578 TRANSACTIONS OF THE AMERICAN INSTITUTE. 



oscillation. Hence it will be readily understood how, by simple machinery, 

 any constant quantity may be added. 



Now suppose we have three wheels, placed one above the other on a ver- 

 tical (shaft) axis, on each of which is inscribed" zero and the nine digits, 

 corresponding with the like number of divisions on their surfaces. If the 

 number 1 on the upper wheel, 3 on the second wheel, and 2 on the third 

 wheel, be brought opposite a fixed or zero point; and the nature of these 

 ■wheels be such that when set in motion by a lever from right to left, the 

 second wheel adds its number to the upper wheel, and by a motion of the 

 lever from left to right, the third wheel adds its number to the second (be- 

 ing in this case constant and always equal to 2); from this arrangement, 

 ■we will be able to compute a table of square numbers. 



We begin by moving the lever from right to left ; when 3 (the number 

 on the second wheel) will be added to 1 (the number on the upper wheel), 

 making 4 the square of 2. On moving the lever back, 2 on the third wheel 

 is added' to 8 on the second wheel, making 5. Moving our lever back again 

 from right to left, 5 is added to 4 on the upper wheel, making 9 the square 

 of 3. Repea'ting the process, we next get 7 on the second tvheel; which 

 added to 9 oh the upper makes 16, the square of 4. 



It is evident from the above statement that the series of squares is do- 

 , veloped by a process of addition, in which the constant significant differ- 

 ence, increased by each preceding first difference, generates the order of 

 first differences ; and with the sum of the preceding first difference and 

 square, produces the series of squares. This principle is true for any se- 

 ries, whatever number n may be. 



It will now be readily understood, should we have a series of wheels, 

 ranged under each other, the nature of which should be such that the series 

 below should add its number to the one above, we could produce any series 

 of numbers in which the fourth order of differences should be equal. 



In case any difference is negative, we get the same result by adding the 

 complement, or difference between the number and 10. Suppose we wish 

 to subtract 2 from 3, our result would be 1. If we take the complement of 

 2 = 8, and add it to 3, we get (10-|-1), or the same result on the first 

 wheel as before. 



' Having given the fundamental principles on which the machine is con- 

 structed, we will add a few particulars. This machine can be used to 15 

 places of figures, of which 8 places are printed, at the time of making the 

 computation. Thirty seconds is the time necessary for a complete result. 



Before starting the machine for any computation, it is necessary to set 

 the proper wheels, after which it needs no further attention; for so long as 

 the last order of differences is constant, it will continue to produce the 

 required numbers. Thus for producing a table of squares, it is only neces- 

 sary to give the machine three numbers, 1, 3 and 2; and from these data 

 ■we can compute the squares of all numbers up to 30 millions. In the same 

 manner, by giving the machine the numbers 1, 7, 6, 6, we can produce a 

 table of cubes, the limit being 15 figures. The same principles apply in 

 the computation of logarithms, or any series of numbers whatever. 



By changing the position of the carrying post on one of the wheels, we 

 at once obtain the nearest whole number, no matter how many decimal 



