then
log 2 = 7a − 2b + 3c, log 3 = 11a − 3b + 5c, log 5 = 16a − 4b + 7c,
and
log 7 = 1⁄2 (39a − 10b + 17c − d) or = 19a − 4b + 8c + e,
and we have the equation of condition,
a − 2b + c = d + 2e.
By means of these formulae Adams calculated the values of loge 2, loge 3, loge 5, and loge 7 to 276 places of decimals, and deduced the value of loge 10 and its reciprocal M, the modulus of the Briggian system of logarithms. The value of the modulus found by Adams is
| Mo = 0.43429 | 44819 | 03251 | 82765 | 11289 |
| 18916 | 60508 | 22943 | 97005 | 80366 |
| 65661 | 14453 | 78316 | 58646 | 49208 |
| 87077 | 47292 | 24949 | 33843 | 17483 |
| 18706 | 10674 | 47663 | 03733 | 64167 |
| 92871 | 58963 | 90656 | 92210 | 64662 |
| 81226 | 58521 | 27086 | 56867 | 03295 |
| 93370 | 86965 | 88266 | 88331 | 16360 |
| 77384 | 90514 | 28443 | 48666 | 76864 |
| 65860 | 85135 | 56148 | 21234 | 87653 |
| 43543 | 43573 | 17253 | 83562 | 21868 |
| 25 |
which is true certainly to 272, and probably to 273, places (Proc. Roy. Soc., 1886, 42, p. 22, where also the values of the other logarithms are given).
If the logarithms are to be Briggian all the series in the preceding formulae must be multiplied by M, the modulus; thus,