rator to the denominator of the given fraction, or made the denominator to the numerator of it, will form the fractional root required.

And this rule will serve, whether the root be finite or infinite, 3. Or reduce the vulgar fraction to a decimal, and extract its root,

4. Mixed numbers may be either reduced to improper fractions, and extracted by the first or second rule, or the vulgar fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted.

By means of the square root also may readily be found the 4th root, or the 8th root, or the 16th root, &c, that is, the root of any power whose index is some power of the number 2; namely, by extracting so often the square root as is denoted by that power of 2; that is, two extractions for the 4th root, three for the 8th root, and so on.

So, to find the 4th root of the number 21035·8, extract the square root two times as follows:

TO EXTRACT THE CUBE ROOT.

I. By the Common Rule*:

1. HAVING divided the given number into periods of three figures each, (by setting a point over the place of units, and also over every third figure, from thence, to the left hand in whole numbers, and to the right in decimals), find the nearest less cube to the first period; set its root in the quotient, and subtract the said cube from the first period; to the remainder bring down the second period, and call this the resolvend.

2. To three times the square of the root, just found, add three times the root itself, setting this one place more to the right than the former, and call this sum the divisor. Then divide the resolvend, wanting the last figure, by the divisor, for the next figure of the root, which annex to the former; calling this last figure e, and the part of the root before found let be called a.

3. Add all together these three products, namely, thrice a square multiplied by e, thrice a multiplied by e square, and cube, setting each of them one place more to the right than the former, and call the sum the subtrahend; which must not exceed the resolvend; but if it does, then make the last figure e less, and repeat the operation for finding the subtrahend, till it be less than the resolvend.

4. From the resolvend take the subtrahend, and to the remainder join the next period of the given number for a new resolvend; to which form a new divisor from the whole root now found; and from thence another figure of the root, as directed in Article 2, and so on.

* The reason for pointing the given number into periods of three figures each, is because the cube of one figure never amounts to more than three places. And, for a similar reason, a given number is pointed into periods of four figures for the 4th root, of five figures for the 5th root, and so on.

And the reason for the other parts of the rule depends on the algebraic formation of a cube: for, if the root consist of the two parts a + b, then its cube is as follows: (a + b)3 = a3 + 3a2b+ 3ab2b3; where a is the root of the first part a3; the resolvend is 3a2b + 3ab2 + b3, which is also, the same as the three parts of the subtrahend; also the divisor is 3a2 + 3a, by which dividing the first two terms of the resolvend 3ab + ab3, gives b for the second part of the root; and so on.

EXAMPLE.

EXAMPLE.

To extract the cube root of 48228.544.

3 x 32 = 27 48228 544 (36'4 root.

Ex. 2. Extract the cube root of 571482.19.
Ex. 3. Extract the cube root of 1628 1582.
Ex. 4. Extract the cube root of 1332.

II. To extract the Cube Root by a short Way*.

1. By trials, or by the table of roots at p. 90, &c, take the nearest rational cube to the given number, whether it be greater or less; and call it the assumed cube.

2. Then

* The method usually given for extracting the cube root, is so exeedingly tedious, and difficult to be remembered, that various other approximating rules have been invented, viz. by Newton, Raphson, Halley, De Lagny, Simpson, Emerson, and several other mathematicians; but no one that I have yet seen, is so simple in its form, or seems so well adapted for general use, as that above given. This rule is the same in effect as Dr. Halley's rational

formula,

2. Then say, by the Rule of Three, As the sum of the given number and double the assumed cube, is to the sum of the assumed cube and double the given number, so is the root of the assumed cube, to the root required, nearly. Or, As the first sum is to the difference of the given and assumed cube, so is the assumed root to the difference of the roots nearly.

3. Again, by using, in like manner, the cube of the root last found as a new assumed cube, another root will be obtained still nearer. And so on as far as we please; using always the cube of the last found root, for the assumed cube.

EXAMPLE.

To find the cube root of 21035.8.

Here we soon find that the root lies between 20 and 30, and then between 27 and 28. Taking therefore 27, its cube is 19683, which is the assumed cube. Then

As 60401.8 : 617546 :: 27 : 27·6047.

27

4322822 1235092

60401-8) 1667374-2 (27.6047 the root nearly.

459338
36525

284
42

formula, but more commodiously expressed; and the first investigation of it was given in my Tracts, p. 49. The algebraic form of it is this:

As P2A: a + 2p :: r: R. Or,

where P is the given number, A the assumed nearest cube, r the cube root of A, and R the root of p sought.

Again, for a second operation, the cube of this root is 21035-318645155823, and the process by the latter method will be thus:

As 63106-43729 : diff. ·481355 :: 27·6047 :

conseq. the root req. is 27-604910560.

Ex. 2. To extract the cube root of 67.

Ex. 3. To extract the cube root of '01.

TO EXTRACT ANY ROOT WHATEVER*.

LET P be the given power or number, n the index of the power, a the assumed power, its root, R the required root of P. Then say,

As the sum of n + 1 times ▲ and n is to the sum of n + 1 times P and n so is the assumed root r, to the required root R.

1 times P,

1 times A;

Or, as half the said sum of n + 1 times A, and n- -1 times , is to the difference between the given and assumed powers, so is the assumed root r, to the difference between the true and assumed roots; which difference, added or subtracted, as the case requires, gives the true root nearly.

That is, n+1. A + n−1. P: n + 1. P. + n − 1. A :: r : R.

Or, n+1. A+ n − 1. {P : P∞ A :: r : Ros r.

And the operation may be repeated as often as we please, by using always the last found root for the assumed root, and its nth power for the assumed power A.

*This is a very general approximating rule, of which that for the cube root is a particular case, and is the best adapted for practice, and for memory, of any that I have yet seen. It was first discovered in this form by myself, and the investigation and use of it were given at large in my Tracts, p. 45, &c.

EXAMPLE.