wv 
 y(iafjififi)v KOI vatTToiv).' 
 
 The early Greek mathematicians, troubled no doubt by 
 these paradoxes of Zeno, and finding the progress of 
 mathematics impeded by their being made a subject of 
 dialectics, seem to have avoided all these difl&culties by 
 banishing from their science the idea of the Infinite — the 
 infinitely small as well as the infinitely great [vt'd. Euclid, 
 Book v., Def. 4). They laid down as axioms that any 
 quantity may be divided ad libitum ; and that, if two spaces 
 are unequal, it is possible to add their difference to itself 
 so often that every finite space can be surpassed.' Accord- 
 ing to this view, there can be no infinitely small difference 
 which being multiplied would never exceed a finite space. 
 
 Hippocrates of Chios, who must be distinguished from 
 his contemporary and namesake, the great physician of 
 Cos, was originally a merchant. All that we know of him 
 is contained in the following brief notices : — 
 
 {a). Plutarch tells us that Thales, and Hippocrates the 
 mathematician, are said to have applied themselves to 
 commerce." 
 
 (3). Aristotle reports of him : It is well known that 
 persons, stupid in one respect, are by no means so in 
 others (there is nothing strange in this : so Hippocrates, 
 though skilled in geometry, appears to have been in other 
 respects weak and stupid; and he lost, as they say, 
 through his simplicity, a large sum of money by the fraud 
 of the collectors of customs at Byzantium (vtto tUsv iv Bu2^av- 
 
 Tllsj TTSlTTIICOffroXoYWv))."' 
 
 {c). Johannes Philoponus, on the other hand, relates that 
 
 ■> Diog. Laert., ix., 47, ed. Cobet, p. ' In Vit. Solonis, ii. 
 
 239. '" Arist., Etk. ad Eud., vii., u. 14, 
 
 8 Archim., De quadr. parab., p. 18, p. 1247, a, 15, ed. Bek. 
 ed. Torelli. 
 
1 86 BE. ALLMAN ON GREEK GEOMETRY 
 
 Hippocrates of Chios, a merchant, having fallen in with a 
 pirate vessel, and having lost everything, went to Athens 
 to prosecute the pirates, and staying there a long time on 
 account of the prosecution, frequented the schools of the 
 philosophers, and arrived at such a degree of skill in 
 geometry, that he endeavoured to find the quadrature of 
 the circle/' 
 
 [d). We learn from Eudemus that CEnopides of Chios 
 was somewhat junior to Anaxagoras, and that after these 
 Hippocrates of Chios, who first found the quadrature of 
 the lune, and Theodorus of Cyrene, became famous in 
 geometry; and that Hippocrates was the first writer of 
 elements." 
 
 [e). He also taught, for Aristotle says that his pupils, 
 and those of his disciple ^schylus, expressed themselves 
 concerning comets in a similar way to the Pythagoreans." 
 
 [/). He is also mentioned by lamblichus, along with 
 Theodorus of Cyrene, as having divulged the geometrical 
 arcana of the Pythagoreans, and thereby having caused 
 mathematics to advance ({Tre'SwicE St ra fiadi'ifiaTa, siru t^evtivi- 
 yOrfaav iiaaoi irpoayovTe, fiaXiara GtoSio/OOc rt 6 KujO»}i/aIoc, Koi 
 'iTTirOKpaTrtQ 6 XTocJ." 
 
 (^). lamblichus goes on to say that the Pythagoreans 
 allege that geometry was made public thus : one of the 
 Pythagoreans lost his property ; and he was, on account 
 of his misfortune, allowed to make money by teaching 
 geometry." 
 
 [k). Proclus, in a passage quoted in the former part of 
 this Paper (Hermathena, vol. iii. p. 197, note), ascribes 
 to Hippocrates the method of reduction [airaywyrt). Proclus 
 
 " Philoponus, Comm. in Arist. phys. 35, ed. Bek. 
 
 ausc, i. 13. Brand., Schol. in Arist., " Iambi, de philcs. Pythag. lib. iii ; 
 
 P- 327. t), 44. Villoison, Anecdota Graeca, ii., p. 216. 
 
 '2 Procl. Comm., ed. Fried., p. 66. " Ibid. ; also Iambi, de Vit. Pyth. 
 
 "Arist., Meteor., i., 6, p. 342, b, c. 18, s. 8g. 
 
FROM THALES TO EUCLID. 187 
 
 defines airaywyri to be a transition from one problem or 
 theorem to another, which being known or determined, the 
 thing proposed is also plain. For example : when the 
 duplication of the cube is investigated, geometers reduce 
 the question to another to which this is consequent, i.e. 
 the finding of two mean proportionals, and afterwards 
 they inquire how between two given straight lines two 
 mean proportionals may be found. But Hippocrates of 
 Chios is reported to have been the first inventor of geo- 
 metrical reduction [aTraywyri) : who also squared the lune, 
 and made many other discoveries in geometry, and who 
 was excelled by no other geometer in his powers of con- 
 struction.'* 
 
 («'). Eratosthenes, too, in his letter to King Ptolemy III. 
 Euergetes, which has been handed down to us by Eutocius, 
 after relating the legendary origin of the celebrated problem 
 of the duplication of the cube, tells us that after geometers 
 had for a long time been quite at a loss how to solve the 
 question, it first occurred to Hippocrates of Chios that 
 if between two given lines, of which the greater is twice 
 the less, he could find two mean proportionals, then the 
 problem of the duplication of the cube would be solved. 
 But thus, Eratosthenes adds, the problem is reduced to 
 another which is no less difiicult." 
 
 {k). Eutocius, in his commentary on Archimedes [Czrc. 
 Dimens. Prop, i), tells us that Archimedes wished to show 
 that a circle is equal to a certain rectilineal area, a thing 
 which had been of old investigated by illustrious philo- 
 sophers." For it is evident that this is the problem con- 
 cerning which Hippocrates of Chios and Antiphon, who 
 carefiiUy searched after it, invented the false reasonings 
 which, I think, are well known to those who have looked 
 
 " Procl. Comm., ed. Fried., p. 212. Oxon. 1792. 
 
 " Archim., ex recens. Torelli, p. 144, " Anaxagoras, for example. 
 
1 88 DR. ALLMAN ON QBEEK GEOMETRY 
 
 into the History of Geometry of Eudemus and the Keria 
 (Kijpi'wv) of Aristotle.'' 
 
 On the passage (/) quoted above, from lamblichus, is 
 based the statement of Montucla, which has been repeated 
 since by recent writers on the history of mathematics,'" 
 that Hippocrates was expelled from a school of Pytha- 
 goreans for having taught geometry for money." 
 
 There is no evidence whatever for this statement, which 
 is, indeed, inconsistent with the passage [g) of lamblichus 
 which follows. Further, it is even possible that the person 
 alluded to in {g) as having been allowed to make money 
 by teaching geometry may have been Hippocrates him- 
 self; for — 
 
 1. He learned from the Pythagoreans ; 
 
 2. He lost his property through misfortune; 
 
 3. He made geometry public, not only by teaching, 
 
 but also by being the first writer of the ele- 
 ments. 
 
 This misapprehension originated, I think, with Fabri- 
 cius, who says : ' De Hippaso Metapontino adscribam adhuc 
 locum lamblichi ^ libro tertio de Philosophia Pythagorica 
 Graece necdum edito, p. 64, ex versione Nic. Scutelli : Hip- 
 pasus (videtur legendum Hipparchus) ejicitur e Pythagorae 
 schola eo quod primus sphaeram duodecivi angulorum (Dode- 
 caedron) edidisset (adeoque arcanum hoc evulgasset), Theo- 
 dorus etiam Cyrenaeus et Hippocrates Chius Geometra ejicitur 
 
 " Archim., exrecens. Torelli, p. 204. 'i Montacla, Histoite des Math., 
 ^o Bretsch., Geom. vor Eukl., p. 93; torn, i., p. 144, I'f ed. 1758; torn, i., 
 Hoefer, Histoire des Math., p. 135. p. 152, nouv. ed. an vii. ; the state- 
 Since the above was written, this state- ment is repeated in p. 155 of this 
 ment has been reiterated by Cantor, edition, and Simplicius is given as the 
 Gesch. der Math., p. 172; and by C. authority for it. lamblichus is, how- 
 Taylor, Geometry of Conies, Prole- ever, referred to by later writers as 
 gomena; 'f. xxviii. the authority for it. 
 
FROM THALES TO EUCLID. i8g 
 
 qui ex geometria quaestum faditabant. Confer Vit. Pyth. 
 c. 34 & 35.'" 
 
 In this passage Fabricius, who, however, had access to 
 a manuscript only, falls into several mistakes, as will be 
 seen by comparing it with the original, which I give 
 here : — 
 
 IIcpi 8' 'Iinratrov \iyov<Ti.v, u)S %v /itv ruiv Tlv6ayoptC<jiv, Sia Si to 
 i^fveyKfiv, koi ypdij/acQai irpioTos <T<j>aipav, t^v (k tu>v SwSexa i^aymvoiv 
 [■jTEvrayuivtov], airoXoiTO Kara daXarrav, <i)S dcrc^S^cras, Sdfav 8i XaySot, is 
 fivoi Se irdvTa tKiivov tov dvSpds" TTpocrayopdiova-i yap ovTio Tov Ilu^ayd- 
 pav, Koi ov Kokovcriv ovo/xari. eire'8<DK£ 8e to. /jMdrjfiaTa, eTrel t^ivrjve- 
 ■)(6r]<xav Sttro-ot irpoayovn, /tiaXio-ra ©cdSuipds t£ 6 Kupjjvaios, koI 
 '\inroKpa.T7)'; o Xtos. Xryovtrt 8c 01 IIii5oydp£i.ot l^cinjve^Oai ytuificTpCav 
 ouT<iis' oTTO^aXeiv riva rr]v ovcriav ruiv TLvOayopcliaV is Sc tovt 
 TjTV)(r](Te, SoOrjvai dtrrcu )(pr]iJux.TC<ra<T0aj, airo yew/xcTptas" t(ca\£tTO §£ 17 
 y^w/jitTpia TTpoi UvOayopov Icrropia.'^ 
 
 Observe that Fabricius, mistaking the sense, says that 
 Hippasus, too, was expelled. Hippocrates may have been 
 expelled by a school of P)rthagoreans with whom he had 
 been associated; but, if so, it was not for teaching geometry 
 for money, but for taking to himself the credit of Pytha- 
 gorean discoveries — a thing of which we have seen the 
 Pythagoreans were most jealous, and which they even 
 looked on as impious (airt/B^dae)." 
 
 As Anaxagoras was born 499 B. C, and as Plato, after 
 the death of Socrates, 399 B.C., went to Cyrene to hear 
 Theodorus {d), the lifetime of Hippocrates falls within the 
 fifth century before Christ. As, moreover, there could not 
 have been much commerce in the ^gean during the first 
 
 22 Jo. Albert! Fabricii BMiotheca concerning Hippocrates, the passage, 
 Graeca, ed. tertia, i., p. 505, Ham- with some modifications, occurs also in 
 burgi 1718. Iambi, de Vit. Pyth., c. 18, ss. 88 and 
 
 23 Iambi, de philos. Pyth. lib. iii. ; 89. 
 \mdaoTX,AnecdotaGraeca;n.,^.2ib. =* See Hkrmathkna, vol. iii., p. 
 With the exception of the sentence 199. 
 
I90 BR. ALLMAN ON GREEK GEOMETRY 
 
 quarter of the fifth century, and, further, as the state- 
 ments of Aristotle and Philoponus [b) and [c] fall in better 
 with the state of affairs during the Athenian supremacy — 
 even though we do not accept the suggestion of Bret- 
 schneider, made with the view of reconciling these incon- 
 sistent statements, that the ship of Hippocrates was taken 
 by Athenian pirates^'* during the Samian war (440 B.C.), in 
 which Byzantium took part — we may conclude with cer- 
 tainty that Hippocrates did not take up geometry until 
 after 450 B.C. "We have good reason to believe that at 
 that time there were Pythagoreans settled at Athens. 
 Hippocrates, then, was probably somewhat senior to So- 
 crates, who was a contemporary of Philolaus and Demo- 
 critus. 
 
 The paralogisms of Hippocrates, Antiphon, and Bryson, 
 in their attempts to square the circle, are referred to and 
 contrasted with one another in several passages of Aris- 
 totle^* and of his commentators — Themistius," Johan. Phi- 
 loponus,^* and Simplicius. Simplicius has preserved in his 
 Comm. to Phys. Ausc. of Aristotle a pretty full and partly 
 literal extract from the History of Geometry of Eudemus, 
 which contains an account of the work of Hippocrates and 
 others in relation to this problem. The greater part of 
 this extract had been almost entirely overlooked by writers 
 on the history of mathematics, until Bretschneider" re- 
 published the Greek text, having careflilly revised and 
 emended it. He also supplied the necessary diagrams, 
 some of which were wanting, and added explanatory and 
 
 "Bretsch., Geom. vor Eukl., p. Schol., i^. 2\i,h, iq. 
 
 98. »8Joh. Philop. f. 25, b, Schol., 
 
 ^^De Sophist. Elench.,ii,pT?.i';i,h, Brand, p. 211, b, 30. Ibid., f. 118, 
 
 and 172, ed. Bek. ; Fhys. Ausc, i., 2, Schol., p. 211, b, 41. Hid., f. 26, b, 
 
 p. 185, a, 14, ed. Bek. Schol., p. 212, a, 16. 
 
 '■' Theinist. f. l6, Schol. in Arist., ^' Bretsch., Geom. vor Eukl., pp. 
 
 Brand., p. 327, b, 33. Ibid., f. 5, 100-121. 
 
FROM THALE8 TO EUCLID. igi 
 
 critical notes. This extract is interesting and important, 
 and Bretschneider is entitled to much credit for the pains 
 he has taken to make it intelligible and better known. 
 
 It is much to be regretted, however, that Simplicius 
 did not merely transmit verbatim what Eudemus related, 
 and thus faithfully preserve this oldest fragment of Greek 
 geometry, but added demonstrations of his own, giving 
 references to the Elements of Euclid, who lived a century 
 and a-half later. Simplicius says : ' I shall now put down 
 literally what Eudemus relates, adding only a short ex- 
 planation by referring to Euclid's Elements, on account 
 of the summary manner of Eudemus, who, according to 
 archaic custom, gives only concise proofs.''" And in 
 another place he tells us that Eudemus passed over the 
 squaring of a certain lune as evident — indeed, Eudemus 
 was right in doing so — and supplies a lengthy demonstra- 
 tion himself.'^ 
 
 Bretschneider and Hankel, overlooking these passages, 
 and disregarding the frequent references to the Elements 
 of Euclid which occur in this extract, have drawn conclu- 
 sions as to the state of geometry at the time of Hip- 
 pocrates which, in my judgment, cannot be sustained. 
 Bretschneider notices the great circumstantiality of the 
 construction, and the long-windedness and the over-ela- 
 boration of the proofs.''' Hankel expresses surprise at the 
 fact that this oldest fragment of Greek geometry— 150 
 years older than Euclid's Elements — already bears that 
 character, typically fixed by the latter, which is so peculiar 
 to the geometry of the Greeks.'' 
 
 Fancy a naturalist finding a fragment of the skeleton 
 of some animal which had become extinct, but of which 
 there were living representatives in a higher state of 
 
 3» Bretsch., Geom. vorEukl., p. 109. '* Ibid., pp. 130, 131. 
 
 31 /5j^_ p. 113. ^ Hankel, Gesch. der Math., p. 112. 
 
192 DR. ALLMAN ON GREEK GEOMETRY 
 
 development ; and fancy him improving the portion of the 
 skeleton in his hands by making additions to it, so that it 
 might be more like the skeleton of the living animal ; then 
 fancy other naturalists examining the improved fragment 
 with so little attention as to exclaim : * Dear me ! how 
 strange it is that the two should be so perfectly alike ! ' 
 
 There is, moreover, much clumsiness, and a want of per- 
 spicuity, in the arrangement of the demonstrations — the con- 
 struction not being clearly stated, but being mixed up with 
 the proof : the proofs, too, which in several instances are 
 plainly supplied by Simplicius — inasmuch as propositions 
 of Euclid's Elements are quoted — are unskilful and weari- 
 some on account of the laboured demonstrations of evident 
 theorems, which are repeated several times under different 
 forms : while, on the other hand, some statements and 
 constructions which stand more in need of explanation 
 are passed over without remark. The conclusion is thus 
 forced on us that Simplicius was but a poor geometer ; and 
 we have greater reason, therefore, to regret that he was 
 not content with transmitting the work of Eudemus un- 
 altered. 
 
 I shall attempt now to restore this fragment by removing 
 from it everything that seems to me not to be the work of 
 Eudemus, and all reference to Euclid's Elements ; and by 
 stating briefly, but at the same time clearly and in order, 
 the several steps of each demonstration. I shall also notice 
 the theorems which are made use of, and the problems 
 whose solution is assumed in it. 
 
 'The difference between false conclusions that can be 
 proved to be siich, and others which cannot, he [Aristotle] 
 shows by some false reasonings in geometry.'* Amongst 
 the many persons who have sought the squaring of the 
 
 3« i)(eu5o7pa^T)/ia, literally a misde- on a faulty diagram, 
 lineation, a false reasoning founded 
 
. FROM THALES TO EUCLID. 193 
 
 circle (that is, to find a square which shall be equal to 
 a circle), both Antiphon and Hippocrates believed that 
 they had found it, and were equally mistaken. Antiphon's 
 mistake, on account of his not having- started from geo- 
 metrical principles, as we shall see, cannot be disproved 
 geometrically. That of Hippocrates, on the other hand, 
 since he was deceived although he clung to geometrical 
 principles, can be disproved geometrically. For we must 
 analyse and examine those reasonings only which, pre- 
 serving the acknowledged principles of the science, lead 
 thus to further conclusions ; but there is no use in examin- 
 ing those in which these principles are set aside.' 
 
 'Antiphon, having drawn a circle, inscribed in it one of 
 those polygons" that can be inscribed : let it be a square. 
 Then he bisected each side of this square, and through the 
 points of section drew straight lines at right angles to 
 them, producing them to meet the circumference; these 
 lines evidently bisect the corresponding segments of the 
 circle. He then joined the new points of section to the 
 ends of the sides of the square, so that four triangles were 
 formed, and the whole inscribed figure became an octagon. 
 And again, in the same way, he bisected each of the sides 
 of the octagon, and drew from the points of bisection per- 
 pendiculars ; he then joined the points where these per- 
 pendiculars met the circumference with the extremities of 
 the octagon, and thus formed an inscribed figure of sixteen 
 sides. Again, in the same manner, bisecting the sides 
 
 35 In Greek mathematical writers, re- poc re ko! laoydvwv, Sec. In Pappus, 
 rpdyavov, as far as I know, always however, these words, though some- 
 means a square. In this oldest geo- times used generally, for the most part 
 metrical writing, iidyuvoi', oKT(i7<oi'or, denote regular figures. The Greeks 
 and iro\6yayov denote regular hexa- could do this, for they had the words 
 gon, octagon, and polygon. This is TeTpoT\cupoi/, i!iVTi.Tr\tvfov, &c., for 
 not the case in the Elements of Euclid, quadrilateral, pentagon, &c. 
 who writes, e.g., -KiVTayavov itriiirAeu- 
 VOL IV. O 
 
194 JDR- ALLMAN ON GREEK GEOMETRY 
 
 of the inscribed polygon of sixteen sides, and drawing 
 straight lines, he formed a polygon of twice as many 
 sides ; and doing the same again and again, until he had 
 exhausted the surface, he concluded that in this manner a 
 polygon would be inscribed in the circle, the sides of 
 which, on account of their minuteness, would coincide 
 with the circumference of the circle. But we can substitute 
 for each polygon a square of equal surface ; therefore we 
 can, since the surface coincides with the circle, construct a 
 square equal to a circle.' 
 
 On this Simplicius observes : * the conclusion here is 
 manifestly contrary to geometrical principles, not, as 
 Alexander maintains, because the geometer supposes as 
 a principle that a circle can touch a straight line in one 
 point only, and Antiphon sets this aside ; for the geometer 
 does not suppose this, but proves it. It would be better 
 to say that it is a principle that a straight line cannot 
 coincide with a circumference, for one without meets the 
 circle in one point only, one within in two points, and not 
 more, and the meeting takes place in single points. Yet, 
 by continually bisecting the space between the chord and 
 the arc, it will never be exhausted, nor shall we ever reach 
 the circumference of the circle, even though the cutting 
 should be continued ad infinitum : if we did, a geometrical 
 principle would be set aside, which lays down that magni- 
 tudes are divisible ad infinitum. And Eudemus, too, says 
 that this principle has been set aside by Antiphon.'* 
 
 'But the squaring of the circle by means of segments, 
 he [Aristotle'**] says, maybe disproved geometrically ; he 
 would rather call the squaring by means of lunes, which 
 Hippocrates found out, one by segments, inasmuch as the 
 
 '8 But Eudemus was a pupU of '** /"Aj-J.^mjc. i.,2, p. iSs^, a, i6, ed. 
 
 Aristotle, and Antiphon was a con- Bek. 
 temporary of Democritus. 
 
FROM TSALES TO EUCLID. 195 
 
 lune is a segment of the circle. The demonstration is as 
 follows : — 
 
 'Let a semicircle a^y be described on the straight line 
 aj3 ; bisect a|3 in S ; from the point S draw a perpendicular 
 87 to a/3, and join 07 ; this will be the side of the square 
 inscribed in the circle of which afiy is the semicircle. On 
 07 describe the semicircle aty. Now, since the square on 
 a/3 is equal to double the square on 07 (and since the squares 
 on the diameters are to each other as the respective circles 
 or semicircles), the semicircle ay/S is double the semicircle 
 aty. The quadrant 078 is, therefore, equal to the semicircle 
 087. Take away the common segment lying between the 
 circumference ay and the side of the square ; then the 
 remaining lune my will be equal to the triangle ayS ; but 
 this triangle is equal to a square. Having thus shown 
 that the lune can be squared, Hippocrates next tries, 
 by means of the preceding demonstration, to square the 
 circle thus : — 
 
 'Let there be a straight line a)3, and let a semicircle 
 be described on it; take 78 double of aj3, and on it also 
 describe a semicircle ; and let the sides of a hexagon, yt, 
 iZ, and ^S be inscribed in it. On these sides describe the 
 semicircles 717?, £0^, 2^kS. Then each of these semicircles 
 described on the sides of the hexagon is equal to the semi ■ 
 circle a/3, for a/3 is equal to each side of the hexagon. The 
 four semicircles are equal to each other, and together are 
 then foiir times the semicircle on a/3. But the semicircle 
 on yS is also four times that on a/3. The semicircle on yS 
 is, therefore, equal to the four semicircles — that on a/3, 
 together with the three semicircles on the sides pf the 
 hexagon. Take away from the semicircles on the sides of 
 the hexagon, and from that on yS, the common segments 
 contained by the sides of the hexagon and the periphery of 
 the semicircle yS; the remaining lunes yije, t0?, and Z^kS, 
 together with the semicircle on a/3, will be equal to the 
 
 o 2 
 
196 LR. ALLMAN ON GREEK GEOMETRY 
 
 trapezium ye, «Z^, 2S. If we now take away from the 
 trapezium the excess, that is a surface equal to the lunes 
 (for it has been shown that there exists a rectilineal figure 
 equal to a lune), we shall obtain a remainder equal to 
 the semicircle aj3 ; we double this rectilineal figure which 
 remains, and construct a square equal to it. That square 
 will be equal to the circle of which a)3 is the diameter, and 
 thus the circle has been squared. 
 
 'The treatment of the problem is indeed ingenious; but 
 the wrong conclusion arises from assuming that as demon- 
 strated generally which is not so ; for not every lune has 
 been shown to be squared, but only that which stands over 
 the side of the square inscribed in the circle ; but the lunes 
 in question stand over the sides of the inscribed^hexagon. 
 The above proof, therefore, which pretends to have squared 
 the circle by means of lunes, is defective, and not conclu- 
 sive, on account of the false-drawn figure ()//t«Soypa0i)/ia) 
 which occurs in it." 
 
 ' Eudemus,^' however, tells us in his History of Geometry, 
 that Hippocrates demonstrated the quadrature of the lune, 
 not merely the lune on the side of the square, but gene- 
 rally, if one might say so : if, namely, the exterior arc of 
 the lune be equal to a semicircle, or greater or less than it. 
 I shall now put down literally (Kara Xi%ivf^ what Eudemus 
 relates, adding only a short explanation by referring to 
 Euclid's Elements, on account of the summary manner of 
 Eudemus, who, according to archaic custom, gives concise 
 proofs. 
 
 'In the second book of his History of Geometry, Eudemus 
 says : the squaring of lunes seeming to relate to an un- 
 
 3' I attribute the above observation 105-109, Bretsch., Geom. vor Eukl. 
 
 on the proof to Eudemus. What fol- ^b im_^ p. jq^. 
 
 lows in Simplicins seems to me not to 39 Simplicius did not adhere to his 
 
 be his. I have, therefore, omitted the intention, or else some transcriber has 
 
 remainder of § 83, and }§ 84, 85, pp. added to the text. 
 
