There remain three kinds of knowledge which should be learnt by freemen—­ arithmetic, geometry of surfaces and of solids, and thirdly, astronomy.  Few need make an accurate study of such sciences; and of special students we will speak at another time.  But most persons must be content with the study of them which is absolutely necessary, and may be said to be a necessity of that nature against which God himself is unable to contend.  ‘What are these divine necessities of knowledge?’ Necessities of a knowledge without which neither gods, nor demigods, can govern mankind.  And far is he from being a divine man who cannot distinguish one, two, odd and even; who cannot number day and night, and is ignorant of the revolutions of the sun and stars; for to every higher knowledge a knowledge of number is necessary—­a fool may see this; how much, is a matter requiring more careful consideration.  ‘Very true.’  But the legislator cannot enter into such details, and therefore we must defer the more careful consideration of these matters to another occasion.  ’You seem to fear our habitual want of training in these subjects.’  Still more do I fear the danger of bad training, which is often worse than none at all.  ‘Very true.’  I think that a gentleman and a freeman may be expected to know as much as an Egyptian child.  In Egypt, arithmetic is taught to children in their sports by a distribution of apples or garlands among a greater or less number of people; or a calculation is made of the various combinations which are possible among a set of boxers or wrestlers; or they distribute cups among the children, sometimes of gold, brass, and silver intermingled, sometimes of one metal only.  The knowledge of arithmetic which is thus acquired is a great help, either to the general or to the manager of a household; wherever measure is employed, men are more wide-awake in their dealings, and they get rid of their ridiculous ignorance.  ‘What do you mean?’ I have observed this ignorance among my countrymen—­they are like pigs—­and I am heartily ashamed both on my own behalf and on that of all the Hellenes.  ‘In what respect?’ Let me ask you a question.  You know that there are such things as length, breadth, and depth?  ‘Yes.’  And the Hellenes imagine that they are commensurable (1) with themselves, and (2) with each other; whereas they are only commensurable with themselves.  But if this is true, then we are in an unfortunate case, and may well say to our compatriots that not to possess necessary knowledge is a disgrace, though to possess such knowledge is nothing very grand.  ‘Certainly.’  The discussion of arithmetical problems is a much better amusement for old men than their favourite game of draughts.  ‘True.’  Mathematics, then, will be one of the subjects in which youth should be trained.  They may be regarded as an amusement, as well as a useful and innocent branch of knowledge;—­I think that we may include them provisionally.  ‘Yes; that will be the way.’  The next