this statement. Setting aside for the present the metaphysical thinkers,
who will occupy us in another chapter, we may safely say that the
earliest mathematicians were the school of Pythagoras, and also that
their work started (so far as they did not start from the highest of
all--pure thinking) from Arithmetic. To this science Pythagoras and his
school attached such importance that they were supposed to hold that
numbers were the essence of the universe. If you think that such a
theory is mere nonsense, I may tell you that I have often heard my
colleagues, distinguished in modern science, discuss a theory, alive at
the present day, that the so-called material universe consists of mere
motion, without anything to be moved! At the root of these speculations
lies the fundamental distinction of form and matter, of the definite and
the indefinite; and the Pythagoreans had got a glimpse of the eternal
truth that it is only through our intuitions of space and time, and
through abstract concepts explicating these, that we can bring the
myriad phenomena of nature under intelligible law. It was an early
anticipation, so far as we can explain it, of the great theory of
Descartes, that all the universe could be reduced to mathematical
relations, and these handled by algebra, which is in its essence but a
very abstract and generalised arithmetic. If therefore all parts of the
world stand in mutual arithmetical relations, of which the chemical law
of definite proportions is the most signal example, the science of
numbers must be the capital of every scientific man.
And remember that in Greek parlance this was the strict meaning of their
_arithmetic_--a pure science, while they used the term _logistic_ (or
computation) for the working of practical rules. At the basis of their
theory of numbers lay of course the one great assumption which makes the
science possible--I mean the absolute equality of the units of any
number used for the purpose of calculation.
This is not merely the abstraction from all their differences, as when I
say that the present audience consists of five hundred people,
regardless of the countless variations existing between the units of
this crowd. It is the assumption of an ideal and accurate identity
between each of the units, as to magnitude, which makes the expression
of geometrical truths arithmetically possible.
The truth that 3² + 4² = 5² applies not only to numbers but to lines,
and probably suggested the geometrical proof to Euclid (1, 47). But it
is only true if the units in the measurement of each line are exactly
equal.
Starting from this first assumption, the Pythagoreans began to speculate
on the peculiarities of the natural series of units in use among men,
and to deduce from these general considerations various theorems, which
they believed might solve the secrets of nature. At the very outset they
were struck with the obvious contrast between odd and even, which Plato,
following them, regarded as a fundamental distinction in nature. Had
they been told that, thousands of years later, men of science would find
that a most primitive and fundamental distinction among animals is
founded on this difference, I mean that of _artio-dactyle_, and
_perisso-dactyle_, actually called by the Greek words, they would have
said that this caused them no surprise, as their arithmetic had long
since laid down the distinction as a law of nature. As simple specimens
of the sort of treatment that the science of numbers received from them,
I may cite the following: The successive additions of the odd numbers
produce the squares of the series of even and odd.[32] The series of
even numbers when added give us no such result, but rather this--that
the addition of even numbers gives us figures which are the products of
successive numbers differing by only one, _e.g._ 2 + 4 = 3 × 2; 2 + 4 +
6 = 4 × 3, and so on. These latter numbers were regarded as rectangles,
when expressed in lines. It was by the discovery of the relation of the
sides to the base of a right-angled triangle that they, so to speak,
stumbled upon irrational numbers. If the two sides are each equal to 1,
the hypothenuse is equal to √2, which is no integral number, but a
problem in itself.[33]
All the results of this Pythagorean research lived through into the days
of Plato and Aristotle and then, as we know from Euclid and Theon, into
the learning of Alexandria. The importance recognised by them in the
numbers ten and twelve was shown by the general adoption of a decimal
system of notation, and of the division of time on a duodecimal system.
You will ask me what symbols the Greeks had which could enable them to
treat arithmetical figures of any complexity, and on this I could give
you now a very definite reply, but the details would lead us away from
our subject, seeing that this notation was lost in the Dark Ages and was
ultimately replaced by the Arabic numerals. But we now know that they
had a very practical system of decimal notation based on the use of the
letters of the alphabet; and the fact that several letters obsolete in
the alphabet of the fifth century B.C. appear as symbols, proves that it
was current as early as Pythagorean days. The sign for 6 is the
_digamma_, that for 90 is the _koph_ of the Phœnician alphabet, which is
still found in Locrian inscriptions; the Phœnician letter known as
_sampi_ is used for 900. We know the practical management of this easy
notation perfectly from the mass of accounts both private and public
found on Egyptian papyri. It can express large numbers far more
compendiously than the Roman system, often more compendiously even than
ours. Suppose you desire to express any large number, say 20,050, here
it is β/ΜΝ; say 47,678, it is δ/ΜΖΧΟΗ, and if there be small gain in
simplicity here, I will give you 800,000 = 10,000 × 80 = π/Μ. But these
are practical matters, though without an easy notation even the most
scientific thinkers could not make large progress.[34]
The next great step was to pass from arithmetic to geometry as the
science of space and to show how far the same laws governed both.
