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GEORGE JOHNSTON ALLMAN, LL.D.,
PROFESSOR OF MATHEMATICS, AND MEMBER OF THE SENATE, OF THE QUEEN’S UNIVERSITY IN IRELAND.
DUBLIN: PRINGED ALT THE UNIVERSITY PRESS, BY PONSONBY AND MURPHY.
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[From “ Hermatuena,” Vol. 117]., No. ΤΥ]
160
DR. ALLMAN ON GREEK GEOMETRY
GREEK GEOMETRY FROM THALES TO EWVCLIB:
N 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 Ahzlosophers. 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
‘It has been frequently observed, and is indeed generally admitted, that the present century is characterized by the importance which is attached to historical researches, and bya widely- diffused taste for the philosophy of his- tory.
In Mathematics, we have evidence of these prevailing views and tastes in two distinct ways :—
1° The publication of many recent
works on the history of Mathematics,
By ρς
Arneth, A., Die Geschichte der reinen Mathematik, Stuttgart, 1852; * Bretschneider, C. A., Die Geometrie
und die Geometer Vor Euklides, Leip- zig, 1870; Suter, H., Geschichte der Mathematischen Wiossenschaften (τοὶ Part), Zurich, 1873; * Hankel, H., Zur Geschichte der Mathematik in Alter- thum und AMfittel-alter, Leipzig, 1874 (a posthumous work); * Hoefer, F., fTistotre des Paris, 1874. (This forms the fifth volume by M. Hoefer on the history of the sciences, all being parts of the Wistoire Uni- verselle, published under the direction of M. Duruy.) In studying the subject of this Paper, I have made use of the works marked thus*. Though the work of M. Hoefer is too metaphysical,
Mathématiques,
FROM THALES TO EUCLID. 101
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 Hzstory 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 Theodosii Sphaericorum libri Tres, even errors, yet I have derived advan- Nizze, Berlin, 1852; Nicomachi Gera- tage from the part which concerns Py- seni Zntvoductiones Arithmeticae, lib. thagoras andhisideas. Hankel’s book 11., Hoche, Lipsiae, 1866 (Teubner) ; contains some fragments ofa great work Boetii De Just. Arithm., &c., ed. ἃ. on the History of Mathematics, which Friedlein, Lipsiae, 1867 (Teubner); was interrupted by the death of the Procli Diadochi zz primum Euclidis author. The part treating of the ma- Llementorum librum commentarii, ex thematics of the Greeks during the first recog. G. Friedlein, Lipsiae, 1873 (Teu- period—from Thales to the foundation bner); Heronis Alexandrini Geometrt- of the School of Alexandria—is fortu- corum et Stereometricorum Reliquiae nately complete. This is an excellent ὁ libris manuscriptis, edidit F. Hultsch, work,and is in many parts distinguished Berolini, 1864; Pappi Alexandrini by its depth and originality. Collectiones quae supersunt e libris The monograph of M. Bretschneider manuscriptis Latina interpretatione is most valuable, and is greatly in ad- et commentariis instruxit F. Hultsch, vance of all that preceded it on the vol. 1, Berolini, 1876: vol. 11, 2, origin of geometry amongst the Greeks. 1877. He has collected with great care, and Occasional portions only of the Greek has set out in the original, the fragments text of Pappus had been published at relating to it, which are scattered in various times (see De Morgan in Dr. W. ancient writers; I have derived much Smith’s Dictionary of Biography). An aid from these citations. Oxford edition, uniform with the great 2° New editions of ancient Mathema- editions of Euclid, Apollonius, and tical works, some of which had become Archimedes, published in the last cen- extremely scarce, ¢. ¢.— tury, has been long looked for.
VOR. It. ἍΜ
162 DR. 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 (καθολικώτερον), and some in a more intuitional or sensible manner (αἰσθητικώτερον). 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 (avAwe καὶ νοερῶς); he also discovered the theory of incommensurable quantities (τῶν ἀλόγων πραγμα- τείαν), 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, Archytas of Tarentum, and Theaetetus of Athens, by all of whom theorems were added or placed 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. (διορισμός)---ι 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 (ταῖς ἀναλύσεσιν. 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
2 Does this mean the cutting of a and synthesis are first used and de- straight line inextreme and meanratio, 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 Zwclid’s schneider, Die Geometrie vor Euklides, Elements, Lib., xiii., the terms analysis p. 168.
M 2
164 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 oe on to say, Euclid was not much younger
“had been ah chee by Theanine further, he substi- tuted inco incontrovertible proofs for the lax demonstrations ‘of his pre predecessors. He lived in the times of the first Ptolemy, by whom, it is said, he was asked whether there
his s Elements, to which he Sia that there was no voyal road to geometry. Euclid then was younger than the dis- ciples of Plato, but elder than Eratosthenes and Archimedes —who were contemporarie ies—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 atonic bodies [the regular solids] as the final aim of his systematizatio systematization of the Ele- ments.*
is
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
3 From these words we infer that the —_ pp. 299, 333, 352, and 379. LfTistory of Geometry by Eudemus is 4 Procli Diadochi zz primum Euclidis most probably referred to, inasmuch as = elementorum librum commentarit. 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 aéstract 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.®
5 The importance, for the present research, of bearing in mind this ab- stract character of Greek science con- sists in this, that it furnishes a clue by means of which we can, in many cases, recognise theorems of purely Greek growth, and distinguish them from those of eastern extraction. The neglect of this consideration has led some recent writers on the early history of geometry greatly to exaggerate the obligations of the Greeks to the Orien- tals; whilst others have attributed to
the Greeks the discovery of truths which were known to the Egyptians. See, in relation to the distinction between ab- stract and concrete science, and its bearing on the history of Greek Ma- thematics, amongst many passages in the works of Auguste Comte, Systéme de Politique Positive, vol. 111., ch. iv., p- 297, and seqg., vol.1., ch. i., pp. 424-- 437; and see, also, Les Grands Types de Humanité, par P. Laffitte, vol. 11., Lecon 15iéme, p. 280, and seg.—Ap- préciation de la Science Antique.
166 DR. ALLMAN ON GREEK GEOMETRY
The following notices of the geometrical work of Thales have been preserved :—
(2). He is reported to have first demonstrated that the circle was bisected by its diameter ;°
(6). 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, /zke (ὁμοῖαι); ᾿Τ
(c). Eudemus attributes to him the theorem that when two straight lines cut each other, the vertically opposite angles are equal ;*
(dz). 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 Apollodorus (ὁ λογιστικός) is one, Say the same of Pythago- pass
(ec). 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. ἘΣ (τ Cobets pio: 1 Ibid, p. 250. 11 6 δὲ Ἱερώνυμος καὶ ἐκμετρῆσαί φησιν 8 7214, p. 299. αὐτὸν Tas πυραμίδας ἐκ τῆς σκιᾶς Tapa-
® Pamphila was a female historian τστηρήσαντα ὅτε ἡμῖν ἰσομεγέθεις εἰσί. who lived at the time of Nero; an Epi- Diog. Laert., I., ον 1, n. 6., ed. Cobet, daurian according to Suidas, an Egyp- ρ. 6. tian according to Photius. 2 Plin. Hist. Nat., xxxvi. 17.
10 Diogenes Laertius, I., c. 1, n. 3,
FROM THALES TO EUCLID. 167
manner by which you measured the height of the pyramid 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 pyramid was to the length of the staff in the same ratio as their respective shadows’’).!°
(72). 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 Eucl. 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 ; ἢ
(9). Proclus, or rather Eudemus, tells us in the passage quoted above 272 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."
13 Plut. Sept. Sap. Conviv. 2.vol. iii, | which it has been stated by Diogenes
p. 174, ed. Didot. Laertius shows that he did not distin- 14 Proclus, ed. Friedlein, p. 352. guish between a problem and a theo- 15 Jbid, 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
168 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 (4)—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 sum 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 tworight 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 (Zwclzd 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 circle, he proposes a problem : for it is possible to inscribe one that is not equilateral. But when anyone asserts that the angles at the base of an isosce- les triangle are equal, he must affirm that he proposes a theorem: for it is not possible that the angles at the base of an isosceles triangle should be un- equal to each other. On which account if anyone, stating it as a problem, should say that he wishes to inscribe a right angle in a semicircle, he must be con- sidered as ignorant of geometry, since
every angle in a semicircle is necessa- rily a right one.”—Taylor’s Proclus, vol. I., p. 110. Procl. ed. Friedlein, pp. 79, 80.
Sir G. C. Lewis has subjected himself to the same criticism when he says— ‘ According to Pamphila, he first solved the problem of inscribing a right-angled triangle in a circle.’—G. Cornewall Lewis, Historical Survey of the Astro- nomy of the Ancients, p. 83.
7 So F. A. Finger, De Primordiis Geometriae apud Graecos, p. 20, Heidel- bergae, 1831.
FROM THALES 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 (Lwcld 1., 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 triangle.’ Further, if a perpen-
18 Proclus, ed. Friedlein, p. 379. 19 Apollonii Conica, ed. Hallejus ,p. 9; Oxon. 1710.
so as to fill a space,’’ is attributed by Proclus to Pythagoras or his school (ἐστι τὸ θεώρημα τοῦτο Πυθαγόρειον.
20 Floors or walls covered with tiles of various colours were common in Egypt. See Wilkinson’s “ Ancient Egyptians,” vol. i1., pp. 287 and 292.
21 Although the theorem that ‘‘ only three kinds of regular polygons—the equilateral triangle, the square and the hexagon—can be placed about a point
Proclus, ed. Friedlein, p. 305), yet it is difficult to conceive that the Egypt- ians—who erected the pyramids—had not a practical knowledge of the fact that tiles of the forms above mentioned could be placed so as to form a con- tinuous plane surface.
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 :— “Ὁ se del mezzo cerchio far si puote Triangol si, ch’un retto non avesse.” Par. ΟΣ χα τοῖς:
Second inference.—The conception 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,
253 Though we are informed 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 (ἐζήτησεν) —_ 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 of geometrical loct, 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 (¢). 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-
23 Montucla, Histoire des Mathéma- tiques, Tome i., p. 183, Paris, 1758. Chasles, Aperpu Historique des Métho- des en Géométrie, p. 5, Bruxelles, 1837. Chasles in the history of geometry be- fore Euclid copies Montucla, and we have aremarkable instance of this here, for Chasles, after Montucla, calls Plato
“ce chef du Lycée.”
24 But we have seen that the account given by Diogenes Laertius of the dis- covery of Thales mentioned by Pam- phila is unintelligible and _ evinces ignorance of geometry on his part.
25 Bretsch. Die Geometrie und Geo- meter vor Euklides, pp. 45, 46.,
DR. ALLMAN ON GREEK GEOMETRY
12
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-
26 [bid, p. 18.
