CAL C ULA TING-MA CHINJES. 



4+5 



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 D^, 

 the first two of the column D,, and the first term, 6, of Dj. The table 

 is then completed by the law given above. 



Fia. 8.— Cubes. 



These explanations are necessary to enable the reader to com- 

 prehend 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 insur- 

 ances, 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 progres- 

 sion 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 



