Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/089.jpg" pagenum="61"/>
                    <arrow.to.target n="note37"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note37"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO XX. PROBLEMA XII.
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                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Circa datum umbilicum Trajectoriam quamvis ſpecie datam deſcribe­
                    <lb/>
                  re, quæ per data puncta tranſibit & rectas tanget pofitione datas.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  1. Dato umbilico
                    <emph type="italics"/>
                  S,
                    <emph.end type="italics"/>
                  deſcribenda ſit Trajectoria
                    <emph type="italics"/>
                  ABC
                    <emph.end type="italics"/>
                  per
                    <lb/>
                  puncta duo
                    <emph type="italics"/>
                  B, C.
                    <emph.end type="italics"/>
                  Quoniam Trajectoria datur ſpecie, dabitur ra­
                    <lb/>
                  tio axis principalis ad diſtantiam
                    <lb/>
                    <figure id="id.039.01.089.1.jpg" xlink:href="039/01/089/1.jpg" number="31"/>
                    <lb/>
                  umbilieorum. </s>
                  <s>In ea ratione cape
                    <lb/>
                    <emph type="italics"/>
                  KB
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  BS,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  LC
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  CS.
                    <emph.end type="italics"/>
                  Cen­
                    <lb/>
                  tris
                    <emph type="italics"/>
                  B, C,
                    <emph.end type="italics"/>
                  intervallis
                    <emph type="italics"/>
                  BK, CL,
                    <emph.end type="italics"/>
                  de­
                    <lb/>
                  ſcribe circulos duos, & ad rectam
                    <lb/>
                    <emph type="italics"/>
                  KL,
                    <emph.end type="italics"/>
                  quæ tangat eoſdem in
                    <emph type="italics"/>
                  K
                    <emph.end type="italics"/>
                  &
                    <lb/>
                    <emph type="italics"/>
                  L,
                    <emph.end type="italics"/>
                  demitte perpendiculum
                    <emph type="italics"/>
                  SG,
                    <emph.end type="italics"/>
                  idemque ſeca in
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  a,
                    <emph.end type="italics"/>
                  ita ut ſit
                    <lb/>
                    <emph type="italics"/>
                  GA
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  AS
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  Ga
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  aS,
                    <emph.end type="italics"/>
                  ut eſt
                    <emph type="italics"/>
                  KB
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  BS,
                    <emph.end type="italics"/>
                  & axe &c.
                    <emph type="italics"/>
                  Aa,
                    <emph.end type="italics"/>
                  verticibus
                    <lb/>
                    <emph type="italics"/>
                  A, a,
                    <emph.end type="italics"/>
                  deſcribatur Trajectoria. </s>
                  <s>Dico factum. </s>
                  <s>Sit enim
                    <emph type="italics"/>
                  H
                    <emph.end type="italics"/>
                  umbilicus
                    <lb/>
                  alter Figuræ deſcriptæ, & cum ſit
                    <emph type="italics"/>
                  GA
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  AS
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  Ga
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  aS,
                    <emph.end type="italics"/>
                  erit di­
                    <lb/>
                  viſim
                    <emph type="italics"/>
                  Ga-GA
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  Aa
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  aS-AS
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  SH
                    <emph.end type="italics"/>
                  in eadem &c. </s>
                  <s>ratione,
                    <lb/>
                  adeoQ.E.I. ratione quam habet axis principalis Figuræ deſcribendæ
                    <lb/>
                  ad diſtantiam umbilieorum ejus; & propterea Figura deſcripta eſt
                    <lb/>
                  ejuſdem ſpeciei cum deſcribenda. </s>
                  <s>Cumque ſint
                    <emph type="italics"/>
                  KB
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  BS
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  LC
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  CS
                    <emph.end type="italics"/>
                  in eadem ratione, tranſibit hæc Figura per puncta
                    <emph type="italics"/>
                  B, C,
                    <emph.end type="italics"/>
                  ut
                    <lb/>
                  ex Conicis manifeſtum eſt. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  2. Dato umbilico
                    <emph type="italics"/>
                  S,
                    <emph.end type="italics"/>
                  deſcribenda ſit Trajectoria quæ rectas
                    <lb/>
                  duas
                    <emph type="italics"/>
                  TR, tr
                    <emph.end type="italics"/>
                  alicubi contingat. </s>
                  <s>Ab umbilico in tangentes demitte
                    <lb/>
                  perpendicula
                    <emph type="italics"/>
                  ST, St
                    <emph.end type="italics"/>
                  & produc ea­
                    <lb/>
                    <figure id="id.039.01.089.2.jpg" xlink:href="039/01/089/2.jpg" number="32"/>
                    <lb/>
                  dem ad
                    <emph type="italics"/>
                  V, v,
                    <emph.end type="italics"/>
                  ut ſint
                    <emph type="italics"/>
                  TV, tv
                    <emph.end type="italics"/>
                  æ­
                    <lb/>
                  quales
                    <emph type="italics"/>
                  TS, tS.
                    <emph.end type="italics"/>
                  Biſeca
                    <emph type="italics"/>
                  Vv
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  O,
                    <emph.end type="italics"/>
                    <lb/>
                  & erige perpendiculum infinitum
                    <lb/>
                    <emph type="italics"/>
                  OH,
                    <emph.end type="italics"/>
                  rectamque
                    <emph type="italics"/>
                  VS
                    <emph.end type="italics"/>
                  infinite pro­
                    <lb/>
                  ductam 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"/>
                  KS
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  Vk
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  kS
                    <emph.end type="italics"/>
                  ut eſt
                    <lb/>
                  Trajectoriæ deſcribendæ axis prin­
                    <lb/>
                  cipalis ad umbilieorum diſtantiam. </s>
                  <s>
                    <lb/>
                  Super diametro
                    <emph type="italics"/>
                  Kk
                    <emph.end type="italics"/>
                  deſcribatur
                    <lb/>
                  circulus ſecans
                    <emph type="italics"/>
                  OH
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  H
                    <emph.end type="italics"/>
                  ; & umbilicis
                    <emph type="italics"/>
                  S, H,
                    <emph.end type="italics"/>
                  axe principali ipſam
                    <lb/>
                    <emph type="italics"/>
                  VH
                    <emph.end type="italics"/>
                  æquante, deſcribatur Trajectoria. </s>
                  <s>Dico factum. </s>
                  <s>Nam biſeca
                    <lb/>
                    <emph type="italics"/>
                  Kk
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  X,
                    <emph.end type="italics"/>
                  & junge
                    <emph type="italics"/>
                  HX, HS, HV, Hv.
                    <emph.end type="italics"/>
                  Quoniam eſt
                    <emph type="italics"/>
                  VK
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  KS
                    <emph.end type="italics"/>
                    <lb/>
                  ut
                    <emph type="italics"/>
                  Vk
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  kS
                    <emph.end type="italics"/>
                  ; & compofite ut
                    <emph type="italics"/>
                  VK+Vk
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  KS+kS
                    <emph.end type="italics"/>
                  ; diviſimque </s>
                </p>
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        </body>
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