PROPOSITION V. THEOREM. If four quantities are in proportion, they will be in proportion by inversion. If we take the reciprocals of both members (A. 7), we have, A C = ; whence, B : A :: D: C; B D which was to be proved. PROPOSITION VI. THEOREM. If four quantities are in proportion, they will be in proportion by composition or division. Assume the proportion, B D A B C D; whence, 1 : = C If we add 1 to both members, and subtract 1 from both whence, by reducing to a common denominator, we have, A B+A: C: D+ C, and, A: B-A :: C: D-C : which was to be proved. PROPOSITION VII. THEOREM. Equimultiples of two quantities are proportional to the quantities themselves. B A Let A and B be any two quantities; then wil' denote their ratio. If we multiply both terms of this fraction by m, its value will not be changed; and we shall have, mB B = whence, mA : mB :: A : B; which was to be proved. PROPOSITION VIII. THEOREM. If four quantities are in proportion, any equimultiples of the first couplet will be proportional to any equimultiples of the second couplet. If we multiply both terms of the first member by. m, and both terms of the second member by n, we shall have, ; whence, mA : mB nC : nD; which was to be proved. PROPOSITION IX. THEOREM. If two quantities be increased or diminished by like parts of each, the results will be proportional to the quantities themselves. If both terms of the first couplet of a proportion be increased or diminished by like parts of each; and if both terms of the second couplet be increased or diminished by any other like parts of each, the results will be in proportion. Since we have, Prop. VIII., mA : mB :: nC : nD; PROPOSITION XI. THEOREM. In any continued proportion, the sum of the antecedents is to the sum of the consequents, as any antecedent to its corresponding consequent. From the definition of a continued proportion (D. 3), B(A + C+E+ G + &c.) = A(B+D+F+II+ &c.) : hence, from Proposition II., À+C+E+G+ &c. : B+D+F+II+&c. :: A : B; which was to be proved. PROPOSITION XII. THEOREM. If two proportions be multiplied together, term by term, the the products will be proportional. Assume the two proportions, A : B C D; whence, and, E: F:: G: H; whence, Multiplying the equations, member by member, we have, Cor. 1. If the corresponding terms of two proportions are equal, each term of the resulting proportion will be the square of the corresponding term in either of the given proportions hence, If four quantities are proportional, their squares will be proportional. Cor. 2. If the principle of the proposition be extended to three or more proportions, and the corresponding terms of each be supposed equal, it will follow that, like powers of proportional quantities are proportionals. |