CALCULUS. 



AC 



AC - 

 ACE 



dT ' 



ACE - 



And fo on. 



AD 

 C 



BA 



PA" 

 B" 



B+A+C 



DA" 



Anteccd<ntat. 



AC+CA , . 

 ~- or &c. &c. 



a a 



AC+CA, or &c. 



ACE+AEC+CEA 

 DF • 



or &c. 



refpedlively when B the Randard or comparifon, becomes 

 I, or arithmetical unity, and when A + N and A — N are 

 to A in relations nearer to that of f-quality than by any 

 given or affigntd magnitude of the fame kind they become 



11 



each of them — .A N. Now if we denote the 

 9 



differential of A by ^ A, and fubftitute this for N we get 



r 7 r 



ACE+AEC+CEA, or Sec. 



I. A 



" (/ A for the differential of A^, and if we call A 



CDA-ADC 

 C' ' 



a 



BA 



a 

 nDA-'-'A 



or &c. 



the fluxion of A and fubftitute it for N we 



r 



get- 



A? 



B' 



a a 



A+C 



-, or &c. &e. 



BA- 



A° 

 B«-» 



B' 



BA- 



nPA i-'A 

 B" ' 



or &c. 



A for the fluxion of A'^. And it is evident that — . 



S 



A ^ </A is a fourth proportional to i (the arithmetical 

 llandard of comparifon, to which the rxprcflions both in 

 the fluxionary and differential calcuh have a reference), 



A -, and - . -^ the differential of the magnitude of the 



A 



A» 

 B- 



&c. &c. 



X f g j, or &c.calhng 



M the magnitude of the ratio of A 

 to B. 

 &c. &c. 

 Mr. Glenie informs us that this calculus firft occurred to 

 him in 1774, and that, as it is purely geometrical and per- 

 fealy fcientific, he has always fince that time ufcd it inllead 

 of the fluxionary and differential calculi. In a paper, which 

 was written in a great meafure at the requcft of iome of 

 his friends, as he informs us, and is publifiied ui the fourth 

 volume of the Tranfaftions of the Royal Socivty of Edin- 

 burgh, explanatory of its principles, to prevent' mifconcep- 

 tions and erroneous opinions refptfling them, he (hews that 

 they are the fame with thofe on which the gtomttric:4 for- 

 mula io the Univerfal Comparifon itfelf are fouiwleii ; tlial 

 no i.idefinitdy fraall or infinitely little magnitudes are fup- 

 pefed in it, but only rr.igiiitudeslcfs than any that may be 

 given or affigned, aid ratios nearer to that of equality than 

 any that may be given or affigned, and proves it to be 

 equally geometrical uith the method of cxhauftionsof the 

 ancientst who never fuppofed lires, furfaces, orfolid'i to he 

 refolved into infinitely imall or nifii>itely httle elements. He 

 fays, indeed, that the expreffion, isfinitL-ly I'f.th nuigmtude, 

 impUes a contradiftion, fmce what has magnitude cannot be 



infinitely httle. , , , r e 



Wc (hall noiv (hew, with what facility both thele other 



cakuh may be derived from the fame principles. 

 The general geometrical expreffions, from 



'-e 



ratio A^-: l to the faid ftandard I ; and that - . A. * 



A is a fourth proportional to i , A', and - 



the fluxion 



of the magnitude of the ratio A '' : i to the fame ftandard 1 , 

 If, therefore, we make ufe of w, y, z, &c. itifttad of the 

 variable magnitudes A. C, E, &c. in the antecedental cat*. 

 cuius as delivered by its inve' tor, the integrals and their 

 differentials will ftand as follou s ; 



vhich the 



A ■? 



anteccdental of 



beeome 



was derived, 



B 



And the fluents and their fluxions will {land in the fol- 

 lowing manner. 



-■>? 



Fluent. 





!1? 



2<c 



A * . N'-f&c. j 



-1 



and 



N- 



2q 



NH — h&c.J 



Fluxion-. 



^x'x 



X' 



fluent. 



