Proclus who headed the Neoplatonic school of Athens during the fifth century AD, wrote a summary of the history of Greek mathematics known as the Eudemian Summary. It is believed that his main source was a prior history authored by Eudemus, one of Aristotle's pupil's, from the third century BC. In his summary, which was part of a commentary on the first book of Euclid, he clearly defines the birth of geometry and the traditional succession of Thales and Pythagoras:
"Since it behooves us to examine the beginnings both of the arts and of the sciences with reference to the present cycle of the universe, we say that according to most accounts geometry was first discovered among the Egyptians, taking its origin from the measurement of areas. For they found it necessary by reason of the rising of the Nile, which wiped out everybody's proper boundaries. Nor is there anything surprising in that the discovery both of this and of the other sciences should have its origin in a practical need, since everything which is in process of becoming progresses from the imperfect to the perfect. Thus the transition from perception to reasoning and from reasoning to understanding is natural. Just as exact knowledge of numbers received its origin among the Phoenicians, by reason of trade and contracts, even so geometry was discovered among the Egyptians for the aforesaid reason.
Thales was the first to go to Egypt and bring back to Greece this study; he himself discovered many propositions, and disclosed the underlying principles of many others to his successors, in some cases his method being more general, in others more empirical. After him Amerstius, the brother of the poet Stesichorus, is mentioned as having touched the study of geometry, and Hippias of Elis spoke of him as having acquired a reputation for geometry. After this Pythagoras transformed this study into the form of a liberal education, examining its principles from the beginning and tracking down the theorems immaterially and intellectually; he it was who discovered the theory of proportionals and the construction of the cosmic figures."
The theory of proportions encompasses the fundamental discovery of ratio in music, mathematics and geometry and will be further enlarged upon below. The three cosmic figures attributed to Pythagoras are the cube, the pyramid and the dodecahedron. To these three, the octahedron and icosahedron were later added by Theaetetus.41 All five came to be known as the Platonic solids or figures and were distinguished by their exact circumscription within a perfect sphere. In Book XI of the Elements, Euclid defines them as follows: (fig. 3)
Definition 25: A cube is a solid figure contained by six equal squares.
Definition 12: A pyramid is a solid figure, contained by planes, which is constructed from one plane to one point. (This open ended definition allows for the construction of two shapes with either square or triangular bases.)
Definition 26: An octahedron is a solid figure contained by eight equal and equilateral triangles.
Definition 27: An icosahedron is a solid figure contained by twenty equal and equilateral triangles.
Pythagoras was from Samos, a Greek island in the Peloponnese off what was then the coast of Ionia (modern Turkey). Historians believe he was born about 560 BC and lived some eighty years (480-490 BC) Like Thales he is thought to have traveled extensively throughout Egypt and Babylonia at an early age, and later, to the other islands of the Peloponnese. In either 530 or 520 BC, he left Samos for Kroton in southern Italy, at that time part of Magna Graecia. There in Kroton, he founded a powerful religious and philosophical sect whose influence extended to neighboring Greek cities of southern Italy and Sicily. In Kroton, the ruling patriarchy came to view the sect as a threat and, in 500 BC, forced Pythagoras and his followers to move to Metapontum, farther up the coast in the heel of Italy, where he died about 480 BC. Later, widespread political upheaval, during which the Pythagoreans were persecuted, forced the sect to the Greek mainland. During this period the group split into two opposing camps, the 'listeners' and more advanced initiates known as 'scientists-mathematicians'. The former embraced Pythagorean teachings as dogma, while the latter believed in an evolving interpretation of Pythagorean thought. The sect is believed to have subsisted in Phleius and Tarentum until the middle of the fourth century BC.
Two later Greek historians who did much to foster Pythagoras' influence were Iamblichus and Porphyry. Iamblichus authored On the Pythagorean Life, a colorful account of Pythagoras and his followers which was preceded by Porphyry's shorter biography. Born in Syria in the mid-third century AD, Iamblichus is said to have come from a landowning family that had the means to educate him in Antioch and Alexandria. He later worked in Sicily or Rome, where he came under the influence of Porphyry (about 232-303 A.D), a pupil of Plotinus who had written a Life of Pythagoras and an anti-Christian polemic. Both men were instrumental to a Greek revival which sought to challenge the increasing hegemony of Christianity in the third century. Although historians today cast a great deal of doubt upon the reliability of these two authors, there is little question that their writings shaped early scholastic views concerning the development of mathematics.
