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.)