Benedetti, Giovanni Battista de, Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]
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EPISTOLAE.
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                <p>
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                    <pb o="251" rhead="EPISTOLAE." n="263" file="0263" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0263"/>
                  terius differentiæ quam ſupra inuenerimus.</s>
                </p>
                <div xml:id="echoid-div501" type="float" level="5" n="1">
                  <figure xlink:label="fig-0262-01" xlink:href="fig-0262-01a">
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                  <figure xlink:label="fig-0262-02" xlink:href="fig-0262-02a">
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                <p>
                  <s xml:id="echoid-s3164" xml:space="preserve">Superius enim dixinon eſſe ponendum polum in
                    <var>.B.</var>
                  eo quod
                    <var>.B.C.</var>
                  ſit gra .89. mi
                    <num value="30">.
                      <lb/>
                    30.</num>
                  vnde nobis prodijſſet triangulus
                    <var>.f.C.D.</var>
                  trium valde paruorum laterum, quorum
                    <lb/>
                  latus
                    <var>.C.D.</var>
                  eſſet gra
                    <var>.o.</var>
                  mi .30. & latus
                    <var>.f.l.</var>
                  gra
                    <var>.o.</var>
                  mi .55. & latus
                    <var>.F.D.</var>
                  gra
                    <var>.o.</var>
                  mi .47. vn-
                    <lb/>
                  de angulus
                    <var>.f.</var>
                  gra .32. min .40. falſus eſſet, qui
                    <reg norm="quidem" type="context">quidẽ</reg>
                  poſtea nobis daret
                    <var>.D.E.</var>
                  gra .45
                    <lb/>
                  minu .16. falſum ſimiliter.</s>
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              <div xml:id="echoid-div503" type="letter" level="4" n="4">
                <head xml:id="echoid-head378" style="it" xml:space="preserve">De paßione circuli bactenus incognita.</head>
                <head xml:id="echoid-head379" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s3165" xml:space="preserve">DVbitandum quidem
                    <reg norm="non" type="context">nõ</reg>
                  eſt quin paſſiones circuli innumerabiles penè ſint, quę
                    <lb/>
                  quidem omnes ferè caſu inueniuntur, vt mihi nunc accidit, quam tibi mitto,
                    <lb/>
                  hæc autem eſt, quòd quadratum lineæ
                    <var>.a.g.</var>
                  in figura hic ſubſcripta ſemper æquale
                    <lb/>
                  eſt ei producto, quod fit ex
                    <var>.a.e.</var>
                  in diametro circuli
                    <var>.g.c.b.</var>
                  ſimul ſumpto cum quadra
                    <lb/>
                  to inſcriptibili in dicto circulo, & ſimul cum quadrato lineæ
                    <var>.a.b.</var>
                    <reg norm="contingentis" type="context">contingẽtis</reg>
                  ipſum
                    <lb/>
                  circulum, ſupponendo
                    <var>.a.g.</var>
                  per centrum ipſius circuli tranſire.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3166" xml:space="preserve">Pro cuius demonſtratione à centro
                    <var>.e.</var>
                  duco ſemidiametrum
                    <var>.e.c.</var>
                    <reg norm="perpendicularem" type="context">perpendicularẽ</reg>
                    <lb/>
                  ipſi
                    <var>.g.a.</var>
                  & à puncto
                    <var>.c.</var>
                  ad
                    <var>.a.</var>
                  duco
                    <var>.c.a.</var>
                  quæ ſecabit circunferentiam ipſius circuli in
                    <reg norm="pum" type="context">pũ</reg>
                    <lb/>
                  cto
                    <var>.d.</var>
                  eo, quod angulus
                    <var>.c.</var>
                  acutus eſt. </s>
                  <s xml:id="echoid-s3167" xml:space="preserve">Nunc ex .35. tertij, productum
                    <var>.c.a.</var>
                  in
                    <var>.a.d.</var>
                  æqua
                    <lb/>
                  le eſt quadrato
                    <var>.a.b.</var>
                  productum autem
                    <var>.a.c.</var>
                  in
                    <var>.d.c.</var>
                  æquale eſt quadrato inſcriptibili in
