Our previous chapter ended with a tantalizingly brief account of Greek religion. That account was too brief for any purpose except to make the reader realize the importance of religion in Greece, the cradle of science. The historian of science, even of Greek science, should never disregard religion. It would not be right to say that the luxuriant development of religion in all its forms, a development that reached a kind of climax during the sixth century, helped science, nor yet that it harmed it. Then as now the two developments, scientific and religious, were parallel, contiguous, interrelated in many ways; they were not necessarily antagonistic; they often took place in the same minds.

One curious aspect of that sixth-century efflorescence of religion is that it occurred in the western part of the Greek domain, rather than in its eastern part as we would expect, but that may be accidental. The Ionian physiologists represented, it is true, a rationalistic wing, but how many were there? or rather, how few? The Oriental Greeks or the Greek Orientals were on the whole religious minded, fond of rites and miracles. When the Persian menace, and later the Persian terror, drove them westward, some of them did not stop in Greece, or at least did not stay there, but continued farther west and found an asylum in the Ionian colonies of Sicily and Magna Graecia.⁴⁹⁹ We have already spoken of one of those eastern refugees, Xenophanes of Colophon; another and more illustrious one was Pythagoras.

What kind of man was he? It is difficult to say because the biographies that have come down to us were written late and are full of impurities. They were composed by Diogenes Laërtios (III–1), Porphyry (III–2), and Iamblichos (IV–1), the work of the latest being the most popular as well as the most spurious. What is more disquieting, the older traditions, such as those of Herodotos, Aristotle, and his pupils, were already fabulous to a degree. For example, Herodotos, the nearest witness in point of time, was already combining Pythagorean ideas with Egyptian, Orphic, and Bacchic ones,⁵⁰⁰ and he mixed up the story of Pythagoras with that of Zalmoxis, thus explaining obscurum per obscurius.⁵⁰¹ According to the story, which he tells with some diffidence (and we should not be more credulous than he was), Zalmoxis was a Thracian who had been a slave of Pythagoras, son of Mnesarchos. Having obtained his freedom, wealth, and some familiarity with the Ionian way of life, he returned to his native country, where he built a great hall and entertained his neighbors. He expounded to them the ideas of immortality and paradise and, in order to convince them, disappeared for three years in an underground chamber. While they were still mourning him he reappeared to them alive in the fourth year, and they could not disbelieve him any longer. This story shows that by the fifth century Pythagoras was almost as mythical as Zalmoxis himself.

Yet there is a small substratum of fact that we may perhaps accept as true. Pythagoras, son of Mnesarchos, was born in Samos and flourished there during the rule of Polycrates (executed in 522). According to Aristoxenos of Tarentum (IV–2 B.C.) — who is not too late a witness as ancient traditions go — he left Samos to escape Polycrates’ tyranny; that is plausible, or he may have been driven away, like so many others, by the fear of the Persians. It would have been natural enough for him to seek refuge in Egypt, where the Samians had many representatives (they had a temple of their own in Naucratis). If we may believe Iamblichos, he first went to Miletos, where Thales recognized his genius and taught him all he knew, then visited Phoenicia, where he remained long enough to be initiated into the Syrian rites. This increased his desire to go to Egypt, which was then considered the fountainhead of esoteric knowledge; he spent there no less than twenty-two years, studying astronomy and geometry as well as the mysteries. When Cambyses conquered Egypt in 525, Pythagoras followed him back to Babylon, where he spent twelve more years studying arithmetic, music, and other disciplines of the Magi. ⁵⁰² He then returned to Samos, being in the fifty-sixth year of his age, but soon resumed his wanderings through Delos, Crete, and Greece proper, and finally reached Croton,⁵⁰³ where he established his famous school. After he had obtained considerable popularity and power, which he may have abused, political enmities or local jealousies drove him out and he spent the last years of his life in Metapontion.⁵⁰⁴

We have told the story at some length, though we attach little credence to Iamblichos. Whether the details are correct or not, the substance is plausible. ⁵⁰⁵Was Pythagoras actually Thales’ disciple or not? Did he devote thirty-four years to graduate studies in Egypt and Babylon? We cannot even be sure that he traveled much on his way from Samos to Croton. The story accounts for the Egyptian and Babylonian roots of Pythagoras’ thought, but a man as intelligent and inquisitive as he was might have gathered a considerable amount of Oriental lore even without visiting those countries, or at any rate without spending there as many years as Iamblichos says he did. Surely Pythagoras did not need thirty-four years to learn what was then to be learned there and was assimilable to his fertile and eager brain. The intention of Iamblichos, or of his informant, was to show that Pythagoras had not simply visited Egypt and Babylonia as many Greeks did, for business or pleasure, but that he had remained in those countries long enough to study with their doctors, to drink deep from their wisdom, and even to be initiated into their mysteries.


