1M ſhall be the Center of Gravity of the remaining proportions by which
the Cone exceeds the inſcribed Figure. Which is impoſſible. Therefore
the Center of Gravity of the Cone is not below the point C. Nor is it
above it. For if it may be, let it be R. And again aſſume L P cut at
pleaſure in N: And as both B C and N P together are to N L, ſo let the
Cone be to X. And let a Figure be, in like manner, circumſcribed about
the Cone, which exceeds the ſaid Cone a leſs quantity than the Solid X.
And let the Line which intercepts bet wixt its Center of Gravity and C,
be leſſer than N P. Now take the circumſcribed Figure, whoſe Center
let be O; the remainder O R ſhall be greater than the ſaid N L. And
becauſe, as both together B C and P N is to N L, ſo is the Cone to X:
And the exceſs by which the circumſcribed exceeds the Cone is leſſer
than X: And B O is leſſer than B C and P N together: And O R grea
ter than L N: The Cone therefore ſhall have much greater proportion to
the remaining proportions by which it was exceeded by the circumſcribed
Figure, than B O to O R. Let it be as M O is to O R. M O ſhall
be greater than B C; and M ſhall be the Center of Gravity of the pro
portions by which the Cone is exceeded by the circumſcribed Figure.
Which is inconvenient. Therefore the Center of Gravity of the Cone is
not above the point C. But neither is it below it; as hath been proved.
Therefore it ſhall be C it ſelf. And ſo in like manner may it be demon
ſtrated in any Pyramid.
the Cone exceeds the inſcribed Figure. Which is impoſſible. Therefore
the Center of Gravity of the Cone is not below the point C. Nor is it
above it. For if it may be, let it be R. And again aſſume L P cut at
pleaſure in N: And as both B C and N P together are to N L, ſo let the
Cone be to X. And let a Figure be, in like manner, circumſcribed about
the Cone, which exceeds the ſaid Cone a leſs quantity than the Solid X.
And let the Line which intercepts bet wixt its Center of Gravity and C,
be leſſer than N P. Now take the circumſcribed Figure, whoſe Center
let be O; the remainder O R ſhall be greater than the ſaid N L. And
becauſe, as both together B C and P N is to N L, ſo is the Cone to X:
And the exceſs by which the circumſcribed exceeds the Cone is leſſer
than X: And B O is leſſer than B C and P N together: And O R grea
ter than L N: The Cone therefore ſhall have much greater proportion to
the remaining proportions by which it was exceeded by the circumſcribed
Figure, than B O to O R. Let it be as M O is to O R. M O ſhall
be greater than B C; and M ſhall be the Center of Gravity of the pro
portions by which the Cone is exceeded by the circumſcribed Figure.
Which is inconvenient. Therefore the Center of Gravity of the Cone is
not above the point C. But neither is it below it; as hath been proved.
Therefore it ſhall be C it ſelf. And ſo in like manner may it be demon
ſtrated in any Pyramid.
PROPOSITION.
If there were four Lines continual proportionals;
and as the leaſt of them were to the exceſs by
which the greateſt exceeds the leaſt, ſo a Line
taken at pleaſure ſhould be to 3/4 the exceſs by
which the greateſt exceeds the ſecond; and as
the Line equal to theſe (viz. to the greateſt,
double of the ſecond, and triple of the third)
is to the Line equal to the quadruple of the
fourth, the quadruple of the ſecond, and the
quadruple of the third, ſo ſhould another Line
taken be to the exceſs of the greateſt above the
ſecond: theſe two Lines taken together ſhall
be a fourth part of the greateſt of the propor
tionals.
and as the leaſt of them were to the exceſs by
which the greateſt exceeds the leaſt, ſo a Line
taken at pleaſure ſhould be to 3/4 the exceſs by
which the greateſt exceeds the ſecond; and as
the Line equal to theſe (viz. to the greateſt,
double of the ſecond, and triple of the third)
is to the Line equal to the quadruple of the
fourth, the quadruple of the ſecond, and the
quadruple of the third, ſo ſhould another Line
taken be to the exceſs of the greateſt above the
ſecond: theſe two Lines taken together ſhall
be a fourth part of the greateſt of the propor
tionals.
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