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

Table of figures

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              <s>PROPOSITION.</s>
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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. </s>
              <s>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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              <s>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. </s>
              <s>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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                <figure id="id.040.01.740.1.jpg" xlink:href="040/01/740/1.jpg" number="63"/>
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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. </s>
              <s>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 </s>
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