Pythagorean Geometry
PYTHAGOREAN GEOMETRY As the introduction of geometry into Greece is by common consent attributed to Thales, so all are agreed that to Pythagoras is due the honour of having raised mathematics to the rank of a science. We know that the early Pythagoreans published nothing, and that, moreover, they referred all their discoveries back to their master (see PHILOLAUS). Hence it is not possible to separate his work from that of his early disciples, and we must therefore treat the geometry of the early Pythagorean school as a whole. We know that Pythagoras made numbers the basis of his philosophical system, as well physical as metaphysical, and that he united the study of geometry with that of arithmetic.
The following statements have been handed down to us. (a) Aristotle (Mela. i. 5, 985) says " the Pythagoreans first applied themselves to mathematics, a science which they improved; and, penetrated with it, they fancied that the principles of mathematics were the principles of all things." (b) Eudemus informs us that " Pythagoras changed geometry into the form of a liberal science, regarding its principles in a purely abstract manner, and investigated its theorems from the immaterial and intellectual point of view (duX&rc nal voepas)." 1 (c) Diogenes Laertius (viii. 1 1) relates that " it was Pythagoras who carried geometry to perfection, after Moeris 2 had first found out the principles of the elements of that science, as Anticlides tells us in the second book of his History of Alexander; and the part of the science to which Pythagoras applied himself above all others was arithmetic." (a) According to Aristpxenus, the musician, Pythagoras seems to have esteemed arithmetic above everything, and to have advanced it by diverting it from the service of commerce and by likening all things to numbers.* (e) Diogenes Laertius (viii. '13) reports on the same authority that Pythagoras was the first person who introduced measures and weights among the Greeks. (/) He discovered the numerical relations of the musical scale (Diog.
1 Proclus Diadochus, In primum Euclidis elemenlorum librum commentarii, ed. Friedlein, p. 65.
4 Moeris was a king of Egypt who, Herodotus tells us, lived 900 years before his visit to that country.
3 Aristox. Fragm. ap. Stob. Eclog. Phys. i. 2, 6.
Laert. viii. n). (g) Proclus 4 says that " the word 'mathematics' originated with the Pythagoreans." (h) We learn also from the same authority 6 that the Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the " how many " (TO vitaov) and the other to the " how much " (r6 Tn\\ixov); and they assigned to each of these parts a twofold division. They said that discrete quantity or the how many " is either absolute or relative, and that continued quantity or the " how much " is, either stable or in motion. Hence they laid down that arithmetic contemplates that discrete quantity which subsists by itself, but music that which is related to another; and that geometry considers continued quantity so far as it is immovable, but that astronomy (17 a<j><upuiri) contemplates continued quantity so far as it is of a self-motive nature, (i) Diogenes Laertius (viii. 25) states, on the authority of Favorinus, that Pythagoras " employed definitions in the mathematical subjects to which he applied himself."
The following notices of the geometrical work of Pythagoras and the early Pythagoreans are also preserved, (i) The Pythagoreans define a point as " unity having position " (Procl. op. cit. p. 95). (2) They considered a point as analogous to the monad, a line to the dyad, a superficies to the triad, and a body to the tetrad (ibid. P- 97)- (3) They showed that the plane around a point is completely filled by six equilateral triangles, four squares, or three regular hexagons (ibid. p. 305). (4) Eudemus ascribes to them the discovery of the theorem that the interior angles of a triangle are equal to two right angles, and gives their proof, which was substantially the same as that in Euclid I. 32 6 (ibid. p. 379). (5) Proclus informs us in his commentary on Euclid I. 44 that Eudemus says that the problems concerning the application of areas where the term "application" is not to be taken in its restricted sense (jrapo/JoXij) , in which it is used in this proposition, but also in its wider signification, embracing vwtpffoXii and 8XX^is, in which it is used in Book VI. Props. 28, 29 are old, and inventions of the Pythagoreans 7 (ibid. p. 419). (6) This is confirmed by Plutarch, 8 who says, after Apollodorus, that Pythagoras sacrificed an ox on finding the geometrical diagram, either the one relating to the hypotenuse, viz. that the square on it is equal to the sum of the squares on the sides, or that relating to the problem concerning the application of an area. 9 (7) Plutarch 10 also ascribes to Pythagoras the solution of the problem, To construct a figure equal to one and similar to another given figure. (8) Eudemus states that Pythagoras discovered the construction of the regular solids (Procl. op. cit. P- 65). (9) Hippasus, the Pythagorean, is said to have 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 inscribed ordinate dodecahedron): 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." 11 (10) The triple interwoven triangle or pentagram star-shaped regular pentagon was used as a symbol or sign of recognition by the Pythagoreans and was called by them " health " (frywio). 12 (n) The discovery of the law of the three 4 Procl. op. cit. p. 45.