One of the aspects of the religious revival that occurred in many places during the sixth century was the growth of communities of people sharing a new revelation and occult doctrines of various kinds. Such communities would naturally take the form of brotherhoods, for the men and women sharing eschatologic secrets would be like members of a family, like brothers and sisters defending their common heritage against outsiders. Pythagoras and his immediate disciples imitated that practice in Croton. Some of their teachings were scientific doctrines that will be explained in the following sections, but others were of a more general nature, and it was probably to these that the order owed its popularity. Pythagoreanism was primarily a way of life.

The Pythagoreans conceived a new kind of holiness, the attainment of which implied ascetic exercises and the observation of taboos, for example, the abstention from certain articles of food, such as meat, fish, beans, and wine, and the avoidance of woolen clothes.⁵⁰⁶ Women were admissible as well as men and seemed to have played an important part in the early community. Members of the order wore distinctive garments, went barefooted, and lived simply and poorly.

They fancied that the soul can leave the body either temporarily or permanently and that it can inhabit the body of another man or of an animal, but whether Pythagoras derived that belief from Hindu or other Oriental sources is of course impossible to say. Granted the intuitive feelings that a soul leaves the body with the last breath and that there is some kinship between men and animals,⁵⁰⁷ feelings shared by many nations, whether primitive or sophisticated, the concept of transmigration of souls might occur (and did occur) independently in many places.⁵⁰⁸

Fig. 50. The “Golden words” and the “symbols” of Pythagoras. In September 1497 the great Venetian publisher Aldo Manucci il Vecchio (1449–1515) published a small folio (30 cm high) containing Iamblichos’ De mysteriis Aegyptiorum, Chaldaeorum et Assyrio-rum, and a dozen other texts translated by the Florentine Platonist Marsilio Ficino (1433–1499). Fewer than three pages of that book concern Pythagoras, but those pages constitute the first Pythagorean printed publication. They contain his “Golden words” (sayings ascribed to him) and his “symbols.” The page that we reproduce shows the end of the “aurea verba” and the beginning of the “sym-bola”; these are chiefly in the nature of taboos. [From the copy in the Harvard College Library.]

The religion of the Pythagoreans was otherworldly to the extent that they regarded this life, the life this side of death, as a kind of exile (apod mia). Like every other religion, it was pure enough at its highest level and the opposite at its lowest. For example, many of their rules (as has already been remarked) were simply taboos,⁵⁰⁹ that is, irrational interdictions due to the fact that certain classes of things were considered sacred and were forbidden because of their purity or impurity; it was unlucky to meddle with them. Those rules were called acusmata and the humblest members of the order, the acusmaticoi, were the poor bigots for whom the taboos took the place of doctrines, for they were unable to understand much else (Fig. 50).⁵¹⁰ On the contrary, the fully initiated attached more importance to eschatology and theology, or to the scientific ideas that constituted the very core of their thinking. It is impossible to know much about those doctrines, or to know them with precision, for the members were pledged to silence (echemythia, echer-rh mosyn ) and even to secrecy.

Political ideas were added gradually to the others, for the order was a little society immersed in a larger one yet kept jealousy separated from it. Conflicts were bound to appear between the groups, and should the little Pythagorean group try to obtain power in order to escape those difficulties, its troubles would increase. It is certain that the Pythagoreans were thwarted and annoyed and that Pythagoras was forced “to leave town” and go to Metapontion. The followers who remained in Croton, Metapontion, and other places suffered greater persecutions after his death and some of them were even massacred (some of those persecutions occurred perhaps as late as 450).

The martyrdom of his disciples increased Pythagoras’ prestige. He was soon regarded as a saint or even (in the Greek way) as a hero, intermediary between gods and men, and the late accounts of his life and deeds were composed in the spirit of hagiography. Under those circumstances, is it at all surprising that the early doctrines are obscure and the founder himself largely unknown? To know the facts about him is as hopeless as to know those concerning St. Gregory the Wonder worker or St. George the Martyr.


