40 PHILOSOPHICAL TRANSACTIONS. [aNNO 16Q5. 



any before me has considered : so that to square any part of this is the same as 

 to square the respective part of the cycloid, or of the trilinear in the cycloid : 

 that which in the cycloid lies between two arcs of the generating circle in 

 different positions, answering to that which, in the restored figure, lies between 

 the respective straight lines. 



And therefore a dv a, =: t d S r, fig. 6, = a f/ d a = t f/ <J t, fig 8, = R-, And 

 Ab ko A,T b k ST,fig.6,= Ab kn a,t b k ST,i\g.8, = sr. And bkd, fig. 6, = bkd, 

 fig. 8, = R- — sR, ibidem, cap. 17, b, p. 756. Where, if b be taken above 

 d k D c, passing through the centre c, these figures are within the cycloid, and 

 within the restored figure, but without them, if b be taken below that line, and 

 adjacent to the curve a b t, ]n both cases. 



By R, I understand the radius of the generating circle ; and by s, the right 

 sine of the arc b a, whose versed sine is v a. And wherever in my whole 

 discourse on the cycloid, or the restored trilinear (which is a figure of arcs, 

 and a figure of versed sines) the arc a is no ingredient in the designation ; 

 such part or portion of them is capable of being geometrically squared. But 

 when I exclude a, I therein exclude ? (for that is an arc also) and J' = a -{- s, 

 and e = fl — s, because a is therein included. 



Mr. Caswell, not being aware that I had squared these figures, had done the 

 same by a method of his own, which he showed me lately, which I would have 

 inserted here, but that he thought it not necessary ; and instead thereof, has 

 given me the quadrature of a portion of the epicycloid, which you will receive 

 with this, and I think it is purely new. 



The Quadrature of a Portion of the Epicycloid. By Mr. Caswell.* N°217, 



p, 1J3. 



Suppose D p V, fig. 9, pi. 1, to be half of an exterior epicycloid, v b its axis, 



V the vertex, v lb half of the generating circle, e its centre, d b the base, c its 



centre ; bisect the arc of the semicircle v b in l, and on the centre c through 



L draw a circle cutting the epicycloid in p: then I say the curvilinear triangle 



VLP will be = BE^into -^ ; that is, the square of the semidiameter of the 



C B ' 



generating circle will be to the curvilinear triangle v l p, as c b the semidiameter 

 of the base to c e, which c k in the exterior epicycloid is the sum of the semi- 

 diameters of the base and generant, but in the interior epicycloid d/j h it is the 

 difterence of the said semidiameters. 



CoroL 1. In the interior epicycloid, if c e be -i^c b, the epicycloid then dege- 



* Of Wadham College, Oxford, and autlior of " A brief, but full. Account of the Doctrine of 

 Trigonometry, both Plane and Spherical ;" printed at the end of Dr. Wallis's Treatise on Algebra. 



