Mathematics are to be cultivated, not as in the Republic with a view to the science of the idea of good,—­though the higher use of them is not altogether excluded,—­but rather with a religious and political aim.  They are a sacred study which teaches men how to distribute the portions of a state, and which is to be pursued in order that they may learn not to blaspheme about astronomy.  Against three mathematical errors Plato is in profound earnest.  First, the error of supposing that the three dimensions of length, breadth, and height, are really commensurable with one another.  The difficulty which he feels is analogous to the difficulty which he formerly felt about the connexion of ideas, and is equally characteristic of ancient philosophy:  he fixes his mind on the point of difference, and cannot at the same time take in the similarity.  Secondly, he is puzzled about the nature of fractions:  in the Republic, he is disposed to deny the possibility of their existence.  Thirdly, his optimism leads him to insist (unlike the Spanish king who thought that he could have improved on the mechanism of the heavens) on the perfect or circular movement of the heavenly bodies.  He appears to mean, that instead of regarding the stars as overtaking or being overtaken by one another, or as planets wandering in many paths, a more comprehensive survey of the heavens would enable us to infer that they all alike moved in a circle around a centre (compare Timaeus; Republic).  He probably suspected, though unacquainted with the true cause, that the appearance of the heavens did not agree with the reality:  at any rate, his notions of what was right or fitting easily overpowered the results of actual observation.  To the early astronomers, who lived at the revival of science, as to Plato, there was nothing absurd in a priori astronomy, and they would probably have made fewer real discoveries of they had followed any other track. (Compare Introduction to the Republic.)

The science of dialectic is nowhere mentioned by name in the Laws, nor is anything said of the education of after-life.  The child is to begin to learn at ten years of age:  he is to be taught reading and writing for three years, from ten to thirteen, and no longer; and for three years more, from thirteen to sixteen, he is to be instructed in music.  The great fault which Plato finds in the contemporary education is the almost total ignorance of arithmetic and astronomy, in which the Greeks would do well to take a lesson from the Egyptians (compare Republic).  Dancing and wrestling are to have a military character, and women as well as men are to be taught the use of arms.  The military spirit which Plato has vainly endeavoured to expel in the first two books returns again in the seventh and eighth.  He has evidently a sympathy with the soldier, as well as with the poet, and he is no mean master of the art, or at least of the theory, of war (compare Laws; Republic), though inclining rather to the Spartan than to the Athenian