In his lost book on the Pythagoreans, Aristotle wrote that “Pythagoras son of Mnesarchos first worked at mathematics and arithmetic and afterwards, at one time, condescended to the wonderworking practiced by Pherecydes.” ⁵¹¹ Aristotle’s hypothesis is plausible, though it does not tally with the traditions concerning Pythagoras’ oriental education. It is possible that Pythagoras’ first independent thinking was centered on mathematics and that the mystical tendencies of his youth reasserted themselves later in life. (He was not by any means the last mathematician to become a mystic in his old age!) At any rate, in order to develop a mystical theory of numbers it was necessary first of all to obtain a sufficient knowledge of them. Pythagoras was in all probability the founder of the great mathematical school bearing his name.

Here are a few examples of the speculations that are of sufficient antiquity to be ascribed to him. The first is the distinction between even numbers (artios) and odd ones (perissos), the former being divisible into two equal parts, the latter, not. That was of immediate value, for one often wishes to divide a group into two smaller ones as fairly and symmetrically as possible. If one builds a temple, the number of columns at the entrance should be even, otherwise a column would face the middle of the door, spoil the inward or outward view, and obstruct the traffic; the number of columns on the sides might be odd or even.⁵¹²

Pythagoras’ arithmetic was based on the use of dots drawn in sand, or of pebbles, which could be grouped easily in different ways. He was then able to make many arithmetic experiments concerning the number of pebbles that would fill a given pattern. If pebbles are arranged in such a manner that they form triangles (Fig. 51), the numbers of pebbles in the triangles (1, 3, 6, 10, ...) are the triangular numbers. Pythagoras probably saw that those numbers were the sums of one or more natural numbers beginning with one. Did he generalize the result?

Probably not, but he experimented far enough to see how these numbers derived each from the preceding one:

the successive additions being made, not with numerals as we have just done, but with pebbles. The fourth triangular number, a triangle with four pebbles on each side, interested Pythagoras specially. This was the so-called quaternion or tetractys (1 + 2 + 3 + 4 = 10), to which the school attached marvelous properties.⁵¹³ The Pythagoreans swore by it!

Fig. 51. Triangular numbers.

Fig. 52. Square numbers.

Square numbers were investigated in the same manner. How does one pass from one to the next? To pass from No. 3 to No. 4, for instance (Fig. 52), one adds a number of pebbles enveloping No. 3 at two sides of one corner. That two-sided series of pebbles, called a gnomon,⁵¹⁴ was necessarily an odd number. Hence the obvious rule: a square plus an odd number makes another square;

n² + (2n + 1) = (n + 1)².

More concretely, consider the series of odd numbers 1, 3, 5, 7, 9, ... The first is also the first square; by adding one by one each of the following odd numbers to it one will obtain all the square numbers:

Hence each square number is the sum of all the odd numbers smaller than twice its root:

This is as beautiful as it is simple. One can imagine Pythagoras’ delight when he discovered these particles of the universal truth; if he had mystical tendencies, such as he would have easily acquired in Egypt and Asia, his growing exaltation was natural.

We have spoken of pebbles, because Pythagoras had no numerals as we have. It is probable that the literal numerals were not yet in use in Pythagoras’ time.⁵¹⁵ If Pythagoras wrote the numbers at all, he probably used decimal symbols similar to the Egyptian ones, but that was simply an adaptation of abacus methods to writing. Let us assume, however, that the literal symbols were already available, for this will give us an opportunity of discussing them.

Fig. 53a

Fig. 53. Tables of Pythagoras. (a) Roman; the Egyptian (Roman) system requires only five different symbols. (b) Greek; the Greek system requires 27 different symbols; the accents following each numeral have been left out. (c) Hindu-Arabic; the Hindu system requires 10 different symbols; its practical value lies in the fact that it adapts abacus methods to script in a deeper manner than the Egyptian.

The three tables are decimal, because no other basis was thought of, except the (Babylonian) sexagesimal for fractions and that only much later (Ptolemy, 11-1), the duodecimal in exceptional cases (division of the day, of the pound), and other oddities in metrology and coinage (such as exist to this day in the English systems); see Isis 23, 206–209 (1935).

