PYTHAGORAS AND PYTHAGOREANS. Pythagoras is one of those figures which have so impressed the imagina-tion of succeeding times that their historical lineaments are difficult to discern through the mythical haze that envelops them. Animated, as it would appear, not merely by the philosophic thirst for knowledge but also by the enthusiasm of an ethico-religious reformer, he became, centuries after his death, the ideal hero or saint of those who grafted a mystical religious asceticism on the doctrines of Plato. Writings were forged in his name. Lives of him were written which gather up in his person all the traits of the philosophic wise man, and surround him besides with the nimbus of the prophet and wonder-worker. He is described by his Neoplatonic biographers as the favourite and even the son of Apollo, from whom he received his doctrines by the mouth of the Delphic priestess. We read that he had a golden thigh, which he displayed to the assembled Greeks at Olympia, and that on another occasion he was seen in Crotona and Metapontum at one and the same time. He is said to have tamed wild beasts by a word and to have foretold the future, while many stories turn upon the knowledge he was reported to retain of his personality and deeds in former states of existence. Thus, as Zeller truly remarks, the information respecting Pythagoreanism and its founder grows fuller and fuller the farther removed in time it is from its subject. The authentic details of Pythagoras's career, on the other hand, are meagre enough and merely approximate in character. He was a native of Samos, and the first part of his life may therefore be said to belong to that Ionian seaboard which had already witnessed the first development of philosophic thought in Greece. The exact year of his birth has been variously placed between 586 and 569 B.C., but 582 may be taken as the most probable date. Some of the accounts make him the pupil of Anaximander ; but such an assertion lies so ready to hand in the circumstances of time and place that we cannot build with any assurance upon the suggested connexion with the Ionic school. It is probable, however, that Pythagoras was aware of their speculations, seeing that he left behind him in Ionia the reputation of a learned and universally informed man. "Of all men Pythagoras, the son of Mnesarchus, was the most assiduous inquirer," says Hera-clitus, and then proceeds in his contemptuous fashion to brand his predecessor's wisdom as only eclectically compiled information or polymathy (iroXv/jiadia). This accumulated wisdom, as well as most of the tenets of the Pythagorean school, was attributed in antiquity to the extensive travels of Pythagoras, which brought him in contact (so it was said) not only with the Egyptians, the Phoenicians, the Chaldasans, the Jews, and the Arabians, but also with the Druids of Gaul, the Persian Magi, and the Brahmans. But these tales are told of too many of the early philosophers to be received implicitly; they represent rather the tendency of a later age to connect the beginnings of Greek speculation with the hoary re-ligions and priesthoods of the East. There is no intrinsic improbability, however, in the statement that Pythagoras visited Egypt and other countries, of the Mediterranean, for travel was then one of the few ways of gathering knowledge. Some of the accounts represent Pythagoras as deriving much of his mathematical knowledge from an Egyptian source. Herodotus traces the doctrine of metempsychosis to Egypt, as well as the practice of burying the dead exclusively in linen garments, but he does not mention any visit of Pythagoras to that country. There is thus little more than conjecture to fill out the first half of the philosopher's life. The historically important part of his career begins with his emigration to Crotona, one of the Dorian colonies in the south of Italy. Nothing is known with certainty of the reasons that led to this step, which he appears to have taken about the year 529 ; perhaps the ethical temper which can be traced in the Pythagorean school attracted the founder towards the sterner Dorian character. At Crotona Pythagoras speedily became the centre of a widespread and influential organization, which seems to have resembled a religious brother-hood or an association for the moral reformation of society much more than a philosophic school. Pythagoras appears, indeed, in all the accounts more as a moral reformer than as a speculative thinker or scientific teacher; and it is noteworthy that the only one of the doctrines of the school which is definitely traceable to Pythagoras himself is the ethico-mystical doctrine of transmigration. The aim of the brotherhood was the moral education and purification of the community; and it seems to have been largely based upon a revival of the Dorian ideal of abstinence and hardi-hood along with certain other traits of a more definitely religious character, which were probably due to the influence of the mysteries. But many details of life and ritual, such as abstinence from animal food and from beans, celibacy, and even community of goods, have been fathered by the organized asceticism of a later period upon the original followers of Pythagoras. Ethics, according to the Greek and especially according to the Dorian conception, being inseparably bound up with the general health of the state, we are not surprised to find the Pythagoreans represented as a political league ; nor is it wonderful that their following was among the aristocracy, and that they formed the staunchest supporters of the old Dorian constitutions. It is unfair, however, to speak of the league as primarily a political organization, wide though its political ramifications must latterly have become. Its entanglement with politics was in the end fatal to its existence. The authorities differ hopelessly in chronology, but according to the balance of evidence the first reaction against the Pythagoreans took place in the lifetime of Pythagoras himself after the victory gained by Crotona over Sybaris in the year 510. Dissensions seem to have arisen about the allotment of the conquered territory, and an adverse party was formed in Crotona under the leadership of Cylon. This was probably the cause of Pythagoras's withdrawal to Metapontum, which an almost unanimous tradition assigns as the place of his death in the end of the 6th or the beginning of the 5th century. The league appears to have continued powerful in Magna Grsecia till the middle of the 5th century, when it was violently trampled out by the successful democrats. The meeting-houses of the Pythagoreans were everywhere sacked and burned; men-tion is made in particular of " the house of Milo" in Crotona, where fifty or sixty of the leading Pythagoreans were surprised and slain. The persecution to which the brotherhood was subjected throughout Magna Grascia was the immediate cause of the spread of the Pythagorean philosophy in Greece proper. Philolaus, who resided at Thebes in the end of the 5th century (cf. Plato, Phaedo, 61D), was the author of the first written exposition of the system. Lysis, the instructor of Epaminondas, was another of these refugees. This Theban Pythagoreanism was not without an important influence upon Plato, and Philolaus had also disciples in the stricter sense. But as a philosophic school Pythagoreanism became extinct in Greece about the middle of the 4th century. In Italy— where, after a temporary suppression, it attained a new importance in the person of Archytas, ruler of Tarentum— the school finally disappeared about the same time.

