II
PYTHAGORAS AND HIS PHILOSOPHY Bygone Beliefs | ||
2. II
PYTHAGORAS AND HIS PHILOSOPHY
IT is a matter for enduring regret that so little is known to us concerning PYTHAGORAS. What little we do know serves but to enhance for us the interest of the man and his philosophy, to make him, in many ways, the most attractive of Greek thinkers; and, basing our estimate on the extent of his influence on the thought of succeeding ages, we recognise in him one of the world's master-minds.
PYTHAGORAS was born about 582 B.C. at Samos, one of the Grecian isles. In his youth he came in contact with THALES—the Father of Geometry, as he is well called,—and though he did not become a member of THALES' school, his contact with the latter no doubt helped to turn his mind towards the study of geometry. This interest found the right ground for its development in Egypt, which he visited when still young. Egypt is generally regarded as the birthplace of geometry, the subject having, it is supposed, been forced on the minds of the Egyptians by the necessity of fixing the boundaries of lands against the annual overflowing of the Nile. But the Egyptians were what is called an essentially practical
One geometrical fact known to the Egyptians was that if a triangle be constructed having its sides 3, 4, and 5 units long respectively, then the angle opposite the longest side is exactly a right angle; and the Egyptian builders used this rule for constructing walls perpendicular to each other, employing a cord graduated in the required manner. The Greek mind was not, however, satisfied with the bald statement of mere facts—it cared little for practical applications, but sought above all for the underlying reason of everything. Nowadays we are beginning to realise that the results achieved by this type of mind, the general laws of Nature's behaviour formulated by its endeavours, are frequently of immense practical importance—of far more importance than the mere rules-of-thumb beyond which so-called
PLATE 3
[Description: FIG. 3. Diagram to illustrate the Theorem of PYTHAGORAS.]
After absorbing what knowledge was to be gained in Egypt, PYTHAGORAS journeyed to Babylon, where he probably came into contact with even greater traditions and more potent influences and sources of knowledge than in Egypt, for there is reason for believing that the ancient Chaldeans were the builders of the Pyramids and in many ways the intellectual superiors of the Egyptians.
At last, after having travelled still further East, probably as far as India, PYTHAGORAS returned to his birthplace to teach the men of his native land the knowledge he had gained. But CRŒSUS was tyrant over Samos, and so oppressive was his rule that none had leisure in which to learn. Not a student came to PYTHAGORAS, until, in despair, so the story runs, he offered to pay an artisan if he would but learn geometry. The man accepted, and later, when PYTHAGORAS pretended inability any longer to continue the payments, he offered, so fascinating did he find the subject, to pay his teacher instead if the lessons might only be continued. PYTHAGORAS no doubt was much gratified at this; and the motto he adopted for his great Brotherhood, of which we shall make the acquaintance in a moment, was in all likelihood based on this event. It ran, "Honour a figure and a step before a figure and a tribolus"; or, as a freer translation renders it:—
Not a figure and a florin."
"At all events, as Mr FRANKLAND remarks, "the motto is a lasting witness to a very singular devotion to knowledge for its own sake."[8]
But PYTHAGORAS needed a greater audience than one man, however enthusiastic a pupil he might be, and he left Samos for Southern Italy, the rich inhabitants of whose cities had both the leisure and inclination to study. Delphi, far-famed for its Oracles, was visited en route, and PYTHAGORAS, after a sojourn at Tarentum, settled at Croton, where he gathered about him a great band of pupils, mainly young people of the aristocratic class. By consent of the Senate of Croton, he formed out of these a great philosophical brotherhood, whose members lived apart from the ordinary people, forming, as it were, a separate community. They were bound to PYTHAGORAS by the closest ties of admiration and reverence, and, for years after his death, discoveries made by Pythagoreans were invariably attributed to the Master, a fact which makes it very difficult exactly to gauge the extent of PYTHAGORAS' own knowledge and achievements. The regime of the Brotherhood, or Pythagorean Order, was a strict one, entailing "high thinking and low living" at all times. A restricted diet, the exact nature of which is in dispute, was observed by all members, and long periods of silence, as conducive to deep thinking, were imposed on novices. Women were admitted to the Order, and PYTHAGORAS' asceticism did not prohibit romance, for we read that one of his fair pupils won her way to his heart, and, declaring her affection for him, found it reciprocated and became his wife.
