order (Fig. 8). We get by direct calculation the first four terms, 1, 8, 27, 64; then, by subtraction, the first three terms of the column D1, the first two of the column D2, and the first term, 6, of D3, The table is then completed by the law given above.
| N. | C. | D1. | D2. | D3. |
| 1 | 1 | 7 | 12 | 6 |
| 2 | 8 | 19 | 18 | 6 |
| 3 | 27 | 37 | 24 | 6 |
| 4 | 64 | 61 | 30 | 6 |
| 5 | 125 | 91 | 36 | 6 |
| 6 | 216 | 127 | 42 | 6 |
| 7 | 343 | 169 | 48 | 6 |
| 8 | 512 | 217 | 54 | |
| 9 | 729 | 271 | ||
| 10 | 1000 |
Fig. 8.—Cubes.
These explanations are necessary to enable the reader to comprehend the function and classification of calculating-machines. The method which is expounded in them appertains to the calculation of differences, and is applicable to all kinds of computations, whether of days' works, tables of interest and annuities, sinking-funds and insurances, tables of logarithms, astronomical tables, or the resolution of numerical equations.
Numeration is based on the theory of geometrical progressions, by which name we call a series of numbers in which each member is equal to the preceding member, multiplied by a fixed number that is called the ratio. Thus the numbers 1, 10, 100, 1,000, 10,000 form a progression of which the ratio is ten, or a decimal progression. The numbers 1, 2, 4, 8, 16, 32, 64, the ratio of which is 2, form a binary progression.
The ancient Tartar hordes communicated with each other by means of sticks notched in an understood manner, so as to indicate the number of men and horses which each camp was expected to furnish. The Inca-Peruvians had knotted cords of various colors, that could be tied in a thousand ways; and the number of knots, their arrangement, the tying of them with sticks, or around a central ring, permitted the expression of a variety of ideas, and of considerable series of numbers. The art of calculation is taught to children in some schools by means of apparatus consisting of a frame with ten rods, on each of which are strung ten counters; and the same kind of an apparatus is managed by the Chinese with much dexterity.
We propose, as an important aid to be used in teaching arithmetical calculation, a vertical checker-board, the squares of which are furnished with pegs on which pawns may be slipped. No distinction is to be made between the white and black squares. "We begin by placing ten black pawns in the squares of the lower horizontal row. Lift the pawns in succession from the right, counting one, two, three, up to nine. This
