BOOK III. THEORY OF PROPORTION. POLYGONS. SIMILAR DEFINITIONS. 227. A Proportion is a statement that two ratios are equal. The statement that the ratio of a to b is equal to the ratio of c to d, may be written in either of the forms 228. The first and fourth terms of a proportion are called the extremes, and the second and third the means. The first and third terms are called the antecedents, and the second and fourth the consequents. Thus, in the proportion a: b = c:d, a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents. 229. If the means of a proportion are equal, either mean is called a mean proportional between the first and last terms, and the last term is called a third proportional to the first and second terms. Thus, in the proportion a: b = b: : c, b is a mean proportional between a and c, and c a third proportional to a and b. 230. A fourth proportional to three quantities is the fourth term of a proportion, whose first three terms are the three quantities taken in their order. Thus, in the proportion a:bc: d, d is a fourth proportional to a, b, and c. PROPOSITION I. THEOREM. 231. In any proportion, the product of the extremes is equal to the product of the means. Multiplying both members of the equation by bd, we have ad = bc. 232. COR. The mean proportional between two quantities is equal to the square root of their product. 233. (Converse of Prop. I.) If the product of two quantities is equal to the product of two others, one pair may be made the extremes, and the other pair the means, of a proportion. Dividing both members of the given equation by bd, PROPOSITION III. THEOREM. 234. In any proportion, the terms are in proportion by ALTERNATION; that is, the first term is to the third as the second term is to the fourth. Let the proportion be a: bc: d. (1) 235. In any proportion, the terms are in proportion by INVERSION; that is, the second term is to the first as the fourth term is to the third. Let the proportion be a: be: d. (1) 236. In any proportion, the terms are in proportion by COMPOSITION; that is, the sum of the first two terms is to the first term as the sum of the last two terms is to the third PROPOSITION VI. THEOREM. 237. In any proportion, the terms are in proportion by DIVISION; that is, the difference of the first two terms is to the first term as the difference of the last two terms is to the third term. in which a is greater than b, and c greater than d. (1) (§ 231.) 238. 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 is to their difference. Let the proportion be a:bc:d, in which a is greater than b, and c greater than d. (1) Dividing the first equation by the second, we have PROPOSITION VIII. THEOREM. 239. In a series of equal ratios, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents. Let To prove = a: b = c d e : f. Let r denote the value of each of the given ratios. 240. Equimultiples of two quantities are in the same ratio as the quantities themselves. Let a and b be any two quantities. 241. In any number of proportions, the products of the corresponding terms are in proportion. |
