THE PYTHAGOREAN SOURCEBOOK AND LIBRARY
IT HAS BEEN SUGGESTED, by Alfred North Whitehead, that "the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. " If such be the case, what might then be said of Pythagoras, to whose philosophy Plato was so greatly indebted? While no definitive answer will be attempted here, it might do well to note that not only did Pythagoras first employ the term philosophy, and define the discipline thereof in the classic sense, but that he bequeathed to his followers, and to the whole of Western civilization, many important studies and sciences which he was instrumental in either formulating or systematizing.
True as this may be, much mystery surrounds the figure of Pythagoras, despite the significant influence of Pythagorean thought in antiquity. Of course, many things can be precisely stated. He was both a natural philosopher and a spiritual philosopher, a scientist and a religious thinker. He was a political theorist, and was even involved in local government. While he may not have been the first to discover the ratios of the musical scale, with which he is credited, there can be no doubt that he did conduct extensive research into musical harmonics and tuning systems. Pythagoras is well known as a mathematician, but few realize that he was also a music therapist having, in fact, founded the discipline. Pythagoras taught the kinship of all living things; hence, he and his followers were vegetarians. Yet, while all these things may be safely stated, quite a bit of mystery still remains.  This is due in large part to the fact that Pythagoras left no writings, although it is said that he wrote some poems under the name of Orpheus . Pythagoras' teaching was of an oral nature. While he seems to have made some speeches upon his arrival in southern Italy to the populace, the true fruits of his philosophic inquiries were presented only to those students who were equipped to assimilate them. Pythagoras no doubt felt, like his later admirer Plato, that philosophic doctrines of ultimate concern should never be published, seeing that philosophy is a process, and that books can never answer questions, nor engage in philosophical enquiry. 
Yet, despite the lack of first-hand writings by Pythagoras himself, we need not be deterred. There is an immense amount of material in the biographies of Pythagoras which goes back to a very early date, and it is certainly possible to sketch an accurate if not complete picture of early Pythagorean philosophy, even being quite specific on many points.  Let us then begin with Pythagoras himself.
THE PRIMARY SOURCES of information about the life of Pythagoras are to be found in this anthology. In addition to containing quite a bit of information about Pythagoras which goes back to an early date, these four biographies also demonstrate in an admirable fashion the high esteem in which the philosopher was held.
Pythagoras was born around 570 B.C.E. to Mnesarchus of Samos, a gemengraver, and his wife, Pythais.
The biographies of Pythagoras are unanimous that at an early age he travelled widely to assimilate the wisdom of the ancients wherever it might be found. He is said by Iamblichus to have spent some 22 years in Egypt studying there with the priests, and is also said to have studied the wisdom of the Chaldeans firsthand. These accounts are generally accepted by most scholars -- as indeed they should be, owing to the high degree of contact between Asia Minor and other cultures -- although it is doubtful, while not impossible, that he travelled to Persia to study the teachings of Zoroaster. In these distant lands Pythagoras not only studied the sciences there cultivated, including mathematical sciences we may safely presume, but was also initiated into the religious mysteries of the "barbarians." As Porphyry succinctly observes, "It was from his stay among these foreigners that Pythagoras acquired the greater part of his wisdom." 
After his studies abroad, Pythagoras returned home to the island of Samos, where he continued his philosophical researches. It is said that he outfitted a cave especially designed for the study of philosophy, and it was there that he made his home. About this time Pythagoras opened his first school, as we are told by Porphyry, yet it probably was not long-lived, as Pythagoras decided to leave Samos at the age of 40, owing to the tyranny of Poly crates which was then flourishing.
From Samos Pythagoras journeyed to South Italy, arriving at Croton, "conceiving that his real fatherland must be the country containing the greatest number of the most scholarly men." 
It would seem that his reputation preceded the philosopher, for he was shortly asked to speak to the populace of Croton -- men, women, and children -- on the proper conduct of life. The essence of these speeches is to be found in the biography by Iamblichus. While obviously not recorded verbatim, it seems quite likely that the content of these talks is genuinely Pythagorean and goes back to Pythagoras himself.  According to the biographies, the populace was enthralled by the wisdom of this man, owing to which he was invited to become involved in local government.
WHERE PYTHAGORAS DEVELOPED his interest in Number we do not know, although it is likely that he was not the first to be concerned with its sacred or metaphysical dimension. What we do know is that a metaphysical philosophy of Number lay at the heart of his thought and teaching, permeating, as we shall see, even the domains of psychology, ethics and political philosophy.
The Pythagorean understanding of Number is quite different from the predominately quantitative understanding of today. For the Pythagoreans, Number is a living, qualitative reality which must be approached in an experiential manner. Whereas the typical modern usage of number is as a sign, to denote a specific quantity or amount, the Pythagorean usage is not, in a sense, even a usage at all: Number is not something to be used; rather, its nature is to be discovered. In other words, we use numbers as tokens to represent things, but for Pythagoreans Number is a universal principle, as real as light (electromagnetism) or sound. As modern physics has demonstrated, it is precisely the numeric, vibrational frequency of electromagnetic energy -- the "wavelength" -- which determines its particular manifestation. Pythagoras, of course, had already determined this in the case of sound.
Because Pythagorean science possessed a sacred dimension, Number is seen not only as a universal principle, it is a divine principle as well. The two, in fact, are synonymous: because Number is universal it is divine; but one could as easily say that because it is divine, it is universal. Hence, the aim of Pythagorean and later Platonic science is different from that of modern "Aristotelian" science: it is not so much involved with the investigation of things, as the investigation of principles. It should be very firmly emphasized, however, that for Pythagoras the scientific and religious dimensions of number were never at odds with each other. Moreover, the Pythagorean approach to Number, for the first time in Greece, elevated mathematics to a study worth pursuing above any purely utilitarian ends for which it had previously been employed.
The Pythagoreans believed that Number is "the principle, the source and the root of all things."  But to make things more explicit: the Monad, or Unity, is the principle of Number. In other words, the Pythagoreans did not see One as a number at all, but as the principle underlying number, which is to say that numbers -- especially the first ten -- may be seen as manifestations of diversity in a unified continuum  To quote Theon of Smyrna:
If One represents the principle of Unity from which all things arise, then Two, the Dyad, represents Duality, the beginning of multiplicity, the beginning of strife, yet also the possibility of logos, the relation of one thing to another:
With the Dyad arises the duality of subject and object, the knower and the known. With the advent of the Triad, however, the gulf of dualism is bridged, for it is through the third term that a Relation or Harmonia ("joining together") is obtained between the two extremes. While Two represents the first possibility of logos, the relation of one thing to another, the Triad achieves that relation in actuality. If this process of emergence is represented graphically as in figure I, we can see that the Triad not only binds together the Two, but also, in the process, centrally reflects the nature of the One in a "microcosmic" and balanced fashion. (See figure 1.)
FIGURE 1. UNlTY, DUALITY AND HARMONY
What we have seen in this example of Pythagorean paradigm, based on the universal principles of pure Number and Form, is the emergence of Duality out of Unity, and the subsequent unification of duality, which in turn results in a dynamic, differentiated image of the One in three parts-a continuum of beginning, middle and end, or of two extremes bound together with a mean term. This process, in fact, is the archaic and archetypal paradigm of cosmogenesis, the pattern of creation which results in the world. As F.M. Corn ford has observed:
Cornford goes on to demonstrate how this universal pattern underlies not only the cosmogonies of Greek myth, but also those of the early Ionian scientific tradition.  It also underlies, as one may suspect, the Pythagorean view of the kosmos, literally "world-order" or "ordered-world," a term that Pythagoras is credited with first applying to the universe. The word kosmos, in addition to its primary meaning of order, also means ornament. The world, according to Pythagoras, is ornamented with order. This is another way of saying that the universe is beautifully ordered.
The idea of order is intimately connected with Limit (peras), the opposite of which is the Unlimited (apeiron), and these are the two most basic, and hence most universal, principles of Pythagorean cosmology. According to the Pythagoreans, the world or cosmos is compounded of these elements, summarized in the famous "Table of Opposites" which has been preserved by Aristotle in his Metaphysics (i. 5 986 a 23):
FIGURE 2. THE PYTHAGOREAN TABLE OF OPPOSITES
Limit is a definite boundary; the Unlimited is indefinite and is therefore in need of Limit. Apeiron also may be translated as Infinite, but it is infinite in a negative sense: that is, it is infinitely or indefinitely divisible, and hence weak, rather than the modern "positive" usage of the term, which is often synonymous with "powerful." To avoid any confusion between the ancient and modern meanings, Apeiron has been translated as either Indefinite or Unlimited in the writings which appear in this book, unless the context suggests otherwise.
Aristotle stated that the Pythagoreans made everything out to be created of numbers; what he means to say is that everything is created out of the elements of number, which include the Limited and the Indefinite, the Odd and the Even.
The Pythagoreans were in the habit of representing arithmetical numbers as geometrical forms, through which they arrived at some interesting insights. In fact, Aristotle makes reference to this very practice:
Aristotle is referring to the following figures:
FIGURE 3. SQUARE NUMBER
FIGURE 4. OBLONG NUMBER
The Greek word gnomon signifies a "carpenter's square." In figure 3 gnomons have been placed around the One, in figure 4 around the Dyad. From this arrangement several patterns arise. In the case of figure 3, each gnomon or band of points is odd, in figure 4 each gnomon is even. From the above diagrams, we can easily see why the Pythagoreans, in the Table of Opposites, identified the Odd and Even with the Square and Oblong respectively. Moreover, the principles of Limit and the Unlimited are also most manifest in these representations, for figure 3 is limited by the stable form of the square, while figure 4 is infinitely variable: with each successive gnomon, the shape and its corresponding lateral to horizontal ratio changes each time, for it is the nature of the Unlimited to be eternally variable and multifarious.
According to the paradigms of ancient cosmology, Matter (the Indefinite) receives and is shaped by Form (Limit); hence, these two principles of peras and apeiron may be seen at the two most universal and essential elements which are absolutely necessary for the manifestation of phenomenal reality. From this perspective it becomes easy to see the logic behind the Pythagorean sentiment that the cosmos is created out of the elements of Number, namely the Limited and the Indefinite. Plato, in fact, takes over this Pythagorean cosmology to the letter. His only change, and a minor one at that, is that he referred to Limit as the One, the Unlimited as the Indefinite Dyad, terms which have even more Pythagorean implications than the originals.