SCHURÉ writes: "By his marriage with Theano, Pythagoras affixed the seal of realization to his work. The union and fusion of the two lives was complete. One day when the master's wife was asked what
PYTHAGORAS was not merely a mathematician. he was first and foremost a philosopher, whose philosophy found in number the basis of all things, because number, for him, alone possessed stability of relationship. As I have remarked on a former occasion, "The theory that the Cosmos has its origin and explanation in Number . . . is one for which it is not difficult to account if we take into consideration the nature of the times in which it was formulated. The Greek of the period, looking upon Nature, beheld no picture of harmony, uniformity and fundamental unity. The outer world appeared to him rather as a discordant chaos, the mere sport and plaything of the gods. The theory of the uniformity of Nature—that Nature is ever like to herself —the very essence of the modern scientific spirit, had yet to be born of years of unwearied labour and unceasing delving into Nature's innermost secrets. Only in Mathematics—in the properties of geometrical figures, and of numbers—was the reign of law, the principle of harmony, perceivable. Even at this present day when the marvellous has become com-
No doubt the Pythagorean theory suffers from a defect similar to that of the Kabalistic doctrine, which, starting from the fact that all words are composed of letters, representing the primary sounds of language, maintained that all the things represented by these words were created by God by means of the twenty-two letters of the Hebrew alphabet. But at the same time the Pythagorean theory certainly embodies a considerable element of truth. Modern science demonstrates nothing more clearly than the importance of numerical relationships. Indeed, "the history of science shows us the gradual transformation of crude facts of experience into increasingly exact generalisations by the application to them of mathematics. The enormous advances that have been made in recent years in physics and chemistry are very largely due to mathematical methods of
The Pythagorean doctrine of the Cosmos, in its most reasonable form, however, is confronted with one great difficulty which it seems incapable of overcoming, namely, that of continuity. Modern science, with its atomic theories of matter and electricity, does, indeed, show us that the apparent continuity of material things is spurious, that all material things consist of discrete particles, and are hence measurable in numerical terms. But modern science is also obliged to postulate an ether behind
According to BERGSON, life—the reality that can only be lived, not understood—is absolutely continuous (i.e. not amenable to numerical treatment). It is because life is absolutely continuous that we cannot, he says, understand it; for reason acts discontinuously, grasping only, so to speak, a cinematographic view of life, made up of an immense number of instantaneous glimpses. All that passes between the glimpses is lost, and so the true whole, reason can never synthesise from that which it possesses. On the other hand, one might also argue —extending, in a way, the teaching of the physical sciences of the period between the postulation of DALTON'S atomic theory and the discovery of the significance of the ether of space—that reality is essentially discontinuous, our idea that it is continuous being a mere illusion arising from the coarseness of our senses. That might provide a complete vindi-
PYTHAGORAS' foremost achievement in mathematics I have already mentioned. Another notable piece of work in the same department was the discovery of a method of constructing a parallelogram having a side equal to a given line, an angle equal to a given angle, and its area equal to that of a given triangle. PYTHAGORAS is said to have celebrated this discovery by the sacrifice of a whole ox. The problem appears in the first book of EUCLID'S Elements of Geometry as proposition 44. In fact, many of the propositions of EUCLID'S first, second, fourth, and sixth books were worked out by PYTHAGORAS and the Pythagoreans; but, curiously enough, they seem greatly to have neglected the geometry of the circle.
The symmetrical solids were regarded by PYTHAGORAS, and by the Greek thinkers after him, as of the greatest importance. To be perfectly symmetrical or regular, a solid must have an equal number of faces meeting at each of its angles, and these faces must be equal regular polygons, i.e. figures whose sides and angles are all equal. PYTHAGORAS, perhaps,
as faces.
The Cube, having six squares as faces.
The Octahedron, having eight equilateral triangles
as faces.
The Dodecahedron, having twelve regular pentagons
(or five-sided figures) as faces.
The Icosahedron, having twenty equilateral triangles
as faces.[13]
Now, the Greeks believed the world to be composed
of four elements—earth, air, fire, water,—
and to the Greek mind the conclusion was inevitable[14]
that the shapes of the particles of the elements were
those of the regular solids. Earth-particles were
cubical, the cube being the regular solid possessed
of greatest stability; fire-particles were tetrahedral,
the tetrahedron being the simplest and, hence,
lightest solid. Water-particles were icosahedral for
exactly the reverse reason, whilst air-particles, as
intermediate between the two latter, were octahedral.
The dodecahedron was, to these ancient mathematicians,
the most mysterious of the solids: it was
by far the most difficult to construct, the accurate
drawing of the regular pentagon necessitating a rather
PLATE 4
[Description: FIGS. 4-8.
Diagrams for constructing the Regular (or Platonic) Solids.]
Music played an important part in the curriculum of the Pythagorean Brotherhood, and the important discovery that the relations between the notes of musical scales can be expressed by means of numbers is a Pythagorean one. 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. "Look how the floor of heaven," says Lorenzo in SHAKESPEARE'S The Merchant of Venice—
Is thick inlaid with patines of bright gold:
There's not the smallest orb which thou behold's"
But in his motion like an angel sings,
Still quiring to the young-eyed cherubins;
Such harmony is in immortal souls;
But whilst this muddy vesture of decay
Doth grossly close it in, we cannot hear it."[17]
Or, as KINGSLEY writes in one of his letters, "When I walk the fields I am oppressed every now and then with an innate feeling that everything I see has a meaning, if I could but understand it. And this feeling of being surrounded with truths which I cannot grasp, amounts to an indescribable awe sometimes!