FROM THALE8 TO EUCLID. igy 
 
 common class of figures was, on account of their relation 
 to the circle, first treated of by Hippocrates, and was 
 rightly viewed in that connection. We may, therefore, 
 more fully touch upon and discuss them. He started 
 with and laid down as the first thing useful for them, that 
 similar segments of circles have the same ratio as the 
 squares on their bases. This he proved by showing that 
 circles have the same ratio as the squares on their dia- 
 meters. Now, as circles are to each other, so are also 
 similar segments ; but similar segments are those which 
 contain the same part of their respective circles, as a semi- 
 circle to a semicircle, the third part of a circle to the third 
 part of another circle.*" For which reason, also, similar 
 Segments contain equal angles. The latter are in all semi- 
 circles right, in larger segments less than right angles, 
 and so much less as the segments are larger than semi- 
 circles ; and in smaller segments they are larger than 
 right angles, and so much larger as the segments are 
 smaller than semicircles. Having first shown this, he 
 described a lune which had a semicircle for boundary, 
 by circumscribing a semicircle about a right-angled isos- 
 celes triangle, and describing on the hypotenuse a seg- 
 
 *" Here T/irJiaa seems to be used for 8ti ivaWdi, ^ apiB/iol Kal ^ ypan/ial 
 sector : indeed, we have seen above koI rj irrcpeo koI ^ XP^""'' &<nrep ^SeiK- 
 that a lune was also called Tfifma. The vvri irore x^C^Sj iySfx^f-fy^y 7* xarci 
 word TOfi€VSj sector, may have been TravTOiV fxia airoSeflci heixBrivai' a\Ao 5ict 
 of later origin. The poverty of the t^ iu] elvai wvoiiaaiiivov ti trivTa ravra 
 Greek language in respect of geo- eV, apiB/xol firiK-q XP^'"^ ffreped, Kal 
 metrical terms has been frequently elSei Sia^fpeiv oAAijXax', x^P^^ i\a/ji.- 
 noticed. For example, they had no fidveTO. vvv Se Ka0i\ov SeiKyvrar ov 
 word for radius, and instead used the yap § ypaixiioi i) p dpifl/iol vmjpxii', 
 periphrasis t] 4k rod Kcirrpov. Again, AAA* ^ toSI, S Ka86\ov inroTiOeyrai 
 Archimedes nowhere uses the word uTrapxtiy- — Aristot., Anal., post., i., 
 parabola; and as to the imperfect' 5> P- 74. ». '7> ed. Bekker. This 
 terminology of the geometers of this passage is interesting in another re- 
 period, we have the direct statement of spect also, as it contains the germ 
 Aristotle, who says : koI ri ai'o^oyoi' of Algebra. 
 
1 98 DR. ALLMAN ON GREEK GEOMETRY 
 
 ment of a circle similar to those cut off by the sides. The 
 segment over the hypotenuse then being equal to the sum 
 of those on the two other sides, if the common part of the 
 triangle which lies over the segment on the base be added 
 to both, the lune will be equal to the triangle. Since the 
 lune, then, has been shown to be equal to a triangle, it can 
 be squared. Thus, then, Hippocrates, by taking for the 
 exterior arc of the lune that of a semicircle, readily squares 
 the lune. 
 
 'Hippocrates next proceeds to square a lune whose 
 exterior arc is greater than a semicircle. In order to do 
 so, he constructs a trapezium" having three sides equal to 
 each other, and the fourth — the greater of the two parallel 
 sides — such that the square on it is equal to three times 
 that on any other side ; he circumscribes a circle about the 
 trapezium, and on its greatest side describes a segment of 
 a circle similar to those cut off from the circle by the three 
 equal sides." By drawing a diagonal of the trapezium, it 
 will be manifest that the section in question is greater 
 than a semicircle, for the square on this straight line sub- 
 tending two equal sides of the trapezium must be greater 
 than twice the square on either of them, or than double 
 the square on the third equal side : the square on the 
 greatest side of the trapezium, which is equal to three 
 times the square on any one of the other sides, is therefore 
 less than the square on the diagonal and the square on the 
 third equal side. Consequently, the angle subtended by 
 
 *' Trapezia, like this, cut off from person from AirorlfleTai to Sfffeis, as 
 
 an isosceles triangle by a line parallel well as by the reference to Euclid, 
 
 to the base, occur in the Papyrus i. 9. A few lines lower there is a gap 
 
 Rhjnd. in the text, as Bretschneider has ob- 
 
 " Then foUows a proof, which I have served ; but the gap occurs in the work 
 omitted, that the circle can be circura- of Simplicius, and not of Eudemus, as 
 scribed about the trapezium. This Bretschneider has erroneously sup- 
 proof is obviously supplied by Simpli- posed.— C^ow;. vor Eukl., p. iii, and 
 cius, as is indicated by the change of note. 
 
FROM TEALES TO EUCLID. 
 
 199 
 
 the greatest side of the trapezium is acute, and the seg- 
 ment which contains it is, therefore, greater than a semi- 
 circle : but this is the exterior boundary of the lune. 
 Simplicius tells us that Eudemus passed over the squaring 
 of this lune, he supposes, because it was evident, and he 
 supplies it himself/' 
 
 'Further, Hippocrates shows that a lune with an ex- 
 terior arc less than a semicircle can be squared, and gives 
 the following constructio.n for the description of such a 
 lune : " — 
 
 'Let a/3 be the diameter of a circle whose centre is k; let 
 78 cut j3k in the point of bisection 7, and at right angles ; 
 through /3 draw the straight line /S^s, so that the part of it, 
 t^t, intercepted between the line 7S and the circle shall be 
 such that two squares on it shall be equal to three squares 
 on the radius ^k;" join kZ, and produce it to meet the 
 
 " Ibid., p. 113, § 88. I have omitted 
 it, as not being the work of Eudemus. 
 
 ** The whole construction, as Bret- 
 schneider has remarked, is quite ob- 
 scure and defective. The main point 
 on which the construction turns is the 
 determination of the straight line pQi, 
 and this is nowhere given in the text. 
 The determination of this line, how- 
 ever, can te inferred from the state- 
 ment in p. 114, Geom. vor Eukl., that 
 'it is assimied that the line ef inclines 
 towards iS ' ; and the further statement, 
 in p. 117, that 'it is assumed that the 
 square on cf is once and a-half the 
 square on the radius.' In order to 
 make the investigation intelligible, I 
 have commenced by stating how this 
 line /3fe is to be drawn. I have, as 
 usual, omitted the proofs of Simphcius. 
 
 Bretschneider, p. 114, notices the 
 archaic manner in which lines and 
 points are denoted in this investiga- 
 
 tion — Ti [ev0€Ta] ^(^* ^ AB, rb {_fft)fiiiov'\ 
 iip' ov K — and infers from it that Eu- 
 dpraus is quoting the very words of 
 Hippocrates. I have found this obser- 
 vation useful in aiding me to separate 
 the additions of Simplicius from the 
 work of Eudemus. The inference of 
 Bretschneider, however, cannot I think 
 be sustained, for the same manner of 
 expression is to be found in Aristotle. 
 *' The length of the line €f can be 
 determined by means of the theorem of 
 Pythagoras (Euchd, i., 47), coupled 
 with the theorem of Thales (Euclid, 
 iii., 31). Then, produce the hne ef 
 thus determined, so that the rectangle 
 under the whole Une thus produced 
 and the part produced shall be equal 
 to the square on the radius; or, in 
 archaic language, apply to the line'cj^^ 
 a rectangle which shall be equal to the 
 square on the radius, and which shall 
 be excessive by a square — a Pytha- 
 
200 DR. ALLMAN ON GREEK GEOMETRY 
 
 straight line drawn through 6 parallel to /3k, and let them 
 meet at t) ; join k£, j3»i (these lines will be equal) ; describe 
 then a circle round the trapezium /Skejj ; also, circumscribe 
 a circle about the triangle t^rj. Let the centres of these 
 circles be X and fi respectively. 
 
 ' Now, the segments of the latter circle on tZ, and Z^ij are 
 similar to each other, and to each of the segments of the 
 former circle on the equal straight lines tic, k3, |3»);" and, 
 since twice the square on tZ, is equal to three times the 
 square on k/3, the sum of the two segments on tZ and ^>) is 
 equal to the sum of the three segments on sk, k/3, /3ij ; to 
 each of these equals add the figure bounded by the straight 
 lines £K, Kj3, /3i), and the arc ij^e, and we shall have the lune 
 whose exterior arc is ek/Sti equal to the rectilineal figure 
 composed of the three triangles J^/Sij, Z^k, ske." 
 
 gorean problem, as Eudemus tells us. 
 (See Hermathena, vol. iii., pp. i8i, 
 196, 197.) If the calculation be made 
 by this method, or by the solution of a 
 quadratic equation, we find 
 
 Bretschneider makes some slip, 
 gives 
 
 2 
 
 and 
 
 ^&- 
 
 ■ifi-'Y 
 
 Geom. vor Eukl., p. 115, note. 
 
 ^* Draw lines from the points €, k, P, 
 and J) to \, the centre of the circle 
 described about the trapezium ; and 
 from € and 7) to fi, the centre of the 
 circle circumscribed about the triangle 
 ffj); it wiU be easy to see, then, that 
 the angles subtended by c/c, kP, and 7)3 
 at \ are equal to each other, and to 
 each of the angles subtended by f( and 
 drj at /I. The similarity of the segments 
 is then inferred ; but observe, that in 
 
 order to bring this under the definition 
 of similar segments given above, the 
 word segment must be used in a large 
 signification; and that farther, it re- 
 quires rather the converse of the defini- 
 tion, and thus raises the difficulty of 
 incommensurability. 
 
 The similarity of the segments might 
 also be inferred from the equality of 
 the alternate angles (fTjf and tikP, for 
 example). In Hermathena, vol. iii., 
 p. 203, I stated, following Bretschnei- 
 der and Hankel, that Hippocrates of 
 Chios did not know the theorem that 
 the angles in the same segment of 
 a circle are equal. But if the latter 
 method of proving the similarity of the 
 segments in the construction to which 
 the present note refers was that used 
 by Hippocrates, the statement in ques- 
 tion would have to be retracted. 
 
 *' A pentagon with a re-entrant angle 
 is considered here : but observe, 1°, that 
 it is not called a pentagon, that term 
 being then restricted to the regular 
 
FROM TEALES TO EVCLIT). 
 
 201 
 
 'That the exterior arc of this lune is smaller than a 
 semicircle, Hippocrates proves, by showing that the angle 
 tKjj lying within the exterior arc of the segment is obtuse, 
 which he does thus : Since the square on iX, is once and 
 a-half the square on the radius |3tc or ks, and since, on 
 account of the similarity of the triangles /3k£ and /32|k, the 
 square on m is greater than twice the square on kZ^," it 
 follows that the square on s^ is greater than the squares on 
 ac and kX, together. The angle ik») is therefore obtuse, and 
 consequently the segment in which it lies is less than a 
 semicircle. 
 
 'Lastly, Hippocrates squared a lune and a circle to- 
 gether, thus : let two circles be described about the centre 
 K, and let the square on the diameter of the exterior be six 
 times that of the interior. Inscribe a hexagon a^-^lvC, in 
 the inner circle, and draw the radii ko, k^, K7, and produce 
 
 pentagon ; and, 2°, that it is described 
 as a rectilineal figure composed of 
 three triangles. 
 
 « It is assumed here that the 
 angle flKt is obtuse, which it evi- 
 dently is. 
 
 Bretschneider points out that in this 
 paragraph the Greek text in the Aldine 
 is corrupt, and consequently obscure : 
 he corrects it by means of some trans- 
 positions and a few trifling additions. 
 (See Geom. vor Eukl., p. u8, note 2.) 
 
202 DR. ALLMAN ON GREEK GEOMETRY 
 
 them to the periphery of the exterior circle ; let them meet 
 it at the points ij, 0, i, respectively, and join tjA, Oi, mi. It 
 is evident that t|0, Qi are sides of the hexagon inscribed 
 in the larger circle. Now, on iji let there be described a 
 segment similar to that cut off by rj0. Since, then, the 
 square on jjj is necessarily three times greater than that on 
 1)0, the side of the hexagon," and the square on rtB six 
 times that on aj3, it is evident that the segment described 
 over rfi must be equal to the sum of the segments of the 
 outer circle over ij0 and Bi, together with those cut off in 
 the inner circle by all the sides of the hexagon. If we now 
 add, on both sides, the part of the triangle ijflj lying over 
 the segment ijt, we arrive at the result that the triangle r]Qi 
 is equal to the lune rjOi, together with the segments of the 
 inner circle cut off by the sides of the hexagon ; and if we 
 add on both sides the hexagon itself, we have the triangle, 
 together with the hexagon, equal to the said lune together 
 with the interior circle. Since, then, these rectilineal 
 figures can be squared, the circle, together with the lune, 
 can also be squared. 
 
 ' Simplicius adds, in conclusion, that it must be admitted 
 that Eudemus knows better all about Hippocrates of Chios, 
 being nearer to him in point of time, and being also a 
 pupil of Aristotle.' 
 
 If we examine this oldest fragment of Greek geometry, 
 we see, in the first place, that there is in it a defini- 
 tion of similar segments of circles ; they are defined to be 
 those which contain the same quotum of their respective 
 circles, as, for instance, a semicircle is similar to a semi- 
 
 *" Then follows the proof of this Plato (Timaeus, 54, D, ed. Stallbaum, 
 statement, which I have omitted, as I vii., p. 228) and Aristotle use it, as we 
 think it was added by Simplicius : the do, for the hypotenuse. It was some- 
 word T] imoTflrouira could scarcely have times used by later writers. Pappus 
 been used by Eudemus in the sense of for example, more generally, as it is 
 sub-tense, as it is in this passage. here. 
 
FROM TEALE8 TO EUCLID. 203 
 
 circle, the third part of one circle is similar to the third 
 part of another circle. 
 
 Next we find the following theorems : — 
 
 [a). Similar segments contain equal angles ;'" 
 
 [b). These in all semicircles are right ; segments which 
 are larger or smaller than semicircles contain, respectively, 
 acute or obtuse angles ; 
 
 (c). The side of a hexagon inscribed in a circle is equal 
 to the radius ; 
 
 [d). In any triangle the square on a side opposite to an 
 acute angle is less than the sum of the squares on the sides 
 which contain the acute angle ; 
 
 [e). In an obtuse-angled triangle the square on the side 
 subtending the obtuse angle is greater than the sum of the 
 squares on the sides containing it ; 
 
 [/). In an isosceles triangle whose vertical angle is 
 double the angle of an equilateral triangle, the square on 
 the base is equal to three times the square on one of the 
 equal sides ; 
 
 [g). In equiangular triangles the sides about the equal 
 angles are proportional ; 
 
 [h). Circles are to each other as the squares on their 
 diameters ; 
 
 [t). Similar segments of circles are to each other as the 
 squares on their bases. 
 
 Lastly, we observe that the solution of the following 
 problems is required : — 
 
 [a). Construct a square which shall be equal to a given 
 rectilineal figure ; 
 
 [b). Find a line the square on which shall be equal to 
 three times the square on a given line ;" 
 
 s" For this, or rather its converse, is AJso, see p. 197. 
 assumed in the demonstration, p. 200. ^' See theorem (/), supra. 
 
204 DR. ALLMAN ON GREEK GEOMETRY 
 
 [c). Find a line such that twice the square on it shall 
 be equal to three times the square on a given line ; 
 
 [d). Being given two straight lines, construct a tra- 
 pezium such that one of the parallel sides shall be equal 
 to the greater of the two given lines, and each of the three 
 remaining sides equal to the less ; 
 
 [e). About the trapezium so constructed describe a 
 circle ; 
 
 (/). Describe a circle about a given triangle ; 
 
 [g). From the extremity of the diameter of a semicircle 
 draw a chord such that the part of it intercepted between 
 the circle and a straight line drawn at right angles to the 
 diameter at the distance of one half the radius shall be 
 equal to a given straight line; 
 
 {h). Describe on a given straight line a segment of a 
 circle which shall be similar to a given one. 
 
 There remain to us but few more notices of the work 
 done by the geometers of this period : — 
 
 Antiphon, whose attempt to square the circle is given by 
 Simplicius in the above extract, and who is also mentioned 
 by Aristotle and some of his other commentators, is most 
 probably the Sophist of that name who, we are told, often 
 disputed with Socrates." It appears from a notice of 
 Themistius, that Antiphon started not only from the 
 square, but also from the equilateral triangle, inscribed 
 in a circle, and pursued the method and train of reasoning 
 above described." 
 
 Aristotle and his commentators mehtion another So- 
 phist who attempted to square the circle — Bryson, of 
 whom we have no certain knowledge, but who was pro- 
 bably a Pythagorean, and may have been the Bryson who 
 is mentioned by lamblichus amongst the disciples of Py- 
 
 s^Xenophon, Memorah. i., 6, \ \ ; *» xhemist., f. i6; Brandis, Sckol. 
 
 Diog. Laert. ii., 46, ed. Cobet, p. 44. in Arist., p. 327, b, 33. 
 
FROM TEALES TO EUCLID. 205 
 
 thagoras." Bryson inscribed a square," or more generally 
 any polygon," in a circle, and circumscribed another of the 
 same number of sides about the circle ; he then argued 
 that the circle is larger than the inscribed and less than 
 the circumscribed polygon, and erroneously assumed that 
 the excess in one case is equal to the defect in the other; 
 he concluded thence that the circle is the mean between the 
 two. 
 
 It seems, too, that some persons who had no know- 
 ledge of geometry took up the question, and fancied, as 
 Alexander Aphrodisius tells us, that they should find the 
 square of the circle in surface measure if they could find 
 a square number which is also a cyclical number" — 
 numbers as 5 or 6, whose square ends with the same 
 number, are called by arithmeticians cyclical numbers.*' 
 On this Hankel observes that 'unfortunately we cannot 
 assume that this solution of the squaring of the circle was 
 only a joke'; and he adds, in a note, that 'perhaps it was 
 of later origin, although it strongly reminds us of the 
 Sophists who proved also that Homer's poetry was a 
 geometrical figure because it is a circle of myths.'" 
 
 That the problem was one of public interest at that 
 time, and that, further, owing to the false solutions of 
 pretended geometers, an element of ridicule had become 
 attached to it, is plain from the reference which Aristo- 
 phanes makes to it in one of his comedies.'" 
 
 In the former part of this Paper (Hermathena, vol. iii. 
 p. 1 85), we saw that there was a tradition that the problem 
 of the quadrature of the circle engaged the attention of the 
 
 " Iambi., Vit. Pyth., c. 23. " Simplicius, in Bretsch. Geom. vor 
 
 " Alex. Aphrod., f. 30 ; Brandis, Eukl., p. 106. 
 
 Schol., p. 306, b. " Ibid. 
 
 56 Themist., f. 5 ; Brandis, Schol., '' Hankel, Geschich. der Math., p. 
 
 p.2ii; Johan.Philop., f. 118; Brandis, 116, and note. 
 
 Schol., p. 211. "" Birds, 1005. 
 
2o6 BR. ALLMAN ON GREEK GEOMETRY 
 
 Pythagoreans. We saw, too [ibid. p. 203), that they pro- 
 bably derived the problem from the Egyptians, who sought 
 to find from the diameter the side of a square whose area 
 should be equal to that of the circle. From their approxi- 
 mate solution, it follows that the Egyptians must have 
 assumed as evident that the area of a circle is propor- 
 tional to the square on its diameter, though they would 
 not have expressed themselves in this abstract manner. 
 Anaxagoras (499-428 B.C.) is recorded to have investigated 
 this problem during his imprisonment.*' 
 
 Vitruvius tells us that Agatharchus invented scene- 
 painting, and that he painted a scene for a tragedy which 
 .^schylus brought out at Athens, and that he left notes on 
 the subject. Vitruvius goes on to say that Democritus 
 and Anaxagoras, profiting by these instructions, wrote on 
 perspective." 
 
 We have named Democritus more than once : it is 
 remarkable that the name of this great philosopher, who 
 was no less eminent as a mathematician," and whose 
 fame stood so high in antiquity, does not occur in the 
 summary of the history of geometry preserved by Proclus. 
 In connection with this, we should note that Aristoxenus, 
 in his Historic Commentaries, says that Plato wished to 
 burn all the writings of Democritus that he was able to 
 collect ; but that the Pythagoreans, Amyclas and Cleinias, 
 prevented him, as they said it would do no good, inasmuch 
 as copies of his books were already in many hands. 
 Diogenes Laertius goes on to say that it is plain that this 
 was the case ; for Plato, who mentions nearly all the 
 ancient philosophers, nowhere speaks of Democritus." 
 
 *' 'AAA" 'kva^ayipas lifv iv t$ Seff- " Cicero, De finiius bonorum et 
 
 liuTiiplif rhv ToS KiK\ov TeTpayavuriihv malorum, i., 6 ; Diog. Laert., ix. 7 
 
 (ypafc. — Plut., De Exil., t. 17, vol.. ed. Cobet, p. 236. 
 iii, p. 734, ed. Didot. « Diog. Laert., Hid., ed. Cobet, 
 
 *^ De Arch., vii., Praef. p. 237. 
 
FROM THALE8 TO EUCLID. 207 
 
 We are also told by Diogenes Laertius that Demo- 
 critus was a pupil of Leucippus and of Anaxagoras, who 
 was forty years his senior;" and further, that he went 
 to Egypt to see the priests there, and to learn geometry 
 from them." 
 
 This report is confirmed by what Democritus himself 
 tells us : ' I have wandered over a larger portion of the 
 earth than any man of my time, inquiring about things 
 most remote ; I have observed very many climates and 
 lands, and have listened to very many learned men ; but 
 no one has ever yet surpassed me in the construction 
 of lines with demonstration ; no, not even the Egyptian 
 Harpedonaptae, as they are called [kol ypafi/jiitov awdimog 
 fUTa aiToSi^iog ovSsig K(i fie irapriXXa^e, oiiS' ol Aiyvirritov 
 KaXco/uffot ' ApTrtSovdiTTai'), with whom I lived five years in 
 all, in a foreign land.'" 
 
 We learn further, from Diogenes Laertius, that Demo- 
 critus was an admirer of the Pythagoreans ; that he seems 
 to have derived all his doctrines from Pythagoras, to such 
 a degree, that one would have thought that he had been 
 his pupil, if the difference of time did not prevent it ; that 
 at all events he was a pupil of some of the Pythagorean 
 schools, and that he was intimate with Philolaus." 
 
 Diogenes Laertius gives a list of his writings : amongst 
 those on mathematics we observe the following : — 
 
 Ilsjoi Sia^opSc yvoijuovoc V Tcpt ^/.avorioe kvkXou koi atpaipriQ 
 (lit., On the difference of the gnomon, or on the contact 
 of the circle and the sphere. Can what he has in view 
 be the following idea: that, the gnomon, or carpenter's 
 rule, being placed with its vertex on the circumference of 
 a circle, in the limiting position, when one leg passes 
 
 "Diog. Laert., is.., 7, ed. Cobet, i.,p.304, ed. Sylburg; MuUach,i^ra£7«. 
 
 p 235. Phil. Graec, p. 370. 
 
 66 Prid., p. 236. °* I^i°g- Laert., ii., 7, ed. Cobet, 
 
 6' Democrit., ap. Clem. AXty.. Strom., p. 236. 
 
2o8 DR. ALLMAN ON GREEK GEOMETRY 
 
 through the centre, the other will determine the tangent ?) ; 
 one on geometry ; one on numbers ; one on incommen- 
 surable lines and solids, in two books; ^ kKTivojpa<pir\ (a 
 description of rays, probably perspective)." 
 
 "We also learn, from a notice of Plutarch, that Demo- 
 critus raised the following question : ' If a cone were cut 
 by a plane parallel to its base [obviously meaning, what 
 we should now call one infinitely near to that plane], what 
 must we think of the surfaces of the sections, that they are 
 equal or unequal ? For if they are unequal, they will show 
 the cone to be irregular, as having many indentations 
 like steps, and unevennesses ; and if they are equal, the 
 sections will be equal, and the cone will appear to have 
 the property of a cylinder, viz., to be composed of equal, 
 and not unequal, circles, which is very absurd.' ■"* 
 
 If we examine the contents of the foregoing extracts, 
 and compare the state of geometry as presented to us 
 in them with its condition about half a century earlier, 
 we observe that the chief progress made in the interval 
 concerns the circle. The early Pythagoreans seem not to 
 have given much consideration to the properties of the 
 circle ; but the attention of the geometers of this period 
 was naturally directed to them in connection with the 
 problem of its quadrature. 
 
 We have already set down, seriatim, the theorems and 
 problems relating to the circle which are contained in the 
 extract from Eudemus. 
 
 Although the attempts of Antiphon and Bryson to 
 square the circle did not meet with much favour from the 
 ancient geometers, and were condemned on account of the 
 paralogisms in them, yet their conceptions contain the 
 first germ of the infinitesimal method : to Antiphon is due 
 
 " Diog. Laert., ix., 7, ed. Cobet, '» Plut., de Co7nm. Not., p. 1321, ed. 
 
 pp. 238 and 239. Didot. 
 