5 Op. cit. p. 35.
' We learn from a fragment of Geminus, which has been handed down by Eutocius in his commentary on the Conies of Apollonius (Apoll. Conica, ed. Halleius, p. 9), that the ancient geometers observed two right angles in each species of triangle, in the equilateral first, then in the isosceles, and lastly in the scalene, whereas later writers proved the theorem generally thus " The three internal angles of every triangle are equal to two right angles."
7 The words of Proclus are interesting. " According to Eudemus the inventions respecting the application, excess and defect of areas are ancient, and are due to the Pythagoreans. Moderns, borrowing these names, transferred them to the so-called conic lines, the parabola, the hyperbola, the ellipse, as the older school, in their nomenclature concerning the description of areas in piano on a finite right line, regarded the terms thus: An area is said to be applied ( jrapa/}a\\cii>) 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 excess (i>xep/3<iXXe') ; 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 (XXjrc). Euclid uses in this way in his sixth book the terms excess and defect. . . . The term application (iropa/SdXXtu'), which we owe to the Pythagoreans, has this signification."
8 Non posse suaviter vivi sec. Epicurum, c. xi.
Eire irp6fl\ripa. irtpl TOV x^pk js irapa/SoX^s. Some authors, rendering the last five words " concerning the area of the parabola," have ascribed to Pythagoras the quadrature of the parabola, which was one of the great discoveries of Archimedes.
10 Symp. viii., Quaest. 2, c. 4.
11 lamblichus, De vit. Pyth. c. 1 8, 88.
12 Lucian, Pro lapsu in salut. 5 ; also schol. on Aristoph. Nub. 6ll. That the Pythagoreans used such symbols we learn from lamblichus (De vit. Pyth. c. 33, 237 and 238). This figure is referred to Pythagoras himself, and in the middle ages was called Pythagorae figura; even so late as Paracelsus it was regarded by squares (Euclid I. 47), commonly called the " theorem of Pythagoras," is attributed to him by many authorities, of whom the oldest is Vitruvius. 1 (12) One of the methods of finding right-angled triangles whose sides can be expressed in numbers (Pythagorean triangles) that setting out from the odd numbers is referred to Pythagoras by Heron of Alexandria and Proclus. 1 (13) The discovery of irrational quantities is ascribed to Pythagoras by Eudemus (Procl. op. cit. p. 65). (14) The three proportions arithmetical, geometrical and harmonical were known to Pythagoras.* (15) lamblichus says, " Formerly, in the time of Pythagoras and the mathematicians under him, there were three means only the arithmetical, the geometrical and the third in order, which was known by the name sub-contrary (vrtvafria), but which Archytas and Hippasus designated the harmonical, since it appeared to include the ratios concerning harmony and melody." (16) The so-called most perfect or musical proportion, e.g. 6 : 8 : : 9 : 12, which comprehends in it all the former ratios, according to lamblichus,' is said to be an invention of the Babylonians and to have been first brought into Greece by Pythagoras. (17) Arithmetical progressions were treated by the Pythagoreans, and it appears from a passage in Lucian that Pythagoras himself had considered the special case of triangular numbers: Pythagoras asks some one, "How do you count?" He replies, "One, two, three, four." Pythagoras, interrupting, says, " Do you see? what you take to be four, that is ten and a perfect triangle and our oath."' (18) The odd numbers were called by the Pythagoreans "gnomons," 7 and were regarded as generating, inasmuch as by the addition of successive gnomons Consisting each of an odd number of unit squares to the original square unit or monad the square form was preserved. (19) In like manner, if the simplest oblong (iTpo^i7j), consisting of two unit squares or monads in juxtaposition, be taken and four unit squares be placed about it after the manner of a gnomon, and then in like manner six, eight . . . unit squares be placed in succession, the oblong form will be preserved. (20) Another of his doctrines was, that of all solid figures the Sphere was the most beautiful, and of all plane figures the circle. 8 (21) According to lamblichus the Pythagoreans are said to have found the quadrature of the circle.' him as a symbol of health. It is said to have obtained its special name from the letters v, y i, 8 ( = ), o having been written at its prominent vertices.