As concerns PYTHAGORAS' ethical teaching, judging from the so-called Golden Verses attributed to him, and no doubt written by one of his disciples,[19] this would appear to be in some respects similar to that of the Stoics who came later, but free from the materialism of the Stoic doctrines. Due regard for oneself is blended with regard for the gods and for other men, the atmosphere of the whole being at once rational and austere. One verse—"Thou shalt likewise know, according to Justice, that the nature of this Universe is in all things alike"[20]—is of particular interest, as showing PYTHAGORAS' belief in that principle of analogy—that "What is below is as that which is above, what is above is as that which is below"—which held so dominant a sway over the
Such, in brief, were the outstanding doctrines of the Pythagorean Brotherhood. Their teachings included, as we have seen, what may justly be called scientific discoveries of the first importance, as well as doctrines which, though we may feel compelled —perhaps rightly—to regard them as fantastic now, had an immense influence on the thought of succeeding ages, especially on Greek philosophy as represented by PLATO and the Neo-Platonists, and the more speculative minds—the occult philosophers, shall I say?—of the latter mediæval period and succeeding centuries. The Brotherhood, however, was not destined to continue its days in peace. As I have indicated, it was a philosophical, not a political, association; but naturally PYTHAGORAS philosophy included political doctrines. At any rate, the Brotherhood acquired a considerable share in the government of Croton, a fact which was greatly resented by the members of the democratic party, who feared the loss of their rights; and, urged thereto, it is said, by a rejected applicant for membership of the Order, the mob made an onslaught on the Brotherhood's place of assembly and burnt it to the ground. One account has it that PYTHAGORAS himself died in
The Pythagorean Order was broken up, but the bonds of brotherhood still existed between its members. "One of them who had fallen upon sickness and poverty was kindly taken in by an innkeeper. Before dying he traced a few mysterious signs [the pentagram, no doubt] on the door of the inn and said to the host: `Do not be uneasy, one of my brothers will pay my debts.' A year afterwards, as a stranger was passing by this inn he saw the signs and said to the host: `I am a Pythagorean; one of my brothers died here; tell me what I owe you on his account.' "[21]
In endeavouring to estimate the worth of PYTHAGORAS' discoveries and teaching, Mr FRANKLAND writes, with reference to his achievements in geometry: "Even after making a considerable allowance for his pupils' share, the Master's geometrical work calls for much admiration"; and, ". . . it cannot be far wrong to suppose that it was Pythagoras' wont to insist upon proofs, and so to secure that rigour which gives to mathematics its honourable position amongst the sciences." And of his work in arithmetic, music, and astronomy, the same author writes: ". . . everywhere he appears to have inaugurated genuinely scientific methods, and to have laid the foundations of a high and liberal education"; adding, "For nearly
See AUGUST EISENLOHR: Ein mathematisches Handbuch der alten Aegypter (1877); J. Gow: A Short History of Greek Mathematics (1884); and V. E. JOHNSON: Egyptian Science from the Monuments and Ancient Books (1891).
Fig. 3 affords an interesting practical demonstration of the truth of this theorem. If the reader will copy this figure, cut out the squares on the two shorter sides of the triangle and divide them along the lines AD, BE, EF, he will find that the five pieces so obtained can be made exactly to fit the square on the longest side as shown by the dotted lines. The size and shape of the triangle ABC, so long as it has a right angle at C, is immaterial. The lines AD, BE are obtained by continuing the sides of the square on the side AB, i.e. the side opposite the right angle, and EF is drawn at right angles to BE.
EDOUARD SCHURÉ: Pythagoras and the Delphic Mysteries, trans. by F. ROTHWELL, B.A. (1906), pp. 164 and 165.
Quoted from a lecture by the present writer on "The Law of Correspondences Mathematically Considered," delivered before The Theological and Philosophical Society on 26th April 1912, and published in Morning Light, vol. xxxv (1912), p. 434 et seq.
Cf. chap. iii., "On Nature as the Embodiment of Number," of my A Mathematical Theory of Spirit, to which reference has already been made.
If the reader will copy figs. 4 to 8 on cardboard or stiff paper, bend each along the dotted lines so as to form a solid, fastening together the free edges with gummed paper, he will be in possession of models of the five solids in question.
In reference to this matter FRANKLAND remarks: "In those early days the innermost secrets of nature lay in the lap of geometry, and the extraordinary inference follows that Euclid's Elements, which are devoted to the investigation of the regular solids, are therefore in reality and at bottom an attempt to `solve the universe.' Euclid, in fact, made this goal of the Pythagoreans the aim of his Elements."—Op. cit., p. 35.
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PYTHAGORAS AND HIS PHILOSOPHY Bygone Beliefs | ||