In the Pythagorean and Platonic cosmology, Limit and the Indefinite, Form and Matter, are woven together through numerical harmony: their offspring, existing in the indefinite receptacle of space, is the phenomenal universe, in which every being is composed of universal constants and local variables. Hence, in his Pythagorean cosmogony of the Timaeus, Plato shows how the fabricator of the cosmos parcels out the stuff of the World Soul according to the numerical proportions of the musical scale. 
PYTHAGORAS IS SAID to have discovered the musical intervals. While the story of the musical smithy is probably a Middle Eastern folk tale,  there can be no doubt that Pythagoras experimented with the monochord (figure 5), a one-stringed musical instrument with a moveable bridge, used to investigate the principles and problems of tuning theory.
FIGURE 5. THE MONOCHORD. String, sounding box and moveable bridge.
The monochord affords an excellent example of how the primary principles of peras and apeiron underlie the realm of acoustic phenomena. Of course, the fact that numerical proportions underlie musical harmony has become a commonplace since the days of Pythagoras; yet there is something about the perfect beauty of these proportions, and their manifestations in the realm of sound, which will exercise a curious fascination over anyone who chooses to actually investigate them on the monochord.
The problem which the monochord presents is that the string can be divided at any point. The string represents an Indefinite continuum of tonal flux which may be infinitely divided. How, then, is it possible to "create" a musical scale at all? The solution, of course, resides in the limiting power of Number.
A curious phenomenon occurs when a string is plucked. First, the string vibrates as a unit. Then, in two parts, then in three parts, four, and so on. As the string vibrates in smaller parts higher tones are produced, this being the so-called harmonic overtone series.  While they are not as loud as the fundamental tone of the entire string vibrating, with practice the overtones can nonetheless be heard.
Through the power of Limit, the most formal manifestation of which is Number, harmonic nodal points naturally and innately exist on the string, dividing its length in halves, thirds, fourths, and so on, as shown in figure 6. Plucking the string at one end, and simultaneously touching one of the nodal points without the bridge, will produce the corresponding overtone vibration. In this fashion, one can play out the overtone series, as far as is practical. However, dampening the string at any other point will just deaden out the string. (See figure 6.)
The overtone series provides, as it were, the architectural foundation of the musical scale, the basic "field" of which is the octave, 1:2, or the doubling of the vibrational frequency, which inversely correlates with a halving of the string. Returning again to the basic question of how one bridges the tonal flux, we know the answer to be Number, but now we can see more clearly that the problem itself is one of mediation or harmonia, through the medium of numerical proportion or logos. The solution, in fact, can be seen as performing a marriage of opposites, linking together the upper and the lower (I :2), in a truly cosmic fashion, which is to say in a manner partaking of both order and beauty.
While the complete ratios of the scale are set out in Appendix IV, "The Ratios and Formation of the Pythagorean Scale," we shall here note the essentials.
In order to arrive at whole number solutions, we will use the octave of 6:12.
1) The first step is one of arithmetic mediation. To find the arithmetic mean we take the two extremes, add them together, and divide by 2. The result is a vibration of 9, which, in relation to 6, is in the ratio of 2: 3. This is the perfect fifth, the most powerful musical relationship.
2) The second form of mediation is harmonic. It is arrived at by multiplying together the two extremes, doubling the sum, and dividing that result by the sum of the two extremes (i.e., 2AB / A+B). The harmonic mean linking together 6 and 12 then is 8. This proportion, 6:8 or 3:4, is the perfect fourth, which is actually the inverse of the perfect fifth.
3) Through only two operations we have arrived at the foundation of the musical scale, the so-called "musical" or "harmonic" proportion, 6:8 :: 9:12, the discovery of which was attributed to Pythagoras. (See figure 7.)
FIGURE 6. THE HARMONIC NODAL POINTS AND OVERTONE SERIES ON THE MONOCHORD. The above figure illustrates the reciprocal relation which exists between string length and vibrational frequency. By stopping the string at the geometrical nodal points the harmonic overtones may be individually emphasized.
FIGURE 7. THE HARMONIC PROPORTION
This arrangement of the perfect consonances of the octave, fifth and fourth needs to be played out, preferably on the monochord, in order to fully appreciate its significance. While we have not "created" a complete musical scale, we have arrived at the architectural foundation on which it is based. By carefully observing the above arrangement, however, we shall discover enough information to complete the scale.
First of all, it would be well to notice the peculiar form of musical and mathematical "dialectic" which is occurring. That is to say, not only is 6:9 a perfect fifth, but 8: 12 is as well; i.e., 6:9 :: 8: 12. Nor is that all, for while 6:8 is a fourth, so too is 9:12; or, 6:8 :: 9:12. Again, the significance of this harmonic symmetry will be fully realized by playing these relations out.  However, not only are the fourth and fifth manifested in these multiple ways, but the ratio of 8:9 defines the whole tone as well.
The tone having been defined, the final creation of the scale is quite simple. The vibration of the tonic C is increased by the ratio 8:9 to arrive at D. D is increased by 8:9 to arrive at E. Now, if E were increased by that ratio, it would overshoot F; hence there we must stop. The ratio between E and F ends up being 243:256, called in Greek the leimma, or "left over," corresponding to our semi-tone.  Ascending from G, the same 8:9 ratio is used to fill up the remaining intervals. Likewise, the interval between B and C is the leimma.
While the fourth and fifth mediate between the two extremes via harmonic and arithmetic proportion, the scale is filled up through the continued geometrical proportion of 8:9; hence, the geometric mean between C and E would be D. All these forms of proportion interpenetrate, cooperate and harmonize with one another to produce the musical scale.
In summary, we can see the paramount importance of the musical scale and its formation in Pythagorean thought. First of all, the experiments conducted by the Pythagoreans on the monochord confirmed the importance of numerical peras as the limiting factor in the otherwise indefinite realm of manifestation. It also suggested for the first time that if a mathematical harmony underlies the realm of tone and music, that Number may account for other phenomena in the cosmic order -- for example, planetary motion, which was also thought of being related to the mathematical harmonia of the scale, this being the famous "Music of the Spheres."  Moreover, the world is full of beings and phenomena which reflect the harmonic principle of dynamic symmetry present in the musical proportion as well. Through their investigation of musical harmony, the Pythagoreans shifted philosophic inquiry away from the materialistic cosmologies of the earlier Ionic tradition to the consideration of Form, which was now to be seen as constituting the world of First Principles. In addition to shifting emphasis from Matter to Form, the Pythagoreans also discovered the principle of harmonia, the fitting together of the high and the low, the hot and the cold, the moist and the dry. From then on, Health was seen as the perfect harmony of the elements comprising the body, disease as that state in which one of the elements becomes too weak or strong, destroying the proper symmetry of the arrangement. Indeed, it was Alcmaeon of Croton, a young man when Pythagoras was old, who first defined health as "the harmonious mixture of the qualities."  This had an inestimable effect on Hippocratic medicine. As is so apparent in their various cultural achievements, the ancient Greeks had a very special affinity with the principles of Form, Symmetry, and Harmony. The Pythagoreans were the inheritors of this affinity, and helped to articulate these principles in new, important ways which have profoundly influenced the arts and sciences of Western civilization.
THE PYTHAGOREANS PERCEIVED another principle of Number, in addition to seeing it as a formative agent active in nature. This is perhaps best exemplified in the figure of the Tetraktys which, as we might say in the present century, stood as a numerical paradigm of whole systems.
As we have observed, the Pythagoreans were accustomed to arranging numbers in geometrical shapes, and there are a variety of descriptions which have come down to us from antiquity of triangular, square, pentagonal, and other figured numbers and their properties.  This way of representing numbers may have well resulted in the discovery of geometrical theorems. Moreover, the observation that the relations between different types of "geometrical numbers" follow certain definite patterns surely furthered the Pythagorean contention that mathematical study is an important route leading, to the perception of universal laws.
The most well known example of such a "figured number" is the famous Pythagorean Tetraktys ("Quaternary"), consisting of the first four integers arranged in a triangle of ten points:
FIGURE 8. THE TETRAKTYS.
For the Pythagoreans the Tetraktys symbolized the perfection of Number and the elements which comprise it. In one sense it would be proper to say that the Tetraktys symbolize, like the musical scale, a differentiated image of Unity; in the case of the Tetraktys, it is an image of unity starting at One, proceeding through four levels of manifestation, and returning to unity, i.e., Ten. In the sphere of geometry, One represents the point , Two represents the line , Three represents the surface , and Four the tetrahedron , the first three-dimensional form. Hence, in the realm of space the Tetraktys represent the continuity linking the dimensionless point with the manifestation of the first body; the figure of the Tetraktys itself also represents the vertical hierarchy of relation between Unity and emerging Multiplicity. In the realm of music, it will be seen that the Tetraktys also contains the symphonic ratios which underlie the mathematical harmony of the musical scale: 1:2, the octave; 2:3, the perfect fifth; and 3:4, the perfect fourth. 
We might further note that the Tetraktys, being a Triangular number, is composed of consecutive integers, incorporating both the Odd and Even, whereas Square number (Limited) is composed of consecutive odd integers, and Oblong number of consecutive even integers (Indefinite). Since the universe is comprised of peras and apeiron woven together through mathematical harmonia, it is easy to see from these considerations why the Tetraktys, or the Decad, was called Kosmos (world-order), Ouranos (heaven), and Pan (the All). In Pythagorean thought the Tetraktys came to represent an inclusive paradigm of the four-fold pattern which underlies different classes of phenomena, as exemplified by Theon of Smyrna in Appendix 1. Not only does a four-fold pattern underlie each class, but each level is in a certain fashion analogous or proportionately similar with that same level in every other class of phenomena. In many respects Pythagorean philosophy is a philosophy of analogia.
