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]