Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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    <archimedes>
      <text>
        <body>
          <chap>
            <subchap1>
              <subchap2>
                <p type="main">
                  <s>
                    <pb xlink:href="039/01/090.jpg" pagenum="62"/>
                    <arrow.to.target n="note38"/>
                  ut
                    <emph type="italics"/>
                  Vk-VK
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  kS-KS,
                    <emph.end type="italics"/>
                  id eſt ut 2
                    <emph type="italics"/>
                  VX
                    <emph.end type="italics"/>
                  ad 2
                    <emph type="italics"/>
                  KX
                    <emph.end type="italics"/>
                  & 2
                    <emph type="italics"/>
                  KX
                    <emph.end type="italics"/>
                  ad
                    <lb/>
                  2
                    <emph type="italics"/>
                  SX,
                    <emph.end type="italics"/>
                  adeoque ut
                    <emph type="italics"/>
                  VX
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  HX
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  HX
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  SX,
                    <emph.end type="italics"/>
                  ſimilia erunt tri­
                    <lb/>
                  angula
                    <emph type="italics"/>
                  VXH, HXS,
                    <emph.end type="italics"/>
                  & propterea
                    <emph type="italics"/>
                  VH
                    <emph.end type="italics"/>
                  erit ad
                    <emph type="italics"/>
                  SH
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  VX
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  XH,
                    <emph.end type="italics"/>
                    <lb/>
                  adeoque ut
                    <emph type="italics"/>
                  VK
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  KS.
                    <emph.end type="italics"/>
                  Habet igitur Trajectoriæ deſcriptæ axis
                    <lb/>
                  principalis
                    <emph type="italics"/>
                  VH
                    <emph.end type="italics"/>
                  eam rationem ad ipſius umbilieorum diſtantiam
                    <emph type="italics"/>
                  SH,
                    <emph.end type="italics"/>
                    <lb/>
                  quam habet Trajectoriæ deſcribendæ axis principalis ad ipſius um­
                    <lb/>
                  bilieorum diſtantiam, & propterea ejuſdem eſt ſpeciei. </s>
                  <s>Inſuper cum
                    <lb/>
                    <emph type="italics"/>
                  VH, vH
                    <emph.end type="italics"/>
                  æquentur axi principali, &
                    <emph type="italics"/>
                  VS, vS
                    <emph.end type="italics"/>
                  a rectis
                    <emph type="italics"/>
                  TR, tr
                    <emph.end type="italics"/>
                    <lb/>
                  perpendiculariter biſecentur, liquet, ex Lemmate XV, rectas illas
                    <lb/>
                  Trajectoriam deſcriptam tangere.
                    <emph type="italics"/>
                  q.E.F.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note38"/>
                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  3. Dato umbilico
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  deſcribenda ſit Trajectoria quæ rect­
                    <lb/>
                  am
                    <emph type="italics"/>
                  TR
                    <emph.end type="italics"/>
                  tanget in puncto dato
                    <emph type="italics"/>
                  R.
                    <emph.end type="italics"/>
                  In rectam
                    <emph type="italics"/>
                  TR
                    <emph.end type="italics"/>
                  demitte perpen­
                    <lb/>
                  dicularem
                    <emph type="italics"/>
                  ST,
                    <emph.end type="italics"/>
                  & produc eandem ad
                    <emph type="italics"/>
                  V,
                    <emph.end type="italics"/>
                  ut ſit
                    <emph type="italics"/>
                  TV
                    <emph.end type="italics"/>
                  æqualis
                    <emph type="italics"/>
                  ST.
                    <emph.end type="italics"/>
                  Junge
                    <lb/>
                    <emph type="italics"/>
                  VR,
                    <emph.end type="italics"/>
                  & rectam
                    <emph type="italics"/>
                  VS
                    <emph.end type="italics"/>
                  infinite productam ſeca in
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  k,
                    <emph.end type="italics"/>
                  ita ut ſit
                    <lb/>
                    <emph type="italics"/>
                  VK
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  SK
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  Vk
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Sk
                    <emph.end type="italics"/>
                  ut Ellipſeos deſcribendæ axis principalis
                    <lb/>
                  ad diſtantiam umbilieorum; circuloque ſuper diametro
                    <emph type="italics"/>
                  Kk
                    <emph.end type="italics"/>
                  de­
                    <lb/>
                  ſcripto, ſecetur producta recta
                    <emph type="italics"/>
                  VR
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  H,
                    <emph.end type="italics"/>
                  & umbilicis
                    <emph type="italics"/>
                  S, H,
                    <emph.end type="italics"/>
                  axe
                    <lb/>
                  principali rectam
                    <emph type="italics"/>
                  VH
                    <emph.end type="italics"/>
                  æquante, deſcribatur Trajectoria. </s>
                  <s>Dico fa­
                    <lb/>
                  ctum. </s>
                  <s>Namque
                    <emph type="italics"/>
                  VH
                    <emph.end type="italics"/>
                  eſſe ad
                    <lb/>
                    <figure id="id.039.01.090.1.jpg" xlink:href="039/01/090/1.jpg" number="33"/>
                    <lb/>
                    <emph type="italics"/>
                  SH
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  VK
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  SK,
                    <emph.end type="italics"/>
                  atque adeo
                    <lb/>
                  ut axis principalis Trajectoriæ
                    <lb/>
                  deſcribendæ ad diſtantiam um­
                    <lb/>
                  bilieorum ejus, patet ex demon­
                    <lb/>
                  ſtratis in Caſu ſecundo, & prop­
                    <lb/>
                  terea Trajectoriam deſcriptam
                    <lb/>
                  ejuſdem eſſe ſpeciei cum deſcri­
                    <lb/>
                  benda; rectam vero
                    <emph type="italics"/>
                  TR
                    <emph.end type="italics"/>
                  qua an­
                    <lb/>
                  gulus
                    <emph type="italics"/>
                  VRS
                    <emph.end type="italics"/>
                  biſecatur, tangere Trajectoriam in puncto
                    <emph type="italics"/>
                  R,
                    <emph.end type="italics"/>
                  patet ex
                    <lb/>
                  Conicis.