Pythagorean Philosophy.

The central thought of the Pythagorean philosophy is the idea of number, the recognition of the numerical and mathematical relations of things. In the naive speculation of an early age the abstract consideration of these relations was tantamount to asserting their essential existence as the causes of phenomena. Hence the Pythagorean thought crystallized into the formula that all things are number, or that number is the essence of everything. "The Pythagoreans seem," says Aristotle, "to have looked upon number as the principle and, so to speak, the matter of which existences consist"; and again, "they supposed the elements of numbers to be the elements of existence, and pronounced the wdiole heaven to be harmony and number." "Number," says Philolaus, '' is great and perfect and omnipotent, and the principle and guide of divine and human life." Fantastical as such a proposition sounds, we may still recognize the underlying truth that prompted it if we reflect that it is number or definite mathematical relation that separates one thing from another and so in a sense makes them things. Without number and the limitation which number brings there would be only chaos and the illimitable, a thought abhorrent to the Greek mind. Number, then, is the principle of order, the principle by which a cosmos or ordered world subsists. So we may perhaps render the thought that is crudely and sensuously expressed in the utterances of the school. They found the chief illustrations, or rather grounds, of their position in the regular movements of the heavenly bodies and in the harmony of musical sounds, the dependence of which on regular mathematical intervals they were apparently the first to discover. The famous theory of the harmony of the spheres combines both ideas : the seven planets are the seven golden chords of the heavenly heptachord.

Immediately connected with their central doctrine is the theory of opposites held by the Pythagoreans. Numbers are divided into odd and even, and from the combination of odd and even the numbers themselves (and therefore all things) seem to result. The odd number was identified with the limited, the even with the un-limited, because even numbers may be perpetually halved, whereas the odd numbers (at least the earlier ones), being without factors, seem to stand in solid singleness. All things, accordingly, were derived by the Pythagoreans from the combination of the limited and the unlimited ; and it is in harmony with the Greek spirit that the place of honour is accorded to the odd or the limited. Following out the same thought, they developed a list of ten fundamental oppositions, which roughly resembles the tables of categories framed by later philosophers :—(1) limited and unlimited ; (2) odd and even ; (3) one and many ; (4) right and left; (5) masculine and feminine ; (6) rest and motion ; (7) straight and crooked ; (8) light and darkness ; (9) good and evil; (10) square and oblong. The arbitrariness of the list and the mingling of mathematical, physical, and ethical contrasts are characteristic of the infancy of speculation. The union of opposites in which consists the exist-ence of things is harmony ; hence the expression already quoted that the whole heaven or the whole universe is harmony. But it is to be noted that interpretations of Pythagoreanism which repre o sent the whole system as founded on the opposition of unity and duality, and suppose this to have been explicitly identified with the opposition of form and matter, of divine activity and passive material, must be unhesitatingly rejected as betraying on the sur-face their post-Platonic origin. Still more is this the ease when in Neoplatonic fashion they go on to derive this original opposition from the supreme Unity or God. The further speculations of the Pythagoreans on the subject of number rest mainly on analogies, which often become capricious and tend to lose themselves at last in a barren symbolism. The decade, as the basis of the numerical system, appeared to them to comprehend all other numbers in itself, and to it are applied, therefore, the epithets quoted above of number in general. Similar language is held of the number "four," because it is the first square number and is also the poten-tial decade (1 + 2 + 3 + 4 = 10); Pythagoras is celebrated as the discoverer of the holy TerpaKTvs, "the fountain and root of ever-living nature." "Seven" is called irapdivos and 'KS-hyn, because within the decade it has neither factors nor product. "Five," on the other hand, signifies marriage, because it is the union of the first masculine with the first feminine number (3 + 2, unity being con-sidered as a number apart). The thought already becomes more fanciful when "one" is identified with reason, because it is un-changeable ; "two" with opinion, because it is unlimited and indeterminate ; "four" with justice, because it is the first square number, the product of equals. More legitimate is their application of number to geometry, according to which "one" was identified with the point, "two" with the line, "three" with the surface, and "four" with the cube. In the history of music the Pytha-gorean school is also of considerable importance from the develop-ment which the theory of the octave owes to its members; according to some accounts the discovery of the harmonic system is due to Pythagoras himself.

