VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 350 



and therefore m = — — - = 4^, and the equation of the curve bp will be 



j : y -.'. a^ : r^i which is the equation of the epicycloid described by the revolu- 

 tion of a circle on a base equal to itself, at the point where the describing point 

 touches the base; which Pascal calls Roberval's snail, and which Delahire con- 

 siders as a conchoid with a circular base, in the Memoirs of the Paris Academy 

 for the year 17O8. All the perpendiculars concur in the point b, and therefore 

 BN = O; whence bp = — — ^ = 2bl. Hence the whole curve bps = 2bs, 



1 — m ' 



or the length of the epicycloid is always double the chord of the corresponding 

 circular arc. ^dly. From the epicycloid let the curve Blls be described, in the 

 same manner as the epicycloid was described from the circle : then in this case 



n = -i-, and m = = ~-. = l, therefore the equation of the curve BITS 



BL + LP 



will he s '. y '.'. a-^ '. rs. The length of the curve will be = |-. bl -I- lp 



= -I-. BL -[- LG, and therefore bII is 4 the sum of the circular arc and its right 

 sine. Now if we take cd = bd, and with radius sd and centre s describe a 

 circle meeting sp in h, and draw hk perpendicular to bs ; then because dh 

 = 4 bl, it will be BIT = dh -{■ hk. Hence the arcs bII are neither com- 

 mensurable to right lines nor to circular arcs, yet the difference of the arcs 

 BIT and dh is the right line hk. The line lg vanishes in the point s, and 

 therefore bIIs = 4 bls; hence the whole curve is a of the semicircle. Yet 

 no part of this assignable curve can be commensurable to the whole, nor is the 

 entire curve divisible in any given ratio, so that the portions may have an 

 assignable ratio to each other, or to the whole. If this curve could be divided 

 geometrically in any given ratio, the quadrature of the circle would be com- 

 pleted. For instance, if it were BIT : bITs :: I : w, and bl : bls :: 1 : w, it 



would be BIT = 



Bns 3bls Swbl , ; , tm.G , nm 



— = •—— = —— = 4. BL + LG; hence bl = , and bls = lg. 



m 2m ^m n—m n—m 



3dly, by the method already explained, from biTs construct the curve br; then 

 because n = -^, it will be m = "— = 4^, and the equation of the curve br 



will he s : y '.'. a^ ; r^. Hence the length of the curve will be 4- x !2bl 4- pii 

 and the whole length of the curve brs = a of the diameter sb. If the con- 

 structions of these curves were continued, there would arise such a series of 

 equations as the following, which it is easy to continue at pleasure: 



