Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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      <text>
        <body>
          <chap>
            <subchap1>
              <subchap2>
                <p type="main">
                  <s>
                    <pb xlink:href="039/01/077.jpg" pagenum="49"/>
                  Sed, punctis
                    <emph type="italics"/>
                  Q
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  coeuntibus,
                    <expan abbr="æquãtur">æquantur</expan>
                  2
                    <emph type="italics"/>
                  PC
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  Gv.
                    <emph.end type="italics"/>
                  Ergo & his pro­
                    <lb/>
                    <arrow.to.target n="note25"/>
                  portionalia
                    <emph type="italics"/>
                  LXQR
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  QT quad.
                    <emph.end type="italics"/>
                  æquantur. </s>
                  <s>Ducantur hæc æqualia in
                    <lb/>
                  (
                    <emph type="italics"/>
                  SPq/QR
                    <emph.end type="italics"/>
                  ) & fiet
                    <emph type="italics"/>
                    <expan abbr="LXSPq.">LXSPque</expan>
                    <emph.end type="italics"/>
                  æquale (
                    <emph type="italics"/>
                  SPq.XQTq/QR
                    <emph.end type="italics"/>
                  ). Ergo (per Corol. </s>
                  <s>1
                    <lb/>
                  & 5 Prop. </s>
                  <s>VI.) vis centripeta reciproce eſt ut
                    <emph type="italics"/>
                    <expan abbr="LXSPq.">LXSPque</expan>
                    <emph.end type="italics"/>
                  id eſt, reci­
                    <lb/>
                  proce in ratione duplicata diſtantiæ
                    <emph type="italics"/>
                  SP. Q.E.I.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note25"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Idem aliter.
                    <emph.end type="italics"/>
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>Cum vis ad centrum Ellipſeos tendens, qua corpus
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  in Ellipſi
                    <lb/>
                  illa revolvi poteſt, ſit (per Corol. </s>
                  <s>I Prop. </s>
                  <s>X) ut
                    <emph type="italics"/>
                  CP
                    <emph.end type="italics"/>
                  diſtantia cor­
                    <lb/>
                  poris ab Ellipſeos centro
                    <emph type="italics"/>
                  C
                    <emph.end type="italics"/>
                  ; ducatur
                    <emph type="italics"/>
                  CE
                    <emph.end type="italics"/>
                  parallela Ellipſeos tan­
                    <lb/>
                  genti
                    <emph type="italics"/>
                  PR:
                    <emph.end type="italics"/>
                  & vis qua corpus idem
                    <emph type="italics"/>
                  P,
                    <emph.end type="italics"/>
                  circum aliud quodvis Ellip­
                    <lb/>
                  ſeos punctum
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  revolvi poteſt, ſi
                    <emph type="italics"/>
                  CE
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  concurrant in
                    <emph type="italics"/>
                  E,
                    <emph.end type="italics"/>
                  erit ut
                    <lb/>
                  (
                    <emph type="italics"/>
                  PE cub./SPq
                    <emph.end type="italics"/>
                  ) (per Corol. </s>
                  <s>3 Prop. </s>
                  <s>VII,) hoc eſt, ſi punctum
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  ſit umbili­
                    <lb/>
                  cus Ellipſeos, adeoque
                    <emph type="italics"/>
                  PE
                    <emph.end type="italics"/>
                  detur, ut
                    <emph type="italics"/>
                  SPq
                    <emph.end type="italics"/>
                  reciproce.
                    <emph type="italics"/>
                  Q.E.I.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Eadem brevitate qua traduximus Problema quintum ad Parabo­
                    <lb/>
                  lam, & Hyperbolam, liceret idem hic facere: verum ob dignita­
                    <lb/>
                  tem Problematis & uſum ejus in ſequentibus, non pigebit caſus ce­
                    <lb/>
                  teros demonſtratione confirmare. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO XII. PROBLEMA. VII.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Moveatur corpus in Hyperbola: requiritur Lex vis centripetæ ten­
                    <lb/>
                  dentis ad umbilicum figuræ.