Fig. 53b

Fig. 53c

The Greek numerals are 27 in number, divided into three groups of nine each, the first nine designating the units from 1 to 9, the second group the decades from 10 to 90, the third group the centuries from 100 to 900. The symbols used are simply the Greek letters (with an accent to the right of each) in their alphabetic order; but since there are only 24 letters in the Greek alphabet, three obsolete letters were added, one in each group, to wit, digamma or stigma for 6, koppa for 90, and swampi for 900. Moreover, the first ten letters (including stigma) were used also to designate the thousands from 1000 to 10000 (in that case the accent is put to the left of the letter, below the line). Not only were the Greeks obliged to remember three times as many symbols as we are, but many simple relations were hidden by that multiplicity. Consider the fundamental distinction between odd and even numbers. It is easy for us to remember that even numbers end in 0, 2, 4, 6, 8. How was it for the Greeks? An odd number might end with any one of the 27 symbols! (Fig. 53).

The table of multiplication that is called in many languages the Pythagorean table (mensula Pythagorae) was certainly not devised by Pythagoras. The earliest example of it known to me occurs in the Arithmetica of Boethius (VI–1) , printed in Augsburg in 1488.⁵¹⁶ There may be earlier ones in manuscripts written possibly with Roman numerals, for the Hindu-Arabic numerals were hardly introduced into the West before the twelfth or thirteenth century, and there was so much resistance to their use that they did not obtain any popularity until much later.

The table of Pythagoras in Hindu numerals is very clear. One sees at once that the lines (or columns) 2, 4, 6, 8, 10 contain only even numbers, that in line (or column) 5 every number ends with 5 or 0 (it is true that in the Greek script half of the numbers end with ε). Neither Pythagoras nor even the latest Pythagorean of antiquity knew of Hindu numerals (or of equivalent ones); hence it is probable that the mensula Pythagorae is but a late medieval creation, perhaps not much older than the printed Boethius.⁵¹⁷

The early Pythagorean ideas on numbers were almost certainly restricted to such as could be illustrated by means, not of numerals, but of counters or pebbles. That simple method brought to light facts of transcendent meaning. Pythagorean arithmetic is the root not at all of our arithmetic or of the art of reckoning, but rather of the present theory of numbers.

The reader, especially the one interested in the sociology of science or in the materialistic interpretation of history, may object that our conclusion does not tally with all that we know of the early and strong proclivities of Greek people to trade. After all, trading and every form of business transaction necessitate plain arithmetic in our sense; from the point of view of sellers and buyers (that is, the whole population), the theory of numbers is a luxury. One might answer that religion, philosophy, and the humanities are also luxuries from the mercantile point of view. Moreover, arithmetic (reckoning) was developed and intensely cultivated by the Greeks, but in an empirical manner. We may be sure that the average Greek dealer knew how to count rapidly and exactly, either in his head or with the help of some kind of abacus.⁵¹⁸ However efficient he might be in that art, he never imagined that he was doing mathematical work; on the other hand, the ancient mathematicians never thought of reckoning as a part of their own field. Even today it is only ignorant people who confuse mathematics with reckoning or accounting, or who mistake tellers for mathematicians.⁵¹⁹

Fig. 54. Angles between parallels.

Fig. 55. Internal angles of a polygon.


Among the geometric achievements of the Pythagorean school that seem early enough to be creditable to Pythagoras himself I would choose the following.

The interior angles of a triangle are equal to two right angles; this could be proved almost immediately if one but knew that when a line cuts two parallels the alternate angles are equal (Fig. 54). If AA′ is parallel to BC, the three angles of the triangle are equal to the two right angles in A. Pythagoras may have extended that proof to polygons of more sides (Fig. 55). In the hexagon ABCDEF, join EA, EB, EC. The sum of the internal angles of the hexagon is equal to that of the internal angles of the four triangles, or eight right angles. More generally, for a polygon of n sides the sum of the internal angles is equal to (2n − 4) right angles. The sum of the external angles (each being the supplement of an internal one) is equal to 2n − (2n − 4) = 4 right angles; it is thus independent of the number of sides.

Common experience in flagging or tile flooring helped to show that the only regular polygons by which the space can be filled without gaps are the equilateral triangle, the square, and the regular hexagon. The proof was easy, for each angle of these regular polygons measures, respectively, two, three, or four thirds of a right angle. The space around a point in one plane, equaling four right angles, can be filled with six triangles, four squares, or three hexagons (Fig. 56).

