a'rab tar-br is less than a®- ab tax, the ratio of a-:c is less than the ratio of a-6+2: +x, and consequently the ratio of amb ; a has been increased by the addition of x to each of its terms. In the same manner it might be shewn that a ratio of greater inequality is INCREASED, and a ratio of lesser inequality is diMINISHED, by SUBTRACTING the same quantity from each of its terms. (106.) On the composition of Ratios. I. Ratios are compounded together by multiplying their antecedents for a new antecedent, and their consequents for a new consequent. Thus, if the ratio of a : 6 be compounded with the ratio of c:d, the resulting ratio is that of ac: bd; or if the ratios 4:3; 5:2; and 7:1, be compounded, there results the ratio of 4 x 5 x 7:3 x 2 vl, or of 140: 6, or of 70: 3, dividing each term ly 2. II. When the same ratio is compounded with itself once, twice, thrice, &c. the resulting ratios are those of u? :bo; us : b*; a: b, &c. &c. The ratio of a : 1,2 is called the duplicate ratio of a :6; the ratio of as: 69 the triplicate ; the ratio of a* : 24 the quadruplicate ; &c. &c.; and as these ratios receive their denominations from the indices of the several powers of a and i, the ratio of va: vb is called the subduplicate ratio of a:b; the ratio of Fa:8, the subtriplicate ; &c. &c. III. When a set of ratios, of which the consequent of the preceding ratio is the same with the antecedent of the succeeding one, is compounded together, the resulting ratio is that of the first antecedent to the last consequent. Thus, when the ratios of a :b; of 6:0; of c:d; of d:e; &c. are compounded together, the resulting ratio is that of abcd, &c.; of érde, &c. or (dividing by lcd) that of a : e, or of the first antecedent : the last consequent. IV. A ratio, compounded with a ratio of greater inequality, will be increased; but, if it be compounded with a ratio of lesser inequality, it will be diminished. Thus, let 1+n: 1 represent a ratio of greater inequality, and let it be compounded with the ratio a:b, the resulting ratio is that of arna : b, which is evidently greater than the ratio of a :0. On the other hand, let 1-n: 1 represent a ratio of lesser inequality, and let it be compounded with the ratio of a:b, then the resulting ratio is that of e-na: 1, which is evidently less than the ratio of a:l. EXAMPLES. 1. Reduce to their lowest terms the ratio of 360: 315, and 1595 667, 2. Reduce the ratio of a'+2ar : q? to its lowest terms. greater? 4. Which is the least of the three ratios, 20:17, 22:18, or 25 : 23 ? and of the three ratios 8:7; 6:5; and 10: 9, which is the greatest ? 5. Which is the greater, the ratio of a+2 : {a+4, or that of a+4:+5? Ans. the ratio of a +4: £a+5. 6. Compound together the ratios of 11 : 3, of 7 : 2, and of 5: 9. Ans. 385 : 54. 7. Compound together the ratios of 15:12, of 6: 7, and of 9:4; and afterwards reduce the resulting ratio to its lowest terms. Ans. 135 : 56. 8. Express in its simplest terms the ratio compounded of a re, :ae, atr: 6 and 6:0-X. Ans. (a+x)? : al. z? y 9. If the ratios of x+y:a, x —y:l, and I : be compounded, shew that the resulting ratio is a ratio of equality. 10. If the ratios of 3a+2: 6a+1, and of 2a+3:2+2, be compounded, is the resulting ratio a ratio of greater or lesser inequality ? Ans. A ratio of greater inequality. 11. What are the least numbers which will represent the ratio compounded of the three following ratios, viz. the ratio 7:5, the duplicate ratio of 4:9, and the triplicate ratio of 3 : 2 ? Ans. 14 and 15. 3, 12. Compound the subduplicate ratio of x* : y', with the quadruplicate ratio of vXiny. Ans.a': y. 8 of Proportion. (107.) The following are the most useful Theorems relating to proportional quantities. (108.) THEOREM 1. If four quantities be proportional, the product of the extremes will be equal to the product of the means. For let a : 6 :: 0:d, then, (by Art. 103) (= â therefore ad = bc. Corol. From hence it follows, that if any three terms of a proportion be known, the fourth may be found; for from the equation bc ad ad bc ad=lc, we have a=7 ; b and d d ;C= ; (109.) THEOREM 2. The converse of the foregoing Theorem 18 also true; viz. If the product of any two quantities be equal to the product of two others, those four quantities will constitute a proportion, provided that the terms of one product be made the MEANS, and a - terms of the other product be made the EXTREMES, of such proportion. Thus, if the four quantities a, b, c, d be such that a'd' = Uc', then ū e ! dividing by b'd', therefore, e' :bc:d, by Art. 103. (110.) THEOREM 3. If three quantities be proportional, the product of the two extremes is equal to the square of the mean ; Or, if a :6:: b:c, then (by Theor. 