More often secondary education was imparted, not in the regular schools by regular, established masters, but by the wandering savants, who taught every conceivable subject, and were all grouped together under the general name of Sophists.[498] From this category the mathematicians and astronomers, who in all respects occupied the same position, are often excluded. This is due to the authority of Plato, who, while detesting the other subjects taught as secondary education, had a great affection for mathematics and astronomy, the only subjects which he prescribes for lads in the _Republic_ and _Laws_. But Aristophanes, taking a more logical position, includes geometry and astronomy among the subjects taught by the burlesque Sophists of the _Clouds_. In point of fact, secondary education included any subject that the lad or his parents desired; and the wandering professors who imparted it, and even established teachers like Isokrates, who kept permanent secondary schools at Athens, were all alike, in the popular view, Sophists. But the more important subjects do naturally fall into two great groups, Mathematics and Rhetoric. Mathematics, as may be seen from the _Republic_, meant, as a part of secondary education, the Science of Numbers, Geometry, and Astronomy, with a certain amount of the theory of Music, which, owing partly to Pythagorean traditions, was classed with mathematics. We have already seen a class learning Astronomy. Plato, in the _Theaitetos_,[499] supplies a sketch of a lesson in more advanced arithmetic, which, by Hellenic custom, was usually expressed in geometrical terms in order to obtain the assistance of a diagram. The lad Theaitetos says to Sokrates that Theodorus of Kurene, the great contemporary mathematician, had been teaching him. “He was giving us a lesson in Roots, with diagrams, showing us that the root of 3 and the root of 5 did not admit of linear measurement by the foot (that is, were not rational). He took each root separately up to 17. There, as it happened, he stopped. So the other pupil and I determined, since the roots were apparently infinite in number, to try to find a single name which would embrace all these roots. “We divided all number into two parts. The number which has a square root we likened to the geometrical square, and called ‘square and equilateral’ (_e.g._ 4, 9, 16). The intermediate numbers, such as 3 and 5 and the rest which have no square root, but are made up of unequal factors, we likened to the rectangle with unequal sides, and called rectangular numbers.” And so on. As the pupils apply the same principle to cubes and cube roots, Theodorus must have initiated them into the mysteries of solid geometry also.