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="040/01/1062.jpg" pagenum="368"/>
                <arrow.to.target n="marg1235"/>
                <lb/>
                <emph type="italics"/>
              that is, H G to N C: and as
                <emph.end type="italics"/>
              (d)
                <emph type="italics"/>
              O H is to H P, ſo is G B to C K; For O H is double
                <lb/>
              to G B, and H P alſo double to G F; that is, to C K; Therefore H G hath the ſame propor­
                <lb/>
              tion to N C, that G B hath to C K: And
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              N C hath to C K the ſame proportion
                <lb/>
              that H G hath to G B.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1232"/>
              (a)
                <emph type="italics"/>
              By
                <emph.end type="italics"/>
              2. Lemma.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1233"/>
              (b)
                <emph type="italics"/>
              By
                <emph.end type="italics"/>
              4. Lemma.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1234"/>
              (b)
                <emph type="italics"/>
              By 19. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1235"/>
              (d)
                <emph type="italics"/>
              By 15. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Then take ſome other Point at pleaſure in the Section, which
                <lb/>
              let be S: and thorow S draw two Lines, the one S T paral­
                <lb/>
              lel to D B, and cutting the Diameter in the Point T; the
                <lb/>
              other S V parallel to A C, and cutting C E in V. </s>
              <s>I ſay
                <lb/>
              that V C hath greater proportion to C K, than T G hath
                <lb/>
              to G B.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For prolong V S unto the Line Q M in X; and from the Point X draw X Y unto the
                <lb/>
              Diameter parallel to B D: G T ſhall be leſſe than G Y, in regard that V S is leße than V X:
                <lb/>
              And, by the firſt Lemma, Y G ſhall be to V C, as H G to N C; that is, as G B to C K, which
                <lb/>
              was demonſtrated but now: And,
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              Y G ſhall be to G B, as V C to C K: But
                <lb/>
              T G, for that it is leſſe than Y G, hath leſſe proportion to G B, than Y G hath to the ſame;
                <lb/>
              Therefore V C hath greater proportion to C K. than T G hath to G B: Which was to be de­
                <lb/>
              monſtrated. </s>
              <s>Therefore a Poſition given G K, there ſhall be in the Section one only Point, to
                <lb/>
              wit M, from which two Lines M E H and M N O being drawn, N C ſhall have the ſame pro­
                <lb/>
              portion to C K, that H G hath to G B; For if they be drawn from any other, that which fall­
                <lb/>
              eth betwixt A C, and the Line parallel unto it ſhall alwayes have greater proportion to C K,
                <lb/>
              than that which falleth betwixt G K and the Line parallel unto it hath to G B. That, there­
                <lb/>
              fore, is manifeſt which was affirmed by
                <emph.end type="italics"/>
              Archimedes,
                <emph type="italics"/>
              to wit, that the Line P I hath unto P H,
                <lb/>
              either the ſame proportion that N
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
                <emph type="italics"/>
              hath to
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
                <emph type="italics"/>
              O, or greater.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1236"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1236"/>
              D</s>
            </p>
            <p type="main">
              <s>Wherefore P H is to H I either double, or leſſe than double.]
                <lb/>
                <emph type="italics"/>
              If leſſe than double, let P T be double to T I: The Centre of Gravity of that part of the
                <lb/>
              Portion that is within the Liquid ſhall be the
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.1062.1.jpg" xlink:href="040/01/1062/1.jpg" number="262"/>
                <lb/>
                <emph type="italics"/>
              Point T: But if P H be double to H I, H ſhall
                <lb/>
              be the Centre of Gravity; And draw H F, and
                <lb/>
              prolong it unto the Centre of that part of the Por­
                <lb/>
              tion which is above the Liquid, namely, unto G,
                <lb/>
              and the reſt is demonſtrated as before. </s>
              <s>And the
                <lb/>
              ſame is to be underſtood in the Propoſition that
                <lb/>
              followeth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>The Solid A P O L, therefore,
                <lb/>
              ſhall turn about, and its Baſe ſhall
                <lb/>
              not in the leaſt touch the Surface
                <lb/>
              of the Liquid.]
                <emph type="italics"/>
              In
                <emph.end type="italics"/>
              Tartaglia's
                <emph type="italics"/>
              Tranſlation it is rendered
                <emph.end type="italics"/>
              ut Baſis ipſius non tangent
                <lb/>
              ſuperficiem humidi ſecundum unum ſignum;
                <emph type="italics"/>
              but we have choſen to read
                <emph.end type="italics"/>
              ut Baſis ipſius
                <lb/>
              nullo modo humidi ſuperficiem contingent,
                <emph type="italics"/>
              both here, and in the following Propoſitions,
                <lb/>
              becauſe the Greekes frequently uſe
                <emph.end type="italics"/>
                <foreign lang="grc">ὡδὲεἶς, ὡδὲ
                  <gap/>
                </foreign>
                <emph type="italics"/>
              pro
                <emph.end type="italics"/>
                <foreign lang="grc">ὠδεὶσ
                  <gap/>
                & οὐδὶν</foreign>
              :
                <emph type="italics"/>
              ſo that
                <emph.end type="italics"/>
                <foreign lang="grc">οὐδἔσινουδείς,</foreign>
              nullus
                <lb/>
              eſt;
                <foreign lang="grc">οὐδ
                  <gap/>
                ὑπ̓ἑρὸς</foreign>
              à nullo,
                <emph type="italics"/>
              and ſo of others of the like nature.
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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