July i i, 1901] 



NATURE 



267 



to look at the machine ; the eyes can be kept on the paper and 

 the series of keys struck the proper number of blows. There 

 is no fear of sliding off on to the next row, as the change from 

 the concave to the level keyheads would at once be felt. 



If the multiplier has more than one digit the second method is 

 still more to be followed. Take, for instance, an example 

 illustrated in one of the pamphlets of the company, 2253x84. 

 You do not, of course, strike the 2, 2. 5 and 3 keys 84 times or 

 the 8 and the 4 keys 2253 times, though if you did the right answer 

 would be found. You get on to the 4 row and strike the last 

 key to the right three times, the next five times and the next two 

 twice each. Then you get on to the S row and, starting at the 

 last key but one to the right, you do the same again. The total 

 number of strokes necessary may be found by adding together 

 the digits in one factor and multiplying the sum by the number 

 of digits in the other. In this case 12 x 2 = 24 strokes. That 

 at, say, 6 strokes a second will be four seconds for the operation. 

 Then the result has to be read and the result wiped off ready for 

 the next. With a greater number of digits the operation is the 

 same. 



It constantly happen? in extended calculations that the result 

 upon the number wheels has to be further operated upon. If the 

 next operation is one of addition or subtraction, the previous result 

 is in the proper place ; the same is true if it is to be divided. But 

 if it has to be multiplied by a new number, the natural thing is to 

 copy it down, wipe it off the machine and multiply in the usual 

 way. This necessity, or supposed necessity, was overcome in 

 Mr. Edmondson's machine by the ingenious method of " working 

 off" results from the machine as distinguished from the usual way 

 of working results on to the machine. That process is impossible 

 in the comptometer, as it is in every other machine except 

 Edmondson's, but instructions are given for a method of 

 multiplying by a figure already on the register without the 

 necessity of wiping it out, which is equally applicable to all 

 arithmometers. It is simply to leave it there and multiply the 

 other factor by a number which is one less than the right one. 

 Then, as the new product by « - i is added to that by one already 

 there, the result is what is wanted. By beginning at the 

 left hand side instead of the right, as explained in the directions, 

 which are abundantly clear, each new figure to be used is read 

 from the undisturbed number wheel most to the left, so that 

 there is no necessity to writedown the intermediate result. Aho, 

 in multiplying long decimals it is best to begin at the left, as in 

 that case a sufficient number of figures can be found on the 

 machine, those discarded having no meaning if the figures 

 operated upon are the results of observations and are not 

 absolute figures. 



Division can, of course, be effected if subtraction can be, 

 for it is merely necessary to go on subtracting the divisor from the 

 earlier digits of the quotient until what is left in those places is 

 less than the divisor, then to shift the place one to the right 

 and start subtracting again. The number of tinges the sub- 

 traction is effected at each place is the figure of the quotient at 

 that place. This, after all, is what every arithmometer does, 

 and the series of indices which record the number of turns of 

 the handle in each place enable the operator to read off the 

 quotient when he has gone as far as may be necessary. 



Now in the comptometer these counting wheels,or their equiva- 

 lent, are absent, and so, unlike arithmometers, it does not leave a 

 record of amultiplicaiion actually effected, but only of the result. 

 If, therefore, a wrong key has been struck, except that the result 

 is wrong there is no means of finding it out, whereas in an 

 arithmometer it is usual to compare the setting and the record 

 of the counting wheels with the figures given, to be sure that the 

 actual operation given to the machine was that intended. If any 

 one or more of the counters indicates a wrong figure it is merely 

 neces.sary to put that place into operation and make so many 

 turns of the handle with the -t- or - gear, or forwards or back- 

 wards, as the case may be, to make the counter read the intended 

 number, when the result will also become right. 



In the comptometer these counters are absent, and there is no 

 kind of record in a multiplication or addition except the result 

 of what the operator really gave to the machine. It would 

 therefore appear that in division there can be no record of 

 what was done, and, therefore, that it would be necessary to 

 write down figure at a time the number of times the set of keys 

 were struck in each place. It is just here that a pleasant sur- 

 prise is met with, and a property of the method of subtracting, 

 by adding the arithmetical complement, is available which I do 

 not think would be foreseen by the arithmometrician in general. 



