TRACT 24.

ROOTS AND RECIPROCALS.

459

then

900 + 920

60

1820

60

91

= 30| the second value of

XT ^ 1 91 » i 2^-1 2 X 9U-1 _ 16561 _

V920. Next make —= —; then — r= -—--—r, — "US" —3 d ’ 9-tin 2x91 x3 546

30*33150183, differing from the truth but by 6 in the tenthplace of figures, the true number being 30*33150117.

And in this way may the square roots, in the tabic at theend of this volume, be easily found.

TRACT XXIV.

TO CONSTRUCT THE SQUARE AND CUBE ROOTS AND THERECIPROCALS OF THE SERIES OF THE NATURAL NUMBERS.

1. For the Square Roots.

Since the square root of a z + n is a + — — — See:

therefore the series of the square roots of a z , a* + I, a z -f- 2,« 2 + 3, &c, and their 1st, 2d, 3d, 4th, &c differences, will beas below i

Nos-

U z

d?+ 1a z + 2a 2 + 3a x -\- 4

Square Roots.

a

1st Diffs.

1 11

2(1 Diffs.

a + — — - 1 - + ~

2a 8a3 1 iOaS

2a 8 ' 16a y

1 3

1 3,7

4#3 $a 6

,2 4.8

a +-—— + —

— - v + ~

1 6

,.3 9 , 27

a + --r + -

1 5 q_ 19

1 9

.4 16 , 64

a + - - r + —

1 7 37

3d Diffs.3

&C.

3

Where, the columns of fractions having in each of them thesame denominator, after the first line, in each class, a dot iswritten in the place of the denominators, to save the too fre-quent repetition of the same quantities. Now it is evidentthat, in every class, both of roots and of every set of differ-ences, the first terms are all alike; and therefore, by thesubtractions, it happens that every class of differences con-