                    <emph.end type="italics"/>
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>Sunto
                    <emph type="italics"/>
                  CA, CB
                    <emph.end type="italics"/>
                  ſemi-axes Hyperbolæ;
                    <emph type="italics"/>
                  PG, KD
                    <emph.end type="italics"/>
                  diametri con­
                    <lb/>
                  jugatæ;
                    <emph type="italics"/>
                  PF, Qt
                    <emph.end type="italics"/>
                  perpendicula ad diametros; &
                    <emph type="italics"/>
                  Qv
                    <emph.end type="italics"/>
                  ordinatim
                    <lb/>
                  applicata ad diametrum
                    <emph type="italics"/>
                  GP.
                    <emph.end type="italics"/>
                  Agatur
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                  ſecans cum diametrum
                    <lb/>
                    <emph type="italics"/>
                  DK
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  E,
                    <emph.end type="italics"/>
                  tum ordinatim applicatam
                    <emph type="italics"/>
                  Qv
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  x,
                    <emph.end type="italics"/>
                  & compleatur pa­
                    <lb/>
                  rallelogrammum
                    <emph type="italics"/>
                  QRPx.
                    <emph.end type="italics"/>
                  Patet
                    <emph type="italics"/>
                  EP
                    <emph.end type="italics"/>
                  æqualem eſſe ſemiaxi tranſ­
                    <lb/>
                  verſo
                    <emph type="italics"/>
                  AC,
                    <emph.end type="italics"/>
                  eo quod, acta ab altero Hyperbolæ umbilico
                    <emph type="italics"/>
                  H
                    <emph.end type="italics"/>
                  linea
                    <lb/>
                    <emph type="italics"/>
                  HI
                    <emph.end type="italics"/>
                  ipſi
                    <emph type="italics"/>
                  EC
                    <emph.end type="italics"/>
                  parallela, ob æquales
                    <emph type="italics"/>
                  CS, CH,
                    <emph.end type="italics"/>
                  æquentur
                    <emph type="italics"/>
                  ES, EI
                    <emph.end type="italics"/>
                  ;
                    <lb/>
                  adeo ut
                    <emph type="italics"/>
                  EP
                    <emph.end type="italics"/>
                  ſemidifferentia ſit ipſarum
                    <emph type="italics"/>
                  PS, PI,
                    <emph.end type="italics"/>
                  id eſt (ob pa­
                    <lb/>
                  rallelas
                    <emph type="italics"/>
                  IH, PR
                    <emph.end type="italics"/>
                  & angulos æquales
                    <emph type="italics"/>
                  IPR, HPZ
                    <emph.end type="italics"/>
                  ) ipſarum
                    <emph type="italics"/>
                  PS,
                    <lb/>
                  PH,
                    <emph.end type="italics"/>
                  quarum differentia axem totum 2
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  adæquat. </s>
                  <s>Ad
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                  de­
                    <lb/>
                  mittatur perpendicularis
                    <emph type="italics"/>
                  QT.
                    <emph.end type="italics"/>
                  Et Hyperbolæ latere recto princi­
                    <lb/>
                  pali (ſeu (2
                    <emph type="italics"/>
                  BCq/AC
                    <emph.end type="italics"/>
                  )) dicto
                    <emph type="italics"/>
                  L,
                    <emph.end type="italics"/>
                  erit
                    <emph type="italics"/>
                  LXQR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  LXPv
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  QR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Pv,
                    <emph.end type="italics"/>
                    <lb/>
                  id eſt, ut
                    <emph type="italics"/>
                  PE
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  AC
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  PC
                    <emph.end type="italics"/>
                  ; Et
                    <emph type="italics"/>
                  LXPv
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  GvP
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  L
                    <emph.end type="italics"/>
                  ad </s>
                </p>
              </subchap2>
            </subchap1>
          </chap>
        </body>
      </text>
    </archimedes>
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