Did Pythagoras know the “Theorem of Pythagoras,” that is, that the square built upon the hypotenuse of a right-angled triangle equals the sum of the squares built upon the two other sides of that triangle?⁵²⁰ Why not? This can be seen almost intuitively in various ways.

For example, suppose we have two unequal squares (Fig. 57) such that the smaller one EF² is inscribed in the larger one AB′ (that is, until its four apices touch the four sides of the larger square). It is clear that the four triangles EAF, . . . outside the smaller square are equal. Draw EE′parallel to AB and FF′ parallel to BC, intersecting in O; we thus divide the square AB² into four parts — two equal rectangles and two squares EO′ and FB². Then the area of the largest square AB² can be expressed in two ways:

But each of the rectangles equals two of the triangles, hence

EF² = EO² + FB² = AF² + AE².


The demonstration is so easy that it may have been made previously and independently by the Egyptians, the Babylonians, the Chinese, and the Hindus. The possibility of Egyptian priority has been considered in Chapter II; we need not consider the other possibilities, for they never come close to certainty. Pythagoras may have been the first to prove the proposition (not simply to see that it was true), or his proof may have been more conscious, or more rigorous, using a method equivalent to Euclid’s. It was said that Pythagoras had sacrificed an ox to celebrate that discovery, or was it perhaps to celebrate the discovery of particular triangles (sides 3n, 4n, 5n ) wherein the geometric proof was easily completed by a numerical verification?

Fig. 56. Fitting of regular plane polygons.

Fig. 57. The Pythagorean theorem.

Fig. 58. The Pythagorean pentagram.

He may have initiated the geometric problems concerned with the finding of an area equal to another (say a square equal to a parallelogram) or with the application of areas (parabol t n ch ri n), the one exceeding the other (hyperbol ) or falling short of it (elleipsis) by a given quantity. In the course of time those problems led to the geometric solution of quadratic equations, and curiously enough the Greek terms just quoted, probably later than Pythagoras, were afterward applied to the three different species of conic sections.

The geometric ideas and theorems that we are tempted to ascribe to Pythagoras could not have been proved easily, in spite of their simplicity, without the use of letters to designate the lines involved. We have used letters in our own explanations, without thinking of it, because it is very difficult to do otherwise. It does not follow that Pythagoras used letters. For example, he might have proved the theorem bearing his name by drawing lines on the sand and pointing to the lines and areas with his fingers. It is only when the proof is written that letters (or other symbols) become indispensable.

According to a tradition of which we hear a late echo in Lucian (120–180), the Pythagoreans used as a symbol of mutual recognition the pentagram,⁵²¹ to which they gave the name “health.” ⁵²² The five letters of that name ( γ ειa, hygieia) were the five apices of that symbol (Fig. 58).⁵²³ This is perhaps the oldest example of the application of letters to various points (or other parts) of a geometric figure. It may be older than the use of letters for ease in geometric demonstrations, or it may have been suggested by that very use.

Pythagoras or his immediate disciples were already acquainted with some regular solids; the cube and pyramid (tetrahedron) were easy enough to conceive or to build, the octahedron not difficult. Their knowledge of the pentagram does not prove that they were able to construct a regular pentagon; but even if they did not know the geometric construction, they could always divide a circumference empirically into five equal parts. Moreover, if, after having built a regular pyramid and a regular octahedron, they continued to play with equilateral triangles and put as many as five together (one apex being common to all five) they thus built one of the solid angles of an icosahedron. Even if they did not complete the icosahedron they must have recognized that the base of that solid angle was a regular pentagon. Playing with regular pentagons, they may have succeeded in building a dodecahedron. There is much guesswork in all this, however, and we shall postpone further discussion of the regular solids, the “Platonic figures,” until later.


We must be at least as discreet when we discuss Pythagorean astronomy as we were for geometry. It cannot be our purpose to catch the new ideas in their embryonic stage, as it were, for that is naturally impossible. It is safer to wait until they have become sufficiently clear and definite. Thus in this section we shall indicate only a few general ideas probably prior to Philolaos (V B.C.), to whom the earliest Pythagorean astronomic writings are ascribed.