The main problem with Iamblichus is that he often contradicts himself and is at times inconsistent with the facts. However, from a modem perspective, this is not reason enough to cast doubt over the whole of his account, for a great deal of his biography is indeed culled from very valuable first sources.44
Iamblichus is much clearer than Proclus about the association between Thales and Pythagoras:
"Thales had helped him [Pythagoras] in many ways, especially in making good use of time. For this reason he had renounced wine, meat and even earlier large meals, and had adjusted to light and adjustable foods. So he needed little sleep, and achieved alertness, clarity of soul, and perfect and unshakable health of body. Then he sailed on to Sidon, aware that it was his birthplace, and correctly supposing that crossing to Egypt would be easier from there. In Sidon he met the descendants of Mochos the natural philosopher and prophet [who was believed to have originated Atomism], and the other Phoenician hierophants [priest and keeper of sacred mysteries], and as initiated in all the rites peculiar to Byblos, Tyre and other districts of Syria. He did not, as one might unthinkingly suppose, undergo this experience from superstition, but far more from a passionate desire for knowledge, and as a precaution lest something worth learning should elude him by being kept secret in the mysteries or rituals of the gods. Besides, he had learnt that the Syrian rites were offshoots of those of Egypt, and hoped to share, in Egypt, in mysteries of purer form, more beautiful and more divine. Awestruck, as his teacher Thales had promised, he crossed without delay to Egypt, conveyed by Egyptian seamen who had made a timely landing on the shore below mount Carmel in Phoenicia,... "
Iamblichus then tells of how the sailors thought they might draw some benefit from their passenger by selling him into slavery. However, Pythagoras avoids this fate through correct composure and disposition. He continues:
"From there [Egypt] he visited all the sanctuaries, making detailed investigations with the utmost zeal. The priests and the prophets he met responded with admiration and affection, and he learned from them most diligently all that they had to teach. He neglected no doctrine valued in his time, no man renowned for understanding, no rite honored in any region, no place where he expected to find some wonder. So he visited all the priests, profited form each one particular wisdom. He spent twenty-two years in the sacred places of Egypt, studying astronomy and geometry, and being initiated - but not just on impulse or as the occasion offered - into all the rites of the gods, until he was captured by the expedition of Kampyses and taken to Babylon. There he spent time with the Magi, to their mutual rejoicing, learning what was holy among them, acquiring perfect knowledge of the worship of the gods and reaching the heights of their mathematics and music and other disciplines. He spent twelve more years with them, and returned to Samos, aged by now about fifty-six."
Upon returning to Samos, Pythagoras acquired his first disciple and resumed his travels around the Greek Islands and the mainland seeking knowledge in Delos, Sparta and Crete. At some point, under circumstances related by Iamblichus, he had a revelation:
"He was once engaged in intense thought about whether he could find some precise scientific instrument to assist the sense of hearing, as compass and ruler and the measurement of angles assist the sight and scales and weights and measures assist touch. Providently, he walked passed a smithy, and heard the hammers beating out the iron on the anvil. They gave out a melody of sounds, harmonious except for one pair. He recognized in them the consonance of octave, fifth and fourth, and saw that what lay between the fourth and fifth was in itself discordant, but was essential to fill out the greater of the intervals."
This led him, Iamblichus writes, to a set of experiments with sound and instruments: "And thus he discovered the sequence from lowest to highest note which proceeds by a kind of natural necessity in the diatonic scale. He also articulated the chromatic and enharmonic scales from the diatonic..."
Pythagoras thereby discovered the relationship between musical harmony and mathematical ratio, which he then introduced into geometry. Reportedly, Pythagoras stretched a string over a straight edge, dividing it into twelve parts. On pressing the string at given intervals or shortening the string from 12 to 6 parts, or from 12 to 8 or 9 parts, he obtained tones that were consecutively an octave, a fifth and a fourth higher. He also found that these tones invariably matched the ratios of 2- to-1, 3-to-2 and 4-to-3. This discovery also led Pythagoras to the definition of three kinds of mean: arithmetic, harmonic and geometric.
What was so significant about this? An arithmetic mean, commonly known as an average, is a simple enough idea. It is the difference in two values, the combination divided by two, or mid-point. More important, it is also a point of equality. With an harmonic mean, by contrast, the point of equality is expressed through a common ratio. (fig. 4).
The mean was therefore seen as a means to bring opposing ends into harmony. Similarly, on a grander scale, the cosmic order resides in constituent elements that are also in dynamic order. Pythagoras understood this as an universal ordering principle whose mathematical expression was the very essence of existence. The mean as a point of equality and harmony was his fundamental insight to the logic of geometry. By exploring ratio, equalities are established and the underlying dynamic of geometrical structure is revealed.
