The cube root of a fraction, when reduced into its lowest terms, is extracted by either extracting the cube root of each term, or by extracting the cube root of the decimal value, when either term is a surd number.

From the facility with which the extraction of roots is performed with Logarithms, by only dividing the logarithm of the given number by the index of its power, and finding the corresponding number of the logarithm of the quotient, it is very seldom that the extraction of any other root than the square root is made by common numbers; and it is therefore here considered unnecessary to give any examples of the extraction of the higher roots, as of the biquadrate root, or the square root of a square root, &c.

Respecting the extraction of the cube root of a fraction, it is proper to observe, that as tables of the roots of numbers under 1000 generally accompany those scientific works in which such calculations occur, the root of each term may be taken from the tables, and the quotient be found for the decimal value.

Or, it may be proper to raise the denominator into a perfect cube, and then extract its root, and apply it as a multiplier to the tabular value of the surd numerator.

Thus, to extract the cube root of 3,

2 7 = 2 4 = 3 + × 321 = √ 21,

and as the cube root of 21 is by the tables 2.758923, one third is 0.919641, which is the value of the cube root of 7-9 ths, or of the decimal 0.777, &c.

In this case the denominator is a square number, but if it is neither a square nor a cube number, the fraction must be multiplied in both terms by the square of the denominator :

but the tabular cube root of 847 is 9.461524, from which the value of 1-11 th is 0.860138, which is the value of the cube root of 7-11 ths.

If we form these principles into a rule for extraction of the cube root of a surd fraction, we have the following;

Rule. Multiply the numerator by the square of the denominator, extract the cube root of the product, and divide the result by the denominator.*

N. B. The cube root of any number which is the exact cube of a whole number of any two figures may be mentally found thus having separated the lower three figures, according às the remaining figures are next above 1, 8, 27, 64, 125, 216, 343, 512, or 729, the tens of the root will be 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Then if the unit figure of the given cube is 1, 4, 5, 6, or 9, the unit figure of the root will be the same figure; but if the unit figure of the cube is 8, 7, 3, or 2, the unit figure of the root will be its complement, or what it wants of 10, that is, 2, 3, 7, or 8.

Thus with 46656, the tens will be 3, because 46 is above 27 and under 64, and 6 being the unit figure, the unit figure of the root will also be 6, making together 36.

Or with 373248, the tens will be 7, because 373 is next above 343, and 8 being the unit of the cube, 2 its complement will be the unit of the root, making together 72.

* The extraction of the square root of a surd fraction may be made in a similar manner: thus, the square root of 5-7 ths equals the square root of 35-49 ths, which is equal to 1-7 th of the square root of 35; that is, the square root of a fraction equals the square root of the product of the multiplication of the numerator by the denominator, divided by the denominator.

RATIOS.

When two arithmetical quantities of the same kind are compared, the number of times or parts which the one is of the other, is called the ratio of the quantity compared to the quantity with which it is compared.

The quantities to be compared by their numbers, must be similar and simple quantities; that is, in one and the same denomination, as 5 pence and 7 pence, 5 yards and 7 yards; for if they are not, either reference must be made to their values in the same denomination, or their relation must be known from some knowledge of the whole values of the two quantities. Thus, to compare 9 pence with 3 shillings, the latter must be considered as 36 pence, and to compare £1 12 6 with £ 3 5 0, we must value them as 32 shillings and 65 shillings, unless we act on our previous knowledge that the latter quantity is double the former.

The same limitations apply to the numbers employed; for if fractions and decimals are used, they must not be only similar in their being fractions of the same integer, but they must be expressed in the same sort of parts; thus, of a shilling cannot, in this form, be compared with of a £; nor of a penny with of a penny; although for the usual purposes of such comparisons, when the fractions are of the same integer, the result of the comparison or the ratio can be found, without reducing them to a common denominator; so also to compare 0.5 with 0.73 we must use both as 100 ths, calling the former 50-100 ths, because the latter is 73-100 ths.