The idea that the earth is a sphere is probably as old as Pythagoras. How did he reach such a bold conclusion, one may wonder? He may have observed that the surface of the sea is not flat but curved, for as a distant ship approaches one first sees the top of its mast and sail and the rest appears gradually. The circular edge of the shadow cast in an eclipse of the Moon would also suggest the spherical shape of the Earth, but that is a more sophisticated kind of observation, implying an understanding of eclipses that had not yet been attained in the sixth century. It is more probable that as soon as the hypothesis of a flat earth had been rejected, the sphericity of the earth was postulated rather wildly, on insufficient experimental grounds. The earth cannot be flat, therefore it ought to be spherical. Was not the starry heaven visibly part of a sphere? Were not the disks of Sun and Moon circular? And was any volume or surface comparable in symmetry and beauty to those of the sphere? This fundamental Pythagorean idea was an act of faith rather than a scientific conclusion. Does not every scientific hypothesis start that way? This hypothesis made the theory of eclipses possible, and, conversely, the development of that theory, the observations that it suggested, repeatedly confirmed the initial assumption.

The dogma of spherical perfection and its cosmologic consequences may be considered the kernel of early Pythagorean science. It was postulated that the celestial bodies are of spherical shape and that they move along circular paths, or as if they were attached to spheres. The earth, naturally enough, was supposed to be immobile in the middle of all, its center being the center of the universe. The movement of all the spheres is uniform, like that of the heavens. How could it be other than uniform?

The Babylonians had been satisfied to describe as accurately as possible the movements of the planets, and to account for them by numerical tables. Pythagoras, acquainted with Milesian physiology, was no longer satisfied with descriptions. He wanted to explain the phenomena, to justify them. The planets cannot be “errant” ⁵²⁴ bodies; they must have circular and uniform movements of their own. That opinion is ascribed to Alcmaion as well as to Pythagoras; whoever held it first, it represents a great step forward in the Pythagorean doctrine. The stars as seen from a position north of the equator move clockwise and with clocklike regularity; the planets (meaning the Sun, Moon, and our planets) do not wander erratically but they have counterclockwise motions of their own. If one could but analyze those complicated motions they would be reduced to uniform circular ones. The whole of Greek astronomy grew out of that arbitrary conviction.⁵²⁵

Another conviction established itself gradually in the same obscure, mystical manner. Out of Milesian monism emerges a new kind of dualism. There is a substantial difference between the celestial world on the one hand, eternal and divine, perfect, unchangeable, the elements of which move in circles without angular acceleration, and the sublunar one (ta hypo sel n n) on the other, subject to endless changes, decomposition, decay, and death, and wherein the motions are capricious and irregular. The superlunar world is the world of the immortal gods and perhaps of the souls. The sublunar one is the abode of things either lifeless or mortal. ⁵²⁶

This Pythagorean dualism influenced scientific thought until the time of Galileo and even later. Its influence upon religion was hardly less important; we shall discuss some aspects of it apropos of the Epinomis later on. It will suffice now to remark that the sidereal religion, which was to be the core of astrology, derived straight from those Pythagorean fancies, added to the Chaldean ones.


The stories told about Pythagoras’ musical experiments are hard to believe, except one. Bearing in mind that in his time the Greeks and other ancient peoples had already acquired considerable familiarity with stringed instruments, his experiments with strings are quite plausible.⁵²⁷ Of course, every citharist would know that he could obtain different sounds and pleasing combinations of sounds by pinching the strings at certain places or changing the length of their vibrating parts. Pythagoras may well have repeated such experiments more methodically and with the detachment of a scientist rather than with the intuitive subjectivity of an artist, and he may have discovered that the uniform strings the lengths of which were in the relation 1:¾: :½ (or 12:9:8:6) produced harmonious sounds. The ratios of the vibration numbers 12:6, 12:8, and 8:6 are the intervals that we call octave, fifth, and fourth (in Greek, diapas n, diapente, and diatessar n).⁵²⁸

That discovery directed Pythagoras’ thought to the ratios themselves, that is, to the theory of means and proportions. Or was it the other way round, and did his familiarity with proportions draw his attention to the musical intervals? Pythagoras was certainly not the first to think of arithmetic means; and geometric means (a:b = b:c) were natural enough to be conceived very early. He was perhaps the introducer of a new kind of mean, called “harmonic” (harmonic analogia) wherein the three terms are such that “by whatever part of itself the first exceeds the second, the second exceeds the third by the same part of the third.” ⁵²⁹ More clearly, if b is the harmonic mean of a and c, we can write a = b + a/p, b = c + c/p; hence a/c = (a–b)/(b–c), or 1/c–1/b = 1/b–1/a. (If b were the arithmetic mean of a and c, we would have a–b = b–c. One sees why the harmonic proportion was also called subcontrary, hypenantla.)

