Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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    <archimedes>
      <text>
        <body>
          <chap>
            <subchap1>
              <subchap2>
                <p type="main">
                  <s>
                    <pb xlink:href="039/01/061.jpg" pagenum="33"/>
                  evaneſcentium ſummas & rationes, primaſque naſcentium, id eſt,
                    <lb/>
                  ad limites ſummarum & rationum deducere; & propterea limitum
                    <lb/>
                  illorum demonſtrationes qua potui brevitate præmittere. </s>
                  <s>His enim
                    <lb/>
                  idem præſtatur quod per methodum Indiviſibilium; & principiis de­
                    <lb/>
                  monſtratis jam tutius utemur. </s>
                  <s>Proinde in ſequentibus, ſiquando
                    <lb/>
                  quantitates tanquam ex particulis conſtantes conſideravero, vel ſi
                    <lb/>
                  pro rectis uſurpavero lineolas curvas; nolim indiviſibilia, ſed eva­
                    <lb/>
                  neſcentia diviſibilia, non ſummas & rationes partium determinata­
                    <lb/>
                  rum, ſed ſummarum & rationum limites ſemper intelligi; vimque
                    <lb/>
                  talium demonſtrationum ad methodum præcedentium Lemmatum
                    <lb/>
                  ſemper revocari. </s>
                </p>
                <p type="main">
                  <s>Objectio eſt, quod quantitatum evaneſcentium nulla ſit ultima
                    <lb/>
                  proportio; quippe quæ, antequam evanuerunt, non eſt ultima, ubi
                    <lb/>
                  evanuerunt, nulla eſt. </s>
                  <s>Sed & eodem argumento æque contendi poſſet
                    <lb/>
                  nullam eſſe corporis ad certum locum pervenientis velocitatem ul­
                    <lb/>
                  timam: hanc enim, antequam corpus attingit locum, non eſſe ulti­
                    <lb/>
                  mam, ubi attingit, nullam eſſe. </s>
                  <s>Et reſponſio facilis eſt: Per velocita­
                    <lb/>
                  tem ultimam intelligi eam, qua corpus movetur neque antequam
                    <lb/>
                  attingit locum ultimum & motus ceſſat, neque poſtea, ſed tunc
                    <lb/>
                  cum attingit; id eſt, illam ipſam velocitatem quacum corpus attin­
                    <lb/>
                  git locum ultimum & quacum motus ceſſat. </s>
                  <s>Et ſimiliter per ulti­
                    <lb/>
                  mam rationem quantitatum evaneſcentium, intelligendam eſſe ratio­
                    <lb/>
                  nem quantitatum non antequam evaneſcunt, non poſtea, ſed qua­
                    <lb/>
                  cum evaneſcunt. </s>
                  <s>Pariter & ratio prima naſcentium eſt ratio qua­
                    <lb/>
                  cum naſcuntur. </s>
                  <s>Et ſumma prima & ultima eſt quacum eſſe (vel
                    <lb/>
                  augeri & minui) incipiunt & ceſſant. </s>
                  <s>Extat limes quem velocitas
                    <lb/>
                  in fine motus attingere poteſt, non autem tranſgredi. </s>
                  <s>Hæc eſt
                    <lb/>
                  velocitas ultima. </s>
                  <s>Et par eſt ratio limitis quantitatum & propor­
                    <lb/>
                  tionum omnium incipientium & ceſſantium. </s>
                  <s>Cumque hic limes
                    <lb/>
                  ſit certus & definitus, Problema eſt vere Geometricum eundem de­
                    <lb/>
                  terminare. </s>
                  <s>Geometrica vero omnia in aliis Geometricis determi­
                    <lb/>
                  nandis ac demonſtrandis legitime uſurpantur. </s>
                </p>
                <p type="main">
                  <s>Contendi etiam poteſt, quod ſi dentur ultimæ quantitatum eva­
                    <lb/>
                  neſcentium rationes, dabuntur & ultimæ magnitudines: & ſic quan­
                    <lb/>
                  titas omnis conſtabit ex Indiviſibilibus, contra quam
                    <emph type="italics"/>
                  Euclides
                    <emph.end type="italics"/>
                  de
                    <lb/>
                  Incommenſurabilibus, in libro decimo Elementorum, demonſtravit. </s>
                  <s>
                    <lb/>
                  Verum hæc Objectio falſæ innititur hypotheſi. </s>
                  <s>Ultimæ rationes
                    <lb/>
                  illæ quibuſcum quantitates evaneſcunt, revera non ſunt rationes
                    <lb/>
                  quantitatum ultimarum, ſed limites ad quos quantitatum ſine limi­
                    <lb/>
                  te decreſcentium rationes ſemper appropinquant; & quas propius
                    <lb/>
                  aſſequi poſſunt quam pro data quavis differentia, nunquam vero </s>
                </p>
              </subchap2>
            </subchap1>
          </chap>
        </body>
      </text>
    </archimedes>
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