Report on the Calculating Machine of M. Scheutz. 543 



The parts move with the utmost facility, in fact, quite loosely. On 

 this accouut no amount of dust which it would reasonably be 

 expected to receive in any moderate time seems likely to interfere 

 with its action. Besides, it can easily be taken to pieces and 

 examined, if need be. Those motions which are not the direct con- 

 sequences of the revolution of the handle acting through a train of 

 rigid bodies ai"e performed in consequence of gravity, no springs 

 being employed in the whole construction except two, the office of 

 which is quite subordinate. When the parts are moved, they remain 

 in their new places either from their weight or from friction, there 

 being nothing to disturb them. This circumstance, which renders 

 a wilful derangement of the machine exceedingly easy, permits of 

 great simplicity and consequent cheapness of construction ; nor does 

 the machine seem likely to get out of order if reasonable care be 

 taken of it. 



The machine is competent to tabulate to any extent a function 

 whose fourth differences are constant, so long as the expression of 

 the numerical value of the function does not involve more than eight 

 digits. The most general form of such a function is of course 



a + bx + cx'^-^dx^ (1) 



Were the machine restricted to such functions, its use would be 

 limited indeed ; its utility must of course depend on its being ap- 

 plicable to functions in general, which, except in singular cases, may 

 be expressed within a limited range of values of the variable a? by a 

 function of the above form. To estimate the capacities of the ma- 

 chine, or rather of a difference engine in general, whatever may be 

 its particular construction, it will be necessary to investigate how 

 soon the quantities neglected begin to tell in the result. 



Now these quantities are of two kinds ; first, the fifth and higher 

 differences; secondly, the decimals of the fifteenth place. The effect 

 of these may be examined separately. We may always suppose the 

 first spindle to rej)resent the first place of decimals, since it will 

 only be necessary to multiply or divide by some power of 10 should 

 that not be the case. 



Suppose the machine set for Ux, and its first four differences (or 

 to speak more exactly, the differences Am^,_,, A*Mj,_,. A^Mj-.j, 

 A''«j,_2), and worked n periods, so as to give what ought to be «,+„. 



We have 



n . , n .n — l.„ , /n\ 



and since the machine would give m^+„ exactly if the fourth differ- 

 ences were constant, the error {E) will be 

 n.«— l.ra^.n — 3.w^ .5 , w.ra — l.w— 2.« — 3.?t — 4.«— 5 ^^^ 

 1.2.3.4.5 1.2.3.4.5.G 



* This expression will not be absolutely exact, since it is Au,..,, A*«,_i, 

 A'''«j._2, A''«^_2, and not Au^., A-u^, A^u^, A-'a^. that are given correctly; but 

 the inaccuracy thus arising in the estimation of the error committed by leaving 

 out the fifth, &c. diflcrenccs will plainly be insignificant. 



