PROPOSITION I. THEOREM. When four quantities are in proportion, the product of the two extremes is equal to the product of the two means X Let A, B, C, D, be four quantities in proportion, and M: N :: P:Q be their numerical representatives; then will M × Q= N Q NxP; for since the quantities are in proportion there. fore N=Mx,or NxP=M×Q. MP P Cor. If there are three proportional quantities (Def. 4.), the product of the extremes will be equal to the square of the mean. PROPOSITION II. THEOREM. If the product of two quantities be equal to the product of two other quantities, two of them will be the extremes and the other two the means of a proportion. Let MxQ=Nx P; then will M: N:: P: Q. For, if P have not to Q the ratio which M has to N, let P have to Q', a number greater or less than Q, the same ratio that M has to N; that is, let M:N:: P: Q'; then MxQ'= NXP NxP NXP (Prop. I.): hence, Q': PROPOSITION III. THEOREM. = ; con If four quantities are in proportion, they will be in proportion when taken alternately. Let M, N, P, Q, be the numerical representatives of four quanties in proportion; so that M:N::P: Q, then will M: P::N: Q. Since MN: P: Q, by supposition, Mx Q=NxP; therefore, M and Q may be made the extremes, and N and P the means of a proportion (Prop. II.); hence, M: P::N: Q. PROPOSITION IV. THEOREM. Let and then will For, by alternation If there be four proportional quantities, and four other proportional quantities, having the antecedents the same in both, the consequents will be proportional. and Q S N=R; or N:Q::R: S. hence Cor. If there be two sets of proportionals, having an antecedent and consequent of the first, equal to an antecedent and consequent of the second, the remaining terms will be proportional. PROPOSITION V. THEOREM. If four quantities be in proportion, they will be in proportion when taken inversely. M:N::P:Q; then will Let For, from the first proportion we have MxQ=Nx P, or NxP=MxQ But the products Nx P and M× Q are the products of the extremes and means of the four quantities N, M, Q, P, and these products being equal, N:M::Q:P (Prop. II.). PROPOSITION VI. THEOREM. If four quantities are in proportion, they will be in proportion by composition, or division. D Let, as before, M, N, P, Q, be the numerical representatives of the four quantities, so that M:N:: P:Q; then will For, from the first proportion, we have Add each of the members of the last equation to, or subtract it from M.P, and we shall have, M.P±N.P=M.P±M.Q; or But M±N and P, may be considered the two extremes, and PQ and M, the two means of a proportion: hence, M±N:M:: P+Q: P. PROPOSITION VII. THEOREM. Equimultiples of any two quantities, have the same ratio as the quantities themselves. Let M and N be any two quantities, and m any integral number; then will m. M: m. N:: M: N. For m. MxN=m. NxM, since the quantities in each member are the same; therefore, the quantities are proportional (Prop. II.); or m. M:m. N::M:N. PROPOSITION VIII. THEOREM. Of four proportional quantities, if there be taken any equimultiples of the two antecedents, and any equimultiples of the two consequents, the four resulting quantities will be proportional. Let M, N, P, Q, be the numerical representatives of four quantities in proportion; and let m and n be any numbers whatever, then will m. M: n. N:: m. P: n. Q. For, since M:N::P: Q, we have MxQ=NxP; hence, m. Mxn. Q=n. N×m. P, by multiplying both members of the equation by mxn. But m. M and n. Q, may be regarded as the two extremes, and n. N and m. P, as the means of a proportion; hence, m. M: n. N::m. P: n. Q. Let For, since And since Therefore, Of four proportional quantities, if the two consequents be either augmented or diminished by quantities which have the same ratio as the antecedents, the resulting quantities and the antecedents will be proportional. or, hence PROPOSITION IX. THEOREM. For, since And since Add : M N P: Q, and let also PROPOSITION X. THEOREM. If any number of quantities are proportionals, any one antecedent will be to its consequent, as the sum of all the antecedents to the sum of the consequents. Let MN: P: Q: R: S, &c. then will MN: P: Q, we have M×Q=NxP and we have, M.N+M.Q+M.S=M.N+N.P+N,R or Mx (N+Q+S)=N× (M+P+R) therefore, MN :: M+P+R: N+Q+S. PROPOSITION, XI. THEOREM, If two magnitudes be each increased,or diminished,by like parts of each, the resulting quantities will have the same ratio as the magnitudes themselves. M N Let M and N be any two magnitudes, and. and be like m m parts of each: then will Let then will and m N For, it is obvious that Mx (N) =Nx (M± m N.M Consequently, the four quan m each is cqual to M.N± tities are proportional (Prop. II.). For, or, and MN: M If four quantities are proportional, their squares or cubes will also be proportional. Let then will and or N m therefore, PROPOSITION XII. THEOREM. M: NP: Q, M2: N2: : P2 : Q2 Q3 therefore, M2: N° :: P2: Q° and M3: N3:: P3 : Q3 MxQ=Nx P, since M: N:: P: Q M2 × Q2=N2 × P2, Cor. In the same way it may be shown that like powers or roots of proportional quantities are proportionals. by squaring both members, If there be two sets of proportional quantities, the products of the corresponding terms will be proportional. PROPOSITION XIII. THEOREM. MN: P: Q R: ST : V MXR NXS :: P×T: Q× V. RxV=SxT, we shall have MXRxQxV=N×S×P×T |