Many of the primary schools of the better sort started courses of study for lads, providing, no doubt, separate class-rooms, or else the younger boys attended at different hours from those at which the elder pupils assembled. Probably some such provision had been made much earlier for those who wished to obtain a more advanced knowledge of literature and music than was offered by the primary schools. But in the time of Sokrates many masters seemed to have held classes for lads as well as for boys. On entering the schools of Dionusios,[495] the master of letters, Sokrates finds a class of lads assembled here.[496] They all belong to noble families: the poor were no doubt unable to afford education of this sort. Two of the lads were busy discussing a point of astronomy, and were quoting the authority of Oinopides[497] and Anaxagoras, for Sokrates catches these two names as he enters the room. They were drawing circles on the ground and imitating the inclination of some orbit or other with their hands. This scene shows a much more advanced sort of study than was usual at the primary school of letters. The Sophists seem to have often lectured in class-rooms.
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.
Here we find a professor lecturing to a select class, in this case of only two lads, and his pupils, as in the class-room of Dionusios, discussing and elaborating among themselves afterwards the subject-matter of the lecture. Theodoros is mentioned as teaching Geometry, Astronomy, and the theory of Music, as well as the Science of Numbers. Geometry by this time included a good number of the easier propositions which were afterwards incorporated in the works of Euclid; the school of Pythagoras, and, later, that of Plato, did much to develop it. The problem of squaring the circle was already occupying attention.[500] Compasses and the rule were the ordinary geometrical implements: diagrams were drawn on the ground, on dust or sand. In Arithmetic surds[501] were a popular subject: but arithmetical problems, being usually expressed in terms of geometry plane or solid, become as a rule a part of the latter science.
To Plato these mathematical studies are alone suitable for secondary education: the philosopher Teles,[502] carrying on the same tradition, makes arithmetic and geometry the special plagues of the lad.[503] But then the philosophers despised Rhetoric.
Rhetoric, from the time of Gorgias onwards, formed a very large part of secondary education Isokrates was its greatest professor. He provided in his school a course of three or four years for lads, to occupy their time till they were epheboi. But the methods, the aims, and the personality of this interesting professor will be discussed later.
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Besides mathematics and rhetoric, there were literary studies. The Axiochos gives κριτικοί among the teachers of a lad. These are the lecturers on literary subjects, who concerned themselves with interpretations, often far-fetched, of the poets; a summary of the literary discussion in the Protagoras may give some idea of such a lesson.
“Protagoras. I consider that it is a most important part of a man’s education to be skilled in poetry; to understand, that is, what is rightly said, and what is not, by the poets. Simonides says to Skopas, son of Kreon the Thessalian, ‘To become indeed a good man is hard, a man foursquare, wrought without blame in hands and feet and mind.’ You know the poem? Do you know then that farther on in the same poem he says, ‘But the saying of Pittakos, wise though he was, seems to me not said aright: he said, “’Tis hard to be noble.”’ Don’t you see that the poet has contradicted himself?”