Likewise the consequents. X. XI. The first and fourth terms may be called extremes. And the second and third, means. XII. XIII. If the ratio of every two adjacent terms is the same, the tion may be said to be continued. XIV. Continued proportion may be called progression. XV. propor Whatever is the ratio of the first to the second, and of the second to the third, the ratio of the first to the third is compounded of the ratio of the first to the second, and of the second to the third. And when the ratio of the second to the third, is the same with that of the first to the second, the ratio of the first to the second may be considered as repeated, and therefore if three quantities are in continued proportion, the first may be said to have to the third, the duplicate ratio of that which it has to the second. XVI. If four, the triplicate ratio, and so on, increasing the denomination still by unity in any number of proportionals. XVII. A ratio may be thus expressed, A: B; as A is to B. XVIII. A ratio may also be expressed by the part which one quantity is of Proportion may be thus expressed; A: B:: C: D. As A is to B, so is C to D. XX. Or the two equal ratios which make up the proportion may be thus A C expressed, A divided by B is equal to C divided by D. PROPOSITION I. THEOREM. If four quantities are proportional, the product of the two extremes is equal to the product of the two means. If four quantities are such that the product of two is equal to the product of the other two, these quantities are proportional. If four quantities are proportional, the product of the means divided by either extreme, will give the other extreme. Because A: B:: C: D, AD=BC; divide by A, and D= BC Α ́ PROP. IV. THEOR. The products of the corresponding terms of two proportions are also proportional. Let A B C : D, and E: F:: G: H; then AE: BF:: CG : DH. : Because A: B :: C : D, and E: F::G: H, COR. If four quantities are proportional, their squares and cubes will likewise be proportional. PROP. V. THEOR. If four quantities, A, B, C, D are proportional; then, Inversely, B: A:: D: C. VI. Alternately, A: C: B: D. VII. Compoundedly, A: A+B :: C: C+D. VIII. Dividedly, A: A-R:: C: C-D. IX. Mixtly, B+A : B—A :: D+C : D—C. A B : XI. By Division, : :: C: D. Because in each case the product of the means is equal to that of the extremes, and therefore the quantities are proportional. PROP. XII. THEOR. In geometrical progression, the product of the extremes is equal to the product of any two terms equidistant from them. Let A, B, C, D, E, F be in progression; then AF=BE or CD. Because the ratio of every two adjacent terms in progression is the same (Def. 13. 4.), A : B :: E: F, and therefore AF-BE; and it may be shewn in the same way, that AF=CD; and therefore AF=BE or CD. PROP. XIII. THEOR. Equal quantities have the same ratio to the same; and the same has the same ratio to equal quantities. Let A and B be equal quantities, and C any other quantity; as A: C: B: C. Because A is equal to B, C is the same part of A that it is of B, A B and therefore = ; or A : C: B: C. Likewise C: A :: C: B. Because A is equal to B, A is the same part of C that B is of C, PROP. XIV. THEOR. Quantities which have the same ratio to the same are equal; and those to which the same quantity has the same ratio are equal. Ratios that are the same to the same ratio, are the same to one another. If there be any number of quantities, and as many others, which taken two and two in order have the same ratio, the first shall have to the last of the first quantities the same ratio which the first of the others has to the last. This is usually cited by the words " ex æquali," or "ex æquo." Because A: B:: D: E, = First, Let there be three quantities A, B, C, and three others D, E, F; and let A B :: D: E, and B : C : : E : F; then A: C :: D: F. A D and because B: C:: E: F, B E A B D E multiply by also by B C' E F B E C F B AB equal to and C BC = DE ; EF Next, Let there be four quantities A, B, C, D, and four others E, F, G, H; and let A: B:: E: F, and B: C:: F: G, and C: D:: G: H; A:D :: E: H. Because A, B, C are three quantities, and E, F, G three others, which taken two and two have the same ratio, by the above case, A: C :: E: G ; but C: D:: G: H, therefore again by the above case, A : D :: E: H; and so on whatever be the number of quantities. PROP. XVII. THEOR. If there be any number of quantities, which taken two and two in a cross order have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. This is usually cited by the words "ex æquali in proportione perturbata," or "ex æquo perturbato." First, Let there be three quantities A, B, C, and D, E, F, and let A: BE: F, and B: C:: D: E; then A: C:: D : F. Next, Let there be four quantities A, B, C, D, and four others E, F, G, H; and let A : B :: G : H, B : C : : F: G, and C :D :: E: F; then A: D:: E: H. Because A, B, C, are three quantities, and F, G, H, three others, which taken two and two in a cross order have the same ratio, by the first case, A: C:: F: H, but C :D :: E : F, wherefore again by the first case, A D : : E : H; and so on whatever be the number of magnitudes. Therefore, If, &c. Q. E. D. : If the first has to the second the same ratio which the third has to the fourth; and the fifth to the second, the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second, the same ratio which the third and sixth together have to the fourth. : : Let A B C : D, and E : B :: F: D; then A+E B:: C+ F: D. : BC+F D. Therefore, If, &c. Q. E. D. and therefore A+E COR. I. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second, as the excess of the |