The Pythagoreans, then, were the first to use numerical and geometrical diagrams as models of cosmic wholeness and the celestial order. This use of arithmetic and geometrical paradigms of whole systems has a long and interesting history, extending from antiquity through Medieval times, through the Renaissance, up until the modern era.  If geometrical principles actually shape the phenomena of nature, why not use those same geometrical forms to illustrate the harmonies and symmetries which exist between natural phenomena? This is no doubt the reasoning behind this symbolic usage of number and geometry, and its appeal seems firmly rooted in the human imagination. In fact, it might be argued that such paradigms possess greater merit than more arbitrary typologies insofar that, being based on the principles of natural order, "Pythagorean" models have more intrinsically in common with the phenomena they seek to classify than other typologies which are of merely human invention. Whereas other models sometimes fail, Pythagorean cosmological symbolism seems particularly well suited in showing how parts relate to a larger whole, thus illustrating the principle of unity underlying diversity.
Likewise, the story is told of how Pythagoras was indeed the first man to call himself a philosopher. Others before had called themselves wise (sophos), but Pythagoras was the first to call himself a philosopher, literally a lover of wisdom.
More importantly, for Pythagoras and his followers philosophy was not merely an intellectual pursuit, but a way of life, the aim of which was the assimilation to God. Even in the days of Plato the surviving Pythagoreans were noted for their distinctive bios Pythagorikos, or Pythagorean way of life, as Plato puts it in the Republic (600a-b).
The school of Pythagoras in Croton appears to have been a religious society centered around the Muses, the goddesses of learning and culture, and their leader Apollo.  Iamblichus' description of the school gives it something of a monastic flavor, and there was indeed a "rule" of life, but while the Pythagoreans gathered together at certain times of the day, most of them did not live together.
Apparently there were different levels within the school. One group, the akousmatikoi or "auditors" (from the verb akouo, to hear), went through a three year probationary period and were limited mainly to hearing lectures. A more advanced group, the mathematikoi or "students," went through a five year period of "silence,"  and held their property in common whereas the akousmatikoi did not; there is, however, nothing to indicate that the mathematikoi took anything like a vow of poverty. Rather, their property was managed by certain members of the society -- the politikoi -- and they received an adequate subsistence in return for its use. 
Pythagoras himself was heavily influenced by Orphism, an esoteric, private religion of ancient Greece, named after the legendary musician Orpheus, "the founder of initiations," which also featured a distinctive way of life. According to Orphism, the soul, a divine spark of Dionysus, is bound to the body (soma) as to a tomb (sema). Mankind is in a state of forgetfulness of its true, spiritual nature. The soul is immortal, but descends into the realm of generation, being bound to the "hard and deeply- grievous circle" of incarnations,  until it is released through a series of purifications and rites, regaining its true nature as a divine being.
Pythagoras fully accepted the Orphic belief in transmigration or "reincarnation" -- in fact, he is said to have possessed the power to remember his previous lives, and the ability to remind his associates of theirs as well. Yet while Pythagoreanism remains closely related to the Orphic thought of the period,  the clearly distinguishing factor between the two is that for the Pythagoreans liberation from the wheel is obtained not through religious rite, but through philosophy, the contemplation of first principles. Hence, philosophia is a form of purification, a way to immortality. As others have observed, whereas the Eleusinian mysteries offered a single revelation, and Orphism a religious way of life, Pythagoras offered a way of life based on philosophy. Burnett notes that this conception lies at the heart of Plato's Phaedo. itself "dedicated, as it were, to a Pythagorean community at Phlious";  moreover, "This way of regarding philosophy is henceforth characteristic of the best Greek thought." 
One may well ask how assimilation to God is possible through philosophy. The answer is to be found in the nature of man:
Man, by comprising a world-order in miniature, contains all of those principles constituting the greater cosmos, of which he is a reflection, including the powers of divinity. The problem is not so much of becoming divine as becoming aware of the divine, universal principles within. It is this end, primarily, toward which the Pythagorean curriculum was focused. Plato alludes to the Pythagorean theory of philosophy in the Republic (500c) when he observes:
Man realizes the divine by knowing the universal and divine principles which constitute the cosmos -- i.e., for the Pythagoreans, Number. To know the cosmos is to seek and know the divine element within, and one must become divine and harmonized since only like can know like. From this perspective it also becomes obvious that philosophy is nothing other, at least in one respect, than the care of the soul.
ACCORDING TO SEVERAL ancient sources, it was from the Pythagoreans that Plato received his doctrine of the tripartite soul, a doctrine which underlies Pythagoras' parable of the three lives: one group of humanity is covetous, another ambitious, and the other curious. As J.L. Stocks has pointed out, "What the division specifies is the three typical motives of human action, and all three motives will be found in operation at different times in every normal human soul."  These motives are the desire for profit, honor, and knowledge.
Plato, apparently in line with the Pythagorean tradition, divides the soul into three parts: one part is reasoning, another part is "spirited," and the last desires the pleasures of nutrition and generation. Unlike certain schools of modern psychology, the Platonic division of the soul is hierarchical: the reasoning part is superior to the other two, and deserves more attention, for it is this dimension of the soul which makes us uniquely human. We might summarize the relation between the levels of the soul and their attendant virtues, or forms of excellence, as shown in figure 9.
FIGURE 9. THE THREE LIVES
Seen in this perspective, it becomes plain that psychic health must result when the three "parts" of the soul are brought into a state of harmony, which is not to say a state of equality. Rather, this state of balance could be seen as a state of attunement, where each part receives what it is due. Psychic disturbance results when each part of the soul tries to go its own separate way; the psyche then becomes a house divided, resulting in dissociation and fragmentation, as opposed to the realization of psychic wholeness.
The grand project behind Plato's Republic is to define the nature of justice. We know that the Pythagoreans identified justice with proportion, especially geometrical proportion, because it is through proportion that "each part receives what it is due."  Following the Pythagorean tradition, Plato observes that in the realm of society justice exists when each part of society receives its due, and is able to achieve the function for which it is truly best suited. Justice, as a universal principle, operates in exactly the same fashion in the realm of the soul. There, "justice is produced in the soul, like health in the body, by establishing the elements concerned in their natural relations of control and subordination, whereas injustice is like disease and means that this natural order is inverted."  As Plato notes, in a magnificently Pythagorean passage:
If Pythagorean philosophy, then, constitutes a care of the soul, of what precisely is that care comprised? The answer to this is to be found in the ethical and educational conceptions of the Pythagoreans, as well as in those special pursuits and studies for which they were renowned.
WE HAVE SEEN that for Pythagoras philosophy represents a "purification," the aim of which is the assimilation to God. The universe is divine because of its order (kosmos), and the harmonies and symmetries which it contains and reflects. These principles make the universe divine for they are the characteristics of divinity, and they also innately subsist within the human soul. The Pythagoreans taught that the soul is a harmony.  If we are to become like God, then according to Pythagorean philosophy the soul must become aware of its harmonic origin, structure and content. Since the source of all harmony and order is the divine principle of Number, we can perhaps come to understand the initially enigmatic statement of Heracleides that, according to Pythagoras, true "happiness consists in knowledge of the perfection of the numbers of the soul." 
In the realm of epistemology the presence of Number is most evident: progress in rational thought depends on a fundamentally dyadic relationship between knower and known, subject and object. Moreover, as certainly as the principle of polarity underlies the world of phenomenal manifestation, so too does the mind depend on dualistic typologies, such as the Table of Opposites, in order to make intellectual progress.  Knowledge itself is the third, harmonic element which conjoins the two poles of subject and object. Knowledge then is unifying, much like the harmonic ratios of the musical scale or the central circle in figure 1. Moreover, as we shall see, to the Pythagoreans the knowledge of divine harmony can be either abstract or experiential or, indeed, both.
Even more immediately evident is the undeniable influence of Number on our psychic state through the medium of music, depending as it does on numerical proportion. Certain musical proportions express a sense of cheerfulness; others, such as the minor third, possess a bittersweet quality that can make us sad. The fact that Number can influence a person's emotional state is indeed mysterious and points toward a dimension of qualitative Number which transcends the merely quantitative.
Related to the question of music and harmony is the principle of resonance: two strings, tuned to the same frequency, will both vibrate if only one is plucked, the unplucked string resonating in sympathy with the first. This, of course, is accomplished through the medium of the vibrating air, but the principle underlying the phenomenon is one of harmonic attunement. If, as the Pythagoreans held, man is a microcosm, and the soul is a harmony, perhaps it is through a form of resonance that we relate so intensely to the archetypal ratios of musical proportion.  Moreover, by experientially investigating and employing the principles of harmony in the external world, one comes to understand and activate those same principles within. This idea in fact underlies the Pythagorean approach to mathematical study.
The Pythagoreans divided the study of Number into four branches which may be analyzed in the following fashion:
Plato, of course, was heavily influenced by the Pythagorean study of number and incorporates the above quadrivium into his own educational curriculum set out in the Republic, adding another branch of study, Stereometry, the investigation of Number in three-dimensional space, which probably relates to the regular "Platonic" solids and other polyhedra. For Plato -- who believed that God geometrizes always  -- "geometry is the knowledge of the eternally existent,"  and the emphasis that he placed on the study is well known from the legendary inscription above the Academy door, "Let no one ignorant of geometry enter here. " 
Within the Platonic curriculum, the purpose of mathematical studies is to purify the eye of the intellect, for mathematical studies have the propensity "to draw the soul towards truth and to direct upwards the philosophic intelligence which is now wrongly turned earthwards."  Number, for Plato, is a transcendent Form to which we must intellectually ascend. For the earlier Pythagoreans, however, the emphasis was clearly on the immanence of Number.
While the Pythagoreans moved the direction of philosophical inquiry from the realm of matter to that of Form and principles, Plato took this movement even further than his predecessors. For Plato mathematical studies are a preparation for the contemplation of divine principles; for the Pythagoreans, mathematical studies are the contemplation of divine principles. As Cornelia de Vogel has lucidly observed, for the Pythagoreans
This particular difference between the earlier Pythagoreans and Plato must have manifested itself in the sphere of praxis. For Plato it was, in a sense, best to pursue mathematical contemplation with as little reference to physical objects as possible: truth must be approached through intellect, and through intellect alone. For the Pythagoreans, truth manifests itself through the world of physical phenomena; for example, the Pythagoreans no doubt felt that through experimentation on the monochord one could experience the divine principles of harmony which underlie the structure of the cosmos.