                    <emph type="italics"/>
                  q.E.F.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  4. Circa umbilicum
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  deſcribenda jam ſit Trajectoria
                    <emph type="italics"/>
                  APB,
                    <emph.end type="italics"/>
                    <lb/>
                  quæ tangat rectam
                    <emph type="italics"/>
                  TR,
                    <emph.end type="italics"/>
                  tranſeatque per punctum quodvis
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  extra
                    <lb/>
                  tangentem datum, quæque ſimilis ſit Figuræ
                    <emph type="italics"/>
                  apb,
                    <emph.end type="italics"/>
                  axe principali
                    <lb/>
                    <emph type="italics"/>
                  ab
                    <emph.end type="italics"/>
                  & umbilicis
                    <emph type="italics"/>
                  s, h
                    <emph.end type="italics"/>
                  deſcriptæ. </s>
                  <s>In tangentem
                    <emph type="italics"/>
                  TR
                    <emph.end type="italics"/>
                  demitte per­
                    <lb/>
                  pendiculum
                    <emph type="italics"/>
                  ST,
                    <emph.end type="italics"/>
                  & produc idem ad
                    <emph type="italics"/>
                  V,
                    <emph.end type="italics"/>
                  ut ſit
                    <emph type="italics"/>
                  TV
                    <emph.end type="italics"/>
                  æqualis
                    <emph type="italics"/>
                  ST.
                    <emph.end type="italics"/>
                  An­
                    <lb/>
                  gulis autem
                    <emph type="italics"/>
                  VSP, SVP
                    <emph.end type="italics"/>
                  fac angulos
                    <emph type="italics"/>
                  hsq, shq
                    <emph.end type="italics"/>
                  æquales; cen­
                    <lb/>
                  troque
                    <emph type="italics"/>
                  q
                    <emph.end type="italics"/>
                  & intervallo quod ſit ad
                    <emph type="italics"/>
                  ab
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  VS
                    <emph.end type="italics"/>
                  deſcribe circu­
                    <lb/>
                  lum ſecantem Figuram
                    <emph type="italics"/>
                  apb
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  p.
                    <emph.end type="italics"/>
                  Junge
                    <emph type="italics"/>
                  sp
                    <emph.end type="italics"/>
                  & age
                    <emph type="italics"/>
                  SH
                    <emph.end type="italics"/>
                  quæ ſit ad
                    <lb/>
                    <emph type="italics"/>
                  sh
                    <emph.end type="italics"/>
                  ut eſt
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  sp,
                    <emph.end type="italics"/>
                  quæque angulum
                    <emph type="italics"/>
                  PSH
                    <emph.end type="italics"/>
                  angulo
                    <emph type="italics"/>
                  psh
                    <emph.end type="italics"/>
                  & angulum
                    <lb/>
                    <emph type="italics"/>
                  VSH
                    <emph.end type="italics"/>
                  angulo
                    <emph type="italics"/>
                  psq
                    <emph.end type="italics"/>
                  æquales conſtituat. </s>
                  <s>Denique umbilicis
                    <emph type="italics"/>
                  S, H,
                    <emph.end type="italics"/>
                    <lb/>
                  & axe principali
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  diſtantiam
                    <emph type="italics"/>
                  VH
                    <emph.end type="italics"/>
                  æquante, deſcribatur ſectio
                    <lb/>
                  Conica. </s>
                  <s>Dico factum. </s>
                  <s>Nam ſi agatur
                    <emph type="italics"/>
                  sv
                    <emph.end type="italics"/>
                  quæ ſit ad
                    <emph type="italics"/>
                  sp
                    <emph.end type="italics"/>
                  ut eſt
                    <emph type="italics"/>
                  sh
                    <emph.end type="italics"/>
                  </s>
                </p>
              </subchap2>
            </subchap1>
          </chap>
        </body>
      </text>
    </archimedes>
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