MACHINES, CALCULATINV.. 



MACHINES, CALCULATING. 



401 



j. of the ouiiiben in Uie fint or Moond series above, on all the 

 wheels, wn> brought by hand to the front of the machine. A* an 

 uampk of the method of uing the machine, lot It be required to adil 

 together the fallowing lunu of money : 



S 9i. M. 

 4 IS C 



By menu of a pointer the whoel connected with the first cylinder ou 

 the right is to be turned till the eighth tooth, ami under it the number 

 8 on that cylinder, arc brought to the front; in like manner the 

 number on the second cylinder and 8 on the third are brought to the 

 front : thu the *orie* of number* in the frontt of all the cylinder* 

 exhibit* the fint of the above sums of money. Then, by mean* of the 

 pointer, the fint wheel and cylinder on the right hand are to be turned 

 ull the sixth tooth (after the eighth) comes to the front, when the 

 cylinder having, fruui iU fint position, performed on iU axle one 



revolution and of another, the number 2, representing 12 + 2, or 



le machine were finished and at work according to the inventor's 

 in. 



In any serie* of numben arranged in line or column, if the difference 

 jetween the first and second, between the aecon i < and so on, 



je taken, there will be formed a line or column of what are called first 



ifiVrencvs : if the difference between the fint and second, between the 

 second and third, and so on, of these hut numben be taken, there will 



M formed a line or column of what are called second differences. 



'roceeding in like manner to form third, fourth, ic. orders of 



ifferenoe*, there will at length be found a aeries of differences whieh 

 are either constant, or to a great extent are very nearly *o. Thus, 



iking the aerie* of numben in the column N below, the several orden 



f difference* will be a* in the succeeding column*; the numben in 



be sixth column being constant : 



A'- 



It 



U. sL, appean in front. Now, by the construction of the wheel-work, 

 OM complete revolution of the fint wheel and cylinder on the right 

 hand cause* the wheel and cylinder on ito left to turn one-twentieth 

 part of a revolution ; by which means, hi this example, the number 9 

 being already in front of the Utter cylinder, the number 10 is now 

 there : thus the one shilling obtained by the addition is, as it were, 

 carried. The second wheel and cylinder being now turned by mean* 

 of the pointer till the fifteenth tooth, after the tenth, is brought to 

 the front, the number in front of the cylinder below is 5, representing 



20 + 5, the cylinder having made 1 + revolution* from its fint posi 



tion; there is thus obtained 12. Si. One revolution of the second 

 wheel and cylinder cause* the next, towards the left, to make one- 

 tenth of a revolution ; and thus the It. is carried, so that the number 

 in front of the third cylinder in now 9 instead of 8. This third wheel 

 and cylinder ore next turned by mean* of the pointer till the fourth 

 tooth, after the ninth, is brought to the front, when there appean on 

 the cylinder below it the number 3, representing 10 + 3 ; and the next 

 cylinder on the left, in front of which was previously zero, has now 1, 

 which may thus be said to have been carried. The subtractions are 

 performed exactly in the same manner as the additions, by means ol 

 the numben in the upper series on the cylinden. 



Subsequently to the time of Pascal, Leibnitz invented a machine by 

 which arithmetical computations could be made : but no account of it 

 appean to have been published. 



The most extraordinary calculating machine ever contrived is that 

 of Mr. Babbage, which has had an eventful history. In 1S22 Mr. 

 Babbage read two papers before the Astronomical Society, descriptive ol 

 a machine which he had devised, for working mathematical questions 

 of some complexity, and even printing its own results by means o: 

 types. An application was mode to the government for pecuniary aid 

 in applying the machine to the construction of mathematical tables 

 and in 1823 the government sought the opinion of the Koyal Society 

 on the merit* of the machine. A committee almost unexampled ii 

 scientific strength met to examine the question ; it comprised the 

 name* of Davy, Henchel, Young. Wollaston, Pond, Kater, Brando 

 Baily, Brunei, Colby, and Davic* Oilbert ; and the highly favourabli 

 report sent in by this committee led the government to promi-i 

 pecuniary aid, in a matter intended wholly for public benefit, and no 

 for the inventor's emolument. Unfortunately, the arrangement \\ . 

