PROPORTION. When two quantities are compared one with another, in respect of their greatness or smallness, the com parison is called ratio, reason, rate, or proportion ; but when more than two quantities are compared, then the com parison is more usually called the propor. tion that they have to one another. The words ratio and proportion are frequent ly used promiscuously. When two quan tities only are compared, the former term is called the antecedent, and the latter the consequent. The relation of two ho mogeneous quantities one to another, may be considered either, 1. By how much the one exceeds the other, which is called their difference. Thus 5 exceeds 3 by the difference 2. Or, 2. What part or parts one is of another, which is called ratio. Thus the ratio of 6 to 3 is 1.=--4, or double ; and the ratio of 3 to 6 is 1=3, er subduple.
When two differences are equal, the terms that compose them are said to be arithmetically proportional. Thus, sup pose the term to be a and b, their differ ence d. If a be the last term, then a+d =b. And it a be the greatest, then a d...b.
But when two ratios are equal, the terms that compose them are said to be geome trically proportional. For suppose a and b to be the terms of any ratio ; if a be the least term, put r=—,then ar=b by equal a multiplication : but if b be the least term, a ' put r=.— then by equal multipli b cation, and by equal division.
Thus the ratio of two quantities, or of two numbers, in geometrical proportion.
is found by dividing the antecedent by the consequent, and the quotient is the expo nent or denominator of the ratio.
If, when four quantities are considered, you find that the first hath as much great ness or smallness in respect to the second,. as the third hath in respect to the fourth : those four quantities are called propor tionals, and are thus expressed As AD .
s 8 218 4 5 that i as A=8 con tains B=2 four times, so C=16 contains D=4, four times ; and, therefore, A has the same ratio to B as C has to D; and, consequently, these four quantities having equal ratios, am proportionals.
Proportion consists of three terms at least, whereof the second supplies the place of two.
When three magnitudes, A, B, C, are proportional, the first, A, has a duplicate ratio to the third, C, of that it hath to the second, B: but when four magnitudes, A, B, C, 1), are proportional, the first, A, has a triplicate ratio to the fourth, D, of what it has to the second, B ; and so al ways in order one more, as the proportion shall be extended.
A A Duplicate ratio is thus expressed, twice ; that is, the ratio of A to C is du. plicate of the ratio of A to 13. For let A-2, B=4, C=8: then the ratio of 2 to 8 is duplicate of the ratio of 2=A to B=4, or as the square of 2 to the square of 4.
A Triplicate ratio is thus thrice ; that is, the ratio of A again = to D = 16, is triplicate of the ratio of A = 2 to II = 4, or as 8 the cube of 2, to 64 the cube of 4. 'Wherefore duplicate ratio is the proportion of squares, and tri plicate that of cubes.
And the ratio of 2 to 8 is compounded of the ratio of that of 2 to 4, and of 4 to 8. From what has been said of the na ture of ratio and proportion, the six ways of arguing, which are often used by ma thematicians, will evidently follow.
1. Alternate proportion is the compar ing of antecedent to antecedent, and con sequent to consequent. As if A : 13 C : D therefore alternate.
2 : 4 :: 8 : 16 ly, or by permutation, as