The differing views between Plato and the earlier Pythagoreans can also be seen in the realm of music. Plato refers to different musical modes throughout his writings, and to the negative effects that some forms of music can have on the soul and on society. The Pythagoreans, however, actually used certain forms of music to pacify and harmonize the psychic state. In the same way that the music of Orpheus enchanted the wild beasts of the field, so too did the Pythagoreans use music to quell and harmonize the irrational passions.
While the Pythagoreans placed emphasis on the immanence of divine Number and Harmonia, they certainly did not ignore the transcendental dimension. This is made clear by their emphasis on peras and apeiron, the elements of Number, which they obviously took to be universal principles of the first order. It seems that, rather than focusing exclusively on either the immanent or transcendent levels of being, the Pythagoreans were intent on unifying all levels of human experience through the principles of harmony. The divine harmony can be grasped through the mind, yet can also be perceived through the senses. The experiential perception of harmony through the senses can lead to its intellectual apprehension. By means of theoria or contemplation the universal and abstract principles of harmony may be perceived, but through praxis they may be felt in the soul, itself a harmonic entity. Yet there is another level, that of therapeia, where harmonic principles can be used to effect changes in the psychic disposition.
Through the use of proper music, diet, and exercise, the early Pythagoreans sought to nurture and maintain the natural harmony of the psychic and somatic faculties. According to Iamblichus, "They took solitary morning walks to places which happened to be appropriately quiet, to temples or groves, or other suitable places. They thought it inadvisable to converse with anyone until they had gained inner serenity, focusing their reasoning powers. They considered it turbulent to mingle in a crowd as soon as they rose from bed, and that is the reason why these Pythagoreans always selected the most sacred spots to walk."  All of these practices can be seen as a form of philosophic "purification" (catharsis) or "practice"(praxis), designed to regulate the body and the emotions. On the intellectual and psychic levels, through their study of mathematics and the natural world, the Pythagoreans approached the principles of harmony experientially through the study of harmonics on the monochord and through geometrical constructions. The Pythagoreans also pursued the study of purely abstract mathematics.
Recalling that the end of all of these pursuits was to follow God, it is interesting to briefly contrast the Pythagorean approach to divinisation with the Christian mysticism of the late Hellenistic period and thereafter. The first stage of "the mystical ascent" consists of the ethical purification of the soul commonly known as praxis. The second stage is contemplation or theoria; in Christianity, however, the contemplation of Nature and universal principles, so characteristic of the Greek philosophic tradition, is replaced predominately by the contemplation of scripture. The final stage, theosis, is the union of the mystic with God. In Hellenistic Christian mysticism of the late antique world, however, the first two stages lose virtually all significance when the final stage is reached. Catharsis and theoria are merely the steps of a ladder; when the summit is reached, the ladder is oftentimes kicked away.
The ancient Pythagorean approach to divinisation would have never sanctioned kicking away the ladder. In early Pythagorean thought there nowhere appears an earnest desire to escape from the world. True, like the Orphics, the Pythagoreans believed in reincarnation, and looked upon the body as limiting the soul. But even so, there is no firm evidence that, like the Orphics, the Pythagoreans sought exemption from the Wheel of Generation. Rather than transcend the world, Pythagorean religiosity held as its goal to exist within the cosmos in a state of emotional repose and intellectual acuteness. Man, while possessing a soul which clearly transcends the limitations of the body, the realm of time and space, is nonetheless a reflection of the entire universe, a microcosm, and is linked together with nature, other living beings, and the Gods through harmony, justice, and proportion. The Pythagorean goal is not to leave the divinely beautiful cosmos behind for a realm of transcendent harmony, but rather to become aware of, and enhance the function of, transcendent harmony in the natural, psychological and social orders. 
THE CENTRAL INSIGHTS of the Pythagoreans concerning the significance of harmonia were applied to political theory as well: in the same way that harmonic proportion underlies the health of the balanced soul, so too does the principle of justice underlie the living structure of a healthy state. In this regard Plato's Republic, which is a study of justice in both the psychic and social realms, appears to be firmly based on earlier Pythagorean conceptions. Plato, via analogia, identifies the three parts of the soul with three different parts of society, and shows how both the soul and society attain their peak of excellence when "each part receives its due" and when each of the three parts fulfils the particular function for which it is best adapted. It is not possible to say whether Plato's tripartite division of society corresponds precisely with an earlier Pythagorean division, although it is known that the Pythagoreans identified justice with proportion, of which they viewed geometrical proportion as being the most perfect. Like Plato the early Pythagoreans were aristocrats, in the authentic sense of the word, believing that the best government will be composed of those best qualified to govern, as opposed to other political systems in which leadership is based on wealth, on power, or on the choice of the populace. Finally, it should be noted that the Pythagoreans were the first philosophical school to concern themselves with such social and political questions, which fell outside the natural philosophy of the earlier Ionian tradition.
As for Pythagorean ethics, little needs to be said, as the entire idea of "proper action" is tied up with the ideas of philosophy as a way of life, and the nature of the soul and the cosmos. As one discovers the structure and nature of the soul, and experiences and begins to understand the principles of harmony, it seems inevitable that such insight will leave a mark on one's personal conduct and dealings with others. Nonetheless, as the writings in this volume demonstrate, the Pythagoreans did not hesitate to make use of aphorisms and other explicit ethical teachings. At all events, however, it will be seen that these teachings reflect and spring from the more universal insights of Pythagorean thought. In short, because each part is linked to the whole through harmonia, every action has its repercussions, either beneficial or not, for which the individual is supremely responsible.
TO SURVEY THE INFLUENCE of Pythagorean thought would expand this introductory essay beyond reasonable boundaries, insofar as the basic conceptions of the Pythagoreans have influenced a long line of thinkers from antiquity reaching up until the present day. Nonetheless, something should be said about the thought of the early Pythagoreans and the Neopythagoreans of the first centuries B.C.E. and C.E., thus helping to place some of the writings here assembled into their proper context.
As the biographical accounts in this volume show, the downfall of the original Pythagorean school had much to do with anti-aristocratic sentiments amongst the populace of south Italy. A revolt was led against the Pythagoreans by Cylon in 500 B.C.E. -- some say because he was rejected admission into the school -- and a period of unrest followed. During the revolt led by Cylon, or during another revolt which followed, various meeting houses were attacked and a good number of Pythagoreans may have perished in the flames. This final attack seems to have been rather successful, and those Pythagoreans that remained alive seem to have migrated to mainland Greece with the exception of Archytas at Tarentum.
Unfortunately, the details concerning the attack on the school are sketchy and little more can be said than the above. The Pythagoreans did, however, carry on in mainland Greece where centers were established at Phlious and Thebes. Echecrates went to Phlious, Xenophilus went to Athens, and the names of Lysis and Philolaus are associated with Thebes, and it was there that Philolaus taught Simmias and Cebes who appear as characters in Plato's Phaedo. Philolaus, who was born around 474 B.C.E., was the first Pythagorean to actually record the teachings of the school in writing; hence his fragments, which are collected in this volume, possess an exceptional value.
Archytas (first half of the fourth century B.C.E.), who was the general of Tarentum and one of the Pythagorean mathematikoi like Philolaus, made contributions to mathematics, geometry, and harmonic theory. He was visited by Plato in 388 B.C.E. and it is possible that Archytas was Plato's model for the so-called "philosopher king."
This brings us to a discussion of Plato himself (428-348 B.C.E.) which is not a topic of minor significance for, as W.K.C. Guthrie has observed, "In general the separation of early Pythagoreanism from the teaching of Plato is one of the historian's most difficult tasks, to which he can scarcely avoid bringing a subjective bias of his own. If later Pythagoreanism was coloured by Platonic influences, it is equally undeniable that Plato himself was deeply affected by earlier Pythagorean belief." 
Many important Pythagorean influences have already been noted on the thought of Plato and perhaps it would be fair to view Plato as the most important Pythagorean thinker in the history of the West. There was quite a bit of interest in Pythagorean thought in the early Academy as well, and it has been suggested that the idea of the Academy was in part due to the inspiration of the earlier Pythagorean school. Whatever the case, some points of contact include the tripartite division of the soul;  Plato's usage of the One and the Indefinite Dyad;  the theory of education in the Republic;  the identification of the One and the Good in "the unwritten doctrine" referred to by Aristotle;  the Pythagorean character of Plato's lecture "On the Good" reported on by Aristotle; the idea that the soul of the philosopher attains order by contemplating those things which possess order in nature;  the idea that "the goodness of anything is due to order and arrangement";  the idea that various beings are linked together through geometrical equality;  a doctrine of idea-numbers in the dogmata agrapha reported by Aristotle;  and various examples of Pythagorean musical symbolism. 
Following Plato in the leadership of the Academy was his nephew Speusippus (407-339 B.C.E.) who was also quite interested in Pythagorean thought: he suggested that there exists a One above being (an important teaching of later Neopythagorean and Neoplatonic thought), and also wrote a work On Pythagorean Numbers about the Tetraktys and numbers comprising the Decad.  This treatise was based on the writings of Philolaus and an interesting fragment of it survives.
Aristotle showed an interest in the Pythagorean school and even wrote an essay On the Pythagoreans which does not survive. One of his students, Aristoxenus of Tarentum, was a music theorist and was in touch with the last surviving generation of Pythagoreans at Phlious. Aristoxenus, who might have been one of the Pythagorean mathematikoi, possessed an antiquarian interest in the school and wrote a biography of Pythagoras which is quoted from by Porphyry, Iamblichus and Diogenes Laertius.
After the time of Aristotle we are left with an uncomfortable gap in the history of Pythagorean thought until the Neopythagorean revival commencing in the first century B.CE. Yet it is precisely during this period that most of the Pythagorean ethical and political tractates contained in the second section of this volume were probably composed. But the question remains, by whom were they written? And why?
Unfortunately, no one is certain even about the date or location of their composition. There now exists a tendency to see these writings as being somewhat earlier than previously thought, and Holger Thesleff suggests that the bulk of them were composed around the third century B.C.E. It will be noted that these writings are attributed to original members of the Pythagorean school, which in fact is actually not the case. This does not mean that these writings are "forgeries" in the modern sense of the word, for it was a fairly common practice in antiquity to publish writings as pseudepigrapha, attributing them to earlier, more-renowned individuals. It was probably out of reverence for their master -- and also perhaps because they were discussing authoritative school traditions -- that certain Pythagoreans who published writings attributed them directly to Pythagoras himself. Even Pythagoras is said to have attributed some poems of his to Orpheus. Other examples which might be cited include the many Jewish pseudepigrapha of the time, Orphic fragments, the Hermetic writings, and even a number of Pauline epistles from the New Testament.
