Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <pb xlink:href="039/01/060.jpg" pagenum="32"/>
                <p type="main">
                  <s>
                    <arrow.to.target n="note15"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note15"/>
                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  5. Et quoniam
                    <emph type="italics"/>
                  DB, db
                    <emph.end type="italics"/>
                  ſunt ultimo parallelæ & in dupli­
                    <lb/>
                  cata ratione ipſarum
                    <emph type="italics"/>
                  AD, Ad:
                    <emph.end type="italics"/>
                  erunt areæ ultimæ curvilineæ
                    <emph type="italics"/>
                  ADB,
                    <lb/>
                  Adb
                    <emph.end type="italics"/>
                  (ex natura Parabolæ) duæ tertiæ partes triangulorum rectili­
                    <lb/>
                  neorum
                    <emph type="italics"/>
                  ADB, Adb
                    <emph.end type="italics"/>
                  ; & ſegmenta
                    <emph type="italics"/>
                  AB, Ab
                    <emph.end type="italics"/>
                  partes tertiæ eo­
                    <lb/>
                  rundem triangulorum. </s>
                  <s>Et inde hæ areæ & hæc ſegmenta erunt in
                    <lb/>
                  triplicata ratione tum tangentium
                    <emph type="italics"/>
                  AD, Ad
                    <emph.end type="italics"/>
                  ; tum chordarum &
                    <lb/>
                  arcuum
                    <emph type="italics"/>
                  AB, Ab.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Scholium.
                    <emph.end type="italics"/>
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>Cæterum in his omnibus ſupponimus angulum contactus nec in­
                    <lb/>
                  finite majorem eſſe angulis contactuum, quos Circuli continent cum
                    <lb/>
                  tangentibus ſuis, nec iiſdem infinite minorem; hoc eſt, curvaturam
                    <lb/>
                  ad punctum
                    <emph type="italics"/>
                  A,
                    <emph.end type="italics"/>
                  nec infinite parvam eſſe nec infinite magnam, ſeu
                    <lb/>
                  intervallum
                    <emph type="italics"/>
                  AJ
                    <emph.end type="italics"/>
                  finitæ eſſe magnitudinis. </s>
                  <s>Capi enim poteſt
                    <emph type="italics"/>
                  DB
                    <emph.end type="italics"/>
                    <lb/>
                  ut
                    <emph type="italics"/>
                  AD
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  :
                    <emph.end type="italics"/>
                  quo in caſu Circulus nullus per punctum
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  inter tangen­
                    <lb/>
                  tem
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  & curvam
                    <emph type="italics"/>
                  AB
                    <emph.end type="italics"/>
                  duci poteſt, proindeque angulus contactus
                    <lb/>
                  erit infinite minor Circularibus. </s>
                  <s>Et ſimili argumento ſi fiat
                    <emph type="italics"/>
                  DB
                    <emph.end type="italics"/>
                    <lb/>
                  ſucceſſive ut
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  4
                    <emph.end type="sup"/>
                  ,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  5
                    <emph.end type="sup"/>
                  ,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  6
                    <emph.end type="sup"/>
                  ,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  7
                    <emph.end type="sup"/>
                  , &c. </s>
                  <s>habebitur ſeries an­
                    <lb/>
                  gulorum contactus pergens in infinitum, quorum quilibet poſte­
                    <lb/>
                  rior eſt infinite minor priore. </s>
                  <s>Et ſi fiat
                    <emph type="italics"/>
                  DB
                    <emph.end type="italics"/>
                  ſucceſſive ut
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  ,
                    <lb/>
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  3/2,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  4/3,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  5/4,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  6/5,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  7/6, &c. </s>
                  <s>habebitur alia ſeries infinita
                    <lb/>
                  angulorum contactus, quorum primus eſt ejuſdem generis cum Cir­
                    <lb/>
                  cularibus, ſecundus infinite major, & quilibet poſterior infinite ma­
                    <lb/>
                  jor priore. </s>
                  <s>Sed & inter duos quoſvis ex his angulis poteſt ſeries
                    <lb/>
                  utrinQ.E.I. infinitum pergens angulorum intermediorum inſeri,
                    <lb/>
                  quorum quilibet poſterior erit infinite major minorve priore. </s>
                  <s>Ut
                    <lb/>
                  ſi inter terminos
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  &
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  inſeratur ſeries
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  (13/6),
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  (1
                    <gap/>
                  /5),
                    <lb/>
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  9/4,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  7/3,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  5/2,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  8/3,
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  (11/4),
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  (14/5),
                    <emph type="italics"/>
                  AD
                    <emph.end type="italics"/>
                  (17/6), &c. </s>
                  <s>Et rur­
                    <lb/>
                  ſus inter binos quoſvis angulos hujus ſeriei inſeri poteſt ſeries no­
                    <lb/>
                  va angulorum intermediorum ab invicem infinitis intervallis diffe­
                    <lb/>
                  rentium. </s>
                  <s>Neque novit natura limitem. </s>
                </p>
                <p type="main">
                  <s>Quæ de curvis lineis deque ſuperficiebus comprehenſis demon­
                    <lb/>
                  ſtrata ſunt, facile applicantur ad ſolidorum ſuperficies curvas &
                    <lb/>
                  contenta. </s>
                  <s>Præmiſi vero hæc Lemmata, ut effugerem tædium dedu­
                    <lb/>
                  cendi perplexas demonſtrationes, more veterum Geometrarum, ad
                    <lb/>
                  abſurdum. </s>
                  <s>Contractiores enim redduntur demonſtrationes per me­
                    <lb/>
                  thodum Indiviſibilium. </s>
                  <s>Sed quoniam durior eſt Indiviſibilium hy­
                    <lb/>
                  potheſis, & propterea methodus illa minus Geometrica cenſetur;
                    <lb/>
                  malui demonſtrationes rerum ſequentium ad ultimas quantitatum </s>
                </p>
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