If we are not well informed upon the beginnings of arithmetic, we are
more fortunate in the case of geometry, and here, if anywhere, the old
Greeks have been the acknowledged teachers of modern Europe. For we have
in the so-called _Elements_ of Euclid, composed most probably at
Alexandria about 300 B.C., a summary of all that had been discovered up
to his day, doubtless with many new things of his own. He had distinctly
built upon his predecessors; he has before him all through his book a
problem discussed in Plato, that of the possible number of regular
polyhedra, and its solution forms the climax of his work. But he begins
from the very beginning and builds up his whole doctrine with such
accuracy that a flaw in the demonstration is hard to be found.
How did this great master attain to such perfection? The form of his
demonstrations does not suggest an intimacy with the logic of his
immediate predecessor Aristotle; but from him he might easily have
obtained the whole notion of a strictly deductive science, which,
starting from the smallest possible number of primary data, proceeds to
derive from these by strict demonstration proposition after proposition.
Philosophers of our own time have often expressed wonder at the
clearness with which these data are laid down. They are three in kind:
first the _common notions_, which apply to all science and all practical
life, such as “the whole is greater than its part”; secondly, the
_axioms_ peculiar to our intuition of space, such as “two right lines
cannot enclose a space”; and thirdly the very simple _postulates_, which
amount to the use of a ruler and compass with a pencil. There are
besides very careful definitions, so careful that they are at first
obscure, because they apply to the ideal construction of the mind in its
intuition of pure space and do not concern themselves about the flaws of
actual figures. Thus his “point which has no parts” is not nothing at
all, but the minimum of definite place; his right line, “which lies in
the same way (όμαλῶς) between any two points taken upon its length,” is
simply unity of direction. Every other line varies in direction in some
of its successive parts. This is a direct appeal to intuition, without
which we can make no beginning in the science of space. Such also is the
axiom about parallel lines. Such is also the proof by superposition, to
show that two triangles, if some of their measurements be the same, must
wholly coincide.
But I must not attempt to give you a lecture on the _Elements_ of
Euclid, of which some of you may have evil recollections. For it is the
misfortune, as well as the glory, of a great work not only to be
repeated for centuries, but to be parroted and travestied by those who
merely accept its greatness from the voice of ages, and who come to
think that the words of inspiration only require blind repetition to
instruct men. So if Euclid has become in many classical schools a sort
of amulet or fetish (which must for common decency be put in the
programme but which may be learned by committing the proofs to memory
without any intelligence) such a misfortune is not the fault of Euclid,
but the most pathetic tribute to his genius. Let me also add, for the
benefit of those of you who have never seen more than six books of the
_Elements_, and who probably thought six more than enough, that these
are but the introduction to the discussion of higher and more complex
questions, which show the large advance made by the Greeks in this
science, and which explain also how in other arts, such as architecture,
there is no defect for want of scientific accuracy. Books VII-X are not
on geometry, but on higher arithmetic, and even treat, as in Book X, of
incommensurable or irrational quantities. With XI he begins to teach
solid geometry, the measurement of pyramids, cones, spheres, and the
like, ending (XIII) with the discussion of the five regular polyhedra,
of which Plato had long since spoken.
From the great sequence of discoverers and teachers of pure mathematics,
I need only here pick out three immortal names: Apollonius of Perga,
living about 200 B.C., whose geometrical treatment of conic sections is,
I am informed, a splendid monument of genius, which would still be the
basis of modern study had not the treatment of these figures by analysis
entirely superseded the geometrical method. Then there is Pappus in the
second century A.D., who gives us in eight books a review of all the
previous masters, with important additions of his own. The third name is
Diophantus, who lived much later, perhaps in the fourth century, and
whose work is considered the first great step toward the science of
Algebra.
All these speculations were developed in the direction of mathematical
physics by Archimedes, Heron, and other great men of the Alexandrian
school. The triumphs of Archimedes in mechanics astonished the Romans,
who, in the defence of Syracuse against their attack, found him equal to
a host. But how little Archimedes confined himself to practical problems
is shown by his famous method of determining the area of a circle by
approximation, by inscribing and circumscribing polygons of a great
number of sides, which can of course be treated and measured as a
complex of triangles. This is still, I am told, the proof admitted by
modern mathematicians as the best.