A careful study of these writings will show that they are deeply imbued with many Pythagorean ideas - what, in fact, could be more Pythagorean than comparing the structure of the family or society to a well-tuned lyre? -- a particularly beautiful and useful simile which appears more than once in the Pythagorica here collected. Yet, alongside the central Pythagorean core of these writings are found strong Academic and Peripatetic influences as well. Thesleff is probably correct in suggesting that these writings were composed as philosophical textbooks for laymen,  but it is unlikely that the exact date or locale of their composition will ever be decisively settled. However, as has been suggested, a careful study of these texts might well provide for some valuable insights into the thought of the early Academy.
The next phase of Pythagorean thought involves the so-called Neopythagorean revival at the beginning of the common era. Due to the Hellenization of the ancient world stemming from the conquests of Alexander the Great, interest in Greek philosophy was no longer limited to one small part of the world. This was especially true of the interest in Pythagorean and Platonic thought, and the names of many Pythagorean philosophers are known from the Hellenistic age. Unfortunately, for some of the most important thinkers the information concerning them is quite fragmentary, which is perhaps one reason why no one has attempted the kind of full scale study that the topic deserves: the study of the Neopythagorean thought of this period is not only significant for its own sake, but also for understanding the thought of Plotinus and the later Neoplatonists who were influenced by a range of Neopythagorean ideas. Actually, as John Dillon has succinctly observed, during this period "Middle Platonism" and "Neopythagoreanism" existed as something of a continuous tradition, with Neopythagoreanism representing "an attitude that might be taken up within Platonism."  Keeping this in mind, it might be useful to mention some of the thinkers during this period who were influenced by Pythagorean thought, for it is only through such a listing that one can get a true feeling for the creative ferment of this period.
The first word that we have concerning a renewed interest in Pythagorean thought comes from Cicero, regarding his friend Nigidius Figulus (98-45 B.C.E.), who was attempting to revive Neopythagoreanism in Rome. It appears however that Nigidius was less interested in abstract philosophy than in integrating astrological, ritual and a variety of occultist beliefs.
Eudorus of Alexandria (fl. 30 B.C.E.) appears to have been influenced by Pythagorean thought, and attempted to show that the Pythagoreans held that a Supreme Principle, the One, existed above the Monad and the Dyad. Whether or not the earlier Pythagoreans actually held such a belief is another question altogether, but the notion of a transcendent One surpassing the principles of Limited and Unlimited is important for the history of later philosophy.
Philo of Alexandria (20 B.C.E.-40 C.E.), a Hellenized Jew who interpreted Jewish scripture in light of Greek philosophy, shows a deep interest in Pythagorean thought, especially arithmology, in his voluminous writings. Philo was primarily a Platonist who subscribed to a emanationist cosmology which he tried to reconcile with Jewish thought, but it is interesting to note that he was referred to by Clement of Alexandria, an early church father, simply as "Philo the Pythagorean." 
We should not fail to mention Apollonius of Tyana, a colorful figure who flourished during the first half of the first century C.E. Apollonius was perhaps more of a Pythagorean wonderworking ascetic than a philosopher, who travelled through the ancient world as something of a pagan missionary, meeting with priests, performing marvels, and restituting cults of worship to their former purity. His life and exploits are chronicled in an entertaining historical novel by Philostratus. Whether or not this biography gives a well-rounded picture of Apollonius remains uncertain. It does seem, however, that Apollonius saw himself as a reincarnation of Pythagoras and also possessed much information on the life of the sage, which he used in compiling a biography, subsequently used by Porphyry and Iamblichus. In addition to the Life of Philostratus, several letters attributed to Apollonius are extant. 
Often overlooked as an important witness to Pythagorean thought is Plutarch of Chaeronea (45-125 C.E.), well known for his famous Lives. What is not so generally well known is that Plutarch was a Platonist with Neopythagorean leanings and was also a priest of Apollo at Delphi. In certain writings of his Moralia he displays a keen interest in the interpretation of myth, the religio-philosophical esoterism of the time, and various bits of Neopythagorean lore, including arithmology. His writings remain a vital resource for understanding the profound, inner dimensions of the contemporary spiritual universe and, like Plotinus, he refers to the Pythagorean interpretation of the name Apollo, which equates Apollo with the One (a = not; pollon = of many). 
About the same time as Plutarch we have Moderatus of Gades (fl. second half of the first century) who has been termed an "aggressive Pythagorean" for the severe criticism he applied to Plato, accusing him of using ideas of Pythagoras without giving proper credit where credit was due. In his cosmology Moderatus taught the existence of three unities: the first and highest, the One above being which he identified with the Good; secondly, a unified, active logos, identified with the intelligible realm; and thirdly, the realm of soul. Needless to say, the resemblance between these ideas and those of Plotinus are quite striking.
Theon of Smyrna brings us into the second century (fl. circa 125 C.E.). He was a Platonist and wrote a work Mathematics Useful for Understanding Plato, of which a good English translation exists, and which is equally useful for understanding aspects of Pythagorean thought.  In addition to discussing the principles of arithmetic, harmonics and astronomy, Theon also treats the symbolism of the first ten integers and various forms of the Tetraktys (see Appendix I). The work also dealt with the principles of geometry, but this section no longer survives.
Working in a similar vein, and not much later, was Nicomachus of Gerasa (active 140-150 C.E.), whose Introduction to Arithmetic,  translated into Latin by Apuleius and Boethius, remained a definitive handbook up until the Renaissance. Also surviving is Nichomachus' Manual of Harmonics, and fragments of his Theology of Arithmetic, a work on Pythagorean arithmology. He also wrote a Life of Pythagoras which was used by the later biographers and an Introduction to Geometry which did not survive. He is known to have been familiar with the practice of gematria,  and it has been suggested that Iamblichus' "Pythagorean encyclopedia" found its inspiration in the wide-ranging works of this scholar.
Numenius of Apamea in Syria (fl. 160 C.E.) was also a Platonist with Neopythagorean leanings. Some fragments of his works remain, but the majority have perished. He wrote On the Good; On the Indestructibility of the Soul; On the Secret Doctrines of Plato; a work called Hoopoe, after the bird of the same name; On Numbers (perhaps a work on arithmology); On Place; and On the Divergence of the Academics from Plato. He had an associate, Cronius, who flourished about the same date and wrote a work On Reincarnation.
With Numenius, the mixture of Middle Platonic and Neopythagorean thought begins to transform itself into Neoplatonism, of which philosophy the most brilliant and beautiful expositor was Plotinus. Plotinus seems to have been influenced to a certain extent by the thought of Numenius, and John Dillon sees Plotinus' direct teacher Ammonius Saccas (fl. circa 230 C.E.) as a being a Platonist of a strongly Neopythagorean cast.
The main feature which differentiates Neoplatonism from Middle Platonism is the Neoplatonic doctrine of the transcendent absolute, the One, which exists above the realm of Being. However, as we have seen, Plato's nephew Speusippus, Eudorus of Alexandria, and Maderatus of Gades all posited the existence of such a transcendent principle; even Plato himself, in the Republic, suggests that the Good. which he identified with the One, exists above being.  As is typical, the "clear cut" distinctions between various schools and periods are not always so sharp as one has been led to believe. This is especially true of the distinctions between Middle Platonism and Neopythagoreanism in the period outlined above.
In addition to the doctrine of the One above being, Plotinus (204-269 CE.) also held that the intelligible realm, which he identified with nous or Mind, exists as a unity-diversity, as a differentiated "image" of the One. Hence, this world of Forms, which contains all the laws and principles of the universe, can be seen as the living union of the Monad and the Indefinite Dyad, with the Monad acting as the limiting and form-giving principle in the realm of nous, while the Indefinite Dyad acts as the "intelligible matter" upon which the Monad acts. The Indefinite Dyad also provides for the element of Infinity which allows for the existence of an unlimited number of forms and souls in the realm of Mind.
Henceforward, Pythagorean ideas played an important role in subsequent Neoplatonic thought. Porphyry (233-305 C.E.), Plotinus' successor, wrote a biography of Pythagoras which was part of his History of Philosophy in Ten Books. Iamblichus (d. circa 330), who was a student of Porphyry and thought of himself as something of a full-fledged Pythagorean, undertook the task of writing a multivolume "Pythagorean encyclopedia" which included The life of Pythagoras here reproduced and a number of other works: The Exhortation to Philosophy, On the Common Mathematical Science, Commentary on Nichomachus' Introduction to Arithmetic, three books On the Natural, Ethical and Divine Conceptions which are Perceived in the Science of Numbers (of which the anonymous Theology of Arithmetic is based on the third book); and three lost works on Pythagorean harmonies, geometry and astronomy, bringing the total number of volumes up to 10, the Pythagorean Perfect Number.
The Neopythagorean component of Neoplatonism did not end with Iamblichus but rather continued through to the closing of the school in Athens, only to resurface in Renaissance Florence with the thinkers associated with the Cosimo de' Medici's Platonic Academy, of which Marsilio Ficino was the head.
This is not to imply that only pagan thinkers were followers of Pythagorean thought. On the contrary, many of the early church fathers held Pythagoras and his teachings in high esteem. Not only that, it became quite fashionable, after the manner of Philo, to enlist the help of Pythagorean number symbolism in the interpretation of scripture. Justin Martyr (100-164 C.E.) was rejected by a Pythagorean teacher on account of his inadequate mathematical knowledge (recalling the words engraved above the Academy door); he turned to Platonism, and then to Christianity, but never gave up his admiration for the Greek sages. Clement of Alexandria (fl. circa 200 CE.) was also an admirer of Hellenistic thought and even applied the ratios of the Harmonic Proportion to the exegesis of the holy writ.  Augustine (354-430), heavily influenced by Neoplatonism, also loved to indulge in numerical exegesis, and he was instrumental in helping to transmit an interest in number symbolism to the Middle Ages.
