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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/891.jpg" pagenum="198"/>
            <p type="head">
              <s>LEMMAI.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let D C be Perpendicular to the Diameter B A; and from the Term
                <lb/>
              B continue forth B E D at pleaſure, and draw a Line from F to B. </s>
              <s>I
                <lb/>
              ſay, that F B is a Mean-proportional be-
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.891.1.jpg" xlink:href="040/01/891/1.jpg" number="136"/>
                <lb/>
                <emph type="italics"/>
              twixt D B and B E. </s>
              <s>Draw a Line from E
                <lb/>
              to F, and by B draw the Tangent B G;
                <lb/>
              which ſhall be Parallel to the former C D:
                <lb/>
              Wherefore the Angle D B G ſhall be equal
                <lb/>
              to the Angle F D B, like as the ſame G B D
                <lb/>
              is equal alſo to the Angle E F B in the al­
                <lb/>
              tern Portion or Segment: Therefore the
                <lb/>
              Triangles F B D and F E B are alike: And,
                <lb/>
              as B D is to B F, ſo is F B to B E.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>LEMMA II.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let the Line A C be greater than D F; and let A B have greater
                <lb/>
              proportion to B C, than D E hath to E F. </s>
              <s>I ſay, that A B is greater
                <lb/>
              than D E. </s>
              <s>For becauſe A B hath to B C
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.891.2.jpg" xlink:href="040/01/891/2.jpg" number="137"/>
                <lb/>
                <emph type="italics"/>
              greater proportion than D E hath to D F,
                <lb/>
              therefore look what proportion A B hath to
                <lb/>
              B C, the ſame ſhall D E have to a Line leſ­
                <lb/>
              ſer than E F; let it have it to E G: And
                <lb/>
              becauſe A B to B C, is as D E, to E G, there­
                <lb/>
              fore, by Compoſition, and by converting the Proportion, as C A is to A B,
                <lb/>
              ſo is G D to D E: But C A is greater than G D: Therefore B A ſhall
                <lb/>
              be greater than D E.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>LEMMA III.</s>
            </p>
            <figure id="id.040.01.891.3.jpg" xlink:href="040/01/891/3.jpg" number="138"/>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let A C I B be the Quadrant of a Circle:
                <lb/>
              and to A C let B E be drawn from B Pa­
                <lb/>
              rallel: And out of any Center taken in the
                <lb/>
              ſame deſcribe the Circle B O E S, touching
                <lb/>
              A B in B, and cutting the Circumference of
                <lb/>
              the Quadrant in I; and draw a Line from
                <lb/>
              C to B, and another from C to I continued
                <lb/>
              out to S. </s>
              <s>I ſay, that the Line C I is alwaies
                <lb/>
              leſſe than C O. </s>
              <s>Draw a Line from A to I;
                <lb/>
              which toucheth the Circle B O E. </s>
              <s>And if
                <lb/>
              D I be drawn it ſhall be equal to D B: And
                <lb/>
              becauſé D B toucheth the Quadrant, the ſaid
                <lb/>
              D I ſhall likewiſe touch it; and ſhall be Per-
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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