r in other writings. He cannot leave his old
enemies, the Sophists, in possession of the field; and therefore he
proposes that youth shall learn by heart, instead of the compositions of
poets or prose writers, his own inspired work on laws. These, and music
and mathematics, are the chief parts of his education.

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