PROPORTION, an equality of ratios; a series in which the first of any even number of terms bears the same mathematical relation to the second term as each other odd-numbered term does to the even-numbered term which immediately follows; thus 4, 12; 5, 15; 6, 18 (or 4:125:15=6:18, or 4:12::5:15::6:18, these forms being read °4 is to 12 as 5 is to 15 as 6 is to 18,) etc.) is a geometrical proportion. The definition given above using the words °same mathematical relation° shows that there may be other than geometrical proportions, in which the relation is the fundamental one of multiplication (or division). The fundamental relation of addition (or subtraction) also gives rise to a proportion, the so-called arithmetical proportion; thus 4, 6; 7, 9; 11, 14, or, generaliz ing, a, a + d; b, b d; c, c d, are in arith metical proportion as each odd-numbered term is related to its corresponding even-numbered term by the difference 3. The distinction be tween geometrical and arithmetical is here the same as in progressions (q.v.), and the propor tion is merely a special case of progression. It is perfectly obvious that the concept of the proportion may be applied elsewhere, the relation for example being not that of multipli cation (or division) nor that of subtraction (or addition), but that of evolution (or involution) ; htus 4, 16; 5 25; 11, 121, is a series in which the analogy between terms 1 and 2, holds in the case of terms 3 and 4, as well as terms 5 and 6, the relation being that of the root to the square. But this general concept of the propor
tion or analogous relation finds small place in mathematics, even the most modern theory be ing little fresher than Euclid's Fifth Book, which was derived in part from Pythagoras, and which is, save for the difficulty in the treatment of incommensurables, far superior, though reckoned more difficult, than the numer ical treatment which came into vogue in the Middle Ages; Euclid's method is concerned with lines, instead of numbers. To all intents and purposes the modern method like Euclid's is concerned merely with geometrical proportions. In a geometrical proportion the odd-numbered terms are called the antecedents, the even the consequents; the first and fourth terms the ex tremes, the second and third, the means. A mean proportion has the second and third terms identical as a: b= b: c, which may more con a b veniently be written —=—. The chief prop .
b c a c erties of a geometrical proportion, say —=—, b d are the following: (1) the product of the means equals the product of the extremes, bc=ad (clear of fractions) ; (2) the terms are in pro a -1- b c+d portion by composition, (add 1 to each member of the original equation) ; (3) a— b the terms are in proportion by division, —= c—d (subtract 1 from each member) ; (4) the terms are in proportion by composition and a + b c+ d division, (divide the equation in a —b c —d (2) by the equation in (3)).