Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/055.jpg" pagenum="27"/>
                  rallelogramma inter ſe ut partes, ſummæ partium ſemper erunt ut
                    <lb/>
                  ſummæ parallelogrammorum; atque adeo, ubi partium & paralle­
                    <lb/>
                  logrammorum numerus augetur & magnitudo diminuitur in infiNI­
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                  tum, in ultima ratione parallelogrammi ad parallelogrammum, id
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                  eſt (per hypotheſin) in ultima ratione partis ad partem. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA V.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Similium Figurarum latera omnia, quæ ſibi mutuo reſpondent, ſunt
                    <lb/>
                  proportionalia, tam curvilinea quam rectilinea; & areæ ſunt in
                    <lb/>
                  duplicata ratione laterum.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA VI.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Si arcus quilibet poſitione datus
                    <emph.end type="italics"/>
                  AB
                    <emph type="italics"/>
                  ſub-
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                    <lb/>
                    <figure id="id.039.01.055.1.jpg" xlink:href="039/01/055/1.jpg" number="8"/>
                    <lb/>
                    <emph type="italics"/>
                  tendatur chorda
                    <emph.end type="italics"/>
                  AB,
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                  & in puncto
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                  aliquo
                    <emph.end type="italics"/>
                  A,
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                  in medio curvaturæ continuæ,
                    <lb/>
                  tangatur a recta utrinque producta
                    <emph.end type="italics"/>
                    <lb/>
                  AD;
                    <emph type="italics"/>
                  dein puncta
                    <emph.end type="italics"/>
                  A, B
                    <emph type="italics"/>
                  ad invicem
                    <lb/>
                  accedant & coëant; dico quod angulus
                    <emph.end type="italics"/>
                    <lb/>
                  BAD,
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                  ſub chorda & tangente conten­
                    <lb/>
                  tus, minuetur in infinitum & ultimo e­
                    <lb/>
                  vaneſcet.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Nam ſi angulus ille non evaneſcit, continebit arcus
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  cum tan­
                    <lb/>
                  gente
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  angulum rectilineo æqualem, & propterea curvatura ad
                    <lb/>
                  ad punctum
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  non erit continua, contra hypotheſin. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  LEMMA VII.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Iiſdem poſitis; dico quod ultima ratio arcus, chordæ, & tangentis
                    <lb/>
                  ad invicem est ratio æqualitatis.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Nam dum punctum
                    <emph type="italics"/>
                  B
                    <emph.end type="italics"/>
                  ad punctum
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  accedit, intelligantur ſemper
                    <lb/>
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  ad puncta longinqua
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  ac
                    <emph type="italics"/>
                  d
                    <emph.end type="italics"/>
                  produci, & ſecanti
                    <emph type="italics"/>
                  BD
                    <emph.end type="italics"/>
                    <lb/>
                  parallela agatur
                    <emph type="italics"/>
                  bd.
                    <emph.end type="italics"/>
                  Sitque arcus
                    <emph type="italics"/>
                  Ab
                    <emph.end type="italics"/>
                  ſemper ſimilis arcui
                    <emph type="italics"/>
                  AB.
                    <emph.end type="italics"/>
                    <lb/>
                  Et punctis
                    <emph type="italics"/>
                  A, B
                    <emph.end type="italics"/>
                  coeuntibus, angulus
                    <emph type="italics"/>
                  dAb,
                    <emph.end type="italics"/>
                  per Lemma ſuperius,
                    <lb/>
                  evaneſcet; adeoque rectæ ſemper ſinitæ
                    <emph type="italics"/>
                  Ab, Ad
                    <emph.end type="italics"/>
                  & arcus interme­
                    <lb/>
                  dius
                    <emph type="italics"/>
                  Ab
                    <emph.end type="italics"/>
                  coincident, & propterea æquales erunt. </s>
                  <s>Unde & hiſce
                    <lb/>
                  ſemper proportionales rectæ
                    <emph type="italics"/>
                  AB, AD,
                    <emph.end type="italics"/>
                  & arcus intermedius
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  </s>
                </p>
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          </chap>
        </body>
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