168 
MR FRANK ROBBINS ON 
effect, the not unreasonable assumption was made that on the average the error 
in one of the fundamental logs would be 2 x 10“ 16 , but no guess could be made as 
to the sign of the error, whether plus or minus. Although some compensation might 
conceivably arise from the balancing of excess and deficit, it seemed clear that 
towards the end of Tables II and III, where the respondent is the sum of, say, 
60 logarithms, some cumulative error must occur. Still more strongly it was felt 
this would be the case in regard to the closing values in Table I, where log 120 ! 
is the sum of 119 logarithms. . But the result was not as expected. The tables 
having been completed, Dr Cargill G. Knott was so good as to make a comparison 
with the treasured MS. tables given to the nation by the daughters of the gifted 
Dr Edward Sang (7). Between expectation and realisation there is always a gap, 
but here the differences, although on the whole arithmetically smaller than expected, 
were puzzling. In the case of Table I, for the first forty values of the argument 
no difference worth mention occurred. From this point differences of two units in 
the fifteenth place appeared (always + in the sense Sang — Bobbins), and growing 
steadily the difference reached a maximum of 5 for argument 79. Thereafter they 
decreased more rapidly, becoming 2 at argument 97 and then tailing off to zero at 
the end of the table. 
Another comparison made by myself with the British Museum copy of the very 
rare Degen (6) fully confirmed Dr Knott’s report. There was the same exact agreement 
at both ends of the table, and a maximum of the same magnitude occurred at the 
same point. Under these circumstances the tables prepared with so much care were 
abandoned, new foundation stones were dug. out of Dr Sang’s quarry, and the 
Tables I, II, and III were rebuilt with twenty-eight decimal places, but cut down on 
completion to eighteen. Not often perhaps has such a severe pruning been given— 
it should put the final results above suspicion. 
Lastly, sixty values of log 2 n from n = 1 to n = 60 were prepared and applied 
by addition to n !, so as to find checks on the sixty values of log A 
The fourth' table contains factorial n as far as n== 50, and in each case to its full 
extent. Every care was taken to secure accuracy, and each value was obtained 
twice, in the first place by pencil and- then on an arithmometer. 
Further, at short intervals other checks were applied such as — 
35 ! = 1190 x 33 ! = 1256640 x 31 ! 
361 = 2 x 7 x 90x 34! 
431 = 1806 x 41! 
45 1 = 146611080x40! 
481 = 8x6x471 = 3x4x4x47! 
501 = 254251200x40! 
The table of factorial n was submitted to the following final test. 
It is easy to see 
50 ! = 2 47 . 3 22 . 5 12 . 7 8 . II 4 . 13 s . 17 2 . 19 2 . 23 2 . 29 . 31 . 37 . 41 . 43 . 47. 
