a point over the units, and then place points over every third, fourth or fifth figure towards the left hand, according as it is the root of the cube, of the 4th or 5th power that is required; and if there be any decimals annexed to the number, point them after the fame manner, proceeding from the place of units towards the right hand. By this means the number will be divided into fo many periods as there are figures in the root required. Then enquire which is the greatest cube, biquadrate, or 5th power in the first period, and the root of that power will give the first figure of the root required. Subtract the greatest cube, biquadrate, or 5th power from the first period, and to the remainder annex the first figure of your fecond period, which fhall give your dividend.

Raife the first figure already found to a power lefs by unit than the power whofe root is fought, that is, to the 2d, 3d, or 4th power, according as it is the cube root, the root of the 4th, or the root of the 5th power that is required, and mul tiply that power by the index of the cube, 4th, or 5th power, and divide the dividend by this product, fo fhall the quotient be the fecond figure of the root required.

Raife the part already found of the root, to the power whofe root is required, and if that power be found less than the two first periods of the given number, the fecond figure of the root is right. But if it be found greater, you must diminish the fecond -figure of the root till that power be found equal to

ΟΣ

or less than thofe periods of the given number. Subtract it, and to the remainder annex the next period; and proceed till you have gone through the whole given number, finding the 3d figure by means of the two first, as you found the fecond by the firft; and afterwards finding the 4th figure (if there be a 4th period) after the fame manner, from the three firft."

Thus to find the cube root of 13824; point it 13824; find the greatest cube in 13, viz. 8, whose cube root 2 is the first figure of the root required. Subtract 8 from 13, and to the remainder 5 annex 8 the firft figure of the second period; divide 58 by triple the fquare of 2, viz. 12, and the quotient is 4, which is the fecond figure of the root required, fince the cube of 24 gives 13824, the number proposed. After the fame manner the cube root of 13312053 is found to be 237.

3 x 4 = 12) 58 (4

Subtract 13824 = 24 × 24 × 24

Rem.

i3312053237

8 = 2 × 2 × 2

12 ) 53 (4 or) 3

Subtract 12167

23 × 23 × 23

3× 23 × 23=1587) 11450 7

Subtract 13312053=237×287 × 237.

Remain.

In extracting of roots, after you have gone through the number propofed, if there is a remainder, you may continue the operation by adding periods of cyphers to that remainder, and find the true root in decimals to any degree of exactness.

СНАР. ІХ.

OF PROPORTION.

§ 58. W

HEN quantities of the fame kind are compared, it may be confidered either how much the one is greater than the other, and what is their difference; or, it may be confidered how many times the one is contained in the other; or, more generally,

what is their quotient. The first relation of quantities is expreffed by their Arithmetical ratio; the fecond by their Geometrical ratio. That term whose ratio is enquired into is called the antecedent, and that with which it is compared is called the confequent.

$59. When of four quantities the difference betwixt the first and second is equal to the difference betwixt the third and fourth, those quantities are called Arithmetical proportionals; as the numbers 3, 7, 12, 16. And the quantities, a, a + b, e, e + b. But quantities form a feries in arithmetical proportion, when they "increafe or decrease by the fame conftant difference.". As these, a, a + b, a + 2 b, a + 3 b, 6 + 4 b, &c; x, x-b, x-2b, &c; or the numbers, 1, 2, 3, 4, 5, &c; and 10, 7, 4, 1, -2,-5, -8, &c.

§ 60. In four quantities arithmetically proportional," the fum of the extremes is equal to the fum of the mean terms." Thus a, a + b, c, e + b, are arithmetical proportionals, and the fum of the extremes (a + e + b) is equal to the fum of the mean terms ( a + b + e). Hence, to find the fourth quantity arithmetically proportional to any three given quantities; " Add the fecond and third, and from their fum fubtract the first term, the remainder fhall give the fourth arithmetical proportional required."

$61. In a series of arithmetical proportionals, "the fum of the first and last term is equal to the fum of any two terms equally diftant from the extremes." If the first terms are a, a + b, a + 2b, &c. and the laft term x, the laft term but one will be x-b, the last but two x 26, the last but three x-3b, &c. So that the first half of the terms, having those that are equally diftant from the last term fet under them, will ftand thus ;

a, a + b, a + 2b, a + 3b, a + 4b, &c. X, X- b, x- 2b, x 3b, x-4b,

a + x, a + x, a + x, a + x, a + x, &c. And it is plain that if each term be added to the term above it, the fum will be a +x equal to the fum of the first term a and the laft term

From which it is plain, that "the fum of all the terms of an arithmetical progreffion is equal to the fum of the first and last taken half as often as there are terms," that is, the fum of an arithmetical progreffion is equal to the fum of the first and last terms multiplied by half the number of terms. Thus in the Thus in the preceding feries, if n be the number of terms, the fum of all the terms will be a + x x

2

§ 62. The common difference of the terms being b, and b not being found in the first term, it is plain that "its coefficient in any

term