CALCULUS. 



ofMN to NO and NR to OP, tlic magnitude LN, wliicU B-B -A to B, as is exprelTed by the number ». And, a, 1 



has to B the ratio compounded of the ratio of A to B. a"d Q^^ may be equal to any magnitude C, of the fame kind Ij"^ 



the ratio of cqu.ility B to B is cxprcffcd by B + A-B. .,.,,„ ^ n n n-l C n n- > n 



1„ hkc manner if PI, RY be parallel to VO and KY to w.th A and B, B- -• C + - -^ B ~ 'l ~r ' 

 NR, and KLbe a fourth prop ortional to Ol , (lj<, and 



* _ 



Q> Q. 



LN. or to B, A-B, and B + A-B, it will be equal to _, +-&C. . . to ± — — - has to B fuch a multiplicate 



y s " B' ' 



-T 77 ^ A — li | gji^j j-j^ ^^.-^11 j,g equal to B + -. ratio of B — C to B as is eKprcfitd by the number n. 



A — U 1 77— I . , 



, , , .—^, That B+--A-B + ? . fL_^l + Sec. Sec 



- -~' ■ A-B) q ? -? B 



A-B+ r -T -JT' , 



Again, if VK be a fourth pixjportional to OP, OR, and has to B fuch a multiplicate ratio of B + - • A-B + |l 



KN, or to B. A-B, and B + -• A-B + -• —j-' f , Czi . fluH' + &c. &c. to B as is exprcfTcd by ' 



A — BV . .„ , 1 . A P J. , A — Bi 1 the number n is thus (hewn, I.'t B + D be equal to B 4- 



, It uill be equal to l\ — L> + 2. — - — ' t ^ 



_!L, ^ L .A^+ L.'SHl. dzii^^ + &c. &c. C. and J 



—^ ' and VN will of courfe be equal to B + J. A-B ^^^.^^^ ^^ ^^^^ ^^j^^^. ^^^ ^.^^j^ ^j. ^^^^ ^^^ homogeneous mag. 



B 



+3' 



rt-B .' + A-B ^^^^^^ ^ 3 .X^B + i.l-' • ritudes whatever). Then B + - ■ D + "- ■ "—1 • 



B B' I I 2 112 



A-B i- . -2. lU.. dZz. A-Bl _ Wherefore the tt + : ' "T" * T" ' »' +^'^•'"7:7:1 has to B (by 



— B~ I2jB' Kizju U 



wliat is above demonftrated) fuch a multiplicate ratio of B + 



magnitude of the fame kind with A and B, which has to B D to B, as is txpreffed by tlie , umber ti. Now if the termj 



the triplicate ratio of A to B, or, which comes to the lame j^ ^^,j^;^.^ A-B is found in the value of D be multiplied by 



thing, the ratio, that arifes by compounding the triplicate ^^ 



ratio of A to B, with the ratio of equality B to B, is geo- 1 . [he co-efficient of D we get — • A-B for the fecond 



5 3 3-1 . * .9 



metrically exprtlTcd by B + -• A — B + - • — —' term of the exprefTion, (in terms of n, r, g, and A — B). And 



"" A — n ^ 



A — Bi" ^ 1 — I 1-- A — bli , , . r ,, , , iftheterm, in which . is found in the valae of D, 



tZ _' J--'- •- •■ • Anduniverfallybythc B 



b ^ I 2 3 ^' 



, . . . r , , T, >' rt — S be multiplied by - • the co-efScient of D, and the term, 

 fame geometrical realoaing it isfound that B -f - ■ -fi- - d ^ i 



n n-l A^Bl' n ti—i n-2 A-B*' in which Z£l is found in the value of — , be multipli. 



+ v~ B" + :' "■~T"-B~ + ^ ^^^ p, 



TT^Bi" , „r , , • ,• • r A . ed by the co-efficient of --» thefe taken together 



j^j._ to fl _' has to B fuch a multiplicate ratio of A to •' i 2 B ° 



B, or of B + A^ to B, as is exprefTed by the number «. »'!» g'^e "« y ' ^^ * ^^^ f"'" ^^^ ^^''^^ t^™- And 



And as A — B may be equal to any magnitude C of the 



fame kind with A and B the expreffion B -f ■^- C + ^ • ''V ^1'"' proceeding we get B -f ^ • D -f 1 • ^' -- 



is a magnitude of the fame kind with C and B, that has to ■ 2L • A — B 4- — • "''~^ • A — B| , g^^ ^^^ 

 B fuch a multiplicate ratio of B -}- C to B as is exprefTed q q 2q B 



by the number n. whatever be the ratio of r to 7. 



The foregoing exprefl-.ons are obtained geome^ically p^.^^;,-^,! j„ ^^^ ^^^^ manner is it proved that the ratio B- 

 Awhen A is greater than B. But it A be lets than B, and 



be reprefented by EN or FN, we have FM equal to "Z. • B— A -f- "- • 21lU • B— A |' _ ^ &(.. : B is 

 B — A, and by proceeding in an exaftly fimilar manner we q f ^? B 



get the geometrical expreffion B — -. B — A -f — • * a multiple by n of the ratio B — - ' B— A + -• 



B^A^' _ ^:lzJ.!L-_^ EIl' + ^c. ...to± cui • lEl^'- -f &c. = b. 



B » * J B' 27 B 

 g"3^i This then being premifed, it can be generally, and at the 

 ■ for the magnitude of the fame kind with A and B, ;. ^ 



B'~' fame time very briefly proved, that B-f- - 'A — B+-" 



that has to fi fuch a multtplicate ratio of A to B, or of ? ? 



IT-? 



