4.-Proportionals are proportionals by composition; that is, the first or second has the same proportion to the sum of the first and second, that the third or fourth has to the sum of the third and fourth. This also follows from the principle of equi-multiples. The two terms are an equi-multiple of either term by the ratio of that term to the other × 1; and, therefore, the sums of the two terms are equi-multiples of the corresponding terms; that is, the first and third, or the second and fourth, and so they must be proportional.

5.-Proportionals are proportionals by separation; that is, the differences of the two terms of two equal ratios, taken in the same order, have the same ratio to the corresponding terms. This is still a matter of equi-multiples, for the difference of the two terms of a ratio is a multiple of either of them by the ratio of that term to the other

1.

6.-Proportionals are proportional ex æquali; that is, when those which are equally distant from each other in one series of proportional quantities are proportional to those which are equally distinct from one another. This is still a matter depending upon the principle of equi-multiples; but it may perhaps require a little more explanation than the former ones.

Of this equality of ratios at equal distances in two series of 'quantities, there are two cases which very much resemble the direct and the inverse statements of common ratios; and these are usually cited in mathematical books by the words ex æquo, from equality, and ex æquo inversely, from equality of cross distance.

Ex æquo. If there are any number of quantities in one series, and as many in another series; and if these taken in the same order have the same proportion to each other two and two, it is assumed for proof that, when an equal number in each series is taken, the first of the last series will have the

same ratio to the last of that series, as the first of the second series has to the last of the second series. Thus let

A, B, C, D, E, &c.

a, b, c, d, e, &c.

be two series of quantities in which a ; B = a : b, в ¦ ¢ = bc, c D = c; d, D; E = d; e; then AE= a;e.

The quantities in each series are continued proportionals, and their ratios taken two and two in order are the same; but we have already seen that in a continued proportion the ratio of the first term to the last is compounded of all the single ratios of the terms taken two and two; the terms in those two series are the same in number, and the ratios of every two terms nearest to each other are the same throughout. These individual ratios are the factors whose product forms the ratio of the first to the last; and those ratios are not only the same in value, but follow each other in the same order in both series; wherefore their products must be equal; that is,

A E α e.

Ex æquo inversely. If there are any number of quantities in one series, and as many in another; and if these taken two and two in a cross order have the same ratios in the one series as in the other, it is assumed for proof that when an equal number in each series is taken, the first of the first series will have the same proportion to the last as the first of the second series has to the last. Thus, as before, let

A, B, C, D, &c.

a, b, c, d, &c.

be two series of quantities in which a¦ B = b ; c, B ; c = ab, B c = c; d, and c: D = bc; then a D = a; d.

This is a case of continued proportion as well as the former; and though the equal ratios in the two series do not follow in the same order, yet they alternate with each other, so that the

whole in the one are still equal, each to each, to the whole in the other. But the ratio of the first quantity of each series to the last, is composed of the product of all the single ratios; and these being equal to each other, it follows that the products must be equal; because the order in which the factors are taken does not alter the value of the product. Consequently the product of those equal ratios arranged in cross order, are exactly the same as if they had been arranged in the same order in the one series as in the other.

The principles which we have now stated contain the whole elements of the doctrine of proportion; and any one who studies them with so much attention, as fully to understand them, can find no difficulty in managing any peculiarities which may arise in particular cases. Indeed, if the changes which can be made on the terms of a ratio without altering the ratio itself, and the condition which determines the equality of ratios, are once clearly understood, the whole doctrine of proportion may be said to be mastered. We shall therefore very briefly recapitulate these, because they are the points which it is essential for the student to bear in mind, in order to profit to the full extent by this most simple, most beautiful, and most useful portion of mathematical science.

1.-A ratio remains the same if both its terms are multiplied, or both divided by equal numbers, whatever may be the value and chararter of those numbers. It also remains the same if equal parts of the terms be added or taken away. In short, if whatever is done to the one term be also done to the other, and be done to both in the proportion of their original values, the relation of the terms, and consequently the ratio, remains unaltered amid all the changes, be they ever so many.

2. Ratios are equal, if the product of the first term of the first, and second term of the second, is equal to the product of

the second term of the first and first term of the second. And this equality holds whether the ratios are considered as original and simple, or as being compounded of any number of equal ratios. Compounding, in the arithmetical sense of the term, means multiplying the one by the other; and in all such cases, it is of no consequence in what order the factors are taken,— which last consideration is evident from the fact, that the 3 times 4 is exactly the same as 4 times 3.

If these two articles, and they are short and simple, are correctly borne in mind, the student will feel little difficulty in the management of proportion; and may and should turn to the Fifth Book of Euclid's Elements, as a most valuable subject of study, not for mathematical purposes only, but for laying a sure foundation for clearness and accuracy in general reasoning,

SECTION XIII.

POWERS AND ROOTS OF QUANTITIES.

A POWER of a quantity is the product which arises from the multiplication of that quantity by itself, the same quantity being both multiplier and multiplicand in the case of one multiplication, and multiplier in every successive multiplication, the product in the previous operation being multiplicand. A power is thus a compound quantity produced by two or any greater number of identical or equal factors; and different powers are distinguished by the number of times that the quantity of which they are the powers occurs as a factor in the composition of them. We have already partially explained some of the simpler properties of powers when treating of the scale of numbers, in the third section, and also in some other

parts of the more elementary portion of this volume. We there mention that the arithmetic of powers gives occasion for a peculiar kind of numbers in arithmetic, and for a peculiar species of notation in algebra; and we shall here revert more particularly to the same subject.

We mentioned also that lines are arithmetically represented by simple or original numbers expressing their lengths in known measures; that surfaces are expressed in numbers by products of two factors, the one expressing the length and the other the breadth, expressed in the same manner, the number expressing the value in squares of the measure; and that solids, or those portions of space in which solids could be contained, are expressed in products of three factors, all in the same measure, the one being length, another breadth, and the third thickness. The number expressing the value of solidity in the last casè consists of cubes, that is of solids bounded by six equal square surfaces, every side in all of which is the same measure as the denomination of lineal measure, in which the length, breadth, and thickness are expressed.

From these circumstances, a power arising from the multiplication of any quantity used twice as a factor, or once multiplied by itself, is called the square of the quantity. There is no objection to the use of this name square, whatever may be the kind of the quantity; because when the quantity is expressed by a number, the number might express the length of a line, and that line might be the side of a square; but it is only when the number actually means the length of a line that the product of it by itself is a square in reality. It is necessary to bear this in mind, in order that when we speak of the square of 3 or any other number, or of the square of a, or any other general quantity, we may not attach the same positive notion of squaring to it which we