1 Ibid, pp. 43, 44.
28 In reference to this I may quote the following passage from Clairaut, Lilémens de Géométrie, pp. 34-35. Paris, 1741.
‘La méthode qu’on vient de don- ner pour mesurer les terrains, dans lesquels on ne s¢auroit tirer de lignes, fait souvent naitre de grandes difficultés dans la pratique. On trouve rarement un espace uni et libre, assez grand pour faire des triangles egaux ἃ ceux du ter- rain dont on cherche la mesure. Et méme quand on en trouveroit, la grande longueur des cétés des triangles pour- roit rendre les opérations trés-difficiles :
abaisser une perpendiculaire sur une ligne du point qui en est éloigné seule- ment de 500 toises, ce seroit un ouvrage extrémement pénible, et peut-étre im-
practicable. Il importe done d’avoir
un moyen qui supplée a ces grandes opérations. Ce moyen s’ offre comme.
Tl vient, &c.”’
*9 Observe that the inventions of the square and level are attributed by Pliny (Wat. Hist., vii., 57) to Theodorus of Samos, who was a contemporary of Thales. They were, however, known long before this period to the Egyptians; so that to Theodorus is due at most the honour of having introduced them into Greece.
de lui-méme.
FROM THALES TO EUCLID.
173
angled triangle is in one case in a vertical, and in the other in a horizontal plane. τ
From what has been said it is plain that there is ἃ 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, 6. 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, Systéme de Poli- thematiques, p. 69. Since this Paper
tique Positive, vol, 111., p. 297.
31 Birch, in Lepsius’ Zectschrift fur Aegyptische Sprache und Alterthums- kunde (year 1868, p. 108). Bret- schneider, Geometrie vor Euklides, p- 16. F. Hoefer, Wistotre des Ma-
was sent to the press, Dr. August Eisenlohr, of Heidelberg, has published this papyrus with a translation and commentary under the title ‘‘ Azz 77α- flandbuch der
thematisches alten
Ligvpter.”
174 DR. 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: “ΝΟ one has ever excelled me in the construction of lines according to certain indications—not even the so-called Egyptian Harpedonaptae.” ”
(0). 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 nxatural law—z. e., the expression of a fixed de- pendence between different 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
32 Mullach, /ragmenta Philosopho- Pos. vol. iii., p. 300).
rum Graecorum, p. 371, Democritus ap. 34 P. Laffitte, Zes Grands Types de Clem. Alex. Strom. I. p. 357, ed. Pot- 2. Humanité, vol. ii., p. 292. ter. 35 Jbid, p. 294.
33 Auguste Comte (Systéme 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.
i:
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 ἡ μεγάλη Ἑλλάς. 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 onehand, and Tarentum and Cnidus on the other, were
36 Herodotus, i. 143. i, p. 141, 1844. 37 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 all* 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
38 Herod., vi. 21, and iii. 138. Listory of Philosophy, Book ii., c. ii., 39 Aristotle, Diogenes Laertius, Pro- where the various dates given by clus, amongst others. scholars are cited.
See ἃ. 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 Pythago- 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 from later writers, especially those
‘Tamblichus, de Vita Pyth.,c.ii.,12. ap. Porphyr., de Vita Pyth., 9. + Tsocrates is the oldest authority for 44 Cicero, de Rep. 11., τῷ; Tusc. Disp., this, Busiris, c. 11. TeeXVIEN 25. 43 Diog. Laert., viii. 3 ; Aristoxenus, VOL. III. N
178 DR. ALLMAN ON GREEK GEOMETRY
of the Neo-Pythagorean 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. Somerecent 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-Pythagoreans had access to ancient and reliable au- thorities which have unfortunately been lost since.”
Though the difficulties to which I refer have been felt chiefly by those who have treated of the Pythagorean p/z- 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- οἵ whom lived in the reign of Justinian. metry, by Eudemus of Rhodes, one of | Eudemusalso wrote a History of Astro- the principal pupils of Aristotle, is omy. 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 THALES TO EUCLID. 179
are under the necessity, therefore, of treating the work of the early Pythagorean School as a whole."
All agree, 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; τ
(6.) Eudemus informs us, in the passage quoted above 77 cxtenso, that Pythagoras 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 //zstory of Alexander ; and the part
46 ἐς Pythagoras and his earliest suc- cessors do not appear to have commit- ted any of their doctrines to wniting. According to Porphyrius (de Vita Pyth. p. 40), Lysis and Archippus collected in a written form some of the principal Pythagorean doctrines, which were handed down as heirlooms in their families, under strict injunctions that they should not be made public. But amid the different and inconsistent accounts of the matter, the first publi- cation of the Pythagorean doctrines is pretty uniformly attributed to Philo-
N 2
laus.”—Smith’s Dictionary, tn v. Phi- lolaus. Philolaus was born at Cro- tona, or Tarentum, and was a contem- porary of Socrates and Democritus. See Diog. Laert. ἐγ Vita Pythag., viii., 1, 15; 22 Vita Empedoclis, viii., i., 2 ; and zz Vita Democriti, ix., vil., 6. See also Iamblichus, de Vita Pythag., ἘΣ 18, Be 58:
“7 Aristot. Met, 1., 5, 985, N. 23, ed. Bekker.
48 Procl. Comm., ed. Friedlein, p. 65.
49 An ancient King of Egypt, who reigned 900 years before Herodotus.
DR. ALLMAN ON GREEK GEOMETRY
180
of the science to which Pythagoras applied himself above all others was arithmetic ; δ᾽
(d.) Pythagoras 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 ; ὃ"
(e.) He was the first person who introduced measures and weights among the Greeks, as Aristoxenus the musi- cian informs us ; δ
(7.) He discovered the numerical relations of the musical scale ;*
(g.) The word mathematics originated with the Pytha- goreans ;*
(λ.) The Pythagoreans made a four-fold division of mathematical science, attributing one of its parts to the how many, τὸ ποσόν, and the other to the how much, τὸ πηλίκον ; 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 ἱτὴν σφαιρικὴν) contemplates continued quantity so far as it is of a self-motive nature ; *
(z.) Favorinus says that he employed definitions on
ὅ0 Diog, Laert., vill. 11, ed, Cobet, εὑρεῖν. Diog. Laert., viii., 11, ed.
p. 207.
51 Aristoxenus, Hragm. ap. Stob. Fclog Leys... il. ὁ. 69... ieeren, vol. I., p. 17.
52 Diog. Laert., viii., 13, ed. Cobet, p- 208.
53 “τῇ ἢ » ~ 5
δύ τόν τε κανόνα τὸν ἐκ μιᾶς χορδῆς
Cobet, p. 207.
54 Procli Com., Friedlein, p. 45.
53 Procli Comm., ed. Friedlein, p. 35- As to the distinction between τὸ πηλίκον, continuous, and τὸ ποσόν. discrete, quantity, see Iambl., 27 Vic. G. Arithm. introd. ed. Ten., p. 148.
FROM THALES TO EUCLID. 181
account of the mathematical subjects to which he applied himself (ὅροις χρήσασθαι διὰ τῆς μαθηματικῆς ὕλης).
As to the particular work done by this school in geo- metry, the following statements have been handed down £0) US === y
(a.) The Pythagoreans define a point as unity having position (μονάδα προσλαϊ[βοῦσαν θέσιν) ;™
(6.) 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 ; δ᾽
(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 (Zuc/. i. 32), and states their method of proving it, which was substantially the same as that of Euclid; δ᾽
(e.) 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 (παραβολή) in which it is used in this proposition, but also in its wider significa- tion, embracing ὑπερβολή and ἔλλειψις, in which it is used in the 28th and 29th propositions of the Sixth Book,—are old, and inventions of the Pythagoreans ; “!
and defect of areas are ancient, and are due to the Pythagoreans. Moderns bor- rowing these names transferred them to
56 Diog. Laert., vili., 25, ed. Cobet, p. 215. 57 Procli Comm. ed. Friedlein, p. 95.
58 Jbid., p. 97. the so-called conic lines—the parabola, 59 Tbid., p. 305. the hyperbola, the ellipse ; as the older b0RGid.. De 79- school in their nomenclature concerning
ΟἹ Joid., p. 419. The words of Pro- the description of areas 272 plano on a
clus are interesting :— “* According to Eudemus, the inven- dions respecting the application, excess,
finite right line regarded the terms thus :— “¢ An areais said to be applted (rapa
DR. ALLMAN ON GREEK GEOMETRY
182
(7) 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 :—
Ἡνίκα Πυθαγόρης τὸ περικλεὲς εὕρετο γράμμα, Κεῖν᾽ ἐφ᾽ ὅτῳ λαμπρὴν ἤγετο βουθυσίην, 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 (εἴτε πρόβλημα περὶ τοῦ χωρίου τῆς Tapa- βολῆς) 5
(g.) One of the most elegant (γεωμετρικωτάτοις) 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 ; 1.) Eudemus, in the passage already quoted from Pro- clus, says Pythagoras discovered the construction of the
regular solids; "
βάλλειν) to a given right line when an area equal in content to some given one is described thereon ; but when the base of the area is greater than the given line, then the area is said to be in ex- cess (ὑπερβάλλειν) ; but when the base is less, so that some part of the given line lies without the described area, then the area is said to be in defect (ἐλλείπειν). Euclid uses in this way, in his Sixth Book, the terms excess and defect. The term afflication (παραβάλλειν), which we owe to the Pythagoreans, has this signification.”’ 8° Plutarch, 202 posse suaviter vivid sec. Epicurum, c. xi. ; Plut., Opera, ed. Didot, vol. iv., p. 1338. Some authors,
rendering περὶ τοῦ χωρίου τῇς παραβολῆς.
“concerning the area of the parabola,’ have ascribed to Pythagoras the qua- drature of the parabola—which was in fact one of the great discoveries of Ar- chimedes ; and this, though Archimedes himself tells us that no one before him had considered the question; and though further he gives in his letter to Dosi- theus the history of his discovery, which, as is well known, was first ob- tained from mechanical considerations, and then by geometrical reasonings.
63 Plutarch, Syp., vili., Quaestio 2, c.4. Plut. Opera, ed. Didot, vol. iv., p. 877.
δ: Procl. Comm., ed. Friedlein, p. 65..
FROM THALES TO EUCLID.
183
(2.) 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 ;”
(7.) The triple interwoven triangle or Pentagram—star- shaped regular pentagon—was used as a symbol or sign of recognition by the Pythagorears, and was called by them Health (ὑγιεία) ; °°
(&.) The discovery of the law of the three squares (Zuc/. Τ., 47), commonly called the Zheorem of Pythagoras, is attributed to him by—amongst others—Vitruvius, ἢ 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;”
(4) One of the methods angles whose sides can be
65 Tambl., de Vit. Pyth., c. 18, 5. 88.
66 Scholiast on Aristophanes, (wd. 611; also Lucian, pro Lapsu in Sa- (ut., 5. 56. That the Pythagoreans used such symbols we learn from Iamblichus (de Vit. Pyth., c. 33, ss. 237 and 238). This figure is referred to Pythagoras himself, and in the middle ages was called Pythagorae figura. It is said to have obtained its special name from his having written the letters v, y, «, θ (Ξ εἰ), a, at its prominent vertices. We learn from Kepler (Opera Omnia, ed. Frisch, vol. v., p. 122) that even so late as Pa-
of finding right-angled tri- expressed in numbers—that
racelsus it was regarded by him as the symbol of health. See Chasles, Yistozre de Géometrie, pp. 477 et seqq.