The numbers 12, 8, 6 quoted above form a harmonic proportion. The cube was called a “geometric harmony” (ge metric harmonia) because it has 12 sides, 8 angles, and 6 faces.⁵³⁰ The theory of means was susceptible of many extensions, which were fully exploited by the Pythagorean arithmeticians in later times.

The idea of harmonic proportion was soon extended to astronomy. The heavenly spheres were supposed to be separated by musical intervals and the planets emitted different notes in harmony. According to Hippolytos (III–1), “Pythagoras maintained that the universe sings and is constructed in accordance with harmony; and he was the first to reduce the motions of the seven heavenly bodies to rhythm and song.” ⁵³¹ St. Hippolytos is a very late witness and not a reliable one. Those mathematical fantasies were potentially in Pythagoras’ mind; it is improbable that he formulated them as neatly as Hippolytos put it, but the formulation took place in the fifth or fourth century, in or before Plato’s time.⁵³²


The earliest medical center of Greece that might be called a school, a theoretical school, was perhaps the one that developed in Croton. Its origin may be prior to Pythagoras, but it more probably synchronized with the Pythagorean school. The writings of the first teacher, Alcmaion of Croton, son of Peirithoos, are lost, but as far as we can judge from the fragments and the doxography, he was a disciple of Pythagoras. Some medical ideas are ascribed to Pythagoras himself, yet it is simpler to consider Alcmaion as the medical teacher of the sect.

The title of Alcmaion’s treatise peri physe s suggests a Milesian influence, and he may have been a Milesian (or Ionian) refugee, like so many of his contemporaries whom the fear of the Persians or local tyrannies had driven out. He investigated sense organs, especially those of vision, and, if we may believe Chalcidius (IV-1), was the first to attempt a surgical operation on the eye.⁵³³ He claimed that the brain was the central sensorium and that there were some pathways or passages (poroi) between it and the sense organs; if those passages were broken or stopped, say by a wound, communication was interrupted. These pregnant views — the first seeds of experimental psychology — were amplified in the following century by Empedocles and the atomists.

Alcmaion may have been the initiator of another psychologic doctrine to which later Pythagoreans attached more and more importance. The souls are comparable to the celestial bodies and thus move eternally in circles. Circularity and immortality are equated. On the other hand, men die because they cannot return to their beginning,⁵³⁴ the cycle of life is not a circle but an unclosed curve. Life, we would interpret, is a running-down process; the stars and souls do not run down but turn eternally around.

Alcmaion’s main medical theory was that health is an equilibrium of forces (isonomia dyname n) in the body; when one of the forces dominates the equilibrium is upset and we have a state of monarchy (monarchia) and disease.

Another physician of Croton, Democedes son of Calliphon, obtained considerable fame. He was for a time in the service of Polycrates, tyrant of Samos (d. 522), and later flourished in Susa at the court of Darios (king of Persia, 521–485). The great king had dislocated his foot in alighting from a horse; Democedes succeeded in healing it after Egyptian physicians had failed to do so and he used his prestige to beg the lives of his unfortunate colleagues who were about to be impaled. Then he cured Darios’ wife, Cyros’ daughter, Atossa,⁵³⁵ who was frightened by a tumor growing on her breast. He took advantage of a political mission forced on him by the king to sail away from Sidon (in Phoenicia) and return to his native country. Persian emissaries tried to persuade the magistrates of Croton to surrender the fugitive in order that they might bring him back to their lord. Democedes was finally allowed to remain because of his wedding with the daughter of the athlete Milon, who was the most illustrious son of Croton.⁵³⁶ It is typical of Greek life to come across this reference to athletics mixed up with the beginnings of nonanony-mous medicine.

The first eleven chapters of the Hippocratic treatise on hebdomads, De hebdo-madibus (Peri hebdomad n) set forth a number of cosmologic, embryologic, physiologic, medical remarks on the importance of the number seven: the embryo takes a human form on the seventh day, some diseases are dominated by hebdomadal cycles, there are seven planets, etc. That text is of early date, not later than the sixth century,⁵³⁷ yet it is not Pythagorean but definitely Ionian (Cnidian?). This would show that mystical extensions of the idea of number were not restricted to Magna Graecia. But why should they be? Mesopotamia might well have been the cradle of such fancies. We should not forget that Pythagoras himself was a Samian.

