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, B is a mean proportional between A and C, and C is 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 antecedents 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. Quantities are proportional by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent. 9. 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. 10. Equimultiples of two or more quantities, are the proJucts obtained by multiplying both by the same quantity. Thus, mA and mB, are equimultiples of A and B. PROPOSITION I THEOREM. If four quantities are in proportion, the product of the means will be equal to the product of the extremes. 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. Assume the proportions, A: B :: C: D; whence, and, A B F: G; whence, :: Cor. If the antecedents, in two proportions, are the same the consequents will be proportional. For, the antecedent of the second couplets may be made the consequents of the first, by alternation (P. III.). PROPOSITION V. THEOREM. If four quantities are in proportion, they will be in proportion by inversion. Assume the proportion, B D A: B :: C: D; whence, Ā C 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, = Α 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. Let A and B be any two quantities; then 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. Assume the proportion, B D A B :: C: D; whence, = If we multiply both terms of the first member by m, and both terms of the second member by n, we shall have, |