Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/253.jpg" pagenum="225"/>
                  dignitatum A
                    <emph type="sup"/>
                  3
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                  , A
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                  3
                    <emph.end type="sup"/>
                  , A
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                  4
                    <emph.end type="sup"/>
                  , A
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                  1/2
                    <emph.end type="sup"/>
                  , A
                    <emph type="sup"/>
                  1/3
                    <emph.end type="sup"/>
                  , A
                    <emph type="sup"/>
                  1/3
                    <emph.end type="sup"/>
                  , A
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                  2/3
                    <emph.end type="sup"/>
                  , A
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                  -1
                    <emph.end type="sup"/>
                  , A
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                  -2
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                  , & A
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                  -1/2
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                  momenta </s>
                </p>
                <p type="main">
                  <s>
                    <arrow.to.target n="note201"/>
                  2
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A, 3
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  , 4
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  , 1/2
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  -1/2
                    <emph.end type="sup"/>
                  , 3/2
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  1/2
                    <emph.end type="sup"/>
                  , 1/3
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  -2/3
                    <emph.end type="sup"/>
                  , 2/3
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  -1/3
                    <emph.end type="sup"/>
                  , -
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  -2
                    <emph.end type="sup"/>
                  ,
                    <lb/>
                  -2
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  -3
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                  , & -1/2
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  -1/2
                    <emph.end type="sup"/>
                  reſpective. </s>
                  <s>Et generaliter, ut dignitatis
                    <lb/>
                  cujuſcunque A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n/m
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  momentum fuerit
                    <emph type="italics"/>
                  n/m a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  (
                    <emph type="italics"/>
                  n-m/m
                    <emph.end type="italics"/>
                  )
                    <emph.end type="sup"/>
                  . </s>
                  <s>Item ut Genitæ
                    <lb/>
                  A
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  B momentum fuerit 2
                    <emph type="italics"/>
                  a
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                  AB+
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                  b
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  ; & Genitæ A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  B
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                  4
                    <emph.end type="sup"/>
                  C
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  momen­
                    <lb/>
                  tum 3
                    <emph type="italics"/>
                  a
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                  A
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  B
                    <emph type="sup"/>
                  4
                    <emph.end type="sup"/>
                  C
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  +4
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  B
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  C
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  +2
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  B
                    <emph type="sup"/>
                  4
                    <emph.end type="sup"/>
                  C; & Genitæ (A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  /B
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  ) ſi­
                    <lb/>
                  ve A
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                  3
                    <emph.end type="sup"/>
                  B
                    <emph type="sup"/>
                  -2
                    <emph.end type="sup"/>
                  momentum 3
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                  a
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                  A
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  B
                    <emph type="sup"/>
                  -2
                    <emph.end type="sup"/>
                  -2
                    <emph type="italics"/>
                  b
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                  A
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                  3
                    <emph.end type="sup"/>
                  B
                    <emph type="sup"/>
                  -3
                    <emph.end type="sup"/>
                  : & ſic in cæteris. </s>
                  <s>
                    <lb/>
                  Demonſtratur vero Lemma in hunc modum. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note201"/>
                  LIBER
                    <lb/>
                  SECUNDUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  1. Rectangulum quodvis motu perpetuo auctum AB,
                    <lb/>
                  ubi de lateribus A & B deerant momentorum dimidia 1/2
                    <emph type="italics"/>
                  a
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                  & 1/2
                    <emph type="italics"/>
                  b,
                    <emph.end type="italics"/>
                    <lb/>
                  fuit A-1/2
                    <emph type="italics"/>
                  a
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                  in B-1/2
                    <emph type="italics"/>
                  b,
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                  ſeu AB-1/2
                    <emph type="italics"/>
                  a
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                  B-1/2
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  A+1/4
                    <emph type="italics"/>
                  ab
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                  ; & quam pri­
                    <lb/>
                  mum latera A & B alteris momentorum dimidiis aucta ſunt, eva­
                    <lb/>
                  dit A+1/2
                    <emph type="italics"/>
                  a
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                  in B+1/2
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                  b
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                  ſeu AB+1/2
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                  a
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                  B+1/2
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  A+1/4
                    <emph type="italics"/>
                  ab.
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                  De hoc rectan­
                    <lb/>
                  gulo ſubducatur rectangulum prius, & manebit exceſſus
                    <emph type="italics"/>
                  a
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                  B+
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  A. </s>
                  <s>
                    <lb/>
                  Igitur laterum incrementis totis
                    <emph type="italics"/>
                  a
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                  &
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                  b
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                  generatur rectanguli incre­
                    <lb/>
                  mentum
                    <emph type="italics"/>
                  a
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                  B+
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                  b
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                  A.
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  2. Ponatur AB ſemper æquale G, & contenti ABC ſeu
                    <lb/>
                  GC momentum (per Cas. </s>
                  <s>1.) erit
                    <emph type="italics"/>
                  g
                    <emph.end type="italics"/>
                  C+
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                  c
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                  G, id eſt (ſi pro G &
                    <emph type="italics"/>
                  g
                    <emph.end type="italics"/>
                    <lb/>
                  ſcribantur AB &
                    <emph type="italics"/>
                  a
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                  B+
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                  b
                    <emph.end type="italics"/>
                  A)
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  BC+
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  AC+
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  AB. </s>
                  <s>Et par eſt ra­
                    <lb/>
                  tio contenti ſub lateribus quotcunque.
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  3. Ponantur latera A, B, C ſibi mutuo ſemper æqualia; &
                    <lb/>
                  ipſius A
                    <emph type="sup"/>
                  2
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                  , id eſt rectanguli AB, momentum
                    <emph type="italics"/>
                  a
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                  B+
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                  b
                    <emph.end type="italics"/>
                  A erit 2
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A, ip­
                    <lb/>
                  ſius autem A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  , id eſt contenti ABC, momentum
                    <emph type="italics"/>
                  a
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                  BC+
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                  b
                    <emph.end type="italics"/>
                  AC
                    <lb/>
                  +
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  AB erit 3
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  . </s>
                  <s>Et eodem argumento momentum dignitatis
                    <lb/>
                  cujuſcunque A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  eſt
                    <emph type="italics"/>
                  na
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -1.
                    <emph.end type="sup"/>
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
                    <emph.end type="italics"/>
                  4. Unde cum 1/A in A ſit 1, momentum ipſius 1/A ductum
                    <lb/>
                  in A, una cum 1/A ducto in
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  erit momentum ipſius 1, id eſt, NI­
                    <lb/>
                  hil. </s>
                  <s>Proinde momentum ipſius 1/A ſeu ipſius A
                    <emph type="sup"/>
                  -1
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                  eſt (-
                    <emph type="italics"/>
                  a
                    <emph.end type="italics"/>
                  /A
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  ). Et ge­
                    <lb/>
                  neraliter cum (1/A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  ) in A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  ſit 1, momentum ipſius (1/A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  ) ductum in A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  </s>
                </p>
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