For bibliography on the Peri hebdomad n see Introduction, vol. 1, p. 97. That text is not available in Greek, except a fragment, but has come down to us in an Arabic translation by Hunain ibn Ish q (IX–2) ⁵³⁸ and in a barbaric Latin translation. The Latin text may be found in Littré, Oeuvres complètes d’Hippocrate (10 vols.; Paris, 1839–1861), vol. 8, pp. 634–673; vol. 9, pp. 433–466. The Arabic text was translated into German by Christian Harder, “Zur pseudohippokratischen Schrift Peri hebdomad n sive To pr ton peri nus n to microteron,” Rheinisches Museum 48, 433–447 (1893), and from German into Italian by Aldo Mieli, Le scuole ionica, pythagorica ed eleata ( Florence, 1916), pp. 93–115 [Isis 4, 347–348 (1921–22)]. See also Joseph Bidez, Eos ( Brussels: Hayez, 1945), pp. 126–133 [Isis 37, 185 (1947)]; the idea of the microcosmos adumbrated in that treatise is probably of Iranian origin.


If one puts together the discoveries that may be ascribed to Pythagoras, or at least to the early brotherhood, in the fields of arithmetic, geometry, astronomy, and music, one is startled by the predominance of numerical concepts. Would one not expect that predominance to be even more startling for those early cogitators than it is for us? And mystically minded as they undoubtedly were, is it very surprising that they finally jumped to a bold and great conclusion? Numbers are immanent in things. To the Ionians, who had postulated a single material basis of nature, and to Anaximandros, who postulated a metaphysical basis, the indefinite, Pythagoras could now triumphantly retort: Numbers are the essence of things. We need not try to investigate that idea more deeply, because it is not likely that Pythagoras had carried it very far, and chiefly because it does not bear analysis. It is valid only as long as it remains in the nebulous form that Pythagoras gave to it. Later Pythagoreans established all kinds of relations between definite numbers and indefinite ideas, but those elaborations were of their nature arbitrary and illusive, while the general concept was (and remains) very impressive.

That numerical philosophy had far-reaching consequences, which are still felt today, in two directions, good and evil. It initiated the quantitative study of nature on the one side, and number mysticism, numerology, on the other. One might claim that the physicists of all ages, the natural philosophers, have been constantly allured by the hope of finding new numerical relations. It is as if they had heard old Pythagoras whisper into their ears: The number is the thing. We would rather say that mathematical relations reflect, if they do not reveal, the essence of reality. As to number mysticism, it is the caricature of the same concept, its reduction to absurdity by the extravagance of ignorant and foolish men.


If numbers are the essence of things, the better we understand them the better shall we be able to understand nature. The theory of numbers is the basis of natural philosophy. It would seem that the Pythagorean brotherhood drew that conclusion early. Vulgar people deal with numbers only because of the need for measuring and counting salable objects and of computing profits, but Pythagoras taught that there was a far deeper reason for being interested in numbers. We should try to penetrate the secrets of nature. Such disinterested efforts raise human life to a higher level, closer to the gods.

The desire for purification and the desire for salvation are innate in the best of men.⁵³⁹ They had been cultivated before Pythagoras’ days in the Orphic mysteries and other religious ceremonies, but Pythagoras was probably the first to associate them or even to confuse them with the desire for knowledge, especially for mathematical knowledge, symmetry, and music. According to the greatest musicologist of antiquity, Aristoxenos of Tarentum (IV–2 B.C.), the Pythagoreans used music to purge the souls even as they used herbs to purge the bodies. We may safely assume that that remark applies to Pythagoras himself or to his earliest (and most scientific) disciples. He went even further when he proclaimed that the pursuit of disinterested knowledge is the greatest purification. The highest kind of life is the theoretical or contemplative.⁵⁴⁰ These views are the seeds of others fully set forth in the Phaid n and in the Nicomachean Ethics; they are also the seeds of pure science. It was the strange destiny of Pythagoras to be at one and the same time the founder of science and the founder of a religion. He was the first to assert that science is valuable, irrespective of its usefulness, because it is the best means of contemplation and understanding. He was the first to connect the love of science with sanctity. He might well be feted as the patron saint of men of science of all ages, the pure theorists, the contemplators.

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