What Have the Greeks Done for Modern Civilisation?

by J. P. Mahaffy

Let me go back to the infancy of Greek science and give you evidence for 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.