PROPOSITION.

If to any Cone or portion of a Cone a Eigure con

ſiſting of Cylinders of equal heights be inſcri

bed and another circumſcribed; and if its Axis

be ſo divided as that the part which lyeth be

twixt the point of diviſion and the Vertex be

triple to the reſt; the Center of Gravity of

the inſcribed Figure ſhall be nearer to the Baſe

of the Cone than that point of diviſion: and

the Center of Gravity of the circumſcribed

ſhall be nearer to the Vertex than that ſame

point.

ſiſting of Cylinders of equal heights be inſcri

bed and another circumſcribed; and if its Axis

be ſo divided as that the part which lyeth be

twixt the point of diviſion and the Vertex be

triple to the reſt; the Center of Gravity of

the inſcribed Figure ſhall be nearer to the Baſe

of the Cone than that point of diviſion: and

the Center of Gravity of the circumſcribed

ſhall be nearer to the Vertex than that ſame

point.

Take therefore a Cone, whoſe Axis is N M. Let it be divided

in S ſo, as that N S be triple to the remainder S M. I ſay, that

the Center of Gravity of any Figure inſcribed, as was ſaid, in

a Cone doth conſiſt in the Axis N M, and approacheth nearer to the Baſe

of the Cone than the point S: and that the Center of Gravity of the

Circumſcribed is likewiſe in the Axis N M, and nearer to the Vertex

than is S. Let a Figure therefore be ſuppoſed to be inſcribed by the Cy

linders whoſe Axis M C, C B, B E, E A are equal. Firſt therefore

the Cylinder whoſe Axis is M C hath

175[Figure 175]

to the Cylinder whoſe Axis is C B the

ſame proportion as its Baſe hath to

the Baſe of the other (for their Alti

tudes are equal.) But this propor

tion is the ſame with that which the

Square C N hath to the Square N B.

And ſo we might prove, that the Cy

linder whoſe Axis is C B hath to the

Cylinder whoſe Axis is B E the ſame

proportion, as the Square B N hath to

the Square N E: and the Cylinder

whoſe Axis is B E hath to the Cylin

der whoſe Axis is E A the ſame pro

portion that the Square E N hath to

the Square N A. But the Lines N C,

N B, E N, and N A equally exceed one

another, and their exceſs equalleth the

leaſt, that is N A. Therefore they are certain Magnitudes, to wit, in

ſcribed Cylinders having conſequently to one another the ſame proporti

on as the Squares of Lines that equally exceed one another, and the ex-

in S ſo, as that N S be triple to the remainder S M. I ſay, that

the Center of Gravity of any Figure inſcribed, as was ſaid, in

a Cone doth conſiſt in the Axis N M, and approacheth nearer to the Baſe

of the Cone than the point S: and that the Center of Gravity of the

Circumſcribed is likewiſe in the Axis N M, and nearer to the Vertex

than is S. Let a Figure therefore be ſuppoſed to be inſcribed by the Cy

linders whoſe Axis M C, C B, B E, E A are equal. Firſt therefore

the Cylinder whoſe Axis is M C hath

175[Figure 175]

to the Cylinder whoſe Axis is C B the

ſame proportion as its Baſe hath to

the Baſe of the other (for their Alti

tudes are equal.) But this propor

tion is the ſame with that which the

Square C N hath to the Square N B.

And ſo we might prove, that the Cy

linder whoſe Axis is C B hath to the

Cylinder whoſe Axis is B E the ſame

proportion, as the Square B N hath to

the Square N E: and the Cylinder

whoſe Axis is B E hath to the Cylin

der whoſe Axis is E A the ſame pro

portion that the Square E N hath to

the Square N A. But the Lines N C,

N B, E N, and N A equally exceed one

another, and their exceſs equalleth the

leaſt, that is N A. Therefore they are certain Magnitudes, to wit, in

ſcribed Cylinders having conſequently to one another the ſame proporti

on as the Squares of Lines that equally exceed one another, and the ex-