Salusbury, Thomas
,
Mathematical collections and translations (Tome I)
,
1667
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>PROPOSITION.</
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The Circle is a Mean-Proportional betwixt any two
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Regular Homogeneal Poligons, one of which is cir
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cumſcribed about it, and the other Iſoperimetral to
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it: Moreover, it being leſſe than all the circumſcri
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bed, it is, on the contrary, bigger than all the Iſoperi
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metral. </
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>And, again of the circumſcribed, thoſe that
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have more angles are leſſer than thoſe that have
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fewer; and on the other ſide of the Iſoperimetral,
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thoſe of more angles are bigger.
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>Of the two like Poligons A and B, let A be circumſcribed
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about the Circle A, and let the other B, be Iſoperime
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tral to the ſaid Circle: I ſay, that the Circle is the Mean
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proportional betwixt them. </
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>For that (having drawn the Semidi
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ameter A C) the Circle being equal to that Right-angled Trian
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gle, of whoſe Sides including the Right angle, the one is equal
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to the Semidiameter A C, and the other to the Circumference:
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And likewiſe the Poligon A being equal to the right angled Tri
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angle, that about the right angle hath one of its Sides equal to
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the ſaid right line A C, and the other to the Perimeter of the ſaid
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Poligon: It is manifeſt, that the circumſcribed Poligon hath the
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ſame proportion to the Circle, that its Perimeter hath to the Cir
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cumference of the ſaid Circle; that is, to the Perimeter of the
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Poligon B, which is ſuppoſed equal to the ſaid Circumference:
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But the Poligon A hath a proportion to the Poligon B, double to
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that of its Perimeter, to the Perimeter of B (they being like Fi
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gures:) Therefore the Circle A is the Mean-proportional be
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tween the two Poligons A and B. </
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>And the Poligon A being
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bigger than the Circle A, it is manifeſt that the ſaid Circle
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A is bigger than the Poligon B, its Iſoperimetral, and conſe
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quently the greateſt of all Regular Poligons that are Iſoperimetral </
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