Pythagoreans

“ The first man—as far as we know—to call himself a ‘philosophos’, or lover of wisdom, was Pythagoras, who applied the label to himself and his followers. Philosophy among the Pythagoreans, with its emphasis on astronomy, geometry, and number, was still decidedly Egyptian; but gradually, from these starry beginnings, the Greek mind built up a vast, independent edifice of thought and language. ”

“ The Pythagoreans, it is said, resolved to keep the existence of incommensurables a profound secret, revealed only to a few of the supreme heads of the sect; and one of their number, Hippasos of Metapontion, is even said to have been shipwrecked at sea for impiously disclosing the terrible discovery to their enemies. It must be remembered that Pythagoras was the founder of a new religion as well as the teacher of a new science: if the science came to be doubted, the disciples might fall into sin, and perhaps even eat beans, which according to Pythagoras is as bad as eating parents' bones. ”

“ Pythagoreans, the men whose names we know, are first of all men of science, and more and more inclined to drop what they doubtless regarded as the superstitious side of the doctrine. In the end they were absorbed in the new philosophical schools which arose at Athens. ”

“ All we can safely say is that, the more primitive any Pythagorean doctrine appears, the more likely it is to be that of Pythagoras himself, and all the more so if it can be shown to have points of contact with views which we know to have been held in his own time or shortly before it. In particular, when we find the later Pythagoreans teaching things that were already something of an anachronism in their own day, we may be pretty sure we are dealing with survivals which only the authority of the master's name could have preserved. ”

“ I think also, it was said by the Pythagoreans, respecting those who teach for the sake of reward, that they show themselves to be worse than statuaries, or those artists who perform their work sitting. For these, when some one orders them to make a 174 statue of Hermes, search for wood adapted to the reception of the proper form; but those pretend that they can readily produce the works of virtue from every nature. ”

“ And, in short, which urges the soul to be untamed by all those passions which are the progeny of the realms of generation, and which draw it to an inferior condition of being. For the exercise and ascent through all these, is the study of the most perfect fortitude. And such are the instances adduced by us of the fortitude of Pythagoras, and the Pythagoreans. ”

“ The world also, which is composed from these tetractys, is perfect, being elegantly arranged in geometrical, harmonical, and arithmetical proportion; comprehending every power, all the nature of number, every magnitude, and every simple and composite body. But it is perfect, because all things are the parts of it, but it is not itself the part of any thing. Hence, the Pythagoreans are said to have first used the before-mentioned oath, and also the assertion that “all things are assimilated to number.” ”

“ We actually have evidence of revisions of the Pythagorean theory. The first step was to get rid of the counter-earth, and some Pythagoreans did this by identifying the counter-earth with the moon. We hear too of a Pythagorean system in which the central fire was not outside the earth but in the centre of the33 earth itself. The descriptions of this system rather indicate that in it the earth was supposed to be at rest, without any rotation, in the centre of the universe. This was practically a return to the standpoint of Pythagoras himself. ”

“ Phytius likewise, Theocles, Elecaon, and Aristocrates, excelled among the Pythagoreans in their studies and manners, which also the cities in those places adopted at those times. In short, it is asserted that Pythagoras was the inventor 95 of the whole of political erudition, when he said that nothing is pure among things that have an existence; but that earth participates of fire, fire of air, air of water, and water of spirit. ”

“ Aristotle lays down the general principle of the Pythagoreans in the following terms: "Number," he says, "is, according to them, the essence of all things; and the organisation of the universe, in its various determinations, is a harmonious system of numbers and their relations.” "The boldness of such an assertion," says Hegel, "impresses us as very remarkable; it is an assertion which strikes down at one blow all that our ordinary representations declare to be essential and true. It displaces sensible existence, and makes thought and not sense to be the criterion of the essence of things. ”

“ They enforced moral self-control. They cultivated the arts and crafts, gymnastics, music, medicine, and mathematics. The development of mathematics in early Greece was largely the work of the Pythagoreans. Pythagoras is said to have discovered the 47th Proposition of Euclid, and to have sacrificed an ox in honour thereof. And there is good reason to believe that practically the whole of the substance of the First Book of Euclid is the work of Pythagoras. ”

“ But he inserted the conceptions of the men in metre, and published in comedies the occult dogmas of Pythagoras.
Of all the Pythagoreans, however, it is probable that many are unknown and anonymous. But the 190 following are the names of those that are known and celebrated: Of the Crotonians, Hippostratus, Dymas, Ægon, Æmon, Sillus, Cleosthenes, Agelas, Episylus, Phyciadas, Ecphantus, Timæus, Buthius, Eratus, Itmæus, Rhodippus, Bryas, Evandrus, Myllias, Antimedon, Ageas, Leophron, Agylus, Onatus, Hipposthenes, Cleophron, Alcmæon, Damocles, Milon, Menon. ”

“ In view of all these properties of numbers, we can easily understand how the Pythagoreans came to “liken all things to numbers” and to find in the principles of numbers the principles of all things (“all things are numbers”) .
We come now to Pythagoras’s achievements in geometry. There is a story that, when he came home from Egypt and tried to found a school at Samos, he found the Samians indifferent, so that he had to take special measures to ensure that his geometry might not perish with him. ”

“ So that since, for the Pythagoreans, the universe is a musical harmony, it follows that the essential character of the universe is number. The study of mathematics confirmed the Pythagoreans in this idea. Arithmetic is the science of numbers, and all other mathematical sciences are ultimately reducible to numbers. For instance, in geometry, angles are measured by the number of degrees. ”

