( 471 ) 
Let a c the Diameter of the greater Circle be equal 
to R, and a b the Diameter of the lefier Circle be e- 
qual to S, 
Let d h the Chord of the Arch d a k be equal to z , 
and fg the Chord of the Arch fag be equal to y, and 
let the Abfcifs a k be equal to x. 
If the Line m n be fuppofed to move till it becomes 
cc incident with the Tangent p a q , the nature of a Cir- 
cle will always give the following ^Equations. 
zz =7 4 Rx *— <\xx. • 
j j = 4 Sx — ^xx. 
When the Line is arrived at the Tangent, z and y 
will become the two Points of Contad, and then 
zz=^Rx and yy—afix, (4 xx being laid afide as Hetero- 
geneous to the reft of the Equation, by reafon of x be- 
ing become infinitely little ) Therefore 
zz .yy :: 4 Rx . 4 Sx : : R. S. 
Therefore z .y : : V R . •/ s. £>.E, D< 
PROP. II. Fig. 2. 
Ihe Point of Contact between a Sphere and a Plane is 
infinitely greater than that between a Circle and a Tan - 
gent. 
Let a be the Point of Contact between the Sphere 
a d qf and the Plane be. About the Sphere deferibe 
the Cylinder n p g m. 
Draw kb to reprefent a Circle parallel to the Plane. 
Let the Circle be fupposM to move, till it becomes co- 
incident with the Plane. The Cylindrical Surface kbgm 
will always be equal (according to Archimedes) to the 
Spherical Surface d a f. 
how 
