L. F. MENABREA ON BABBAGE’S ANALYTICAL ENGINE. 687 
a single one (Ad + Ba) a” +4, For this purpose, the cards may 
order m+ q and n + pto be transferred into the mill, and 
there subtracted one from the other; if the remainder is nothing, 
as would be the case on the present hypothesis, the mill will order 
other cards to bring to it the coefficients A 4 and B a, that it may 
add them together and give them in this state as a coefficient 
_ for the single term 2” +? = 2” +4. 
___ This example illustrates how the cards are able to reproduce 
_ all the operations which intellect performs in order to attain a 
determinate result, if these operations are themselves capable 
_ of being precisely defined. 
Let us now examine the following expression :— 
92, 42, 62. 82. 10?..... (20)? 
17, 37. 5. 72. 97..... (2m—1)?. (2Qn+1)” 
DAP 
_ which we know becomes equal to the ratio of the circumference 
to the diameter, when x is infinite. We may require the ma- 
chine not only to perform the calculaiton of this fractional ex- 
_ pression, but further to give indication as soon as the value be- 
comes identical with that of the ratio of the circumference to the 
diameter when » is infinite, a case in which the computation 
would be impossible. Observe that we should thus require of 
the machine to interpret a result not of itself evident, and that 
this is not amongst its attributes, since it is no thinking being. 
Nevertheless, when the cos of n = O has been foreseen, a card 
_ may immediately order the substitution of the value of 7 (x being 
_ the ratio of the circumference to the diameter), without going 
through the series of calculations indicated. This would merely 
_ require that the machine contain a special card, whose office it 
should be to place the number z in a direct and independent 
manner on the column indicated to it. And here we should in- 
_ troduce the mention of a third species of cards, which may be 
called cards of numbers. There are certain numbers, such as 
those expressing the ratio of the circumference to the diameter, 
the Numbers of Bernoulli, &c., which frequently present them- 
selves in calculations. To avoid the necessity for computing 
them every time they have to be used, certain cards may be 
combined specially in order to give these numbers ready made 
into the mill, whence they afterwards go and place themselves on 
those columns of the store that are destined for them. Through 
this means the machine will be susceptible of those simplifica- 
VOL, III. PART XII, 2 2, 