FHOM THALES TO EUCLID. 209 
 
 the merit of having first got into the right track by intro- 
 ducing for the solution of this problem — in accordance 
 with the atomic theory then nascent — the fundamental 
 idea of infinitesimals, and by trying to exhaust the circle 
 by means of inscribed polygons of continually increasing 
 number of sides ; Bryson is entitled to praise for having 
 seen the necessity of taking into consideration the circum- 
 scribed as well as the inscribed polygon, and thereby 
 obtaining a superior as well as an inferior limit to the 
 area of the circle. Bryson's idea is just, and should be 
 regarded as complementary to the idea of Antiphon, which 
 it limits and renders precise. Later, after the method of 
 exhaustions had been invented, in order to supply demon- 
 strations which were perfectly rigorous, the two limits, 
 inferior and superior, were always considered together, 
 as we see in Euclid and Archimedes. 
 
 We see, too, that the question which Plutarch tells us 
 that Democritus himself raised involves the idea of infini- 
 tesimals ; and it is evident that this question, taken in con- 
 nection with the axiom in p. 185, must have presented real 
 difficulties to the ancient geometers. The general question 
 which underlies it was, as is well known, considered and 
 answered by Leibnitz : ' Caeterum aequalia esse puto, non 
 tantum quorum differentia est omnino nulla, sed et quo- 
 rum differentia est incomparabiliter parva ; et lic^t ea Nihil 
 omnino dici non debeat, non tamen est quantitas compara- 
 bilis cum ipsis, quorum est differentia. Quemadmodum 
 si lineae punctum alterius lineae addas, vel superficiei 
 lineam, quantitatem non auges. Idem est, si lineam qui- 
 dem lineae addas, sed incomparabiliter minorem. Nee 
 uUa constructione tale augmentum exhiberi potest. Sci- 
 licet eas tantum homogeneas quantitates comparabiles 
 esse, cum EucUde, lib. v., defin. 5, censeo, quarum una 
 numero, sed finito, multiplicata, alteram superare potest. 
 Et quae tali quantitate non differunt, aequalia esse statue, 
 
 VOL. IV. P 
 
2 10 BR. ALLMAN ON GREEK GEOMETRY 
 
 quod etiam Archimedes sumsit, aliique post ipsum omnes. 
 Et hoc ipsum est, quod dicitur difFerentiam esse data 
 qua vis minorem. Et Archimedeo quidem processu res 
 semper deductione ad absurdum confirmari potest.'" 
 Further, we have seen that Democritus wrote on the 
 contact of the circle and of the sphere. The employment 
 of the gnomon for the solution of this problem seems to 
 show that Democritus, in its treatment, made use of the 
 infinitesimal method ; he might have employed the 
 gnomon either in the manner indicated above, or, by 
 making one leg of the gnomon pass through the centre of 
 the circle, and moving the other parallel to itself, he 
 could have found the middle points of a system of parallel 
 chords, and thus ultimately the tangents parallel to them. 
 At any rate this problem was a natural subject of inquiry 
 for the chief founder of the atomic theory, just as Leibnitz 
 — the author of the doctrine of monads and the founder of 
 the infinitesimal calculus — was occupied with this same 
 subject of tangency. 
 
 We observe, further, that the conception of the irra- 
 tional (aXoyov), which had been a secret of the Pythagorean 
 school, became generally known, and that Democritus 
 wrote a treatise on the subject. 
 
 We have seen that Anaxagoras and Democritus wrote 
 on perspective, and that this is not the only instance in 
 which the consideration of problems in geometry of three 
 dimensions occupied the attention of Democritus. 
 
 On the whole, then, we find that considerable progress 
 had been made in elementary geometry; and indeed the 
 appearance of a treatise on the elements is in itself an 
 indication of the same thing. We have further evidence 
 of this, too, in the endeavours of the geometers of this 
 period to extend to the circle and to volumes the results 
 
 ■" Leibnitii Opera Omnia, ed. L. Dutens, torn. iii. p. 328. 
 
FROM THALES TO EUCLID. 211 
 
 which had been arrived at concerning rectilineal figures 
 and their comparison with each other. The Pythagoreans, 
 as we have seen, had shown how to determine a square 
 whose area was any multiple of a given square. The 
 question now was to extend this to the cube, and, in 
 particular, to solve the problem of the duplication of the 
 cube. 
 
 Proclus (after Eudemus) and Eratosthenes tell us [h and 
 t, p. 187) that Hippocrates reduced this question to one 01 
 plane geometry, namely, the finding of two mean propor- 
 tionals between two given straight lines, the greater of 
 which is double the less. Hippocrates, therefore, must 
 have known that if four straight lines are in continued 
 proportion, the first has the same ratio to the fourth that 
 the cube described on the first as side has to the cube 
 described in like manner on the second. He must then 
 have pursued the following train of reasoning : — Suppose 
 the problem solved, and that a cube is found which is 
 double the given cube ; find a third proportional to the 
 sides of the two cubes, and then find a fourth proportional 
 to these three lines ; the fourth proportional must be double 
 the side of the given cube : if, then, two mean propor- 
 tionals can be found between the side of the given cube 
 and a line whose length is double of that side, the problem 
 will be solved. As the Pythagoreans had already solved 
 the problem of finding a mean proportional between two 
 given lines — or, which comes to the same, to construct a 
 square which shall be equal to a given rectangle — it was 
 not unreasonable for Hippocrates to suppose that he had 
 put the problem of the duplication of the cube in a fair 
 way of solution. Thus arose the famous problem of finding 
 two mean proportionals between two given lines — a prob- 
 lem which occupied the attention of geometers for many 
 centuries. Although, as Eratosthenes observed, the diffi- 
 culty is not in this way got over ; and although the new 
 
 P 2 
 
212 DB. ALLMAN ON GREEK GEOMETRY 
 
 problem cannot be solved by means of the straight line 
 and circle, or, in the language of the ancients, cannot 
 be. referred to plane problems, yet Hippocrates is entitled 
 to much credit for this reduction of a problem in stereo- 
 metry to one in plane geometry. The tragedy to which 
 Eratosthenes refers in this account of the legendary origin 
 of the problem is, according to Valckenaer, a lost play of 
 Euripides, named IloXuEtSoc:" if this be so, it follows that 
 this problem of the duplication of the cube, as well as that 
 of the quadrature of the circle, was famous at Athens at 
 this period. 
 
 Eratosthenes, in his letter to Ptolemy III., relates that 
 one of the old tragic poets introduced Minos on the stage 
 erecting a tomb for his son Glaucus ; and then, deeming 
 the structure too mean for a royal tomb, he said 'double 
 it, but preserve the cubical form': /xiKpov j tXt^ae ^aat- 
 Xeikov ai\K.ov Taipov, im\a.aiog iarw. tov Se tov kv(3ov ftij 
 •K^aXEit,'." Eratosthenes then relates the part taken by 
 Hippocrates of Chios towards the solution of this problem 
 as given above (p. 187), and continues : 'Later [in the time 
 of Plato], so the story goes, the Delians, who were suffer- 
 ing from a pestilence, being ordered by the oracle to 
 double one of their altars, were thus placed in the same 
 difiiculty. They sent therefore to the geometers of the 
 Academy, entreating them to solve the question.' This 
 problem of the duplication of the cube — henceforth known 
 as the Delian Problem — may have been originally sug- 
 gested by the practical needs of architecture, as indicated 
 in the legend, and have arisen in Theocratic times ; it 
 
 '* See Reimer, Historia problematis kenaer shows that these words of Era- 
 
 de cubi duplicationc, p. 20, Gottingae, tosthenes contain two verses, which he 
 
 1798; and Biering, Historia proble- thus restores : — 
 
 matis cubi duplicandi, p. 6, Hauniae, Mlkp'ov y iXefos ^aaiXimi injicbi. Ti.j)ou- 
 
 1844.. AtTrAacrios eOT6j, Tou Kv^ov fie ja?) tT^aXrj<i. 
 
 '^ Archim., ed. ToreUi, p. 144. Vale- See Reimer, /. c. 
 
FEOM THALES TO EUCLID. 213 
 
 may subsequently have engaged the attention of the Py- 
 thagoreans as an object of theoretic interest and scientific 
 inquiry, as suggested above. 
 
 These two ways of looking at the question seem suited 
 for presenting it to the public on the one hand and to 
 mathematical pupils on the other. From the consideration 
 of a passage in Plutarch,'* however, I am led to believe 
 that the new problem — to find two mean proportionals 
 between two given lines— which arose out of it, had a 
 deeper significance, and that it must have been regarded 
 by the Pythagorean philosophers of this time as one of 
 great importance, on account of its relation to their 
 cosmology. 
 
 In the former part of this Paper (Hermathena, vol. 
 iii. p. 194) we saw that the Pythagoreans believed that 
 the tetrahedron, octahedron, icosahedron, and cube cor- 
 responded to the four elements of the real world. This 
 doctrine is ascribed by Plutarch to Pythagoras himself;" 
 Philolaus, who lived at this time, also held that the 
 elementary nature of bodies depended on their form. The 
 tetrahedron was assigned to fire, the octahedron to air, 
 the icosahedron to water, and the cube to earth ; that is 
 to say, it was held that the smallest constituent parts 
 of these substances had each the form assigned to it." 
 This being so, what took place, according to this theory, 
 when, under the action of heat, snow and ice melted, or 
 water became vapour ? In the former case, the elements 
 which had been cubical took the icosahedral form, and 
 
 ^* Symp., viii., Quaestio 2, c. 4 ; t^jv tov iravrhs txtpaipaj/. 
 Plut. Opera, ed. Didot, vol. iv., p. 877. nXiiraiv S« koX iv rovTois vvSayopi^fi. 
 
 " Tlu8ay6pas, ircVre o'xw'''"'' oyruv Plut. Plac, ii., 6, 5 & 6; Opera, ed. 
 
 <mpi£>v, avep KoKftrai Kal fiaBT/i/iaTuca, Didot, vol. iv., p. 1081. 
 cK fiev TOV Kv$ov ^Tjirl yfyovevai r^v ^^ Stob. Eclog. ab Heeren, lib. i., 
 
 yriv, ex 8t Trjs irvpafiiSos rh iivp, iic 5e p. 10. See also Ze\leT,Die P/u'los. der 
 
 ToC oKTaeSpou rhp aepa, €K Si tov ci/to- Griechen, Erster Theil, p. 376, Leip- 
 
 ffae'Sfioi/ xi vSap, ix 5e ToiJ SwSeKaeSpov zig, 1876. 
 
214 DR. ALLMAN ON GREEK GEOMETRY 
 
 in the latter the icosahedral elements became octahedral. 
 Hence would naturally arise the following geometrical 
 problems : — 
 
 Construct an icosahedroh which shall be equal to a 
 given cube ; 
 
 Construct an octahedron which shall be equal to a 
 given icosahedron. 
 
 Now Plutarch, in 'h.is,Symp.,viu., Quaestio ii. — Ilwc IlXarwv 
 tXs7£ Tov 6eov au ytwfUTpuv, 3 & 4" — accepts this theory of 
 Pythagoras and Philolaus, and in connection with it points 
 out the importance of the problem : ' Given two figures, to 
 construct a third which shall be equal to one of the two 
 and similar to the other' — which he praises as elegant, and 
 attributes to Pythagoras (see Hermathena, vol. iii.p. 182). 
 It is evident that Plutarch had in view solid and not plane 
 figures ; for, having previously referred to the forms of the 
 constituent elements of bodies, viz., air, earth, fire, and 
 water, as being those of the regular solids, omitting the 
 dodecahedron, he goes on as follows : ' What,' said Dio- 
 genianus, 'has this [the problem — given two figures, to 
 describe a third equal to one and similar to the other] to 
 do with the subject?' 'You will easily know,' I said, 'if 
 you call to mind the division in the Timaeus, which di- 
 vided into three the things first existing, from which the 
 Universe had its birth ; the first of which three we call 
 God [9£oe, the arranger], a name most justly deserved ; the 
 second we call matter, and the third ideal form. . . . God 
 was minded, then, to leave nothing, so far as it could be 
 accomplished, undefined by limits, if it was capable of 
 being defined by limits ; but [rather] to adorn nature 
 with proportion, measurement, and number : making 
 some one thing [that is, the universe] out of the ma- 
 terial taken all together ; something that would be 
 
 " Plut. Oj>era, ed. Didot, vol. iv, pp. 876, 7. 
 
FROM THALES TO EUCLID. 215 
 
 like the ideal form and as big as the matter. So having 
 given himself this problem, when the two were there, 
 he made, and makes, and for ever maintains, a third, 
 viz., the universe, which is equal to the matter and like 
 the model.' 
 
 Let us now consider one of these problems — the 
 former — and, applying to it the method of reduction, see 
 what is required for its solution. Suppose the problem 
 solved, and that an icosahedron has been constructed 
 which shall be equal to a given cube. Take now another 
 icosahedron, whose edge and volume are supposed to be 
 known, and, pursuing the same method which was followed 
 above in p. 211, we shall find that, in order to solve the 
 problem, it would be necessary — 
 
 1 . To find the volume of a polyhedron ; 
 
 2. To find a line which shall have the same ratio to a 
 given line that the volumes of two given polyhedra have 
 to each other ; 
 
 3. To find two mean proportionals between two given 
 lines ; and 
 
 4. To construct on a given line as edge a polyhedron 
 which shall be similar to a given one. 
 
 Now we shall see that the problem of finding two mean 
 proportionals between two given lines was first solved by 
 Archytas of Tarentum — ultimus Pythagoreorum — then by 
 his pupil Eudoxus of Cnidus, and thirdly by Menaechmus, 
 who was a pupil of Eudoxus, and who used for its solu- 
 tion the conic sections which he had discovered : we shall 
 see further that Eudoxus founded stereometry by showing 
 that a triangular pyramid is one-third of a prism on the 
 same base and between the same parallel planes ; lastly, 
 we shall find that these great discoveries were made 
 with the aid of the method of geometrical analysis which 
 either had meanwhile grown out of the method of reduc- 
 tion or was invented by Archytas. 
 
2i6 DR. ALLMAN ON GREEK GEOMETRY 
 
 It is probable that a third celebrated problem — the 
 trisection of an angle — also occupied the attention of the 
 geometers of this period. No doubt the Egyptians knew 
 how to divide an angle, or an arc of a circle, into two 
 equal parts ; they may therefore have also known how to 
 divide a right angle into three equal parts. We have seen, 
 moreover, that the construction of the regular pentagon 
 was known to Pythagoras, and we infer that he could have 
 divided a right angle into five equal parts. In this way, 
 then, the problem of the trisection of any angle — or the 
 more general one of dividing an angle into any number 
 of equal parts — would naturally arise. Further, if we 
 examine the two reductions of the problem of the tri- 
 section of an angle which have been handed down to 
 us from ancient times, we shall see that they are such 
 as might naturally occur to the early geometers, and that 
 they were quite within the reach of a Pythagorean — one 
 who had worthily gone through his noviciate of at least 
 two years of mathematical study and silent meditation. 
 For this reason, and because, moreover, they furnish good 
 examples of the method called airaywyrt, I give them here. 
 
 Let us examine what is required for the trisection of 
 an angle according to the method handed down to us by 
 Pappus." 
 
 Since we can trisect a right angle, it follows that the 
 trisection of any angle can be effected if we can trisect an 
 acute angle. 
 
 Let now a(3y be the given acute angle which it is 
 required to trisect. 
 
 From any point a on the line a^, which forms one leg 
 of the given angle, let fall a perpendicular 07 on the other 
 leg, and complete the rectangle ayj3S. Suppose now that 
 the problem is solved, and that a line is drawn making 
 
 '8 Pappi Alex. Collect., ed. Hultsch, vol. i. p. 274. 
 
FROM TEALES TO EUCLID. 217 
 
 with /Sy an angle which is the third part of the given 
 angle ajSy ; let this line cut 07 in l„ and be produced until 
 it meet Sa produced at the point i. Let now the straight 
 line Cs be bisected in ij, and art be joined ; then the lines 
 tri, rj£, OT), and /3a are all evidently equal to each other, 
 and, therefore, the line Zi is double of the line aj3, which is 
 known. 
 
 The problem of the trisection of an angle is thus 
 reduced to another : — 
 
 From any vertex |3 of a rectangle jSSay draw a line 
 /32e, so that the part ^t of it intercepted between the two 
 opposite sides, one of which is produced, shall be equal 
 to a given line. 
 
 This reduction of the problem must, I think, be referred 
 to an early period : for Pappus " tells us that when the 
 ancient geometers wished to cut a given rectilineal angle 
 into three equal parts they were at a loss, inasmuch as 
 the problem which they endeavoured to solve as a plane 
 problem could not be solved thus, but belonged to the 
 class called solid ; ^'' and, as they were not yet acquainted 
 with the conic sections, they could not see their way : 
 but, later, they trisected an angle by means of the conic 
 sections. He then states the problem concerning a rect- 
 angle, to which the trisection of an angle has been just 
 now reduced, and solves it by means of a hyperbola. 
 
 The conic sections, we know, were discovered by 
 
 '9 Ibid., vol. i. p. 270, et seq. one or more conic sections were called 
 
 s" The ancients distingiiished three solid, inasmuch as for their construc- 
 
 kinds of problems— ^/ane, solid, and tion we must use the superficies of solid 
 
 linear. Those which could be solved figures— to wit, the sections of a cone, 
 
 by means of straight lines and circles A third kind, caUed linear, remains, 
 
 were caUed plane ; and were justly so which required for their solution curves 
 
 called, as the lines by which the prob- of a higher order, such as spirals, 
 
 lems of this kind could be solved have quadratrices, conchoids, and cissoids. 
 
 their origin inplano. Those problems See Pappi Collect.,, ed. Hultsch, 
 
 whose solution is obtained by means of vol. i. pp. 54 and 270. 
 
2i8 BR. ALLMAN ON GREEK GEOMETRY 
 
 Menaechmus, a pupil of Eudoxus (409-356 B.C.), and the 
 discovery may, therefore, be referred to the middle of the 
 fourth century. 
 
 Another method of trisecting an angle is preserved 
 in the works of Archimedes, being indicated in Prop. 8 
 of the Lemmata '^ — a book which is a translation into 
 Latin from the Arabic. The Lemmata are referred to 
 Archimedes by some writers, but they certainly could not 
 have come from him in their present form, as his name 
 is quoted in two of the Propositions. They may have been 
 contained in a note-book compiled from various sources by 
 some later Greek mathematician,*^ and this Proposition 
 may have been handed down from ancient times. 
 
 Prop. 8 of the Lemmata is : ' If a chord AB of a 
 circle be produced until the part produced BC is equal 
 to the radius ; if then the point C be joined to the centre 
 of the circle, which is the point D, and if CD, which cuts 
 the circle in F, be produced until it cut it again in E, the 
 arc AE will be three times the arc BE.' This theorem 
 suggests the following reduction of the problem : — 
 
 With the vertex A of the given angle BAC as centre, 
 and any lines AC or AB as radius, let a circle be de- 
 scribed. Suppose now that the problem is solved, and 
 that the angle EAC is the third part of the angle BAC ; 
 through B let a straight line be drawn parallel to AE, and 
 let it cut the circle again in G and the radius CA produced 
 in F. Then, on account of the parallel lines AE and FGB, 
 the angle ABG or the angle BGA, which is equal to it, 
 will be double of the angle GFA; but the angle BGA 
 is equal to the sum of the angles GFA and GAF ; the 
 
 *i Archim. ex recens. Torelli, p. ' Itaque puto, haec lemmata e plurium 
 
 358. mathematicorum operibus esse ex- 
 
 "- See ibid., Praefatio J. Torelli, cerpta, neque definiri jam potest, 
 
 pp. xviii. and xix. See also Heiberg, quantum ex lis Archimedi tribuendum 
 
 Quaest. Aixliim., p. 24, who says : sit.' 
 
FROM THALES TO EUCLID. 219 
 
 angles GFA and GAF are, therefore, equal to each other, 
 and consequently the lines GF and GA are also equal. The 
 problem is, therefore, reduced to the following : From B 
 draw the straight line BGF, so that the part of it, GF, 
 intercepted between the circle and the diameter CAD 
 produced shall be equal to the radius." 
 
 For the reasons stated above, then, I think that the 
 problem of the trisection of an angle was one of those 
 which occupied the attention of the geometers of this 
 period. Montucla, however, and after him many writers 
 on the history of mathematics, attribute to Hippias of 
 Elis, a contemporary of Socrates, the invention of a 
 transcendental curve, known later as the Quadratrix of 
 Dinostratus, by means of which an angle may be divided 
 into any number of equal parts. This statement is made 
 on the authority of the two following passages of 
 Proclus : — 
 
 ' Nicomedes trisected every rectilineal angle by means 
 of the conchoidal lines, the inventor of whose particular 
 nature he is, and the origin, construction, and properties 
 of which he has explained. Others have solved the same 
 problem by means of the quadratrices of Hippias and 
 Nicomedes, making use of the mixed lines which are 
 called quadratrices ; others, again, starting from the 
 spirals of Archimedes, divided a rectilineal angle in a 
 given ratio.'** 
 
 ' In the same manner other mathematicians are accus- 
 tomed to treat of curved lines, explaining the properties 
 of each form. Thus, Apollonius shows the properties of 
 each of the conic sections ; Nicomedes those of the con- 
 
 83 See F. Vietae Opera Maihema- given by Montucla, but he did not 
 
 tica, studio F. a Schooten, p. 245, give any references. See Hist, des 
 
 Lugd. Bat. 1646. These two reduc- Math., torn. i. p. 194, !'"<■ ed. 
 tions of the trisection of an angle were ^ Prod. Comm., ed. Fried., p. 272. 
 
2 20 DB. ALLMAN ON GREEK GEOMETRY 
 
 choids ; Hippias those of the quadratrix, and Perseus 
 those of the spirals' ((riretpiKoiv)/* 
 
 Now the question arises whether the Hippias referred 
 to in these two passages is Hippias of Elis. Montucla 
 believes that there is some ground for this statement, for 
 he says : ' Je ne crois pas que 1' antiquity nous fournisse 
 aucun autre geometre de ce nom, que celui dont je parle.''° 
 Chasles, too, gives only a qualified assent to the statement. 
 Arneth, Bretschneider, and Suter, however, attribute the 
 invention of the quadratrix to Hippias of Elis without any 
 qualification." Hankel, on the other hand, says that surely 
 the Sophist Hippias of Elis cannot be the one referred to, 
 but does not give any reason for his dissent.*' I agree 
 with Hankel for the following reasons : — 
 
 1. Hippias of Elis is not one of those to whom the 
 progress of geometry is attributed in the summary of the 
 history of geometry preserved by Proclus, although he is 
 mentioned in it as an authority for the statement con- 
 cerning Ameristus [or Mamercus].'' The omission of his 
 name would be strange if he were the inventor of the 
 quadratrix. 
 
 2. Diogenes Laertius tells us that Archytas was the 
 first to apply an organic motion to a geometrical dia- 
 gram;'" and the description of the quadratrix requires 
 such a motion. 
 
 •5 Procl. Comm., ed. Fried., p. 356. di Biiliografia e di Storia delle Scienze 
 
 86 Montucl., Hist, des Math., torn. i. Matematiche e Fisiche, says : • A pag. 
 
 p. 181, nouvle ed. 31 (lin. 3-6), Hippias, I'inventore della 
 
 8' Chasles, Histoire de la Geam., quadratrice, 6 identificato col Sofista 
 
 p. 8; KxneMv, Gesch.der Math.,-^.()y, Hippias, il che veramente avea gia 
 
 Bretsch., Geom. vor Eukl., p. 94 ; fatto il Bretschneider (pag. 94, lin. 39- 
 
 Suter, Gesch. der Math. Wissenschaft., 42), ma senza dame la minima prova.' 
 
 P- 32- Bullet., &c., torn. v. p. 297. 
 
 ^^Haiikel, Gesck. der Math., 'p.\e,l, 89 procl. Comm., ed. Fried., p. 
 
 note. Hankel, also, in a review of Suter, 65. 
 
 GeschichtederMathematischen Wissen- »» Diog. Laert., viii. c. 4, ed. Cobet, 
 
 .uhaften, published in the Bullettino p. 224. 
 
FROM THALE8 TO EUCLID. 
 
 221 
 
 3. Pappus tells us that: 'For the quadrature of a circle 
 a certain line was assumed by Dinostratus, Nicomedes, 
 and some other more recent geometers, which received its 
 name from this property : it is called by them the qua- 
 dratrix.'" 
 
 4. With respect to the observation of Montucla, I may 
 mention that there was a skilful mechanician and geo- 
 meter named Hippias contemporary with Lucian, who 
 describes a bath constructed by him.'" 
 
 I agree, then, with Hankel that the invention of the 
 quadratrix is erroneously attributed to Hippias of Elis. 
 But Hankel himself, on the other hand, is guilty of a 
 still greater anachronism in referring back the Method of 
 Exhaustions to Hippocrates of Chios. He does so on 
 grounds which in my judgment are quite insufl&cient. 
 
 " Pappi, Collect., ed. Hultsch, vol. i. 
 pp. 250 and 252. 
 