As already mentioned, the movements of the heavenly bodies formed for the Pythagoreans an illustration on a grand scale of the truth of their theory. Their cosmological system is also interest-ing on account of peculiarities which mark it out from the current conceptions of antiquity and bring it curiously near to the modern theory. Conceiving the universe, like many early thinkers, as a sphere, they placed in the heart of it the central fire, to which they gave the name of Hestia, the hearth or altar of the universe, the citadel or throne of Zeus. Around this move the ten heavenly bodies—farthest off the heaven of the fixed stars, then the five planets known to antiquity, then the sun, the moon, the earth, and lastly the counter-earth (_____), which revolves between the earth and the central fire and thus completes the sacred decade. Revolving along with the earth, the last-mentioned body is always interposed as a shield between us and the direct rays of the central fire. Our light and heat come to us indirectly by way of reflexion from the sun. When the earth is on the same side of the central fire as the sun, we have day ; when it is on the other side, night. This attribution of the changes of day and night to the earth's own motion led up directly to the true theory, as soon as the machinery of the central fire and the counter-earth was dispensed with. The counter-earth became the western hemisphere, and the earth re-volved on its own axis instead of round an imaginary centre. But, as appears from the above, the Pythagorean astronomy is also remarkable as having attributed a planetary motion to the earth instead of making our globe the centre of the universe. Long after-wards, when the church condemned the theory of Copernicus, the indictment that lay against it was its heathen and " Pythagorean " character.

The doctrine which the memory of mankind associates most closely with Pythagoras's name is that of the transmigration of souls—METEMPSYCHOSIS (q. v.). Though evidently of great importance for Pythagoras himself, it does not stand in any very obvious connexion with his philosophy proper. He seems to have adopted the idea from the Orphic Mysteries. The bodily life of the soul, according to this doctrine, is an imprisonment suffered for sins committed in a former state of existence. At death the soul reaps what it has sown in the present life. The reward of the best is to enter the cosmos, or the higher and purer regions of the universe, wdiile the direst crimes receive their punishment in Tartarus. But the general lot is to live afresh in a series of human or animal forms, the nature of the bodily prison being determined in each case by the deeds done in the life just ended. This is the same doctrine of retribution and purificatory wandering which meets us in Plato's mythical descriptions of a future life. They are borrowed by him in their substance from the Pythagoreans or from a common source in the Mysteries. In accordance with this religious view of life as a stage of probation were the ethical precepts of the school, inculcat-ing reverence towards the gods and to parents, justice, gentleness, temperance, purity of life, prayer, regular self-examination, and the observance of various ritual requirements.
Connecting its ethics in this way with religion and the idea of a future life, the Pythagorean societies had in them from the be-ginning a germ of asceticism and contemplative mysticism which it was left for a later age fully to develop. The Pythagorean life Was destined to survive the peculiar doctrines of the Pythagorean philosophy and to be grafted on later philosophic ideas. The asceticism which characterized it appears in the 4th century B.C. in close connexion with the Orphic Mysteries ; and the "Pytha-goreans " of that time are frequently the butts of the New Athenian Comedy. In the Alexandrian period the Pythagorean tradition struck deeper roots ; in Alexandria and elsewhere schools of men arose calling themselves Pythagoreans, but more accurately dis-tinguished by modern criticism as Neopythagoreans, seeing that their philosophical doctrines are evidently derived in varying pro-portions from Plato, Aristotle, and the Stoics. In general it may be said that they develop the mystic side of the Platonic doctrine ; and only so far as this is connected with the similar speculations of Pythagoras can they claim to be followers of the latter. Hence men like Plutarch, who personally prefer to call themselves Platonists, may also be considered as within the scope of this Pythagorean revival. The link that really connects these Neopythagoreans with the Samian philosopher and distinguishes them from the other schools of their time is their ascetic ideal of life and their preoccupation with religion. In religious speculation they paved the way for the Neoplatonic conception of God as immeasur-ably transcending the world ; and in their thirst for prophecies, oracles, and signs they gave expression to the prevalent longing for a supernatural revelation of the divine nature and will, The asceticism of the Jewish sect of the Essenes seems, as Zeller contends, to be due to a strong infusion of Neopythagorean elements. At a still later period Neopythagoreanism set up Pythagoras and Apollonius of Tyana not only as ideals of the philosophic life but also as prophets and wonder-workers in immediate communication with another world, and in the details of their " lives " it is easy to read the desire to emulate the narrative of the Gospels. The Life of Apollonius by Philostratus, which is for the most part an historical romance, belongs to the 3d Christian century.

