VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 47 



In the next place he shows, how it was possible for Bellini to find that the 

 quantity of perspirable matter, emitted every minute from a villous or ex- 

 cretory portion of the skin, weighing a scruple, is equal to the liOOth part of 

 a scruple. 



He rejects the use of ferments, both in accounting for the production of 

 diseases, and the business of secretion ; which, in his opinion, no person 

 can pretend to explain who is not thoroughly acquainted with the philosophy 

 of Newton. 



A general Proposition for measuring all Cycloids and Epicycloids, &c. By 

 E. Halley. N°218, p. 125. Translated from the Latin. 



The Prop. — The area of a cycloid or an epicycloid, whether it be primary, or 

 contracted, or prolate, is to the area of the generating circle ; and also the 

 areas of the generating parts in the same curves, to the areas of the analogous 

 segments of the circle ; as the sum of the double velocity of the centre and 

 the velocity of the circular motion, is to the velocity of the circular motion. 



Demonslr. — Let any epicycloid ypsqvb (fig. 10, pi. 1,) be described, by 

 the revolution of the circle v l b on the circular base y m n b. Let the centre 

 of the generating circle be c, and drawing c m k, let the circle stand on the 

 base at the point m, and let s be the delineating point. Now, separating the 

 motions, let the point s be transferred to r by the circular motion that the 

 arc M s may be increased by the indivisible particle r s ; then let the centre c 

 proceed forward to c. By this motion the segment r s m being transferred to 

 the situation a t n, the point a will meet the curve. It appears that the tri- 

 angle R s M is the moment or fluxion of the area of the circular segment : and 

 the trapezium q s m n is the fluxion of the area of the curvilinear space gene- 

 rated in the same time. Now since SM, rm, sm, are supposed to differ from 

 one another only by a point, conceive the areola a s m n to consist of three 

 sectors r m s, r m q, m q n ; then the areola r m s is to the areola a s m n, as the 

 angle r m s is to the sum of the three angles KMs-|-RMa-|-MaN. But the 

 angles r m a -{- m a n are equal to the angles m c n -f- m k n, or to the angle 

 CMC; because of the lines r m, a n being inclined to each other in an angle 

 equal to m K n, and because of the angle m a n being the half of m c n, by 

 Eulc. iii. "20. Therefore the angle k m s is to the angles r m s -)- c m c, that is, 

 by the same Prop, as the arc -^r s to the two arcs c c -|- 4^r s ; or r s la 2 c c -|- 

 R s, as the areola r s m to the areola a s m n, or as the moment of the circular 

 segment a t n, to the moment a s y m n of the epicycloidal segment generated 

 in the same time. And as these moments are always in the same ratio, where- 

 ever the point a is takenj it follows that the areas themselves a t n, a s y m n. 