 '2 Hippias, seu Balneum. Since 
 the above was written I find that 
 Cantor, Varies, iiber Gesck. der Math., 
 p. 165, e* seq., agrees with Montucla in 
 this. He says : ' It has indeed been 
 sometimes doubted whether the Hip- 
 pias referred to by Proclus is really 
 Hippias of Elis, but certainly without 
 good grounds.' In support of his 
 view Cantor advances the following 
 reasons : — 
 
 I. Proclus in his commentary fol- 
 lows a custom from which he never 
 deviates — he introduces an author whom 
 he quotes with distinct names and sur- 
 names, but afterwards omits the latter 
 when it can be done without an injury 
 to distinctness. Cantor gives instances 
 of this practice, and adds : ' If, then, 
 Proclus mentions a Hippias, it must be 
 Hippias of Elis, who had been already 
 once distinctly so named in his Com- 
 mentary.' 
 
 2. Waiving, however, this custom of 
 Proclus, it is plain that with any author, 
 especially with one who had devoted 
 such earnest study to the works of 
 Plato, Hippias without any further 
 name could be only Hippias of Elis. 
 
 3. Cantor, having quoted passages 
 from the dialogues of Plato, says : 
 ' We think we may assume that Hip- 
 pias of EUs must have enjoyed reputa- 
 tion as a teacher of mathematics at 
 least equal to that which he had as a 
 Sophist proper, and that he possessed 
 all the knowledge of his time in natural 
 sciences, astronomy, and mathematics.' 
 
 4. Lastly, Cantor tries to reconcile 
 the passage quoted from Pappus with 
 the two passages from Proclus : ' Hip- 
 pias of Elis discovered about 420 B. c. 
 a curve which could serve a double pur- 
 pose — trisecting an angle and squaring 
 the circle. From the latter application 
 it got its name, Quadratrix (the Latin 
 translation), but this name does not seem 
 to reach further back than Dinostratus. 
 
222 BR. ALLMAN ON GREEK GEOMETRY 
 
 Hankel, after quoting from Archimedes the axiom — ' If 
 two spaces are unequal, it is possible to add their diffe- 
 rence to itself so often that every finite space can be 
 surpassed,' see p. 185 — quotes further: 'Also, former geo- 
 meters have made use of this lemma ; for the theorem that 
 circles are in the ratio of the squares of their diameters, &c., 
 has been proved by the help of it. But each of the theo- 
 rems mentioned is by no means less entitled to be accepted 
 than those which have been proved without the help of 
 that lemma ; and, therefore, that which I now publish 
 must likewise be accepted.' Hankel then reasons thus : 
 'Since, then, Archimedes brings this lemma into such 
 connection with the theorem concerning the ratio of the 
 areas of circles, and, on the other hand, Eudemus states 
 that this theorem had been discovered and proved by 
 Hippocrates, we may also assume that Hippocrates laid 
 down the above axiom, which was taken up again by 
 Archimedes, and which, in one shape or another, forms 
 the basis of the Method of Exhaustions of the Ancients, 
 t. e. of the method to exhaust, by means of inscribed and 
 circumscribed polygons, the surface of a curvilinear figure. 
 For this method necessarily requires such a principle 
 in order to show that the curvilinear figure is really 
 exhausted by these polygons.'" Eudemus, no doubt, 
 stated that Hippocrates showed that circles have the same 
 ratio as the squares on their diameters, but he does not 
 give any indication as to the way in which the theorem 
 was proved. An examination, however, of the portion of 
 the passage quoted from Archimedes which is omitted by 
 Hankel will, I think, show that there is no ground for 
 his assumption. 
 
 The passage, which occurs in the letter of Archimedes 
 to Dositheus prefixed to his treatise on the quadrature of 
 
 '' Hankel, Gesch. der Math., pp. 12 1-2. 
 
FROM THALES TO EUCLID. 221 
 
 the parabola, ruhs thus : ' Former geometers have also 
 used this axiom. For, by making use of it, they proved 
 that circles have to each other the duplicate ratio of their 
 diameters ; and that spheres have to each other the tripli- 
 cate ratio of their diameters ; moreover, that any pyramid 
 is the third part of a prism which has the same base and 
 the same altitude as the prism ; also, that any cone is the 
 third part of a cylinder which has the same base and the 
 same altitude as the cone : all these they proved by assum- 
 ing the axiom which has been set forth.' '* 
 
 We see now that Archimedes does not bring this axiom 
 into close connection with the theorem concerning the 
 ratios of the areas of circles alone, but with three other 
 theorems also ; and we know that Archimedes, in a sub- 
 sequent letter to the same Dositheus, which accompanied 
 his treatise on the sphere and cylinder, states the two 
 latter theorems, and says expressly that they were dis- 
 covered by Eudoxus." We know, too, that the doctrine of 
 proportion, as contained in the Fifth Book of Euclid, is 
 attributed to Eudoxus.'* Further, we shall find that the 
 invention of rigorous proofs for theorems such as Euclid, 
 vi. I, involves, in the case of incommensurable quantities, 
 the same diiEculty which is met with in proving rigorously 
 the four theorems stated by Archimedes in connection with 
 this axiom ; and that in fact they all required a new 
 method of reasoning — the Method of Exhaustions — which 
 must, therefore, be attributed to Eudoxus. 
 
 The discovery of Hippocrates, which forms the basis of 
 his investigation concerning the quadrature of the circle, 
 has attracted much attention, and it may be interesting to 
 
 9* Archim. ex recens. Torelli, p. 18. see Eucl. Elem., Graece ed. ab. 
 
 95 Ibid., p. 64. August, pars ii., p. 329 ; also Unter- 
 
 9« We are told so in the anonymous suchungen, Sec, Von Dr. J. H. 
 
 scholium on the Elements of Euclid, Knoche, p. 10. Cf. Hermathena, 
 
 which Knoche attributes to Proclus : vol. ui. p. 204, and note 105. 
 
224 DR- ALLMAN ON GREEK GEOMETRY 
 
 inquire how it might probably have been arrived at. It 
 appears to me that it might have been suggested in the 
 following way : — Hippocrates might have met with the 
 annexed figure, excluding the dotted lines, in the arts of 
 decoration ; and, contemplating the figure, he might have 
 completed the four smaller circles and drawn their diame- 
 ters, thus forming a square inscribed in the larger circle, 
 as in the diagram. A diameter of the larger circle being 
 then a diagonal of the square, whose sides are the diame- 
 ters of the smaller circles, it follows that the larger circle is 
 equal to the sum of two of the smaller circles. The larger 
 circle is, therefore, equal to the sum of the four semicircles 
 included by the dotted lines. Taking away the common 
 parts — sc. the four segments of the larger circle standing 
 on the sides of the square — we see that the square is equal 
 to the sum of the four lunes. 
 
 This observation — concerning, as it does, the geometry 
 of areas— might even have been made by the Egyptians, 
 who knew the geometrical facts on which it is founded, and 
 who were celebrated for their skill in geometrical construc- 
 tions. See Hermathena, vol. iii. pp. i86, 203, note loi. 
 
 In the investigation of Hippocrates given above we meet 
 with manifest traces of an analytical method, as stated in 
 Hermathena, vol. iii. p. 1 97, note 9 1 . Indeed, Aristotle— 
 and this is remarkable— after having defined aTraytoyri, evi- 
 
FROM THALES TO EJTCLID. 225 
 
 dently refers to a part of this investigation as an instance 
 of it : for he says, ' Or again [there is reduction], if the 
 middle terms between -y and |3 are few ; for thus also there 
 is a nearer approach to knowledge. For example, if 8 
 were quadrature, and t a rectilineal figure, and Z a circle ; 
 if there were only one middle term between t and Z, viz., 
 that a circle with lunes is equal to a rectilineal figure, 
 there would be an approach to knowledge.'" See p. 195, 
 above. 
 
 In many instances I have had occasion to refer to the 
 method of reduction as one by which the ancient geometers 
 made their discoveries, but perhaps I should notice that in 
 general it was used along with geometrical constructions : '^ 
 the importance attached to these may be seen firom the 
 passages quoted above from Proclus and Democritus, 
 pp. 178, 207; as also from the fact that the Greeks had a 
 special name, \ptvSoypa<prifia, for a faulty construction. 
 
 The principal figure, then, amongst the geometers of 
 this period is Hippocrates of Chios, who seems to have 
 attracted notice as well by the strangeness of his career as 
 by his striking discovery of the quadrature of the lune. 
 Though his contributions to geometry, which have been 
 set forth at length above, are in many respects important, 
 yet the judgment pronounced on him by the ancients is 
 certainly, on the whole, not a favourable one — ^witness 
 the statements of Aristotle, Eudemus, lamblichus, and 
 Eutocius. 
 
 How is this to be explained? The faulty reasoning 
 
 " fi iraKtv [hrayayh ^<rTi] €j o\lya xi ed. Bek. Observe the expressions rh 
 
 fitaa tS>v /S-y- Kol yh-f ovrws iyyinepov ^ i(j>' $ t tieiyfaii/iov, &c., here, and 
 
 ToC e'lSfvai. oTov ei rh S fiv -rerpayuvl- see p. 199, note 44. 
 
 feffflai, Ti S- i,l>' $ e ^i,eiypa^i^iov, rh S' »' Concerning the importance of 
 
 i<l>' $ C nixKos- el ToS ef fv n6pov tit) geometrical constructions as a process 
 
 liiaov. Til iLi-rh. uTiviiTKav iaov yiveaSai of deduction, see P. Laffitte, Les 
 
 eievypd/xfif 7-iK KiK\ov, iyyiii Sc ^7) Grands Types de V Humaniti, vol. ii. 
 
 Tou eiSeVoi. Anal. Prior, ii. 25, p. 69, a, p. 329. 
 
 VOL. IV. Q 
 
226 DR. ALLMAN ON GREEK GEOMETRY 
 
 into which he is reported to have fallen in his pretended 
 quadrature of the circle does not by itself seem to me to 
 be a sufi&cient explanation of it : and indeed it is difiBcult 
 to reconcile such a gross mistake with the sagacity shown 
 in his other discoveries, as Montucla has remarked.'- 
 
 The account of the matter seems to me to be simply 
 this : — Hippocrates, after having been engaged in com- 
 merce, went to Athens and frequented the schools of the 
 philosophers — evidently Pythagorean — as related above. 
 Now we must bear in mind that the early Pythagoreans 
 did not commit any of their doctrines to writing""" — their 
 teaching being oral : and we must remember, further, 
 that their pupils (oKouirrtKo/) were taught mathematics for 
 several years, during which time a constant and intense 
 application to the investigation of difficult questions was 
 enjoined on them, as also silence — the rule being so 
 stringent that they were not even permitted to ask ques- 
 tions concerning the difficulties which they met with : '"' 
 and that after they had satisfied these conditions they 
 passed into the class of mathematicians {fiaOtjuariKoi), being 
 freed from the obligation of silence ; and it is probable 
 that they then taught in their turn. 
 
 Taking all these circumstances into consideration, we 
 may, I think, fairly assume that Hippocrates imperfectly 
 understood some of the matter to which he had listened ; 
 and that, later, when he published what he had learned, he 
 did not faithfully render what had been communicated to 
 him. 
 
 If we adopt this view, we shall have the explanation of — 
 I . The intimate connection that exists between the work 
 of Hippocrates and that of the Pythagoreans ; 
 
 99 Montucla, Histoire des recherches there. 
 
 sur la Quadrature du Cercle, p. 39, wi gee A. Ed. Chaignet, Pythagore 
 
 nouv. ed., Paris, 1831. et la Philosophie Pythagoricienne, vol. 
 
 wo See Hermathena, vol. iii. p. i. p. 115, Paris, 1874; see also Iambi., 
 
 179, note, and the references given de Fit. Pyth., c. 16, s. 68. 
 
FROM TEALES TO EUCLID. 227 
 
 2. The paralogism into which he fell in his attempt 
 to square the circle: for the quadrature of the lune on 
 the side of the inscribed square may have been exhibited 
 in the school, and then it may have been shown that the 
 problem of the quadrature of the circle was reducible to 
 that of the lune on the side of the inscribed hexagon ; 
 and what was stated conditionally may have been taken 
 up by Hippocrates as unconditional ; ""* 
 
 3. The further attempt which Hippocrates made to 
 solve the problem by squaring a lune and circle together 
 (see p. 201) ; 
 
 4. The obscurity and deficiency in the construction 
 given in p. 199; and the dependence of that construction 
 on a problem which we know was Pythagorean [see 
 Hermathena, vol. iii. p. i8i [e), and note 61) ;''>^ 
 
 5. The passage in lamblichus, see p. 186 (/) ; and, gene- 
 rally, the unfavourable opinion entertained by the ancieiits 
 of Hippocrates. 
 
 This conjecture gains additional strength from the fact 
 that the publication of the Pythagorean doctrines was first 
 
 102 In reference to this paralogism tions of the question, 
 
 of Hippocrates, Bretschneider (Geom. "' Referring to the application of 
 
 Tjor Eukl., p. 122) says, 'It is diffi- areas, Mr. Charles Taylor, An Intro- 
 
 cult to assume so gross a mistake on Suction to the Ancient and Modern 
 
 the part of such a good geometer,' Geometry of Conies, Prolegomena, p. 
 
 and he ascribes the supposed error to a xxv., says, ' Although it has not been 
 
 complete misunderstanding. He then made out wherein consisted the im- 
 
 gives an explanation similar to that portance of the discovery in the hands 
 
 given above, with this difference, that of the Pythagoreans, we shall see that 
 
 he supposes Hippocrates to have stated it played a great part in the system of 
 
 the matter correctly, and that Aristotle Apollonius, and that he was led to 
 
 took it up erroneously ; it seems to me designate the three conic sections by 
 
 more probable that Hippocrates took the Pythagorean terms Parabola, Hy- 
 
 up wrongly what he had heard at perbola, EUipse.' 
 
 lecture than that Aristotie did so I may notice that we have an instance 
 
 on reading the work of Hippocrates. of these problems in the construction 
 
 Further, we see from the quotation referred to above : for other applica- 
 
 in p. 225, from Anal. Prior., that tions of the method see Hermathena, 
 
 Aristotle fully understood the condi- vol. iii. pp. 196 and 199. 
 
228 DR. ALLMAN ON GREEK GEOMETRY, ETC. 
 
 made by Philolaus, who was a contemporary of Socrates, 
 and, therefore, somewhat junior to Hippocrates : Philolaus 
 may have thought that it was full time to make this pub- 
 lication, notwithstanding the Pythagorean precept to the 
 contrary. 
 
 The view which I have taken of the form of the 
 demonstrations in geometry at this period differs alto- 
 gether from that put forward by Bretschneider and 
 Hankel, and agrees better not only with what Simplicius 
 tells us ' of the summary manner of Eudemus, who, 
 according to archaic custom, gives concise proofs' (see 
 p. ig6), but also with what we know of the origin, develop- 
 ment, and transmission of geometry : as to the last, what 
 room would there be for the silent meditation on difficult 
 questions which was enjoined on the pupils in the Pytha- 
 gorean schools, if the steps were minute and if laboured 
 proofs were given of the simplest theorems ? 
 
 The need of a change in the method of proof was 
 brought about at this very time, and was in great mea- 
 sure due to the action of the Sophists, who questioned 
 everything. 
 
 Flaws, no doubt, were found in many demonstrations 
 which had hitherto passed current ; new conceptions arose, 
 while others, which had been secret, became generally 
 known, and gave rise to unexpected difficulties; new 
 problems, whose solution could not be effected by the old 
 methods, came to the front, and attracted general atten- 
 tion. It became necessary then on the one hand to recast 
 the old methods, and on the other to invent new methods, 
 which would enable geometers to solve the new problems. 
 
 I have already indicated the men who were able for 
 this task, and I propose in the continuation of this Paper 
 to examine their work. 
 
 GEORGE J. ALLMAN. 
 
GREEK GEOMETRY, 
 
 FROM 
 
 THALES TO EUCLID. 
 
 Part III. 
 
 BY 
 
 GEORGE JOHNSTON ALLMAN, 
 
 LL.D. DUBLIN; D.Sc. Q. UNIV. ; F.R.S. ; 
 
 PROFESSOR or MATHEMATICS IN QUEEN'S COLLEGE, GALWAY ; 
 
 AND 
 MEMBER OF THE SENATE OF THE ROYAL UNIVERSITY OF IRELAND. 
 
 DUBLIN : 
 
 PRINTED AT THE UNIVERSITY PRESS, 
 
 BY PONS<^NBY AND WELDRICK. 
 
 1884. 
 
\_From " Hermathena," A^o. X.. Vol. F.] 
 
186 DR. ALLMAN ON GREEK GEOMETRY 
 
 GREEK GEOMETRY FROM THALES TO 
 EUCLID.* 
 
 IV. 
 
 DURING the last thirty years of the fifth century 
 before the Christian era no progress was made 
 in geometry at Athens, owing to the Peloponnesian War, 
 which, having broken out between the two principal States 
 of Greece, gradually spread to the other States, and for 
 the space of a generation involved almost the whole of 
 Hellas. Although it was at Syracuse that the issue was 
 really decided, yet the Hellenic cities of Italy kept aloof 
 from the contest,' and Magna Graecia enjoyed at this time 
 
 * In the preparation of this part of 
 my Paper I have again made use of 
 the worlcs of Bretschneider and Hankel, 
 and have derived much advantage from 
 the great work of Cantor — Vorlesungen 
 uher Geschichte der MatheTnatik. I 
 have also constantly used the Index 
 Graecitatis appended by Hultsch to 
 vol. iii. of his edition of Pappus ; which, 
 indeed, I have found invaluable. 
 
 The number of students of the his- 
 tory of mathematics is ever increasing ; 
 and the centres in which this subject 
 is cultivated are becoming more nume- 
 rous. 
 
 I propose to notice at the end of 
 this part of the Paper some recent pub- 
 lications on the history of Mathematics 
 and new editions of ancient mathema- 
 
 tical works, which have appeared since 
 the last part was published. 
 
 ' At the time of the Athenian expe- 
 dition to Sicily they were not received 
 into any of the Italian cities, nor were 
 they allowed any market, but had only 
 the liberty of anchorage and water — 
 and even that was denied them at Ta- 
 rentum and Locri. At Rhegium, how- 
 ever, though the Athenians were not 
 received into the city, they were allowed 
 a market without the walls ; they then 
 made proposals to the Rhegians, beg- 
 ging them, as Chalcideans, to aid the 
 Leontines. ' To which was answered, 
 that they would take part with neither, 
 but whatever should seem fitting to the 
 rest of the Italians that they also would 
 do.' Thucyd. vi. 44. 
 
FROM THALES TO EUCLID. 
 
 187 
 
 a period of comparative rest, and again became flourishing. 
 This proved to be an event of the highest importance : 
 for, some years before the commencement of the Pelopon- 
 nesian War, the disorder which had long prevailed in the 
 cities of Magna Graecia had been allayed through the 
 intervention of the Achaeans,' party feeling, which had 
 run so high, had been soothed, and the banished Pytha- 
 goreans allowed to return. The foundation of Thurii 
 (443 B.C.), under the auspices of Pericles, in which the 
 different Hellenic races joined, and which seems not to 
 have incurred any opposition from the native tribes, may 
 be regarded as an indication of the improved state of 
 affairs, and as a pledge for the future.' It is probable that 
 
 ' ' The political creed and peculiar 
 form of goveminent now mentioned 
 also existed among the Achaeans in 
 former times. This is clear from many 
 other facts, but one or two selected 
 proofs will suffice, for the present, to 
 make the thing believed. At the time 
 when the Senate-houses (avviSpia) of the 
 Pythagoreans were burnt in the parts 
 about Italy then called Magna Graecia, 
 and a universal change of the form of 
 government was subsequently made (as 
 was likely when all the most eminent 
 men in each State had been so unex- 
 pectedly cut off), it came to pass that 
 the Grecian cities in those parts were 
 inundated vdth bloodshed, sedition, and 
 every kind of disorder. And when em- 
 bassies came from very many parts of 
 Greece with a view to effect a cessation 
 of differences in the various States, the 
 latter agreedin employing the Achaeans, 
 and their weE-known integrity, for the 
 removal of existing evUs. Not only at 
 this time did they adopt the system of 
 the Achaeans, but, some time after, they 
 
 set about imitating their form of govern- 
 ment in a complete and thorough man- 
 ner. For the people of Crotona, Sybaris 
 and Caulon sent for them by common 
 consent ; and first of all they esta- 
 blished a common temple dedicated to 
 Zeus, ' the Giver of Concord,' and a 
 place in which they held their meet- 
 ings and deliberations : in the second 
 place, they took the customs and laws 
 of the Achaeans, and applied them- 
 selves to their use, and to the manage- 
 ment of their pubUc affairs in accordance 
 with them. But some time after, being 
 hindered by the overbearing power of 
 Dionysius of Syracuse, and also by the 
 encroachments made upon them by the 
 neighbouring natives of the country, 
 they renounced them, not voluntarily, 
 but of necessity.' Polybius, ii. 39. 
 Polybius uses avviSpiov for the senate 
 at Rome : there would be one in each 
 Graeco-Italian State — a point which, 
 as will be seen, has not been sufficiently 
 noted. 
 
 2 The foundation of Thurii seems to 
 
188 
 
 DR. ALLMAN ON GREEK GEOMETRY 
 
 the pacification was effected by the Achaeans on condition 
 that, on the one hand, the banished Pythagoreans should 
 be allowed to return to their homes, and, on the other, 
 that they should give up all organised political action/ 
 Whether this be so or not, many Pythagoreans returned 
 to Italy, and the Brotherhood ceased for ever to exist as 
 a political association.'* Pythagoreanism, thus purified, 
 
 have been regarded as an event of high 
 importance ; Herodotus was amongst 
 the first citizens, and Empedocles vi- 
 sited Thurii soon after it was founded. 
 The names of the tribes of Thurii show 
 (he pan-Hellenic character of the foun- 
 dation. 
 
 * Chaignet, Pythagore et la Philoso- 
 phic Pythagorienm.^ i. p. 93, says so, 
 but does not give his authority ; the 
 passage in Polybius, ii. 39, to which he 
 refers, does not contain this statement. 
 
 ' There are so many conflicting ac- 
 counts of the events referred to here 
 that it is impossible to reconcile them 
 (cf. Hermathena, vol. iv., p. 181). 
 The view which I have adopted seems 
 to me to fit best with the contemporary 
 history, with the history of geometry, 
 and with the balance of the authorities. 
 Zeller, on the other hand, thinks that 
 the most probable account is ' that the 
 first public outbreak must have taken 
 place after the death of Pythagoras, 
 tliough an opposition to him and his 
 friends may perhaps have arisen during 
 his lifetime, and caused his migration 
 to Metapontum. The party struggles 
 with the Pythagoreans, thus begun, 
 may have repeated themselves at dif- 
 ferent times in the cities of Magna 
 Graecia, and the variations in the state- 
 ments may be partially accounted for 
 as recollections of these different facts. 
 The burning of the assembled Pytha- 
 
 goreans in Crotona and the general 
 assault upon the Pythagorean party 
 most likely did not take place untQ the 
 middle of the fifth century ; and, lastly, 
 Pythagoras may have spent the last 
 portion of his life unmolested at Meta- 
 pontum.' (Zeller, Pre-Socratic Philo- 
 sophy, vol. i., p. 360, E. T.). 
 
 Ueberweg takes a similar view : — 
 ' But the persecutions were also 
 several times renewed. In Crotona, as 
 it appears, the partisans of Pythagoras 
 and the Cylonians were, for a long time 
 after the death of Pythagoras, living 
 in opposition as political parties, tiU 
 at length, about a century later, the 
 Pythagoreans were surprised by their 
 opponents, while engaged in a delibe- 
 ration in the ' house of Milo ' (who him- 
 self had died long before), and the 
 house being set on fire and surrounded, 
 all perished, with the exception of 
 Archippus and Lysis of Tarentum. 
 (According to other accounts, the burn- 
 ing of the house, in which the Pytha- 
 goreans were assembled, took place on 
 the occasion ofthe first reaction against 
 the Society, in the lifetime of Pytha- 
 goras.) Lysis went to Thebes, and 
 was there (soon after 400 B.C.) a teacher 
 of the youthful Epaminondas.' (Ueber- 
 weg, History of Philosophy, vol. i., 
 p. 46, E. T.) 
 
 ZeUer, in a note on the passage 
 quoted above, gives the reasons on 
 
FROM in ALES TO EUCLID. 
 