81 De Arch., ix., Praef. 5, 6, and 7.
68 Where the same couplet from Apollodorus as that in (/) is found, except that κλεινὴν ἤγαγε occurs in place of λαμπρὴν ἤγετο. Diog. Laert., Viil., 11, p. 207, ed. Cobet.
69 Procli Comm., Ὁ. 426, ed. Fried- lein.
70 De Is. et Osir.,c. 56. Plut. Of., vol. iii., p. 457, Didot.
184 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 ;”
(z.) The three proportions—arithmetical, geometrical, and harmonical, were known to Pythagoras ;*
(ο.) 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 ὑπεναντία, but which Archytas and Hippasus designated the harmonical, since it appeared to include the ratios concerning harmony and melody (μετακληθεῖσα ὅτι τοὺς κατὰ TO ἁρμοσμένον καὶ ἐμμελὲς ἐφαίνετο λόγους περιέχουσα);
(2.) With reference to the means corresponding to these proportions, Iamblichus 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 ;*
71 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, Iambl., 271
7 Procli Comm., ed. Friedlein, p.65. Nicom. Arithm, ed. Ten., pp. 172 et
“3 Nicom. G. Zntrod, Ar. c. xxil., ed. _ seq.)
R. Hoche, p. 122.
Hankel, commenting on this pas- 74 Tamblichus zz Nicomachi Arith-
sage of Iamblichus, says: “* What we meticam aS. Tennulio, p. 141. are to do with the report, that this
79 [bid., Ὁ. 168. proportion was known to the Baby-
τὸ /bid., 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.’’—Geschichie arithmetical means between two num- der Mathematik, p. 105. In another bers forming a geometrical proportion ρατί of his book,, however, after refer-
FROM THALES TO EUCLID. 185
(g.) The doctrine of arithmetical progressions is attribu- ted to Pythagoras aut
(7.) It would appear that he had considered the special case of ¢vzangular numbers. Thus Lucian:—IY@. Eir’ ἐπὶ τουτεοῖσιν ἀριθμέειν. IYO. Πῶς NYO. ὍὉρᾶς; ἃ σὺ δὸο- κέεις τέτταρα, ταῦτα δέκα ἐστὶ καὶ τρίγωνον ἐντελὲς καὶ ἡμέτερον ὥρκιον.᾽5
(5.) Another of his doctrines was, that of all solid figures the sphere was the most beautiful; and of all plane figures, the circle.”
(4) Also Jamblichus, 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 (z)—for some of them are found in the Egyp- tian architecture ; in the problems concerning the applica- tion of areas (δ); and lastly, in the law of the three
re ~ > ~ AT. Οἶδα καὶ viv ἀριθμεῖν. U «ριθμέεις ; AT. “Ev, δύο, τρία, τέτταρα.
ring to two authentic documents of the Babylonians which have come down to us, he says: ‘‘We cannot, therefore, doubt that the Babylonians occupied themselves with such progressions [arithmetical and geometrical]; and a Greek notice that they knew propor- fions, nay, even invented the so-called perfect or musical proportion, gains thereby in value.””—J/dzd., p. 67.
τ Theologumena Arithmetica, p. 153, ed. F. Ast, Lipsiae, 1817.
78 Lucian, Βίων πρᾶσις, 4, vol. 1., p- 317, ed. Ὁ. Jacobitz.
19 Kal τῶν σχημάτων τὸ κάλλιστον σφαῖραν εἶναι κύκλον, Diog. Laert., ἐγε Vita Pyth., viii., 19.
80 Simplicius, Comment., &c., ap. Bretsch., Die Geometrie vor Euklides,
p. 108.
τῶν στερεῶν
186 DR. ALLMAN ON GREEK GEOMETRY
squares (£), coupled with the rule given by Pythagoras for the construction of right-angled triangles in num- bers (ὦ.
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 containing 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 (ἀρχὴ), Isis as the receptive principle (ὑποδοχή), and Horus as the product (ἀποτέλεσμα).᾽᾿ ἢ
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 of 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
80a Plutarch, De Js. et Osir. c. 56, rately with squares before the figures vol. ili., p. 457, ed. Didot. were introduced. See Wilkinson’s 51 It was the custom of the Egyp- Ancient Egyptians, vol. ii., pp. 265, tians, where a subject was to be drawn, 207. to rule the walls of the building accu-
FROM THALES TO EUCLID. 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 (γνώμων), 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, vzcc zversd. 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 Tyduwv means that by which any- thing is known, or criterion; its oldest concrete signification seems to be the carpenter’s square (7orma), by which a right angle is known. Hence, it came to denote a perpendicular, of which, indeed, it was the archaic name, as we learn from Proclus on Euclid, i., 12 :—Todto τὸ πρόβλημα πρῶτον Oivo- πίδης ἐζήτησεν χρήσιμον αὐτὸ πρὸς ἀστρολογίαν οἰόμενος" ὀνομάζει δὲ τὴν κάθετον ἀρχαϊκῶς κατὰ γνώμονα, διότι καὶ ὃ γνώμων πρὸς ὀρθάς ἐστι τῷ ὁρίζοντι (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, u., 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 gzomonic numbers, which were also looked on as male, or generating.
83 Wilkinson’s Axcient Egyptians, vol. ii., p. IIT.
188 DR. ALLMAN ON GREEK GEOMETRY
odd numbers,** and include, therefore, all odd squares. Suppose, now, two squares are given, one consisting of 16 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 12 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.”
8: It may be observed here that we first count with cowzters, as is indicated by the Greek ψηφίζειν and the Latin calculare. The counters might be
in the table of principles attributed by Aristotle to certain Pythagoreans (iJZe- taph.,i., 5, 986 a, ed. Bekker).
The odd—or gromontc—numbers are finite ; the even, infinite. Odd num- bers were regarded also as male, or generating. Further, by the ad- dition of successive gnomons—con- sisting, as we have seen, each of an
equal squares, as well as any other like objects. There is an indication that the odd numbers were first regarded in this manner in the name gzvomonic numbers, which the Pythagoreans ap-
plied to them, and that term was used in the same signification by Aristotle, and by subsequent writers, even up to Kepler. See Arist. Phys., lib. iii., ed. 3ekker, vol. i. p. 203; Stob., Zclog., ab Heeren, vol. i., p. 24, and note; Kepleri Opera Omnia, ed. Ch. Frisch, vol. viii., J/athematica, pp. 164 et seq. 85 This seems to me to throw light on some of the oppositions which are found
odd number of units—to the original unit square or monad, the square form is preserved. On the other hand, if we start from the simplest oblong (érepo- μήκε5), consisting of two unit squares, or monads, in juxtaposition, and place about it, after the manner of a gnomon ~-and gnomon, as we have seen, was used in this more extended sense also at a later period—4 unit squares, and
FROM THALES TO EUCLID.
189
It is another matter to see that the triangle formed by 3, 4, and 5 units is right-angled, and this I think the
then in succession in like manner 6, §, ... unit squares, the oblong form ἑτερο-μήκες will be preserved. The elements, then, which generate a square are odd, while those of which the ob- long is madeup areeven. 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 :-- Ἔτι δὲ τῇ μονάδι τῶν ἐφεξῆς περισσῶν γνωμόνων περιτι- θεμένων, ὃ γινόμενος Gel τετράγωνός ἐστι. τῶν δὲ ἀρτίων ὁμοίως περιτιθέμενων, ἕτε- ρομήκεις καὶ ἄνισοι πάντες ἀποβαίνουσιν. ἴσον δὲ ἰσάκις οὐδείς.
“ Explicanda haec sunt ex antiqua Pythagoricorum terminologia, Γνώμονες nempe de quibus hic loquitur auctor, vocabantur apud eos omnes numeri im- pares, Yoh. Philop. ad Aristot. Phys., 1. 1Π., p. 131: Καὶ of ἀριθμητικοὶ δὲ γνώμονας καλοῦσι πάντας τοὺς περιττοὺς ἀριθμούς. Causam adjicit Simplicius ad eundem locum, Γνώμονας δὲ ἐκάλουν τοὺς περιττοὺς of Πυθαγόρειοι διότι προσ- τιθέμενοι τοῖς τετραγώνοις, τὸ αὐτὸ σχῆμα φυλάττουσι, ὥσπερ καὶ οἱ ἐν yew- μετρίᾳ γνώμονες. 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 prodire nume- res quadratos, τ΄. c. positis gnomonibus 3,5, 7,9 primum I + 3 = 22, tunc porro 14+3 005 & 4)+5=3%, 94+7=4, 16+ 9 -- 5", et sic porro, cf. Ziedem.
Geist der Speculat. Philos., pp. 107, 108. Reliqua expedita sunt.’ Stob. Eclog. ab Heeren, lib. 1., p. 24 and note.
The passage of Aristotle referred to is—onpetov δ᾽ εἶναι τούτου τὸ συμβαῖνον ἐπὶ τῶν ἀριθμῶν. περιτιθεμένων γὰρ τῶν γνωμόνων περὶ τὸ ἕν καὶ χωρὶς ὅτὲ μὲν ἄλλο ἀεὶ γίγνεσθαι τὸ εἶδος. Phys., 111... 4, p. 2039, 14.
Compare, ἀλλ᾽ ἔστι τινα αὐξανόμενα ἃ οὐκ ἀλλοιοῦνται, οἷον τὸ τετράγωνον γνώμονος περιτεθέντος ηὔξηται μέν, ἀλ- λοιότερον δὲ οὐδὲν γεγένηται. Cat. 14, 158, 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 will 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 under the image of square and rectangle, which, in the extraction of the square root, have led precisely to those ideas ?’’ Geschichte der Mathematik, p. 110, note.
Hankel also says—‘‘ Upon what the comparison of the odd with the limited may have been based, and whether upon the theory of the gnomons, can scarcely be made out now.” bid. p- 109, note.
May not the gnomon be looked on as framing, as it were, or limiting the squares ἢ
το zDR. ALLMAN ON GREEK GEOMETRY
Egyptians may have first arrived at by an induction founded on direct measurement, 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 13 units as the sum of those constructed on 12 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 Pythagoras 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 from this number g 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 THALES TO EUCLID. ΙΟΙ
4, then subtracting 1 he gets 3, and by adding 1 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 2 square together with the η΄} gnomon is the (7 + 1)" square ; if the 2 gnomon contains 7” unit squares
: ; 7222 —1 m being an odd number, we have 272+ 1 -- 2,15, .". 71 = — :
᾽ 2
hence the rule of Pythagoras. Similarly the sum of two successive gnomons contains an even number of unit squares, and may therefore consist of 7* unit squares, where # is an even number; we have then (2 2 -- 1) + (2 2
m\* : 1) - 7... or z= (2 : 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 Mundt.