NO. 1654, YOL. 64] 



The property is this. If the arithmetical complement is added 

 to the group of digits to the left of the dividend that would be 

 first used in ordinary division, and if the push is not put into 

 operation to prevent the carrying, then when the addition has 

 been effected the right number of times the digit on the result 

 wheels which has received these carryings will itself be the same 

 as the number of additions, and the figures to the right of it will 

 have become less than the divisor. All the operator has to do, 

 therefore, is to watch this wheel and count I, 2, 3, &c., every 

 time he strikes the proper keys ; when this wheel reads the same 

 number as his count he then looks at the figures to the right ; 

 if they are more than the divisor he goes on striking and count- 

 ing until they are less. The counting here is not necessary, 

 but it is safe. As soon as they are less the wheel receiving the 

 carryings records the corresponding figure of the quotient, the 

 same number, in fact, that he will have counted. 



This operation is best explained by the aid of an example. 

 Divide 365 by 52. 365 is first set on the result wheels as far to 

 the left as possible. Then the keys carrying the red numbers 

 5 and I in the columns over 6 and 5 are struck, while the 

 operator watches the wheel at first showing 3 and counts i for 

 each time he strikes the 5 and I keys. These really add 48 

 each time. 



The series of numbers indicated below will then one by one 

 appear : — 



Count I 413 



2 461 



.> 3 509 



.. 4 557 



.. 5 605 



,> 6 653 



.. 7 701 



The operator watches the 3 gradually getting larger while he 

 counts. When he has counted 6 it also will read 6, but the 

 next two figures, 5 3, are more than the divisor, so he goes on. The 

 next count, 7, then necessarily agrees with the indication of the 

 wheel which receives the carryings, and the operative wheels to 

 the right show i as the temporary remainder, so the answer at 

 present is 7 and I over. If a long decimal answer is requireH 

 the figures are made to slide along the keys on the rows on 

 which they find themselves, in this case two places at first and 

 then one place at a time, and are pressed down, the fingers 

 alternately and simultaneously rising and falling, while the 

 operator counts and watches the wheel receiving the carryings, 

 and thus each new figure of the quotient is found, the time 

 necessary for a figure varying from two to five seconds according 

 as it is low or high. This is the time I require after no regular 

 practice. I expect a skilled operator would require but little 

 more than half as much. It seems strange at first that the mere 

 process of addition should, where necessary, lead to a long 

 decimal quotient, but, as explained above, such a result must 

 follow. 



The gradual and irregular change of the wheel receiving the 

 carryings until it agrees with the count, so as to give a figure of 

 the quotient, also seems mysterious. The manufacturers do 

 not think it necessary to explain to users why this is so, but 

 they give the following somewhat wholesome advice. "Do 

 not worry about why the above process brings the answer. It 

 is simply an arbitrary rule by which any and all examples in 

 division can be computed on the comptometer, and, once under- 

 stood, is so simple that it cannot be forgotten. All there is to 

 I it is that you strike the divisor on the keys just as many times 

 I as indicated by the figure in the ' next place to the left in the 

 I register,' and then, if the remainder is larger than the divisor, 

 I strike the keys again once or more times until the remainder 

 becomes smaller than the divisor." 



There is no occasion for much w'orry any way, for the mystery 

 may be explained quite easily. Let na + ;- be the dividend and 

 a the divisor : then n is the quotient and r is "over." What 

 is done by the machine is to add i- a n times, counting up to 

 «. When this has been done the result will be na ~ k+ k(i - <j) 

 = «, and r is " over." 



The operation described is quite simple, easy and quick where 

 the divisor has two figures only, and is not inconvenient with as 

 many as four, for then two fingers of each hand may be used 

 and the keys struck without looking at them. When the 

 divisor has more than four figures the process is modified in an 

 ingenious way, but in such cases the comptometer is, in my 