Also important in the transmission of the Pythagorean ideas to the following age were the pagan encyclopediasts of the late antique world. Macrobius (first part of the fifth century) discussed Pythagorean thought in his Commentary on the Dream of Scipio, and Martianus Capella (fl. 410-429) in his allegorical work on the seven liberal arts, The Marriage of Philology and Mercury, discussed arithmology in Book VII. Another important source for the medievals was Boethius (480-525), especially his works On Arithmetic and On Music.
Having arrived at the close of the ancient world we have also arrived at the end of our survey. Pythagorean ideas continued to be transmitted in the work of Christian thinkers and applied in the realm of sacred architecture by groups of medieval masons. Insofar as Pythagorean thought had been Christianized, it had been changed, yet nonetheless many important conceptions -- such as the ideas of celestial harmony and the significance of Number as a cosmic paradigm -- remained unaltered. There was a brief and beautiful Renaissance of Pythagorean thought at the cathedral school of Chartres in France during the 12th century, due in part to a Latin translation of Plato's Timaeus, and of course there was a renewed interest in Pythagorean thought with the rediscovery of classical writings during the rebirth of learning in Renaissance Italy.
IT SHOULD COME AS NO SURPRISE that the figure of Pythagoras appealed strongly to those scholars of Renaissance Italy, as he has invariably appealed to the more universal thinkers of every age. The appeal lies in his important and influential conceptions, which in many ways seem to directly reveal important principles existing at the sacred "root of Nature's fount," but also in the man's character, for Pythagoras himself seems to well represent the possibility of an integrated approach to the study of Nature as a philosophical way of life.
As we have seen, the central focus of Pythagorean thought is in many respects placed on the principle of harmonia. The Universe is One, but the phenomenal realm is a differentiated image of this unity -- the world is a unity in multiplicity. What maintains the unity of the whole, even though it consists of many parts, is the hierarchical principle of harmony, the logos of relation, which enables every part to have its place in the fabric of the all.
Because of Pythagoras' approach, integrating mathematics, psychology, ethics, and political philosophy into one comprehensive whole, it would be quite inappropriate to end this essay without devoting a few words to the contemporary significance of the Pythagorean approach. Pythagoras would have never wished that his insights remain the focus of a merely antiquarian interest, and so we shall honor his intentions by inquiring into what value Pythagorean thought might possess in the contemporary world, addressing these matters in quite general terms.
Pythagoras, no doubt, would have disapproved of the radical split which occurred between the sciences and philosophy during the 17th century "enlightenment" and which haunts the intellectual and social fabric of Western civilization to this day. In retrospect perhaps we can see that man is most happily at home in the universe as long as he can relate his experiences to both the universal and the particular, the eternal and the temporal levels of being.
Natural science takes an Aristotelian approach to the universe, delighting in the multiplicity of the phenomenal web. It is concerned with the individual parts as opposed to the whole, and its method is one of particularizing the universal. Natural science attempts to quantify the universal, through the reduction of living form and qualitative relations to mathematical and statistical formulations based on the classification of material artifacts.
By contrast, natural philosophy is primarily Platonic in that it is concerned with the whole as opposed to the part. Realizing that all things are essentially related to certain eternal forms and principles, the approach of the natural philosopher strives to understand the relation that the particular has with the universal. Through the language of natural philosophy, and through the Pythagorean approach to whole systems, it is possible to relate the temporal with the eternal and to know the organic relation between multiplicity and unity.
If the scientific spirit is seen as a desire to study the universe in its totality, it will be seen that both approaches are complementary and necessary in scientific inquiry, for an inclusive cosmology must be equally at home in dealing with the part or the whole. The great scientists of Western civilization -- Kepler, Copernicus, Newton, Einstein, and those before and after -- were able to combine both approaches in a valuable and fruitful way.
It is interesting that the split between science and philosophy coincides roughly with the industrial revolution -- for once freed from the philosophical element, which anchors scientific inquiry to the whole of life and human values, science ceases to be science in a traditional sense, and is transformed into a servile nursemaid of technology, the development and employment of mechanization. Now machines are quite useful as long as they are subservient to human good, in all the ramifications of that word -- but as it turned out, the industrial revolution also coincided with a mechanistic conceptualization of the natural order, which sought to increase material profit at the expense of the human spirit. This era, which gave rise to the nightmare of the modern factory -- William Blake's "dark satanic mills" -- gained its strength through the naive premise that the human spirit might be elevated and perfected through the agency of the machine.
Today, in many circles, to a large part fueled by the desire for economic reward, science has nearly become confused with and subservient to technology, and from this perspective it might be said that the ideal of a universal or inclusive science has been lost. This is because the ideal scientist is also a natural philosopher who is interested in relating his discoveries to a larger universal framework, whereas the dull-minded technologist, if he has any interest in universal principles at all, limits that interest to their specific mechanistic applications rather than their intrinsic worth of study. Yet, those who study universal principles as principles-in-themselves, often find that these principles have many applications in a wide variety of fields.
While Pythagoras would have taken a dim view of this artificial and dangerous split between science and philosophy, the negative consequences of this rupture have not gone unnoticed. Yet with his emphasis on the unity of all life, Pythagoras would have been in an excellent position to foresee the negative consequences: ecological imbalance, materialism, the varied effects of personal greed, the disintegration of human values, the decline of the arts, a lack of interest in personal excellence and achievement. In a sense these problems, not necessarily unique to this age, result from a lack of balance and an ability to see the parts in relation to the whole. As the poet Francis Thompson said, "You cannot move a flower without troubling a star," and so it is with every individual and collective action. Pythagoras correctly observed that all things are linked together proportionately, by justice, harmony -- call it what you will. By cultivating an awareness of harmonic forming principles and working within the bounds set by necessity, mankind possesses the potential to become a sacred steward of the earth and co-creator with Nature; but the inevitable corollary is that humanity also has every power to create and inhabit a hell of its own making. The simple fact remains that the scales of justice are inexorable -- it is a principle of Nature, and not merely of human morals, that each should receive his due. If we poison our rivers, we poison ourselves; if we act in stupidity, it is only appropriate that we suffer the consequences. If there is a moral to the story it is simply that individuals and societies are far less likely to run into trouble should they possess an awareness of these principles and relationships. And if one would like to cultivate the innate human ability to see things as they are, in whole-part relations, there is scarcely a better guide than the Pythagorean sciences. There has been much talk among the avant garde of "whole systems," the "philosophy of holism," etc., but few have realized that it is actually Pythagoras who is the tutelary genius and founder of the philosophy of whole systems.
We have mentioned the split between science and philosophy because it is an easy and self-evident example. Yet Pythagoras would also have something to say about the structure of our educational system as well. It has become fashionable to create ever more specialized disciplines -- a Ph.D. thesis is considered proportionately better the fewer the number of people that can understand it. This is not to imply that specialized knowledge lacks value, but rather to say that a danger exists in the self- inflicted alienation of academia and the sciences. Great things cannot fail to happen when minds get together and one mind fertilizes another -- when disciplines inspire one another. Pythagoras would say that, from the standpoint of natural philosophy, a superfluous multiplicity of facts and compartmentalized data is useless in a higher sense unless one can determine their relation to the whole, or the universal patterns which underlie all creation.
Interestingly, it is the modern-day physicists who have come most closely to approximating Pythagorean conceptions. Hell-bent on proving the mechanistic notions of 18th-century materialism, physicists have discovered that the deeper they push into matter the more it looks like the cosmos of the Pythagoreans and Platonists. Each atom is a Pythagorean universe, the sight of eternity in a grain of sand, consisting of an arithmetic number of particles, geometrically distributed in space, dancing and vibrating like a miniature solar system to the music of the spheres. A modern physicist would have little difficulty comprehending the teaching of the Orphic theologians that "the essence of the Gods" -- that is to say the formative principles -- "is defined by Number." 
Matter and energy are but different aspects of one, underlying continuum. Advancing to the subatomic level, quantity becomes quality, energy becomes information. Many physicists, proceeding from particulars to universals, are now on the verge of recognizing the essential truth of the statement, common to all spiritual traditions, that "Through consciousness the universe is but one single thing; all is interdependent with all."  The science of physics, proceeding from matter to energy, from energy to intelligence (i.e., pattern, logos), and from intelligence to Nous, has all but discovered the deus absconditus of the alchemists, the God hidden in matter. If the alchemical poeticism be allowed, even matter, if properly tortured, slain and resurrected, contains the innate potentiality of revealing the Hermetic mercury of eternal being. With the atomic accelerator at their disposal, modern physicists indeed have the capability to change lead into gold. They have taught the world that flesh, coal and diamond are made of the same basic stuff (carbon), driving home the reality that soul and Form is the essential component of all things.
One important Pythagorean insight which possesses ramifications for both the sciences and human behavior is the observation that the phenomenal universe is a mixture, a synthesis of Limited and Unlimited elements. Plato, drawing upon this notion in the Timaeus, compares the limited world of the Forms to a father, and the unlimited Receptacle of Space to a mother, the "nurse of becoming" as he puts it.  From their conjunction is born an offspring: the visible, phenomenal universe, the world of eternal principles manifesting in time and space.
The significance of this observation lies in the fact that it paints a picture of the phenomenal realm existing as a manifestation of what might be called "ordered chaos" -- we exist in an intermediate realm. Platonism, which posits the existence of an extratemporal and extraspatial world of perfect form, recognizes that the universe in which we live mirrors this perfection in an imperfect way. Hence Plato notes in the Timaeus that the receptacle of becoming, which we inhabit, was initially "filled with powers that were neither alike nor evenly balanced."  The receptacle might be compared to a sea, in which various currents provide for an ambient randomness. Stated in the terms of contemporary physics, one might observe that in the intelligible realm light, as principle, travels in a perfectly straight line, while in the realm of manifestation its path -- and the fabric of space itself -- is warped to a degree by gravitational mass.
While the Pythagoreans identified the principle of Limit with the Good, it should also be observed that without the principle of the Unlimited all manifestation would be impossible.  Moreover, working in conjunction with its partner, the principle of Unlimitedness is equally responsible for the organic beauty of the phenomenal realm. All trees of the same species more-or-less follow the same laws of growth, but at each juncture of growth there exists an indefinite number of possibilities. It is precisely the unlimited element which makes for the beauty of a forest, which would be much less beautiful if each tree were exactly the same. A musical composition also relies on order and randomness (change); should either element come to predominate it ceases to be beautiful.
