429 
of Edinburgh, Session 1874-75. 
essential character of the denary system of logarithms, that from 
which it derives all its advantages, is this, — that the mantissa of 
the logarithm of such a number as 10010 is an exact copy of that 
for 1001 ; but this number 1001 has had its logarithm already 
computed to nineteen places, so that we have only to collate the 
two in order to verify this much of the work. That is to say, this 
great unwieldy gap of 200 intervals had already been divided into 
twenty smaller gaps, and the labour of the interpolation has been 
uselessly augmented at least one hundred times. 
And yet further, in the accompanying second scheme I have 
put the logarithms of the first ten numbers on the first sheet of 
the Cadastre manuscript, true all to the fourteenth place, with 
their differences of the first, second, and third orders. Differences 
of the fourth order only make their appearance in the sixteenth 
decimal place, and are here awanting. Fourteen place logarithms, 
then, for numbers above 10,000 can have no differences of the 
fourth, fifth, and sixth orders; and all this display of high orders 
of differences and of additional places of decimals has been a matter 
of pure supererogation. 
N. 
Log. 
1st Diff. 
2d. 
3d. 
10000. 
•00000 00000 0000 
4 34272 7686 
43 4207 
86 
01 
04 34272 7686 
4 34229 3479 
43 4121 
86 
02 
08 68502 1165 
4 34185 9358 
43 4035 
89 
03 
13 02688 0523 
4 34142 5323 
43 3946 
85 
04 
17 36830 5846 
4 34099 1377 
43 3861 
87 
10005 
21 70929 7223 
4 34055 7516 
43 3774 
87 
06 
26 04985 4739 
4' 34012 3742 
43 3687 
87 
07 
30 38997 8481 
4 33969 0055 
43 3600 
86 
08 
34 72966 8536 
4 33925 6455 
43 3514 
87 
09 
39 06892 4991 
4 33882 2941 
43 3427 
86 
10010 
11 
12 
43 40774 7932 
47 74613 7446 
52 0S409 3619 
4 33838 9514 
4 33795 6173 
43 3341 
Here the third differences are,- as it were, constant; the irregu- 
larities shown by them are, as every calculator knows, due to the 
neglect of the farther decimal parts. The third differences, if 
absolutely true, should show a slight steady diminution, and thus 
we may expect, without any calculation, that the two succeeding 
third differences should be from 88 to 85. Now the logarithm 
