PROPOSITION.
The Circle is a Mean-Proportional betwixt any two
Regular Homogeneal Poligons, one of which is cir
cumſcribed about it, and the other Iſoperimetral to
it: Moreover, it being leſſe than all the circumſcri
bed, it is, on the contrary, bigger than all the Iſoperi
metral. And, again of the circumſcribed, thoſe that
have more angles are leſſer than thoſe that have
fewer; and on the other ſide of the Iſoperimetral,
thoſe of more angles are bigger.
Regular Homogeneal Poligons, one of which is cir
cumſcribed about it, and the other Iſoperimetral to
it: Moreover, it being leſſe than all the circumſcri
bed, it is, on the contrary, bigger than all the Iſoperi
metral. And, again of the circumſcribed, thoſe that
have more angles are leſſer than thoſe that have
fewer; and on the other ſide of the Iſoperimetral,
thoſe of more angles are bigger.
Of the two like Poligons A and B, let A be circumſcribed
about the Circle A, and let the other B, be Iſoperime
tral to the ſaid Circle: I ſay, that the Circle is the Mean
proportional betwixt them. For that (having drawn the Semidi
ameter A C) the Circle being equal to that Right-angled Trian
gle, of whoſe Sides including the Right angle, the one is equal
63[Figure 63]
to the Semidiameter A C, and the other to the Circumference:
And likewiſe the Poligon A being equal to the right angled Tri
angle, that about the right angle hath one of its Sides equal to
the ſaid right line A C, and the other to the Perimeter of the ſaid
Poligon: It is manifeſt, that the circumſcribed Poligon hath the
ſame proportion to the Circle, that its Perimeter hath to the Cir
cumference of the ſaid Circle; that is, to the Perimeter of the
Poligon B, which is ſuppoſed equal to the ſaid Circumference:
But the Poligon A hath a proportion to the Poligon B, double to
that of its Perimeter, to the Perimeter of B (they being like Fi
gures:) Therefore the Circle A is the Mean-proportional be
tween the two Poligons A and B. And the Poligon A being
bigger than the Circle A, it is manifeſt that the ſaid Circle
A is bigger than the Poligon B, its Iſoperimetral, and conſe
quently the greateſt of all Regular Poligons that are Iſoperimetral
about the Circle A, and let the other B, be Iſoperime
tral to the ſaid Circle: I ſay, that the Circle is the Mean
proportional betwixt them. For that (having drawn the Semidi
ameter A C) the Circle being equal to that Right-angled Trian
gle, of whoſe Sides including the Right angle, the one is equal
63[Figure 63]to the Semidiameter A C, and the other to the Circumference:
And likewiſe the Poligon A being equal to the right angled Tri
angle, that about the right angle hath one of its Sides equal to
the ſaid right line A C, and the other to the Perimeter of the ſaid
Poligon: It is manifeſt, that the circumſcribed Poligon hath the
ſame proportion to the Circle, that its Perimeter hath to the Cir
cumference of the ſaid Circle; that is, to the Perimeter of the
Poligon B, which is ſuppoſed equal to the ſaid Circumference:
But the Poligon A hath a proportion to the Poligon B, double to
that of its Perimeter, to the Perimeter of B (they being like Fi
gures:) Therefore the Circle A is the Mean-proportional be
tween the two Poligons A and B. And the Poligon A being
bigger than the Circle A, it is manifeſt that the ſaid Circle
A is bigger than the Poligon B, its Iſoperimetral, and conſe
quently the greateſt of all Regular Poligons that are Iſoperimetral