based on Napier's Rods. 193 



6 + 3. In the process the elements of the product came out 

 in the following order: — 



With F at 12 appeared (8, 4), 



„ „ 11 „ (8, 3), (7, 4), 



„ „ 10 „ (8, 2), (7, 3), (6, 4), 



„ „ 9 „ (8, 1), (7, 2), (6, 3), (5, 4), 



„ „ 8 „ (7, 1), (6, 2), (5, 3), (4, 4), 



and so on ; at each step all the elements of the product which 

 have the rank indicated by F. 



4. It is evident how the apparatus may be made to work 

 ap + bq + . . . , so as to require no writing except to record the 

 answer, the number of slides on the frame and places on the 

 board being supposed great enough. For example, with an 

 apparatus having the range shown in fig. 1, we can write down 

 the sum of the squares of two numbers of 5 figures, to do 

 which in the ordinary way would require about 75 figures 

 more. 



5. Division. — Let the multiplicand (divisor) be set up, as 

 before described, on the board, and the product (dividend) 

 written on paper, which it would be convenient to have ruled 

 to correspond with the scale S S. Then the question is — how 

 to get the slides a, b, c, &c. set to the successive digits of the 

 multiplier (quotient). 



The apparatus may be made to do this by following the 

 ordinary process of division; and it is obvious what advantages 

 that plan would have*. But it is much better to follow a 

 method analogous to Horner's synthetic division in Algebra f, 

 for which the apparatus is very well adapted. 



In this plan the slides are carried by the frame left to right, 

 and each takes its proper setting when it arrives at a column 

 agreed on, usually column 1 or column 2. Until it has passed 

 this column let it be called a trial slide, afterwards a collecting 

 slide. 



6. Division, using three trial slides. 



First stage, fixing a. — Advance F to 3, so that c, b, a come 

 over columns 0, 1, 2. Bring c,b, a down together, seeking 

 somewhat less than the first three figures of the dividend, and 

 subtract the number found J. Put back c, b ; move F on to 

 4, and append a new figure, from the dividend. 



Second stage, fixing b. — Subtract the element shown by a 



* It would "be this : — Having found by inspection (at any stage of the 

 process) what row is to be subtracted, set a slide to that, and move the 

 frame towards the left, subtracting each element as it presents itself. 



t Todhunter's i Algebra/ p. 805. 



% Not the sum of the elements ; for the elements shown by the trial 

 slides are of different ranks. 



