been shown hence, the triangles are equal in all their parts (P. V.); and consequently, the alternate angles ADB and DBC are equal. The sides BC and AD are, therefore, parallel, and the figure is a parallelogram; which was to be proved. Cor. If two points be taken at equal distances from a line, and on the same side of it, the line joining them will be parallel to the given line. PROPOSITION XXXI. THEOREM. The diagonals of a parallelogram divide each other into equal parts, or mutually bisect each other. Let ABCD be a parallelogram, and B AC, BD, its diagonals: then will AE be equal to EC, and BE to ED. For, the triangles BEC and AED, have the angles EBC and ADE equal (P. XX., C. 2), the angles ECB and DAE equal, and the included sides BC and AD equal: hence, the triangles are equal in all of their parts (P. VI.); consequently, AE is equal to EC, and BE to ED; which was to be proved. Scholium. In a rhombus, the sides AB, BC, being equal, the triangles AEB, EBC, have the sides of the one equal to the corresponding sides of the other; they are, therefore, equal: hence, the angles AEB, BEC, are equal, and therefore, the two diagonals bisect each other at right angles. BOOK II. RATIOS AND PROPORTIONS. DEFINITIONS. 1. THE RATIO of one quantity to another of the same kind, is the quotient obtained by dividing the second by the first. The first quantity is called the ANTECEDENT, and the second, the CONSEQUENT. 2. A PROPORTION is an expression of equality between two equal ratios. Thus, expresses the fact that the ratio of A to B is equal to the ratio of C to D. In Geometry, the proportion is written thus, A : B :: C : D, and read, A is to B, as C is to D. which severa 3. A CONTINUED PROPORTION is one in which ratios are successively equal to each other; as, A : B :: C : D:: E : F:: G : H, &c. 4. There are four terms in every proportion. The first and second form the first couplet, and the third and fourth, the second couplet. The first and fourth terms are called extremes; the second and third, means, and the fourth term, a fourth proportional to the other three. When the second term is equal to the third, it is said to be a mean proportional between the extremes. In this case, there are but three different quantities in the proportion, and the last is said to be a third proportional to the other two. Thus, if we have, A : B :: B : C, B is a mean proportional between A and C, and Cis a third proportional to A and B. 5. Quantities are in proportion by alternation, when antecedent is compared with antecedent, and consequent with consequent. 6. Quantities are in proportion by inversion, when antesedents are made consequents, and consequents, antecedents. 7. Quantities are in proportion by composition, when the sum of antecedent and consequent is compared with either antecedent or consequent. 8. Two varying quantities are reciprocally or inversely proportional, when one is increased as many times as the other is diminished. In this case, their product is a fixed quantity, as xy = m. 9. Equimultiples of two or more quantities, are the products obtained by multiplying both by the same quantity. Thus, mA and mB, are equimultiples of A and B. and PROPOSITION I. THEOREM. If four quantities are in proportion, the product of the means will be equal to the product of the extremes. Cor. If B is equal to C, there will be but three proportional quantities; in this case, the square of the mean is equal to the product of the extremes. PROPOSITION II. THEOREM. If the product of two quantities is equal to the product of two other quantities, two of them may be made the means, and the other two the extremes of a proportion. If we have, AD = BC, by changing the members of the equation, we have, PROPOSITION III. THEOREM. If four quantities are in proportion, they will be in pro portion by alternation. If one couplet in each of two proportions is the same, the other couplets will form a proportion. Cor. If the antecedents, in two proportions, are the same, the consequents will be proportional. For, the antecedents of the second couplets may be made the consequents of the first, by alternation (P. III.). |