The works of Heron show not only an excellent practical knowledge of
mechanics, but of hydrostatics, from which he deduces a number of most
ingenious inventions, such as our penny in the slot, and even the
construction of a whole scene acted by marionettes moving by a most
elaborate hidden machinery. It[35] is a fine specimen of his ingenuity
in using the ordinary mechanical contrivances. He postulates a tall
hollow basis, adorned with pilasters, and having an architrave, with
boards covering its upper surface. Over this stands a little round
temple, visible from all sides, with six pillars. It is covered with a
conical roof, and on the apex is a figure of Victory with outspread
wings and holding in her right hand a garland. Under the centre of the
roof stands a figure of Bacchus, holding a thyrsus in his left hand, and
a cup in his right. At his feet lies a little stuffed panther. Before
and behind Bacchus, and outside the temple, stands an altar with dry
shavings of wood. Also on each side, outside the temple, a Bacchante, in
a proper costume and attitude. The whole concern being set up at some
suitable spot, the exhibitor will retire, and the automatic machine will
presently move forward to a fixed spot. The moment it stops, the altar
fire in front of Bacchus will light up, and from his thyrsus will flow
milk or water, and from his cup wine will be poured out on the panther
beneath him, the pilasters beneath will be adorned with garlands, the
Bacchantes will dance round the temple; drums and cymbals will be heard.
When this noise ceases, the figure of Bacchus will turn round to the
other altar and all the movements be repeated in the other direction. As
soon as this has happened the second time, the show is over, and the
whole machine will return to its original place. We have felt bound, he
adds, to make the measurements (which he gives) small, for if made
large, the suspicion naturally arises in the audience that there is a
man inside the machine producing all the movements. This precaution,
then, should be observed in making any automatic machine.
He then proceeds to give in great detail the construction of this
machine. It is as ingenious as any construction of the present day, but
cannot be presented to you without a series of figures, which are given
in his book. Any of you may read it in the Greek (Teubner text), to
which is added an excellent German translation. It will be enough to
mention that the lighting of the altar fires is done by concealing a
lamp inside the altar immediately under the wood, and by withdrawing a
metal plate which separates them. The flowing of milk and wine is
produced by concealing two little reservoirs in the summit of the
building, and leading the liquor by pipes down the inside of the
pillars, and up the inside of the figure of Bacchus, so that, when the
cocks are turned by machinery, the milk and wine flow and rise to the
level of the thyrsus and the cup, which are set underneath the level of
the cisterns. It is evident enough that people who could do these
things were capable of inventing the _sakia_ now in use throughout
Egypt, where a horizontal wheel worked round a capstan by oxen moves
another set perpendicularly, at right angles to it, furnished with jars,
which get filled below and, when they pass over the highest point of
their revolution, are emptied into a water course, and so irrigate a
higher level. This is well known to have been the invention of these
Alexandrian mechanicians, whose theory had long preceded their practice,
and whose applications of science they never valued so highly as their
pure speculations.
Perhaps before leaving the subject I should tell you what was the moving
force in the automatic machinery. It was a weight suspended in the air
by a rope over a pulley, which, as soon as it was allowed to sink from
its support, made the rope, wrapped round the axle of a large wheel,
move the wheel, that was in its turn connected with other wheels. With
very great and ingenious contrivance, as the machinery was all carefully
concealed, the exhibitor could take his seat among the spectators, and
make the ignorant believe that the whole effect was produced by some
magic.
Nor were the laws of optics and the correction of the illusions of sight
neglected. Euclid wrote a work on the subject which is now lost; but the
praise of it by competent men of the Alexandrian school shows that it
was on a level with his other scientific productions. To our educated
public, the work of the Greeks in most fields is known at least by
hearsay; the great library of Greek mathematics, scores of volumes, some
of which are only quite recently published, is, except for Euclid,
absolutely unknown. Yet from it is derived not only the scanty knowledge
of science that filtered through the Romans into Western Europe, but
also that adopted by the Arabs, and which in translations from Arabic
versions came from them into awakening Italy and Germany and France. But
let me add that now, when their discoveries in pure mathematics are
being weighed by the light of expert knowledge, we are assured by all
those really competent to judge that in no field of learning have the
old Greeks shown their amazing originality and acuteness more signally
than in higher arithmetic and in higher geometry.