 badly organised, and has given satisfaction to no one. The plan of the 

 machine underwent changes, the drawing* prepared were very elabo 

 rate, and artisan* had to be specially educated to the work. By the 

 year 1828, a sum of SOOOf. had been advanced by the government 

 parliament complained, and the government complained, because n< 

 machine wan yet forthcoming. A second committee of the Koya 

 Society made a report, countenancing a further application of publi 

 money to this purpose. A sum of 80004. more was advanced, and then 

 0001.. in 1829 ; bcaide* a huge expenditure out of the private purse 

 of Mr. Babbage himself. In 1830 a third committee investigated th 

 subject at the request of the government ; and the report of this 

 committee led to a further advance of funds, not only for making th 

 machine iteslf, but also for building a house to contain it. When 

 17.0001. had been spent, an irreooncileable dispute took place between 

 the government and Mr. Babbage ; no further money was given : anc 

 the operation* were suspended. Mr. Babbage developed a new plan 

 17 which hi* machine would |rform higher mathematical problems 

 hi* fint be called a difference engine ; the second would, if ever com 

 pWted, be an anutflifal engme ; but no fund* being forthcoming, th 

 operation* wen never rammed. In 1848 the government placed th 

 Ufferance engine, *o far a* it had been constructed, in the museum o 

 King's College, London ; and there it remains to the present tim 

 (I860), an unfinished memento of a splendid conception. The machin 

 1* Wrely a yard in height, by half a yard in width and a foot in depth 

 and it certainly appean *urpri*ing that no greater result than this has 

 been presented for so enormous a turn of money. 



.5 ^f t.. ". P~ ble ' we h " no describe the principle an 

 action of Babbages difference engine in a brief form : treating it as 



1-J'J 



Ac. 



Now it is evident that, having any one of the numbers in the first 

 column, and the numbers corresponding to it in the several columns of 

 lifferences ; all the succeeding numbers of the series may be found by 

 mere additions. It may happen however that while the numbers in 

 he original series increase, the numbers in some of the columns of 

 difference may decrease ; and then, iu forming the terms of the series, 

 subtractions might take place : in such a case, the arithmetical com- 

 >lementa of the numbers to be subtracted being taken, the operations 

 nay still be performed by additions only. 



Babbage's Difference Engine accomplishes these additions by the 

 movements of a number of cylinders having on the convex surface of 

 each the series of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The opera- 

 . are thus distributed : by the fint, the additions are made ; and 

 >y the second, there is introduced the 1 which should be carried to the 

 di'i place every time that the sum of two numbers is greater than 10. 

 Let it be imagined that there are several vertical axles, on each of 

 which are several cylinders one above another ; and that these axle* 

 with their cylinders are capable of being turned by wheelwork, so that 

 any one of the ten figures may be made to stand on the face of the 

 machine, and immediately under a fixed index. Let it be further 

 magined that, when the moving power (the hand of the operator 

 applied to a winch) has made one quarter of a revolution, the first, 

 third, fifth, &c., axles may turn, causing the cylinders on them to make 

 parts of a revolution or allowing them to remain at rest as the case 

 inay be, while all the cylinders on the second, fourth, &c., axles remain 

 at rest. Again, when the moving power has made the second qu.-uier 

 of a revolution, let it be imagined that certain, only, of the cyl: 

 on the first, third, fifth, &c. axles make one-tenth of a revolution, so 

 that, whatever figure be under the index of each of these moving 

 cylinders, the next figure, in increasing order, may be brought under 

 that index ; the cylinders on the second, fourth, Ac., axles still rein lin- 

 ing at rest. Let it next be imagined that, when the moving p. -\\ir- 

 deacribea the third quarter of a revolution, the cylinders ou the 

 second, fourth, &c., axles describe parts of a revolution or remain 

 at rest, the cylinders ou the other axles being unmoved : and lastly, 

 when the moving power lias described the fourth quarter of a revo- 

 lution, let certain cylinders only ou the second, fourth, &c., axles 

 make one-tenth of a revolution, the other cylinders remaining at rest. 

 Then it may be understood that by the first and third of these 

 motions, numbers are added together: while, by the second and 

 fourth, 1 is carried to each figure which ought to be increased in 

 consequence of a sum of two figures being greater than 10. 



As an example, let it be required to obtain the numbers of the 

 series in the fint column above, beginning with 3126. The figure* 

 composing the numbers ore brought under the indices in front of 

 the cylinders on the fint vertical axis, towards the le.it hnnd ; and the 

 figures composing the several orden of differences in front of the 



cylinders on the other axis, towards the right, as in this second 

 table. Then the moving power being turned one quarter of a revo- 

 lution, each cylinder in the firat, third, and fifth vertical column* 

 will be turned through as many tentln of a revolution a* are 

 expressed by the number in front of the cylinder immediately on its 

 right hand : thus, the cylinder having :! in front (at the top of the fint 

 vertical column) will moro through four-tenths of a revolution, nn.l the 

 number 7 will be brought under its index in front. The cylinder hav- 

 ing 1 in front (in the fint vertical column) will be turned through 

 six-tenths of a revolution, which will bring 7 under ita index in front ; 



