the Sum of the Semidiameters of the Bafe and Gene- 
rant, but in the Interior Epicycloid D p u, 'tis the diffe- 
rence of the faid Semidiameters. 
COROLLARY. 
In the Interior Epicycloid, if C E 'iS{C B, the Epi- . 
cycloid then degenerating into a right Line, the Quadra^ 
ture of the Triangle Ipu will be in efFeii the fame with 
the Quadrature of Hippocrates Chins. 
COROL, II. 
If the Semidiameter of the Bafe is fuppofed infinite, 
the Epicycloid then being the common Cycloid, the 
Area of the faid Triangle will be equal to the Square of 
the Radius of theGenerant, and fo it falls in with that 
Theorem which Lalovera found, and calls MirahiU. 
Though I do not think the abovefaid Quadrature can 
eafily be deduced from what has been yet publiflied of 
the Epicycloid, I have not added the Demonft ration ; 
but think it enough to name a general Propofition from 
whence I deduced it, viz. The Segments of the Gene- 
rant Circle are to the Correfpondent Segments of the 
Epicycloid, as C g to zCE-^CB, For Example, fup- 
pofe Fm6 the Pofition of part of the Generant when 
the point F of the Exterior Epicycloid was defi^ned , 
then the Segment FmGn is to the Segment D FnGi: 
as C B to 2C£ + CB. 
And confequently the whole Epicycloid to the whole 
Generant in the fame Proportion : Which is the only 
cafe demonftrated by Monfieur De la Hire. 
It follows alfo that in the Vulgar Cycloid, its Seg- 
ments are triple of the Correfpondent Seders of the Ge- 
nerant, which was fir ft fhewn'by Dr . Wallts, 
A Demonjlration hereof^ with a General Propofition for 
all Curves of this kind, fhaU he given in the next Tran- 
faction. V. An 
