3go rHiLosopHicAL TRANSACTIONS. [anno l6g7 . 



of the ordinate bf, the fluxion of the curve af, and the right Hne fr, will be 

 continual proportionals. 



Let the little right line f/ be produced, till it meets the next ordinate Wip 

 in o. And because by the hypothesis Fs=fw, it will be oJ=Ff, and therefore 

 o^ will be the fluxion of fs, that is, the fluxion of the fluxion of the ordinate. 

 Moreover the triangles Ofj/and/FR are equiangular, because Oi^is equal to the 

 alternate /pfr, and/o p = f/i- = f/r, because their difi'erence r/t vanishes in 

 respect of either of them, since Rr is nothing in comparison of //-. Therefore 

 it is 0(p : (pfy.J'v : fr. But <p/ and /f are equal, since they only differ by the 

 fluxion of each. Therefore otp -.Jf -.-.J'f : fr. q. e. d. 



Prop. 6. Proh. — To find the curve Kv, by the evolution of which the cate- 

 naria afq is described. 



Make a b = x, and b f = y, as before. Then by Prop. 2 of this, it is 



it = — = , or laxiiy -\-xxyii:=aal\v. Then, by Newton's method now 



in common use, it is 'laly'^- -^ Aaxyii •^'Ixxij- ■^2x'^ ifi)-=Q.d- l\v-=iO ; for .r 

 = 0, because x is a constant quantity. Therefore y = „ ^ / '^ = 



a_ X •x.ax , substituting instead of ii its value . " ; for the 



sign — prefixed to the quantity y only shows, that the place of the point r in 

 respect of f, is opposite to the place of the point f in respect of b, since the 

 curve AFa is concave towards the axis ab. And by the second prop, of this, 



F f = - ",-j!i^ . Wherefore by the foregoing lemma, f r = ^^ = q + jI'xa-^ 



■^ ^/'lax + xx J i= o y 2ax + xx 



Su.r + J.rx A/'2a.v + ,T,c a+iX \^'iax + xx a • u c i.\ • i ^ i i 



X — — = — — — . Again, because of the right-angled 



triangles fs/ and fkv, having equal angles /fs and vfr, because vf* is the 



complement of each to a right angle, it is f* : ^:: fr : vr, or .v : . \\ 

 'vlax+xx 



a^xy.'^^ax^xx . ^^^ ^^\{x^h. therefore is equal to a-\-x. Therefore this is the 



a 



nature of the curve kv, that if ab be called x, it will be fr = ^ — : ^x lax+xx 



a ' 



and vR=a-\-x. q. e. i. 



Carol. 1. — AC : CB :: bh : PR. For this is the property of the right line fr 

 found above. 



Corol. 1. — The right line cb is equal to bi or vk : for each of them is equal 

 to a-\-x, 



Corol. 3.— The evolving right line ve is a third proportional to the lines ac 