1 De arch. ix. ; Praef. 5, 6, 7. Amongst other authorities are Diogenes Laertius (viii. n), Proclus (op. cit., p. 426), and Plutarch (ut supra, 6). Plutarch, however, attributes to the Egyptians the knowledge of this theorem in the particular case where the sides are 3, 4, and 5 (De Is. et Osir. c. 56).
* Heron Alex. Geom. et stereom. rel., ed. F. Hultsch, pp. 56, 146; Procl. op. cit. p. 428. The method of Pythagoras is as follows: he took an odd number as the lesser side; then, having squared this number and diminished the square by unity, he took half the remainder as the greater side, and by adding unity to this number he obtained the hypotenuse, e.g. 3, 4, 5; 5, 12, 13.
1 Nicom. Ger. Intrpd. Ar. c. xxii.
* In Nicomachi arithmeticam, ed. S. Tennulius, p. 141.
1 Op. cit. p. 168. As an example of this proportion Nicomachus and, after him, lamblichus give the numbers 6, 8, 9, 12, the harmonical and arithmetical means between two numbers forming a geometric proportion with the numbers themselves (a:-^pr :^:b\ .
lamblichus further relates (loc. cit.) that many Pythagoreans made use of this proportion, as Aristaeus of Crotona, Timaeus of Locri, Philolaus and Archytas of Tarentum and many others, and after them Plato in his Timaeus (see Nicom. Inst. arithm., ed. Ast, p. 153, and Animadversiones, pp. 327-329; and Iambi, op. cit. p. 172 seq.).
1 Biwv n-pSaa, 4, i. 317, ed. C. Jacobitz.
' ryciyiuy means that by which anything is known or " criterion "; its oldest concrete signification seems to be the carpenter's square (norma) by which a right angle is known. Hence it came to denote a perpendicular, of which, indeed, it was the archaic name (Proclus, op. cit. p. 283). Gnomon is also an instrument for measuring 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 ofa square or rectangle, together with the two complements, on account of the resemblance of the figure to a carpenter's square; and then, more generally, the similar figure with regard to any parallelogram, as defined by Euclid II. def. 2. Again, in a still more general signification, it means the figure which, being added to any figure, preserves the original form. See Heron, Definitiones (59). When gnomons are added successively in this manner to a square monad, the first gnomon may be regarded as that consisting of three square monads, and is indeed the constituent of a simple Greek fret; the second of five square monads, etc.; hence we have the gnomonic numbers.
1 Diag. Laert. De vit. Pyth. viii. 19.
* Simplicius, In Aristotelis physicorum libros quattuor priores commentaria, ed. H. Diels, p. 60.
On examining the purely geometrical work of Pythagoras and his early disciples, as given in the preceding extracts, 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 (3) concerning the filling up a plane with regular figures for floors or walls covered with tiles of various colours were common in Egypt; in the construction of the regular solids (8), for some of them are found in Egyptian architecture; in the problems concerning the application of areas (5) ; and lastly, in the theorem of Pythagoras (i i), coupled with his rule for the construction of rightangled triangles in numbers (12). We learn from Plutarch that the Egyptians were acquainted with the geometrical fact that a triangle whose sides contain three, four and five parts is rightangled, and that the square of the greatest side is equal to the squares of the sides containing the right angle. It is probable too that this theorem was known to them in the simple case where the right-angled triangle is isosceles, inasmuch as it would be at once suggested by the contemplation of a floor covered with square tiles the square on the diagonal and the sum of the squares on the sides contain each four of the right-angled triangles into which one of the squares is divided by its diagonal. It is easy now to see how the problem to construct a square which shall be equal to the sum of two squares could, in some cases, be solved numerically. From the observation of a chequered board it would be perceived that the element in the successive formation of squares is the gnomon or carpenter's square. Each gnomon consists of an odd number of squares, and the successive gnomons correspond to the successive odd numbers, and include, therefore, all odd squares. Suppose, now, two squares are given, one consisting of sixteen and the other of nine unit squares, and that it is proposed to fomi from them another square. It is evident that the square consisting of nine unit squares can take the form of the fourth gnomon, which, being placed round the former square, will generate a new square containing twenty-five unit squares. Similarly it may have been observed that the twelfth gnomon, consisting of twenty-five unit squares, could be transformed into a square each of whose sides contains five 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 twelve units as base, would produce the square of thirteen units, and so on. This method required only to be generalized in order to enable Pythagoras to arrive at his rule for finding right-angled triangles whose sides can be expressed in numbers, which, we are told, sets out from the odd numbers. The nth square together with the nth gnomon forms the (n + i)th square; if the nth gnomon contains nf unit squares, m being an odd number, we have 2n+l = m*,. .n = J(m 1 i), which gives the rule of Pythagoras.
