Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/059.jpg" pagenum="31"/>
                  concurrentes in
                    <emph type="italics"/>
                  G
                    <emph.end type="italics"/>
                  ; dein accedant puncta
                    <emph type="italics"/>
                  D, B, G,
                    <emph.end type="italics"/>
                  ad puncta
                    <emph type="italics"/>
                  d, b, g,
                    <emph.end type="italics"/>
                    <lb/>
                  ſitque
                    <emph type="italics"/>
                  J
                    <emph.end type="italics"/>
                  interſectio linearum
                    <emph type="italics"/>
                  BG, AG
                    <emph.end type="italics"/>
                  ultimo facta ubi puncta
                    <emph type="italics"/>
                  D, B
                    <emph.end type="italics"/>
                    <lb/>
                  accedunt uſque ad
                    <emph type="italics"/>
                  A.
                    <emph.end type="italics"/>
                  Manifeſtum eſt quod diſtantia
                    <emph type="italics"/>
                  GJ
                    <emph.end type="italics"/>
                  minor
                    <lb/>
                  eſſe poteſt quam aſſignata quævis. </s>
                  <s>Eſt autem (ex natura circulorum
                    <lb/>
                  per puncta
                    <emph type="italics"/>
                  ABG, Abg
                    <emph.end type="italics"/>
                  tranſeuntium)
                    <emph type="italics"/>
                  ABquad.
                    <emph.end type="italics"/>
                    <lb/>
                    <figure id="id.039.01.059.1.jpg" xlink:href="039/01/059/1.jpg" number="12"/>
                    <lb/>
                  æquale
                    <emph type="italics"/>
                  AGXBD,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  Ab quad.
                    <emph.end type="italics"/>
                  æquale
                    <emph type="italics"/>
                  AgXbd,
                    <emph.end type="italics"/>
                    <lb/>
                  adeoque ratio
                    <emph type="italics"/>
                  AB quad.
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Ab quad.
                    <emph.end type="italics"/>
                  compo­
                    <lb/>
                  nitur ex rationibus
                    <emph type="italics"/>
                  AG
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Ag
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  bd.
                    <emph.end type="italics"/>
                    <lb/>
                  Sed quoniam
                    <emph type="italics"/>
                  GJ
                    <emph.end type="italics"/>
                  aſſumi poteſt minor longitu­
                    <lb/>
                  dine quavis aſſignata, fieri poteſt ut ratio
                    <emph type="italics"/>
                  AG
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  Ag
                    <emph.end type="italics"/>
                  minus differat a ratione æqualitatis quam
                    <lb/>
                  pro differentia quavis aſſignata, adeoque ut ra­
                    <lb/>
                  tio
                    <emph type="italics"/>
                  AB quad.
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Ab quad.
                    <emph.end type="italics"/>
                  minus differat a ra­
                    <lb/>
                  tione
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  bd
                    <emph.end type="italics"/>
                  quam pro differentia quavis
                    <lb/>
                  aſſignata. </s>
                  <s>Eſt ergo, per Lemma 1, ratio ultima
                    <lb/>
                    <emph type="italics"/>
                  AB quad.
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Ab quad.
                    <emph.end type="italics"/>
                  æqualis rationi ultimæ
                    <lb/>
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  bd. </s>
                  <s>
                    <expan abbr="q.">que</expan>
                  E. D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  2. Inclinetur jam
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  in angulo
                    <lb/>
                  quovis dato, & eadem ſemper erit ratio ultima
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  bd
                    <emph.end type="italics"/>
                  quæ
                    <lb/>
                  prius, adeoque eadem ae
                    <emph type="italics"/>
                  AB quad.
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  Ab quad. </s>
                  <s>
                    <expan abbr="q.">que</expan>
                  E. D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  3. Et quamvis angulus
                    <emph type="italics"/>
                  D
                    <emph.end type="italics"/>
                  non detur, ſed recta
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                  ad da­
                    <lb/>
                  tum punctum convergente, vel alia quacunque lege conſtituatur;
                    <lb/>
                  tamen anguli
                    <emph type="italics"/>
                  D, d
                    <emph.end type="italics"/>
                  communi lege conſtituti ad æqualitatem ſemper
                    <lb/>
                  vergent & propius accedent ad invicem quam pro differentia qua­
                    <lb/>
                  vis aſſignata, adeoque ultimo æquales erunt, per Lem. I & prop­
                    <lb/>
                  terea lineæ
                    <emph type="italics"/>
                  BD, bd
                    <emph.end type="italics"/>
                  ſunt in eadem ratione ad invicem ac prius.
                    <lb/>
                    <emph type="italics"/>
                    <expan abbr="q.">que</expan>
                  E. D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  1. Unde eum tangentes
                    <emph type="italics"/>
                  AD, Ad,
                    <emph.end type="italics"/>
                  arcus
                    <emph type="italics"/>
                  AB, Ab,
                    <emph.end type="italics"/>
                  & eo­
                    <lb/>
                  rum ſinus
                    <emph type="italics"/>
                  BC, bc
                    <emph.end type="italics"/>
                  fiant ultimo chordis
                    <emph type="italics"/>
                  AB, Ab
                    <emph.end type="italics"/>
                  æquales; erunt
                    <lb/>
                  etiam illorum quadrata ultimo ut ſubtenſæ
                    <emph type="italics"/>
                  BD, bd.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  2. Eorundem quadrata ſunt etiam ultimo ut ſunt arcuum
                    <lb/>
                  ſagittæ quæ chordas biſecant & ad datum punctum convergunt. </s>
                  <s>
                    <lb/>
                  Nam ſagittæ illæ ſunt ut ſubtenſæ
                    <emph type="italics"/>
                  BD, bd.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  3. Ideoque ſagitta eſt in duplicata ratione temporis quo
                    <lb/>
                  corpus data velocitate deſcribit arcum. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  4. Triangula rectilinea
                    <emph type="italics"/>
                  ADB, Adb
                    <emph.end type="italics"/>
                  ſunt ultimo in tripli­
                    <lb/>
                  cata ratione laterum
                    <emph type="italics"/>
                  AD, Ad,
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                  inque ſeſquiplicata laterum
                    <emph type="italics"/>
                  DB,
                    <lb/>
                  db
                    <emph.end type="italics"/>
                  ; utpote in compoſita ratione laterum
                    <emph type="italics"/>
                  AD,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  DB, Ad
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  db
                    <emph.end type="italics"/>
                    <lb/>
                  exiſtentia. </s>
                  <s>Sic & triangula
                    <emph type="italics"/>
                  ABC, Abc
                    <emph.end type="italics"/>
                  ſunt ultimo in triplicata
                    <lb/>
                  ratione laterum
                    <emph type="italics"/>
                  BC, bc.
                    <emph.end type="italics"/>
                  Rationem vero Seſquiplicatam voco tri­
                    <lb/>
                  plicatæ ſubduplicatam, quæ nempe ex ſimplici & ſubduplicata com­
                    <lb/>
                  ponitur, quamque alias Seſquialteram dicunt. </s>
                </p>
              </subchap2>
            </subchap1>
          </chap>
        </body>
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    </archimedes>
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