428 
Proceedings of the Royal Society 
the accuracy of his work. Also, an error in the determination of 
the first difference of the sixth order is augmented 82,472,326,300 
times in the final logarithm. Perhaps it is on this account that 
the differences are carried two places further at each step. 
I do not understand in what way the two additional figures 
belonging to one order of difference are disposed of when sub- 
tracted from the difference of the next lower order, and M. Lefort 
gives us no information on this most perplexing subject. Putting 
the embarrassment therewith connected out of view, the perform- 
ance of these twelve hundred operations, subject at every step to the 
prodigious accumulation of early error, is a task which, I venture 
to affirm, was never successfully accomplished by any computer. 
This, however, is but a small part of the difficulty. In order to 
make the matter clear, I have placed in the accompanying scheme 
the logarithims of the first ten numbers, with their differences 
arranged in the manner described, as they ought to be found on the 
first sheet, each of them true to the last figure. 
N. 
Log. 
1st Diff. 
2d Diff. 
3d Diff. 
4th. 
5th. 
6th. 
10000 
•00000 00000 0000 
4 34272 76862 7 
43 42076 382 
868 19823 
26036 83 
10 4100 
520 2 
01 
04 34272 7686 
4 34229 34786 3 
43 41208 184 
867 93786 
26026 42 
10 4048 
519 9 
02 
08 68502 1165 
4 34185 93578 1 
43 40340 246 
867 67760 
26016 02 
10 3996 
519 6 
03 
13 02688 0523 
4 34142 53237 9 
43 39472 568 
867 41744 
26005 62 
10 3945 
519 2 
04 
17 36830 5846 
4 34099 13765 3 
43 38605 151 
867 15738 
25995 23 
10 3893 
519 0 
10005 
06 
07 
08 
09 
10010 
21 70929 7223 
26 04985 4739 
30 38997 8481 
34 72966 8536 
39 06892 4991 
43 40774 7932 
4 34055 75160 1 
4 34012 37422 1 
4 33969 00551 0 
4 33925 64546 6 
4 33882 29408 5 
43 37737 994 
43 36871 096 
43 36004 459 
43 35138 081 
866 89743 
866 63758 
866 37783 
25984 84 
25974 45 
10 3841 
There we find that the sixth difference is rapidly becoming less ; 
at the top of the page it is 5202, and at the end of the 200 opera- 
tions it should shrink to 4619. Now the third class computer, to 
whom the ruled sheet with only the first line inscribed on it was 
delivered, must necessarily have carried the same sixth difference 
all the way down, and therefore, even on the most favourable suppo- 
sition that all the differences had been carried out to the twenty- 
sixth place, the result must have been egregiously erroneous. 
Again, we come inevitably to the number 10010. Now the 