The great fathers of the exact sciences are therefore in arithmetic the
Pythagoreans, whose history is too obscure to mention from it any single
name before Archytas, Euclid, and Theon of Smyrna; in geometry, Euclid;
in mechanics, Archimedes; in conic sections, Apollonius of Perga; in
hydrostatics, Heron; in astronomy, Eudoxus and Hipparchus; last, but not
least, in higher arithmetic and algebra, Diophantus; all of these were,
moreover, men who did not confine themselves to any single department,
but promoted accurate thinking in many. These, and others hardly less
great, have left a record and a legacy to posterity second to none in
its mighty consequences.
But among them all Aristotle stands out as the “master of those that
knew”--the man who attained in the Middle Ages such celebrity and
authority that he narrowly escaped being canonised as a saint in the
Roman calendar. If that distinction really belonged to the benefactors
of mankind, I know not that any man ever lived who had a better claim to
it. For his life and activity mark an epoch not only in the progress of
many sciences, but in the general culture of the human mind, to which I
know no parallel. He was brought up under the influence of the Socratic
method of inquiry as perfected by Plato, but, though in some popular
works (now lost) he adopted the dialogue as the correct method of
teaching, there can be no doubt that the sober and practical tone of his
mind made him despise all the delays and delights of character-drawing,
and of spinning out the subject, for what we have from him is
pre-eminently plain and scientific in form. There is seldom an
unnecessary sentence; if there be a metaphor, it is a mere flash of
colour across the cold severity of his argument. He writes like a man
who had no time to waste and a vast world of subjects to teach. If it
was still an age when the sciences had not entered upon the path of
observation and experiment, but were philosophical speculations,
Aristotle did more than any man to establish a separation between
philosophy and science, while fully recognising, what in our day most
scientists ignore, that positive science without a sound knowledge of
philosophy is apt to run into fatal mistakes.
Of course this immense programme which Aristotle set before him could
not be carried out without large collaboration, and so we know that, as
Plato seems to have underrated such collaboration, and thus have failed
in fruitfulness among his pupils, Aristotle, who was not chosen as his
successor by the school (I suppose as usual there were jealousies among
the commonplace and docile pupils toward the great original thinker),
formed and stimulated a band of helpers, who gathered special
observations in botany, mineralogy, zoölogy, physics as the science of
nature, and others who put into shape his views on rhetoric and on
poetry, on ethics and on theology. We have, in my opinion, a new
specimen of such delegated work in the now famous _Constitution of
Athens_, which was known and quoted as Aristotle’s through later
antiquity, but which is rather the work of a pupil and not a brilliant
one. But then we know that Aristotle either wrote or brought out 158 of
these tracts on Greek constitutions. To this I shall return in a
subsequent lecture.
Theophrastus, Eudemus, and Aristoxenus are among the best-known names of
these helpers, and from these we have valuable work extant. Physical
geography was entrusted to Dikæarchus. All these researches were carried
out in the same spirit, and with that unity of purpose that marks a
school. There was apparently but one division of all the domain of
science in which Aristotle did no original work, and yet his
contribution to it is not to be underrated. This was the field of pure
mathematics. For we know that he entrusted to his ablest pupil, Eudemus
the Rhodian, the task of writing the history of what other men had done
in this field. These books on the history of arithmetic, of geometry,
and of astronomy (then called astrology) were well known and valued, and
the modern critics declare that whatever is now known about the earlier
development of mathematics was derived from this pure and rich source.
Still more remarkable is it that this, the part of the edifice to which
Aristotle himself did not contribute, should have been the only one that
took root and flourished without any period of corruption or decay. As
to Aristotle’s personal competence in this matter, I am assured by the
best mathematicians that his not infrequent allusions to mathematics, by
way of metaphor or illustration, show a clear and sound understanding of
the subject. It is not, therefore, the vagary of an idle admirer, but
the deliberate expression of a weighty judge, when we learn from him in
his Discussion on Beauty--which he, being a Greek, of course seeks in
form, symmetry, and proportion--that the highest and noblest examples of
earthly beauty are to be found in mathematics.
Euclid was almost the contemporary of Aristotle, and so the Peripatetic
Mathematics found at Alexandria a new home and a mighty development,
which lasted for centuries and is not stayed to this day. But the rest
of the vast system of Aristotle seems, after about two generations, to
have fallen into incompetent hands. The activity of the Greek intellect
passed into other channels and became again purely philosophical and
ethical instead of scientific, as I shall show when I speak of the Stoic
and Epicurean systems.