This notion was best expressed many years later, in a dialogue of Eratosthenes, a major contributor to mathematics and geometry of the Alexandrian era. In The Platonicus, an imagined conversation between two mathematicians, Pappus and Theon, Pappus is given to say:
"Proportionality is composed from ratio and equality is the origin of all ratios. Geometric mediety (mean) indeed has its first origin in equality; it established itself and also the other medieties (means). It shows us, as says divine Plato, that proportionality is the source of all harmonies and ordered existence...."50
At some point, Iamblichus says, Pythagoras decided that his civic duties on Samos interfered with his practice of philosophy:
"[So] he left for Italy, resolved to take as his homeland a country fertile in people who were well-disposed to learning. On his first visit to the famous city of Kroton, he made many disciples (it is reported that he had there six hundred people who were not only inspired to study his philosophy, but actually become 'coenobites' according to his instructions. These were the students of philosophy: the majority were listeners, whom the Pythagoreans call 'acousmatic.'"
By this time, Pythagoras and the Pythagoreans had a central doctrine based on the principles outlined above and supported by various teachings and an ascetic way of life. Central to this ideology was the tetraktys, the initial number series 1,2, 3, 4. This series was arranged in the form of an equilateral triangle of dots which added to the number 10. 1 stood for point, 2 for line, 3 for plane and 4 for solid. The numbers totaled 10, a divine, universal number. In other words, God was number and the universe consequentially rational. The first four numbers were seen to generate the harmonic ratios, linking music with the cosmos, and thus also generating the music of the spheres.
While Iamblichus provides the most complete biography of the Pythagorean movement, it is a prior author of the 1st Century AD, Nichomachus of Gerasa, who most thoroughly summarizes Pythagorean mathematics. Originally from Geresa in Palestine, Nichomachus is thought to have written several books, including an introduction to geometry, a work on astronomy and possibly a Life of Pythagoras that may have been a reference for Iamblichus and Porphyry a few centuries later. None of these have survived. What has is his Manual on Harmonics, a highly Pythagorean work, and his Introduction to Mathematics, which gives a complete explanation of Pythagorean arithmetic and its principles of number, proportion and ratio.
The second book's passages about Pythagorean number theory make abundantly clear how the early Greeks achieved the insights necessary to develop geometry. They devised a new number system which expressed numbers in a highly visual and physical form, through the arrangement of dots. (fig. 5)
While 1 is expressed as a unit, and 2 as two units, three is a triangle, four is a square, five is a pentagon, six a hexagon, seven a septagon, eight an octagon. Nine is another square, and ten is a large equilateral triangle with a three-unit side. From this, it followed that higher numbers, those above ten, could be grouped by shapes derived from the original series. Thus, there were square numbers, triangular numbers and pentagonal numbers. This same sequencing could also be used to express a third dimension, in which numbers represented cubes, pyramids and other solids.
With these essentially geometrical identities came a host of symbolic meanings that very quickly became bases for a highly evolved symbolic system, or numerology. This began with the numbers 1 and 2, which were not understood to be numbers per say but expressions of being. One symbolized sameness and unity, 2 represented opposition and divisibility. True numbering commenced only with 3, for 3 had a beginning, middle and end.
Thus, mathematics was a preexisting pattern in the mind of the creator, who in the act of creation reflected this pattern. Of course, with such associations, God or mind were associated with the monad, as were Apollo and the sun. The dyad such was associated to certain properties such as equality and deficiency, and the deities Zeus, Artemis and Aphrodite. The triad was the marriage number, through the combination of the odd and the even, masculinity and femininity. This, too, was also the number for Hecate and Athena. Four, the tetrad, reflected the musical harmonies and was associated with Hermes and Dionysus. The pentad, 5, connoted the heavens and was the number for justice. Six, the hexad, was a perfect number, being the product of 2 and 3, the odd and even and its own denominators. The sphere and the soul shared its qualities. Seven, an odd prime number, neither generating or generated, was associated with virginity, Athena and the Moon. Eight was unlucky. Nine connoted boundary, because after it all numbers repeated themselves. As an expression of cube (3x3 x3) it was associated with Rhea-Cybele, as well as Hera, Prometheus and Ares. Which brings us to ten.
The decad, embracing all numbers and all numerical forms, was deemed the perfect number. It was the sum of the tetrad elements (1 + 2 + 3 + 4Ñ10) and represented the tetraktys. The Pythagoreans viewed it as the All, the Cosmic, the Universal, Sun, Memory and unending God. It was the number they swore by. Needless to say, the wealth of associations often contradicted each other and were subject to mutation and permutation.
