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the quotient, and its square under the said left hand term, and subtract. III. Bring down the two next terms to the remainder for a dividend, and set double the root on its left for a divisor. IV. Divide the dividend by the divisor, and subjoin the result to both the quotient and the divisor. V. Multiply the divisor (so increased) by the term last put in the quotient, subtract the product from the dividend, and bring down the two next terms to the remainder for a new dividend. IV. Bring down the divisor, with its last term doubled, to the left of the new dividend, for a divisor; divide ; subjoin the result to both quotient and divisor; multiply; subtract; bring down two terms, &c. proceeding in this manner until all the terms in the dividend are brought down, and the work is finished. The operations may be proved either by addition or involution ". i

* Evolution being the converse of involution, necessarily requires a converse operation, which as far as it respects simple quantities is sufficiently plain; but when the roots of compound quantities are to be extracted, the mode of operating is not so obvious : indeed it does not appear that the rule here given is capable of an investigation a priori, but that it owes its origin to the mechanical process of trial or experiment. Thus it was known by involution, that the square of a + b is a 4 + 2 ab - b%, but the difficulty was, how to frame a rule whereby the square being given the root could be obtained from it : the first member a of the root evidently arises by taking the root of a *, the first member of the square; but the next inquiry was, how the other member b of the root could be had from 2 at 4-bo (the remaining members of the square) without remainder. By trials it was found, that the first member a of the root being doubled, and the second member b added to it, the whole will form a divisor of 2 ab # b”, which will exactly give b (without remainder) in the quotient; and this method (continued according to the precepts given above) will give the roots of all other compound quantities, as is proved by the converse process of involution, from whence alone the truth of the rule is manifest. In the same manner rules for finding the roots of higher powers were discovered, which, as they are not wanted in this place, are reserved for the exercise of the rules of approximation, given in the following part of the work,

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Earplanation.

The greatest square in art is r" ; I put its root r* in the quotient, subtract the square, and bring down 4 x 3 + 6 ar”; I next double the root w?, making 2 r", which I place on the left for a divisor; I divide 4 aro by 2 w", and place the result 2 a both in the quotient and divisor, making the latter 2 w" + 2 r ; this I multiply by 2 r, making 4 r" + 4 ro, which, subtracted from the quantities above, leaves 2 w" ; to this I bring down 4 x + 1, making 2 r * + 4 + + 1 for a new dividend; to the left of this I bring down the divisor, after doubling its last term, by which it becomes 2 r * + 4 a ; I divide the new dividend by this, placing the quotient figure 1 both in the quotient and divisor: then I multiply the divisor by it, place the product below the dividend, and the work is finished. The proof by addition, as well as that by involution, will be sufficiently obvious; the former being the sum of the several products, or lower lines, and the latter the root involved to the square.

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* That this part of the subject may not appear imperfect, the following rule for extracting the roots of powers in general is here given, as it may be useful for finding the cube root; but it is too laborious for roots of a higher denomination.

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58. A surd, or irrational quantity, is a quantity under a radi

RULE I. Find the root of the first term, and place it in the quotient. II. Subtract its power from that term, and bring down the second term for a dividend. III. Involve the root last found to the next lower power, and multiply it by the index of the given power, for a divisor. IV. Divide the dividend by the divisor, and the quotient will be the next term of the root. V. Involve the whole root, subtract, divide, and proceed as before, until the whole is finished. ExAMPLEs. 1. Required the cube root of aro +6a 3 +15+4+20x2 + 152* +6++ l2 are +6 as + 1524 +20 a 3 + 15 a.” +6a + 1 (a 2 + 2 x + 1 the root. & 6 . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = cube of w?.

3a:4) 6 ro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = second term ––3 wo. a 6 + 6 a 5 + 12 w" -H 8 a 3 . . . . . . . . . . . . . . . . = cube of ar” + 2 a. 3.c4) 3 a “ . . . . . . . . . . . . . . . . . . . . . . = remainder —-3 a 4.

a 6 + 6 as +15++ +20 a 3 + 15a, 2 + 6 a + 1 = cube of a " +2 a + 1.

2. Extract the cube root of a 3 + 3 a 2 b + 3 abo + b 3. Root a + b. 3. Required the cube root of a 3 – 15 a 2 +75 a -1252 Root a -5. 4. Required the cube root of a 6 +8 a 5–40 a 3 +96 r–64 2 Root a " + 24 –4. 5. What is the fifth root of a 5–10 w +40 as -80.w' + 80 r—32? Root r—2. The following method is convenient for some of the higher roots. For the biquadrate root, extract the square root of the square root. For the sixth root, .......... the square root of the cube root. For the eighth root, .......... the square root, of the square root, of the [square root. For the ninth root, ... . . . . . . . . the cube root of the cube root, &c. &c. And, in general, all powers whose indices are any powers or products of 2 and 3, may be unfolded by the rules for the square and cube root; but the 5th, 7th, &c. roots, must be found directly by the above rule, or by some of the methods of approximation. Another method is, by finding the roots of some of the most simple terms, (Art. 56.) and connecting them by the sign + or —; then involve this compound root to the required power, which, if equal to the given power, the root is found; but if it differ in some of the signs, let the signs of one or more terms of the root be changed from + to —, or from — to +, till its power agrees in all respects with the given power. k If the learner should find what is here given on surds too difficult, he may omit either the whole, or any part, for the present; he should however resume the subject as soon as he has passed through quadratic equations.

cal sign or fractional index, the root of which cannot be exactly obtained.

Simple surds are such as are expressed by one single term, as A/2, * v5, 3 al+, &c.

Compound surds are such as consist of two or more simple surds, connected by the sign + or —; thus w/3+ v2r, "V7– * v2, "A/3 a + v5, &c. are compound surds: the latter is called an universal root'.

REDUCTION OF SURDS".
59. To reduce a rational quantity to the form of a surd.

RULE. Involve the given quantity to a power equivalent to that of the surd, over which place the index of the surd, and it will be the form required".

ExAMPLEs.

1. Reduce 3 to the form of the square root. Thus, 3]*=3x3=9. Wherefore yo is the answer required.

2. Reduce 2 to the form of the cube root, and 3 a to the form of the fourth root.

1 A surd is a quantity incommensurate to unity, or that is inexpressible in rational numbers by any known method of notation, otherwise than by its proper radical sign or fractional index; hence these numbers are called irrational or incommensurable numbers, and sometimes imperfect powers. When it is proposed to extract any root from a quantity which is a complete power of the same name with the root, such root can be exactly obtained ; but if the given quantity be not a complete power of that name, then the proposed root (which cannot be exactly found) is denoted by placing the sign or index over the quantity: this expression (as we have observed) is what is properly called a surd. The fractional index is mostly to be preferred in practice to the radical sign, because all the rules of fractions may be conveniently applied to fractional indices, whereby the operations are rendered extremely perspicuous and easy. * Reduction of surds does not alter their value, it merely changes them from one form to another; a process which is frequently necessary to prepare them for operations, and for estimating their value. * The reason of this rule is extremely plain, for if any quantity be involved to a power, that quantity is (not only equal to, but is) the root of the power;

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