this property is universally true, whether the proportion be direct or inverse. When a b::c: d, these are proportionals when taken inversely Also when the four quantities are of the same kind, or abstract numbers, they are proportionals when taken alternately, as 6. Prop. When four quantities are proportionals, (1) the sum of the first and second is to the second, as the sum of the third and fourth is to the fourth. (2) The difference of the first and second, is to the second, as the difference of the third and fourth is to the fourth. (3) The sum of the first and second is to their difference, as the sum of the third and fourth is to their difference. (Euc. V. 17, 18.) or the sum of the first and second is to the second, &c. or the difference of the first and second is to the second, &c. Divide the former of these equals by the latter; a+b a b c+d c-d a+b c+d or and a+ba-b:: c+d: c-d or the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference. 7. Prop. If four quantities of the same kind be proportionals, the first quantity being the greatest, the sum of the extremes is greater than the sum of the means. (Euc. V. 25.) Let a, b, c, d be proportionals, then ad = bc, and Subtract 1 from these equals, a b -= с d but a being the greatest of the four quantities, is greater then b, ... also a—c>b—d, and adding c+d to these unequals, .. a+d>b+c, or the sum of the extremes is greater then the sum of the means. 8. Prop. If a, b, c, d be proportionals, there is no number which can be added to each term, so that the four terms so increased shall be proportionals. Since a, b, c, d are proportionals, ... ad = bc. Let any number x be added to each term. And if possible, let (a+x)(b+x) = (b+x)(c+x), or ad+(a+d)x+x2 = bc+(b+c)x+x2, and a+d=b+c, that is, the sum of the first and fourth terms of a proportion is equal to that of the second and third. But it has been shewn that a+d is greater than b+c. Hence it follows, that there is no number which can be added to the terms of a proportion, so that the terms so increased shall be proportional. 9. Prop. If a and b are prime to each other, they are the least quantities Then = = a с = in which c and d are less than a and b. ad, and therefore ad is divisible by b, but a is prime to b, ... d is divisible by b, which is impossible, since d is less than b; therefore cannot be equal to whence c and d are respectively less than a and b; that is, a and b are the least in every proportion in which they appear as antecedent and consequent. 10. Prop. If any number of quantities are proportionals, as any antecedent is to its consequent, so is the sum of all the antecedents to all the consequents. (Euc. VII. 12.) Suppose these quantities a, b, c, d, e, f to be proportionals so that abcd::e: f. Then since a : b : : c : d, .'. ad = bc, and a be: f, af = be, and ab = ba. By addition, ab+ad+af = ba+bc+be, The same is true for four, five, or any number of proportions. 11. Prop. If the first of six magnitudes has the same ratio to the second as the third to the fourth, and the third to the fourth the same ratio as the fifth to the sixth; then the first shall have to the second the same ratio as the fifth has to the sixth. (Euc. V. 22.) Let a, b, c, d, e, ƒ be six quantities, such that a:b::c:d, and c:d::e: since a b c : d, .. a с с e d-f and a : b :: e : ƒ. 12. Prop. If the first of six quantities has the same ratio to the second as the third to the fourth, and the second to the fifth the same ratio as the fourth to the sixth; then shall the first have the same ratio to the fifth as the third has to the sixth. or the first is to the fifth as the third is to the sixth. 13. Prop. If four quantities be proportionals, then any equimultiples or equisubmultiples of the first and second, shall also be proportionals to any other equimultiples or equisubmultiples of the third and fourth. Or, The equimultiples or equisubmultiples of the first and second, &c. 14. Prop. If four quantities be proportionals, and if equimultiples or equisubmultiples of the first and third be taken, and any other equimultiples or equisubmultiples of the second and fourth be also taken, then these four quantities shall be proportionals. Or, If equimultiples or equisubmultiples, &c. It is also manifest from Art. 6 that ma+nbb::mc+nd: d, ma+nb: b:: mc-nd: d, ma+nb : ma—nb :: mc+nd: me-nd. 15. Prop. If four quantities be proportionals, the same powers and the same roots of these quantities are also proportionals. Since a, b, c, d are proportionals, That is, equal powers and equal roots of proportionals, are also proportionals. Also a+bbm :: cm+dTM : dTM, am-bm: bm :: cm. -dm: dm, aTM+bTM : am—bTM :: cm+dm : cm—dm. The following may also be shewn to be true: a+b": a" -b" :: c"+d" : cTM- d". 16. Prop. If there be two or any number of sets of proportionais, the products of their corresponding terms are proportionals. Also a e k с g m aek cgm - or n bfl-dhn and ack bfl cgm: dhn. And so, for any number of proportionals. 17. Prop. If the corresponding terms of two sets of proportionals be divided, the quotients will form a proportion. 18. Prop. If three quantities a, b, c be in continued proportion, the first is to the third as the square of the first is to the square of the second. (Euc. VIII. 11, 26.) 19. Prop. If four quantities a, b, c, d be in continued proportion, the first is to the fourth as the cube of the first is to the cube of the second.* (Euc. VIII. 13, 27.) From the following illustration, it will be seen that the duplicate and triplicate ratios in Geometry correspond to the ratios of the squares and cubes in Algebra :— 1. Similar triangles are to one another in the duplicate ratio of their homologous sides. (Euc. VI. 19.) Let a, b, c; a', ', c' represent the sides of two similar triangles; Then, because the triangles are similar, their homologous sides are proportional, a b с that is = α' p = C ; And if p, p′ be the perpendiculars from the vertices on the sides a, a', then a b с = = a'b' ¿' Also if A, A' denote the areas of the triangles, Aap, and 4'=fa'p'. (Euc. I. 41.) |