358 PHILOSOPHICAL TRANSACTIONS. [aNNO I7I8. 



tion is sought, the equation of the curve bl will be s : i/ :: a' : r"; and the 



equation of the curve dp, s : y :: a""'"' : r""*"' ; and the angle vsp : angle ls/ :: 



^ : — or :: — : — or Ccallino; sp, x. and sl, r) ::-:-, that is, (because x = 

 «P SL SP SL ^ & * ' ' ' x r ' ^ 



nt^) :: !i±lr : -, or :: n + 1 : 1. Hence (fig. 2) bsp : bsl :: 77 + 1 : 1 ; 

 by which the curve bp may be more easily drawn without the tangents. If the 

 angle bsp be taken to bsl in the ratio of n + 1 to 1, and lp be perpendi- 

 cular to sp, the concourse of the perpendicular with sp, will be in the curve 

 BP, before described by means of the tangents. 



3. Having shown how, from one, an infinite series of curves may be de- 

 duced, I shall now proceed to show how the length of each may be known, 

 from the length of that and one other being given. Since the angle spp = 

 sl/, and ls/ : psp :: 1 : n -|- 1, it will be l/ : pjb :: sl : n + I sp, or (because 

 SL : SP :: l/ : loj as l/ : n + 1 ^0, and therefore vp = n "{- I lo; but lo = 

 /ra — on = /w —■ LN -f- Nw, therefore vp = n -\- 1 X /w— LN-fNn. But in •— 

 LN is the moment of ln perpendicular to sl, also Fp the moment of the curve 

 BP, and N72 the moment of the curve bn: and since bp, bn, bl vanish together 

 in B, they will be in the ratio of their moments; and therefore bp = n -f- l x 

 BN + or — ln. Hence the curve bp is to the sum or difference of the last 

 curve but one in the series, and of its tangent intercepted by the intermediate 

 curve, as w -f 1 to J ; or, putting m for the index of the equation of the 



curve bp (because m = -— ,) as 1 to 1 — m. 



^ n-\- 1' 



Hence, 1st, in the infinite series of curves described above, if there be given 

 the lengths of two that are next each other, the lengths of all will be given ; 

 for the measure of any one depends always on the measure of the last but one 

 in the series, and therefore one pair will suffice for measuring all. If one curve 

 be commensurable or incommensurable to right lines, then half the series will 

 be commensurable or incommensurable to right lines. Hence 2dly, though 

 the curves bp and bn should be incommensurable to right lines, yet the 

 difference of the curve bp from the n -\- I times of the curve bn, would be 

 equal to an assignable right line. 3dly, If the curve pass through s, the line 



LN vanishing in s, it will be bps = . 



1 — m 



4. Of all the curves we have treated of, viz. of which * : ^ :: a" : r", the 



circle is the chief, s being in the circumference, whose equation is * : ^r :: a : r, 



as appears from the similarity of the triangles lo/, bls (fig. 3) ; therefore n = I, 



