in the tens' place, and b of that in the units'. Then a is the greatest number whose cube is contained in 405000, that is, 70; subtract its cube from the whole quantity, and the remainder is 62224; divide this remainder by 3a, or 14700, and the quotient 4, or b, is the second term in the root: then subtract the cube of 74 from the original number, and as the remainder is nothing, 74 is the cube root required. Observe that the ciphers may be omitted in the operation; and that as a was at first subtracted, if from the first remainder 3a'b+Sab2+ b3 be taken, the whole cube of a+b will be taken from the original quantity.

158. In extracting the cube root of a decimal care must be taken that the decimal places be three, or some multiple of three, before the operation is begun, by annexing ciphers to the right (Art. 41); because there are three times as many decimal places in the cube as there are in the root (Art. 46).

3a13467..)11134910 second remainder.

The new value of a is 670, or, omitting the cipher, 67; and 3a2, the new divisor, is 13467..hence 8 is the next figure in the root; and

107736..=3a2b
12864.=3ab2

512-63

10902752 subtrahend

232158 the third remainder.

It appears from the pointing, that there is one decimal place in the root; therefore 67.8 is the root required nearly. If three more

It will be clearer to read "a is the greatest multiple of 10 &c."

ciphers be annexed to the decimal, another decimal place is obtained in the root; and thus approximation may be made to the true root of the proposed number to any required degree of accuracy.

159. Since the first remainder is 3a2b+3ab2+ b3, the exact value of b is not obtained by dividing by 3a; and if upon trial the subtrahend be found to be greater than the first remainder, the value assumed for b is too great, and a less number must be tried. The greater a is with respect to b, the more nearly is the true value obtained by division.

For the first remainder divided by 3a gives b+

b2

--

tient; and if this be adopted for b, the error == value of b is evidently less as a is greater.

b3 + for the quo

a За

which for a given

160. In extracting the square or cube root of a vulgar fraction the rule stated in Art. 150 may be followed; but it is generally preferable to convert the vulgar fraction into a decimal, and then extract the root.

Thus let the cube root of 54, or be required.

11
2

Now, if the rule of Art. 150 be applied to this case, the cube root of 11, and the cube root of 2, must be found to a certain number of places of decimals, and then the long division of the one root by the other must be effected: whereas, if 5 be, first of all, converted into a decimal, viz. 5.5, one single extraction of the cube root completes the whole process.

Another method is, to multiply the numerator and denominator by such a quantity as will make the latter a perfect cube, and then apply

the rule of Art. 150.

161. In extracting either the square or cube root of any number, when a certain number of figures in the root have been obtained by the common rule, that number may be nearly doubled by division only.

I. The square root of any number may be found by using the common Rule for extracting the square root until one more than half the number of digits in the root is obtained; then the rest of the digits in the root may be determined by Division.

For, let N represent the number whose square root, consisting of 2n+1 digits, is required;

a...

........ the first n+1 digits of the root found by the common

Rule, with n ciphers annexed;

that is, N-a, (which is the remainder after n+1 digits of the root are found) divided by 2a will give the rest of the root required, x, increased Now, since contains n digits, a' has 2n at most (Art. 154). But, by the supposition, a is a number of 2n+1 digits, and 2a has 2n+1 digits at least; therefore

by

2a

x2 2a

x2<2a, or is a proper fraction, or<1;

that is, if the quotient of (N-a2)÷2a be taken for x, the error is less

than 1.

Hence it appears, that if n+1 digits of a square root are obtained by the common Rule, n digits more may be correctly obtained by Division only.

Ex. Required the square root of 2 to 6 places of decimals.

2.0000... (1.414

1

24)100

96

281)400

281

2824)11900
11296

2828)604000(213

5656

3840

2828

10120

8484

1636

..the root required is 1·414213......

When only one figure in the root has been obtained, a, which represents the part already obtained, may be as small as 10, and x, the next

, may therefore

digit, may be as great as 9; the error in the quotient 2a'

be easily greater than 1, unless a be as great as 50, i. e. the first figure in the root as great as 5, and we should then obtain too large a quotient; this is not unfrequently observed to happen at the first division, but from the foregoing proposition it appears that this error cannot take place in any subsequent division.

II. In the extraction of a cube root, when n+1 digits have been found by the ordinary rule, n more can be correctly obtained by dividing by the trial divisor.

Let a+b be the cube root,

when a consists of n+1 digits, and ciphers,

... b ......... n digits.

a3+3a2b+3ab2+63 the quantity whose root is required.

Then after a has been found, we have

remainder = 3a2b+3ab2+b2;

trial divisor = 3a2;

..the greatest possible error will be when a and b have the above values;

The error therefore is always <1, i.e. the n last digits can be correctly obtained by ordinary division.

From this it will be seen that as in square root, it is only at the first division that too large a quotient can be obtained for the next digit in the

root.

THEORY OF INDICES.

162. The subject of Indices deserves a separate and distinct consideration. It is proposed to bring together here all that has been defined or proved with respect to them in the preceding pages-to shew that the several Definitions are not in practice inconsistent with each other—and to supply the proofs still wanting in order that the Rules may be extended to all possible cases which can occur. .

(1) The primary Definition was given in Art. 63, whereby we agreed to represent a.a.a. &c. to n factors by a", where n expresses the number of factors, and therefore can only be a positive integer.

(2) The next Definition was given in Art. 65, whereby we agreed to represent by a"; but at that stage we could only consider it as a short

է

since a negative quantity can in no sense express a

(3) The last Definition was given in Art. 70, whereby we agreed to represent the nth root of a by a", and the nth root of the mth power of a by a". Here again we felt the restriction that a fraction can in no sense express a number of factors multiplied together, that is, a power of a, in the true sense of the word.

(4) From (1) it is strictly proved in Arts. 91, 95, 138, and 144, that aTM×a"=a"+",

These are the fundamental Rules for the Multiplication, Division, Involution, and Evolution of powers and roots.

(5) When a negative value is given to either m or n, or both of them, and restricted to the meaning pointed out in (2), it is proved in Arts. 132, 134, and 138, that these Rules still hold true. Hence it appears, that no error can arise from using negative powers according to the second Definition.

(6) It remains, however, to be proved, that the 3rd Definition is generally admissible, that is, that the above Rules hold true when positive or negative fractions are treated as indices of the powers or roots of any quantity, with the meaning assigned to them by the Definition. This being done, the Rules for the treatment of Indices will have been shewn to apply generally to all cases whatever, whether the indices be positive or negative, whole or fractional.