"He no longer used musical instruments or songs to create order in himself: through some unutterable, almost inconceivable likeness to the gods, his hearing and his mind were intent upon the celestial harmonies of the cosmos. It seemed as if he alone could hear and understand the universal harmony and music of the spheres and of the stars which move within them, uttering a song more complete and satisfying than any human melody, composed of subtly varied sounds of motion and speeds and sizes and positions, organized in a logical and harmonious relation to each other, and achieving a melodious circuit of subtle and exceptional beauty."
Hence, the harmonic ratios governed the music of the spheres and governed astronomical theory. This theory held that the Earth, the Moon and the stars, which were perfect spheres, along with the planets rotated around a central, invisible fire. Moreover, all distances were understood to have a harmonic proportion or ratio. Both these ideas would greatly influence scientific thinking many centuries later.
The Pythagoreans believed in reincarnation, the transmigration of souls. They did not eat meat or practice animal sacrifice. They also, on entering the brotherhood, gave up all private effects and held only communal property. The brotherhood also included a sisterhood of wives and independent women. Iamblichus counts the original Kroton members at 218 men and 17 women.
Initiation was a lengthy process:
"The person he [Pythagoras] had examined was sent away and ignored for three years, to test his constancy and his genuine love of learning, and to see whether he had the right attitude and reputation and was able to despise status. After this, he imposed a five year silence on his adherents, to test their self-control: control of the tongue, he thought, is the most difficult type of self-control, a truth made apparent to us by those who established the mysteries. During this time each one's property was held in common, entrusted to the particular students who were called 'civil servants' and who managed the finances and made the rules. If the candidates were found worthy to share in the teachings, judging by their life and general principles, then after the five year silence they joined the inner circle: now within the veil, they could both hear and see Pythagoras.
Before this they were outside the veil: they never saw Pythagoras and shared his discourses only through hearing, and their character was tested over a long period. If one failed the test, he was given double his property and his fellow-hearers... built a grave-mound for him as if he were dead."
Thus, secrecy was of paramount importance to the Pythagoreans. Unfortunately, it resulted in later historians' great reluctance to attribute anything to Pythagoras or his followers. But secrecy was a practice commonly associated with oral traditions that sought to impart sacred truth, whether in Greece or elsewhere in the ancient world.
A final passage from On the Pythagorean Life suggests what the Pythagoreans may have been sensitive about:
"The first man to reveal the nature of symmetry and asymmetry to those unworthy to share the teachings was so much detested, they say, that not only was he excluded from their common life and meals, but they built him a tomb, as if their former companion had left human life behind. Some say the supernatural power took revenge on those who published Pythagoras' teachings. The man who revealed the construction of the 'twenty-angle shape' was drowned at sea like a blasphemer. (He told how to make a dodecahedron, one of the 'five solid figures', into a sphere). Some say his fate befell the man who told about irrationality and incommensurability."
This particular passage, as well as other Greek fragments, point to a certain crisis in the school. It is hard to precisely determine when and how this came about, but clearly at some point this new cosmology was severely shaken. The problem emerged with the examination of the square and the geometric mean.
The objective of the Pythagoreans and of Greek mathematics was to define an ordered cosmos, a rational universe, where everything was in harmony and relationships among elements were expressible in ratios. But there were certain instances where mathematical relationships defied ordinary ratios.
In order to exemplify the problem it is first necessary to turn our attention to two other points. The first is that while geometry evolved into an extremely sophisticated methodology, it was initially predicated on constructions that depended on the use of three very basic tools: a ruler, a set-square and a compass. It is therefore possible to reconstruct the problems the Pythagoreans encountered at an early stage with a number of simple constructions.
The second point is a more sophisticated idea that is traditionally associated with Pythagoras, commonly referred to as the 'Pythagoras Theorem' and identified with Proposition 47 of Book I of Euclid's Elements. To be sure, Proposition 47 is a very sophisticated proof which is actually attributed to Euclid (by Proclus) who lived 250 years after Pythagoras. So what can actually be attributed to Pythagoras or his School?
Again, what might be reasonably attributed is a rule which is easily seen with the following constructions. (fig. 6)
Now, the observation that the square on the longer side of a triangle is equal in area to the sum of the squares of the other two sides, was probably taken from the Babylonians or Egyptians, who knew as much in terms of certain right-angled triangles. However, with the Greeks, this property became a rule applied to all right-angled triangles and was finally refined into a proof that covered all right-angled triangles. So what might be attributed to Pythagoras is the establishment of a general rule. The proof would come later.
