Part 3

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, may be credited with the great discovery that there are only five such solids. These are as follows:--

The Tetrahedron, having four equilateral triangles 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.(1)

(1) 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.

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(2a) 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 elaborate application of PYTHAGORAS' great theorem.(1) Hence the conclusion, as PLATO put it, that "this (the regular dodecahedron) the Deity employed in tracing the plan of the Universe."(2b) Hence also the high esteem in which the pentagon was held by the Pythagoreans. By producing each side of this latter figure the five-pointed star (fig. 9), known as the pentagram, is obtained. This was adopted by the Pythagoreans as the badge of their Society, and for many ages was held as a symbol possessed of magic powers. The mediaeval magicians made use of it in their evocations, and as a talisman it was held in the highest esteem.

(2a) Cf. PLATO: The Timaeus, SESE xxviii--xxx.

(1) 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.

(2b) Op. cit., SE xxix.

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--

"... Look how the floor of heaven 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."(1)

(1) Act v. scene i.

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! Everything seems to be full of God's reflex, if we could but see it. Oh! how I have prayed to have the mystery unfolded, at least hereafter. To see, if but for a moment, the whole harmony of the great system! To hear once the music which the whole universe makes as it performs His bidding!"(1) In this connection may be mentioned the very significant fact that the Pythagoreans did not consider the earth, in accordance with current opinion, to be a stationary body, but believed that it and the other planets revolved about a central point, or fire, as they called it.

(1) CHARLES KINGSLEY: His Letters and Memories of His Life, edited by his wife (1883), p. 28.

As concerns PYTHAGORAS' ethical teaching, judging from the so-called Golden Verses attributed to him, and no doubt written by one of his disciples,(2) 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"(3)--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 minds of ancient and mediaeval philosophers, leading them--in spite, I suggest, of its fundamental truth--into so many fantastic errors, as we shall see in future excursions. Metempsychosis was another of the Pythagorean tenets, a fact which is interesting in view of the modern revival of this doctrine. PYTHAGORAS, no doubt, derived it from the East, apparently introducing it for the first time to Western thought.

(2) It seems probable, though not certain, that PYTHAGORAS wrote nothing himself, but taught always by the oral method.

(3) Cf. the remarks of HIEROCLES on this verse in his Commentary.

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 mediaeval 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 conflagration, a sacrifice to the mad fury of the mob. According to another account--and we like to believe that this is the true one--he escaped to Tarentum, from which he was banished, to find an asylum in Metapontum, where he lived his last years in peace.

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.'"(1)

(1) EDOUARD SCHURE: Op. cit., p. 174.

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 a score of centuries, to the very close of the Middle Ages, the four Pythagorean subjects of study--arithmetic, geometry, astronomy, music--were the staple educational course, and were bound together into a fourfold way of knowledge--the Quadrivium."(1) With these words of due praise, our present excursion may fittingly close.

(1) Op. cit., pp. 35, 37, and 38.

III. MEDICINE AND MAGIC

THERE are few tasks at once so instructive and so fascinating as the tracing of the development of the human mind as manifested in the evolution of scientific and philosophical theories. And this is, perhaps, especially true when, as in the case of medicine, this evolution has followed paths so tortuous, intersected by so many fantastic byways, that one is not infrequently doubtful as to the true road. The history of medicine is at once the history of human wisdom and the history of human credulity and folly, and the romantic element (to use the expression in its popular acceptation) thus introduced, whilst making the subject more entertaining, by no means detracts from its importance considered psychologically.

To whom the honour of having first invented medicines is due is unknown, the origins of pharmacy being lost in the twilight of myth. OSIRIS and ISIS, BACCHUS, APOLLO father of the famous physician AESCULAPIUS, and CHIRON the Centaur, tutor of the latter, are among the many mythological personages who have been accredited with the invention of physic. It is certain that the art of compounding medicines is extraordinarily ancient. There is a papyrus in the British Museum containing medical prescriptions which was written about 1200 B.C.; and the famous EBERS papyrus, which is devoted to medical matters, is reckoned to date from about the year 1550 B.C. It is interesting to note that in the prescriptions given in this latter papyrus, as seems to have been the case throughout the history of medicine, the principle that the efficacy of a medicine is in proportion to its nastiness appears to have been the main idea. Indeed, many old medicines contained ingredients of the most disgusting nature imaginable: a mediaeval remedy known as oil of puppies, made by cutting up two newly-born puppies and boiling them with one pound of live earthworms, may be cited as a comparatively pleasant example of the remedies (?) used in the days when all sorts of excreta were prescribed as medicines.(1)