 189 
 
 continued as a religious society and as a philosophic 
 School ; further, owing to this purification and to the 
 members being thus enabled to give their undivided atten- 
 tion and their whole energy to the solution of scientific 
 questions, it became as distinguished and flourishing as 
 ever : at this time, too, remarkable instances of devoted 
 friendship and of elevation of character are recorded of 
 
 which his suppositions are chiefly based. 
 Chaignet, Pyth. et la Phil. Pyth. vol. i., 
 p. 88, and note, states Zeller's opi- 
 nion, and, while admitting that the 
 reasons advanced by him do not want 
 force, says that they are not strong 
 enough to convince him : he then gives 
 his objections. Chaignet, further on, 
 p. 94, n. , referring to the name Italian, 
 by which tl\e Pythagorean philosophy 
 is known, says : ' C'est meme ce qui 
 me fait croire que les luttes intestines 
 n'ont pas eu la duree que suppose M. 
 Zeller ; car si les pythagoriciens avaient 
 ete exil& pendant pr6s de soixante- 
 dix ans de I'ltaUe, comment le nom de 
 r Italic serait-il devenu ou reste attache 
 i leur ecole .' ' Referring to this ob- 
 jection of Chaignet, ZeUer says ' I know 
 not with what eyes he can have read a 
 discussion which expressly attempts to 
 show that the Pythagoreans were not 
 expelled tiU 440, and returned before 
 406' {loc. cit. p. 363, note). 
 
 To the objections urged by Chaignet 
 I would add — 
 
 1. Nearly all agree in attributing the 
 origin of the troubles in Lower Italy to 
 the events which followed the destruc- 
 tion of Sybaris. 
 
 2. The fortunes of Magna Graecia 
 seem to have been at their lowest ebb 
 at the time of the Persian War ; this 
 appears from the fact that, before the 
 battle of Salamis, ambassadors were 
 
 sent by the Lacedemonians and Athe- 
 nians to Syracuse and Corcyra, to in- 
 vite them to join the defensive league 
 against the Persians, but passed by 
 Lower Italy. 
 
 3. The revival of trade consequent 
 on the formation of the Confederacy of 
 Delos, 476 B. c, for the protection of 
 the Aegean Sea, must have had a bene- 
 ficial influence on the cities of Magna 
 Graecia, and the foundation of Thurii, 
 443 B. C, is in itself an indication that 
 the settlement of the country had been 
 already effected. 
 
 4. The answer of the Rhegians to 
 Nicias, 415 B. C, shows that at that time 
 there existed a good understanding be- 
 tween the Italiot cities. 
 
 5. Zeller's argument chiefly rests on 
 the assumption that Lysis, the teacher 
 of Eparainondas, was the same as the 
 Lysis who in nearly all the statements 
 is mentioned along with Archippus as 
 being the only Pythagoreans who 
 escaped the slaughter. Bentley had 
 long ago suggested that they were not 
 the same. Lysis and Archippus are 
 mentioned as having handed down Py- 
 thagorean lore as heir-looms in their 
 families (Porphyry, devitaPyth. p. loi, 
 Didot). This fact is in my judgment 
 decisive of the matter ; for when Lysis, 
 the teacher of Epaminondas, lived, 
 there were no longer any secrets. See 
 HERM.vrHENA, vol. iii., p. 179, n. 
 
190 LB. ALLMAN ON QREEE GEOMETRY 
 
 some of the body. Towards the end of this and the begin- 
 ning of the following centuries encroachments were made 
 on the more southerly cities by the native populations, and 
 some of them were attacked and taken by the elder Dio- 
 nysius :° meanwhile Tarentum, provided with an excellent 
 harbour, and, on account of its remote situation, not yet 
 threatened, had gained in importance, and was now the 
 most opulent and powerful city in Magna Graecia. In 
 this city, at this time, Archytas — the last great Pytha- 
 gorean — grew to manhood. 
 
 Archytas of Tarentum' was a contemporary of Plato 
 (428-347 B. C), but probably senior to him, and was said 
 by some to have been one of Plato's Pythagorean teachers* 
 when he visited Italy. Their friendship'' was proverbial, 
 and it was he who saved Plato's life when he was in danger 
 of being put to death by the younger Dionysius (about 
 361 B.C.). Archytas was probably, almost certainly, a 
 pupil of Philolaus.'" We have the following particulars 
 of his life : — 
 
 • In 393 B. c. a league was formed ' Iambi., de Vit.Pyth. 127, p. 48, ed. 
 
 by some of the cities in order to pro- Didot. ' Verum ergo illud est, quod a 
 
 tact themselves against the Lucanians Tarentino Archyta, ut opinor, dici soli- 
 
 and against Dionysius. Tarentum ap- turn, nostros senes coramemorare audivi 
 
 pears not to have joined the league tiU ab aliis senibus auditum : si quis in 
 
 later, and then its colony Heraclea was caelum ascendisset naturamque mundi 
 
 the place of meeting. The passage in gt pulchritudinem siderum perspexisset, 
 
 Thucydides, quoted above, shows, how- insuavem iUam admirationem ei fore, 
 
 ever, that long before that date a good quae jucundissuma fuisset, si aliquem 
 
 understanding existed between the cities cui narraret habuisset. Sic natura so- 
 
 of Magna Graecia. litarium nihil amat, semperque ad ali- 
 
 ' See Diog. Laert. viii. c. 4. See qupd tamquam adminiculum adnititur, 
 
 also J. Navarro, Tentamen de Archytae quod in amicissimo quoque dulcissimum 
 
 Tarentini vita atque operihus. Pars est.' Cic. De Amic. %x 87. 
 
 Prior. Hafniae, 1819, and authorities "Cic.rf^Oraiore, Lib.m. xxxiv.139, 
 
 given by him. aut Philolaus Archytam Tarentinum ? 
 
 8 Cic. de Fin. v. 29, 87 ; Rep. i. The common reading Philolaum Ar- 
 
 10, 16; de Senec. 12,41. Val. Max. cA/^aj- 7a?-^«//K«j, which is manifestly 
 
 ^■'"- 7- wrong, was coirected by Orellius. 
 
FROM THALES TO EUCLID. 191 
 
 He was a great statesman, and was seven times" ap- 
 pointed general of his fellow-citizens, notwithstanding the 
 law which forbade the command to be held for more than 
 one year, and he was, moreover, chosen commander-in- 
 chief, with autocratic powers, by the confederation of the 
 Hellenic cities of Magna Graecia;''' it is further stated that 
 he was never defeated as a general, but that, having once 
 given up his command through being envied, the troops he 
 had commanded were at once taken prisoners : he was cele- 
 brated for his domestic virtues, and several touching anec- 
 dotes are preserved of his just dealings with his slaves, and 
 of his kindness to them and to children.*' Aristotle even 
 mentions with praise a toy that was invented by him for 
 the amusement of infants : " he was the object of universal 
 admiration on account of his being endowed with every 
 virtue;" and Horace, in a beautiful Ode,'" in which he re- 
 fers to the death of Archytas by shipwreck in the Adriatic 
 Sea, recognises his eminence as an arithmetician, geo- 
 meter, and astronomer. 
 
 In the list of works written by Aristotle, but unfortu- 
 nately lost, we find three books on the philosophy of 
 Archytas, and one j^to ik tou Tifiaiov kqi tCiv ^ ApyyTtitDV a] ; 
 these, however, may have been part of his works" on the 
 
 '1 Diog. Laert. loc. cit. .^lian, Var. accordance ■with Pythagorean princi- 
 
 Hist. via. 14, says six. pies, see Iambi, de mt. Pyth. xxxi. 197, 
 
 i* ToS Koivoxi I'k Twv 'ItoKiihtSiv irpoe- PP- 66, 67, ed. Did. ; Plutarch, de ed. 
 
 (rrv, arpaTnyhs atpeBeh axnoKpaTup imh ^"^- iH-, P- >2, ed. Did. ; as to the lat- 
 
 tSiv TroKiTav Kal twv irepl iKilvov rhv ter, see Athenaeus, xii. 16; Aelian, 
 
 r6Trov'Y.\Ki)vuiv. Suidas, to5 w. This Var. Hist. -x^ it,. 
 
 title (rrpar. OUT. was conferred on Nicias " Aristot. Pol. V. (8), c. vL See 
 
 and his colleagues by the Athenians also Suidas. 
 
 when they sent their great expedition '^ edavfuaCfTo Si koI iraph to?i voK- 
 
 to Sicily : it was also conferred by the \oir iirl iraari apcTi), Diog. Laert. loc. 
 
 Syracusans on the elder Dionysius : cit. 
 
 Diodorus, xiii. 94. See Arnold, Hist. '* i. 28. 
 
 of Rome, I. p. 448, u. 18. " Diog.Laert. v. i, ed.Cobet,p.u6. 
 
 13 As to the former, which was in This, however, could hardly have been 
 
192 BR. ALLMAN ON GREEK GEOMETRY 
 
 Pythagoreans which occur in the same list, but which also 
 are lost. Some works attributed to Archytas have come 
 down to us, but their authenticity has been questioned, 
 especially by Griippe, and is still a matter of dispute:'^ 
 these works, however, do not concern geometry. 
 
 He is mentioned by Eudemus in the passage quoted 
 from Proclus in the first part of this Paper (Herm ATHENA, 
 vol. iii. p. 162) along with his contemporaries, Leodamas of 
 Thasos and Theaetetus of Athens, who were also contem- 
 poraries of Plato, as having increased the number of 
 demonstrations of theorems and solutions of problems, 
 and developed them into a larger and more systematic 
 body of knowledge. '' 
 
 The services of Archytas, in relation to the doctrine of 
 proportion, which are mentioned in conjunction with those 
 of Hippasus and Eudoxus, have been noticed in Herma- 
 THENA, vol iii. pp. 184 and 201. 
 
 One of the two methods of finding right-angled tri- 
 angles whose sides can be expressed by numbers — the 
 Platonic one, namely, which sets out from even numbers — 
 is ascribed to Architas [no doubt, Archytas of Tarentum] 
 by Boethius:'" see Hermathena, vol. iii. pp. igo, 191, 
 and note 87. I have there given the two rules of Pytha- 
 
 so, as one book only on the Pythago- I'arithmetique, et dont le nom, qui ne 
 
 reans is mentioned,' and one against serait du reste, ni grec ni latin, aurait 
 
 them. totalement disparu avec ses oeuvres, i. 
 
 ^8 Gruppe, Ueber die Fragmente des I'exception de quelque passages dans 
 
 Archytas und der dlteren Pythagoreer. Boece.' The question, however, still 
 
 Berlin, 1840. remains as to the authenticity of the 
 
 '5 Procl. Comm., ed. Fried., p. 66. Ars Geo?netriae. Cantor stoutly main- 
 
 ^^ Boet. Geom., ed. Fried., p. 408. tains that the Geometry of Boethius is 
 
 Heiberg, in a notice of Cantor's ' His- genuine : Friedlein, the editor of the 
 
 tory of Mathematics,' Revue Critique edition quoted, on the other hand, dis- 
 
 d^Histoire et de Litterature, i6 Mai, sents ; and the great majority of philo - 
 
 1 88 1, remarks, 'II est difficile de logists agree in regarding the question 
 
 croire a I'existence d'un auteur romain as still sub judice. See Rev. Crit. loc. 
 
 nomme Architas, qui aurait ecrit sur cit. 
 
FROM THALE8 TO EUCLID. 193 
 
 goras and Plato for finding right-angled triangles, whose 
 sides can be expressed by numbers ; and I have shown 
 how the method of Pythagoras, which sets out from odd 
 numbers, results at once from the consideration of the 
 formation of squares by the addition of consecutive gno- 
 mons, each of which contains an odd number of squares. 
 I have shown, further, that the method attributed to Plato 
 by Heron and Proclus, which proceeds from even numbers, 
 is a simple and natural extension of the method of Pytha- 
 goras : indeed it is difficult to conceive that an extension 
 so simple and natural could have escaped the notice of his 
 successors. Now Aristotle tells us that Plato followed the 
 Pythagoreans in many things ; " Alexander Aphrodisiensis, 
 in his Commentary on the Metaphysics, repeats this state- 
 ment;" Asclepius goes further and says, not in many 
 things but in everything." Even Theon of Smyrna, a 
 Platonist, in his work ' Concerning those things which in 
 mathematics are useful for the reading of Plato,' says that 
 Plato in many places follows the Pythagoreans." All this 
 being considered, it seems to me to amount almost to a 
 certainty that Plato learned his method for finding right- 
 angled triangles whose sides can be expressed numerically 
 from the Pythagoreans ; he probably then introduced it 
 into Greece, and thereby got the credit of having invented 
 his rule. It follows also, I think, that the Architas refer- 
 red to by Boethius could be no other than the great Pytha- 
 gorean philosopher of Tarentum. 
 
 The belief in the existence of a Roman agrimensor 
 named Architas, and that he was the man to whom Boe- 
 thius— or the pseudo-Boethius — refers, is founded on a 
 
 "Arist., Met. i. 6, p. 987, a, ed. " Asclep. Schol. 1. i.., p. 548, a, 
 
 Bek. 35- 
 
 « Alex. Aph.5cAo/.!«^m^, Brand., ''i Theon. Smym. A rithm., ed. de 
 
 p. 548, a, 8. Gdder, p. i;. 
 
 vdf,. v. iJ 
 
194 DR. ALLMAN ON GREEK GEOMETRY 
 
 remarkable passage of the ^^-^ Geometriae^' which, I think, 
 has been incorrectly interpreted, and also on another pas- 
 sage in which Euclid is mentioned as prior to Architas.*" 
 The former passage, which is as follows : — ' Sed jam tem- 
 pus est ad geometri calls mensae traditionem ab Archita, 
 non sordido hujus disciplinae auctore, Latio accommo- 
 datam venire, si prius praemisero,' &c., is translated by- 
 Cantor thus : ' But it is time to pass over to the communi- 
 cation of the geometrical table, which was prepared for 
 Latium by Architas, no mean author of this science, when 
 I shall first have mentioned,' &c.:" this, in my opinion, is 
 not the sense of the passage. I think that * ab Archita ' 
 should be taken with trdditiotiem, and not with acconimo- 
 dataiii, the correct translation being — ' But it is now time 
 to come to the account of the geometrical table as given 
 by Architas (" no mean authority" in this branch of learn- 
 ing), as adapted by me to Latin readers ; when,' &c. Now 
 it is remarkable — and this, as far as I know, has been over- 
 looked — that the author of the A rs Geometrtae, whoever he 
 may have been, applies to Architas the very expression 
 applied by Archytas to Pythagoras in Hor. Od. i. 28 : 
 
 ' iudice te, non sordidus auctor 
 ' naturae verique.' 
 
 The mention of Euclid as prior to Archytas is easily 
 explained, since we know that for centuries "Euclid the 
 geometer was confounded with Euclid of Megara,^* who 
 was a contemporary of Archytas, but senior to him. 
 
 We learn from Diogenes Laertius that he was the first 
 to employ scientific method in the treatment of Mechanics, 
 
 -' Boet. ed. Fried., p. 393. with Valerius Maximus (viii. 12), au 
 
 -'' Id., p. 412. author probably of the time of the 
 
 -'• Cantor, Gesch. derMath., p. 493. emperor Tiberius, and was current in 
 
 -'■ This eiTor seems to have originated the middle ages. 
 
FROM THALES TO EUCLID. 195 
 
 by introducing the use of mathematical principles; and 
 was also the first to apply a mechanical motion in the 
 solution of a geometrical problem, while trying to find 
 by means of the section of a semi-cylinder two mean 
 proportionals, with a view to the duplication of the 
 cube."' 
 
 Eratosthenes, too, in his letter to Ptolemy III., having 
 related the origin of the Delian Problem (see Herjia- 
 THENA, vol. iv. p. 212), tells us that 'the Delians sent a 
 deputation to the geometers who were staying with Plato 
 at Academia, and requested them to solve the proble n for 
 them. While they were devoting themselves without stint 
 of labour to the work, and trying to find two mean propor- 
 tionals between the two given lines, Archytas of Taren- 
 tum is said to have discovered them by means of his 
 semi-cylinders, and Eudoxus by means of the so-called 
 'Curved Lines' (Sio tu)v KaXovfxivwv KUfiTrvXoiv ypafijxwv). 
 It was the lot, however, of all these men to be able to 
 solve the problem with satisfactory demonstration ; while 
 it was impossible to apply their methods practically so 
 that they should come into use; except, to some small 
 extent and with difficulty, that of Menaechmus.' ^° 
 
 ''^ ovTos irpuTos rci nvxafM^ rats jxa- This seems to be the meaning of the 
 
 eTiiiaTMats irpo(rxpi;(r<£/*«''o» apxais fu- passage ; but Mechanics, or rather Sta- 
 
 e^Scvae, Kol irpuTos Kivniriv opyayiK^y tics, was first raised to the rank of a 
 
 Siaypifi/iari yiw/ifTpM^ irpoai\yay(, iia. demonstrative science by Archimedes, 
 
 TTJi To/i7)i To5 TiiiMv\lvSpo\) 5iJo yu^traj who founded it on the principle of the 
 
 ova \6yov Xa^fiv CrrrSiv (h rhv toE lever. Archytas, however, was a prac- 
 
 Kil^ou 5iirAa(riairM<!>'- Diog. Laert. loc. tical mechanician, and his wooden flying 
 
 cit., ed. Cobet, p. 224. dove was the wonder of antiquity. Fa- 
 
 That is, he first propounded the vorinus, see Aul. Gell. Nodes Atticae, 
 
 affinity and comiexion of Mechanics x. 12. 
 
 and Mathematics with one another, by "" Archimedis, ex recens. Torelli, 
 
 applying Mathematics to Mechanics, p. 144; Archimedis, Optra Omnia, 
 
 and mechanical motion to Mathema- ed. J. L. Heiberg, vol. iii. pp. 104, 
 
 tics. 106. 
 
 02 
 
196 BR. ALLMAN ON GREEK GEOMETRY 
 
 There is also a reference to this in the epigram which 
 closes the letter of Eratosthenes." 
 
 The solution of Archytas, to which these passages 
 refer, has come down to us through Eutocius, and is as 
 follows : — 
 
 ' The invention of Archytas as Eudemus relates it?"^ 
 ' Let there be two given lines aS, 7 ; it is required to 
 find two mean proportionals to them. Let a circle aj3S? 
 
 be described round the greater line aS; and let the line 
 
 lirjhi Mei'c;^^eiovs KbtvoTOfitZv rpiaStis 
 6t^i,at, ^T)S' et Tt OeovSeos EvBofoio 
 
 Ka/XTTvAoi' ^v ypati-tiali ei6os avaypa4t^Ttii.. 
 
 Archim., ex. rec. Torelli, p. 146 ; 
 Archim., Opera, ed. Heiberg, vol. iii. 
 p. H2. 
 
 ^* Ibid., ex. rec. Tor. p. 143 ; Ibid., 
 ed. Heib. vol. iii. p. 98. 
 
FROM TRALE8 TO EUCLID. 197 
 
 afi, equal to y, be inserted in it ; and being produced let it 
 meet at the point tt, the line touching the circle at the 
 point S : further let jSt^ be drawn parallel to ttS. Now let 
 it be conceived that a semicylinder is erected on the semi- 
 circle aj3S, at right angles to it : also, at right angles to it, 
 let there be drawn on the line aS a semicircle lying in the 
 parallelogram of the cylinder. Then let this semicircle be 
 turned round from the point 8 towards /3, the extremity a 
 of the diameter remaining fixed ; it will in its circuit cut 
 the cylindrical surface and describe on it a certain line. 
 Again, if, the line a8 remaining fixed, the triangle aTrS be 
 turned round, with a motion contrary to that of the semi- 
 circle, it will form a conical surface with the straight line 
 air, which in its circuit will meet the cylindrical line [i. e. the 
 line which is described on the cylindrical surface by the 
 motion of the semicircle] in some point ; at the same time 
 the point j3 will describe a semicircle on the surface of the 
 cone. Now, at the place" of meeting of the lines, let the 
 semicircle in the course of its motion have a position SVa, 
 and the triangle in the course of its opposite motion a 
 position SXa ; and let the point of the said meeting be k. 
 Also let the semicircle described by j3 be /3yuZ|, and the 
 common section of it and of the circle /3S^a be /3^ : now 
 from the point k let a perpendicular be drawn to the plane 
 of the semicircle j3Sa ; it will fall on the periphery of the 
 circle, because the cylinder stands perpendicularly. Let it 
 fall, and let it be w ; and let the line joining the points ; 
 and a meet the line /3^ in the point Q ; and let the right 
 line aX meet the semicircle /S/x? in the point fx ; also let the 
 lines K.^, fjLi, txB be drawn. 
 
 ' Since, then, each of the semicircles S'ko, ^/i^ is at right 
 angles to the underlying plane, and, therefore, their common 
 
 ■'^ e'xeVu 8f) Biaiv Kara rhv Tdwov TJjj cru^n-Twcrtius twv ypanfiuv rh fxiv kwuu- 
 jjLiVov TfixiKmKiov ir t^v tov AKA., &c. 
 
198 DR. ALLMAN ON GREEK GEOMETRY 
 
 section fiO is at right angles to the plane of the circle ; 
 so also is the line fxd at right angles to /3S. Therefore, the 
 rectangle under the lines 0/3, Q^ ; that is, under da, 9i ; is 
 equal to the square on fiQ. The triangle ajjn is therefore 
 similar to each of the triangles ^id, fiad, and the angle 
 ifia is right. But the angle S'ko is also right. Therefore, 
 the lines kS', fii are parallel. And there will be the propor- 
 tion : — As the line S'a is to ok, i. e. ro to ai, so is the line la 
 to ayi, on account of the similarity of the triangles. The 
 four straight lines S'a, ok, ai, oju are, therefore, in continued 
 proportion. Also the line a\k is equal to y, since it is equal 
 to the line aj3. So the two lines ah, j being given, two 
 mean proportionals have been found, viz. aic, ai.' 
 
 Although this extract from the History of Geometry of 
 Eudemus seems to have been to some extent modernized 
 by the omission of certain archaic expressions such as 
 those referred to in Part II. of this Paper (Hermathena, 
 vol. iv. p. 1 99, n. 44), yet the whole passage appears to me 
 to bear the impress of Eudemus's clear and concise style : 
 further, it agrees perfectly with the report of Diogenes 
 Laertius, and also with the words in the letter of Eratos- 
 thenes to Ptolemy JIL, which have been given above. If 
 now we examine its contents and compare them with 
 those of the more ancient fragment, we shall find a re- 
 markable progress. 
 
 The following theorems occur in it : — 
 
 [a]. If a perpendicular be drawn from the vertex of a 
 right- ingled triangle on the hypotenuse, each side is a 
 mear, proportional between the hypotenuse and its ad- 
 jacent segment.'* 
 
 [b). The perpendicular is the mean proportional be- 
 
 '* The whole investigation is, in fact, based on this theorem. 
 
FROM THALES TO EUCLID. 190 
 
 tween the segments of the hypotenuse ; " and, conversely, 
 if the perpendicular on the base of a triangle be a mean 
 proportional between the segments of the base, the ver- 
 tical angle is right. 
 
 {c). If two chords of a circle cut one another, the rect- 
 angle under the segments of one is equal to the rectangle 
 under the segments of the other. This was most probably 
 obtained by similar triangles, and, therefore, required the 
 following theorem, the ascription of which to Hippocrates 
 has been questioned. 
 
 [d). The angles in the same segment of a circle are 
 equal to each other. 
 
 [e). Two planes which are perpendicular to a third 
 plane intersect in a line which is perpendicular to that 
 plane, and also to their lines of intersection with the third 
 plane. 
 
 Archytas, as we see from his solution, was familiar 
 with the generation of cylinders and cones, and had also 
 clear ideas on the interpenetration of surfaces ; he had, 
 moreover, a correct conception of geometrical loci, and of 
 their application to the determination of a point by means 
 of their intersection. Further, since by the theorem of 
 Thales the point fi must lie on a semicircle of which ck is 
 the diameter, we shall see hereafter that in the solution of 
 Archytas the same conceptions are made use of and the 
 same course of reasoning is pursued, which, in the hands 
 of his successor and contemporary Menaechmus, led to the 
 discovery of the three conic sections. Such knowledge 
 and inventive power surely outweigh in importance many 
 special theorems. 
 
 Cantor, indeed, misconceiving the sense of the word 
 roTToe, supposes that the expression 'geometrical locus' 
 
 ■'5 The solutions of the Dalian problem attributed to Plato, ami iiy >rc- 
 naechmus, are founded on thi^ theorem. 
 
200 BB ALLMAN ON GREEK GEOMETRY 
 
 occurs in this passage. He says : ' In the text handed 
 down by Eutocius, even the word tottoc, geometrical locus, 
 occurs. If we knew with certainty that here Eutocius 
 reports literally according to Eudemus, and Eudemus lite- 
 rally according to Archytas, this expression would be 
 very remarkable, because it corresponds with an import- 
 ant mathematical conception, the beginnings of which we 
 are indeed compelled to attribute to Archytas, whilst we 
 find it hard to believe in a development of it at that time 
 which has proceeded so far as to give it a name. In 
 our opinion, therefore, Eudemus, who was probably fol- 
 lowed very closely by Eutocius, allowed himself, in j his 
 report on the doubling of the cube by Archytas, some 
 changes in the style, and in this manner the word " locus" 
 which in the meanwhile had obtained the dignity of a 
 technical term, has been inserted. This supposition is 
 supported by the fact that the whole statement of the pro- 
 cedure of Archjrtas sounds far less antique than, for in- 
 stance, that of the attempts at quadrature of Hippocrates 
 of Chios. Of course we only assume that Eudemus has, 
 to a certain extent, treated the wording of Archytas freely. 
 The sense he must have rendered faithfully, and thus the 
 conclusions we have drawn as to the stereometrical know- 
 ledge of Archytas remain untouched.' ** 
 
 This reasoning of Cantor is based on a misconception 
 of the meaning of the passage in which the word tottoc 
 occurs ; tottoc in it merely means place, as translated above. 
 Though Cantor's argument, founded on the occurrence of 
 the word tottoc, is not sound; yet, as I have said, the 
 solution of Archytas involves the conception oi geometrical 
 loci, and the determination of a point by means of their 
 intersection — not merely ' the beginnings of the concep- 
 tion,' as Cantor supposes ; for surely such a notion could 
 
 '^ Cantor, Vorlesungen iiber Geschichte der Mathematik, p. 197. 
 
