There are over four hundred theorems in Euclid, and the Pythagoras Theorem is so pivotal that Euclid employs it explicitly thirty-eight times throughout The Elements. It is hard to estimate the number of times it is used implicitly.
The Golden Section reemerges, in a striking coda, in Book XIII, The Elements' final book. The book begins with a group of propositions concerning the Golden Section's properties. It then describes the inscription of the five Platonic solids in a sphere. It is in the last two of these figures, the icosahedron and the dodecahedron, that Golden Section comes into play. And it is in Proposition 8, Book XIII, that identifies its conspicuous and potent role in the pentagon: If in an equilateral and equiangular pentagon straight lines subtend two angles taken in order, they cut one another in extreme and mean ratio, and their greater segments are equal to the side of the pentagon.82 (fig. 32)
Here, with the construction of the Platonic solids within the perfect form of the sphere, Euclidean geometry culminates. The geometry is logical, pure, ethereal and abstract, and one may wonder what, if any, practical value it all has. However, four hundred and fifty years later, in 150 AD, Ptolemy, the last of the great Alexandrian mathematicians, supplied an answer when he mined Book XIII's construction of the pentagon and decagon to complete the first complete table of chords.
The very first theorem in Ptolemy's Almagest, antiquity's final word on astronomy, uses the Golden Section and the Theorem of Pythagoras repeatedly to determine the section of chord subtended by 36- and 72- degree arcs. Using these two values Ptolemy went on to assemble a table of chords from half a degree to 159 1/2 degrees. Not only was this the beginning of trigonometry, it marked the beginning of mathematical astronomy, since with this table of chords it was possible to fix precisely the longitude and latitude of celestial constellations. This development would make it possible to later navigate the oceans and eventually leave the planet.
The same author tells us, as I have already mentioned, that he received his doctrines from Themiclea at Delphi. Hieronymus says, that when he descended into the shades below, he saw the soul of Hesiod bound to a brazen pillar, and gnashing it's teeth; and that of Homer suspended from a tree, snakes around it, as a punishment for the things that they had said of the Gods. Those who refrain from commerce with their wives also were punished and that on account of this he was greatly honored at Crotona. Aristippus of Cyrene, in his Account of Natural Philosophers, says that Pythagoras derived his name from the fact of his speaking (agoreuein), truth no less than the God at Delphi (touputhieu).
He used to admonish his disciples to repeat these lines to themselves whenever they returned home to their houses:
"In what have I transgressed? What Have I done? What that I should have done have I omitted?"
He used to forbid them to offer victims to the Gods, ordering them to worship only at those altars which were unstained with blood. He also forbad to swear by the Gods, saying, "That every man ought so to exercise himself as to be worthy of belief without an oath. He also taught men that it behoved them to honor their elders, thinking most honorable that which was precedent inpoint of time; just as in the world, the rising of the sun was more so than the setting; in life, the beginning more so than the end; and in animals, production more than destruction.
Another of his rules was that men should honor the Gods above the geniuses, and heroes above men and of all men, parents were those entitled to more honor. Another, that people should associate with each other in such a way as not to make their friends enemies, but to render their enemies friends. Another was that they should not think anything exclusively their own. Another was to assist the law, and to make war upon lawlessness. Not to destroy or injure a cultivated tree, nor any animal which does not injure man. Modesty and decorum consisted in never yielding to laughter, without looking stern. Men should avoid eating too much flesh, and in travelling should let rest and exertion alternate; that they should exercise memory, nor ever say or do anything in anger, not pay respect to every kind of divination, should sing songs accompanied by the lyre, and should display a reasonable gratitude to the Gods and eminent men by hymns.
His disciples were forbidden to eat beans, because, as they were flatulent, they greatly partook of animal properties; (that their stomachs would be kept in much better order by avoiding them), and that such abstinence would make the visions that appear in one's sleep gentle and free from agitation.
Alexander, in his Successions of Philosophers, reports the following doctrines as contained in Pythagoras's Commentaries: the Monad is the beginning of everything. From this proceeds an indefinite duad, which is subordinate to the monad, as to its cause. From the monad and the indefinite duad proceed numbers. From numbers proceed signs. From these, lines, of which plane figures consist. From these plane figures are derived solid bodies. From solid bodies are derived sensible bodies, of which last there are four elements, fire, water, earth and air. The world, which is endued with life and intellect, and which is of a spherical figure, in its centre containing the earth, which is also spherical, and inhabited all over, results from a combination of these elements, and from them derives its motion. There are antipodes, and what to us is below, is to them above, He also taught that light and darkness, cold and heat, dryness and moisture, were equally divided in the world; and that, while heat was predominant in summer, so when cold prevailed, it was winter; when dryness prevailed, it was spring; and when moisture preponderated, autumn. The loveliest season of the year was when all these qualities were equally balanced; of which the flourishing spring was the most wholesome, and the autumn, the most pernicious. Of day, the most flourishing period was the morn while the evening was the fading one, and the least healthy.
Another of his theories was that the air around the earth was immovable, and pregnant disease, and that in it everything was mortal while the upper air was in perpetual motion, and salubrious; and that in it everything was immortal, and on that account divine. The sun, moon and the stars were all Gods; for in them dominates the principle which is the cause of. The moon derives its light from the sun. There a relationship between men and the Gods, because men partake of the divine principle; on which count, therefore, God exercises his providence for our advantage. Fate is the cause of the arrangement of the world, both in general and in particular. From the sun a ray penetrates both the cold aether, which is the air, aer and the dense aether, pachun aithera, which is the sea and moisture. This ray descends into the depths and in this way vivifies everything. Everything which partakes of the principle of heat lives, which account, also, plants are animated beings but that not all living beings necessarily have souls. The soul is something torn off from the aether, both warm and cold, from its partaking of the cold aether. The soul is something different from life. It is immortal, because of the immortality of that from which it was torn off.
