Pythagoras

“ Special emphasis was laid upon the mystery of numbers, and it is possible that the Neo-Pythagoreans had a far wider knowledge of the true teachings of Pythagoras than is available today. Even in the first century Pythagoras was regarded more as a god than a man, and the revival of his philosophy was resorted to apparently in the hope that his name would stimulate interest in the deeper systems of learning. But Greek philosophy had passed the zenith of its splendor; the mass of humanity was awakening to the importance of physical life and physical phenomena. ”

“ 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. ”

“ Pythagoras is said to have explained the conception of God in terms of mathematics and to have been willing to accept the personages of the popular polytheism on condition of finding their true mathematical equation.89.1 But this philosopher stands apart from the other leaders of this first period of Hellenic free thought. The mathematical mind is often a prey to mysticism. And Pythagoras was the most powerful champion and apostle of Orphism, the founder of those secret societies that threatened the secular and the intellectual freedom of Hellas. ”

“ The disciples of Pythagoras were required to pass through a preliminary training, consisting in a moral and religious preparation for life, which included the elements of music and mathematics. In fact Pythagoras made the science of numbers the basis of his philosophy in the belief that accurate measurement was essential to the definition of form, and consequently that the entire universe was founded upon a numerical basis. ”

“ But after these, Pythagoras changed that philosophy, which is conversant about geometry itself, into the form of a liberal doctrine, considering its principles in a more exalted manner; and investigating its theorems immaterially and intellectually; who likewise invented a treatise of such things as cannot be explained [109] in geometry, and discovered the constitution of the mundane figures. ”

“ 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. ”

“ If so, the innovation of Pythagoras was as little heeded in this regard in a subsequent age as was his theory of the motion of the earth; for to strike a swinging blow from the hip, rather than from the shoulder, is a trick which the pugilist learned anew in our own day. ”

“ Owing to the Greek mode of representing numbers by patterns of dots, the notions of number and of geometrical configuration are less separated than with us. Also Pythagoras, without doubt, included the shape-iness of shape, which is an impure mathematical entity. So to-day, when Einstein and his followers proclaim that physical facts, such as gravitation, are to be construed as exhibitions of local peculiarities of spatio-temporal properties, they are following the pure Pythagorean tradition. In a sense, Plato and Pythagoras stand nearer to modern physical science than does Aristotle. ”

“ It was in existence at least in the still dark field of vision of Pythagoras, which became illumined one day when he took such a view of the number-ratios 3—4—5 that an exact intuition could actually come about. It is erroneous to assume that a creative act suddenly called up before his soul as if by magic the figure with the three squares drawn externally on the sides of a triangle. ”

“ In his lectures, Pythagoras taught the mathematics, as a medium whereby to prove the existence of God from observation and by means of reason; grammar, rhetoric, and logic, to cultivate and improve that reason, arithmetic, because he conceived that the ultimate benefit of man consisted in the science of numbers, and geometry, music, and astronomy, because he conceived that man is indebted to them for a knowledge of what is really good and useful. ”

“ When, from out of this feeling, Pythagoras evolved and expressed the decisive formula, number had come, for him, to be an optical symbol—not a measure of form generally, an abstract relation, but a frontier-post of the domain of the Become, or rather of that part of it which the senses were able to split up and pass under review. By the whole Classical world without exception numbers are conceived as units of measure, as magnitude, lengths, or surfaces, and for it no other sort of extension is imaginable. ”

“ 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. ”

“ There is, however, no reason for regarding Pythagoras as an impostor, because experience seems to show, that while in certain ages it is not difficult for a man to persuade others that he is inspired, it is still less difficult for him to contract the same belief himself.
Looking at the general type of Pythagoras, as conceived by witnesses in and nearest to his own age,—Xenophanês, Hêrakleitus, Herodotus, Plato, Aristotle, Isokratês, [721] —we find in him [p. 395] chiefly the religious missionary and schoolmaster, with little of the politician. ”

“ He believed that the laws by which the universe is governed are in accordance with the doctrines of mathematics. The same opinion was shared by Pythagoras, the great founder of the science, whose main formula was that number is the essence or first principle of all things. No thinkers ever surpassed the Greeks in originality and profundity; and mathematics, being highly prized by them, were carried to the greatest perfection their method would allow. ”

“ 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. ”

“ Of course a late writer who read of Pythagoras making extracts from books would assume that he put them into a book of his own, just as people did in his own days. For the rest, I understand ἱστορίη of science, which is contrasted with the κακοτεχνίη which Pythagoras derived from the συγγραφαί of men like Pherekydes of Syros.
336. The word κόσμος must mean “world” here, not merely “order;” for only the world could be identified with fire. This use of the word is Pythagorean, and there is no reason to doubt that Herakleitos may have known it. ”

