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-
zoom in
zoom out
zoom area
full page
page width
set mark
remove mark
get reference
digilib