Newton, Isaac, Philosophia naturalis principia mathematica, 1713

Table of figures

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                  <s>
                    <pb xlink:href="039/01/447.jpg" pagenum="419"/>
                  Quadraturis, & ad
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  demittatur perpendiculum
                    <emph type="italics"/>
                  GH
                    <emph.end type="italics"/>
                  : erit
                    <lb/>
                    <arrow.to.target n="note448"/>
                    <emph type="italics"/>
                  AH
                    <emph.end type="italics"/>
                  ſinus Inclinationis quæſitæ. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note448"/>
                  LIBER
                    <lb/>
                  TERTIUS.</s>
                </p>
                <p type="main">
                  <s>Nam
                    <emph type="italics"/>
                  GEq
                    <emph.end type="italics"/>
                  æquale eſt
                    <emph type="italics"/>
                  GHq+HEq=BHD+HEq=
                    <lb/>
                  HBD+HEq-BHq=HBD+BEq
                    <emph.end type="italics"/>
                  -2
                    <emph type="italics"/>
                  BHXBE=
                    <lb/>
                  BEq
                    <emph.end type="italics"/>
                  +2
                    <emph type="italics"/>
                  ECXBH
                    <emph.end type="italics"/>
                  =2
                    <emph type="italics"/>
                  ECXAB
                    <emph.end type="italics"/>
                  +2
                    <emph type="italics"/>
                  ECXBH
                    <emph.end type="italics"/>
                  =2
                    <emph type="italics"/>
                  ECXAH.
                    <emph.end type="italics"/>
                    <lb/>
                  Ideoque cum 2
                    <emph type="italics"/>
                  EC
                    <emph.end type="italics"/>
                  detur, eſt
                    <emph type="italics"/>
                  GEq
                    <emph.end type="italics"/>
                  ut
                    <emph type="italics"/>
                  AH.
                    <emph.end type="italics"/>
                  Deſignet jam
                    <emph type="italics"/>
                  AEg
                    <emph.end type="italics"/>
                    <lb/>
                  duplicatam diſtantiam Nodorum à Quadraturis poſt datum ali­
                    <lb/>
                  quod momentum temporis completum, & arcus
                    <emph type="italics"/>
                  Gg.,
                    <emph.end type="italics"/>
                  ob datum
                    <lb/>
                    <figure id="id.039.01.447.1.jpg" xlink:href="039/01/447/1.jpg" number="218"/>
                    <lb/>
                  angulum
                    <emph type="italics"/>
                  GEg,
                    <emph.end type="italics"/>
                  erit ut diſtantia
                    <emph type="italics"/>
                  GE.
                    <emph.end type="italics"/>
                  Eſt autem
                    <emph type="italics"/>
                  Hh
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Gg
                    <emph.end type="italics"/>
                    <lb/>
                  ut
                    <emph type="italics"/>
                  GH
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  GC,
                    <emph.end type="italics"/>
                  & propterea
                    <emph type="italics"/>
                  Hh
                    <emph.end type="italics"/>
                  eſt ut contentum
                    <emph type="italics"/>
                  GHXGg,
                    <emph.end type="italics"/>
                    <lb/>
                  ſeu
                    <emph type="italics"/>
                  GHXGE
                    <emph.end type="italics"/>
                  ; id eſt, ut
                    <emph type="italics"/>
                  (GH/GE)XGEq
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  (GH/GE)XAH,
                    <emph.end type="italics"/>
                  id eſt,
                    <lb/>
                  ut
                    <emph type="italics"/>
                  AH
                    <emph.end type="italics"/>
                  & ſinus anguli
                    <emph type="italics"/>
                  AEG
                    <emph.end type="italics"/>
                  conjunctim. </s>
                  <s>Igitur ſi
                    <emph type="italics"/>
                  AH
                    <emph.end type="italics"/>
                  in
                    <lb/>
                  caſu aliquo ſit ſinus Inclinationis, augebitur ea iiſdem incremen­
                    <lb/>
                  tis cum ſinu Inclinationis, per Corol. </s>
                  <s>3. Propoſitionis ſuperioris,
                    <lb/>
                  & propterea ſinui illi æqualis ſemper manebit. </s>
                  <s>Sed
                    <emph type="italics"/>
                  AH
                    <emph.end type="italics"/>
                  ubi
                    <lb/>
                  punctum
                    <emph type="italics"/>
                  G
                    <emph.end type="italics"/>
                  incidit in punctum alterutrum
                    <emph type="italics"/>
                  B
                    <emph.end type="italics"/>
                  vel
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  huic ſinui
                    <lb/>
                  æqualis eſt, & propterea eidem ſemper æqualis manet.
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>In hac demonſtratione ſuppoſui angulum
                    <emph type="italics"/>
                  BEG,
                    <emph.end type="italics"/>
                  qui eſt du­
                    <lb/>
                  plicata diſtantia Nodorum à Quadraturis, uniformiter augeri. </s>
                  <s>
                    <lb/>
                  Nam omnes inæqualitatum minutias expendeve non vacat. </s>
                  <s>Con­
                    <lb/>
                  cipe jam angulum
                    <emph type="italics"/>
                  BEG
                    <emph.end type="italics"/>
                  rectum eſſe, & in hoc eaſe
                    <emph type="italics"/>
                  Gg
                    <emph.end type="italics"/>
                  eſſe
                    <lb/>
                  augmentum horarium duplæ diſtantiæ Nodorum & Solis ab invi­
                    <lb/>
                  cem; & Inclinationis Variatio horaria in eodem caſu (per Corol. </s>
                  <s>
                    <lb/>
                  3. Prop. </s>
                  <s>noviſſimæ) erit ad 33′. </s>
                  <s>10′. </s>
                  <s>33
                    <emph type="sup"/>
                  iv
                    <emph.end type="sup"/>
                  . </s>
                  <s>ut contentum ſub In­
                    <lb/>
                  clinationis ſinu
                    <emph type="italics"/>
                  AH
                    <emph.end type="italics"/>
                  & ſinu anguli recti
                    <emph type="italics"/>
                  BEG,
                    <emph.end type="italics"/>
                  qui eſt dupli­
                    <lb/>
                  cata diſtantia Nodorum a Sole, ad quadruplum quadratum radii;
                    <lb/>
                  id. </s>
                  <s>eſt, ut mediocris Inclinationis ſinus
                    <emph type="italics"/>
                  AH
                    <emph.end type="italics"/>
                  ad radium quadru­
                    <lb/>
                  plicatum; hoc eſt (cum Inclinatio illa mediocris ſit quafi 5
                    <emph type="sup"/>
                  gr.
                    <emph.end type="sup"/>
                  8′1/2)
                    <lb/>
                  ut ejus ſinus 896 ad radium quadruplicatum 40000, ſive ut 224
                    <lb/>
                  ad 10000. Eſt autem Variatio tota, ſinuum differentiæ
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                    <lb/>
                  reſpondens, ad Variationem illam horariam ut diameter
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  ad </s>
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