The third quarter of a turn will produce the addition of the third and fifth rows to the second and fourth, omitting the carriages; which it will do by causing the dials of the second and fourth rows to turn through as many divisions as are expressed by the numbers at the indices immediately below them.
The fourth and last quarter of a turn will cause the carriages consequent on the previous addition, to be made by moving the proper dials forward one division.
This evidently completes one calculation, since all the rows except the first have been respectively added to all the rows except the last.
To illustrate this: let us suppose the table to be computed to be that of the fifth powers of the natural numbers, and the computation to have already proceeded so far as the fifth power of 6, which is 7776. This number appears, accordingly, in the highest row, being the place appropriated to the number of the table to be calculated. The several differences as far as the fifth, which is in this case constant, are exhibited on the successive rows of dials in such a manner, as to be adapted to the process of addition by alternate rows, in the manner already explained. The process of addition will commence by the motion of the dials in the first, third, and fifth rows, in the following manner: The dial A, [fig. 1], must turn through one division, which will bring the number 7 to the index; the dial B must turn through three divisions, which will 0 bring to the index; this will render a carriage necessary, but that carriage will not take place during the present motion of the dial. The dial C will remain unmoved, since 0 is at the index below it; the dial D must turn through nine divisions; and as, in doing so, the division between 9 and 0 must pass under the index, a carriage must subsequently take place upon the dial to the left; the remaining dials of the row T, [fig. 1], will remain unmoved. In the row D2 the dial A2 will remain unmoved, since 0 is at the index below it; the dial B2 will be moved through five divisions, and will render a subsequent carriage on the dial to the left necessary; the dial C2 will be moved through five divisions; the dial D2 will be moved through three divisions, and the remaining dials of this row will remain unmoved. The dials of the row D4 will be moved according to the same rules; and the whole scheme will undergo a change exhibited in [fig. 2]; a mark (*) being introduced on those dials to which a carriage is rendered necessary by the addition which has just taken place.
Fig. 2.
The second quarter of a turn of the moving axis, will move forward through one division all the dials which in [fig. 2] are marked (*), and the scheme will be converted into the scheme expressed in [fig. 3].