At this juncture we can turn our attention to the discovery of the geometrical mean. Now the mean does have an arithmetic expression which will return to below. However, having explored the origin of the Pythagoras theorem, we can now produce a similar construction which goes along way to also explaining the origin of the third form of mean traditionally associated with the early Pythagoreans.
The construction shows that another rule can be derived between a triangle and the areas formed off its sides. In this instance, it can be shown that the square on one of the sides is equal to a rectangle formed on the adjacent side. (fig. 7)
This construction which shows the equality of two areas, a square and a rectangle, can now be turned into the following ratio: AB/AC = AD/AB, where AB is the geometric mean between the line segments AD and AC. This rule was also later formalized into a construction in the Elements. Referred to as Proposition 13, Book VI, it shows how to find the geometrical mean between two given line segments. It uses a similar construction in which the proof is the reverse of what is shown. (fig. 8)
Armed with these two rules, we can now proceed to a number of very simple constructions which involve a compass and the construction of a square:
The first construction requires dropping a perpendicular onto a horizontal line. (fig. 9)
The second requires building a square on the given base line with the help of the perpendicular. (fig. 10)
The third requires generating a rectangle with the diagonal of the square. (fig. 11)
The fourth requires generating a new square with the excess of the diagonal over the side of the first square. (fig. 12)
The fifth requires producing a succession of smaller squares with each new diagonal. (fig. 13)
What becomes immediately apparent with the last construction, (fig. 13), is an endless sequence of squares which one may imagine extending for perpetuity in either direction. We have met the ineffable.
In (fig. 14), we return to the initial square and construct a succession of ever-increasing rectangles.56
By placing a set square on the first diagonal of the square and marking off a length equal to the side of the initial square, one can produce a spiral. (fig. 15)
Each spoke becomes the diagonal of each successive rectangle in (fig. 15). 57 The spiral will eventually turn into an endless smooth curve. This too is ineffable.
If we return to our initial square first shown in (fig. 11), which is again shown as (fig. 16a),
(fig. 16a) (fig. 16b)
and place the compass point at a mid-point on the base line instead of the corner of the square and produce (fig. 16b), this too can be used to generate another sequence of diminishing squares (fig. 17a).
(fig. 17a) (fig. 17b)
However, (fig. 17b) is similar to (fig. 7) and (fig. 8). It is as if the triangle in (fig. 7) and (fig. 8) has collapsed onto its base line. We have encountered another expression for the geometric mean, so that the square on AB is equal to the rectangle on AC. (Fig 18)
This in turn means that on the line segment AC, we have the following ratio: AB/AC = BC/AB.
What was the explanation for these endless sequences? What was the ineffable that they had encountered? If we return to figs. 16a & 16b, and strip them down, we are left with figs. 19a & 19b.
Now if we then set the shorter segment AB from each construction as an unit length, i.e.: AB = 1, and measure it against the longer segment, AC and CB, in both cases we are left with a remainder ED. (fig. 20)
If we take the remainder as a new unit length, and measure it off against AB, we are again left with a remainder FG. By the third operation it becomes apparent that the unit length will never quite measure out the remainder. It will never generate a whole integer or fraction (3 or a 1/2.)
The Greeks had encountered what they would eventually term 'the irrational'. If two lines could not be used to measure each other or did not have a common measure, they were deemed 'incommensurable'. If one line could be used to divide the other into a solid integer or approximate fraction they were said to be 'commensurable'.
It should be noted, if not stressed, that the Greeks did have a very sophisticated theory of numbers or integers, known as the 'theory of the odd and the even,' which allowed for fractions. And oddly enough it is through the 'theory of the odd and even' that a proof was finally devised to show that the side and diameter of a square are incommensurable. This proof came much later and is provided in Aristotle's Metaphysics, Book IX. But numerically, what would be necessary here, is the notion of decimals. But decimals require Arabic numbers which were not introduced until the 13th century by Fibonacci. And, decimals themselves were not explored until the 17th century. Thus, when Archimedes (200 BC) discovered Pi (3.14), he expressed it in terms of two limiting approximations, less than 3 1/7 and more than 3 10/71.
However, at this point we can review a number of things numerically. Reviewing the relationship between the side of a square and its diameter, according to the theorem of Pythagoras, the square of the diagonal's length equals the sum of the squares of the two sides. However, if the sides have a length of 1, then the formula becomes 1 squared + 1 squared = 2 squared, and there is no whole number or rational number that when multiplied by itself is equal to 2. The answer is provided numerically with the figure: 1.414....
Returning to the extremes 6-12, and the three means, where the arithmetic is given as 9 and the harmonic as 8; the geometric mean is provided by the square root of 6 x 12 = 72, or 8.485.... The geometrical solution is provided in (fig. 21).