456

POWERS OP NUMBERS.

TRACT 22.

5. Again, if m 3 , n 3 , p 3 , be three adjacent cubes ; then

n m — 3m + 3m + 1 ) a nd the differences of these firstp 3 -n 3 = 3n* + 3» + 1 »

differences is 3(« 2 — m 2 ) +- 3(?t — wt) = 6(wi + l), the 2d differ-ence. In like manner, the next 2d difference will be 6(re+1).Then the dif. of these 2d differences is 6(n — m) = 6 the 3ddifference, which therefore is constant. Now, supposing theseries of cubes to begin from 0, the first of each of the severalorders of differences will be found by making m ~ 0, in thegeneral expression for each order; thus, 6(»n+l) becomes6for the first of the 2d differences; and 3w*+3wi+ 1 becomes1 for the first of the 1st differences. And hence is found allthe others, as in this table.

3d diIs. 6, 6, 6, 6, 6, 6, 6, 6, 6, &c.

2d difs. 6, 12, 18, 24, SO, 36, 42, 48, 54, &c.

1st difs. 1, 7, 19 , 37, 61, 91, 127, 169, 217, &c.

cubes 0, 1, 8, 27, 64, 125, 216, 343, 512, &c.

And thus may all the powers of the series of natural num-bers 1, 2, 3, 4, 5, &c, be found, by addition only, addingcontinually the numbers throughout the several orders of dif-ferences. And here it is remarkable, that the number of theorders of differences, will be the same as the index of thepowers to be formed ; that is, in the series of squares, there,are two orders of differences; in the cubes, three; in the 4thpowers, four, 8tc: or, which is the same thing, of the squares,the 2d differences are equal to each other; of the cubes, the3d differences are equal; of the 4th power, the 4th diffs. areequal; 8tc. Further, the 2d diffs. in the squares are 1.2 = 2;the 3d diffs. in the cubes 1.2.3 = 6; the 4th diffs. in the 4thpowers 1.2.3.4 = 24; and so on. And from these propertieswere found, by continual additions only, all the series ofsquares and cubes in the table at the end of this volume, andin my large Table of the Products and Powers of Numbers,published in 1781, by the Board of Longitude.