Animals are born from one another by seeds and that it is impossible for there to be any spontaneous production by the earth. Seed is a drop from the brain which in itself contains a warm vapor; and that when this is applied to the womb, it transmits moisture, virtue, and blood from the brain, from which flesh, sinews, bones and hair, and the whole body are produced. From the vapor is produced the soul and also sensation. The infant first becomes a solid body at the end of forty days; but, according to the principles of harmony, it is not perfect till seven, or perhaps nine; or at most ten months, and then it is brought forth. In itself it contains all the principles of life which are all connected together, and by their union and combination form a hormonious whole, each of them developing itself at the appointed time.
In general the senses, and especially sights, are a vapor of intense heat, on which account a man is said to see through air, or through water. For the hot principle is opposed by the cold one; since, if the vapor in the eyes were cold, it would have the same temperature as the air, and so would be dissipated. As it is, in some passages he calls the eyes the gates of the sun. In a similar manner he speaks of hearing, and of the other senses. He also says that the soul of man is divided into three parts; into intuition (nous), reason (phren), and mind (thumos); and that the first and last divisions are found also in other animals, but that the middle one, reason, is found in man only. The chief abode of the soul is in those parts of the body which are between the heart and the brain. The mind abides in the heart, while the intuition (or deliberation) and reason reside in the brain.
The senses are drops from them; and the reasoning sense is immortal, while the others are mortal. The soul is nourished by the blood, and reasons are the winds of the soul. The soul is invisible, and so are its reasons, inasmuch as the aether itself is invisible. The links of the soul are the arteries, veins and nerves. When the soul is vigorous, and is by itself in a quiescent state, then its links are words and actions. When it is cast forth upon the earth, it wanders about, resembling the body. Mercury is the steward of the souls, and that is the reason of his name Conductor, Commercial, and Infernal, since it is he who conducts the souls from their bodies, and from earth, and sea; and that he conducts the pure souls to the highest region, and that he does not allow the impure ones to approach them nor to come near one another; committing them to be. bound in indissoluble fetters by the Furies. The Pythagoreans also assert that the whole air is full of souls, and that these are those that Page 169 DIOGENES LAERTES BIOGRAPHY XIX are accounted geniuses or heroes. They are the ones that send down among men dreams, and tokens of disease and health; the latter not being reserved to human beings, but being sent also to sheep and other cattle. They are concerned with purifications, expiations, and all kinds of divinations, oracular predictions, and the l(ike).
Man's most important privilege is to be able to persuade his soul to be either good or bad. (Men) are happy when they have a good soul; yet they never quiet, never long retaining the same mind. An oath is justice; and on that account Jupiter is called Jupiter of Oaths. Virtue is harmony, health, universal good and God; on which account everything owes its existence and preservation to harmony. Friendship is a harmonious quality. Honors to Gods and heroes should not be equal. The Gods should be honored at all times, extolling them with praises, clothed in white garments, and keeping one's body chaste; but that to the heroes such honors should not be payed till after noon.
A state of purity is brought about by purifications, washings and sprinklings; by a man's purifying himself from all funerals, concubinage, or any kind of pollution; by abstaining from all flesh that has either been killed or died of itself, from mullets, from melanuri, from eggs, from such animals as lay eggs, from beans, and from other things that are prohibited by those who have chared of the mysteries in the temples.
In his treatise on Beans, Aristotle says that Pythagoras's reason for demanding abstention from them on the part of his disciples, was that either they resemble parts of the human body, or because they are like the gates of hell Ñ they are the only plants without parts; -- or because they dry up other plants, or because they are representatives of universal nature, or because they are used in elections in oligarchical governments. He also forbade his disciples to pick up what fell from the table, for the sake of accustoming them to eat moderately, or else because such things belong to the dead. Aristophanes, indeed said that what fell belonged to the heroes, in his heroes singing, "Never taste the things which fall, From the table on the floor."
He also forbade his disciples to eat white poultry, because a cock of that color was sacred to the god Month, and was also a suppliant. He was also accounted a good animal (?) and he was sacred to the god Month, for he indicates the time.
The Pythagoreans were also forbidden to eat of all fish that was sacred, on the ground that the same animals should not be served up before both gods and men, just as the same things do not belong to both freemen and slaves. Now white is an indication of a good nature, and black of a bad one.
Another of the precepts of Pythagoras was never to break bread; because in ancient, times friends used to gather around the same loaf, as they even now do among the barbarians. Nor would he allow men to divide bread which unties them. Some think that he laid down this rule in reference to the judgment which takes place in hell; some because this practice engenders timidity in war. According to others, the refence is to the Union, which presides over the government of the Universe.
Another one of his doctrines was that of all solid figures the sphere was the most beautiful; and of all plane figures, the circle. That old age, and all diminution was similar, and also; all increase and youth. That health was the permanence of form, and disease, its destruction. He thought salt should be set before people as a reminder of justice; for salt preserves everything which it touches, and is composed of the purest particles of water and the sea.
These are the doctrines which Alexander asserts that he discovered in th