FROM THALE8 TO EUCLID. 201 
 
 not first arise with a curve of double curvature. The first 
 beginning of this notion has been referred to Thales in the 
 first part of this Paper" (Hermathena, vol. iii. p. 170). 
 
 Further, Archytas makes use of the theorem of Thales — 
 the angle in a semicircle is right. He shows, moreover, 
 that fiQ is a mean proportional between ai) and Qi, and 
 concludes that the angle ifxa is right : it seems to me, there- 
 fore, to be a fair inference from this that he must have seen 
 that the point ju may lie anywhere on the circumference of 
 a circle of which ai is the diameter. Now Eutocius, in his 
 Commentaries on the Conies of Apollonius,'' tells us what 
 the old geometers meant by Plane Loci, and gives some 
 example of them, the first of which is this very theorem. 
 It is as follows : — 
 
 ' A finite straight line being given, to find a point from 
 which the perpendicular drawn to the given line shall be a 
 mean proportional between the segments. Geometers call 
 such a point a locus, since not one point only is the solu- 
 tion of the problem, but the whole place which the circum- 
 ference of a circle described on the given line as diameter 
 occupies : for if a semicircle be described on the given line, 
 whatever point you may take on the circumference, and 
 draw from it a perpendicular on the diameter, that point 
 will solve the problem.' 
 
 Eutocius then gives a second example — ' A straight 
 line being given, to find a point without it from which the 
 
 3' Speaking of the solution of the buiscono alia scuola di Platone ; G. 
 
 ' Delian Problem' by Menaechmus, Johnston Allman {Greek Geometry 
 
 Favaro observes: 'Avvertiamo es- froyn Thales to Euclid. Dublin, 1877, 
 
 pressamente che Menecmo non fu egli p. 171) la fa risalire a Talete, appog- 
 
 stesso Tinventore di questa dottrina giando la sua argumentazione con va- 
 
 [dei luoghi geometrici]. Montucla lide ragioni.' Antonio Favaro, Notizie 
 
 {Histoire des Mathematiques, nouvelle Siorico-Criticke Sulla Costruzione delle 
 
 edition, tome premier. A Paris, An. Equazioni. Modena, 1878, p. 21. 
 vii. p. 171), e Chasles (Aperfu Histo- ^* Apollonius, Conic, ed. Halleius, 
 
 rique. Bruxelles, 1837, p. 5) la attri- p. 10. 
 
202 BR. ALLMAN ON GREEK GEOMETRY 
 
 straight lines drawn to its extremities shall be equal to 
 each other ' — on which he makes observations of a similar 
 character, and then adds : 'To the same effect ApoUonius 
 himself writes in his Locus Resolutus, with the subjoined 
 [figure] : 
 
 " Two points in a plane being given, and the ratio 
 of two unequal lines being also given, a circle can be 
 described in the plane, so that the straight lines in- 
 flected from the given points to the circumference of the 
 circle shall have the same ratio as the given one." ' 
 
 Then follows the solution, which is accompanied with a 
 diagram. As this passage is remarkable in many respects, 
 I give the original : — 
 
 To Se TpiTOV tZv KuiviKwv irtpiiy^ii, <j>rjo-\, TroXXa koX TrapdSo^a 6e<jiprj- 
 fiara ■^■qa'ifxa irpos ras cruvSecrsts tSv arepeZv tottojv. 'ETrtTrcSous 
 TOTTOvs Wo'! Tots waXatots y£(u/x£T/3ais Xcy€tv, ore tu>v ■jrpo^X-qp.a.rwv ovk 
 d(f) ivbi a-qixiiov fj,6vov, aXX' oltto TrXeidvcuv ytVerai to iroirfp.a' oTov iv 
 eTTiTafet, t-^s eu^etas SoOeiarji Treirepaa-p.ivq's evpelv n cry]p.eiov a<f}' ov rj 
 ayfOticra KofitToi im rrjv SoOelcrav p-ccrr] dvaXoyov ytVcTat toiv T/j.Tj/j.dTwv. 
 
 ToTTOV KaXoiKTi TO TOtOUTOV, OV p,0VOV yap "cV O"qjX€l0V €tTTt TO TTOtOVV TO 
 
 irpoPXyjpa, aXKh. tottos oXo's ov i\ei fj Trepc^ip^ia toD Trept hidpL^Tpov T-qv 
 Soduo'av evOeiav kvkXov iav -yap IttI tijs 8o6eicrr;s eifletas rjpLiKVKXiov 
 ypa(f>yj, OTrep av Ittl t^s Trept^epft'as Xd/Sy; (rqpelov, Koi aTV avTov 
 KadeTov aydyrjs ctti T-qv SidpeTpov, Troi-qa-ft to irpopX-qOlv . . . ofX,oiov 
 Koi ypa<^£i avTO% ' AiToXXd)VLO<; iv tuj ai'aA.uo^u.eva) tottui, IttI tov VTroKei- 
 /xcVov.^' 
 
 Avo So$ivTOiv crr/peiisiv iv cTrtTrcSo) Kal Aoyov So^evTos dvto'cov evOeiZv 
 SvvaTov i(TTLV Iv TiS cTTtTreSo) ypdxpaL kvkXov S><tt€ Tas diro twv SoOevTwv 
 crr]pei.wv iirl ttjv Trcpit^ipnav toS kvkXov KXix>p.iva% ei^cias Xdyov tvetv 
 
 TOV aVTOV T<U So^CVTt. 
 
 It is to be observed, in the first place, that a contrast is 
 
 59 Heiberg, in his Litterargeschicht- virondnivov, a statement which is not 
 
 licke Studien titer Eui/i'd, p. yo, reads correct. I have interpreted Halley's 
 
 rh {r7roKfi/j.fvov, and adds in a note that reacUng as referring to the subjoined 
 
 Halley has iiroKufievif:, in place of rh diagram. 
 
FROM THALE8 TO EUCLID. 203 
 
 here made between Apollonius and the old geometers (o? 
 TToXaioi jtwfitTpai), the same expression which, in the second 
 part of this Paper (Hermathena, vol. iv. p. 217), we found 
 was used by Pappus in speaking of the geometers before the 
 time of Menaechmus. Secondly, on examination it will 
 be seen that loct, as, e. g., those given above, partake of a 
 certain ambiguity, since they can be enunciated either as 
 theorems or as problems ; and we shall see later that, about 
 the middle of the fourth century B. c, there was a discus- 
 sion between Speusippus and the philosophers of the Aca- 
 demy on the one side, and Menaechmus, the pupil and, no 
 doubt, successor of Eudoxus, and the mathematicians of 
 the school of Cyzicus, on the other, as to whether every- 
 thing was a theorem or everything a problem : the mathe- 
 maticians, as might be expected, took the latter view, and 
 the philosophers, just as naturally, held the former. Now 
 it was to propositions of this ambiguous character that the 
 term pori'sm, in the sense in which it is now always used, 
 was applied — a signification which was quite consistent 
 with the etymology of the word." Lastly, the reader will 
 not fail to observe that the first of the three loci given above 
 is strikingly suggestive of the method of Analytic Geo- 
 metry. As to the term roVoc, it may be noticed that Aris- 
 taeus, who v/as later than Menaechmus, but prior to Euclid, 
 wrote five books on Solid Loct [ol crTepeoX tottoi)." In conclu- 
 sion, I cannot agree with Cantor's view that the passage 
 has the appearance of being modernized in expression : 
 
 *« iropifco-flai, to procure. The ques- Amongst the ancients the word J>oi-ism 
 
 tion is in a theorevi, to prove some- had also another sijfiiification, that of 
 
 thing ; in 3. problem, to construct some- corollary. See Heib., Lilt. Stud, rider 
 
 thing; ia 3. porism, to find something. Eukl., pp. 56-79, where the obscure 
 
 So the conclusion of the theorem is, subject of porisms is treated with re- 
 
 Sirep «Sei Berfai, Q.E. D., of the pro- markable clearness, 
 blem, iiirep ^Sei iro.77(ra<, Q. E.F., and " Pappi, Collect., ed. Hultsch, vol. 
 
 of the porism, Sttep eSti fvpe7v, Q. E. I. ii. p. 672. 
 
204 DR. ALLMAN ON GREEK GEOMETRY 
 
 there is nothing in the text from which any alteration in 
 phraseology can be inferred, as there can be in the two 
 solutions of the 'Delian Problem' by Menaechmus, in 
 which the words parabola and hyperbola occur. 
 
 The solution of Archytas seems to me not to have been 
 duly appreciated. Montucla does not give the solution, 
 but refers to it in a loose manner, and says that it was 
 merely a geometrical curiosity, and of no practical import- 
 ance." Chasles, who, as we have seen (Hermathena, 
 vol. iii. p. 171), in the history of Geometry before Euclid, 
 copies Montucla, also says that the solution was purely 
 speculative ; he even gives an inaccurate description of the 
 construction — taking an arete of the cylinder as axis of the 
 cone" — in which he is followed by some more recent 
 writers." Flauti, on the other hand, gives a clear and full 
 account of the method of Archytas, and shows how his 
 solution may be actually constructed. For this purpose it 
 is necessary to give a construction for finding the intersec- 
 tion of the surface of the semi-cylinder with that of the 
 tore generated by the revolution of the semicircle round 
 the side of the cylinder through the point a as axis ; and 
 also for finding the intersection of the surface of the same 
 semi-cylinder with that of the cone described by the revo- 
 lution of the triangle qttS : the intersection of these curves 
 gives the point k, and then the point i, by means of which 
 the problem is solved. Now, in order to determine the 
 point K, it will be sufficient to find the projections of these 
 two curves on the vertical plane on aS, which contains the 
 axes of the three surfaces of revolution concerned, and 
 which Archytas calls the parallelogram of the cylinder. 
 
 " 'Mais ce n'etoit-ia qu'une curio- torn. i. p. 188. 
 
 site geometrique, uniquement propre a *' Chasles, Histoire de la Geonte- 
 
 satisfaire I'esprit, et dont la pratique trie, p. 6. 
 
 ne sfauroil tirer aucun secours.'— *^ e.g. Hosier, Ifistoire des Mai/i., 
 
 Montucla, Histoire des Mathdmatiques, p. 133. 
 
FHOM THALES TO EUCLID. 205 
 
 The projection on this plane of the curve of intersection of 
 the tore and semi-cylinder can be easily found : the pro- 
 jection of the point k, for example, is at once obtained by 
 drawing from the point i, which is the projection of the 
 point K on the horizontal plane o|3S, a perpendicular t? on 
 aS, and then at the point ^ erecting in the vertical plane a 
 perpendicular l,n equal to ik, the ordinate of the semicircle 
 okS', corresponding to the point i ; and in like manner for 
 all other points. The projection on the same vertical plane 
 of the curve of intersection of the cone and semi-cylinder can 
 also be found : for example, the projection of the point k, 
 which is the intersection of ok and ik, the sides of the cone 
 and cylinder, on the vertical plane, is the intersection of 
 the projections of these lines on that plane ; the latter pro- 
 jection is the line ^t(, and the former is obtained by draw- 
 ing in the vertical plane, through the point t, a line tv 
 perpendicular to aS and equal to dfi, the ordinate of the 
 semicircle /3ju2^, and then joining av, and producing it to 
 meet ??} ; and so for all other points on the curve of inter- 
 section of the cone and cylinder." So far Flauti. 
 
 Each of these projections can be constructed by 
 points : — 
 
 To find the ordinate of the first of these curves cor- 
 responding to any point ^, we have only to describe a 
 square, whose area is the excess of the rectangle under the 
 line aS and a mean proportional between the lines aS and 
 a?, over the square on the mean : the side of this square is 
 the ordinate required." In order to describe the projection 
 of the intersection of the cone and cylinder, it will be suffi- 
 cient to find the length, a?, which corresponds to any ordi- 
 
 ■15 Flauti, Geometria di Sito, terza Again, since aS : ai ; : oi : o|, 
 
 edizione. Napoli, 1842, pp. 192-194. we have also 
 
 *« i-if = ik" = 01 . 18' = 01 . (oy - oi) ; 
 
 but a5'=oS; therefore, |T)^=a6 . 01- ai^. ^r)- = oS. (Va5 . oj-aj). 
 
206 
 
 DR. ALLMAN ON GREEK GEOMETRY 
 
 nate, ^t/ (= jk), supposed known, of this curve ; and to effect 
 this we have only to apply to the given line ae a rectangle, 
 which shall be equal to the square on the line %r}, and 
 which shall be excessive by a rectangle similar to a given 
 one, namely, one whose sides are the lines aS and ae — 
 i. e. the greater of the two given lines, between which the 
 two mean proportionals are sought, and the third propor- 
 tional to it and the less." 
 
 Now 6fj. = f c, and er : ^t; : : ae : a£ ; 
 we have, therefore, 
 
 liencc 
 
 "l^ 
 
 ,86 
 
 = e«' - ( 
 
 ee^ 
 
 W = —,•<-—=■«£' 
 
 /3e- 
 
 :;•«!'- 'I-. 
 
 since Se : i| : : ae : af. 
 
 But i|2 = o? . (a5 - a|) ; 
 
 hence we get 
 
 a;82 
 
 l')^ 
 
 . a|- — aS . a£ ; 
 
 and, finally, since 
 
 a^ : a^ : : a^ : as, 
 we have 
 
 The equations of these projections 
 can, as M. Paul Tannery has shown 
 (Sur Us Solutions du ProMeme de Delos 
 par Archytas et par Eudoxe, Memoires 
 de la Societe des Sciences Physiques et 
 Naturelles de Bordeaux, 2e serie, tome 
 ii. p. 277), be easily obtained by ana- 
 lytic geometry. Taking, as axes of co- 
 ordinates, the line aB, the tangent to 
 the circle o35 at the point a, and the 
 side of the cylinder through the point 
 
 a, the equations of the three surfaces 
 are : — 
 
 the cyhnder, x'- + y''- = ax\ 
 the tore. 
 
 ^' + / + 22 = a^lx'^^y- ; 
 the cone, 
 
 x"- ^ y"- ^■ z^ -^ --x», 
 
 where a and S are the lines o5 and o3, 
 between which the two mean propor- 
 tionals are sought. 
 
 We easily obtain from these three 
 equations : 
 
 A- 
 
 ^Ix-^y"' = yab'^, 
 
 first mean proportional between h 
 and a ; 
 
 second mean proportional between 5 
 and a. 
 
 We also obtain easily the projections 
 on the plane of zx of the curves of in- 
 tersection of the cylinder and tore — 
 
 z- = a\/x {Va- •Jx); 
 
 and of the cylinder and cone, 
 
 a' 
 z-=-x--ax. 
 
 These results agree with those ob- 
 tained above geometrically. 
 
FROM THALE8 TO EUCLID. 207 
 
 So much ingenuity and ability are shown in the treat- 
 ment of this problem by Archytas, that the investigation 
 of these projections, in itself so natural,*' seems to have 
 been quite within his reach, especially as we know that 
 the subject of Perspective had been treated of already by 
 Anaxagoras and Democritus (see Hermathena, vol. iv., 
 pp. 206, 208). It may be observed, further, that the con- 
 struction of the first projection is easily obtained ; and as 
 to the construction of the second projection, we see that 
 it requires merely the solution of a problem attributed to 
 the Pythagoreans by Eudemus, simpler cases of which we 
 have already met with (see Hermathena, vol. iii., pp. 181, 
 196, 197; and vol. iv., p. 199, et sq.). On the other hand, 
 it should be noticed — 1° that we do not know when the 
 description of a curve by points was first made; 2" that 
 the second projection, which is a hyperbola, was obtained 
 later by Menaechmus as a section of the cone ; 3° and, 
 lastly, that the names of the conic sections — parabola, hyper- 
 bola, and ellipse — derived from the problems concerning the 
 application, excess, and defect of areas, were first given to 
 them by ApoUonius." 
 
 Several authors give Archytas credit for a knowledge 
 of the geometry of space, which was quite exceptional 
 and remarkable at that time, and they notice the pecu- 
 liarity of his making use of a curve of double curvature 
 — the first, as far as we know, conceived by any geome- 
 ter ; but no one, I believe, has pointed out the importance 
 of the conceptions and method of Archytas in relation to 
 
 "' La recherche des projections sui- rique.' P. Tannery, /oc. aV. p. 279. 
 les plans donnes des mtersections " See Hermathena, vol. iii. p. 181, 
 
 deux a deux des surfaces auxiliaires est, and n. 61 : see, also, Apollonii Conica, 
 
 ;\ cet egard, si natureHes que, si I'on ed. Halleius, p. 9, also pp. 31, 33, 35 ; 
 
 peut s'etonner d'une chose, c'est pre- and Pappi Collect., ed. Hultsch, vol. 
 
 cisemeut qu' Archytas ait conserve a ii. p. 674; and ProcU Comm., ed. 
 
 sa solution une forme pui-ement theo- Friedlein, p. 419. 
 
208 DR. ALLMAN ON GREEK GEOMETRY 
 
 the invention of the conic sections, and the filiation of ideas 
 seems to me to have been completely overlooked. 
 
 Bretschneider, not bearing in mind what Simplicius 
 tells us of Eudemus's concise proofs, thinks that this solu- 
 tion, though faithfully transmitted, may have been some- 
 what abbreviated. He thinks, too, that it must belong to 
 the later age of Archytas — a long time after the opening 
 of the Academy — inasmuch as the discussion of sections of 
 solids by planes, and of their intersections with each other, 
 must have made some progress before a geometer could 
 have hit upon such a solution as this ; and also because 
 such a solution was, no doubt, possible only when Analysis 
 was substituted for Synthesis.^" 
 
 Bretschneider even attempts to detect the particular 
 analysis by which Archytas arrived at his solution, and, 
 as Cantor thinks, with tolerable success.^' The latter 
 reason goes on the assumption, current since Montucla, 
 that Plato was the inventor of the method of geometrical 
 analysis — an assumption which is based on the following 
 passages in Diogenes Laertius and Proclus : — 
 
 He [Plato] first taught Leodamas of Thasos the ana- 
 lytic method of inquiry.'*^ 
 
 Methods are also handed down, of which the best is 
 that through analysis, which brings back what is required 
 to some admitted principle, and which Plato, as they say, 
 transmitted to Leodamas, who is reported to have become 
 thereby the discoverer of many geometrical theorems.^' 
 
 "■" Bretsch. Geom. vor Eukl., pp. ^3 Mt'floSoi St g/tois iropaSiSoi/Toi' koK- 
 
 151, 152. \i<TTri fihv T) Sict Trjs dvaKiaeas «V dpxh" 
 
 ^^ Cantor, Geschichte der Matkema' dfioXoyovfievtjj/ dvdyovffa rh ^ijTovfi.ftfoVj 
 
 tik, p. Ig8. %v KoX b TVKdTuiv, Sis ^ain, AeaiSd/iavri 
 
 ^^ Kal TTpwros rhv kotcc t^jv avdKvffLV irap4SuKev. dtp* ^s koX iKe'ivos iroWup 
 
 TTJs ^7}r7i(T€ws Tp6nou ela-qyfuTaro Aew- Kara yetitfj-erpiav eitper^s iffToprjTai 76- 
 
 Safxayrt t:2 @a(ri<i>. Diog. Laert. iii. 24, viaSai. — Procl. Comm., ed. Fried., 
 
 ed. Cobet, p. 74. p. 211, 
 
FROM THALES TO EUCLID. 209 
 
 Some authors, on the other hand, think, and as it seems 
 to me with justice, that these passages prove nothing more 
 than that Plato communicated to Leodamas of Thasos this 
 method of analysis with which he had become acquainted, 
 most probably, in Cyrene and Italy." It is to be remem- 
 bered that Plato — who in mathematics seems to have been 
 painstaking rather than inventive — has not treated of this 
 method in any of his numerous writings, nor is he reported 
 to have made any discoveries by means of it as Leodamas 
 and Eudoxus are said to have done, and as we know 
 Archytas and Menaechmus did. Indeed we have only to 
 compare the solution attributed to Plato of the problem of 
 finding two mean proportionals — which must be regarded 
 as purely mechanical, inasmuch as the geometrical theo- 
 rem on which it is based is met with in the solution of 
 Archytas — with the highly rational solutions of the same 
 problem by Archytas and Menaechmus, to see the wide 
 interval between them and him in a mathematical point of 
 view. Plato, moreover, was the pupil of Socrates, who 
 held such mean views of geometry as to say that it might 
 be cultivated only so far as that a person might be able to 
 distribute and accept a piece of land by measure." We 
 know that Plato, after his master's death, went to Cyrene 
 to learn geometry from Theodorus, and then to the Pytha- 
 goreans in Italy. Is it likely, then, that Plato, who, as far 
 as we know, never solved a geometrical question, should 
 have invented this method of solving problems in geometry 
 
 ** J. J. de Gelder quotes these pas- celeberrimum Geometram, quem hanc 
 
 sages of Diogenes Laertius and Proclus, rationem reducendi quaestiones ad sua 
 
 and adds : ' Haec satis testantur doc- principia iguoravisse, non vero simile 
 
 tissimum Montucla methodi analyticae est (Bruckeri, Hist. Crit. Phil., torn. i. 
 
 inventionem perperara Platoni tribuere. p. 642) ' — De Gelder, Theonis Smymaei 
 
 Bruckerura rectius scripsisse existimo ; Arithm., Praemonenda, p. xlix. Lugd. 
 
 scilicet eos, qui Platonem hanc me- Bat. 
 
 thodum invenisse volunt, non cogitare, " Xenophon, Metnorab., iv. 7 ; Diog. 
 
 ilium audivisse Theodorum Cyrcnaeum, Laert., ii. 32, p. 41, ed. Cobet. 
 VOL. V. P 
 
210 DR. ALLMAN ON GREEK GEOMETRY 
 
 and taught it to Archytas, who was probably his teacher, 
 and who certainly was the foremost geometer of that time, 
 and that thereby Archytas was led to his celebrated solu- 
 tion of the Delian problem ? 
 
 The former of the two reasons advanced by Bret- 
 schneider, and given above, has reference to and is based 
 upon the following well-known and remarkable passage of 
 the Republic of Plato. The question under consideration 
 is the order in which the sciences should be studied : 
 having placed arithmetic first and geometry — i. e. the 
 geometry of plane surfaces — second, and having proposed 
 to make astronomy the third, he stops and proceeds : — 
 
 " 'Then take a step backward, for we have gone wrong 
 in the order of the sciences.' 
 
 ' What was the mistake ? ' he said. 
 
 ' After plane geometry,' I said, ' we took solids in revo- 
 lution, instead of taking solids in themselves ; whereas 
 after the second dimension the third, which is concerned 
 with cubes and dimensions of depth, ought to have fol- 
 lowed.' 
 
 ' That is true, Socrates ; but these subjects seem to be 
 as yet hardly explored.' 
 
 • Why, yes,' I said, ' and for two reasons : in the first 
 place, no government patronises them, which leads to a 
 want of energy in the study of them, and they are difiicult ; 
 in the second place, students cannot learn them unless 
 they have a teacher. But then a teacher is hardly to be 
 found ; and even if one could be found, as matters now 
 stand, the students of these subjects, who are very con- 
 ceited, would not mind him. That, however, would be 
 otherwise if the whole State patronised and honoured 
 this science ; then they would listen, and there would be 
 continuous and earnest search, and discoveries would be 
 made ; since even now, disregarded as these studies are 
 
FROM TEALES TO EUCLID. 211 
 
 by the world, and maimed of their fair proportions, and 
 although none of their votaries can tell the use of them, 
 still they force their way by their natural charm, and very 
 likely they may emerge into light.' 
 
 ' Yes,' he said, ' there is a remarkable charm in them. 
 But I do not clearly understand the change in the order. 
 First you began with a geometry of plane surfaces ?' 
 
 ' Yes,' I said. 
 
 ' And you placed astronomy next, and then you made a 
 step backward ?' 
 
 ' Yes,' I said, ' the more haste the less speed ; the 
 ludicrous state of solid geometry made me pass over this 
 branch and go on to astronomy, or motion of solids.' 
 
 ' True,' he said. 
 
 ' Then regarding the science now omitted as supplied, 
 if only encouraged by the State, let us go on to astro- 
 nomy.' 
 
 ' That is the natural order,' he said."" 
 