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 equiangular triangles, attri- buted above to Thales, yet there is no indication whatever of his having arrived at it in that deductive manner. On the
86 Procli Comm., ed. Friedlein, p.
capable of further extension, e. ¢. : the 428. Hero, Geom., ed. Hultsch, pp.
sum of 9 (an odd square number) suc-
56, 57. 87 This rule is ascribed to Architas {no doubt, Archytas of Tarentum] by Boetius, Geom., ed. Friedlein, p. 408. 88 Hoefer, Histoire des Alath., p. 112. 89 Kepleri Opera Omnia, ed. Frisch, vol. vil., pp. 163 et seq. It be observed that
may
this method is
cessive gnomons may contain an odd number (say 49 x 9) of square units ; hence we obtain a right-angled triangle in numbers, whose hypotenuse exceeds one side by 9 units—the three sides being 20, 21, and 29. Plato’s method may be extended in like marner.
192 DR. 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 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 Pythagoras, . - - 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. δ
- 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
ight-angled triangles can be placed so that a vertex of
ach shall be in one of the corners of the square in such a
ay that a greater and less side are in continuation. The original square is thus dissected into the four triangles as ‘
90 Procli Comm.ed. Friedlein, p.426. Camerer, Zuclidis Element., vol. i., p- 91 Bretsch., Die Geometrie vor Eu- 444, and references given there. klides, p» 82. This proof is old: see
FROM THALES TO EUCLID. 193
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 may be 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
81 This theorem (Euclid, i. 43) Bret- | gnomon was not used init 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
WOR anit. 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 10 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.” δ᾽
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.
88 Hankel says it cannot be ascer- nate dodecahedron was known to them. tained with precision how far the Py- Hankel, Geschichte der Mathematik, thagoreans had penetrated into this p. 95, note. theory, namely, whether the construc- 89 Plato, 7i72., c. 20, 5. 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 known 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
90 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., 10). 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 excesszve by a square. Now it is to be observed that the problem is solved in this manner by Euclid (vi., 30, 1st 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 (é).”
The statements, then, of Iamblichus concerning Hip- pasus (z)—that he divulged the sphere with the twelve pentagons; and of Lucian and the scholiast on ‘Aristo- phanes (7)—that the pentagram was used as a symbol of
recognition importance. thagoreans
amongst the Pythagoreans, become of greater We learn too from Jamblichus that the Py- made use of signs for that purpose.”
Further, the discovery of irrational magnitudes is ascribed to Pythagoras in the same passage of Eude-
91 Tt may be objected that this reason- ing presupposes a knowledge, 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 will appear, was taught by the Pythagoreans the method of reduction (ἀπαγωγή), ἃ Sys- tematization,,as it seems to me, of the manner of reasoning that was sponta- neous with Pythagoras. Proclus de- fines ἀπαγωγή to be ““ ἃ 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, z.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 (ἀπα- yon) : and made many other discoveries in
who also squared the lunule,
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.
2Tambl. de Pyth. Vita, cxxxiil., Ρ. 77, ed. Didot.
198 DR. ALLMAN ON GREEK GEOMETRY
mus (7), 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 reductio 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.
% Aristoteles, Analyt. Prior.,1.,c.23, for its historical interest only, since the 41, a, 26, 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.—Hankel, Geschichte der Math., 117. Hankel thinks he didso probably ρ. 102, note.
FROM THALES TO EUCLID. 199
The existence of the irrational, as well as that of the regular 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 Iamblichus 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: Tocut a straight line internally, or externally, so that the rectangle under the segments shall be equal to a given rectilineal figure. Onexamination 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 Pythagoras. 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. Untersuchungen iiber die neu auf- Dr. Joachim Heinrich Knoche, Her- gefundenen Scholien des Proklus Dia- ford, 1865, pp. 20 and 23. dochus zu Euclid’s Elementen, von
DR. ALLMAN ON GREEK GEOMETRY
200
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 (z) and Iambli- chus (0 and 2).
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 ὑπεναντία.
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 gzofzent. 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, 4, c, be taken. If -- ὁ -- ὁ -- ε, the three magnitudes are in arithmeti- cal proportion; but τα : δ: : ὁ : ο, they are in geometrical proportion.” In the latter case, it follows at once, from the
95 In dimes we may have c = a — ὦ, or a:6:a-—6, This particular case, in which the geometrical and arithmetical ratios both occur in the same propor-
tion, is worth noticing. The line a is
then the sum of the other two lines, and is said to be cut in extreme and mean ratio. This section, as we have seen, has arisen out of the construction
of the regular pentagon, and we learn
FROM THALES TO EUCLID.
201
second theorem of Thales (Euclid, vi., 4), thata-2:4-c :: @: 6, whereas in the former case we have plainly a -- 6 :6-c::a:a. This might have suggested the considera- tion of three magnitudes, so taken thata-6:b-c::a:¢; 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—
@:¢::b6-¢:a-—6 ... thecontrary of the harmonical ;
b:¢::b6-c:a-b6
eet The discovery of these proportions is attributed * to Hip- pasus, Archytas, and Eudoxus.
We have seen also (2) 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—
. . the contrary of the geometrical.
*. 2ab ἜΝ αν
We have also seen (9) 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 /soferimetry by proving that of all figures having the same perimeter the circle is the greatest, and
from Kepler that it was called by the moderns, on account of its many won- derful properties, sectio divina, et pro- portio divina. He sees in it a fine image of generation, since the addition to the line of its greater part produces a new line cut similarly, and so on. See Kepleri Opera Ommnia, ed. Frisch,
vol. v., pp. 90 and 187 (Harmonia Mundz) ; also vol. i. p. 377 (Literae de Rebus Astrologicis). The {pentagram might be taken as the image of all this, as each of its sides and part of a side are cut in this d7vzze proportion.
96 Tambl. in Nic. A7zth., pp. 142, 159, 163. See above, p. 163.
DR. ALLMAN ON GREEK GEOMETRY
202
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 (a\oyov), and the construction of the regular solids; the
97 **Suivant Diogéne, dont le texte est ici fort corrompu, et probable- ment transposé, il ébaucha aussi la doctrine des Isopérimétres, en démon- trant que de toutes les figures de méme contour, parmi les figures planes, c’est le cercle qui est la plus grande, et par- mi les solides, la sphére.’’-—Montucla,
Histoire des Mathématiques, tom. 1., p. 113:
98. Timaeus, 33, B., vol. vii., ed. Stallbaum, p. 129.
99 See Bretschneider, Die Geometrie vor Euklides, pp. 89, 90.
100 See above, p. 182, note.
FROM THALES TO EUCLID. 203
a
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 (/) 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 gwad- 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-
101 This problem is considered in the Papyrus Rhind, pp. 97, 98, 117. The point of view from which it was regarded by the Egyptians was different from that of Archimedes. Whilst he made it to depend on the determination of the ratio of the circumference to the dia- meter, they sought to find from the
diameter the side of a square whose area should be equal to that of the circle. Their approximation was as follows :—The diameter being divided into nine equal parts, the side of the equivalent square was taken by them to consist of eight of those parts.
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- dern 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 (σύνδεσμον) of mathematics.” 1”
Weare told, too, in an anonymous scholium on the Ele- ments of Euclid, which Knoche attributes to Proclus, that the fifth 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
102 Bretschneider (Die Geometrie vor 104 Procl. Comm.,ed. Freidlein, p. 43. Euklides, p. 74) and Hankel (Ge- 105 Ruclidis Elem. Graece ed. ab schichte der Mathematik, p.104)doso, E. F. August, pars ii., p. 328, Berolini, although they are treating ofthe history 1829. Untersuchungen tiber die neu of Greek geometry, which is clearly a aufgefundenen Scholien des Proklus xu mistake. Euclid’s Elementen, von Dr. J. H.
103 On this see A. Comte, Politigue Knoche, p. το, Herford, 1865. Positive, vol. iii., ch. iv., p. 390.
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 «« Rt quidem geometriae theoreti- cae initio hujus tractatus duas fecimus
partes, unam de magnitudinibus, qua-
- tenus fiunt figurae, alteram de compara- tione figurarum et quantitatum, unde proportio existit.
‘* Hae duae scientiae, arithmetica et
geometria speculativa, mutuas tradunt operas nec ab invicem separari possunt, quamvis et arithmetica sit principium cognitionis.’’—Kepleri Opera Omnia, ed. Dr. Ch. Frisch, vol. vili., p. 160, Francofurti, 1870.
206 DR. 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 fruit in due season, so Pythagoras devoted himself to the forma- tion of a society of é/z¢e, 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 generations.
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 ; but not a figure and three obolt (σχᾶμα καὶ Bawa, ἀλλ᾽ ov σχᾶμα καὶ τριώ-
[3oXov).2"
107 Procli 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- aenigma among the Pythagoric symbols = ment. This symbol occurs in Iambli-
FROM THALES TO EUCLID. 207
Such, then, were the men by whom the first steps in mathematics—the first steps ever the most difficult—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 Iambl., Adhortatio ad Philosophiam, ed. Wiessling, Symb. EXXVi., Cap. xxi., p. 317; also Axl.
Ρ. 374. Td δὲ προτίμα τὸ σχῆμα καὶ βῆμα τοῦ σχῆμα καὶ τριώβολον.
GEORGE Jj; ALEMAR:
Ἐπ ΕΠ GHOMETRY,
FROM
ΠΑ ΕΞ TOR EUCLID:
Pak T ar.
BY
GEORGE JOHNSTON ALLMAN, ΠῚ De OF TRINITY COLLEGE, DUBLIN ;
PROFESSOR OF MATHEMATICS, AND MEMBER OF THE SENATE, OF THE QUEEN’S UNIVERSITY IN IRELAND 3; MEMBER OF THE SENATE OF THE ROYAL UNIVERSITY OF IRELAND.
ὌΝ Δ Δ ΔΝ Δ ΔΝ
DUBLIN : PRINTED AT THE UNIVERSITY PRESS" BY PONSONBY AND WELDRICK. 1881.
ΧΡ ahi RY ὴ sep 2.0 672
τῷ ἢ ΝΣ “A
% yy, , Οὐ 2 ὡς τ ϑιτΥ af 1θῦ
[ Prom “ HERMATHENA,” Vol. IV., No. VII.)
180 DR. ALLMAN ON GREEK GEOMETRY
GREEK GEOMETRY FROM THALES TO EUGEID:*
[Continued from Vol. 117., No. V.]
11.
