430 Proceedings of the Royal Society 
of 5006 had been computed to nineteen places, and if we write 
the logarithm of 2 on the edge of a piece of paper, and then 
appose it to that of 5006, we shall obtain at a glance the logarithm 
of 10012. However, we shall not even take the trouble of this 
addition, but shall content ourselves with the fourteenth figure, 
which we find to be 9. Let us write this 9 in its proper place, 
which -is two lines below. All that remains for us is to discover 
what two digits must be written in the last place of the third 
difference in order to produce this digit 9, knowing also that these 
digits should be about 6. 
The strict investigation is simplicity itself. Let us put a , b, c, cZ, 
for the last numbers in the successive columns a lt b iy r 15 d u for 
those which are to succeed a 2 , b.. y c 2 , d. 2 , for the next again, and 
we find 
a 2 — a - 2b + 3c = 3 d 1 + d 2 . 
In resolving this equation we need attend only to the last figures, 
so that, applying it to the example before us, 
• • • 9 — 2 — 2 + 2 = 3d y + d. 2% or 
7 = Sd ± + d % - 
Recollecting that d x and d. 2 must be absolutely or nearly alike, 
we may put for trial 
7=4 d lt 
whence d 1 is nearly 7, and we have actually d x = 7, d. 2 = 6; so that 
the succeeding pair of differences must be 87, 86. Writing these 
in their places, we readily compute the next pair of logarithms as 
shown by the slender figures. 
By a computation performed mentally in a few seconds, without 
putting pen to paper, we thus compute the third differences for the 
next pair of logarithms, with the certainty that the alternate one 
is absolutely true in the last place, and with a very slight uncer- 
tainty as to the intermediate. The whole magnificent array of 
fourth, fifth, and sixth differences, with decimals to the twenty- 
sixth place, disappears. 
This amounts to the recognition of the fact that the large gap 
of 200 intervals had already been supplied with 100 stepping- 
stones, a fact of which neither Prony nor any one of his regiment 
