TRACT 24.
OF NUMBERS.
461
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, 84, £CC, Roots. Squares. Diffs.as in the margin; in order to find the 32 1024
intermediate or irrational roots, to any 33 1089
proposed extent in decimals. The roots 34 1156
will be obtained true to different num- 35 1225
bers of figures, according to the number 36 1296
of the orders of differences emploj-ed.
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.
65
67
69
71
First, To find the Diffs.
1 __
2 a “— 1 _8 ns “+ 1 _Ida 2 * 4
0*015625. . . -38147+18a
1st dif.
0-0156211S7
1 __
0-000007629
— 3
-
8a 5
2d dif.
0-000007618
Then for the "Roots.
2d Difs.
1st Difs.
•00000762
•01562119
761
Oi 561337
760
'01560596
758
•01359836
757
•01539078
756
•01538321
756
■01357565
754
•01556809
753
01556055
752
■01555302
750
•01554550
750
■01553S00
•01553050
2. For the Cube Roots.
Roots.32-0000000032-0156211932-0312347632 0468407232-0624390832-0780298632-0936130732-1091887232-124756S132-1403173632-1558703832-1714158832-186953S8
In the series and contrivances for constructing a table ofcube roots of numbers, the process is exactly similar to that
for the square roots, just above explained, in every respect,differing only in the terms of the general series by which theroot of the binomial is expressed, viz, the series for’/(a’ + w),instead of the series for 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 maj*here be dispensed with, and we shall only need t.o set down