Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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    <archimedes>
      <text>
        <body>
          <chap>
            <subchap1>
              <subchap2>
                <p type="main">
                  <s>
                    <pb xlink:href="039/01/097.jpg" pagenum="69"/>
                  vertetur in geminas Rectas, quarum una eſt recta illa in quam pun­
                    <lb/>
                    <arrow.to.target n="note45"/>
                  ctum
                    <emph type="italics"/>
                  p
                    <emph.end type="italics"/>
                  incidit, & altera eſt recta qua alia duo ex punctis quatuor jun­
                    <lb/>
                  guntur. </s>
                  <s>Si Trapezii anguli duo oppoſiti ſimul ſumpti æquentur
                    <lb/>
                  duobus rectis, & lineæ quatuor
                    <emph type="italics"/>
                  PQ, PR, PS, PT
                    <emph.end type="italics"/>
                  ducantur ad
                    <lb/>
                  latera ejus vel perpendiculariter vel in angulis quibuſvis æqualibus,
                    <lb/>
                  ſitque rectangulum ſub duabus ductis
                    <emph type="italics"/>
                  PQXPR
                    <emph.end type="italics"/>
                  æquale rectangu­
                    <lb/>
                  lo ſub duabus aliis
                    <emph type="italics"/>
                  PSXPT,
                    <emph.end type="italics"/>
                  Sectio conica evadet Circulus. </s>
                  <s>Idem
                    <lb/>
                  fiet ſi lineæ quatuor ducantur in angulis quibuſvis & rectangulum
                    <lb/>
                  ſub duabus ductis
                    <emph type="italics"/>
                  PQXPR
                    <emph.end type="italics"/>
                  ſit ad rectangulum ſub aliis duabus
                    <lb/>
                    <emph type="italics"/>
                  PSXPT
                    <emph.end type="italics"/>
                  ut rectangulum ſub ſinubus angulorum
                    <emph type="italics"/>
                  S, T,
                    <emph.end type="italics"/>
                  in quibus
                    <lb/>
                  duæ ultimæ
                    <emph type="italics"/>
                  PS, PT
                    <emph.end type="italics"/>
                  ducuntur, ad rectangulum ſub ſinubus angu­
                    <lb/>
                  lorum
                    <emph type="italics"/>
                  Q, R,
                    <emph.end type="italics"/>
                  in quibus duæ primæ
                    <emph type="italics"/>
                  PQ, PR
                    <emph.end type="italics"/>
                  ducuntur. </s>
                  <s>Cæteris
                    <lb/>
                  in caſibus Locus puncti
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  erit aliqua trium figurarum quæ vulgo
                    <lb/>
                  nominantur Sectiones Conicæ. </s>
                  <s>Vice autem Trapezii
                    <emph type="italics"/>
                  ABCD
                    <emph.end type="italics"/>
                  ſub­
                    <lb/>
                  ſtitui poteſt Quadrilaterum cujus latera duo oppoſita ſe mutuo in­
                    <lb/>
                  ſtar diagonalium decuſſant. </s>
                  <s>Sed & e punctis quatuor
                    <emph type="italics"/>
                  A, B, C, D
                    <emph.end type="italics"/>
                    <lb/>
                  poſſunt unum vel duo abire ad infinitum, eoque pacto latera fi­
                    <lb/>
                  guræ quæ ad puncta illa convergunt, evadere parallela: quo in
                    <lb/>
                  caſu Sectio Conica tranſibit per cætera puncta, & in plagas paralle­
                    <lb/>
                  larum abibit in infinitum. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note45"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA XIX.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Invenire
                    <expan abbr="punctũ">punctum</expan>
                    <emph.end type="italics"/>
                  P,
                    <emph type="italics"/>
                  a quo ſi rectæ
                    <emph.end type="italics"/>
                    <lb/>
                    <figure id="id.039.01.097.1.jpg" xlink:href="039/01/097/1.jpg" number="42"/>
                    <lb/>
                    <emph type="italics"/>
                  quatuor
                    <emph.end type="italics"/>
                  PQ, PR, PS, PT,
                    <lb/>
                    <emph type="italics"/>
                  ad alias totidem poſitione da
                    <lb/>
                  tas rectas
                    <emph.end type="italics"/>
                  AB, CD, AC, BD,
                    <lb/>
                    <emph type="italics"/>
                  ſingulæ ad ſingulas in datis
                    <lb/>
                  angulis ducantur,
                    <expan abbr="rectangulũ">rectangulum</expan>
                    <lb/>
                  ſub duabus ductis,
                    <emph.end type="italics"/>
                  PQXPR,
                    <lb/>
                    <emph type="italics"/>
                  ſit ad rectangulum ſub aliis
                    <lb/>
                  duabus,
                    <emph.end type="italics"/>
                  PSXPT,
                    <emph type="italics"/>
                  in data ra­
                    <lb/>
                  tione.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Lineæ
                    <emph type="italics"/>
                  AB, CD,
                    <emph.end type="italics"/>
                  ad quas rectæ duæ
                    <emph type="italics"/>
                  PQ, PR,
                    <emph.end type="italics"/>
                  unum rectan­
                    <lb/>
                  gulorum continentes ducuntur, conveniant cum aliis duabus poſi­
                    <lb/>
                  tione datis lineis in punctis
                    <emph type="italics"/>
                  A, B, C, D.
                    <emph.end type="italics"/>
                  Ab eorum aliquo
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  age
                    <lb/>
                  rectam quamlibet
                    <emph type="italics"/>
                  AH,
                    <emph.end type="italics"/>
                  in qua velis punctum
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  reperiri. </s>
                  <s>Secet ea
                    <lb/>
                  lineas oppoſitas
                    <emph type="italics"/>
                  BD, CD,
                    <emph.end type="italics"/>
                  nimirum
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  H
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  CD
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  I,
                    <emph.end type="italics"/>
                  & ob
                    <lb/>
                  datos omnes angulos figuræ, dabuntur rationes
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PA
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PA
                    <emph.end type="italics"/>
                  </s>
                </p>
              </subchap2>
            </subchap1>
          </chap>
        </body>
      </text>
    </archimedes>
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