 Cantor, too, says that ' stereometry proper, notwith- 
 standing the knowledge of the regular solids, seems on 
 the whole to have been yet [at the time of Plato] in a very 
 backward state,'" and in confirmation of his opinion quotes 
 part of a passage from the Laws.^^ This passage is very 
 important in many respects, and will be considered later. 
 It will be seen, however, on reading it to the end, that the 
 ignorance of the Hellenes referred to by Plato, and de- 
 nounced by him in such strong language, is an ignorance — 
 not, as Cantor thinks, of stereometry — but of incommensu- 
 
 rables. 
 
 We do not know the date of the Republic, nor that of 
 the discovery of the cubature of the pyramid by Eudoxus, 
 
 ^^ Plato, Ref. vii. 528 ; Jowett, The tik, p. 193. 
 Dialogues of Plato, vol. ii. pp. 363, 58 piato, Zef«j, vii. 819, 820; Jowett, 
 
 ,^. The Dialogues of Plato, vol. iv. pp. 
 
 " Cantor, Geschichte der Mathema- 333, 334. 
 
 P 2 
 
212 DR. ALLMAN ON GREEK GEOMETRY 
 
 which founded stereometry," and which was an important 
 advance in the direction indicated in the passage given 
 above : it is probable, however, that Plato had heard from 
 his Pythagorean teachers of this desideratum ; and I have, 
 in the second part of this Paper (Herm ATHENA, vol. iv., 
 pp. 213, el sq.), pointed out a problem of high philosophical 
 importance to the Pythagoreans at that time, which re- 
 quired for its solution a knowledge of stereometry. Fur- 
 ther, the investigation given above shows, as Cantor re- 
 marks, that Archytas formed an honourable exception to 
 the general ignorance of geometry of three dimensions 
 complained of by Plato. It is noteworthy that this diflH.- 
 cult problem — the cubature of the pyramid — was solved, 
 not through the encouragement of any State, as suggested 
 by Plato, but, and in Plato's own lifetime, by a solitary 
 thinker — the great man whose important services to geo- 
 metry we have now to consider. 
 
 V. 
 
 Eudoxus of Cnidus"" — astronomer, geometer, physician, 
 lawgiver — was born about 407 B. C, and was a pupil of 
 Archytas in geometry, and of Philistion, the Sicilian [or 
 Italian Locrian], in medicine, as Callimachus relates in his 
 Tablets. Sotion in his Successions, moreover, says that he 
 also heard Plato ; for when he was twenty -three years of 
 
 's It should be noticed, however, /ucrpfov. — Prodi Comm., ed. Fried., 
 
 that with the Greeks, Stereometry had p. 39 : see also ibid.., pp. 73, 
 
 the wider signification of geometry of 116. 
 
 three dimensions, as may be seen from *" Diog. Laert., viii. c. 8; A. Boeckh, 
 
 the following passage in Proclus : t) Ueher die vierjdhrigen Sonnenkreise 
 
 fiiv yfaficrpla itaipi'iTai ir<£\iv (Is re der Alten, vorziiglich den Eudoxischen, 
 
 T^v iiritreBov Oeaplav Ka\ rijy ffTfpeO' 'Berlin, 1863. 
 
FROM THALE8 TO EUCLID. 213 
 
 age and in narrow circumstances, he was attracted by the 
 reputation of the Socratic school, and, in company with 
 Theomedon the physician, by whom he was supported, he 
 went to Athens, where — or rather at Piraeus — he remained 
 two months, going each day to the city to hear the lectures 
 of the Sophists, Plato being one of them, by whom, how- 
 ever, he was coldly received. He then returned home, 
 and, being again aided by the contributions of his friends, 
 he set sail for Egypt with Chrysippus — also a physician, 
 and who, as well as Eudoxus, learnt medicine from Philis- 
 tion — bearing with him letters of recommendation from 
 Agesilaus to Nectanabis, by whom he was commended to 
 the priests. When he was in Egypt with Chonuphis of 
 Heliopolis, Apis licked his garment, whereupon the priests 
 said that he would be illustrious (iVSolov), but short-lived." 
 He remained in Egypt one year and four months, and 
 composed the Ociaeterts^'' — an octennial period. Eudoxus 
 then — his years of study and travel now over — took up 
 his abode at Cyzicus, where he founded a school (which 
 became famous in geometry and astronomy), teaching there 
 and in the neighbouring cities of the Propontis ; he also 
 went to Mausolus. Subsequently, at the height of his 
 reputation, he returned to Athens, accompanied by a great 
 
 6' Boeckh thinks, and advances Plato, vol. i., pp. 120-124. 
 weighty reasons for his opinion, that *' The Octaeteris was an intercalary 
 the voyage of Eudoxus to Egypt took cycle of eight years, which was formed 
 place when he was still young— that is, with the object of estabUshing a cor- 
 about 378 B. C. ; and not in 362 B. c, respondence between the revolutions of 
 in which year it is placed by Letronne the sun and moon ; eight lunar years of 
 and others. Boeckh shows that it is 354 days, together with three months 
 probable that the letters of recommen- of 30 days each, make up 2922 days : 
 dation from Agesilaus to Nectanabis, this is precisely the number of days in 
 which Eudoxus took with him, were of eight years of 365! days each. This 
 a much earUer date than the miUtary period, therefore, presupposes a know- 
 expedition of Agesilaus to Egypt. In ledge of the true length of the solar 
 this view Grote agrees. See Boeckh, year : its invention, however, is attri- 
 Sonnenkreise, pp. 140-148; Grote, buted by Censorinus to Cleobtratus. 
 
214 DR. ALLMAN ON GREEK GEOMETRY 
 
 many pupils, for the sake, as some say, of annoying Plato, 
 because formerly he had not held him worthy of attention. 
 Some say that, on one occasion, when Plato gave an enter- 
 tainment, Eudoxus, as there were many guests, introduced 
 the fashion of sitting in a semicircle."' Aristotle tells 
 us that Eudoxus thought that pleasure was the summum 
 lonum ; and, though dissenting from his theory, he praises 
 Eudoxus in a manner which with him is quite unusual : — 
 ' And his words were believed, more from the excellence 
 of his character than for themselves ; for he had the repu- 
 tation of being singularly virtuous, adjcjipwv : it therefore 
 seemed that he did not hold this language as being a 
 friend to pleasure, but that the case really was so. ' " On 
 his return to his own country he was received with great 
 honours — as is manifest, Diogenes Laertius adds, from the 
 decree passed concerning him — and gave laws to his fel- 
 low-citizens ; he also wrote treatises on astronomy and 
 geometry, and some other important works. He was 
 accounted most illustrious by the Greeks, and instead of 
 Eudoxus they used to call him Endoxus, on account of the 
 brilliancy of his fame. He died in the fifty-third year of 
 his age, cz'rc. 354 B. c. 
 
 The above account of the life of Eudoxus, with the ex- 
 ception of the reference to Aristotle, is handed down by 
 Diogenes Laertius, and rests on good authorities." Un- 
 fortunately, some circumstances in it are left undetermined 
 as to the time of their occurrence. I have endeavoured to 
 present the events in what seems to me to be their natural 
 
 «' Is this the foundation of the state- librarian of the Ubrary of Alexandria ; 
 
 ment in Grote's Plato, vol. i., p, 124 — he held this office from about 250 B. c. 
 
 ' the two then became friends ' ? mitil his death, about 240 b. c. Her- 
 
 6* Aristot. £th. Nic, x. 2, p. 1172, mippus of Smyrna. Sotion of Alex- 
 
 ed. Bek. andria flourished at the close of the 
 
 ^ Callimachus of Cyrene ; he was in- third century B. c. Apollodorus of 
 
 vited by Ptolemy II. Philadelphus, to Athens flourished about the year 143 
 
 a place in the Museum ; and was chief B. c. — Smith's Dictionary. 
 
FROM THALES TO EUCLID. 215 
 
 sequence. I regret, however, that in a few particulars as to 
 their sequence I am obliged to differ from Boeckh, who 
 has done so much to give a just view of the life and career 
 of Eudoxus, and of the importance of his work, and of the 
 high character of the school founded by him at Cyzicus. 
 Boeckh thinks it likely that Eudoxus heard Archytas in 
 geometry, and Philistion in medicine, in the interval be- 
 tween his Egyptian journey and his abode at Cyzicus/" 
 
 Grote, too, in the notice which he gives of Eudoxus, 
 takes the same view. He says : — ' Eudoxus was born in 
 poor circumstances ; but so marked was his early promise, 
 that some of the medical school at Knidus assisted him to 
 prosecute his studies — to visit Athens, and hear the So- 
 phists, Plato among them — to visit Egypt, Tarentum 
 (where he studied geometry with Archytas), and Sicily 
 (where he .studied ra iarpiKa with Philistion). These facts 
 depend upon the DtvaKec of Kallimachus, which are good 
 authority ' (Diog. L. viii. 86)." 
 
 Now I think it is much more likely that, as narrated 
 above, Eudoxus went in his youth from Cnidus to Taren- 
 tum — between which cities, as we have seen, an old com- 
 mercial intercourse existed °* — and there studied geometry 
 under Archytas, and that he then studied medicine under 
 the Sicilian [or Italian Locrian] Philistion. In support of 
 this view, it is to be observed that — 
 
 i". The narrative of Diogenes Laertius commences with 
 this statement, which rests on Callimachus, who is good 
 authority ; 
 
 2°. The life of Eudoxus is given by Diogenes Laertius 
 in his eighth book, which is devoted exclusively to the 
 Pythagorean philosophers : this could scarcely have been 
 so, if he was over thirty years of age when he heard Archy- 
 tas, and that, too, only casually, as some think ; 
 
 ts'SoecVh, Sonnenkreise,'p. n<). ss Hermathena, vol. iii. p. 175: 
 
 6' Grote, Plato, vol. i. p. 123, «. Herod., iii. 13-8. 
 
216 BR. ALLMAN ON GREEK GEOMETRY 
 
 3°. The statement that he went from Tarentum to 
 Sicily [or the Italian Locri] to hear Philistion, who pro- 
 bably was a Pythagorean — for we know that medicine was 
 cultivated by the Pythagoreans — is in itself credible ; 
 
 4''. Chrysippus, the physician in whose company Eu- 
 doxus travelled to Egypt, was also a pupil of Philistion in 
 medicine, and Theomedon, with whom Eudoxus went to 
 Athens, was a physician likewise ; in this way might arise 
 the relation between Eudoxus and some of the medical 
 school of Cnidus noticed by Grote. 
 
 The statement of Grote, that ' these facts depend on the 
 UivaKiQ of Kallimachus,' is not correct ; nor is there any 
 authority for his statement that Eudoxus was assisted by 
 the medical school of Cnidus to visit Tarentum and Sicily : 
 the probability is that he became acquainted with some 
 physicians of Cnidus as fellow-pupils of Philistion. 
 
 The geometrical works of Eudoxus have unfortunately 
 been lost ; and only the following brief notices of them 
 have come down to us : — 
 
 [a]. Eudoxus of Cnidus, a little younger than Leon, and 
 a companion of Plato's pupils, in the first place, increased 
 the number of general theorems, added three proportions 
 to the three already existing, and also developed further 
 the things begun by Plato concerning the section [of a 
 line], making use, for the purpose, of the analytical 
 method ; ^' 
 
 [b). The discovery of the three later proportions, re- 
 ferred to by Eudemus in the passage just quoted, is at- 
 tributed by lamblichus to Hippasus, Archytas, and 
 Eudoxus ; '" 
 
 [c). Proclus tells us that Euclid collected the elements, 
 and arranged much of what Eudoxus had discovered." 
 
 89 Procl. Comm., ed. Fried., p. 67 : nulius, pp. 142, 159, 163. 
 see Hermathena, vol. iii. p. 163. ■" Procl. Comm., ed. Fried., p. 68 : 
 
 '" Iambi, in Nic. Arithm., ed. Ten- see Hermathena, vol. iii. p. 164. 
 
FROM THALE8 TO EUCLID. 217 
 
 (d). We learn further from an anonymous scholium on 
 the Elements of Euclid, which Knoche attributes to Pro- 
 clus, that the fifth book, which treats of proportion, is com- 
 mon to geometry, arithmetic, music, and, in a word, to all 
 mathematical science ; and that this book is said to be the 
 invention of Eudoxus (EiiSogou nvoc tov nXarwvof SiSaa- 
 KaXov) ; " 
 
 {e). Diogenes Laertius tells us, on the authority of the 
 Chronicles of Apollodorus, that Eudoxus was the disco- 
 verer of the theory of curved lines {tvpilv re ra nepl rac koju- 
 
 (/). Eratosthenes says, in the passage quoted above, 
 that Eudoxus employed these so-called curved lines to 
 solve the problem of finding two mean proportionals be- 
 tween two given lines ; ''* and in the epigram which con- 
 cludes his letter to Ptolemy III., Eratosthenes associates 
 him with Archytas and Menaechmus;" 
 
 {g). In the history of the ' Delian Problem ' given by 
 Plutarch, Plato is stated to have referred the Delians, who 
 implored his aid, to Eudoxus of Cnidus, or to Helicon of 
 Cyzicus, for its solution ; " 
 
 [k). We learn from Seneca that Eudoxus first brought 
 back with him from Egypt the knowledge of the motions 
 of the planets ; " and from Simplicius, on the authority of 
 Eudemus, that, in order to explain these motions, and in 
 particular the retrograde and stationary appearances of 
 the planets, Eudoxus conceived a certain curve, which he 
 called the hippopede;''^ 
 
 "Euclidis.£'/e»«.,ed. August., vol.ii. Heib., iii. p. 112. Some writers trans- 
 
 p. 328 ; Knoche, Untersuchungen, &c., late fleouSe'oi in this epigram by ' divine,' 
 
 p. 10 : see Hermathena, vol. iii. but the true sense seems to be ' God- 
 
 p. 204. fearing, pious' : see Arist., p. 214, sup. 
 
 ■" Diog. Laert., viii. c. 8, ed. Cobet, '^ Plutarch, de Gen. Soc. i. Opera, 
 
 p. 226. ed. Didot, vol. iii. p. 699. 
 
 " Archlm., ed. Torelli, p. 144; ed. " Seneca, Quaest. Nat., vii. 3. 
 
 Heiberg., iii. p. 106. '* Brandis, Scholia in Arisiot., p. 
 
 « Archim., ed. Tor., p. 146 ; ed. 500, «, 
 
218 DR. ALLMAN ON GREEK GEOMETRY 
 
 [i). Archimedes tells us expressly that Eudoxus disco- 
 vered the following theorems : — 
 
 Any pyramid is the third part of a prism which 
 
 has the same base and the same altitude as the 
 
 pyramid ; 
 Any cone is the third part of a cylinder which has 
 
 the same base and the same altitude as the 
 
 cpne." 
 
 [j). Archimedes, moreover, points out the way in which 
 these theorems were discovered : he tells us that he himself 
 obtained the quadrature of the parabola by means of the 
 following lemma : — ' If two spaces are unequal, it is pos- 
 sible to add their difference to itself so often that every 
 finite space can be surpassed. Former geometers have 
 also used this lemma ; for, by making use of it, they proved 
 that circles have to each other the duplicate ratio of their 
 diameters, and that spheres have to each other the tripli- 
 cate ratio of their diameters ; further, that any pyramid is 
 the third part of a prism which has the same base and the 
 same altitude as the pyramid ; and that any cone is the 
 third part of a cylinder which has the same base and the 
 same altitude as the cone.' *" 
 
 Archimedes, moreover, enunciates the same lemma for 
 lines and for volumes, as well as for surfaces." And the 
 fourth definition of the fifth book of Euclid — which book, 
 we have seen, has been ascribed to Eudoxus — is some- 
 what similar.^^ It should be observed that Archimedes 
 
 " Archim., ed. Torelli, p. 64 ; ed. variv 4ariv iirepix^'y iravThs tov irpoa- 
 
 Heib., vol. i. p. 4. redevTos Ttov Trphs &K\T]Ka Keyofievwv, 
 
 80 Archim., ed. Tor. p. l8;ed. Heib., Archim., ed. Tor., p. 65 ; ed. Heib., 
 
 vol. ii. p. 296. vol. i. p. 10. 
 
 8' 'Eti 5^ TaJc aviaav ypafiiiMv Ka\ tSiv ^^ This definition is — 
 
 aviffup ^TTKpayeicip Kal twv aviawv are- A6yov ex^tv irphs &?i^rj\a ^cyeBT] 
 
 piSiv rb /neifoi' ToE i\a(T(rovos vnepexf"' Xeyerai, & Sivarai iro\\ttirAaina(6neva 
 
FROM THALE8 TO EUCLID. 219 
 
 does not say that the lemma used by former geometers was 
 exactly the same as his, but like it : his words are : — bfioiov 
 Ti^ npotiprifiivtff \rififia ti XafifiavovTEQ iypa<j>ov. 
 
 Concerning the three new proportions referred to in [a) 
 and {6), see the first part of this Paper (Hermathena, vol. 
 iii., pp. 200, 201). In Proclus they are ascribed to Eudoxus ; 
 whereas lamblichus reports that they are the invention of 
 Archytas and Hippasus, and says that Eudoxus and his 
 school (ot iTipl EuSo^ov paOtifiaTiKoi) only changed their 
 names. The explanation of these conflicting statements, 
 as Bretschneider has suggested, probably lies in this — that 
 Eudoxus, as pupil of Archytas, learned these proportions 
 from his teacher, and first brought them to Greece, and 
 that later writers then believed him to have been the in- 
 ventor of them.*^ 
 
 For additional information on this subject, and with 
 relation to the further development of this doctrine by later 
 Greek mathematicians, who added four more means to the 
 six existing at this period, the reader is referred to Pappus, 
 Nicomachus, lamblichus, and also to the observations of 
 Cantor with relation to them.^* 
 
 The passage {a) concerning the section (irtpt rriv Toj-triv) 
 was for a long time regarded as extremely obscure : it was 
 explained by Bretschneider as meaning the section of a 
 straight line in extreme and mean ratio, sech'o aurea, and 
 in the first part of this Paper (Hermathena, vol. iii., 
 p. 163, note) I adopted this explanation. Bretschneider' s 
 interpretation has since been followed by Cantor in his 
 classical work on the History of Mathematics,^^ and may 
 now be regarded as generally accepted. 
 
 A proportion contains in general four terms ; the second 
 and third terms may, however, be equal, and then three 
 
 83 Bretsch. Geom. vorEukl.ip. 164. p. 70. Cantor, Gesch. derMath. p. 206. 
 
 " Pappi Collect., ed. Hultsch, vol. i. ^ Ibid. p. 208. 
 
220 DR. ALLMAN ON GREEK GEOMETRY 
 
 magnitudes only are concerned : further, if the magnitudes 
 are lines, the third term may be the difference between the 
 first and second, and thus the geometrical and arithmetical 
 ratios may occur in the same proportion : the greatest line 
 is then the sum of the two others, and is said to be cut in 
 extreme and mean ratio. The construction of the regu- 
 lar pentagon depends ultimately on this section — which 
 Kepler says was called sectio aurea, sectio divina, and pro- 
 portio divina, on account of its many wonderful properties. 
 This problem, to cut a given straight line in extreme and 
 mean ratio, is solved in Euclid ii. ii, and vi. 30; and 
 the solution depends on the application of areas, which 
 Eudemus tells us was an invention of the Pythagoreans. 
 Use is made of the problem in Euclid iv. 10-14; and the 
 subject is again taken up in the Thirteenth Book of the 
 Elements. 
 
 Bretschneider observes that the first five propositions 
 of this book are treated there in connexion with the ana- 
 lytical method, which is nowhere else mentioned by Euclid ; 
 and infers, therefore, that these theorems are the property 
 of Eudoxus.*° Cantor repeats this observation of Bret- 
 schneider, and thinks that there is much probability in the 
 supposition that these five theorems are due to Eudoxus, 
 and have been piously preserved by Euclid." Heiberg, in 
 a notice of Cantor's Vorlesungen Hber Geschichte der Mathe- 
 matik, already referred to, has pointed out that these ana- 
 lyses and syntheses proceed from a scholiast:** the reason- 
 ing of Bretschneider and Cantor is, therefore, not con- 
 clusive. 
 
 88 Bretsch., Geom. vor Eukl. p. i68. elements d'Euclide (xiii. 1-5). EUes 
 8' Cantor, Gesch. der Math., p. 208. proviennent d'un scholiaste, ce qui res- 
 ^^ Rev. Crit., Sec, 16 Mai, 1881, p. sort, d'ailleurs, de ce que, dans les 
 380. 'P. iSget surtout, p. 236, M. C. manuscrits, elles se trouvent tantot 
 parait accepter pour authentiques les juxtaposees aux theses une a une, tan- 
 syntheses et analyses inserSes dans les tot reunies apres le chap. xiii. 5.' 
 
FROM TMALES TO EUCLID. 221 
 
 There is, however, I think, internal evidence to show 
 that these five propositions are older than Euclid, for— 
 
 1. The demonstrations of the first four of these theo- 
 rems depend on the dissection of areas, and use is made 
 in them of the gnomon — an indication, it seems to me, of 
 their antiquity. 
 
 2. The first and fifth of these theorems can be obtained 
 at once from the solution of Euclid ii. 1 1 ; and of these two 
 theorems the third is an immediate consequence ; the solu- 
 tion, therefore, of this problem given in Book ii. must be 
 of later date. 
 
 These theorems, then, regard being had to the passage 
 of Proclus quoted above, may, as Bretschneider and Cantor 
 think, be due to Eudoxus; it appears to me, however, to 
 be more probable that the theorems have come down from 
 an older time ; but that the definitions of analysis and 
 synthesis given there, and also the aWtuq (or aliter proofs), 
 in which the analytical method is used, are the work of 
 Eudoxus.*' 
 
 As most of the editions of the Elements do not contain 
 the Thirteenth Book, I give here the enunciations of the 
 first five propositions: — 
 
 Prop. I. If a straight line be cut in extreme and mean 
 ratio, the square on the greater segment, increased by half 
 of the whole line, is equal to five times the square of half of 
 the whole line. 
 
 Prop. II. If the square on a straight line is equal to 
 five times the square on one of its segments, and if the 
 
 «« I have since learned that Dr. Hei- that these definitions are due to Eu- 
 
 berg takes the same view ; he thinks doxus is probable. Zeitschrift fur 
 
 that Cantor's supposition— or rather, as Math, und Phys., p. 20 ; 29. Jahrgang, 
 
 he should have said, Bretschneider's— i. Heft. 30 Dec. 1883. 
 
222 DR. ALLMAN ON GREEK GEOMETRY 
 
 double of this segment is cut in extreme and mean ratio, 
 the greater segment is the remaining part of the straight 
 line first proposed. 
 
 Prop. III. If a straight line is cut in extreme and mean 
 ratio, the square on the lesser segment, increased by half 
 the greater segment, is equal to five times the square on 
 half the greater segment. 
 
 Prop. IV. If a straight line is cut in extreme and 
 mean ratio, the squares on the whole line and on the 
 lesser segment, taken together, are equal to three times the 
 square on the greater segment. 
 
 Prop. V. If a straight line is cut in extreme and mean 
 ratio, and if there be added to it a line equal to the greater 
 segment, the whole line will be cut in extreme and mean 
 ratio, and the greater segment will be the line first pro- 
 posed. 
 
 From the last of these propositions it follows that, if a 
 line be cut in extreme and mean ratio, the greater seg- 
 ment will be cut in a similar manner by taking on it a 
 part equal to the less ; and so on continually ; and it re- 
 results from Prop. III. that twice the lesser segment ex- 
 ceeds the greater. If now reference be made to the Tenth 
 Book, which treats of incommensurable magnitudes, we 
 find that the first proposition is as follows : — ' Two unequal 
 magnitudes being given, if from the greater a part be taken 
 away which is greater than its half, and if from the re- 
 mainder a part greater than its half, and so on, there will 
 remain a certain magnitude which will be less than the 
 lesser given magnitude ' ; and that the second proposition 
 is — ' Two unequal magnitudes being proposed, if the lesser 
 be continually taken away from the greater, and if the 
 remainder never measures the preceding remainder, these 
 
FEOM THALES TO EUCLID. 223 
 
 magnitudes will be incommensurable '; lastly, in the third 
 proposition we have the method of finding the greatest 
 common measure of two given commensurable magni- 
 tudes. Taking these propositions together, and consider- 
 ing them in connexion with those in the Thirteenth Book, 
 referred to above, it seems likely that the writer to whom 
 the early propositions of the Tenth Book are due had in 
 view the section of a line in extreme and mean ratio, out 
 of which problem I have expressed the opinion that the 
 discovery of incommensurable magnitudes arose (see Her- 
 MATHENA, vol. iii., p. 1 98). 
 
 This, I think, affords an explanation of the place occu- 
 pied by Eucl. x. I in the Elements, which would otherwise 
 be difficult to account for : we might rather expect to find 
 it at the head of Bookxii., since it is the theorem on which 
 the Method of Exhaustions, as given by Euclid in that 
 book, is based, and by means of which the following theo- 
 rems in it are proved : — 
 
 Circles are to each other as the squares on their 
 
 diameters, xii. 2 ; 
 A pyramid is the third part of a prism having the 
 
 same base and same height, xii. 7 ; 
 A cone is the third part of a cylinder having the 
 
 same base and same height, xii. 10 ; 
 Spheres are to each other in the triplicate ratio 
 
 of their diameters, xii. 18. 
 
