" If the difference between the ante cedent and consequent of a ratio be small when compared with either of them, the double of the ratio, or the ratio of their squares, is nearly Obtained by doub ling this difference." Let a. x: a be the proposed ratio, where xis small when compared with a ; then a' -F 2 ax : a' is the ratio of the squares of the antecedent and conse quent; but since xis small when compar ed with a, x' or x X x is small when compared with 2 a X x, much small Pt than a X a ; therefore, a' + 2 ar :a., or a + 2 x : a will nearly express the ra tio of a' + 2 ax : Thus the ratio of the square of 1001 to the square of 1000 is nearly 1002: 1000 ; the real ratio is 1002.001 : 1000, in which the antecedent differs from its approxi mate value, only by one thousandth part of an unit.
Hence the ratio of the square root of a + 2 x to the square root of a is the ra tio a X x: a, that is, if the differ ence of two quantities be small with res pect to either of them, the ratio of their square roots is nearly obtained by halving their difference.
In the same manner, a+3x: a; a 4-4x: a:a+mx:a:are nearly equal to the ratios a3; a + x4 ; am ; if nix be small when com pared with a.
Or we may treat the subject different ly, thus ; ratio is that relation of homoge neous things which determines the quantity of one from the quantity of another, without the intervention of a third. Two numbers, lines, or quantities, A and B, being proposed, their relations one to another may be considered under one of these two heads : 1. Flow much A exceeds B, or B exceeds A ; and this is found by taking A from B, or B from A, and is called arithmetic reason, or ratio. 2. Or how many times and parts of a time, A contains B, or B contains A ; and this is called geometric reason, or ratio ; (or, as Euclid defines it, it is the mutual habitude, or respect, of two mag nitudes of the same kind, according to quantity ; that is, as to how often the one contains, or is contained, in the other ;) and is found by dividing A by B, or B by A • and here note, that the quantity which is referred to another quantity, is called the antecedent of the ratio ; and that to which the other is referred is called the consequent of the ratio ; as, in the ratio of A to B, A is the antecedent, and B the consequent. Therefore any
quantity, as antecedent, divided by any quantity as a consequent, gives the ratio of that antecedent to the consequent. Thus the ratio of A to B is - A but the ratio of B to A ; in numbers, the ratio of 12 to 4 is 12 = 3, or triple ; but 1 , _ the ratio of 4 to 12 is 4 -- — 3 or sub triple.
The quantities, thus compared, must be of the same kind ; that is, such which, by multiplication, may be made to ex ceed one the other, or as these quantities are said to have a ratio between them, which, being multiplied, may be made to exceed one another. Thus a line, how short soever, mly be multiplied, that is, produced so long as to exceed in length any given right line, and consequently these may be compared together, and the ratio expressed ; but as a line can never, by any multiplication whatever, be made to have breadth, that is, to be made equal to a superficies, how small soever ; these can therefore never be compared together, and consequently have no ratio or respect one to another, according to quantity ; that is, as to how often the one contains, or is contained in the other.