BOOK III. PROPORTIONAL LINES AND SIMILAR THE THEORY OF PROPORTION. 292. A proportion is an expression of equality between two equal ratios. A proportion may be expressed in any one of the following forms: α C b=2; a:b=c:d; a:b::c:d; and is read, "the ratio of a to b equals the ratio of c to d." 293. The terms of a proportion are the four quantities compared; the first and third terms are called the antecedents, the second and fourth terms, the consequents; the first and fourth terms are called the extremes, the second and third terms, the means. 294. In the proportion a: b = c: d, d is a fourth proportional to a, b, and c. In the proportion a: bbc, c is a third proportional to a and b. In the proportion a:bb: c, b is a mean proportional between a and c. PROPOSITION I. 295. In every proportion the product of the extremes is equal to the product of the means. 296. A mean proportional between two quantities is equal to the square root of their product. In the proportion a: b = b : c, b2 = ac, § 295 (the product of the extremes is equal to the product of the means). Whence, extracting the square root, 297. If the product of two quantities is equal to the product of two others, either two may be made the extremes of a proportion in which the other two are made the means. or, Divide both members of the given equation by bd. PROPOSITION IV. 298. If four quantities of the same kind are in proportion, they will be in proportion by alternation; that is, the first term will be to the third as the second to the fourth. 299. If four quantities are in proportion, they will be in proportion by inversion; that is, the second term will be to the first as the fourth to the third. PROPOSITION VI. 300. If four quantities are in proportion, they will be in proportion by composition; that is, the sum of the first two terms will be to the second term as the sum of the last two terms to the fourth term. 301. If four quantities are in proportion, they will be in proportion by division; that is, the difference of the first two terms will be to the second term as the difference of the last two terms to the third term. Let a:bc: d. PROPOSITION VIII. 302. In any proportion the terms are in proportion by composition and division; that is, the sum of the first two terms is to their difference as the sum of the last two terms to their difference. 303. In a series of equal ratios, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. Let a:bc:d=e:f=g: h. To prove a+c+e+g:b+d+f+h=a: b. Denote each ratio by r. or, Whence, a=br, c=dr, e=fr, g=hr. Add these equations. Then a+c+e+g= (b+d+f+h)r. a+c+e+g:b+d+f+ha: b. Q. E. D. |