A : C B D 2 : 8 :: 4 : 165 2. Inverse ratio, is when the consequent is taken as the antecedent, and so compar ed to the antecedent as the consequent. As A : B:: C : D ; therefore inversely as S B : A :: D : C 4 : 2 :: 16 : 85 3. Compound ratio, is when the antece dent and consequent, taken both as one, are compared to the consequent itself As A : B :: C: D ; therefore by composi tion, as A + B : B :: C + D : D : in numbers, as 2 + 4 = 6, is to 4, so is 8 + 16 = 24, to 16.
4. Divided ratio, is when the excess wherein the antecedent exceedeth the consequent is compared to the conse quent. As A : B :: C : D ; therefore by division, A—B : B :: C—I) : D in num bers, as 16: 8 :: 12: 6: thet etbre us 16 8=8, is to 8, so is 12-6=6 to 6.
When of several quantities the differ. ence or quotient of the first and second is the same with that of the second and third, they are said to be in a continued arithmetic or geometric proportion.
Thus 5 a, a+d, a±2 d, 0+3 ti, a±4 d?_ Z a, a—d, a-2 d, a-3 d, v-4 d &c. is a series of continued arithmetical proportionals, whose common difference is d.
a, ar, arr, arrr, a rrrr, art Anda, n, a, a / r rr 7177' 71 &c. is a series of continued geometric proportionals, whose common multiplier is 1 r 1 —or —, or whose ratio is that of 1 to r, or r to 1.
Pilo PORTION of figures. To find the proportion that one rectangle bath to another, both length and breadth must be considered. For rectangles are to each other, as the products of their respective lengths multiplied by their breadths.— Thus, if there be two rectangles, the for mer of which bath its length five feet, and its breadth three; and the latter bath its length eight feet, and its breadth tour.—
Then the rectangles will be to each other as 3 X 5 (= 15), is to 4 x 8 (---- 32) ; that is, as 15 : 32, so that all the rectan gles are to one another in a ratio com pounded of that of their sides.
When rectangles have their sides pro A 8: D: portionable, so that - 8 : H 4 : 4 E F, then is the rectangle A, to the rec 2 tangle B, in a duplicate proportion to the ratio of the sides. For the ratio of A to B, is compounded of the ratio of A B to E H, and of the ratio of A D to E F. And therefore the proportion of A to B, being compounded of equal ratios, must be du plicate of the ratio of their sides to each other ; that is, duplicate of the ratio of A B: H, or of A D: E F.
Hence all triangles, parallelograms, prisms, parallelopipeds, pyramids, cones, and cylinders, are to one another respec tively compared, in a proportion com pounded of that of their heights and ba ses. All triangles, and parallelograms, pyramids, prisms, and parallelopipeds ; al so all cones and cylinders, each kind com pared among themselves; if they have equal altitudes, are in the same propor tion as their bases ; if they have equal bases, are as their heights.
For the bases, or heights, will severally be common efficients or multipliers ; and therefore most make the products be in the same proportion as the multiplicand was before.
Thus, if the equal altitude of' any two triangles, parallelopipeds, cones, &c. be called A, and their unequal bases B and D: then it will be as 11 :D::AB:A D.