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="040/01/1075.jpg" pagenum="381"/>
                <emph type="italics"/>
              that A Z hath to Z D; by the fourth Propoſition of
                <emph.end type="italics"/>
              Archimedes, De quadratura Para­
                <lb/>
              bolæ:
                <emph type="italics"/>
              But A Z is Seſquialter of Z D; for it is as three to two, as we ſhallanon demon-
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1326"/>
                <lb/>
                <emph type="italics"/>
              ſtrate: Therefore D B is Seſquialter of B V; but D B and B K are Seſquialter:
                <lb/>
              And, therefore, the Lines
                <emph.end type="italics"/>
              (c)
                <emph type="italics"/>
              B V and B K are equall: Which is imposſible:
                <lb/>
              Therefore the Section of the Right-angled Cone A E I, ſhall paſs thorow the Point K; which
                <lb/>
              we would demonstrate.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1326"/>
              (c)
                <emph type="italics"/>
              By 9 of the
                <lb/>
              fifth,
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>In regard, therefore, that the three
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortions A P O L, A E I
                <lb/>
                <arrow.to.target n="marg1327"/>
                <lb/>
              and A T D are contained betwixt Right Lines and the Sections
                <lb/>
              of Right-angled Cones, and are Right, alike and unequall,
                <lb/>
              touching one another, upon one and the ſame Baſe.]
                <emph type="italics"/>
              After theſe words,
                <emph.end type="italics"/>
                <lb/>
              upon one and the ſame Baſe,
                <emph type="italics"/>
              we may ſee that ſomething is obliterated, that is to be
                <lb/>
              deſired: and for the Demonſtration of theſe particulars, it is requiſite in this place to
                <lb/>
              premiſe ſome things: which will alſo be neceſſary unto the things that follow.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1327"/>
              M</s>
            </p>
            <p type="head">
              <s>LEMMA. I.</s>
            </p>
            <p type="main">
              <s>Let there be a Right
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine A B; and let it be cut by two
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ines,
                <lb/>
              parallel to one another, A C and D E, ſo, that as
                <emph type="italics"/>
              A B
                <emph.end type="italics"/>
              is to
                <lb/>
              B D. ſo
                <emph type="italics"/>
              A C
                <emph.end type="italics"/>
              may be to D E. </s>
              <s>I ſay that the Line that con­
                <lb/>
              joyneth the Points C and B ſhall likewiſe paſs by E.</s>
            </p>
            <figure id="id.040.01.1075.1.jpg" xlink:href="040/01/1075/1.jpg" number="274"/>
            <p type="main">
              <s>
                <emph type="italics"/>
              For, if poſſible, let it not paſs by E, but either
                <lb/>
              above or below it. </s>
              <s>Let it first paſs below it,
                <lb/>
              as by F. </s>
              <s>The Triangles A B C and D B F ſhall
                <lb/>
              be alike: And, therefore, as
                <emph.end type="italics"/>
              (a)
                <emph type="italics"/>
              A B is to B D,
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1328"/>
                <lb/>
                <emph type="italics"/>
              ſo is A C to D F: But as A B is to B D, ſo was
                <lb/>
              A C to D E: Therefore
                <emph.end type="italics"/>
              (b)
                <emph type="italics"/>
              D F ſhall be equall to
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1329"/>
                <lb/>
                <emph type="italics"/>
              D E: that is, the part to the whole: Which is
                <lb/>
              abſurd. </s>
              <s>The ſame abſurditie will follow, if the
                <lb/>
              Line C B be ſuppoſed to paſs above the Point E:
                <lb/>
              And, therefore, C B muſt of necesſity paſs thorow
                <lb/>
              E: Which was required to be demonſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1328"/>
              (a)
                <emph type="italics"/>
              By 4. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1329"/>
              (b)
                <emph type="italics"/>
              By 9. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>LEMMA. II.</s>
            </p>
            <p type="main">
              <s>Let there be two like
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortions, contained betwixt Right Lines,
                <lb/>
              and the Sections of Right-angled Cones; A B C the great­
                <lb/>
              er, whoſe Diameter let be B D; and E F C the leſser, whoſe
                <lb/>
              Diameter let be F G: and, let them be ſo applyed to one
                <lb/>
              another, that the greater include the leſser; and let their
                <lb/>
              Baſes A C and E C be in the ſame Right Line, that the ſame
                <lb/>
              Point C, may be the term or bound of them both: And,
                <lb/>
              then in the Section A B C, take any Point, as H; and draw
                <lb/>
              a Line from H to C. </s>
              <s>I ſay, that the Line H C, hath to that
                <lb/>
              part of it ſelf, that lyeth betwixt C and the Section E F C, the
                <lb/>
              ſame proportion that A C hath to C E.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Draw B C, which ſhall paſs thorow F, For, in regard, that the Portions are alike, the
                <lb/>
              Diameters with the Baſes contain equall Angles: And, therefore, B D and F G are parallel
                <lb/>
              to one another: and B D is to A C, as F G it to E C: and,
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              B D is to F G, as
                <lb/>
              A C is to C E; that is,
                <emph.end type="italics"/>
              (a)
                <emph type="italics"/>
              as their halfes D C to C G; therefore, it followeth, by the
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1330"/>
                <lb/>
                <emph type="italics"/>
              preceding Lemma, that the Line B C ſhall paſs by the Point F. Moreover, from the Point
                <lb/>
              H unto the Diameter B D, draw the Line H K, parallel to the Baſe A C: and, draw a Line
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
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      </text>
    </archimedes>
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