Whereas the Platonism before the Renaissance possessed a tendency to focus on the transcendent world of forms, the first Pythagoreans seemed to concentrate more on the incarnate manifestations of universal principles. After all, not only did they study harmonia as a universal principle, seeing it reflected on all levels of the beautiful cosmos, they incorporated the principle into the fabric of their daily lives as well. But already with Aristotle we find a lack of insight into the Pythagorean view -- he cannot understand how the Pythagoreans view Number as possessing "magnitude."  This could well represent a misunderstanding of the incarnationalist dimension of Pythagorean thought, and perhaps reveals an overly literalist interpretation on Aristotle's part as well.
The Pythagorean view of the universe as a living, harmonic mixture is not only indispensable as a scientific concept, but it beautifully articulates the position of man in the cosmos as well. If, along with Plato, we view time as a moving image of eternity,  then each generation of humanity stands poised between the present moment and the timeless immensity of the eternal. Rather than being a worthless speck meaninglessly situated in the infinite expanse of space, each person, according to the Pythagorean view, is a microcosm, a complete image of the entire cosmos, with one foot located in the realm of eternal principles and the other foot rooted in a particular world of manifestation. Poised as he is between time and eternity, matter and spirit, man possesses an incredible freedom to learn, create and know, limited only by those principles on which creation is based. From this vantage point, humanity is engaged in a never-ceasing dialectic between time and eternity, possessing the ability to incarnate eternal principles in time (and in this sense mirror the creative work of Nature), yet also possessing the ability to elevate the particular to the universal through conscious understanding.
In relation to this theme, one final observation is in order: the creative endeavors of humanity seem to attain their peak of excellence precisely at that point when the intermediate nature of humanity is actively recognized. For with this recognition comes the realization that one must actively integrate the particular and universal aspects of being. Hence, the best science will once again embrace its sister, philosophia: both deal with universal principles and particular phenomena -- and together they will not attempt to build up a system of thought either from "the top down," deduced from purely a priori formulations, nor will they dare to start exclusively from "the bottom up" abstracting observations only from particular phenomena while ignoring universal principles. This approach I believe is fundamentally Pythagorean: the harmonic proportion, according to legend discovered by Pythagoras, exists as a purely universal principle, but it would have never been discovered without empirical experimentation on the monochord. The value of the harmonic proportion lies in both its universal nature as well as the significance and usefulness of its particular applications. Through the creative dialectic between the temporal and the eternal, there necessarily occurs a form of integration between otherwise purely theoretic and pragmatic approaches. Another benefit of this realization -- the realization that all things are composed of constants and variables -- is that, if seriously embraced, it actively encourages honest inquiry, rendering the twin dangers of Fundamentalism and Relativism equally impotent, for universal justice has its own means of dealing with individuals who mistakenly believe that they possess the Absolute Truth, or, conversely, think that "everything is relative."
One final point needs to be made about the Pythagorean approach, and that concerns the topic of value. There are many occasions where it is useful to take a divisive approach to Nature for the purposes of abstract analysis, yet there are also times when it becomes expedient to stress the unity of all being. For Aristotle Number was merely an abstraction as opposed to an innately existing a priori principle, so it is easy to see how he might become confused. by the notion that abstract number possesses "magnitude." Yet, if Number acts as a geometrical forming principle in the sphere of natural phenomena, as some of the studies cited in the bibliography abundantly demonstrate, it seems unwise to deny its immanent efficacy. Likewise, Aristotle was equally bewildered by the Pythagorean symbolism which equated certain archetypal number forms with principles such as "justice." Yet the truth of the matter is that it is precisely through the Pythagorean approach that quantity (number) and quality are discovered to be integrally related. As Ernst Levy has pointed out in an important article, "The Pythagorean Concept of Measure," this is shown to be especially true in the realm of music where each tone is actually a number, yet also a qualitative phenomenon possessing value.  It is particularly true in the realms of music and what has been called "sacred geometry" that one can gain insight into the Pythagorean conception of Number as both creative paradigm and qualitative relation. Levy suggests that in order to once again benefit from a unified scientific and philosophical synthesis that "a new mental attitude is required which many among us will be reluctant to assume, because it is contrary to the scientifically determined mind. The definition of that attitude is simple enough. It consists in this, that we must be willing to ascribe equal reality and equal importance to quality and quantity."  It is worth observing that the Pythagorean approach, while realizing the necessity of employing antithetical pairs of opposites in a conceptual sense, always arrives at a position which emphasizes the unity of the all.
To conclude that Number, in the most Pythagorean sense of the term, and the cosmos itself possesses a dimension of meaning is, within the context of mechanistic "science" or modern reductionistic "philosophy," perhaps the ultimate heresy; yet, for the traditional scientist and philosopher such a realization is only the starting point. If Pythagoras had but one imperative for the present age -- or any age -- it would be, as F.M. Cornford has suggested, this:
Seek truth and beauty together; you will never find them apart. 
-- DAVID R. FlDELER
References to writings appearing in this volume follow Guthrie's chapter divisions.
1) It will not be my intention in this introductory essay to attempt to pierce through the various mysteries surrounding the figure of Pythagoras or to embark on the overwhelming task of textual criticism. Nor do I desire to write much about the life of Pythagoras, seeing that all of the primary source material is presented in this volume. Rather, I will attempt to briefly sketch out the history of the Pythagorean school and its influence, discussing those doctrines which are generally agreed upon, and to provide some type of context into which the writings of this sourcebook may be placed by the general reader. Indeed, I have tried to keep the general reader in mind throughout the introduction, and have limited more specialized comments to these notes, which also include references for further reading on topics which can only be touched upon here.
2) The source for this is Ion of Chios, quoted by Diogenes Laertius, Life of Pythagoras, chapter 5. For a discussion of Pythagorean Orphica see West, The Orphic Poems, Oxford University Press, 1984, 7- 15.
3) For Plato's views on writing about matters of ultimate concern see his Seventh Letter.
4) Since Pythagoras left no writings, this presents the historian with some difficulties. What, in fact, can safely be attributed to Pythagoras? Moreover, what do we know about his life? At a very early date a body of legends grew up around Pythagoras; many of these beautiful and amusing stories are recorded in the biographies, which constitute the first part of this book. For a long time, due to the nearly miraculous accounts, certain scholars dismissed the biographies as "late" and "unreliable." However, Aristotle in his lost monograph On the Pythagoreans emphasized how Pythagoras was seen at two places at once, how he showed his golden thigh, how he was thought to be the Hyperborean Apollo, and how he was addressed by a certain river. Obviously these stories are not "late Neopythagorean inventions" but go back to the time of Plato or before. Another source of these accounts was The Tripod of Andron of Ephesus, who was roughly contemporary with Aristotle. While the interpretive dimension of Iamblichus' biography is certainly colored by later Neopythagorean and Neoplatonic influence, it is now taken for granted that the biographies contain a great deal of early information about Pythagoras and his school, and much of the information is taken from older authorities whose work has since perished. Some of the ealiest authorities include Timaeus of Tauromenium (circa 352-256 B.C.E.) who wrote a History of Sicily which contained information of the Pythagoreans and the speeches of Pythagoras, and Dicaearchus of Messina (fourth century B.C.E.), a pupil of Aristotle who wrote a comprehensive study of Greek history which also treated the Pythagoreans. Another student of Aristotle, Aristoxenus of Tarentum, wrote several works on the Pythagoreans, used by the later biographers, which drew on early sources and his first-hand contact with members of the Pythagorean school.
5) Porphyry, Life of Pythagoras, chapter 12.
6) Iamblichus, Life of Pythagoras, chapter 5. For all we know, Pythagoras may have been invited to go to Croton. While this, to my knowledge, has not been previously suggested, it seems unlikely that he would have moved his teaching activities to a distant city without having some contact with and knowledge of the inhabitants. If this is correct, it would help explain his rapid acceptance by the populace.
7) See Vogel, Pythagoras and Early Pythagoreanism, for an analysis of these speeches. Iamblichus' source for these is Timaeus of Tauromenium.
8) Theon of Smyrna, Mathematics Useful for Understanding Plato, 12.
9) This concept -- that the One or principle of unity is the source of all numbers -- is easily grasped if one envisions "the One" as a circle in which various polygons are inscribed; the polygons, or numbers contained within the circle, may then be seen as various manifest aspects of the underlying unity. Another analogy is found in Pythagorean harmonics: the monochord, symbolizing unity (1/1), innately contains the entire overtone series (2/1, 3/1, 4/1, 5/1, etc.), which is manifested when the string is plucked.
10) Theon of Smyrna, Mathematics Useful for Understanding Plato, 66.
11) Ibid, 66.
12) Cornford, "Science and Mysticism in the Pythagorean Tradition," part 2, 3.
14) Aristotle, Physics 203 a 10.
15) Plato, Timaeus 35b f.
16) See Levin, The Harmonics of Nichomachus and the Pythagorean Tradition, chapter 6.
17) Different musical instruments emphasize different overtones. For example, the clarinet emphasizes the odd numbered overtones, thus accounting for its peculiar timbre.
18) This relation has been defined by Flora R. Levin in her Harmonics of Nicomachus and the Pythagorean Tradition, 1, as the "metaphysical octave," the characteristic feature of which is "a perfect fusion of parts (3:4 and 2:3) into a whole (2: 1)." Richard L. Crocker in his article "Pythagorean Mathematics and Music," 330, observes: "This construction, dividing as it does the first consonance by the second and third in a curious, interlocking way, has every right to be called the harmony. Here the inner affinity of whole-number arithmetic and music finds its most congenial expression."
19) The leimma is the excess of the fourth over the double tone: 4/3 : 9/8 x 9/8 = 4/3 x 64/81 = 256/243.
20) Needless to say, the "Music of the Spheres" is one of the most influential conceptions of Pythagoras. For a complete discussion which deals with its long and interesting history, as well as the significance of the concept as a poetic fact, see Joscelyn Godwin's The Harmonies of Heaven and Earth.
21) Alcmaeon of Croton in Freeman, Ancilla to the Pre-Socratic Philosophers, 41 (24 DK 4).
