The exact age at which arithmetic was taught to boys at Athens involves a somewhat complicated inquiry. The arrangements which Plato makes in the _Republic_ and _Laws_ defer this subject till the age of sixteen. In the _Laws_[268] he says: “It remains to discuss, first the question of Letters, and secondly that of the Lyre and practical arithmetic――by which I mean so much as is necessary for purposes of war and household management and the work of government.” His citizens will also require, he thinks, enough astronomy to make the calendar intelligible to them. In this passage he distinctly couples practical arithmetic with music; and when he proceeds to detail, he makes the study of the lyre last from thirteen to sixteen, and then deals with arithmetic, the weights and measures, and the astronomical calendar, studies which terminate with the seventeenth year. This course is designed for all the free boys in his State: it is to be noticed that it is eminently practical, elementary, and concrete. In the _Republic_ he is educating a few picked boys: before they are eighteen they are to have gone through a course of abstract and theoretical mathematics, the Theory of Numbers, Plane and Solid Geometry, Kinetics and Harmonics. Thus he mentions two sorts of mathematics, the one practical and concrete, called by the Hellenes λογιστική [logistikê],[269] whose object is mainly mercantile, and the other theoretical and abstract, which they called ἀριθμητική [arithmêtikê]. Both sorts are to be learned in the period next before the eighteenth year. But it must not be assumed that this was the case at Athens. The philosopher is dealing with an ideal State, where education can be arranged in the theoretically best way, not with the real Athens, where the boy might be called away to the counting-house or the farm at any moment, and many did not stay at school after they had once learned to read and write. Moreover Plato, as a good Pythagorean, saw a peculiar appropriateness in making numbers follow music, and his Dorian sympathies made him divide up education into clearly marked periods, in each of which only one subject was taught. This arrangement, I have already shown, did not find favour at Athens.