BOOK II. OF RATIOS AND PROPORTIONS. Definitions. 1. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind. Thus, if A and B represent quantities of the same kind, the ratio of A to B is ex The ratios of magnitudes may be expressed by numbers, either exactly or approximatively; and in the latter case, the approximation may be brought nearer to the true ratio than any assignable difference. Thus, of two magnitudes, one of them may be considered to be divided into some number of equal parts, each of the same kind as the whole, and one of those parts being considered as an unit of measure, the magnitude may be expressed by the number of units it contains. If the other magnitude contain a certain number of those units, it also may be expressed by the number of its units, and the two quantities are then said to be commensurable. If the second magnitude do not contain the measuring unit an exact number of times, there may perhaps be a smaller unit which will be contained an exact number of times in each of the magnitudes. But if there is no unit of an assignable value, which shall be contained an exact number of times in each of the magnitudes, the magnitudes are said to be incommensurable. It is plain, however, that the unit of measure, repeated as many times as it is contained in the second magnitude, would always differ from the second magnitude by a quantity less than the unit of measure, since the remainder is always less than the divisor. Now, since the unit of measure may be made as small as we please, it follows, that magnitudes may be represented by numbers to any degree of exactness, or they will differ from their numerical representatives by less than any assignable quantity. Therefore, of two magnitudes, A and B, we may conceive A to be divided into M number of units, each equal to A': then A=M × A': let B be divided into N number of equal units, each equal to A'; then B-Nx A'; M and N being integral numbers. Now the ratio of A to B, will be the same as the ratio of M× A' to N× A'; that is the same as the ratio of M to N, since 'A' is a common unit. In the same manner, the ratio of any other two magnitudes C and D may be expressed by Px C' to QxC', P and Q being also integral numbers, and their ratio will be the same as that of P to Q. 2. If there be four magnitudes A, B, C, and D, having such B D values that is equal to then A is said to have the same ratio A C' to B, that C has to D, or the ratio of A to B is equal to the ratio of C to D. When four quantities have this relation to each other, they are said to be in proportion. To indicate that the ratio of A to B is equal to the ratio of C to D, the quantities are usually written thus, A: B:: C:D, and read, A is to B as C is to D. The quantities which are compared together are called the terms of the proportion. The first and last terms are called the two extremes, and the second and third terms, the two means. 3. Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents; and the last is said to be a fourth proportional to the other three taken in order. 4. Three quantities are in proportion, when the first has the same ratio to the second, that the second has to the third; and then the middle term is said to be a mean proportional between the other two. 5. Magnitudes are said to be in proportion by inversion, or inversely, when the consequents are taken as antecedents, and the antecedents as consequents. 6. Magnitudes are in proportion by alternation, or alternately, when antecedent is compared with antecedent, and consequent with consequent. 7. Magnitudes are in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent. 8. Magnitudes are said to be in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent. 9. Equimultiples of two quantities are the products which arise from multiplying the quantities by the same number: thus, mx A, mx B, are equimultiples of A and B, the common multiplier being m. 10. Two quantities A and B are said to be reciprocally proportional, or, invasely proportional, when one of them is proportional to unity divided by the other, in which case their product is always equal to a constant quantity. PROPOSITION I. THEOREM. When four quantities are in proportion, the product of the two extremes is equal to the product of the two means. Let A, B, C, D, be four quantities in proportion, and M : N :: P:Q be their numerical representatives; then will M × Q NxP; for since the quantities are in proportion Q fore N=Mxor N× P=M× Q. N Q there M 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 Mx N=Px Q' (Prop. I.): hence, Q'= quently Q-Q', and the four quantities are proportional; that is M:N:: P: Q. MxN ; but Q: MxN ; conse PROPOSITION III. THEOREM. 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 M: N:: P: Q, by supposition, MxQ=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. If there be four proportional quantities, and four other proportional quantities, having the antecedents the same in both, the consequents will be proportional. 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 two quantities be in proportion, they will be in proportion when Let taken inversely. M:N::P:Q; then will For, from the first proportion we have MxQ=Nx P, or NxP=MxQ But the products Nx P and MxQ 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. Ꭰ 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 MxQ=NxP, or NxP=MxQ; 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 (MN) x P (P±Q) × M. But M±N and P, may be considered the two extremes, and P+Q 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. Nx M, 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. M×n. 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. |