Their scorner hobble here with shackled feet.
The exact age at which arithmetic was taught to boys at Athens involves a somewhat complicated inquiry. The arrangements which Plato makes in the Republic and Laws defer this subject till the age of sixteen. In the Laws[268] he says: “It remains to discuss, first the question of Letters, and secondly that of the Lyre and practical arithmetic—by which I mean so much as is necessary for purposes of war and household management and the work of government.” His citizens will also require, he thinks, enough astronomy to make the calendar intelligible to them. In this passage he distinctly couples practical arithmetic with music; and when he proceeds to detail, he makes the study of the lyre last from thirteen to sixteen, and then deals with arithmetic, the weights and measures, and the astronomical calendar, studies which terminate with the seventeenth year. This course is designed for all the free boys in his State: it is to be noticed that it is eminently practical, elementary, and concrete. In the Republic he is educating a few picked boys: before they are eighteen they are to have gone through a course of abstract and theoretical mathematics, the Theory of Numbers, Plane and Solid Geometry, Kinetics and Harmonics. Thus he mentions two sorts of mathematics, the one practical and concrete, called by the Hellenes λογιστική,[269] whose object is mainly mercantile, and the other theoretical and abstract, which they called ἀριθμητική. Both sorts are to be learned in the period next before the eighteenth year.
But it must not be assumed that this was the case at Athens. The philosopher is dealing with an ideal State, where education can be arranged in the theoretically best way, not with the real Athens, where the boy might be called away to the counting-house or the farm at any moment, and many did not stay at school after they had once learned to read and write. Moreover Plato, as a good Pythagorean, saw a peculiar appropriateness in making numbers follow music, and his Dorian sympathies made him divide up education into clearly marked periods, in each of which only one subject was taught. This arrangement, I have already shown, did not find favour at Athens.
His system must, then, be received with caution. It is inherently far more probable that the simpler, practical arithmetic would be taught at the elementary schools of letters, which all citizens, including future tradesmen and artisans, attended, not at some later date in a separate school. But can any evidence be found for such an arrangement? Yes, Plato himself in the Laws[270] declares that the future builder ought to play with toy bricks and learn weights and measurements when he is a child. His builder, at any rate, cannot wait to learn arithmetic till he is sixteen. Then, in the same work, he quotes the instance of Egypt, where “a very large number of children learn practical arithmetic simultaneously with their letters,” and he goes on to commend the methods by which it was taught. Now Egypt in the Laws is represented as the home of ideal education, a sort of Utopia. Again, in Plato[271] Protagoras blames his brother Sophists for “leading their pupils back, much against their wish, and casting them again into the sciences from which they have escaped, practical arithmetic and astronomy and geometry and music.” How could the Sophists[272] be described as “leading them back and casting them again” into studies from which they had escaped? Where had they learnt these subjects before they were fourteen? It could only have been at school. But what the Sophists taught must have been new to the boys, or they would not have paid to learn it. It was new, because the Sophists taught the advanced and theoretical stages, which appear in the Republic, and the elementary schoolmasters taught the simpler and concrete elements of arithmetic, weights and measures, and the calendar, described in the Laws, which were necessary to every Athenian citizen. From all this it may be assumed that the Athenian boys, like Plato’s Egyptian boys, learnt simple arithmetic, weights and measures, and perhaps the calendar, “simultaneously with their letters.”
Now there are two passages in Xenophon which seem to suit this view. They are not conclusive in themselves, but they give a valuable hint. In the first[273] it is stated that any one who knows his letters could say how many letters there are in “Sokrates,” and in what order they occur. In the second,[274] in the course of an argument, two illustrations are used, in close connection with one another. The passage runs:—“Take the case of Letters. Suppose some one asks you how many letters there are in ‘Sokrates,’ and which are they?… Or take the case of Numbers. Suppose some one asks what is twice five?” These two quotations certainly make simple counting a part of learning letters, with which study the second passage also closely connects the multiplication table. It would seem that it was part of a spelling lesson to answer such questions as “How many letters in ‘Sokrates’?” Answer, “Eight.” “Where does R come?” Answer, “Fourth.” It may be noticed also that the symbols of the numerals in ancient Hellas were, with one or two exceptions, identical with the current alphabet. The games with cubic dice and knucklebones, to which the boys were much addicted, must also have needed some arithmetical skill. The natural conclusion is that simple arithmetic, with, probably, the weights and measures, and the outlines of the calendar, were taught by the letter-master: the practice of music by the music-master: while the theory of numbers, of astronomy, and of music were taught by the Sophists to μειράκια.
Simple counting was done on the fingers. “Reckon on your fingers,” says a character in Aristophanes,[275] “not with pebbles.” A common word for counting was πεμπάζειν, “to reckon on the five fingers”; the division of the month into three periods of ten days can be traced to the same custom. But by various devices it was possible to count up to very large numbers on the fingers. Pebbles were also employed to assist in arithmetic. In the case of complicated accounts a reckoning board (ἄβακος or ἄβαξ) was used, on which the pebbles varied in value according to their position. Such boards go back to early days at Athens, for Solon compared the life of a courtier to a pebble upon them, since he was now worth much and now little.[276] A character in a fourth-century comedy[277] sends for an abacus and pebbles, in order that he may do his accounts. The pebbles were arranged in grooves,
being worth one or ten or a hundred and so forth, according to the groove in which they were placed. If they were put on the left-hand side of the board, their value was multiplied by five.[278] The various games of πεσσοί, which somewhat resembled chess, were played on a somewhat similar board to this, and these chess-boards were known as ἄβακες. Now the art of playing with πεσσοί is more than once coupled by Plato with arithmetic or mathematics generally in such a way as to show that the game must have involved mathematical skill.[279] As was usual in Athens, instruction went hand in hand with amusement, and, in playing games, the boys learned arithmetic willingly. A similar value seems to have attached to the game of knucklebones, which the boys in the Lusis are found playing during their whole holiday. Each boy carried a large basket of knucklebones, and the loser in each game paid so many of them over to the winner. The art of playing this game is also coupled with mathematics by Plato;[280] so it must at any rate have encouraged the study of arithmetic, in his opinion. In the school scene of the British Museum amphora, a little bag, usually supposed to contain knucklebones, is figured: so they may even have been used in schools for teaching arithmetic. In another school scene this bag is present with a lyre and ruler; so it was evidently part of the school furniture.
PLATE III.