“ We saw, in dealing with Pythagoras, that we could not be sure as to precisely what he himself taught, but there is no question that the idea of the world's motion became from an early day a so-called Pythagorean doctrine. While all the other philosophers, so far as we know, still believed that the world was flat, the Pythagoreans out in Italy taught that the world is a sphere and that the apparent motions of the heavenly bodies are really due to the actual motion of the earth itself. ”

“ The eternal essence of number is the most providential principle of the universe, &c.
The following account of the manner in which the Pythagoreans philosophized about numbers, is extracted from my Theoretic Arithmetic, and the information contained in it is principally derived from the great Syrianus.
“The Pythagoreans, turning from the vulgar paths, and delivering their philosophy in secret to those alone who were worthy to receive it, exhibited it to others through mathematical names. Hence, they called forms, numbers, as things which are the first separated from impartible union ”

“ It is evident also, that those which are in the same order, are contained through mutual comparison, in sameness rather than in difference, but that those which are in different orders are conversant with much diversity, through the dominion of difference.
“Again, the Pythagoreans asserted that nature produces sensibles by numbers ”

“ It must have seemed to its discoverer—as, in a sense, it indeed is—a striking confirmation of the numerical theory of the Cosmos. The Pythagoreans held that the positions of the heavenly bodies were governed by similar numerical relations, and that in consequence their motion was productive of celestial music. This concept of "the harmony of the spheres" is among the most celebrated of the Pythagorean doctrines, and has found ready acceptance in many mystically-speculative minds. ”

“ But as the Pythagoreans define a point to be unity having position, let us consider what they mean. That numbers, indeed, are more immaterial and more pure than magnitudes, and that the principle of numbers is more simple than the principle of magnitudes, is manifest to every one: but when they say that a [122] point is unity endued with position, they appear to me to evince that unity and number subsist in opinion: I mean monadic number [131] . On which account, every number, as the pentad and the heptad, is one in every soul, and not many; ”

“ And indeed a greater good never came, nor ever will come to mankind, than that 20 which was imparted by the Gods through this Pythagoras. Hence, even now the proverb of the long-haired Samian, is applied to the most venerable man. But Aristotle relates, in his Treatise On the Pythagoric Philosophy, that such a division as the following was preserved by the Pythagoreans among their principal arcana; viz. that of rational animals one kind is a God, another man, and another such as Pythagoras. ”

“ The doctrine of transmigration which they had adopted was in harmony with such views, and, if it does not imply the absolute immortality of the soul, at least asserts its existence after the death of the body, for the disembodied spirit becomes incarnate again as soon as it finds a tenement which fits it. To their life after death the Pythagoreans added a doctrine of retributive rewards and punishments, and, in this respect, what has been said of animals forming a penitential mechanism in the theology of India and Egypt, holds good for the Pythagoreans too. ”

“ We also gather that the dots were supposed to represent pebbles (ψῆφοι) , and this throws light on early methods of what we still call calculation.
48.Triangular, square and oblong numbers. That Aristotle refers to this seems clear, and is confirmed by the tradition that the great revelation made by Pythagoras to mankind was precisely a figure of this kind, the tetraktys, by which the Pythagoreans used to swear, [77] and we have the authority of Speusippos for holding that the whole theory was Pythagorean. ”

“ First of all Thales thought that water was the primordial substance of all things. Heraclitus of Ephesus, surnamed by the Greeks [Greek: skoteinos] on account of the obscurity of his writings, thought that it was fire. Democritus and his follower Epicurus thought that it was the atoms, termed by our writers "bodies that cannot be cut up," or, by some, "indivisibles." The school of the Pythagoreans added air and the earthy to the water and fire. ”

“ Plato seems also to hint at the possibility of other applications of arithmetical or mathematical proportions, such as we employ in chemistry and natural philosophy, such as the Pythagoreans and even Aristotle make use of in Ethics cx and Politics, e.g. ”

“ Pythagoras studied science in Egypt and first became familiar with Egyptian and Babylonian mathematics and geometry. He also studied the Milesian cosmological philosophy. On his return to Greece from his foreign studies he sought to discover a principle of homogeneity in the universe more acceptable than any suggested by the earlier philosophers. He had noticed numerous relationships between numbers and natural phenomena, and believed that the true basis of philosophy was to be found in numbers. In seeking data to sustain this thesis, he discovered harmonic progression. ”

“ Pythagoras affirmed this to be a sacred quaternion, source of everlasting nature, [7] having, as it were, roots in itself; and that from this number all the numbers receive their originating principle. For eleven, and twelve, and the rest, partake of the origin of existence [8] from ten. Of this decade, the perfect number, there are termed four divisions,—namely, number, monad, [9] square, (and) cube. And the connections and blendings of these are performed, according to nature, for the generation of growth completing the productive number. ”

“ There is so much of truth in the philosophy of Pythagoras, that when we consider the universe as composed of properties that can be measured, and that by measure all properties are reduced to number, then all properties can be considered as number. ”

“ Pythagoras, by his skill in mathematics, discovered that there can be no more than five regular solid figures, terminated by plane surfaces which are all similar and equal; to wit, the tetrahedron, the cube, the octahedron, the dodecahedron, and the eicosihedron. As nature works in the most simple and regular way, he thought that all elementary bodies must have one or other of those regular figures; and that the discovery of the properties and relations of the regular solids must be a key to open the mysteries of nature. ”

“ These four numbers which, united by addition produce the number 10, constituted the Being, as much universal as particular; so that the Quaternary, which is as its virtue, could become the emblem of all beings, since there is none which may not recognize the principles, and which does not manifest itself by faculties more or less perfect, and which may not enjoy an existence universal or relative; but the being to which Pythagoras applied it most commonly was Man. Man, as I have said, manifests himself as does the Universe, under the three principal modifications of body, soul, and spirit. ”