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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 people, and
their geometrical knowledge did not extend beyond a few empirical rules
useful for fixing these boundaries and in constructing their temples.
Striking evidence of this fact is supplied by the AHMES papyrus, compiled
some little time before 1700 B.C. from an older work dating from about
3400 B.C.,(1) a papyrus which almost certainly represents the highest
mathematical knowledge reached by the Egyptians of that day. Geometry is
treated very superficially and as of subsidiary interest to arithmetic;
there is no ordered series of reasoned geometrical propositions given—nothing,
indeed, beyond isolated rules, and of these some are wanting in accuracy.
(1) 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).
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 practical minds never advance. The
classic example of the utility of seemingly useless knowledge is afforded
by Sir WILLIAM HAMILTON'S discovery, or, rather, invention of
Quarternions, but no better example of the utilitarian triumph of the
theoretical over the so-called practical mind can be adduced than that
afforded by PYTHAGORAS. Given this rule for constructing a right angle,
about whose reason the Egyptian who used it never bothered himself, and
the mind of PYTHAGORAS, searching for its full significance, made that
gigantic geometrical discovery which is to this day known as the Theorem
of PYTHAGORAS—the law that in every right-angled triangle the square
on the side opposite the right angle is equal in area to the sum of the
squares on the other two sides.(1) The importance of this discovery can
hardly be overestimated. It is of fundamental importance in most branches
of geometry, and the basis of the whole of trigonometry—the special
branch of geometry that deals with the practical mensuration of triangles.
EUCLID devoted the whole of the first book of his Elements of Geometry
to establishing the truth of this theorem; how PYTHAGORAS demonstrated it
we unfortunately do not know.
(1) 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.
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 CROESUS 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:—
"A figure and a step onward 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."(1)
(1) W. B. FRANKLAND, M.A.: The Story of Euclid (1902), p. 33
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.
SCHURE 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 length of time
elapsed before a woman could become pure after intercourse with a man, she
replied: 'If it is with her husband, she is pure all the time; if with
another man, she is never pure.'" "Many women," adds the writer, "would
smilingly remark that to give such a reply one must be the wife of
Pythagoras, and love him as Theano did. And they would be in the right,
for it is not marriage that sanctifies love, it is love which justifies
(1) EDOUARD SCHURE: Pythagoras and the Delphic Mysteries, trans. by
F. ROTHWELL, B.A. (1906), pp. 164 and 165.
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 commonplace, that property of right-angled
triangles... already discussed... comes to the mind as a remarkable and
notable fact: it must have seemed a stupendous marvel to its discoverer,
to whom, it appears, the regular alternation of the odd and even numbers,
a fact so obvious to us that we are inclined to attach no importance to
it, seemed, itself, to be something wonderful. Here in Geometry and
Arithmetic, here was order and harmony unsurpassed and unsurpassable. What
wonder then that Pythagoras concluded that the solution of the mighty
riddle of the Universe was contained in the mysteries of Geometry? What
wonder that he read mystic meanings into the laws of Arithmetic, and
believed Number to be the explanation and origin of all that is?"(1)
(1) A Mathematical Theory of Spirit (1912), pp. 64-65.
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
interpreting and co-ordinating facts experimentally revealed, whereby
further experiments have been suggested, the results of which have
themselves been mathematically interpreted. Both physics and chemistry,
especially the former, are now highly mathematical. In the biological
sciences and especially in psychology it is true that mathematical methods
are, as yet, not so largely employed. But these sciences are far less
highly developed, far less exact and systematic, that is to say, far less
scientific, at present, than is either physics or chemistry. However, the
application of statistical methods promises good results, and there are
not wanting generalisations already arrived at which are expressible
mathematically; Weber's Law in psychology, and the law concerning the
arrangement of the leaves about the stems of plants in biology, may be
instanced as cases in point."(1)
(1) 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.
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 these atoms, an ether which is wholly continuous, and hence
transcends the domain of number.(1) It is true that, in quite recent
times, a certain school of thought has argued that the ether is also
atomic in constitution—that all things, indeed, have a grained
structure, even forces being made up of a large number of quantums or
indivisible units of force. But this view has not gained general
acceptance, and it seems to necessitate the postulation of an ether beyond
the ether, filling the interspaces between its atoms, to obviate the
difficulty of conceiving of action at a distance.
(1) Cf. chap. iii., "On Nature as the Embodiment of Number," of my A
Mathematical Theory of Spirit, to which reference has already been
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
vindication of the Pythagorean view; but a better vindication, if not of
that theory, at any rate of PYTHAGORAS' philosophical attitude, is
forthcoming, I think, in the fact that modern mathematics has transcended
the shackles of number, and has enlarged her kingdom, so as to include
quantities other than numerical. PYTHAGORAS, had he been born in these
latter centuries, would surely have rejoiced in this, enlargement, whereby
the continuous as well as the discontinuous is brought, if not under the
rule of number, under the rule of mathematics indeed.
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)
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
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.
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