Zeller's discussion of Pythagoreanism, in his Philosophie d. Griechen, book i., is very full; he also deals at considerable length in the last volume of the work with the Neopythagoreans, considered as the precursors of Neoplatonism and the probable origin of the Essenes. The numerous monographs dealing with special parts of the subject are there examined and sifted. (A. SE.)





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 (Met., 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 (_____)." (c) Diogenes Laertius (viii. 11) relates that "it was Pythagoras who carried geometry to perfection, after Mceris had first found out the prin-ciples 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." (d) According to Aris-toxenus, 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. Laert., viii. 11). (g) Proclus says that " the word ' mathematics' originated with the Pythagoreans." (h) We learn also from the same author-ity that the Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the "how many" (_____) and the other to the "how much" (_____) o 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 con-tinued quantity or the "how much" is either stable or in motion. Hence they laid down that arithmetic contem-plates 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 (_____) 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 mathema-tical subjects to which he applied himself."

1 We learn, however, from a fragment of Gemiuus, which has been handed down by Eutocius in his commentary on the Conies of Apol-lonius (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."

The following notices of the geometrical work of Pytha-goras and the early Pythagoreans are also preserved. (1) 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 duad, a superficies to the triad, and a body to the tetrad (16., p. 97). (3) They showed that the plane around a point is completely filled by six equilateral triangles, four squares, or three regular hexagons (ib., 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 1 (ib., p. 379). (5) Proclus informs us in his comment-ary 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 (___), in which it is used in this proposition, but also in its wider signification, embracing _____ and ___, in which it is used in Book VI. Props. 28, 29 —are old, and inventions of the Pythagoreans (ib., p. 419). (6) This is confirmed by Plutarch, who says, after Apollodorus, that Pythagoras sacrificed an ox on finding the geometrical diagram, 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 application of an area. (7) Plutarch 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 know-ledge 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."8 (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" (_____).1 (11) The discovery of the law of the three squares (Euclid I. 47), commonly called the "theorem of Pythagoras," is attributed to him by many authorities, of whom the oldest is Vitruvius.8 (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.9 (13) The discovery of irrational quantities is ascribed to Pythagoras by Eudemus (Procl., op. cit., p. 65). (14) The three proportions—arithmetical, geometrical, and harrnonical—were known to Pythagoras.10 (15) Iamblichus11 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 (virevavrCa), but which Archytas and Hippasus designated the harrnonical, 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 Iamblichus,12 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 1" he replies, " One, two, three, four." Pythagoras, interrupting, says, " Do you see 1 what you take to be four, that is ten and a perfect triangle and our oath."13 (18) The odd numbers were called by the Pytha-goreans "gnomons,"14 and asmuch as by the addition of successive gnomons—consist-ing 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 (_____), 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. (21) According to Iamblichus the Pythagoreans are said to have found the quadrature of the circle.





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 thé problems con-cerning the application of areas (5) ; and lastly, in the theorem of Pythagoras (11), coupled with his rule for the construction of right-angled triangles in numbers (12). We learn from Plutarch that the Egyptians were acquainted with the geometrical fact that a triangle whose sides contain three, foul', and five parts is right-angled, 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 form 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 i">und 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 whoso 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 (ji + l)th square ; if the mth gnomon contains m? unit squares, m being an odd number, we have 2n + l = m , . _. n = ——K~ , which gives the rule of Pythagoras.

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, &c. ; hence we have the gnomonic numbers.