HE first twenty years of the fifth century before the Christian era was a period of deep gloom and despondency throughout the Hellenic world. The Ionians 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 Atégean 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. 160, note) I acknowledged my obligations to the works of Bretschneider and Hankel: I have again made use of them in the preparation of this part. Since it was written, I have received from Dr. Moritz Cantor, of Heidelberg, the portion of his History of Mathematics which treats of the Greeks (Vorlesun- gen uber Geschichte der Mathematik, von Moritz Cantor, Erster Band. Von den altesten Zeiten bis zum Jahre 1200 n. Chr. Leipzig, 1880 (Teubner) ). To the list of new editions of ancient
referred to above, I have to add: Theonis Smyrnaei Lxfositio rerum Mathematicarum ad legendum Fla- tonem utilium. WRecensuit Eduardus Hiller, Lipsiae, 1878 (Teubner) ; Pappi Alexandrini Collectionis quae super- sunt, &c., instruxit F. Hultsch, vol. iii., Berolini, 1878; (to the latter the editor has appended an /udex Graeci- tatis, a valuable addition; for as he remarks, ‘Mathematicam Graecorum dictionem nemo adhuc in lexici formam redegit.’ Praef., vol. iii., tom. ii.) ; Archimedis Opera omnia cum com-
FROM THALES TO EUCLID. 181
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 Ionians, 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 dAtgean 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 memorabie in the history of the
world.
is historical.
! The names Jonzan Sea, and Jonian Zsles, still bear testimony to the inter- course between these cities and Ionia.
mentarits Hutocit. E codice Florentino recensuit, Latine vertit notisque illus- travit J. L. Heiberg, Dr. Phil. Vol.i., Lipsiae, 1880 (Teubner). Since the
above was in type, the following work has been published: An Jntroduction to the Ancient and Modern Geometry of Conics : being a geometrical treatise on the Conic Sections, with a collection of Problems and Historical Notes, and Prolegomena. By Charles Taylor, M.A., Fellow of St. John’s College, Cam- bridge. Cambridge, 1881. The matter of the Prolegomena, pp. xvii.—Ixxxviii.,
The writer of the article in Smith’s Dictionary of Geography thinks that the name Ionian Sea was derived from Ionians residing, in very early times, on the west coast of the Peloponnesus. Is it not more probable that it was so called from being the highway of the Ionian ships, just as, now-a-days, in a provincial town we have the Lozedon road ?
182 DR. ALLMAN ON GREEK GEOMETRY
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 (czvc. 450 B.C.), the invention of dialectics by Zeno, and his famous polemic against multiplicity and
* #.g. the Asclepiadae. See Curtius, I have adopted. See a fine chapter of History of Greece, Engl. transl., vol. ii. his Gesch. der Math., pp. 115 et seq., p- 510. from which much of what follows is
8’ Not so Hankel, whose views as to taken. the influence of the Eleatic philosophy
OO μμμνυνμνδιννδον. ΟΝ ΝΟ δ υυνυνυυ αὐπυυκυνμνυννδινυυνυτυννσων
FROM THALES 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 zxfinitwm: 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 wy =a, x andy may both
il est renfermé; on n’en sera pas sur- pris. Ce Géométre avoit a convaincre des Sophistes obstinés, qui se faisoient gloire de se refuser aux vérités les plus
4 This influence is noticed by Clairaut, Elémens de Géométrie, Pref. p. x., Paris, 1741: ‘Qu’ Euclide se donne la peine de démontrer, que deux cercles qui se
coupent n’ont pas le méme centre, qu'un triangle renfermé dans un autre a la somme de ses cOtés plus petite que celle des cétés du triangle dans lequel
évidentes: il falloit done qu’alors la Géométrie efit, comme la Logique, le secours des raisonnemens en forme, pour fermer la bouche a Ja chicanne.’
184 DR. 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 (ἀτόμοι) in finite numbers: and indeed Aristotle— who, a century later, wrote a treatise on Jndivisible Lines (περὶ ἀτόμων γραμμῶν), in order to show their mathematical and logical impossibility—tells us that Zeno’s disputation was taken as compelling such a view.> We shall see, too, that in Antiphon’s attempt to square the circle; τὸ 15 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 division, 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 insecab. lineis, p. 968, a, δ Vid. Bretsch., Geom. vor Eufl., ed. Bek. Pp. IOI, e¢ wnfra, p. 194.
FROM THALES TO EUCLID. 185
wrote on incommensurable lines and solids (περὶ ἀλόγων γραμμῶν καὶ ναστῶν).ἦ
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 difficulties by banishing from their science the idea of the Infinite—the infinitely small as well as the infinitely great (vzd. Euclid, Book v., Def. 4). They laid down as axioms that any quantity may be divided ad /zbztum ; 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. ΑἹ] 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.’
(ὁ). 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 (ὑπὸ τῶν ἐν BuZav- τίῳ πεντηκοστολόγων) )."° (c). Johannes Philoponus, on the other hand, relates that
7 Diog. Laert., ix., 47, ed. Cobet, p. 9 In Vit. Solonts, 11.
239. 10 Arist., Lik. ad Eud., νὰ.» c. 14, 8 Archim., De qguadr. parab., p. 18, ρ. 1247, a, 15, ed. Bek.
ed. Torelli.
186 DR. 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.”
(4). 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.”
(ὁ). He also taught, for Aristotle says that his pupils, and those of his disciple A®schylus, expressed themselves concerning comets in a similar way to the Pythagoreans.”
(7). He is also mentioned by Iamblichus, along with Theodorus of Cyrene, as having divulged the geometrical arcana of the Pythagoreans, and thereby having caused mathematics to advance (ἐπέδωκε δὲ τὰ μαθήματα, ἐπεὶ ἐξενηνέ- χθησαν δισσοὶ προαγόντε, μάλιστα Θεόδωρός τε ὃ Κυρηναῖος, καὶ ἹἽἹπποκράτης ὃ Xtoc).™
(g). Iamblichus 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.”
(z). 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 (ἀπαγωγή). Proclus
11 Philoponus, Comm.in Arist. phys. 35, ed. Bek.
ausc., f. 13. Brand., Schol. in Arist, 14Tambl. de philos. Pythag. lib. 111; p- 327, Ὁ, 44. Villoison, Azecdota Graeca, ii., p. 216. 2 Procl. Comm., ed. Fried., p. 66. 15 [bid.; also Iambl. de Vit. Pyth.
3 Arist., Jeteor., 1., 6, p. 342, b, Ὁ 18, s. 89.
FROM THALES TO EUCLID. 187
defines araywyy 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, Ζ.6. 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 (ἀπαγωγή): 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."
(z). Eratosthenes, too, in his letter to King Ptolemy ITI. 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 difficult.”
(2). Eutocius, in his commentary on Archimedes (C7z7c. Dimens. Prop. 1), telis 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 carefully searched after it, invented the false reasonings which, I think, are well known to those who have looked
16 Procl. Comm., ed. Fried., p. 212. Oxon. 1792. 1 Archim., ex recens. Torelli, p. 144, 18 Anaxagoras, for example.
188 DR. ALLMAN ON GREEK GEOMETRY
into the Azstery of Geometry of Eudemus and the Kerza (Κηρίων) of Aristotle."
On the passage (72) quoted above, from Iamblichus, 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 Iamblichi é libro tertio de Philosophia Pythagorica Graece necdum edito, p. 64, ex versione Nic. Scutelli: Hzp- pasus (videtur legendum Hipparchus) ezcetur ὁ Pythagorae schola e0 quod primus sphacram duodecim angulorum (Dode- caedron) edzdisset (adeoque arcanum hoc evulgasset), Zheo- dorus etiam Cyrenaeus et Hippocrates Chius Geometra ejicitur
19 Archim., ex recens. Torelli, p. 204. 21 Montucla, Hzstotre des Math., 20 Bretsch., Geom. vor Hukl., p.93; tom. i., p. 144, 1¢ ed. 1758; tom.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. Iamblichus is, how- Taylor, Geometry of Conics, Prole- ever, referred to by later writers as
gomena, Ὁ. XXVill. the authority for it.
FROM THALES TO EUCLID. 189
qui ex geometria quaestum factitabant. 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 :—
Περὶ δ᾽ Ἱππάσου λέγουσιν, ὡς ἣν μὲν τῶν Πυθαγορείων, διὰ δὲ τὸ
3 “A Ν / 6 a a Ἀ ΕἸ ἴον ὃ 00 ε , ἐξενεγκεῖν, καὶ γράψασθαι πρῶτος σφαῖραν, τὴν ἐκ τῶν δώδεκα ἑξαγώνων [πενταγώνων], ἀπόλοιτο κατὰ θάλατταν, ὡς ἀσεβήσας, δόξαν δὲ λάβοι, ὡς > \ ΄ > ΄ “a 3 ὃ (da , \ ως Ν ΄ εἶναι δὲ πάντα ἐκείνου τοῦ ἀνδρός: προσαγορεύουσι γὰρ οὕτω τὸν Πυθαγό- A > δι -“ 3 ’, > ὃ δὲ Ἂν θ fe > NS , ραν, kal ov καλοῦσιν ὀνόματι. ἐπέδωκε δὲ τὰ μαθήματα, ἐπεὶ ἐξενην έ- θ 8 \ ΄ aN Θ aN , ε Κ Ξ \ χθησαν δισσοὶ προαγόντε, μάλιστα Θεόδωρός te ὁ Κυρηναῖος, kat Ἵπποκράτης ὃ Χῖος. λέγουσι δὲ οἱ Πυθαγόρειοι ἐξενηνέχθαι γεωμετρίαν
ράτη μετρ οὕτως" ἀποβαλεῖν τινα τὴν οὐσίαν τῶν Πυθαγορείων: ὡς δὲ τοῦτ᾽ > , 67 3 - γ θ 3 Ν 4 a 5 x Lal δέ ec ἠτύχησε, δοθῆναι ἀυτῷ χρηματίσασθαι ἀπὸ γεωμετρίας" ἐκαλεῖτο δέ ἡ
γεωμετρία πρὸς Πυθαγόρου ἱστορία."
Observe that Fabricius, mistaking the sense, says that Hippasus, too, was expelled. Hippocrates may have been expelled by a school of Pythagoreans 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 (ἀσεβήσας).""
As Anaxagoras was born 499 B.C., and as Plato, after the death of Socrates, 399 B.C., went to Cyrene to hear Theodorus (4), the lifetime of Hippocrates falls within the fifth century before Christ. As, moreover, there could not have been much commerce in the A’gean during the first
Jo. Alberti Fabricii Bibliotheca concerning Hippocrates, the passage, Graeca, ed. tertia, i., p. 505, Ham- with some modifications, occurs also in
burgi, 1718. Jambl. de Vit. Pyth., c. 18, ss. 88 and 23Tambl. de philos. Pyth. lib. iii.; 89. Villoison, Anecdota Graeca, ii., p. 216. 24. See HERMATHENA, vol. ili., p.
