This mode of representation will be found particularly advantageous when the analysis is pursued to a still greater number of factors; which should be done gradually, proceeding from three to four, five, and six, factors, in the order pointed out in the following table: The pupil having found out all the different cases of analysis, according to the order of this table, may next be called upon to classify the numbers with reference to their general capability of being resolved into factors. That classification would, when completed, present the whole of the numbers gone through, in the following order: THE LAW OF TRANSPOSITIONS. 327 The attention of the pupil may now be called to the sorts of factors of which each number is composed when analysed to the utmost extent of which it is capable. He will then find, that in this respect he has three sorts of numbers in the table, viz. 1. Such as are composed only of one sort of factors. (a) of twos, 4, 8, 16, 32, 64. (b) of threes, 9, 27, 81. (c) of fives, 25. 2. Such as are composed of two sorts of factors. (c) of threes and fives, 15, 45, 75. 3. Such as are composed of three sorts of factors. 30, 60, 90. After this the pupil may pursue the analysis of each of these sorts independently, as far as the teacher may think it necessary, taking the numbers of each sort in the order pointed out in the tables on p. 289 and 290. Before, however, he be allowed to proceed much farther, he ought to be led to investigate the law of transpositions, a knowledge of which will greatly facilitate the analysis of the higher numbers. For this purpose counters, or wafers, of different colours, will be found very serviceable, the pupil being thus enabled, in a manner the most striking to his eye, to survey the different arrangements of which a given number of objects, some similar and some dissimilar, or all dissimilar, is capable. For instance, if the question be, how many transpositions are possible of three objects of one kind and two of another, the pupil ought to be supplied with a sufficient number of red and yellow wafers, to set out his problem in the following manner: 328 THE LAW OF TRANSPOSITIONS. This problem once solved, will supersede the necessity of his going through all the changes possible with any set of factors which he may meet with in his analysis, analogous to the given set of wafers. For instance, taking the case in hand for an example, the pupil would find among the numbers already analysed the following set of factors. In the analysis of 72: 2 x 2 x 2 x 3 x 3. Proceeding with the numbers beyond 100, he would find, in the analysis of 128, 2 x 2 x 2 x 4 x 4; in the analysis of 288, 2 × 2 × 2 × 6 × 6; in the analysis of 432, 3 x 3 x 3 x 4 x 4; and in each of these cases he would at once know, that the given set of factors is capable of ten changes. In this manner the law of transpositions ought to be connected throughout with the analysis of numbers, which will afford the teacher numberless opportunities of exercising the ingenuity, and drawing forth the minds of his pupils. Into this subject, however, we cannot, without swelling a single chapter to the size of a volume, enter any farther, and we must content ourselves for the present with appropriating the little space we have left, to a few hints on the manner of teaching fractions. For this purpose we would strongly recommend the use of the fractional squares, described in the extract which we have given from Pestalozzi, with this difference only, that instead of placing them in mechanical succession, halfs, thirds, fourths, &c., as was the case in Pestalozzi's fraction tables, we would arrange them in an order similar to that which we have observed with the integers. Thus, for instance, we would exercise the pupil, first in the division of the square into halves, fourths, eighths, &c., and lead him to compare those different fractions, with a view to discover the proportions which they bear to each other. |