86 Scientific Proceedings, Royal Dublin Society. 



specialTneclianism which cannot be described in this paper, and are included 

 in the tables merely to render them complete. 



The sum of two products is obtained by retaining the first product in 

 the mill until the second product is found — the mill will then indicate 

 their sum. By reversing the direction of rotation of the mill before the 

 second product is obtained, the difference of the products results. Conse- 

 quently, by making the multiplier unity in each case, simple addition and 

 subtraction may be performed. 



In designing a calculating machine it is a matter of peculiar difficulty and 

 of great importance to provide for the expeditious carrying of tens. In 

 most machines the carryings are performed in rapid succession ; but Babbage 

 invented an apparatus (of which I have been unable to ascertain the details) 

 by means of which the machine could " foresee " the carryings and act on the 

 foresight. After several years' work on the problem, I have devised a method 

 in which the carrying is practically in complete mechanical independence of 

 the adding process, so that the two movements proceed simultaneously. By 

 my method the sum of m numbers of n figures would take 9m + n units of 

 time. In finding the product of two numbers of twenty figures each, forty 

 additions are required (the units' and tens' figures of the partial products being 

 added separately). Substituting the values 40 and 20 for m and n, we get 

 9 X 40 + 20 = 380, or 9| time- units for each addition — the time-unit being 

 the period required to move a figure- wheel through -^-^ revolution. With 

 Variables of 20 figures each the quantity n has a constant value of 20, which 

 is the number of units of time required by the machine to execute any 

 carrying which has not been performed at the conclusion of an indefinite 

 number of additions. Now, if the carryings were performed in succession, 

 the time required could not be less than 9 + n, or 29 units for each addition, 

 and is, in practice, considerably greater.' 



In ordinary calculating machines division is accomplished by repeated 

 subtractions of the divisor from the dividend. The divisor is subtracted 

 from the figures of the dividend representing the higher powers of ten until 

 the remainder is less than the divisor. The divisor is then moved one place 

 to the right, and the subtraction proceeds as before. The number of 

 subtractions performed in each case denotes the corresponding figure of the 

 quotient. This is a very simple and convenient method for ordinaiy 

 calculating machines ; but it scarcely meets the requirements of an Analytical 

 Machine. At the same time, it must be observed that Babbage used this 

 method, but found it gave rise to many mechanical complications. 



^ For further notes on the problem of the carrying of tens, see C. Babhage : " Passages from the 

 Life of a Philosopher," p. 114, &c, 



