17. A distinction is drawn, and very naturally and properly drawn, by the later Greek mathematicians between ὰριθμητική and λογιστική, by the former of which they designated the ‘science of numbers,’ by the latter, the ‘art of calculation.’ An opposition between these terms occurs much earlier and is frequently used by Plato, but though λογιστική can hardly mean anything but ‘calculation,’ it is not quite clear whether ὰριθμητική then bore the sense which it had undoubtedly acquired by the time of Geminus (say b.c. 50). That it did so, however, is rendered pretty certain by many circumstances. It is probable, in the first place, that the Pythagoreans would have required some variety of terms to distinguish the exercises of schoolboys from their own researches into the genera and species of numbers. In Aristotle a distinction, analogous to that between the kinds of arithmetic, is drawn between γεωδaισίa, the practical art of land-surveying, and the philosophical γεωμετρίa. Euclid, who is said to have been a Platonist and who lived not long after Plato, collected a large volume of the theory of numbers, which he calls ὰριθμητική only and in which he uses exactly the same nomenclature and symbolism as we find in those passages where Plato draws a philosophical illustration from arithmetic. It may therefore be assumed that λογιστική and ὰριθμητική covered, respectively, the same subjectmatter in Plato's time, as afterwards and since he uses these terms casually, with no hint that they were novel, we may infer that the distinction between them dates from a very early time in the history of Greek science and philosophy.