LuDGATE — On a Proposed Analytical Machine. 87 



My method of dividiug is based on quite different principles, and to explain 



it I must assume that the machine can multiply, add, or subtract any of its 



Variables ; or, in other words, that a formula-paper can be prepared which 



could direct the macliiiie to evaluate any specified function (which does not 



contain the sign of division or its equivalent) for given values of its variables. 



P 

 Suppose, then, we wish to find the value of - for particular values 



of p and q which have been communicated to the machine. Let the first three 

 figures of q be represented by/, and let A be the recipi'oeal oif, where A is 

 expressed as a decimal of 20 figures. Multiplying the numerator and 



An 

 denominator of the fraction by A, we have ~, where Aq must give a number 



g 



of the form 100 . . . because Aq = -. On placing the decimal point after the 



unit, we have unity plus a small decimal. Represent this decimal by x : 

 then — 



^ = i^ or Apa + x)" 

 q 1 + X 



Expanding by the binomial theorem— 



(1) - = Ap (l~x + x'^-x^ + x^-xi^ + &c.), 



(2) "- = Ap (1 - x) (1 + x') (1 + x") (1 + ««), &c. 



The series (1) converges rapidly, and by finding the sum as far as «'" we 

 obtain the correct result to at least twenty figures ; whilst the expression (2 

 gives the result correctly to at least thirty figures. The position of the decimal 

 point in the quotient is determined independently of tliese formulae. As the 

 quantity A must be the reciprocal of one of the numbers 100 to 999, it has 

 900 possible values. The machine must, therefore, have the power of 

 selecting the proper value for the quantity A, and of applying that value in 

 accordance with the formula. For this purpose the 900 values of A are stored 

 in a cylinder — the individual figures being indicated by holes of from one 

 to nine units deep in its periphery. When division is to be performed, this 

 cylinder is rotated, by a simple device, until the number A (represented on 

 the cylinder by a row of holes), which is the reciprocal of the first three 

 figures of the divisor, comes opposite to a set of rods. These rods then 

 transfer that number to the proper shuttle, whence it becomes an ordinary 

 Variable, and is used in accordance with the formula. It is not necessary 

 that every time the process of division is required the dividiug formula 