With the exception of the sentence 199.
190 DR. ALLMAN ON GREEK GEOMETRY
quarter of the fifth century, and, further, as the state- ments of Aristotle and Philoponus (6) 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 Hzstory 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 carefully revised and emended it. He also supplied the necessary diagrams, some of which were wanting, and added explanatory and
25 Bretsch., Geom. vor Eukl., p. Schol., Ὁ. 211, b, 19. 98. 28 Joh. Gehilopy ats 25a" by, Sezor..
26. De Sophist. Elench., 11,pp.171,b, Brand. p. 211, b, 30. Jdid., f. 118, and 172, ed. Bek.; Phys. Ausc.,i.,2, Schol., p. 211, b, 41. Lbzd., f. 26, b, p. 185, a, 14, ed. Bek. Schol., p. 212, a, 16.
21 Themist. f. 16, Schol. in Arist., 29 Bretsch., Geom. vor Eukl., pp. Brand., p. 327, b, 33. 71... f..5, 100=121.
FROM THALES TO EUCLID. ΙΟῚ
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 verdatzm 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
30 Bretsch., Geom. vor Lukl., p. 109. 32 Tbid., pp. 130, 131. sl 7bid., p. 1132. 33 Hankel, Gesch. der Math., p. 112.
192 DR. 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 (HERMATHENA, vol. ili. 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 111. 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. 190, 191, and note 87. I have there given the two rules of Pytha-
so, as ove book only on the Pythago- reans is mentioned, and ove against them.
18 Gruppe, Ueber die Fragmente des Archytas und der alteren Pythagoreer. Berlin, 1840.
19 Procl. Comm., ed. Fried., p. 66.
20 Boet. Geom., ed. Fried., p. 408. Heiberg, in a notice of Cantor’s ‘ His- tory of Mathematics,’ Revue Critique d’ Histoire et de Littérature, 16 Mai, 1881, remarks, ‘Il est difficile de croire a l’existence d’un auteur romain nommé Architas, qui aurait écrit sur
Varithmétique, et dont le nom, qui ne serait du reste, ni grec ni latin, aurait totalement disparu avec ses ceuvres, a Vexception de quelque passages dans Boéce.’ The question, however, still remains as to the authenticity of the Ars Geometriae, Cantor stoutly main- tains that the Geometry of Boethius is genuine: Friedlein, the editor of the edition quoted, on the other hand, dis- sents; and the great majority of philo- logists agree in regarding the question as still sub judice. See Rev. Crit. loc. cit.
a
FROM THALES 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
21 Arist., Jet. 1. 6, p. 987, a, ed. 23 Asclep. Schol. 1. c., p. 548, a, Bek. 35.
2 Alex. Aph. Schol. in Arist., Brand., *4Theon. Smyrn. Arvithm., ed. de p- 548, a, ὃ. Geldersip., 17:
VOL. V. O
194 DR. ALIMAN 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 zxjfinttum: if we did, a geometrical principle would be set aside, which lays down that magni- tudes are divisible ad zzjfinztum. 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, may be disproved geometrically ; he would rather call the squaring by means of lunes, which Hippocrates found out, one by segments, inasmuch as the
36 But Eudemus was a pupil of 36% Phys, Ausc. i., 2, Ὁ. 185, a, 16, ed. Aristotle, and Antiphon was a con- Bek. temporary of Democritus.
FROM THALES TO EUCLID. 195
lune is a segment of the circle. The demonstration is as follows :—
‘Let a semicircle ay be described on the straight line a; bisect a in 6; from the point ὃ draw a perpendicular oy to a3, and join ay; this will be the side of the square inscribed in the circle of which afy is the semicircle. On ay describe the semicircle azy. Now, since the square on af} is equal to double the square on ay (and since the squares on the diameters are to each other as the respective circles or semicircles), the semicircle αγβ is double the semicircle aey. The quadrant ayo is, therefore, equal to the semicircle ary. Take away the common segment lying between the circumference ay and the side of the square; then the remaining lune acy will be equal to the triangle ayé; 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 αἴ, and let a semicircle be described on it; take yé double of a, and on it also describe a semicircle; and let the sides of a hexagpn, γε, eC, and Zé be inscribed in it. On these sides describe the semicircles γηε, εθζ, xd. Then each of these semicircles described on the sides of the hexagon is equal to the semi. circle a3, for αβ is equal to each side of the hexagon. The four semicircles are equal to each other, and together are then four times the semicircle on af3. But the semicircle on γὸ is also four times that on a3. The semicircle on yd is, therefore, equal to the four semicircles—that on αβ, together with the three semicircles on the sides of the hexagon. Take away from the semicircles on the sides of the hexagon, and from that on yd, the common segments contained by the sides of the hexagon and the periphery of the semicircle yd; the remaining lunes γηε, εθζ, and Zxé, together with the semicircle on αβ, will be equal to the
O02
196 DR. ALLMAN ON GREEK GEOMETRY
trapezium γε, «2, Zo. 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 af3 ; we double this rectilineal figure which remains, and construct a square equal to it. That square will be equal to the circle of which af 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 (ψευδογράφημα) which occurs in it.”
‘Eudemus,®* however, tells us in his H/zstory 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 (κατὰ λέξιν)" 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 Hzstory of Geometry, Eudemus says: the squaring of lunes seeming to relate to an un-
37 I attribute the above observation 105-109, Bretsch., Geom. vor Eukl. on the proof to Eudemus. What fol- 88 χα.» p. 109. lows in Simplicius seems to me not to 39 Simplicius did not adhere to his be his. Ihave, therefore, omitted the intention, or else some transcriber has remainder of §83, and §§84, 85, pp. added to the text.
FROM THALES TO EUCLID. 197 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-
oe
40 Here τμῆμα seems to be used for sector: indeed, we have seen above
that a lune was also called τμῆμα. The
word τομεύς, sector, may have been of later origin. The poverty of the Greek language in respect of geo- metrical terms has been frequently noticed. For example, they had no word for radius, and instead used the periphrasis 7 ἐκ τοῦ κέντρου. Again, Archimedes nowhere uses the word parabola; and as to the imperfect terminology of the geometers of this period, we have the direct statement of Aristotle, who says: kal τὸ ἀνάλογον
ὅτι ἐναλλάξ, ἣ ἀριθμοὶ καὶ ἣ γραμμαὶ καὶ 7 στερεὰ καὶ ἣ χρόνοι, ὥσπερ ἐδείκ- νυτό ποτε χωρίς, ἐνδεχόμενόν γε κατὰ πάντων μιᾷ ἀποδείξει δειχθῆναι" ἀλλὰ διὰ τὸ μὴ εἶναι ὠνομασμένον τι πάντα ταῦτα ἕν, ἀριθμοί μήκη χρόνος στερεά, καὶ εἴδει διαφέρειν ἀλλήλων, χωρὶς ἐλαμ- βάνετο. νῦν δὲ καθόλου δείκνυται: οὐ γὰρ ἣ γραμμαὶ ἢ ἣἧ ἀριθμοὶ ὑπῆρχεν, ἀλλ᾽ ἧ τοδί, ὃ καθόλου ὑποτίθενται imdpxew.—Aristot., Anal., post., 1., ἘΡ 4a) aan) 17... εα- bekicers) sdihis passage is interesting in another re- spect also, as it contains the germ
of Algebra.
198 Dk. 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
1 Trapezia, like this, cut off from an isosceles triangle by a line parallel to the base, occur in the Papyrus Rhind.
42 Then follows a proof, which I have omitted, that the circle can be circum- scribed about the trapezium. This proof is obviously supplied by Simpli- cius, as is indicated by the change of
person from ὑποτίθεται to δείξεις, as well as by the reference to Euclid, 1.9. A few lines lower there is a gap in the text, as Bretschneider has ob- served ; but the gap occurs in the work of Simplicius, and not of Eudemus, as Bretschneider has erroneously sup- posed.—Geom. vor Eukl., p. 111, and note.
FROM THALES 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 construction for the description of such a lune : “—
‘Let af be the diameter of a circle whose centre is «; let y8 cut Bx in the point of bisection y, and at right angles; through 3 draw the straight line βζε, so that the part of it, Ze, intercepted between the line yé and the circle shall be such that two squares on it shall be equal to three squares on the radius Bx;* join «2, and produce it to meet the
43 Jbid., p. 113, $88. I have omitted it, as not being the work of Eudemus.
44 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 B¢e, and this is nowhere given in the text. The determination of this line, how- ever, can be inferred from the state- ment in p. 114, Geom. vor Ewkl., that ‘it is assumed that the line e¢ inclines towards 8’; and the further statement, in p. 117, that ‘it is assumed that the square on ε 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 B¢e is to be drawn. I have, as usual, omitted the proofs of Simplicius.
Bretschneider, p. 114, notices the archaic manner in which lines and points are denoted in this investiga-
tion—7 [εὐθεῖα] ἐφ᾽ ἣ AB, τὸ [σημεῖον] ἐφ᾽ οὗ K—and infers from it that Eu- demus 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. 45 The length of the line e¢ can be determined by means of the theorem of Pythagoras (Euclid, i., 47), coupled with the theorem of Thales (Euclid, iii., 31). Then, produce the line e¢ thus determined, so that the rectangle under the whole line thus produced and ‘the part produced shall be equal to the square on the radius; or, in archaic language, apply to the line e¢ a rectangle which shall be equal to the square on the radius, and which shall be excessive by a square—a Pytha-
DR. ALLMAN ON GREEK GEOMETRY
200
straight line drawn through « parallel to Bx, and let them meet at n; join κε, Bn (these lines will be equal) ; describe then a circle round the trapezium βκεη; also, circumscribe a circle about the triangle «Zn. Let the centres of these circles be X and μ respectively.
‘Now, the segments of the latter circle on εζ and 2» are similar to each other, and to each of the segments of the former circle on the equal straight lines ex, «8, βη;" and, since twice the square on εζ is equal to three times the square on κβ, the sum of the two segments on εζ and ζη is equal to the sum of the three segments on εκ, κβ, Bn; to each of these equals add the figure bounded by the straight lines ex, «3, Bn, and the arc ηζε, and we shall have the lune whose exterior arc is «x(3n equal to the rectilineal figure
composed of the three triangles Bn, ζβκ, Ze.”
gorean problem, as Eudemus tells us. (See HERMATHENA, vol. ili., pp. 181, 196, 197.) If the calculation be made by this method, or by the solution of a quadratic equation, we find
_ Bk 3 II ἘΠῚ
Bretschneider makes some slip, and gives
Geom. vor Eukl., p. 115, note.
