126. Proportion is written not only with the sign = but, more often, with the sign :: between the ratios. Thus, A B:: X: Y, expresses a proportion, and is read, The ratio of A to B is equal to the ratio of X to Y; or, A is to B as X is to Y. 127. The first and third terms of a proportion are called the ANTECEDENTS; the second and fourth, the CONSEQUENTS. The first and fourth are also called the EXTREMES, and the second and third the MEANS. Thus, in the proportion A: B:: C: D, A and C are the antecedents; B and D are the consequents; A and D are the extremes; and B and C are the means. The antecedents are called homologous or like terms, and so also are the consequents. 128. All the terms of a proportion are called PROPORTIONALS; and the last term is called a FOURTH PROPORTIONAL to the other three taken in their order. Thus, in the proportion A: B:: C: D, D is the fourth proportional to A, B, and C. 129. When both the means are the same magnitude, either of them is called a MEAN PROPORTIONAL between the extremes; and if, in a series of proportional magnitudes, each consequent is the same as the next antecedent, those magnitudes are said to be in CONTINUED PROPORTION. Thus, if we have A: B:: B:C::C:D:D: E, B is a mean proportional between A and C, C between B and D, D between C and E; and the magnitudes A, B, C, D, E are said to be in continued proportion. 130. When a continued proportion consists of but three terms, the middle term is said to be a MEAN PROPORTIONAL between the other two; and the last term is said to be the THIRD PROPORTIONAL to the first and second. Thus, when A, B, and C are in proportion, A: B:: B: C; in which case B is called a mean proportional between A and C; and C is called the third proportional to A and B. 131. Magnitudes are in proportion by INVERSION, or INVERSELY, when each antecedent takes the place of its consequent, and each consequent the place of its antecedent. Thus, let A: B::C:D; then, by inversion, B: A:: D: C. 132. Magnitudes are in proportion by ALTERNATION, or ALTERNATELY, when antecedent is compared with antecedent, and consequent with consequent. Thus, let A: B:: D:C; then, by alternation, A: D:: B: C. 133. Magnitudes are in proportion by COMPOSITION, when the sum of the first antecedent and consequent is to the first antecedent, or consequent, as the sum of the second antecedent and consequent is to the second antecedent, or consequent. Thus, let A: B:: C: D; then, by composition, A+BA:: C+D: C, or A+B: B:: C+D: D. 134. Magnitudes are in proportion by DIVISION, when the difference of the first antecedent and consequent is to the first antecedent, or consequent, as the difference of the second antecedent and consequent is to the second antecedent, or consequent. Thus, let A: B:: C: D; then, by division, A-B: A:: C-D: C, or A-B: B::C-D: D. PROPOSITION I.THEOREM. 135. If four magnitudes are in proportion, the product of the two extremes is equal to the product of the two means. Let A B C D; then will A X DBX C. : : For, since the magnitudes are in proportion, C A and reducing the fractions of this equation to a common denominator, we have or, the common denominator being omitted, AX DBX C. PROPOSITION II.-THEOREM. 136. If the product of two magnitudes is equal to the product of two others, these four magnitudes form a proportion. Let A X DBX C; then will A: B::C: D. For, dividing each member of the given equation by BX D, we have and, by Prop. I., AX D BX C = which, reduced to the lowest terms, gives A C 1=0 B D Whence A: B:: C: D. PROPOSITION III.-THEOREM. 137. If three magnitudes are in proportion, the product of the two extremes is equal to the square of the mean. : Let A B B: C; then will A X C B2. = AX CBX B, or AXC B2. = PROPOSITION IV.-THEOREM. 138. If the product of any two quantities is equal to the square of a third, the third is a mean proportional between the other two. whence Let AXC B2; then B is a mean proportional between A and C. For, dividing each member of the given equation by BX C, we have = A B B C' A: BB: C. PROPOSITION V.-THEOREM. 139. If four magnitudes are in proportion, they will be in proportion when taken inversely. Let A B C D; then will B: A:: D: C. : : For, from the given proportion, by Prop. I., we have AX DBX C, or BX CAX D. Hence, by Prop. II., B: A:: D: C. = PROPOSITION VI. THEOREM. 140. If four magnitudes are in proportion, they will be in proportion when taken alternately. Let A B C D; then will A: C:: B : D. : : For, since the magnitudes are in proportion, C D A B A X B = and multiplying each member of this equation by have which, reduced to the lowest terms, gives A B C D A: C:: B: D. whence = PROPOSITION VII.-THEOREM. 141. If four magnitudes are in proportion, they will be in proportion by composition. Let A: B::C: D; then will A+B: A::C+D: C. For, from the given proportion, by Prop. I., we have BXC=A × D. Adding A XC to each side of this equation, we have and resolving each member into its factors, Hence, by Prop. II., have A+BA::C+D: C. PROPOSITION VIII.-THEOREM. 142. If four magnitudes are in proportion, they will be in proportion by division. : : Let A B C D; then will A-B: A :: C—D : C. For, from the given proportion, by Prop. I., we have BX CA × D. Subtracting each side of this equation from A × C, we AXC-BX C=AXC-AX D, and resolving each member into its factors, Hence, by Prop. II., A-BA:: C-D: C. |