tract 24.
OF NUMBERS.
464
The computation may begin at 1024, for the series ofsquares 1024, 1089, 1156, &c, their differences being 65, 67,69, &c, and their roots 32, 33, 34, £cc, Roots,as in the margin ; in order to find the 32intermediate or irrational roots, to any 33proposed extent in decimals. The roots 34will be obtained true to different num- 35bers of figures, according to the number 36of the orders of differences employed.
The first differences only will give the roots true to 5 placesof figures, in commencing with the square 1024; the 2ddifferences will give the roots true to 9 places; the 3d dif-ferences to 12 places; and so on, as here below.
Squares.
1024
1089
1156
1225
1296
Diffs.
65
67
69
71
First, To find the Diffs.
= 0-015625. . . -38147.+18-J-
-18nS '+ 116a 5
1st dif. 0-015621187-i = 0-000007629
4a3
.—
2d dif. 0-0000076IS
8 a 4
= 3d dif. ..11
2. For the Cube Roots.
Then for the Roots.
3d Dif.• 0 7 1 +
2d Difs.
1st Difs.
Roots.
■00000762
•01562119
32-00000000
761
■015613.57
32 01562119
760
01560596
32-03123476
758
•01559836
32 04684072
757
■01559078
32-06243908
756
•01558321
32-07802986
756
‘01557565
32-09361307
754
■01556809
32-10918872
753
01556055
32-12475681
752
•01555302
32-14031736
750
•01554550
32-15587033
750
•01553800
32-17141588
■01553050
32-18695388
In the series and contrivances for constructing a table ofcube roots of numbers, the process is exactly similar to thatfor the square roots, just above ex-plained, in every respect,differing only in the terms of the general series by which theroot of the binomial is expressed, viz, the series for ’/(«* + n),instead of the series for ^(a^ + n). So that, all the explana-tion, and forms of process, being the same here, as in theformer case, for the square roots, the repetition of these mavhere be dispensed with, and we shall only need to set down