46 Draw lines from the points e, κ, β, and ἡ to A, the centre of the circle described about the trapezium; and from ε and 7 to yp, the centre of the circle circumscribed about the triangle εζη; it will be easy to see, then, that the angles subtended by ex, κβ, and ηβ at A are equal to each other, and to each of the angles subtended by e¢ and (m at μ. 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 further, 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 (en¢ and 7x8, for example). In HERMATHENA, vol. iil., 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.
47 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 THALES TO EUCLID.
201
‘That the exterior arc of this lune is smaller than a semicircle, Hippocrates proves, by showing that the angle exn lying within the exterior arc of the segment is obtuse, which he does thus: Since the square on εζ is once and a-half the square on the radius Bx or xe, and since, on account of the similarity of the triangles Bxe and Bl«, the square on κε is greater than twice the square on xZ,** it follows that the square on εζ is greater than the squares on ex and κζ together. The angle exn 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 x, and let the square on the diameter of the exterior be six times that of the interior. Inscribe a hexagon αβγδεζ in the inner circle, and draw the radii xa, «3, cy, and produce
Bretschneider points out that in this paragraph the Greek text in the Aldine
pentagon ; and, 2°, that it is described as a rectilineal figure composed of
three triangles.
48 Τί is assumed here that the angle Bre is obtuse, which it evi- dently is.
is corrupt, and consequently obscure : he corrects it by means of some trans- positions and a few trifling additions. (See Geom, vor Eukl., p. 118, note 2.)
202 DR. 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 Apollonius 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 :—
Ν a an / Ν Ν \ To δὲ τρίτον τῶν κωνικῶν περιέχει, φησὶ, πολλὰ Kal παράδοξα θεωρή- ἴω ἴων / > ματα χρήσιμα πρὸς τὰς συνθέσεις τῶν στερεῶν τόπων. ᾿Επιπέδους ΄ 5" an a , “ an ΄ 9 τόπους ἔθος τοῖς παλαιοῖς γεωμέτραις λεγειν, ὅτε τῶν προβλημάτων οὐκ ΔΙ ΘΝ ΘΝ ΄ ΄ 3 a7 oN 4 ΄ Ν ΄ὔ @ ς ἀφ ἑνὸς σημείου μόνον, GAN ἀπὸ πλειόνων γίνεται τὸ ποίημα" οἷον ἐν > / a > , Υ̓͂ ΄ ς = a > 3 aot ἐπιτάξει, τῆς εὐθείας δοθείσης πεπερασμένης εὑρεῖν TL σημεῖον ἀφ᾽ οὗ ἡ > a , DLN EEN A ΄ S07, ΄ a , ἀχθεῖσα κάθετος ἐπὶ τὴν δοθεῖσαν μέση ἀνάλογον γίνεται TOV τμημάτων. Τόπον καλοῦσι τὸ τοιοῦτον, οὐ μόνον γὰρ ἕν σημεῖόν ἐστι τὸ ποιοῦν τὸ / 3 Ἂς ’ σ΄ Δ 3 ε / “Ὁ ‘\ 4 Ν πρόβλημα, ἀλλὰ τόπος ὅλος ὃν ἔχει ἡ περιφέρεια τοῦ περὶ διάμετρον τὴν A 3 “A , oN Ν > \ ip ἊΝ Up > “4 ἐς , δοθεῖσαν εὐθεῖαν κύκλου" ἐὰν yap ἐπὶ τῆς δοθείσης εὐθείας ἡμικύκλιον lal fal a Ν γραφῇ, ὅπερ ἂν ἐπὶ τῆς περιφερείας λάβῃς σημεῖον, καὶ ἀπ᾽ αὐτοῦ , > ΄ ΚΝ τς Ν , ΄ Ν ΄, 7 κάθετον ayayns ἐπὶ τὴν διάμετρον, ποιήσει TO προβληθέν. .. . ὅμοιον Ἂς ΄ Β΄ ΕῸΝ, 3 ΄, > ALS / / Shaw im le 4 kat γράφει αὑτὸς Ἀπολλώνιος ἐν τῷ ἀναλυομένῳ τόπῳ, ἐπὶ τοῦ ὑποκει- μένου. Δύο δοθέντων σημείων ἐν ἐπιπέδῳ καὶ λόγου δοθέντος ἀνίσων εὐθειῶν ἡ 4 > > “9 / 4 / σ Ν 5 A nr 7 . a ὶ δυνατόν ἐστιν ἐν τῷ ἐπιπέδῳ γράψαι κύκλον ὥστε τὰς ἀπὸ τῶν δοθέντων * ΄, aL Xx ἈΝ Ψ a iid , 3 ,, ΄ ΝΜ σημείων ἐπὶ τὴν περιφέρειαν τοῦ κύκλου κλωμένας εὐθείας λόγον ἔχειν
τὸν αὐτὸν τῷ δοθέντι. It is to be observed, in the first place, that a contrast is
39 Heiberg, in his Litterargeschicht- ὑποκείμενον, a statement which is not liche Studien uber Euklid, p.70,reads correct. I have interpreted Halley’s τὸ ὑποκείμενον, and adds ina note that reading as referring to the subjoined Halley has ὑποκειμένῳ, in place of τὸ diagram.
_ FROM THALES TO EUCLID. 203
here made between Apollonius and the old geometers (οἱ παλαιοὶ γεωμέτραι), 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 /ocz, as, ¢. 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 forzsm, 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 /ocz given above is strikingly suggestive of the method of Analytic Geo- metry. As to the term τόπος, it may be noticed that Aris- taeus, who was later than Menaechmus, but prior to Euclid, wrote five books on Solzd Loct (οἱ στερεοὶ τόποι). In conclu- sion, I cannot agree with Cantor’s view that the passage has the appearance of being modernized in expression:
40 πορίζεσθαι, to procure. The ques- Amongst the ancients the word porism
tion is—in a ¢heorem, to prove some- thing ; in a problem, to construct some- thing ; ina porism, to find something. So the conclusion of the theorem is, ὅπερ ἔδει δεῖξαι, Ὁ. E. D., of the pro- blem, ὅπερ ἔδει ποιῆσαι, Q. E.F., and
of the porism, ὅπερ ἔδει εὑρεῖν, Q. E. I.
had also another signification, that of corollary. See Heib., 2. Stud. tiber Eukl., pp. 56-79, where the obscure subject of porisms is treated with re- markable clearness.
41 Pappi, Collect., ed. Hultsch, vol. ii. p. 672.
224 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 ;
(4). 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 ;
(ce). About the trapezium so constructed describe a circle ;
(f). 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;
(2). 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 mention 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 Iamblichus amongst the disciples of Py-
52 Xenophon, AZemorad. i., 6, ὁ τ; 53 Themist., f. 16; Brandis, Schol. Diog. Laert. ii., 46, ed. Cobet, p. 44. in Artst., p. 327, b, 35.
Eee EEE eee
FROM THALES 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. 185), we saw that there was a tradition that the problem of the quadrature of the circle engaged the attention of the
54 Tambl., Vit. Pyth., c. 23. 57 Simplicius, in Bretsch. Geom. vor 55 Alex. Aphrod., f. 30; Brandis, ΖΚ. p. 106.
Schol., p. 306, b. 58 Jbid. 56 Themist., f. 5; Brandis, Schol., 59 Hankel, Geschich. der Math., p.
p- 211; Johan. Philop., f.118; Brandis, 116, and note. Schol., p. 211. 9 Birds, 1005.
206 DR. ALLMAN ON GREEK GEOMETRY
Pythagoreans. We saw, too (zdzd. 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 4éschylus brought out at Athens, and that he left notes on the subject. Vitruvius goes on to say that Democritus and Anaxagoras, profiting ee 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 //zstor7ic Commentarzes, 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.”
51 ᾿Αλλ᾽ ᾿Αναξαγόρας μὲν ἐν τῷ δεσ- 65 Cicero, De finibus bonorum et μωτηρίῳ τὸν τοῦ κύκλου τετραγωνισμὸν malorum, i., 6; Diog. Laert., ix., 7, éypape.—Plut., De Exil., c. 17, vol. ed. Cobet, p. 236.
ill., p. 734, ed. Didot. 64 Diog. Laert., zbid., ed. Cobet,
52: De Arch., vil., Praek. p37
a
FROM THALES TO EVCLID. 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 (kat γραμμέων συνθέσιος μετὰ ἀποδέξιος οὐδείς Kw μὲ παρήλλαξε, οὐδ᾽ οἱ Αἰγυπτίων καλεόμενοι ᾿Αρπεδονάπται"), 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 :—
Περὶ διαφορῆς γνώμονος ἢ περὶ ψαύσιος κύκλου καὶ σφαίρης (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
8 Diog. Laert., ix., 7, ed. Cobet, 1.,}. 304, ed. Sylburg; Mullach, Fragm. Ῥ- 225- Phil. Graec., p. 370.
86 7614., Ὁ. 236. femDiog: Laert:, ix, ἡ, eds ‘Cobet,
87 Democrit., ap. Clem. Alex. Strom., Ρ. 236.
208 DR. ALLWAN ON GREEK GEOMETRY
through the centre, the other will determine the tangent ἢ: one on geometry; one on numbers; one on incommen- surable lines and solids, in two books; ᾿Ακτινογραφίη (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, serzatzm, 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
S9eDioo? laert.snix., 7 88. Gobet, 70 Plut., de Comm. Not., p. 1321, ed. pp. 238 and 239. Didot.
FROM 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 licét 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. Nec ulla constructione tale augmentum exhiberi potest. Sci- licet eas tantum homogeneas quantitates comparabiles esse, cum Luclide, lib. v., defin. 5, censeo, quarum una numero, sed finito, multiplicata, alteram superare potest. Et quae tali quantitate non differunt, aequalia esse statuo,
VOL. UV. P
210 DR. ALTMAN ON GREEK GEOMETRY
quod etiam Archimedes sumsit, aliique post ipsum omnes. Et hoc ipsum est, quod dicitur differentiam esse data quavis minorem. Et <Avrchzmedeo 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 gzomon 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 enomon 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 (ἄλογον), 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
71 Leibnitii Ofera Omnia, ed. 1,. Dutens, tom. 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 (ἢ and 2, p. 187) that Hippocrates reduced this question to one ot 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
JE
212 DR. ALIMAN 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 [oAveidoce:” 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’: μικρόν γ᾽ ἔλεξας Baor- λεικοῦ σηκὸν τάφου, διπλάσιος ἔστω. τοῦ δὲ τοῦ κύβου μὴ σφαλεὶς. 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 difficulty. 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
72 See Reimer, Historia problematis kenaer shows that these words of Era- de cubi duplicatione, p. 20, Gottingae, tosthenes contain two verses, which he 1798; and Biering, Historia proble- thus restores :— matis cubt duplicandi, p- 6, Hauniae, Μικρὸν γ᾽ ἔλεξας βασιλικοῦ σηκὸν τάφου" 1844. Διπλάσιος ἔστω, τοῦ κύβου δὲ μὴ σφαλῆς.
