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Def. 1.—Quantities of the same kind may be compared with each other with respect to their magnitudes, by means of the number of equal parts or units they each contain ; and ratio is the relation which two such quantities have to each other, determined by considering what fractional part one is of the other when both quantities are numerically expressed; as one line is compared with another, by means of the number of equal units, as inches, &c., and one surface with another surface, by the number of square or superficial units which they respectively contain. If the number of equal units contained in 2 geometrical quantities of the same kind be represented by a and b, the ratio of the quantities is therefore expressed by the fraction. The meaning of which is, that one quantity being divided into as many equal parts as a expresses, the other is divisible into a number of the same parts equal to b.

Geometrical quantities may also be so related to each other, as to admit of no numbers that will strictly represent them, as the side and diagonal of a square, which by Prop. 39, are as I to J2, or 1 to 1,414, &c.; since they do not each contain an exact number of inches, or equal parts of an inch, without a remainder; yet by the subdivision of the inch (or whatever unit of measurement is used) into any number of equal parts, the ratio of the lines may be obtained to any assigned degree of accuracy. By the continued subdivision of

the unit, the remainder, or undivided part of any quantity may be conceived to be so diminished, as to be of no assignable value compared with the other part, and as there are no conditions which limit this subdivision, the ratio of these quantities may be expressed as before by the fraction

Dep. 2.- A proportion consists of 2 equal ratios. Or if, of four quantities; the first contains the second, or part of the second, as often as the third contains the fourth, or like part of the fourth; they are said to be proportionals. That is, if a, b, c, d be the numerical values of 4 geometrical quantities; the pro

a c portion is represented by the equation =; which is usually expressed by saying, a is to b as c is to d. In this proportion, a and c are called the antecedents, and b and d the consequents. The proportion is also expressed by a:b::c:d, or by a:b=c:d as most convenient ; in which a and d are called the extremes, and b and c the means. And in any proportion if the means are equal to each other, i. e. if a : 6::6:c, then 6 is said to be a mean proportional between a and b.

DEF. 3.-Similar rectilineal figures are those which have their respective angles equal to each other, and the sides about the equal angles proportional.

PROP. LXV. THEOR. 16. 6 Eu.

If four geometrical quantities, numerically represented by a, b, c, d, be proportionals; that is, if a:b::c:d; then will ad=bc.

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If there be any number of proportional quantities, represented numerically as a:b:: c:d::e:f, &c.; then will one antecedent be to its consequent, as the sum of all the antecedents to the sum of all the consequents; or a:6::a+cte, &c. : b+d+f, &c.

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