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. A square can be dissected into the sum of two squares and two equal rectangles, as in Euclid II. i ; these two rectangles can, by draw-ing 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 bo 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 iu their cosmology played an essential part as the fundamental forms of the elements of the universe. "We can trace the origin of theso mathematical speculations in the theorem (3) that "the plane 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 equi-lateral triangle ; and out of these two figures arise the solids which correspond with the four elements of the real world, the tetra-hedron, 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 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 wdiat 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 iu 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 re-mained 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, which is reduced to Euclid II. 11, which problem is reduced to the following: To produce a given straight line so that the rect-angle 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 wdiich 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 bo 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 Iamblichus concerning Hippasus (9)— that he divulged the sphere with the twelve pentagons—and of Lucian and the scholiast on Aristophanes (10)—that the penta-gram was used as a symbol of recognition amongst the Pythagoreans —become of greater importance.
4 The dodecahedron was assigned to the fifth element, quinta pars,, either, or, as some think, to the universe. (See PHILOLAUS. )

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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 incommen- surable ; 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 absurd-: uni, showing that, if the diagonal be commensurable with the ! side, it would follow that the same number would be odd and \ even —makes it more probable, however, that this was accom- j plished by his successors. The existence of the irrational as j 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 Iamblichus of Hippasus is narrated of the person who first published the idea of the irrational, namely, that he suffered shipwreck, &c. - j

Eudemus ascribes the problems concerning the application of I figures to the Pythagoreans. The simplest cases of the problems, Euclid VI. 28, 29—those, namely, in which the given parallelogram is a square—correspond to the problem: To cut a 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 homo-logous 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 ascrib-ing 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. That we owe the foundation and development of the doctrine of proportion to Pythagoras and his school is confirmed by the testimony of Nieomachus (14) and Iamblichus (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 representa-tion. 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 isojierimetry, by proving that of all figures having the same peri-meter 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.

Let us now see what conclusions can be drawn from the foregoing examination of the mathematical work of Pythagoras and his school, and thus 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 (âXoyov) 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 combination of arithmetic with geometry. The notions of an equation and a propor-tion—which are common to both, and contain the first germ of algebra—were introduced among the Greeks by Thaïes. These notions, especially the latter, were elabo-rated 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 C. A. Bretschneider, Die Geometrie u. die Geometer vor Euklides (Leipsic, 1870) ; H. Hankel, Zur Geschichte der Mathematik (Leipsic, 1874) ; F. Hoefer, Histoire des Mathématiques (Paris, 1874); G. J. Allman, "Greek Geometry from Thaïes to Euclid," in Hermathena, Nos. v., vii., and x. (Dublin, 1877, 1881, and 1884) ; M. Cantor, Vorlesungen über Geschichte der Mathematik (Leipsic, 1880). The recently published Short History of Greek Mathematics by James Gow (Cambridge, 1884) will be found a convenient compilation. (G. J. A.)


Footnotes

5 Symp. viii., Quaest. 2, c. 4.

7 Lucian, Pro Lapsn in Saint., s. 5; also schol. on Aristoph., Nub., 611. 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 Pytlmcjoree figura ; even so late as Paracelsus it was regarded by him as a symbol of health. It is said to have obtained its special name from the letters v,y, i, 6 ( = ei), a having been written at its prominent vertices.

8 De Arch., ix., Proef., 5, 6, 7. Amongst other authorities are Diogenes Laertius (viii. 11), Proclus (op. cit, p. 426), and Plutarch (ut sup., 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).

9 Heron Alex., Geom. et Slereom. Ret, 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.

10 Nicom. Ger., Introd. Ar., c. xxii.

11 In Nicomachi Arithmeticam, ed. S. Tennulius, p. 141.

12 Op. cit, p. 168. As an example of this proportion Nicomachus and, after him, Iamblichus give the numbers 6, 8, 9,12, the harrnonical and arithmetical means between two numbers forming a georsetrio proportion with the numbers themselves (a: : :~^-* Iamblichus further relates (I.e.) that many Pythagoreans made use of this proportion, as Aristaeus of Crotona, Timants of Locri, Philolaus and Archytas of Tarentum, and many others, and after them Plato in his Timwus (see Nicom., Inst. Arithm., ed. Ast, p. 153, and Animacl-versiones, pp. 327-329 ; and Iambi., op. cit, p. 172 sq.).

13 ____, 4, vol. i. p. 317, ed. C. Jacobitz.

14 _____ 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 per- pendicular, 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 sun-dial, 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 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,



The above article was written by:

Philosophy
Andrew Seth, M.A., Professor of Logic and Philosophy, University College of South Wales

Geometry
Prof. G. J. Allman, LL.D.