RATIO. 



And it may be further confuincd by multiplying the pro- 

 pofed feries by a — h, for the produft will be a, all the 

 terms except the fird dellroyiiig each other. 



We may alfo obfervc, that if every term ot the foregoing 

 feries be multiplied by any number c, it will become c -f- 



bc b'-c Pc „ r • , • 



1 -I + &c. ; a leries liavnig the lame ratio, 



rt a' a' 



but whofe firft term is c ; and it is therefore evident, that, in 

 this cale, to lind the fum of all the terms, wcmuft multiply 



the former limit by c, whence it will become . 



Before we begin to confider ratios, it may not be amifs to 

 caution the reader againil confounding the terms of a ratio 

 with the ratio itlelf: the terms of a ratio may vary in fomc 

 cafes through all degrees of magnitude, and yet the ratio re- 

 main conftant or invariable. In other inftances, varying the 

 terms in infinitum, likewife varies the ratio in injinitum ; 

 while in others, though varying the terms may alfo vary the 

 ratio, yet the laft ratio can never exceed certain limits. 



Let X be any varying quantity ; make 4*" f 3 .v = A, 

 and 2 .r' + jc =: B, then will A and B alfo be varying 

 quantities, as depending upon .v ; when .v vanifhes, A and 

 B will both vanifh ; when x is infinite, they will be both in- 

 finite. I fay then the ratio of A to B, while x decreafes 

 /',■; injinitum, approximates to the ratio of 3 to i. 



For, firll, A is to B, as 4 x' + 3 x is to 2 x' + x, or as 

 4.i;-f-3to2K+i; where it is obvious, that as .v decreafes, 

 the quantities 4 .v and 2 .v alfo decreafe, and confequently 

 the ratio of 4 .v -|- 3 to 2 .t- + I, or of A to B, approaches 

 to that of 3 to I. Secondly, the ratio of A to B can 

 r.ever exceed 3 to i . For 6 x +3 x is to 2 .v' -|- x, as 3 to I ; 

 but 4 X- + 3 .V is a lefs quantity than 6 x + 3 x, therefore 

 4 x"' -f- 3 .V is to 2 .V + X, or A is to B in a lefs ratio than 

 6 x'' -f 3 .■« to 2 .v' + X ; that is, lefs than the i-atio of 3 to i . 

 Laftly, the ratio of A to B will approach nearer to the 

 ratio of 3 to I, than any affigned difference. For in the 

 terms of this ratio, 4.r + 3 to 2 .v + I, the varying parts 

 4.\- and 2 x, by diminithing x, may become lefs than any 

 affigned quantity, while the other parts, 3 and i, remain the 

 i'ame ; therefore the ratio of A to B will approach nearer 

 to the ratio of 3 to i, than by any affigned difference. 



In like manner the ratio of A to B, while x increafes in 

 infinitum, approximates to the ratio of 2 to i, as its limit. 



3 

 4 .V + 3 to 2 X + I, or as 4 x -f- — 



For fuice A is to B 



to the ratio of 4 to 2, or 2 to i, than the affigned difference. 

 Hence then we fee that though diminifhing x, and confe- 

 quently diminifhing the terms A and B, we increafe their 

 ratio, and on the contrary incrcafing thefe terms, by increaf- 

 ing the quantity x, we decreafe their ratio, yet there is a 

 limit both to the increafe and decreafe of this ratio, al- 

 though there be none to the terms themfelvcs which com- 

 pofe it. 



The ratio of 3 to i, which limits the ratio of A to B, when 

 thefe terms decreafe in injinitum, is called the ultimate ra'iooi 

 the evanefcent quantities A and B. The ratio of 2 to i 

 which is their otiier limit, is called the ultimate ratio of the " 

 quantities A and B increafing in injinitum. 



Another example of a fimilar kind we have as follows. 

 Let X be a varying quantity, and d a conftant one, then 

 will X -|- d and x be two varying quantities capable of all 

 degrees of magnitude, I fay that the ratio of x + d \.o x, 

 while X increafes, will continually decreafe, but not beyond 

 a certain limit, which is the limit of equality. On the 

 contrary, if x decreafe, the ratio of x + </ to x will con- 

 tinually decreafe more and more ad infinitum, and never 



come to a limit. For x 4- d is to .v, as i + — to i ; now 



X 



as X increafes, the fraftion — decreafes, and may become 



X 



lefs than any affigned fraftion ; but the number 1, which is 

 the other part of the antecedent of this ratio, remains 

 the fame, as does likewife the confequent, therefore the 



d ■ u ■ 



ratio 01 1 -f — to I, continually approximates to a 



ratio of equ.ility. Secondly, it can never reach that ratio, 



becaufe — has always fome magnitude, and confequently 



d 



I H always greater than i. It therefore can never reach 



the ratio of equality, and much lefs can it pafs it, fo as to be- 

 come a ratio mlnorls Imequa/ltatis, or a ratio in which the an- 

 tecedent is lefs than the confequent. Laftly, the varying 



fraftion — , as x increafes, will become lefs than any affigned 



fraftion, while the other part of the antecedent, and like- 

 wife the confequent of this ratio, remain the fame. There- 



i 



it is obvious, that as .v increafes, the quantities ^°''^^^ "^'° of i + - to ., will approach nearer to the 



•-^- and — decreafe, and confequently the ratio of 4 + — 



X K X 



to 2 -1 , or of A to B, approaches to the ratio of 4 to 2, 



X 



or of 2 to I . For 4 .<- + 2 x is to 2 x- + x, as 2 is to i ; 

 but 4 x' + 3 X, is a greater quantity than 4 x' -t- 2 x, there- 

 fore 4 X- -(- 3 X is to 2 x' + X, or A is to B always in a 

 greater ratio than 2 to i. Laftly, the ratio of A to B 

 will approach nearer to that of 2 to i , than by any affigned 



3 I 



difference. For in the terms of this ratio, 4 -f - and 2 -) , 



X X 



3 I 



the variable parts -^ and — , by increafing x, may-become 



X iC 



lefs tlian any affigned fraftion, while the parts 4 and 2 remain 

 the fame ; therefore the ratio of A to B will approach nearer 



ratio of equality, than by any ratio that can be affigned, 

 however fmall fuch ratio may be ; therefore the ratio of 



equality is the limit of the ratio of i -f — to i, and 



X 



confequently the limit of the ratio of x + </ to x, which 

 continually d'-creafes, while the terms which compofe this 

 ratio continually iacreafe in Infinitum. 



Ifx decreafe then — will increafe, and may become greater 



X 



than any affigned number, while the other part of the ante- 

 cedent, and likewife the confequent, remain invariable ; 



therefore the ratio of 1 -j to i, and confequently the 



ratio of x + d \o x, as x decreafes, will become greater than 

 any affigned ratio whatever, having i>o limit to its increafe. 



We 