                    <lb/>
                  circulo
                    <var>.g.c.b.</var>
                  ex .130. primi Vitellionis,
                    <reg norm="in" type="wordlist">ĩ</reg>
                  qua propoſitione ipſe Vitellio ſupplet pro
                    <lb/>
                  eo, quod in quinta propoſitione libri de lineis ſpirabilibus Archimedis deſideratur,
                    <lb/>
                  ſed quadratum
                    <var>.a.c.</var>
                  æquale eſt ijs duobus productis. per .2. ſecundi Eucli. ergo qua-
                    <lb/>
                  dratum
                    <var>.a.c.</var>
                  æquale erit quadrato inſcriptibili in circulo
                    <var>.d.c.g.</var>
                  & quadrato
                    <var>.a.b.</var>
                  ſed
                    <lb/>
                  quadratum lineæ
                    <var>.a.c.</var>
                  æquale eſt duobus quadratis, hoc eſt lineæ
                    <var>.a.e.</var>
                  & lineæ
                    <var>.e.c.</var>
                  ex
                    <lb/>
                  pitagorica, </s>
                  <s xml:id="echoid-s3168" xml:space="preserve">quare ex communi conceptu duo quadrata lineæ
                    <var>.a.e.</var>
                  & lineę
                    <var>.e.c.</var>
                  hoc eſt
                    <lb/>
                  lineæ
                    <var>.e.g.</var>
                  quod idem eſt, æqualia erunt duobus iam dictis, hoc eſt inſcriptibili,
                    <lb/>
                  & ei, quod fit ex
                    <var>.a.b.</var>
                  ſed quadratum lineæ
                    <var>.a.g.</var>
                  æquale eſt quadrato lineæ
                    <var>.a.e.</var>
                  & qua
                    <lb/>
                  drato quod fit ex
                    <var>.e.g.</var>
                  & duplo illius quod fit ex
                    <var>.a.e.</var>
                  in
                    <var>.e.g.</var>
                  hoc eſt producto
                    <var>.a.e.</var>
                  in
                    <lb/>
                  diametrum. </s>
                  <s xml:id="echoid-s3169" xml:space="preserve">Quare quadratum lineæ
                    <var>.a.g.</var>
                  æquale eſt quadrato circunſcriptibili, &
                    <lb/>
                  quadrato lineæ
                    <var>.a.b.</var>
                  & producto lineæ
                    <var>.a.e.</var>
                  in diametrum circuli
                    <var>.d.c.g</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3170" xml:space="preserve">Breuiori etiam methodo demonſtrare poſſu
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0263-01a" xlink:href="fig-0263-01"/>
                  mus quadrata lineæ
                    <var>.a.e.</var>
                  et
                    <var>.e.g.</var>
                  æqualia eſ-
                    <lb/>
                  ſe quadrato circunſcriptibili, & quadrato lineæ
                    <var>.
                      <lb/>
                    a.b.</var>
                  ducendo lineam
                    <var>.e.b.</var>
                  quæ æqualis eſt lineæ
                    <var>.
                      <lb/>
                    e.g.</var>
                  tali methodo, hoc eſt, conſiderando, quod
                    <lb/>
                  quadratum inſcriptibile ſemper duplum eſt qua
                    <lb/>
                  drato ſemidiametri, vel medietati circumſcri-
                    <lb/>
                  ptibili, quod quidem nihil aliud eſt, niſi æquale
                    <lb/>
                  eſſe ijs duobus quadratis, hoc eſt lineæ
                    <var>.e.b.</var>
                  & li-
                    <lb/>
                  neæ
                    <var>.e.g.</var>
                  ſed quadratum lineæ
                    <var>.a.e.</var>
                  æquale eſt iis
                    <lb/>
                  duobus quadratis, hoc eſt lineæ
                    <var>.a.b.</var>
                  & lineæ
                    <var>.b.e.</var>
                  vnde quadrat um lineæ
                    <var>.a.e.</var>
                  cum
                    <lb/>
                  quadrato lineæ
                    <var>.e.g.</var>
                  æquale eſt quadrato circunſcriptibili, ſimul collecto cum qua-
                    <lb/>
                  drato lineæ
                    <var>.a.b</var>
                  .</s>
                </p>
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                  <figure xlink:label="fig-0263-01" xlink:href="fig-0263-01a">
                    <image file="0263-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0263-01"/>
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