PRO 



PRO 



C B : A :: D 



14, :2 ::16 



ed to the antecedent as the consequent. 

 As A : B :: C : D ; therefore inversely as 



G 



2 :: 16 : 8$ 



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 -f- 4 = 6, is to 4, so is 8 

 -f 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) : U in num- 

 bers, as 16 : 8 :: 12 : 6 ; therefore as 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 $ o,fl+ rf .<H-2f/,4-3 </ c H- 4d 2 



18 I a, a d, a 2 d, a 3 d, a 4 d $ 



&c. is a series of continued arithmetical 



proportionals, whose common difference 



is d. 



Ca, ar t arr, arrr, amr, flr5"^ 



s a a a a a C 



J a,_, , , , _ C 



\- r rr rrr JVJT n j 



&c. is a series of continued geometric 

 proportionals, whose common multiplier is 



or , or whose ratio is that of 1 to r, or 



r to 1. 



Puo PORTION of fgnres. To find the 

 proportion that one rectangle hath 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. 

 Tims, if there be two rectangles, the for- 

 mer of which hath its length five feet, and 

 its breadth three ; and the latter hath its 

 length eight feet, and its breadth four. 

 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 B: : E H :: A D: 



portionable, so that 



And 



E F, 



I 



8 :: 4 : : 4 

 then is the rectangle A, to the rec- 



** 



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 tr> 

 E H, and of the ratio of A D to E F. And 



therefore the proportion of A to li, 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 : E 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 ail 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 must make the products be in 

 the same proportion as the multiplicand 

 was before. 



Thus, if the equal altitude of any two 

 triangles, parallelepipeds, cones, &c. be 

 called A, and their unequal bases B and 

 D: then it will be as B : D : : A B : A D. 



PROPORTION, hai-manic, is when three 

 terms are so disposed, that as the differ- 

 ence of the first and second : the differ- 

 ence of the second and third :: first : 

 trnrd; and they are said to be harmonical- 

 ly proportional. Thus, 10, 15, 30, are 

 harmonically proportional. For as the 

 difference of 10 and 15, is to the differ- 

 ence of 15 and 30, so is 10 to 30. Also, 

 12, 6, 4, are harmonically proportional ; 

 for 12 6 : 6 4 :: 12 : 4. So h> -f 3 

 h n 4- 2 n j , b- H- 2 h n, /i 1 -f h n t are har- 

 monically proportional. For h n -f- 2 n 1 : 

 hn : : h 1 -f 3 h n + 2 n 1 : h* -f h n. 

 Whence, if the two first terms of an bar* 

 monic proportion be given, the third is 

 readily found. 



For if A, B, C, be harmonically propor- 

 tional. Then A B : B C : : A : C, 

 and A C B C = A B A C. There- 

 fore AB = 2A BxC, and B C = 

 2 C B X A. Consequently C = 



AB BC 



and A= -. Again, when 



2 A B'~ ~2C B' 



four terms are so disposed, that as the 

 difference of the 1st and 2d : the differ- 

 ence of the 3d and 4th : : 1st : 4th, they 

 are also harmonically proportional. As 

 10, 16, 24, 60 ; for as'lO 16 : 24 60 : : 

 10 : 60. Whence, if the three first terms 

 of such an harmonic proportional be given, 

 the 4th is easily found. 



For if a, if, c, d, be harmonic propor- 

 tionals, then a b i c d;:a;d; and a 

 d 6</s= ad, therefore d = 



