48 PHILOSOPHICAL TRANSACTIONS. " [aNNO \6Q5. 



generated by these moments, have the same constant ratio, viz, that of the 

 velocity of the circular motion r s, to double the velocity of the centre added 

 to the circular motion, or 2 c c + R s. Also as the area v b z to the area 

 Q V B N, so is therefore the area of the semicircle v lb to the curvilineal space 

 VQYNB. Hence the proposition appears. And there is no difference in the 

 manner of demonstrating when the generating semicircle moves on the arc 

 of a concave base, except that the angle c m c, in this case, is the difference of 

 the angles m c n, m k n. But if the base be a right line, then m k n vanishing, 

 and RM, QN being parallel, the proof becomes easier. Now in all these curves 

 there are quadrable portions, analogous to those portions, which in the 

 primary cycloid Dr. Wallis found to be perfectly quadrable, as easily follows 

 from the premises. 



With the centre k, through the point a describe the circular arc a z, and draw 

 z B cutting off the segment z l b equal to the segment a t n ; then bisect the 

 semicircle v b in l, and through the point l, from the same centre k, describe 

 the arc p l, catting the epicycloid in p, the generating circle in t, and the 

 chords Q N, z B, in y and x. Now put the arc v z = o, and its sine = s, the 

 generating radius =: r, and the radius of the base =: r ; and let the arc c e or 

 motion of the centre = m. It is plain, that the sector c k e has the same ratio 

 to the spacex YN b, as the square of k e, to the difference of the squares of 

 K L and KB; orasRR + 2R?-j- ?rto2Rr-f "^rr; that is, as r + r to 2 r, 

 or K E to Bv; and therefore the rectangle b e X c e, or r7n, is equal to the 

 space X Y N b. But the space v z b is equal to the rectangle -i-ar + Js/- ; and 

 therefore, according to our proposition, it will be as a to 2 m, so is ,;«r -}- 4rsr 



to """ ' , which is equal to the curvilinear space qvzlbnq : from this 



subduct the space xynb = rm, and there will remain the space qvzxy 

 And since the spaces zxl, qyt are equal to each other, the 

 space QVLTQ is also equal to — ^. Therefore, whenever a is to m, or 



the circular motion to the progressive motion of the centre, in a given ratio, 

 then there will also be given the perfect quadrature of the curvilinear spaces 

 a V L T a : also the whole space v r l, to the square of the radius b e, will be in 

 the same ratio of the motions 7n to a, that is, in every primary epicycloid, in 

 the ratio of the radii k e to k B ; which is Mr. Caswell's proposition. But the 

 less spaces a v l t a will be to one another, as the sines of the arcs v z ; and 

 the triangular spaces q t p, by the same way of reasoning, will be as the versed 

 sines of the arcs « t or z l ; and are therefore quadrable. In like manner it 

 may be proved that the spaces PAT, plu, /; a T are always to the square of the 



711 r s 

 a 



