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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IO. BABPT. BENED.
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            <div xml:id="echoid-div586" type="section" level="3" n="21">
              <div xml:id="echoid-div586" type="letter" level="4" n="1">
                <head xml:id="echoid-head450" xml:space="preserve">ELIPSIM PROPOSITAM QVALITER</head>
                <head xml:id="echoid-head451" xml:space="preserve">quadrare valeamus.</head>
                <head xml:id="echoid-head452" style="it" xml:space="preserve">Illuſtri Uiro Franciſco Mendo Zzæ</head>
                <p>
                  <s xml:id="echoid-s3730" xml:space="preserve">QVod antea tuo nomine fecerat Marcus Antonius amicus noſter ſufficie-
                    <lb/>
                  bat. </s>
                  <s xml:id="echoid-s3731" xml:space="preserve">Sed quia, quæ nunc à me petis, talia ſunt, vt ſine tripartita
                    <reg norm="aequa- liter" type="simple">ęqua-
                      <lb/>
                    liter</reg>
                  aliqua data proportione non poſſit aliquis exactè intentum perfice-
                    <lb/>
                  re, nihilominus, ſuppoſita di
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0314-01a" xlink:href="fig-0314-01"/>
                  cta diuiſione, reliqua facilia
                    <reg norm="erunt" type="context">erũt</reg>
                  . </s>
                  <s xml:id="echoid-s3732" xml:space="preserve">
                    <reg norm="Primum" type="context">Primũ</reg>
                    <lb/>
                  enim eſt. </s>
                  <s xml:id="echoid-s3733" xml:space="preserve">Propoſitam Ellipſim qua-
                    <lb/>
                  drare.</s>
                </p>
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                  <figure xlink:label="fig-0314-01" xlink:href="fig-0314-01a">
                    <image file="0314-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0314-01"/>
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                <p>
                  <s xml:id="echoid-s3734" xml:space="preserve">Sit
                    <reg norm="igitur" type="simple">igit̃</reg>
                  Ellipſis propoſita
                    <var>.a.b.d.c.</var>
                  cu-
                    <lb/>
                  ius axes ſint
                    <var>.a.b.</var>
                  et
                    <var>.d.c.</var>
                  dati, ſeu
                    <reg norm="reperti" type="simple">reꝑti</reg>
                  ex
                    <lb/>
                  47.
                    <reg norm="ſecundi" type="context">ſecũdi</reg>
                  Pergei,
                    <reg norm="ſintque" type="simple">ſintq́;</reg>
                  duo circuli
                    <var>.a.e.
                      <lb/>
                    b.f.</var>
                  et
                    <var>.g.d.h.c.</var>
                  circa eaſdem diametros,
                    <lb/>
                    <reg norm="tunc" type="context">tũc</reg>
                  proportio
                    <var>.a.b.</var>
                  ad
                    <var>.d.c.</var>
                    <reg norm="dimidium" type="context">dimidiũ</reg>
                  erit
                    <lb/>
                  proportionis circulorum ex .2. 12. Eu-
                    <lb/>
                  clid. </s>
                  <s xml:id="echoid-s3735" xml:space="preserve">ſed proportio
                    <var>.a.b.</var>
                  ad
                    <var>.d.c.</var>
                  æqualis
                    <lb/>
                  eſt proportioni maioris circuli ad Elli
                    <lb/>
                  pſim .ex .5. Archimedis in lib. de cono­
                    <lb/>
                  idalibus, quapropter proportio Elli-
                    <lb/>
                  pſis ad minorem circulum altera me-
                    <lb/>
                  dietas erit totius proportionis circulo-
                    <lb/>
                  rum, hoc eſt maioris ad minorem, qua
                    <lb/>
                  re Ellipſis media proportionalis erit
                    <lb/>
                  inter eos circulos. </s>
                  <s xml:id="echoid-s3736" xml:space="preserve">Nunc verò cum
                    <lb/>
                  ex Archimede repertę fuerint duæ fi-
                    <lb/>
                  guræ rectilineæ æquales duobus circu
                    <lb/>
                  lis iam dictis, & inter has, reperta fue
                    <lb/>
                  rit alia media proportionalis propoſi-
                    <lb/>
                  tum obtinebimus.</s>
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              <div xml:id="echoid-div588" type="letter" level="4" n="2">
                <head xml:id="echoid-head453" style="it" xml:space="preserve">Spheroidem propoſitam cubare.</head>
                <head xml:id="echoid-head454" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s3737" xml:space="preserve">PRopoſita ſphæroides erit, aut prolata, aut oblonga, ſit prius prolata,
                    <reg norm="ſitque" type="simple">ſitq́;</reg>
                    <var>.a.b.</var>
                    <lb/>
                  diameter circuli, qui eam per æqualia ſecat, circa quam
                    <var>.a.b.</var>
                  vt circa axem in-
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                  telligatur ſphæroides oblonga, cuius ſpiſſitudo ſit
                    <var>.d.c.</var>
                  axis prolatæ, cogitemus
                    <reg norm="nunc" type="context">nũc</reg>
                    <lb/>
                  duas ſphæras
                    <var>.a.e.b.f.</var>
                  et
                    <var>.g.d.h.c.</var>
                  circa dictos axes. </s>
                  <s xml:id="echoid-s3738" xml:space="preserve">Vnde quatuor corpora habebi-
                    <lb/>
                  mus, hoc eſt duas ſphæras, & duas ſphæroides, quas probabo continuas proportio-
                    <lb/>
                  nales inuicem eſſe.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3739" xml:space="preserve">Conſideremus igitur duos conos rectos, quorum
                    <var>.a.b.</var>
                  diameter ſit eorum baſium,
                    <lb/>
                  altitudo autem maioris, æqualis ſit ſemidiametro majori, hoc eſt medietati
                    <var>.a.b.</var>
                  al- </s>
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