Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667

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1
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.
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

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-