CALCULUS. 



r-q A-F' , r r-q r^tq A-B]' . . 



■ — rr + - • • i • —5-— + &C. 



J? B 72? 37 ^ 



has to B fiich a ratio, as has to the ralio of B + A — B to B, 

 or of A to B, the ratio of r to g, be the ratio of r to ^ what 

 it may, by taking any multiples whatfoever of thefe ratio* 

 and the fame nuiltiplts of r and g. For the fake of perfpi- 

 cuity let thefe m:igiiitudes, and their multiples, ftand in the 

 following manner : 



The magnitudes. 



ift r 



jd g 



3d. B + - • B-B + - ■ 



7 7 



-7 A-B" 



+ &c. : B 



Their multiples. 



Of the ift nr 



Of the id rtj 



3d. BH- — •A-1J + -• 

 7 _ 7 



=::i:2.A-Br+8,e..B 



:? B 



27 B 



<!<h. . . . B4-A-B: B 



"i'- + &c. : B 

 z 13 



Now it is evident, that if nr he equal to rnq. by fubftitut- 



. , . „ nr nr ni — g 



ine m<7 for nr, the ratio D + — • A — B+ — • ' 



7 7 27 



B 



+ &c. : B is equal to, or becomes the ratio B + 



'" • A-B + - 



I X 



:^' 



B 



— + 



to 



?E3'\b. 



B"— ' 



It is equally erldent, that if nr be greater than mq, the ar- 



T> "'' A "5 "'■ nr — g A— ^" 



*ecedent B + — • A-B + — • 1- • fi — Tl + 



7 7 2y B 



kc. is greater than B + 



m 

 I 



K=^y 



+ &c, to 



A-B")" 



• A-B + - 

 fince . . . 



nr nr ni — q 



m »i — I 



• &c. 



2? 



- n- '"9 '"? ""7 — 7 5 V 

 *re then rclpcMively greater than — ' — * 7-' i'-^- r 



or than 



-' &;:. 



indthe ratio B-f --A — B + — ' • — j; — 



Q 7 ^7 •" 



+ &c. : B is, of courfe, a greater ratio of greater ihequali- 



m X "* m—i A — B|' 



ty than the ratio B-f— ■ A — B+ - • • — g— 



-f &c. :Bis. And it is no Icfs evident that, if nr be lefs than 



„ nr , - nr nr — q A— wj 

 /no, tlie antecedent B + — • A— B-f - • — • 

 9 7 



2q B 



-f &c. is Ufs than the anttcedent B -f — • A — B -f — 



2 ' ~ir" 



are then refpcftively lefs than 



- nr nr m — g 

 + «C. Cnoc ...-'— • !' 



7 7 ^7 



mq mg _ mg — g 



7 7 



m m 



S:c. 



or than . . . 



=7 r 



2 J 



and the ratio B -f— -A — B -f — 



nr ni — < 



'9 



A-W' 



— jj — + &c. : B is then, of courfe, a lcf» ratio of greater 



inequality than the ratio B -f 



A-B-f - 

 I 



m — I 



1 



A-Bl" 



B 



+ &c. 



Here then are four magnitudes, and any equimultiples be- 

 ing taken of the ill and 3d, and alfo any equimultiples be- 

 ing taken of the id and 4th, it is proved that, if the multiple 

 of the ill be equal to the multiple of the id, the multiple of 

 the 3! IS alfo equal to the multiple of the 4tli ; if greater, 

 greater; and if 1 is, lei's. Wherefore (5. Def. E. J.) the 

 magnitudes thcmfclvcs are proportional, or the ratio B -f 



Tj + L.'-JZl . ^pl' ^ &c. : B is to the 



?_ ^7 B 



B as r : 7 whatever be the ralio of r 



B -f A 



-• A- 

 7 



ratio 

 to g. 



Hiving thus fully explained the doftrine of proportion, 

 on which this calculus is grounded, we will brielly illullrate 

 Mr. Gleiur's derivation of it from the fame, and then ihew, 

 with how much cafe and lac lity both the difTcrcntial and 

 fluxionary calculi may be derived from the fame fuurcc, and 

 in a manner too altogether unexcep'ioiiablc. 



U tor R and Q^we lubllilute r and g in the ill and 5th 

 formula;, in the ,d theorem of liis Uiuveiftl Companion, 



'■-7 '' ' 



they become rtfpeflivcly A -f 



>;- 7 



27 



A^By 



B' 

 B 



A-h\ 



B' ' 



-f &c. and B -f 



+ - 

 9 



r-9 

 -7 



^ r-g 



• !:^ -A. 



37 



7 27 



-f &c. the 



J7 B' 



lall of which is immediately derived from the firfl, by fub- 



liiuiting tor A in it, where it Hands unconncded with B by 



the lign minus, B -f A — B, which is equal 10 A, and is 



the very magnitude, which wc have juft prov, d to have to 



B a ritio, whicli has to the ratio of B -f A — IV: E or of 



A to B the ratio of r : q, even when r ha-; to g the ratio of 



any two h ■'mogepeoas magnitudes whatfoever. 



Mr. G'.enie, however, takes the ill of thefe two formulae, 



and fuppufcs A-f N to be in the firit place fuhllituted in it 



every where tor A. By this ful)ftitution he gets A-f N-f 



A— '' + N ■ 



A-f .^( . r— for the magnitude, which has to B the 



duplicate r.itio of A-fN to B, exceeding the magnitude 



A-f A ■ 5-» which, has to B the duplicate ratio of 



„ . 2AN-fN' , ,., 

 A to 0, by — ■ • In like manner he gets the ex- 



B 



5 A » 



ceft 