73 Archim., ed. Torelli, p. 144. Valc- | See Reimer, Z. c.
FROM 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
74 Symp., Viil., Quaestio 2, c. 4; Plut. Ogera, ed. Didot, vol. iv., p. 877.
7 Πυθαγόρας, πέντε σχημάτων ὄντων στερεῶν, ἅπερ καλεῖται καὶ μαθηματικὰ, 3 δι a / / ἐκ μὲν τοῦ κύβου φησὶ γεγονέναι τὴν γῆν, ἐκ δὲ τῆς πυραμίδος τὸ πῦρ, ἐκ δὲ
a 3 / b Ty! > \ ~ 3 τοῦ ὀκταέδρου τὸν ἀέρα, ἐκ δὲ TOU εἰκο-
σαέδρου τὸ ὕδωρ, ἐκ δὲ τοῦ δωδεκαέδρου
τὴν τοῦ παντὸς σφαῖραν.
Πλάτων δὲ καὶ ἐν τούτοις πυθαγορίζει. Plut. Plac., ii., 6, 5 & 6; Opera, ed. Didot, vol. iv., p. 1081.
76 Stob. Eclog. ab Heeren, lib. i., p.-10. See also Zeller, Die Philos. der Griechen, Erster Theil, p. 376, Leip- 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 icosahedron which shall be equal to a given cube;
Construct an octahedron which shall be equal to a given icosahedron.
Now Plutarch, in his Sywzp., viii., Quaestzo ii. Noe Πλάτων ἔλεγε τὸν θεὸν ἀεὶ γεωμετρεῖν, 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. 111. 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 | Θεός, the arranger ],a name most justly deserved ; the second we call matter, and the third zdeal 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
7 Plut, Opera, ed, Didot, vol. iv. pp. 876, 7.
FROM THALES TO EUCLID. 215
like the zdeal form and as big as the matter. So having given himself this problem, when the /wo were there, he made, and makes, and for ever maintains, a Zhzrd, 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—wluimus 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.
216 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 ἀπαγωγή, 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 αβγ 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 ay on the other leg, and complete the rectangle αγβδ. Suppose now that the problem is solved, and that a line is drawn making
78 Pappi Alex. Codlect., ed. Hultsch, vol. i. p. 274.
FROM THALES TO EUCLID. 217
with By an angle which is the third part of the given angle aj3y ; let this line cut ay in Z, and be produced until it meet da produced at the point ε. Let now the straight line Ze be bisected in yn, and an be joined; then the lines én, ne, an, and (3a are all evidently equal to each other, and, therefore, the line Ze is double of the line a3, which is known.
The problem of the trisection of an angle is thus reduced to another :—
From any vertex β of a rectangle βδαγ draw a line (Ze, so that the part Ze 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
79 Jbrd., vol. i. p. 270, et seq. one or more conic sections were called
80 The ancients distinguished three kinds of problems—flane, solid, and linear. ‘Those which could be solved by means of straight lines and circles were called plane; and were justly so called, as the lines by which the prob- lems of this kind could be solved have their origin 271 Zlano. Those problems whose solution is obtained by means of
solid, inasmuch as for their construc- tion we must use the superficies of solid figures—to wit, the sections of a cone. A third kind, called linear, remains, which required for their solution curves of a higher order, such as spirals, quadratrices, conchoids, and cissoids, See Pappi Collect., ed. Haultsch, vol. i. pp. 54 and 270.
218 DR. 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 FE, the arc AE will be three times the arc BF.’ 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
St Archim. ex recens. Torelli, p. ‘Itaque puto, haec lemmata e plurium 358. mathematicorum operibus esse ex-
8 See zbid., Praefatio J. Torelli, cerpta, neque definiri jam _ potest, pp. xviii. and xix. See also Heiberg, quantum ex iis Archimedi tribuendum Quaest. Archim., p. 24, who says: - sit.
FROM THALES TO EVCLID. 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 Mathema- given by Montucla, but he did not tica, studio F. ἃ Schooten, p. 245, give any references. See ist. des Lugd. Bat. 1646. These two reduc- Math., tom. i. p. 194, Here ed. tions of the trisection of an angle were 8: Procl. Comm., ed. Fried., p. 272.
DR. ALLMAN ON GREEK GEOMETRY
220
choids; Hippias those of the quadratrix, and Perseus those of the spirals’ (σπειρικῶν).δ5
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 l’antiquité nous fournisse aucun autre géométre de ce nom, que celui dont je parle.’ δ 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.
85 Procl. Comm., ed. Fried., p. 356.
86 Montucl., Hist. des Math., tom. i. p- 181, nouvle ed.
87 Chasles, Histoire de la Géom., p-8; Ameth, Gesch. der Math., p. 95; Bretsch., Geom. vor Eukl., p. 94; Suter, Gesch. der Math. Wissenschaft., ps2:
88 Hankel, Gesch. der Math., p. 151, note. Hankel, also, in a review of Suter, Geschichte der Mathematischen Wissen- schaften, published in the Budlettino
di Bibliografia e di Storia delle Scienze Matematiche e Fisiche, says: ‘A pag. 31 (lin. 3-6), Hippias, l’inventore della quadratrice, ἃ identificato col Sofista Hippias, il che veramente avea gia fatto il Bretschneider (pag. 94, lin. 39- 42), ma senza darne la minima prova.’ Bullet., &c., tom. v. p. 297.
GE) lerdoyel ἘΠῚ ἘΠΕ. ip: 65.
99 Diog. Laert., viii. Ὁ: 4, ed. Cobet, Pp. 224.
Comm.,
FROM THALES 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 insufficient.
91 Pappi, Collect., ed. Hultsch, vol. i. pp- 250 and 252.
9 Hippias, seu Balneum. Since the above was written I find that Cantor, Vorles. ber Gesch. der Math., p. 165, εὖ seg., 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 :—
1. 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 Elis 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—risecting 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 DR. 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,’ sce Ὁ. 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, 2.6. 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. 121-2.
FROM THALES TO EUCLID. 223
the parabola, runs 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. 1, involves, in the case of incommensurable quantities, the same difficulty 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
35: Archim., ex recens. Torelli, p. 18. see Eucl. Zlem., Graece ed. ab.
96 Jbid., p. 64. August, pars ii., p. 329; also Unter-
96 Weare toldsointhe anonymous suchungen, &c., Von Dr. J. H. scholium on the Elements of Euclid, Knoche, p. 10. Cf. HERMATHENA, which Knoche attributes to Proclus: vol. iii. 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. 186, 203, note 101.
In the investigation of Hippocrates given above we meet with manifest traces of an analytical method, as stated in HERMATHENA, Vol. iii. p. 197, note g1. Indeed, Aristotle— and this is remarkable—after having defined ἀπαγωγή, evi-
FROM THALES TO EUCLID. 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 ὃ were quadrature, and « a rectilineal figure, and Z a circle; if there were only one middle term between ε and ζ, 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 from the passages quoted above from Proclus and Democritus, pp. 178, 207; as also from the fact that the Greeks had a special name, Ψψευδογράφημα, 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, Iamblichus, and Eutocius.
How is this to be explained? The faulty reasoning
7 ἢ πάλιν [ἀπαγωγή ἐστι] εἰ ὀλίγα Ta ed. Bek. Observe the expressions τὸ
μέσα τῶν By’ καὶ yap οὕτως ἐγγύτερον ᾿ ἐφ᾽ ᾧ ε εὐθύγραμμον, &c., here, and τοῦ εἰδέναι. οἷον εἰ τὸ ὃ εἴη τετραγωνί- see p. 199, note 44. nm ζεσθαι, τὸ δ᾽ ἐφ᾽ ᾧ ε εὐθύγραμμον, Td δ᾽ 88 Concerning the importance of
ἐφ᾽ @ ¢ κύκλος" εἰ TOU ες ἕν μόνον etn geometrical constructions‘as a process μέσον, τὸ μετὰ μηνίσκων ἴσον γίνεσθαι οἵ deduction, see P. Laffitte, Les εὐθυγράμμῳ τὸν κύκλον, ἐγγὺς ἂν εἴη Grands Types de l’ Humanité, vol. ii. τοῦ εἰδέναι. Anal. Prior. ii. 25, p.69,a, p. 329.
VOL, LV. QO
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 sufficient explanation of it: and indeed it is difficult 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 (ἀκουστικοί) 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 (μαθηματικοί), 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—
1. The intimate connection that exists between the work of Hippocrates and that of the Pythagoreans ;
9 Montucla, Aistotre des recherches there.
sur la Quadrature du Cercle, p. 39, 101 See A. Ed. Chaignet, Pythagore nouv. ed., Paris, 1831. et la Philosophie Pythagoricienne, vol. 00 See HERMATHENA, vol. iii. p. i. p. 115, Paris, 1874; see also Iambl.,
179, note, and the references given de Vit. Pyth., c. 16, 5. 68.
FROM THALES TO EUCLID.
237
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. Ὁ. 181 (¢), and note 61) ; 1°
5. The passage in Iamblichus, see p. 186(/); and, gene- rally, the unfavourable opinion entertained by the ancients
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 of Hippocrates, Bretschneider (Geom. vor Hukl., p. 122) says, ‘It is difh- cult to assume so gross a mistake on the part of such a good geometer,’ and he ascribes the supposed error to a complete misunderstanding. He then gives an explanation similar to that given above, with this difference, that he supposes Hippocrates to have stated the matter correctly, and that Aristotle took it up erroneously ; it seems to me more probable that Hippocrates took } up wrongly what he had heard at lecture than that Aristotle did so on reading the work of Hippocrates. Further, we see from the quotation in p. 225, from Amal. Prior., that Aristotle fully understood the condi-
tions of the question.
103 Referring to the application of areas, Mr, Charles Taylor, An /ntro- duction to the Ancient and Modern Geometry of Conics, Prolegomena, p. xxv., says, ‘ Although it has not been made out wherein consisted the im- portance of the discovery in the hands of the Pythagoreans, we shall see that it played a great part in the system of Apollonius, and that he was led to designate the three conic sections by the Pythagorean terms Parabola, Hy- perbola, Ellipse.’
I may notice that we have an instance of these problems in the construction referred to above: for other applica- tions of the method see HERMATHENA, 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- 196), 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.
/ ae Cheng lak em oes “Lic FES le Be yar
[From “ Hermatuena,” Wo. X., Vol. γι]
186
DR. ALLMAN ON GREEK GEOMETRY
GREEK GEOMETRY FROM _ THALES, ΤῸ RUCEIDS
TW.
URING 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 works of Bretschneider and Hankel, and have derived much advantage from the great work of Cantor— Vorlesungen uber Geschichte der Mathematik. I have also constantly used the “dex Graecitatis appended by Hultsch to vol. 111. 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.
1 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
2 «The political creed and peculiar form of government 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 (συνέδρια) 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 with 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 well-known integrity, for the removal of existing evils. 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