22) For more on figured numbers and their properties, see Heath, A History of Greek Mathematics, vol. 1, chapter 3, "Pythagorean Arithmetic." Also see the first volume of Ivor Thomas' Greek Mathematical Fragments.
23) Relating to the musical ratios of the Tetraktys is the Pythagorean saying "What is the Oracle at Delphi?" The answer is: "The Tetraktys, the very thing which is the Harmony of the Sirens" (Iamblichus, Life of Pythagoras, chapter 18). Nicomachus of Gerasa also identifies the harmonic ratio of 6:8: :9: 12 as a version of the Tetraktys in his Manual of Harmonics, vii, 10.
24) One modern proponent of this approach was Buckrninster Fuller. Some important sources for the study of this approach and geometrical forming principles include Keith Critchlow's Order in Space and Islamic Patterns, Ghyka's The Geometry of Art and Life, and The Geometrical Basis of Natural Structure by Robert Williams.
25) Iamblichus, The Life of Pythagoras, chapter 12.
26) Pythagoras was thought to be associated in some way with the God Apollo. This is only natural since Apollo is related to the celestial principles of harmonic order and logos, these being also the principles with which Pythagoras was most concerned. This connection is even made plain by the name of the philosopher -- for Pythios is the name of Apollo at Delphi, his most sacred shrine from which his oracles were delivered. This obvious etymological connection led Diogenes Laertius to interpret the name as meaning that he spoke (agoreuein) the truth no less than Apollo (Pythios) at Delphi.
27) This period of silence may have only been a ritual matter required during the religious ceremonies of the society, not during everyday life. "The ceremonies are conducted by Pythagoras behind a veil or curtain. Those who have passed this five-year test may pass behind the curtain and see him face to face during the ceremonies; the others must merely listen." Minar, "Pythagorean Communism," 39.
28) This is Minar's analysis in "Pythagorean Communism."
29) From an Orphic gold funerary plate, translated in Freeman, Ancilla to the Pre-Socratic Philosophers, 5. (l DK 18)
30) A fragment quoted by Iamblichus maintains that the nature of the Gods is related to Number (Iamblichus, Life of Pythagoras, chapter 28), and there was even an Orphic "Hymn to Number," portions of which are found in Kern, Orphicorum Fragmenta, Berlin, Weidemann, 1922.
Cameron, in his important study of Pythagorean thought observes that harmonia in Pythagorean thought inevitably possesses a religious dimension. He goes on to note that both harmonia -- there is no "h" in the Greek spelling -- and arithmos appear to be descended from the single root ar. This seems to "indicate that somewhere in the unrecorded past, the Number religion, which dealt in concepts of harmony or attunement, made itself felt in Greek lands. And it is probable that the religious element belonged to the arithmos-harmonia combination in prehistoric times, for we find that ritus in Latin comes from the same Indo-European root." (Alister Cameron, The Pythagorean Background of Recollection, 26.)
31) Burnet, Early Greek Philosophy, 83.
33) The Life of Pythagoras preserved by Photius, chapter 15.
34) Stocks, "Plato and the Tripartite Soul," 210-11.
35) For a good discussion of the Pythagorean view of justice as proportion see John Robinson, An Introduction to Early Greek Philosophy, 81-83.
36) Plato, Republic 444d (Comford translation, 143).
37) Plato, Republic 443d f. (Cornford translation, 141-2).
38) For sources and a discussion of the soul as a harmonia see Guthrie, A History of Early Greek Philosophy, vol. 1, 307 f.
39) Quoted by Clement of Alexandria, Stromateis ii, 84.
40) Certain scholars, appealing primarily to the "Table of Opposites," have argued that the orientation of Pythagoreanism was essentially dualistic. Such a simplistic view overlooks the fact that every philosophical system employs dualistic typologies and that "as a religious philosophy, Pythagoreanism unquestionably attached central importance to the idea of unity, in particular the unity of all life, divine, human, and animal, implied in the scheme of transmigration." (F.M. Cornford, Plato and Parmenides, 4.)
41) After writing this section I came across the observation of Vilctor Goldschmidt: "Our capacity to apprehend the outside world may be explained thus, that there are processes in our mind (microcosm) which are analogous to those in nature. These psychological processes we call natural laws. " (Quoted by Ernst Levy, "The Pythagorean Concept of Measure," 53.)
42) Heath, A History of Greek Mathematics, vol. 1, 10.
43) Plato, Republic 527b (Cornford translation, p. 244).
44) Tzetzes, Chiliad, viii. 972.
45) Plato, Republic 527b (Cornford translation, p. 244).
46) Vogel, Pythagoras and Early Pythagoreanism, 197.
47) Iamblichus, Life of Pythagoras, chapter 21.
48) Within this context it might be noted that Pythagorean metaphysics has, from ancient Greece to the present day, had an influence on the arts. The lesser can lead to the greater, and natural beauty, which is relative, can lead to the apprehension of transcendent Beauty which is absolute. This is, of course, a Platonic sentiment, but it is foreshadowed in the structure of Pythagorean thought. Perhaps, however, it would be more accurate in Pythagorean thought, with its immanent metaphysics, to suggest that the universal is realized through the particular. The use of mathematical and geometrical harmonies in sacred architecture, for example, can lead to the perception of, and resonance with, universal harmony. For example, through the medium of "Pythagorean geometry," a sacred edifice has the potential to become a celestial mediator: through the harmonic nature of its structure, the heavenly principles of harmonic form are reflected on earth; yet through the effect that this harmony has on those that are receptive to its beauty, the particular may be exalted to perceive the universal. This two- fold principle is applicable not only in architecture, however, but indeed in all the arts.
49) Guthrie, A History of Early Greek Philosophy, vol. 1, 170.
50) Plato, Republic 434d -44lc; also see J.L. Stocks, "Plato's Tripartite Soul."
51) Aristotle, Metaphysics i 6, 987 a 29 f.
52) Plato, Republic, particularly 398c -403c and 52lc -531c.
53) This was stated in Plato's lecture "On the Good." See the note below.
54) Aristotle enjoyed telling a story about Plato's lecture "On the Good": "Everyone went there with the idea that he would be put in the way of getting one or other of the things in human life which are usually accounted good, such as Riches, Health, Strength, or, generally, any extraordinary gift of fortune. But when they found that Plato discoursed about mathematics, arithmetic, geometry, and astronomy, and finally declared the One to be the Good, no wonder they were altogether taken by surprise; insomuch that in the end some of the audience were inclined to scoff at the whole thing, while others objected to it altogether." (Aristoxenus, Harmonica ii ad init., quoted by Heath, A History of Greek Mathematics, vol. 1, 24.)
55) Republic 500c.
56) Gorgias 506e.
57) Gorgias 508a.
58) Aristotle, Metaphysics xiii 7; see also the discussion in Dillon and Vogel (below).
59) For Pythagorean musical symbolism in Plato see Ernest McClain, The Pythagorean Plato. For more on similarities and differences between Plato and the Pythagoreans, and the characteristics of post-Platonic Pythagoreanism, see the valuable discussion in chapter 8 of Vogel's Pythagoras and Early Pythagoreanism. For a brief discussion of the dogmata agrapha and the early Academy see Dillon, The Middle Platonists, chapter 1.
60) A translation of the fragment appears in volume I of Ivor Thomas' Greek Mathematical Fragments. Dillon gives a succinct overview of the thought of Speusippus in chapter 1 of The Middle Platonists; Taran's Speusippus of Athens is a comprehensive study of the remaining Greek fragments and the thought of Speusippus.
61) Holger Thesleff, An Introduction to the Pythagorean Writings of the Hellenistic Period, 72.
62) Dillon in A.H. Armstrong, ed., Classical Mediterranean Spirituality, NY. Crossroad, 1987. 226.
63) Clement of Alexandria, Stromateis, i, 15.
64) Philostratus' Life of Apollonius of Tyana, which includes the letters, appears in the Loeb Classical Library.
65) For the equation of Apollo with the One see Plutarch, Moralia 270f, 393c and 436b. For his Pythagorizing tendencies see "On the Mysteries of Isis and Osiris" and his Delphic essays, collected in volume 5 of Plutarch's Moralia, Harvard. 1936.
66) Theon of Smyrna, Mathematics Useful for Understanding Plato, translated by Robert and Deborah Lawlor, San Diego, Wizards Bookshelf, 1979.
67) Nicomachus of Gerasa, Introduction to Arithmetic, translated by M.L. D'Ooge. New York, MacMillan, 1926.
68) Dillon, The Middle Platonists, 359. Each letter of the Greek alphabet possesses a numerical value and hence each word may also be represented as a number. The science of gematria involves the conscious use of this numerical symbolism and is not to be confused with either arithmology or numerology. A well known example is found in the name of the Gnostic divinity Abraxas, the numerical value of which is 365, the number of days in a solar year. The Babylonian divinities were represented by whole numbers and a Babylonian clay tablet indicates that Sargon II (fl. 720 B.C.E.) ordered that the wall of Khorsabad be constructed to have a length of 16,283 cubits, the numerical value of his name. The present writer has researched the history of gematria in some depth and has discovered an extremely strong body of evidence that gematria was utilized in Greece prior to the Hellenistic period.
69) Plato, Republic 509b.
70) In book ix, chapter II of the Stromateis, which deals with "The Mystical Meanings in the Proportions of Number, Geometrical Ratios, and Music." Clement also refers to the sage from Samos as "Pythagoras the great." (Stromateis i, 21.)
71) Iamblichus, The Life of Pythagoras, chapter 28.
72) R.A. Schwaller de Lubicz, Nature Word, West Stockbridge, MA, Lindisfarne Press, 1982, 99.
73) Plato, Timaeus 49.
74) Plato, Timaeus 52e.
75) Plato's nephew Speusippus thought that it was inappropriate to apply ethical values to the principles of peras and apeiron.
76) Aristotle, Metaphysics 1080 b 16; 1080 b 31; 1083 b 9; and 1090 a 20.
77) Plato, Timaeus 37d. Plato adds that time is a moving image of eternity "according to number."
78) Ernest Levy, "The Pythagorean Concept of Measure," 53.
80) Francis M. Cornford, "The Harmony of the Spheres." 27.