 Now two of the foregoing theorems are attributed to 
 Eudoxus by Archimedes; and the lemma, which Archi- 
 medes tells us former geometers used in order to prove 
 these theorems, is substantially the same as that assumed 
 by Euclid in the prbof of the first proposition of his Tenth 
 Book : it is probable, therefore, that this proposition also is 
 due to Eudoxus. 
 
224 DR. ALLMAN ON GREEK GEOMETRY 
 
 Eudoxus, therefore, as I have said (Hermathena, 
 vol. iv., p. 223), must be regarded as the inventor of the 
 Method of Exhaustions. We know, too, that the doctrine 
 of proportion, as contained in the Fifth Book of Euclid, is 
 attributed to him. I have, moreover, said (Hermathena, 
 loc. cti.) that ' the invention of rigorous proofs for theorems 
 such as Euclid vi. i, involves, in the case of incommen- 
 surable quantities, the same difficulty which is met with in 
 proving rigorously the four theorems stated by Archimedes 
 in connexion with this axiom.' °° In all these cases the 
 difficulty was got over, and rigorous proofs supplied, in 
 the same way — namely, by showing that every supposition 
 contrary to the existence of the properties in question led, 
 of necessity, to some contradiction, in short by the redudio 
 ad ahsurdum^'^ {airayivyri tie a^vvaTov). Hence it follows 
 that Eudoxus must have been familiar with this method of 
 reasoning. Now this indirect kind of proof is merely a 
 case of the Analytical Method, and is indeed the case in 
 which the subsequent synthesis, that is usually required as 
 a complement, may be dispensed with. In connexion with 
 this it may be observed that the term used here airaywyii 
 is the same that we met with (Hermathena, vol. iii., 
 p. 197, n.) on our first introduction to the analytical me- 
 
 3° 'C'etait encore par la reduction a latter contains the theory of proportion 
 
 I'absurde que les anciens etendaient for numbers and for commensurable 
 
 aux quantites incommensurables les magnitudes. It is easy to see, then, 
 
 rapports qu'ils avaient decouverts entre that this theorem can be proved in a 
 
 les quantites commensurables ' (Camot, general manner — so as to include the 
 
 Riflexions sur la Mitaphysique du case where the bases are incommensu- 
 
 Calcul Infinitisimal, p. 137, second rable — by the method of reductio ad 
 
 gdition : Paris, 1813). dbsurdum by means of the axiom used 
 
 If the bases of the triangles are com- in EucUd i. i, which has been attri- 
 
 mensurable, this theorem, Euclid vi. r, buted above to Eudoxus: see pp. 2i8 
 
 can be proved by means of the First and 223. 
 Book and the Seventh Book, which " Camot, ibid., p. 135. 
 
FROM THALE8 TO EUCLID. 225 
 
 thod ; this indeed is natural, for analysis, as Duhamel re- 
 marked, is nothing else but a method of reduction." 
 
 Eutocius, in his Commentary on the treatise of Archi- 
 medes On the Sphere and Cylinder, in which he has handed 
 down the letter of Eratosthenes to Ptolemy III., and in 
 which he has also preserved the solutions of the Dalian 
 Problem by Archytas, Menaechmus, and other eminent 
 mathematicians, with respect to the solution of Eudoxus 
 merely says : 
 
 ' We have met with the writings of many illustrious 
 men, in which the solution of this problem is professed; 
 we have declined, however, to report that of Eudoxus, 
 since he says in the introduction that he has found it by 
 means of curved lines, KafiirvXwv ypafifxwv : in the proof, how- 
 ever, he not only does not make any use of these curved 
 lines, but also, finding a discrete proportion, takes it as 
 a continuous one ; which was an absurd thing to con- 
 ceive — not merely for Eudoxus, but for those who had to 
 do with geometry in a very ordinary way.'" 
 
 As Eutocius omitted to transmit the solution of Eudoxus, 
 so I did not give the above with the other notices of his 
 geometrical work. It is quite unnecessary to defend 
 Eudoxus from either of the charges contained in this 
 passage. I will only remark, with Bretschneider, that it 
 is strange that Eutocius, who had before him the letter of 
 Eratosthenes, did not recognise in the complete corruption 
 of the text the source of the defects which he blames.'* 
 
 We have no further notice of these so-called curved 
 lines : it is evident, however, that they could not have been 
 any of the conic sections, which were only discovered later 
 by Menaechmus, the pupil of Eudoxus. 
 
 "2 'L'analyse n'est done autre chose p. 41). 
 
 qu'une methode de reduction' (Du- 93 Archim. ed. Tor., p. 135, ed. Hei- 
 
 hamel, Des Methodes dans les Sciences berg, vol. iii. p. 66. 
 
 de Raismnement, Premiere Partie, si Bretsch. 6><??«. z'or .£■««. p. 16C. 
 vol.. V. Q 
 
226 BR. ALLMAN ON GREEK GEOMETRY 
 
 There is a conjecture, however, concerning them, which 
 is worth noticing : M. P. Tannery thinks that the term 
 KafiirvXai jpafifioi has, in the text of Eratosthenes, a par- 
 ticular signification, and that, compared with, e. g. the 
 xajuTTuAa ro^a of Homer, it suggests the idea of a curve 
 symmetrical to an axis, which it cuts at right angles, and 
 presenting an inflexion on each side of this axis. Tannery 
 conjectures that these curves of Eudoxus are to be found 
 amongst the projections of the curves used in the solution 
 of his master, Archytas ; and tries to find whether, amongst 
 these projections, any can be found to which the denomi- 
 nation in question can be suitably applied. We have seen 
 above, p. 204, that Flauti has shown how the solution of 
 Archytas could be constructed by means of the projections, 
 on one of the vertical planes, of the curves employed in 
 that solution. I have further shown that the actual con- 
 struction of these projections can be obtained by the aid of 
 geometrical theorems and problems known at the time of 
 Archytas ; though we have no evidence that he completed 
 his solution in this way. Tannery has considered these 
 curves, and shown that the term k. 7., in the sense which 
 he attaches to it, does not apply to either of them, nor to 
 the projections on the other vertical plane; but that, on 
 the contrary, the term is quite applicable to the projection 
 of the intersection of the cone and tore on the circular 
 base of the cylinder." 
 
 The astronomical work of Eudoxus is beyond the scope 
 of this Paper, and is only referred to in connexion with 
 the hippofede {h). I may briefly state, however, that he 
 was a practical observer, and that he ' may be considered 
 as the father of scientific astronomical observation in 
 Greece'; further, that 'he was the first Greek astronomer 
 who devised a systematic theory for explaining the periodic 
 
 "» Tannery, Sur les Solutions du Prollhne de Dilos par Archytas et par 
 Eudoxe. 
 
FROM THALE8 TO EUCLID. 227 
 
 motions of the planets';'* that he did so by means of geo- 
 metrical hypotheses, which later were submitted to the test 
 of observations, and corrected thereby; and that hence 
 arose the system of concentric spheres which made the 
 name of Eudoxus so illustrious amongst the ancients. 
 
 Although this theory was substantially geometrical, 
 and is in the highest degree worthy of the attention of the 
 students of the history of geometry, yet to render an 
 account of it which would be in the least degree satisfac- 
 tory would altogether exceed the limits prescribed to me ; 
 I must, therefore, refer my readers to the excellent and 
 memorable monograph" of Schiaparelli, who with great 
 ability and with rare felicity has restored the work of 
 Eudoxus. In this memoir the nature of the spherical curve, 
 called by Eudoxus the hippopede, was first placed in a clear 
 light : it is the intersection of a sphere and cylinder ; and 
 
 on account of its form, which resembles the figure R, it is 
 called by Schiaparelli a spherical lemniscate^^ A passage 
 in Xenophon, De re equestri, cap. 7, explains why the 
 name hippopede was given to this curve, and also to one of 
 the spirics (17 'nriroTzSit], \xia rdiv airiifUKwv ovaaY^ of Perseus, 
 which also has the form of a lemniscate. 
 
 I have examined the work of Eudoxus, and pointed out 
 the important theorems discovered by him ; I have also 
 dwelt on the importance of the methods of inquiry and 
 
 96 Sir George Comewall Lewis, A «' Procl. Comm. ed. Fried., p. 127. 
 
 Historical Survey of the Astronomy of With respect to the spiric lines, see 
 
 the Ancients, p. 147, et sg. : London, Knoche and Maerlcer, Ex Prodi suc- 
 
 jgg2 cessoris in Euclidis elementa commen- 
 
 9' G. V. Schiaparelli, Le Sfere Onto- tariis definitionis quartae expositio7iem 
 
 centriche di Eudosso, di Calippo e di qucie de recta est linea et sectionibus, 
 
 Aristotele (Ulrico Hoepli : Milano, spiricis commentati sunt J. H. Kno- 
 
 1875). chius et F. y. Maerkerus, Herfordiae, 
 
 89 See Schiaparelli, loc. cit., sec- 1856. 
 tion V. 
 
228 DE. ALLMAN ON GREEK GEOMETRY 
 
 proof which he introduced. In order to appreciate this 
 part of his work, it seems desirable to take a brief retro- 
 spective glance at the progress of geometry as set forth in 
 the two former parts of this Paper, and the state in which 
 it was at the time of Eudoxus, and also to refer to the phi- 
 losophical movement during the last generation of the fifth 
 century B. c. : — 
 
 In the first part (Hermathena, vol. iii., p. 171) I attri- 
 buted to Thales the theorem that the sides of equiangular 
 triangles are proportional ; a theorem which contains the 
 beginnings of the doctrine of proportion and of the simila- 
 rity of figures. It is agreed on all hands that these two 
 theories were treated at length by Pythagoras and his 
 School. It is almost certain, however, that the theorems 
 arrived at were proved for commensurable magnitudes 
 only, and were assumed to hold good for all. We have 
 seen, moreover, that the discovery of incommensurable 
 magnitudes is attributed to Pythagoras himself by Eude- 
 mus : this discovery, and the construction of the regular 
 pentagon, which involves incommensurability, depending 
 as it does on the section of a line in extreme and mean ratio, 
 were always regarded as glories of the School, and kept 
 secret ; and it is remarkable that the same evil fate is said 
 to have overtaken the person who divulged each of these 
 secrets — secrets, too, regarded by the brotherhood as so 
 peculiar that the pentagram, which might be taken to re- 
 present both these discoveries, was used by them as a sign 
 of recognition. It seems to be a fair inference from what 
 precedes, that the Pythagoreans themselves were aware 
 that their proofs were not rigorous, and were open to 
 .serious objection:'™ indeed, after the invention of dialec- 
 
 ^"o A similar view of the subject is Societe des Sciences physiques et na- 
 
 taken by P.. Tannery, De la solution turelles de Bourdeaux, t. iv. (2= serie), 
 
 geomitrique des probUmes du second p. 406. He says: — ' La decouverte de 
 
 Jegre avant Euclide. Memoires de la I'incommeusurabilit^ de certaines Ion- 
 
FROM THALES TO EUCLID. 229 
 
 tics by Zeno, and the great effect produced throughout 
 Hellas by his novel and remarkable negative argumenta- 
 tion, any other supposition is not tenable. Further, it is 
 probable that the early Pythagoreans, who were naturally 
 intent on enlarging the boundaries of geometry, took for 
 granted as self-evident many theorems, especially the con- 
 verses of those already established. The first publication 
 of the Pythagorean doctrines was made by Philolaus ; and 
 Democritus, who was intimate with him, and probably his 
 pupil, wrote on incommensurables. 
 
 Meanwhile the dialectic method and the negative mode 
 of reasoning had become more general, or, to use the words 
 of Grote : — 
 
 ' We thus see that along with the methodised question 
 and answer, or dialectic method, employed from hencefor- 
 ward more and more in philosophical inquiries, comes out 
 at the same time the negative tendency — the probing, test- 
 ing, and scrutinising force — of Grecian speculation. The 
 negative side of Grecian speculation stands quite as pro- 
 minently marked, and occupies as large a measure of the 
 intellectual force of their philosophers, as the positive side. 
 It is not simply to arrive at a conclusion, sustained by 
 a certain measure of plausible premise — and then to pro- 
 claim it as an authoritative dogma, silencing or disparag- 
 ing all objectors — that Grecian speculation aspires. To 
 unmask not only positive falsehood, but even affirmation 
 without evidence, exaggerated confidence in what was 
 only doubtful, and show of knowledge without the reality — 
 to look at a problem on all sides, and set forth all the diffi- 
 culties attending its solution — to take account of deduc- 
 tions from the affirmative evidence, even in the case of 
 conclusions accepted as true upon the balance — all this 
 
 gueurs entre elles, et avant tout de la des lors, etre un veritable scaadale 
 diagonale du carre a son cote, qu'elle logique, uiie redoutable pierre d'achop- 
 soit due au Maitre ou aux disciples, dut, pement.' 
 
230 DR. ALLMAN ON GREEK GEOMETRY 
 
 will be found pervading the march of their greatest think- 
 ers. As a condition of all progressive philosophy, it is not 
 less essential that the grounds of negation should be freely 
 exposed than the grounds of affirmation. We shall find 
 the two going hand in hand, and the negative vein, in- 
 deed, the more impressive and characteristic of the two, 
 from Zeno downward, in our history.''"' 
 
 As an immediate consequence of this it would follow 
 that the truth of many theorems, which had been taken for 
 granted as self evident, must have been questioned ; and 
 that, in particular, doubt must have been thrown on the 
 whole theory of the similarity of figures and on all geome- 
 trical truths resting on the doctrine of proportion : indeed 
 it might even have been asked what was the meaning of 
 ratio as applied to incommensurables, inasmuch as their 
 mere existence renders the arithmetical theory of propor- 
 tion inexact in its very definition."^ 
 
 Now it is remarkable that the doctrine of proportion is 
 twice treated in the Elements — first, in a general manner, 
 so as to include incommensurables, in Book v., which tradi- 
 tion ascribes to Eudoxus, and then arithmetically in Book 
 vii., which probably, as Hankel has supposed, contains the 
 treatment of the subject by the older Pythagoreans."" The 
 twenty-first definition, of Book vii. is — ^Apidfioi avaXoyov 
 j((T(v, orav 6 npwTog rov Sivripov koi 6 rpiTOQ tov rnapTov 
 lauKiQ p TToXXaTrXdaiog, r) to avru fiepog, 7] to. aura fiipr). 
 
 Further, if we compare this definition with the third, 
 fourth, and fifth definitions of Book v., I think we can see 
 evidence of a gradual change in the idea of ratio, and of a 
 development of the doctrine of proportion — 
 
 I. The third definition, which is generally considered 
 
 "" Grote, History of Greece, vol. vi. and Ratio in. the English Cyclopaedia, 
 p. 48. i»3 Hankel, Gesch. der Math., p. 
 
 "" See the Articles on Proportion 390. 
 
FROM THALE8 TO EUCLID. 231 
 
 not to belong to Euclid,'"^ seems to be an attempt to bridge 
 over the difficulty which is inherent in incommensurables, 
 and may be a survival of the manner in which the subject 
 was treated by Democritus. 
 
 2. The fourth definition is generally regarded as having 
 for its object the exclusion of the ratios of finite magni- 
 tudes to magnitudes which are infinitely great on the one 
 side, and infinitely small on the other : it seems to me, 
 however, that its object may have been, rather, to include 
 the ratios of incommensurable magnitudes; moreover, since 
 the doctrine of proportion by means of the apagogic me- 
 thod of proof can be founded on the axiom which is con- 
 nected with this definition, and which is the basis of the 
 method of 'exhaustions, it is possible that the subject may 
 have been first presented in this manner by Eudoxus. 
 
 3. Lastly, in the fifth definition his final and systematic 
 manner of treating the subject is given.^"^ 
 
 Those who are acquainted with the history of Greek 
 philosophy know that a state of things somewhat similar to 
 that represented above existed with respect to it also, and 
 that a problem of a similar character, also requiring a new 
 method, proposed itself for solution towards the close of 
 the fifth century B. C. ; and, further, that this problem was 
 solved by Socrates by means of a new philosophic me- 
 thod — the analysis of general conceptions. This must 
 have been known to Eudoxus, for we are informed that he 
 
 101 Aifyo J iirrX Suo ixeyfOuv Sfjioye- nothing geometrical by means of arith - 
 
 vHv ri KBTci TTri\tK6TriTa vphs HWjiKa metic. . . . Where the subjects are so 
 
 iroio ffxeVij. See Camerer, Euclidis different as they are in arithmetic and 
 
 elementorum libri sex priores, tom .ii. geometry we cannot apply the arith- 
 
 p. 74, et sq., Berolini, 1824. metical sort of proof to that which 
 
 105 In connexion with what precedes, belongs to quantities in general, unless 
 
 we are reminded of the aphorism of these quantities are numbers, which 
 
 Aristotle — ' We carmot prove anything can only happen in certain cases.' 
 
 by starting from a different genus, e.^. Anal. post. i. 7, p. 75, a, ed. Bek. 
 
232 BR. ALLMAN ON GREEK GEOMETRY 
 
 was attracted to Athens by the fame of the Socratic School. 
 Now a service, similar to that rendered by Socrates to phi- 
 losophy, but of higher importance, was rendered by Eu- 
 doxus to geometry, who not only completed it by the 
 foundation of stereometry, but, by the introduction of new 
 methods of investigation and proof, placed it on the firm 
 basis which it has maintained ever since. 
 
 This eminent thinker — one of the most illustrious men 
 of his age, an age so fruitful in great men, the precur- 
 sor, too, of Archimedes and of Hipparchus — after having 
 been highly estimated in antiquity,^"" was for centuries un- 
 duly depreciated ; "' and it is only within recent years that, 
 owing to the labours of some conscientious and learned 
 men, justice has been done to his memory, and liis reputa- 
 tion restored to its original lustre."^ 
 
 Something, however, remained to be cleared up, espe- 
 cially with regard to his relations, and supposed obligations, 
 to Plato."' I am convinced that the obligations were quite 
 
 '"^ E.g. Cicsro, de Div. a. ^2, 'Ad enorme proposizione.' Equally moii- 
 
 Chaldaeorum monstra veniamus : de strous is the foJlomng : — ' it is only 
 
 quibus Eudoxus, Platonis auditor, in in the first capacity [astronomer and 
 
 astrologia judicio doctissimorum homi- not geometer] that his fame has de- 
 
 num facile princeps, sic opinatur, id scended to our day, and he has more 
 
 quod scriptum reliquit : Chaldaeis in of it than can be justified by any ac- 
 
 praedictione et in notatione cujusque count of his astronomical science now 
 
 vitae ex natali die, minime esse creden- in existence.' De Morgan, in Smith's 
 
 dum': Plutarch, koh j>osse suwv. vivi Dictionarj-. 
 
 sec. Epic. c. xi. T.\iU(,<f 5e Kol 'Apx'" '''^ Ideler, ueler Eudoxus, Abh. der 
 
 M-iySei Koi 'lirirapxif avvivBovawiifv. Berl. Akad. v. J. 1828 and 1830: Le- 
 
 II" As evidence of this depreciation tronne, sur les ecrits et les tra-vaux 
 
 I may notice— Delambre, Bisioire de d'Eudoxe de Cnide, d'apres M. Lud. 
 
 VAsironcmie ancienne 'L'Astrono- !»;;§- /rf^/^r, journal des Savants, 1840: 
 
 mie n'a ete cultivee veritablement qu'en Boeckh, Sonnenkreise der Alien, 1863 : 
 
 Grece, et presque uniquement par deux Schiaparelli, le Sfere Omocentriche, 
 
 homraes, HipparqueetPtolemee' (tom. &c., 1875. 
 
 i. p. 325) : ' Rien ne prouve qu'U [Eu- lo" Even those, by whom the fame of 
 
 doxe] fut geometre ' (tom. i. p. 131). Eudoxus has been revived, seem to 
 
 Well may Schiaparelli say — ' Questa acquiesce in this. 
 
FROM THALE8 TO EUCLID. 233 
 
 in the opposite direction, and that Plato received from 
 Eudoxus incomparably more than he gave. As to his 
 solving problems proposed by Plato, the probability is 
 that these questions were derived from the same source — 
 Archytas and the Pythagoreans. Yet I attach the highest 
 importance to the visit of Eudoxus to Athens ; for although 
 he heard Plato for two months only, that time was suf- 
 ficient to enable Eudoxus to become acquainted with the 
 Socratic method, to see that it was indispensable to clear 
 up some of the fundamental conceptions of geometry, and, 
 above all, to free astronomy from metaphysical mystifica- 
 tions, and to render the treatment of that science as real and 
 positive as that of geometry. To accomplish this, how- 
 ever, it was incumbent on him to know the celestial phe- 
 nomena, and for this purpose — inasmuch as one human life 
 was too short — he saw the necesssity of going to Egypt, to 
 learn from the priests the facts which an observation con- 
 tinued during many centuries had brought to light, and 
 which were there preserved. 
 
 I would call particular attention to the place which 
 Eudoxus filled in the history of science — with him, in fact, 
 an epoch closed, and a new era, still in existence, opened.'^" 
 He was geometer, astronomer, physician, lawgiver, and 
 was also counted amongst the Pythagoreans, and versed 
 in the philosophy of his time. He was, however, much 
 
 "» This has been pointed out by lement philosophe, Eudoxe de Cnide 
 
 AugusteComte:— 'Celle-ci[laseconde fut le dernier theoricien embrassant, 
 
 evolution scientifique de la Grece] com- avec un egal succes, toutes les specu- 
 
 men9a pourtant, avec tous ses carac- lations accessibles a I'esprit mathe- 
 
 t^res propres, pendant la generation matique. II servit pareillement la 
 
 anterieure a cette ere [la fondation du geomStrie et I'astronomie, tandis que, 
 
 Musee d'Alexandrie], chez un savant bientot apr^s lui, la specialisation de- 
 
 trop meconnu, qui fournit une transi- vint deja telle que ces deux sciences ne 
 
 tion normale entre ces deux grandes purent plus etre notablement perfec- 
 
 phases theoriques, composees chacune tionneespar lesmemesorganes.' Poli- 
 
 d'environ trois siecles. Quoique nul- tique Positive, iii., p. 316, Paris, 1853. 
 
 VOL. V. R 
 
234 I)R. ALLMAN ON GREEK GEOMETRY 
 
 more ihe man of science, and, of all the ancients, no one 
 was more imbued with the true scientific and positive 
 spirit than was Eudoxus : in evidence of this, I would point 
 to— 
 
 i". His work in all branches of the geometry of the 
 day — founding new, placing old on a rational basis, and 
 throwing light on all — as presented above. 
 
 2°. The fact that he was the first who made observation 
 the foundation of the study of the heavens, and thus be- 
 came the father of true astronomical science. 
 
 3''. His geometrical hypothesis of concentric spheres, 
 which was conceived in the true scientific spirit, and which 
 satisfied all the conditions of a scientific research, even 
 according to the strict notions attached to that expression 
 at the present day. 
 
 4°. His ' practical and positive genius, which was averse 
 to all idle speculations.' '" 
 
 5°. The purely scientific school founded by him at 
 Cyzicus, and the able mathematicians who issued from 
 that school, and who held the highest rank as geometers 
 and astronomers in the fourth century B. C. 
 
 We see, then, in Eudoxus something quite new — the 
 first appearance in the history of the world of the man of 
 science ; and, as in all like cases, this change was effected 
 by a man who was thoroughly versed in the old system.' '- 
 
 1" Ideler, and after him Schiapa- instance of this is found in Plutarch 
 
 relli : this appears from the fact testi- {non posse suav. -viv. sec. Epic, cxi., 
 
 fied by Cicero (Dili. jK^?-a, II. 106), that vol. iv., p. 1138, ed. Didot), who re- 
 
 Eudoxus had no faith in the Chaldean lates that he, instead of speculating, as 
 
 astrology which was then coming into others did, on the nature of the sun, 
 
 fashion among the Greeks ; and also contented himself with saying that ' he 
 
 from this — that he did not, like many would wUlingly undergo the fate of 
 
 of his predecessors and contemporaries, Phaeton if, by so doing, he could 
 
 give expression to opinions upon things ascertain its nature, magnitude, and 
 
 which were inaccessible to the observa- form.' 
 tions and experience of the time. An '" Eudoxus may even be regarded as 
 
FROM THALE8 TO EUCLID. 235 
 
 It is not without significance, too, that Eudoxus se- 
 lected the retired and pleasant shores of the Propontis as 
 the situation of the school which he founded for the trans- 
 mission of his method. Among the first who arose in this 
 school was Menaechmus, whose work I have next to con- 
 sider.* 
 
 in a peculiar manner uniting in him- rian cities ; lie then went to study under 
 
 sell and representing the previous phi- the Pythagoreans in Italy ; and, sub- 
 
 losophic and scientific movement; for sequently, he went to Athens, being 
 
 — though not an Ionian — he was a attracted by the reputation of the 
 
 native of one of the neighbourinL; Do- Socratic school. 
 
 * [The Bibliographical Note, reTeiTed to in page i86, will be given in ll'.e 
 next No. of HeRMATHENA.] 
 
 GEORGE J. ALLMAN. 
 
SEP 19 1991