178 
MINUTES OF PROCEEDINGS OF 
(i) 
Nos. 
log 
Ai 
a 2 
Correc¬ 
tions. 
log 
1280 
107 210 
+ 3380 
+ 3353 
+ 3318 
+ 3313 
+ 3278 
+ 3254 
+ 3228 
+ 3205 
+ 3182 
107 210 
1290 
110 590 
- 27 
110 590 
1300 
113 943 
- 35 
+ 10 
113 943 
1310 
117 261 
- 5 
- 20 
117 271 
1320 
120 574 
- 35 
+ 10 
120 574 
1330 
1340 
123 852 
127 106 
- 24 
- 26 
- 1 
+ 2 
123 852 
127 105 
1350 
130 334 
- 23 
- 1 
130 334 
1360 
133 539 
- 23 
133 539 
1370 
136 721 
136 721 
(n) 
Ai 
+ 3380 
+ 3353 
+ 3328 
+ 3303 
+ 3278 
+ 3253 
+ 3229 
+ 3205 
+ 3182 
- 27 
- 25 
- 25 
- 25 
- 25 
- 24 
- 24 
It is at once evident from (I) that a correction of + 10 applied to 117 261, 
and a correction of —1 applied to 127 106, will introduce regularity into 
the column of second differences A a . These corrections are introduced in 
(II), where the differencing has been repeated. We cannot apply the common 
rule of proportional parts to calculate the logarithms of intermediate num¬ 
bers, because the column of first differences A x is so far from being constant. 
We can, however, find by interpolation the logarithms of 1335, 1345, 
1355, &c., thus :— 
(m) 
Nos. log 
Ai A 2 
1330 
123 
852 
5 
125 
481 
1340 
127 
105 
5 
128 
722 
1350 
130 
334 
5 
131 
939 
1360 
133 
539 
5 
135 
133 
1370 
136 
721 
+ 1629 
+ 1624 
+ 1617 
+ 1612 
+ 1605 
+ 1600 
+ 1594 
+ 1588 
- 5 
- 7 
- 5 
- '7 
- 5 
- 6 
- 6 
Still the column of first differences A x is far from constant, and there¬ 
fore the rule of proportional parts cannot be applied as (III) now stands. 
Let the interpolation be carried further, as below :— 
(iy) 
Nos. log 
Ai A 2 
1345 
6 
7 
8 
9 
1350 
1 
2 
3 
4 
1355 
128 722 
129 045 
129 368 
129 690 
130 012 
130 334 
130 656 
130 977 
131 298 
131 619 
131 939 
+ 323 
+ 323 
+ 322 
+ 322 
+ 322 
+ 322 
+ 321 
+ 321 
+ 321 
+ 320 
On referring to a table of seven-figure logarithms it will be found that 
the correct value of log 1351 is 130 6553, whilst th^ value found above is 