2,) ac = 62. From hence also it follows, that a mean proportional between any two quantities. is equal to the square root of their product; for let x be a mean proportional between a and c, then a :x :: x:C, therefore x = ac, and a = vac. (111.) THEOREM 4. If four quantities be proportional, they will also be proportional when taken inversely or alternately; Thus, if a : 6::c:d, then g = a; but invert the fractions, and then ; therefore b:a:d:c. Again, since (hy Theor. 1,) ad bc 6 ad=bc, then we have crcd ; dividing by cd, therefore d 2:6:b:d, and b:d :: 4:6. (112.) THEOREM 5. If there be six proportional quantities, and the first be to the second as the third is to the fourth; and the third to the fourth as the fifth to the sixth ; then will the first be to the second as the fifth to the sixth. q and a or (113.) THEOREM 6. If four quantities be proportional, then the SUM OF DIFFERENCE of the first and second will be to the second as the sum or DIFFERENCE of the third and fourth is to the fourth. atb atd For let a : 0 :: 0 :d, then i = 2; add or subtract 1 from each side of the equation ; then i +1=4+1, therefore, consequently, a+b: 0 :: c+d: d, by Art. 103. (114.) THEOREM 7. If four quantities be proportional, the FIRST is to the sum or DIFFERENCE of the first and second, as the third to the sum or LIFFERENCE of the third and fourth. For (by TABOREM 6,) a +b:6::c+d:d, and alternately @ +6:c+d::b:d; but (by THBOREM 4,) b:d :: 0 :c; hence, (by Theorem 5,) a +b:c+d:: 0 :c, and alternacely a+b:a ::c+d:c, .. inversely a : a+b::C:c+d. (115.) THEOREM 8. If four quantities be proportional, then the sum of the first and second is to their DIFFERENCE, as the Sum of the third and fourth is to their difference. atb ctd a-bc-d For (by THEOREM 6,) and invert the 6 d' 6 d ctd ca di d Ն ato hence two last fractions, then d х a-6 6 e- 0 d cod ato c+d therefore (by Art. 103), a+b:a-6::c+d:c-d. or a ;d-تا (116.) Theorem 9. If four quantities be proportional, and any EQUIMULTIPLE OF EQUAL PARTS whatever be taken of the first and second, and also of the third and fourth; then will the resulting quantities, taken in the same order, be still proportional. For let a :/:; c:d; then (by Case I. Art. 104,) the ratio of ma: mb is the same with the ratio of a :b; and for the same reason, the ratio of nc : nd is the same with the ratio of c:d; hence (Art. 101), ma : mb :: nc : nd, where m and n may be any quantities whatever, either integral or fractional. (117.) THEOREM 10. The same theorem is true if any EQUIMULTIPLE OF EQUAL PARTs whatever be taken of the first and THIRD, and also of the second and FOURTH; For since = , multiply each side of the equation by : then ma mc :. ma : nb :: mc : nd, where m and n may be any quannl nd' tities whatever, either integral or fractional. (118.) THEOREM 11. If four quantities be proportional, any POWERS or ROOTS of those quantities will also be proportional. :' For since = i.a" : {": [": d", where n may ē a bonda' be any number, either integral or fractional. (1.9.) THEOREM 12. If the corresponding terms of two sets me с a" we have proportionals be multiplied together, or divided by each other, the resulting quantities taken in order will still be proportionai. For let ad: Again (by THEOREM 1), ad=bc, and eh=fg; .. eh fg henco C:d (120.) THEOREM 13. If there be two rows of proportional quantities, whereof the SECOND and FOURTH of the first row are the same with the first and third of the second row, then will the remaining quantities, taken in order, be proportional ; For, let a : 6 and b:e :: d:f, then (by Theorem 12), ab : be :: cd : df, or (reducing each ratio to its lowest terms) a : e:: c:f. This is Euclid's er æquali proportion. (121.) THEOREM 14. If there be a set of proportional quantities, a : b :: c:d :: e:f::g: h, &c. &c. then will the first be to the SECOND as the sum OF ALL THE ANTECEDENTS to the SUM OF ALL THE CONSEQUENTS. :: e Again, since a+c:b+d :: a : 1,or :: c:dand c:d :: e: fi :. by THEOREM 5, atc:6+d :.fi alternately, a +cie :: 6+d :f, By THEOREM 6, atcte: e :: 6+d+f:fi :, alternately, atcte:b+d+f :: e 6 :fi therefore a :/ :: acte : 1+3+5; And so on for any number of these proportions. :f. But a : :: e (122.) THEOREM 15. If there be a set of quantities, a, b, c, d, e, : in CONTINUED proportion ; |