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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="040/01/812.jpg" pagenum="120"/>
              D A hath to A Z; which is the ſame that the Rectangle K E hath to
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              the Rectangle A G, their heights A K and K L being equal. </s>
              <s>There­
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              fore the proportion that the Square E D hath to the Square Z G;
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              that is, the Square L A hath to the Square A K, the Rectangle K E
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              hath likewiſe to the Rectangle K Z. </s>
              <s>And in the ſelf-ſame manner
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              we might prove that the other Rectangles L F, M H, N I, O B are
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              to one another as the Squares of the Lines M A, N A, O A, P A.
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              </s>
              <s>Conſider we in the next place, how the Circumſcribed Figure is
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              compounded of certain Spaces that are to one another as the
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              Squares of the Lines that exceed with Exceſſes equal to the leaſt,
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              and how the Rectangle C P is compounded of ſo many other Spa­
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              ces each of them equal to the Greateſt, which are all the Rectan­
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              gles equal to O B. Therefore, by the Lemma of
                <emph type="italics"/>
              Archimedes,
                <emph.end type="italics"/>
              the
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              Circumſcribed Figure is more than the third part of the Rectangle
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              C P: But it was alſo leſſe, which is impoſſible: Therefore the
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              Mixt-Triangle is not leſſe than one third of the Rectangle C P.
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              </s>
              <s>I ſay likewiſe, that it is not more: For if it be more than one
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              third of the Rectangle C P, ſuppoſe the Space X equal to the ex­
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              ceſſe of the Triangle above the third part of the ſaid Rectangle
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              C P, and the diviſion and ſubdiviſion of the Rectangle into Rect­
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              angolets, but alwaies equal, being made, we ſhall meet with ſuch as
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              that one of them is leſſer than the Space X; which let be done:
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              and let the Rectangle
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              O be leſſer than X; and, having deſcribed
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              the Figure as before, we ſhall have inſcribed in the Mixt-Triangle
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              a Figure compounded of the Rectangles V O, T N, S M, N L, Q K,
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              which yet ſhall not be leſs
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                <figure id="id.040.01.812.1.jpg" xlink:href="040/01/812/1.jpg" number="72"/>
                <lb/>
              than the third part of the
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              great Rectangle C P, for
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              the Mixt Triangle doth
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              much leſſe exceed the In­
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              ſcribed Figure than it doth
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              exceed the third part of
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              the Rectangle C P; Be­
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              cauſe the exceſſe of the
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              Triangle above the third part of the Rectangle C P is equal to
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              the Space X which is greater than the Rectangle
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              O, and this al­
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              ſo is conſiderably greater than the exceſſe of the Triangle above
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              the Inſcribed Figure: For to the Rectangle
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              O, all the Rectan­
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              golets A G, G E, E
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              F,
                <emph.end type="italics"/>
              F H, H I, I
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              are equal, of which the Ex­
                <lb/>
              ceſſes of the Triangle above the Inſcribed
                <emph type="italics"/>
              F
                <emph.end type="italics"/>
              igure are leſſe than
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              half: And therefore the Triangle exceeding the third part of the
                <lb/>
              Rectangle C P, by much more (exceeding it by the Space X)
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              than it exceedeth its inſcribed
                <emph type="italics"/>
              F
                <emph.end type="italics"/>
              igure, that ſame
                <emph type="italics"/>
              F
                <emph.end type="italics"/>
              igure ſhall alſo
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              be greater than the third part of the Rectangle C P:
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              ut it is leſſer,
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              by the Lemma preſuppoſed:
                <emph type="italics"/>
              F
                <emph.end type="italics"/>
              or that the Rectangle C P, as being </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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