From ENGINEERING Aug. 20, 1909.pages 256-257
By name, at any. rate, Babbage's famous analytical engine is known to all. It was intended to be a machine for the arithmetical solution of all problems in mathematical physics. Such solutions are generally, perhaps always, feasible, but in most cases when the computations have to be effected by direct human agency, they are so extremely tedious as to be practically, if not theoretically, impossible. Every operation in arithmetic can be reduced to addition, subtraction, multiplication, and division, and, indeed, the two latter operations can be regarded as mere extensions of the two former. The analytical engine was a machine by which these four operations could be performed in any desired sequence ; moreover, a number of partial operations could be combined, and the final results automatically tabulated for any required values of the variable. As is well known, though many years' labour was spent on the machine, it was never even partially completed, Mr. Babbage's scheme being far too ambitious for a first effort. He wished, indeed, to tabulate values to 50 significant figures, thus enormously complicating the mechanism and augmenting the cost of the experiment. In a paper read not long ago before the Royal Dublin Society, Mr. Percy E. Ludgate has revived again the idea of constructing such a machine: As proposed by him, the machine differs from that of Babbage in some fundamental details, though, as in its predecessor, Jacquard cards will be used to control the sequence of operations. Thus if, for instance, a number of values of the series
y = x - x222 + x322.32 - x422.32.42 + &c.
were required, the appropriate card would be placed in the machine, which would then, for different values of x, calculate each term of the series, add all the positive terms together, subtract from this sum all the negative, and print the result. For a different series a different card would be used.
In Babbage's engine it was proposed to effect multiplication by successive additions, and divisions by successive subtractions, just as is now done in the case of the ordinary arithmometer. Mr. Ludgate, in his engine, proposes to effect these operations on entirely different principles. Multiplication is effected by a series of index numbers analogous to logarithms;
The arrangement is shown diagrammatically in Fig. 1. Here the number 813,200 is to be multiplied by 9247. The arrow under 8 represents a slide, which to denote 8 is set at 38 above the zero or starting line. The slide representing 1 lies on the starting line, whilst that representing 3 stands 78 in. above this line, and that corresponding to the number two 18 in. above. On the other hand, the slides representing zero are set 50 eighths above the. starting line. The number of units above the starting line corresponding to each digit of the multiplicand are known as index numbers, and a complete table of these has been drawn up by Mr. Ludgate. All the slides aforementioned are mounted in a frame, and to multiply by. 9, this frame is moved up over another frame divided with another series of index numbers. Thus, as shown, the distance between the lower frame and the starting line is such that the top of this lower frame lies on the index number corresponding to 9; that is, 14 eighths below the starting line. The lower end of the No. 8 slide, represented by the black circle, rests then, it will be seen, on a line marked "72," which is "the product of 8 and 9. The digits 7 and 2 appear accordingly on the register below. Similarly; the tail of the No. 1 slide rests on the No. 9 line, that of the No: 3 slide on the No. 27 line, and that of the No. 2 slide on 18, corresponding to the partial products 9 × 1; 9 × 3 ; and 9 × 2. The tails of the zero slides rest on no line in the lower frame, and hence zero is registered for these. All these partial products are registered in the mill below, as indicated. In a final operation these partial products are added together as indicated, giving 7,318,800. If now the frame is moved to the index number below the starting line marked "2," it will be found, on trial with a piece of tracing paper, that the tail of the No. 8 slide rests now on the line marked "16" — i.e., 8 × 2. That of the No. 1 slide on the index-line marked "2," that of slide 3 on the line marked "15," and that of the No. 2 slide on the line marked "10." These partial products will then appear on the mill and be added together, giving the result of the multiplication of 813,200 by 2. The process is repeated for the remaining figures of the multiplier, and the whole added together so as to give the product of 813,200 × 9247. Mr. Ludgate proposes to give such products to twenty significant figures, the time required being, he states, about 10 seconds.
To divide one number by another he proceeds in a different fashion. He notes that the expression pq, where p and q are any two numbers, can always be expressed in the form—
pq = A p1 + x ,
where x is a small quantity, and A is the reciprocal of some number between 100 and 999.
The above expression can also obviously be written
pq = A p( 1 - x + x2 - x3 + x4 - x5 + &c.),
the series being very rapidly convergent, the first eleven terms give the value of 11 + x correct to at least twenty figures.
He proposes to perform division, therefore, by making the machine first calculate the value of this series, after which it will multiply A p by the value thus found. As a maximum, he considers that this operation giving the result correct to twenty figures might require 112 minutes.