CH. XIIl] CALCULATING PRODIGIES 289 



£70 before mentioned, made £92. 15s. Od Next the 5s. due at 

 the end of the second year (as interest on the £5 due at the end 

 of the first year) produced in the same way an annual interest 

 of 3d All these three-pences amount to (12/3) (13/2) (14) (3) 

 pence, i.e. £4. lis. Od. which, added to the previous sum of 

 £92. 15s. Od., made £97. 6s. Od To this we have similarly 

 to add (11/4) (12/3) (13/2) (14) (3/20) pence, i.e. 12s. 6d, which 

 made a total of £97. 18s. 6d To this again we have to add 

 (1.0/5) (11/4) (12/3) (13/2) (14) (3/400) pence, i.e. Is. 3d, which 

 made a total of £97. 19s. 9d. To this again we have to add 

 (9/6) (10/5) (11/4) (12/3) (13/2) (14) (3/8000) pence, i.e. Id, 

 which made a total of £97. 19s. lOd The remaining sum to 

 be added cannot amount to a farthing, so he at once gave the 

 answer as £97. 19s. lOd The work in this particular example 

 did in fact occupy him less than one minute — a much shorter 

 time than most mathematicians would take to work it by aid 

 of a table of logarithms. It will be noticed that in the course 

 of his analysis he summed various series. 



In the ordinary notation, the sum at compound interest 

 amounts to £(1"05)" x 100. If we denote £100 by P and -05 

 by r, this is equal to P(l+r) 14 or P(l + 14r + 91r 2 + ...), 

 which, as r is small, is rapidly convergent. Bidder in effect 

 arrived by reasoning at the successive terms of the series, and 

 rejected the later terms as soon as they were sufficiently small. 



In the course of this lecture Bidder remarked that if his 

 ability to recollect results had been equal to his other intel- 

 lectual powers he could easily have calculated logarithms. 

 A few weeks later he attacked this problem, and devised a 

 mental method of obtaining the values of logarithms to seven 

 or eight places of decimals. He asked a friend to test his 

 accuracy, and in answer to questions gave successively the 

 logarithms of 71, 97, 659, 877, 1297, 8963, 9973, 115249, 

 175349, 290011, 350107, 229847, 369353, to eight places of 

 decimals ; taking from thirty seconds to four minutes to make 

 the various calculations. All these numbers are primes. The 

 greater part of the answers were correct, but in a few cases 

 there was an error, though generally of only one digit : such 



b. r. 19 



