Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <pb xlink:href="039/01/053.jpg" pagenum="25"/>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA II.
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                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Si in Figura quavis
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                  AacE,
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                  rectis
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                  Aa, AE
                    <emph type="italics"/>
                  & curva
                    <emph.end type="italics"/>
                  acE
                    <emph type="italics"/>
                  com
                    <lb/>
                  prehenſa, inſcribantur parallelogramma quotcunque
                    <emph.end type="italics"/>
                  Ab, Bc, Cd
                    <lb/>
                  &c.
                    <emph type="italics"/>
                  ſub baſibus
                    <emph.end type="italics"/>
                  AB, BC, CD, &c.
                    <emph type="italics"/>
                  æqualibus, & lateribuſ
                    <emph.end type="italics"/>
                    <lb/>
                  Bb, Cc, Dd, &c.
                    <emph type="italics"/>
                  Figuræ lateri
                    <emph.end type="italics"/>
                  Aa
                    <emph type="italics"/>
                  pa­
                    <lb/>
                  rallelis contenta; & compleantur paral-
                    <emph.end type="italics"/>
                    <lb/>
                    <figure id="id.039.01.053.1.jpg" xlink:href="039/01/053/1.jpg" number="6"/>
                    <lb/>
                    <emph type="italics"/>
                  lelogramma
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                  aKbl, bLcm, cMdn, &c.
                    <lb/>
                    <emph type="italics"/>
                  Dein horum parallelogrammorum lati­
                    <lb/>
                  tudo minuatur, & numerus augeatur
                    <lb/>
                  in infinitum: dico quod ultimæ rationes,
                    <lb/>
                  quas habent ad ſe invicem Figura in­
                    <lb/>
                  ſcripta
                    <emph.end type="italics"/>
                  AKbLcMdD,
                    <emph type="italics"/>
                  circumſcripta
                    <emph.end type="italics"/>
                    <lb/>
                  AalbmcndoE,
                    <emph type="italics"/>
                  & curvilinea
                    <emph.end type="italics"/>
                  AbcdE,
                    <lb/>
                    <emph type="italics"/>
                  ſunt rationes æqualitatis.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Nam Figuræ inſcriptæ & circumſcriptæ differentia eſt ſumma pa­
                    <lb/>
                  rallelogrammorum
                    <emph type="italics"/>
                  Kl, Lm, Mn, Do,
                    <emph.end type="italics"/>
                  hoc eſt (ob æquales om­
                    <lb/>
                  nium baſes) rectangulum ſub unius baſi
                    <emph type="italics"/>
                  Kb
                    <emph.end type="italics"/>
                  & altitudinum ſumma
                    <lb/>
                    <emph type="italics"/>
                  Aa,
                    <emph.end type="italics"/>
                  id eſt, rectangulum
                    <emph type="italics"/>
                  ABla.
                    <emph.end type="italics"/>
                  Sed hoc rectangulum, eo quod
                    <lb/>
                  latitudo ejus
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  in infinitum minuitur, fit minus quovis dato. </s>
                  <s>Er­
                    <lb/>
                  go (per Lemma 1) Figura inſcripta & circumſcripta & multo magis
                    <lb/>
                  Figura curvilinea intermedia fiunt ultimo æquales.
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA III.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Eædem rationes ultimæ ſunt etiam rationes æqualitatis, ubi paral­
                    <lb/>
                  lelogrammorum latitudines
                    <emph.end type="italics"/>
                  AB, BC, CD, &c.
                    <emph type="italics"/>
                  ſunt inæquales,
                    <lb/>
                  & omnes minuuntur in infinitum.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Sit enim
                    <emph type="italics"/>
                  AF
                    <emph.end type="italics"/>
                  æqualis latitudini maximæ, & compleatur paralle­
                    <lb/>
                  logrammum
                    <emph type="italics"/>
                  FAaf.
                    <emph.end type="italics"/>
                  Hoc erit majus quam differentia Figuræ in­
                    <lb/>
                  ſcriptæ & Figuræ circumſcriptæ; at latitudine ſua
                    <emph type="italics"/>
                  AF
                    <emph.end type="italics"/>
                  in infinitum
                    <lb/>
                  diminuta, minus fiet quam datum quodvis rectangulum.
                    <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. Hinc ſumma ultima parallelogrammorum evaneſcentium
                    <lb/>
                  coincidit omni ex parte cum Figura curvilinea. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  2. Et multo magis Figura rectilinea, quæ chordis evaneſ-</s>
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