“ The mind that could conceive the world as a sphere and that interested itself in weights and measures was, obviously, a mind of the visualizing type. It is characteristic of this type of mind to be interested in the tangibilities of geometry, hence it is not surprising to be told that Pythagoras "carried that science to perfection." ”

“ Plutarch, of all the philosophers who have spoken of Pythagoras, had a better opportunity of entering into the ideas of that great man, and has explained them better than any one besides. Pliny, Macrobius, and Censorinus, have also spoken of the harmony which Pythagoras observed to reign in the course of the planets; but Plutarch makes him say, that it is probable that the bodies of the planets, their distances, the intervals between their spheres, the celerity of their courses and revolutions, are not only proportionable among themselves, but to the whole of the universe. ”

“ For if they assert that the reason why the doctrine of Pythagoras has not prevailed always and universally is, that Pythagoras was but a man, and had not power to secure this, can they also affirm that in the age and in the countries in which his philosophy flourished, all who had the opportunity of hearing him were found willing to believe and follow him? ”

“ In Pythagoras occurs also, perhaps for the first time, an idea which had an extremely important influence on ancient and mediaeval astronomy. Not only were the stars supposed to be attached to a crystal sphere, which revolved daily on an axis through the earth, but each of the seven planets (the sun and moon being included) moved on a sphere of its own. The distances of these spheres from the earth were fixed in accordance with certain speculative notions of Pythagoras as to numbers and music ”

“ It is specially true of science that it is the first steps which are the most difficult, and Pythagoras left a sufficient achievement in mathematics behind him for others to elaborate. The Greeks took less than three centuries to complete the edifice, and that was chiefly due to Pythagoras, who had laid the foundations truly and well.
We have now seen how the two great conceptions of Matter [68] and Form were reached; the next problem Greek philosophy had to face was that of Motion. At first the fact of movement had simply been taken for granted. ”

“ Pythagoras and his disciples understood almost the same thing by their numbers, that Plato did by his ideas. The Pythagorists expressed themselves with regard to numbers in the same terms as Plato uses, calling them “τα ὁντως ὁντα, real existences, the only things truly endowed with essence, eternally invariable.” They give them also the appellation of incorporeal entities, by means of which all other beings participate of existence. ”

“ All that Pythagoras and Plato ever wrote of noble and elevating truths, are merely flashes of that primeval light, in the full flood of which, man, in his more perfect antediluvian state, delighted to dwell; and it is remarkable in the case of Pythagoras, Anaxagoras, Thales, and so many other of the Greek philosophers, that the further we trace them back, we come nearer to the divine truth, which, in the systems of Epicurus, Aristippus, Zeno, or the shallow or cold philosophers of later origin, altogether disappears. ”

“ Of propositions attributed to him the most famous is, of course, the theorem that in a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the sides about the right angle (Eucl. I., 47) ; and, seeing that Greek tradition universally credits him with the proof of this theorem, we prefer to believe that tradition is right. This is to some extent confirmed by another tradition that Pythagoras discovered a general formula for finding two numbers such that the sum of their squares is a square number. ”

“ It is well known that all of the eminent men, as many among the ancients as among the moderns, all the savants commendable for their labours or their learning, are agreed in regarding the precepts of Pythagoras as symbolical, that is, as containing figuratively, a very different meaning from that which they would seem to offer literally. ”

“ These men were arithmeticians and geometricians to whom especially it seems Pythagoras first supplied the principles (of their arts) . And they took the first beginnings (of things) , discovered apparently by reason alone, from the [115] numbers which can always proceed to infinity by multiplication and the figures (produced by it) . For the beginning of geometry, as may be seen, is an indivisible point; but from that point the generation of the infinite figures from p. 132. the point [203] is discovered by the art. ”

“ It was a maxim with them that a right education is not only of importance to the individual, but also to the interests of the state. Pythagoras himself, as is well known, paid much attention to the determination of [118] extension and gravity, the ratios of musical tones, astronomy, and medicine. He directed his disciples, in their orgies or secret worship, to practise gymnastics, dancing, music. ”

“ Pythagoras probably does not yield to any one of these in the evidences of true intellectual greatness. In his school, in the followers he trained resembling himself, and in the salutary effects he produced on the institutions of the various republics of Magna Graecia and Sicily, he must be allowed greatly to have excelled them. His discoveries of various propositions in geometry, of the earth as a planet, and of the solar system as now universally recognised, clearly stamp him a genius of the highest order. ”

“ 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. ”