The Greeks do not parade their logic in their writings, though we know they were fond of subtleties; there are indeed examples of it in the _Sophist_ of Plato, where this sort of thing is ridiculed in his travesty of two professional educators. But there are two great and solid proofs of the power which strict Logic had upon their minds. The first comes out in their literature. Wherever they undertake to argue an issue, whether political, social, or religious, their reasoning is clear and easily followed. They of course often start from traditional beliefs, which may not now command assent, but they always reason from these with clear and sober thinking. There was no more important cause for the permanence of that great literature. Its sound thinking has kept it from all extravagance and made it acceptable to educated men of all ages and nations. The second proof is my chief subject to-day: it is the peculiarly logical character of Greek mathematics which has made this too the model of the scientific thinking of the world. 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.