The general proof of Euclid I. 47 is attributed to Pythagoras, but we have the express statement of Proclus (op. cit. p. 426) that this theorem was not proved in the first instance as it is in the Elements. The following simple and natural way of arriving at the theorem is suggested by Bretschneider after Camerer. 10 A square can be dissected into the sum of two squares and two equal rectangles, as in Euclid II. 4; these two rectangles can, by drawing their diagonals, be decomposed into four equal right-angled triangles, the sum of the sides of each being equal to the side of the square; again, these four right-angled triangles can be placed so that a vertex of each shall be in one of the corners of the square in such a way that a greater and less side are in continuation. The original square is thus dissected into the four triangles as before and the figure within, which is the square on the hypotenuse. This square, therefore, must be equal to the sum of the squares on the sides of the right-angled triangle.
It is well known that the Pythagoreans were much occupied 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 speculations in the theorem (3) that " the pMane around a point is completely filled by six equilateral triangles, four squares, or three regular hexagons." Plato also makes the Pythagorean Timaeus explain " Each straight-lined figure consists of triangles, but all triangles can be dissected into rectangular 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 " " (Timaeus, 53, 54- 55)- The construction of the regular solids is distinctly ascribed to Pythagoras himself by Eudemus (8). Of these five 10 See Bretsch. Die Geom. vor Euklides, p. 82; Camerer, Euclidis elem. i. 444, and the references given there.
11 The dodecahedron was assigned to the fifth element, quinta pars, aether, or, as some think, to the universe. (See PHILOLAUS.)
solids three the tetrahedron, the cube and the octahedron were known to the Egyptians and are to be found in their architecture. Let us now examine what is required for the construction of the other two solids the icosahedron and the dodecahedron. 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 coincident; and it was known that if six such triangles were placed round a common vertex with their adjacent sides coincident, they would lie in a plane, and that, therefore, no solid could be formed in that manner from them. It remained, then, to try whether five such equilateral triangles could be placed at a common vertex in like manner; on trial it would be found that they could be so placed, and that their bases would form a regular pentagon. The existence of a regular pentagon would thus become 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 regular* hexagons were placed round a point as common vertex with adjacent sides coincident, they would form a plane. It remained in this case, too, 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 that the construction of the regular pentagon is required for the formation of each of these two regular solids, and that, therefore, it must have been a discovery of Pythagoras. If we examine now what knowledge of geometry was required for the solution of this problem, we shall see that it depends on Euclid IV. 10, whith is reduced to Euclid II. n, which problem is reduced to the following: To produce a given straight line so that the rectangle under the whole line thus produced and the produced part shall be equal to the square on the given line, or, in the language of the ancients, To apply to a given straight line a rectangle which shall be equal to a given area in this case the square on the given line and which shall be excessive by a square. Now it is to be observed that the problem is solved in this manner by Euclid (VI. 30, 1st method), and that we know on the authority of Eudemus that the problems concerning the application of areas and their excess and defect are old, and inventions of the Pythagoreans (5). Hence the statements of lamblichus concerning Hippasus (9) that he divulged the Sphere with the twelve pentagons and of Lucian and the scholiast on Aristophanes (10) that the pentagram was used as a symbol of recognition amongst the Pythagoreans become of greater importance.
Further, the discovery of irrational magnitudes is ascribed to Pythagoras by Eudemus (13), and this discovery has been ever regarded as 1 , 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 the present writer, however, more probable that the discovery of incommensurable 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 without end. On the other hand, if a similar method be adopted in the case of any two lines which can be represented numerically, the process would end. Hence would arise the distinction between commensurable and incommensurable quantities. A reference to Euclid X. 2 will show that the method above is the one used to prove that two magnitudes are incommensurable; and in Euclid X. 3 it will be seen that the greatest common measure of two commensurable magnitudes is found by this process of continued subtraction. It seems probable that Pythagoras, to whom is attributed 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, failing 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 diagonal. The nature of the old proof which consisted of a reductio ad absurdum, showing that, if the diagonal be commensurable with the side, it would follow that the same number would be odd and even 1 makes it more probable, however, that this was accomplished by his successors. The existence of the irrational as well as that of the regular dodecahedron appears to have been regarded by the school as one of their chief discoveries, and to have been preserved as a secret; it is remarkable, too, that a story similar to that told by lamblichus of Hippasus is narrated of the person who first published the idea of the irrational, viz. that he suffered shipwreck, etc. 2 Eudemus ascribes the problems concerning the application of figures to the Pythagoreans. The simplest cases of the problems, 1 For this proof, see Euclid X. 117; see also Aristot. Analyt. Pr. i. c. 23 and c. 44.