The first of the two terms of comparison is called the antecedent, and the second the consequent; and the latter is considered to be produced by the multiplication of the former by the ratio of the consequent to the antecedent.

The ratios of similar quantities are formed by comparing them through their common integer. Thus, to find the ratio of 5 pence to 7 pence, as 1 d. is their common integer, and as 5 pence are 5 times 1 penny, and also as 1 penny is the 7 th part of 7 d., we say, that 5 pence are 5-7 ths of 7 pence; or, on the reverse, 7 pence are 7 times the 5 th part of 5 pence; and the fractions and are called the mutual ratios of 5 pence and 7 pence.* If therefore any quantity is given, and the ratio of any other quantity to this given quantity is also given, the other quantity may be found by multiplying the given quantity by the ratio; as 5 pence X = 7 pence.

To distinguish these ratios they are called direct and inverse, or reciprocal to the same terms used in the same order: thus, the fraction 5-7 ths is called the direct ratio of 5 pence to 7 pence, and the inverse of this fraction, or 7-5 ths, is called the inverse or reciprocal ratio of 5 pence to 7 pence.*

Compound ratios are those which are formed from the multiplication of two or more simple ratios: thus, from the ratio of 5 pence to 7 pence, which is, and the ratio of 11 oz. to 12 oz., which is, is formed the compound ratio

When the ratios are compounded of the same or equal ratios, the results, with reference to a single ratio, are termed duplicate, triplicate, &c., ratios; according to the number of the terms employed; the duplicate ratios being the squares, and the triplicate ratios the cubes, of the original ratios: thus, with the ratio ths,

욕 of 2 = 籍 is called the duplicate ratio of 2,

of 2 of 2

125 343

the triplicate ratio of &;

pence

and in the using of such ratios we say, for example, that 25 have to 49 pence, the duplicate ratio of 5 pence to 7 pence. Same or equal ratios are those of which the results are equal; as the ratio of 5 pence to 7 pence is equal to the ratio of 20 oz. to 28 oz., for = 28.

The reciprocal of any number may be expressed by inverting the fractional form of that number, or by making it the denominator having unity for the numerator; as for 5, 4, or 4, the reciprocals are, †, and = }.

PROPORTIONS.

Proportion is the relation of quantities, which taken two and two have the same or equal ratios.

Thus, as the relation of 5 pence to 7 pence is equal to the relation of 20 oz. to 28 oz. these quantities form a proportion, or are in proportion; and they are expressed as such by saying that 5 pence are to 7 pence as 20 oz. to 28 oz.; or, as 5 pence are to 7 pence, so are 20 oz. to 28 oz. ; or by characters, thus, 7 d.:: 20 oz. : 28 oz.*

As 5 d.

In this order they are

said to be in direct proportion, but if one of the pair of the terms is inverted, as in this arrangement,

5 d. 7 d. :: 28 oz.: 20 oz.

they are said to form an inverse, or more correctly a reciprocal, proportion; and they are read, as 5 d. are to 7 d. so are 28 oz. inversely to 20 oz.

When four quantities are proportional, the two terms in each set may be multiplied or divided by the same number, without altering the equality of their relation, or making any alteration in its value.

As multiplying the first set of the above terms by 3,
15 d. 21 d. :: 20 oz. : 28 oz.

Or dividing the same set of terms by 2,

5 d. 7 d. :: 10 oz. : 14 oz.

And if either the powers or the roots be taken of the numbers of all the terms of a proportion, these powers or roots will be proportional; as of the above taking the squares,

25 : 49: 100: 196.

If the powers be taken of the numbers of only one set of terms, they have then the duplicate, triplicate, &c. ratios of the other terms; as with

100 and 196,

have to the fourth, the duplicate 5 being to 7 as 100 to 140, and

When the first three terms of a proportion are given to find a fourth, and they are made a form of question to be worked by the usual Rule of Three, if they are not all three similar quantities, the above characters should not be used; for the meaning of these characters, as arithmetically used, is, that the quantity placed on one side of this character is produced from or compared with that placed on the other side: and the character :: is used to link together the different sets of terms.