460

ROOTS AND RECIPROCALS

TRACT 24.

tains one term fewer than the one immediately preceding it.These differences are to be employed in constructing tablesof square roots ; and the extent to which the orders of differ-ences are to be continued, must be regulated by the numberof decimal figures to which the roots in the table are to becarried. In the above specimen the differences are continuedas far as the 3d order, where the common first term is whichmay be sufficiently small for constructing all the precedingorders of differences, and then the series of roots themselves,as far as to 7 places of decimals in each, when we commence•with the number 1024, for the first square a % the root of whichis 32. After this, the squares 1025, 1026, 1027, &c, conti-nually increasing, their roots 32+ , 8cc, proceed increasingalso; but the series of numbers, in every order of differences,are all in a decreasing progression ; so that the followingorders are all found by taking each latter difference from theone immediately above it. Then, to construct the table ofroots, having found the first term of each order of differences,as far as necessary, suppose to the 3d order; subtract thatcontinually from the first of the 2d differences, which willcomplete the series of this order of differences. Then thesebeing taken each from the first difference, the successive re-mainders will form the whole series of first differences.—Lastly, these first differences added continually with the firstsquare root, will form the whole series of roots, from thefirst rational root, suppose 32, the root of the square num-ber 1024, to he continued to the next rational root 33, orroot of the next square number 1089. Then begin again,from this last square number, in like manner, with a newseries of roots and differences, which are to he continuedto the third square number 1156, the root of which is thenext rational root 34. Then the like process is to be re-peated a train, and continued from the 3d to the 4th squarenumber. And so on, continuing from each successive squarenumber, to the next, following one, ns far as necessary; thelast of each scries of roots and differences always verifyingthe whole series from square to square.