* Knoche, Untersuchungen uber die neuaufgefundenen Scholien des Proklus Diadochus zu Euclids Elementen, pp. 20 and 23 fHerford, 1865).
Euclid VI. 28, 29 those, viz. in which the given parallelogram is a square correspond to the problem: To cut a given straight line internally or externally so that the rectangle under the segments shall be equal to a given rectilineal figure. The solution of this problem in which the solution of a quadratic equation is implicitly contained depends on the problem, Euclid II. 14, and the theorems, Euclid II. 5 and 6, together with the theorem of Pythagoras. It is probable that the finding of a mean proportional between two given lines, or the construction of a square which shall be equal to a given rectangle, is due to Pythagoras himself. The solution of the more general problem, Euclid VI. 25, is also attributed to him by Plutarch (7). The solution of this problem depends on that of the particular case and on the application of areas; it requires, moreover, a knowledge of the theorems: Similar rectilineal figures are to each other as the squares on their homologous sides (Euclid VI. 20) ; and, If three lines are in geometrical proportion, the first is to the third as the square on the first is to the square on the second. Now Hippocrates of Chios, about 440 B.C., who was instructed in geometry by the Pythagoreans, possessed this knowledge. We are justified, therefore, in ascribing the solution of the general problem, 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 and the similarity of figures. 3 That we owe the foundation and development of the doctrine of proportion to Pythagoras and his school is confirmed by the testimony of Nicomachus (14) and lamblichus (15 and 16). From these passages it appears that the early Pythagoreans were acquainted not only with the arithmetical and geometrical means between two magnitudes, but also with their harmonical mean, which was then called " subcontrary." The Pythagoreans were much occupied with the representation of numbers by geometrical figures. These speculations originated with Pythagoras, who was acquainted with the summation of the natural numbers, the odd numbers and the even numbers, all of which are capable of geometrical representation. See the passage in Lucian (17) and the rule for finding Pythagorean triangles (12) and the observations thereon supra. On the other hand, there is no evidence to support the statement of Montucla that Pythagoras laid the foundation of the doctrine of isoperimetry, by proving that of all figures having the same perimeter the circle is the greatest, and that of all solids having the same surface the Sphere is the greatest. 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 noticed the misconception which gave rise to this erroneous inference.
Certain conclusions may be drawn from the foregoing examination of the mathematical work of Pythagoras and his school, which enable us to form an estimate of the state of geometry about 480 B.C. First, as to matter. It forms the bulk of the first two books of Euclid, and includes a sketch of the doctrine of proportion which was probably limited to commensurable magnitudes 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 latter requiring the description of certain regular polygons the foundation, in fact, of the fourth book of Euclid. Secondly, 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 between discrete and continuous quantities, and regarded geometry as a branch of mathematics, of which they made the fourfold division that lasted to the middle ages the quadrivium (fourfold way to knowledge) of Boetius and the scholastic philosophy. And it may be observed that the name of " mathematics," as well as that of " philosophy," is ascribed to them. Thirdly, as to method. One chief characteristic of the mathematical work of Pythagoras was the * It is agreed on all hands that these two theories were treated at length by Pythagoras and his school. It is almost certain, however, that the theorems arrived at were proved for commensurable magnitudes only, and were assumed to hold good for all. The Pythagoreans themselves seem to have been aware that their proofs were not rigorous, and were open to serious objection ; in this we may have the explanation of the secrecy which was attached by them to the idea of the incommensurable and to the pentagram which involved, and indeed represented, that idea. Now it is remarkable that the doctrine of proportion is twice treated in the Elements of Euclid first, in a general manner, so as to include incommensurables, in book v., which tradition ascribes to Eudoxus, and then arithmetically in book vii., which, as Hankel has supposed, contains the treatment of the subject by the older Pythagoreans.
combination of arithmetic with geometry. The notions of an equation and a proportion which are common to both, and contain the first germ of algebra were introduced among the Greeks by Thales. These notions, especially the latter, were elaborated by Pythagoras and his school, so that they reached the rank of a true scientific method in their theory of proportion. 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.
See G. J. Allman, Greek Geometry from Thales to Euclid (Cambridge, 1889); M. Cantor, Vorlesungen liber Geschichte der Mathematik (Leipzig, 1894); James Gow, Short History of Greek Mathematics (Cambridge, 1884). (G. J. A.)
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)
