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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            <div xml:id="echoid-div518" type="section" level="3" n="8">
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                <pb o="265" rhead="EPISTOLAE." n="277" file="0277" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0277"/>
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              <div xml:id="echoid-div524" type="letter" level="4" n="3">
                <head xml:id="echoid-head396" style="it" xml:space="preserve">De inuentione axis propoſite portionis datæ ſphæræ.</head>
                <head xml:id="echoid-head397" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s3308" xml:space="preserve">VTaxem propoſitæ alicuius datæ ſphæræ inuenire poſſis ita tibi operandum eſt
                    <lb/>
                  vt gratia exempli. </s>
                  <s xml:id="echoid-s3309" xml:space="preserve">Propoſita nobis eſt ſphæra
                    <var>.c.i.e.t.</var>
                  diametri cognitæ. </s>
                  <s xml:id="echoid-s3310" xml:space="preserve">pro
                    <lb/>
                  poſita etiam eſt nobis eius portio
                    <var>.n.e.u.</var>
                  axis
                    <var>.e.a.</var>
                  cognitæ minoris ſemidiametro, da-
                    <lb/>
                  ta etiam nobis eſt proportio alterius portionis minoris hemiſphærio
                    <var>.i.e.t.</var>
                  ad por-
                    <lb/>
                  tionem
                    <var>.n.e.u.</var>
                  quæritur nunc quantus ſit axis
                    <var>.e.x.</var>
                  ſecundæ portionis hoc eſt deſidera-
                    <lb/>
                  mus cognoſcere proportionem
                    <var>.e.x.</var>
                  ad
                    <var>.e.a.</var>
                  vel ad diametrum ipſius ſpheræ.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3311" xml:space="preserve">Cuius gratia reperiatur primò proportio
                    <reg norm="circunferentiæ" type="context">circũferentiæ</reg>
                  maioris circuli ipſius
                    <reg norm="ſphae­ ræ" type="simple">ſphę­
                      <lb/>
                    ræ</reg>
                  adeius diametrum, quæ ferè eſt vt .22. ad .7. ex Archimede.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3312" xml:space="preserve">Quo facto, inueniatur quantitas ſuperficialis huiuſmodi maioris circuli, quæ ſem-
                    <lb/>
                  per æqualis eſt producto quod fit ex ſemidiametro in dimidium circunferentiæ ip-
                    <lb/>
                  fius circuli, ex eodem Archimede. </s>
                  <s xml:id="echoid-s3313" xml:space="preserve">Et ſic cognoſcemus quartam partem ſuperficiei
                    <lb/>
                  ſphæricæ ſphærę propoſite ex .31. primi lib. de ſphæra, & cyllindro Archimedis.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3314" xml:space="preserve">Deinde ſumatur tertia pars producti, quod fit ex ſemidiametro in ſuperficiem
                    <lb/>
                  maioris circuli, & habebimus conum, cuius baſis erit circulus maior, altitudo verò
                    <lb/>
                  ſemidiameter propoſitæ ſphæræ ex .9. duodecimi Eucli.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3315" xml:space="preserve">Quadruplum poſtea huiuſmodi coni, erit quantitas ſoliditatis, ſeu corporeitas to
                    <lb/>
                  tius ſphærę ex .32. dicti lib. Archimedis.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3316" xml:space="preserve">Imaginemur poſtea
                    <reg norm="in" type="wordlist">ĩ</reg>
                  ſphærica portione
                    <var>.n.e.u.</var>
                    <reg norm="lineam" type="context">lineã</reg>
                    <var>.e.u.</var>
                  à
                    <reg norm="summitate" type="context">sũmitate</reg>
                  ad
                    <reg norm="extremitatem" type="context">extremitatẽ</reg>
                    <lb/>
                  baſis, cuius
                    <var>.e.u.</var>
                  quantitatem cognoſcemus, hoc modo ſcilicet, fumendo
                    <reg norm="radicem" type="context">radicẽ</reg>
                  qua-
                    <lb/>
                  dratam producti
                    <var>.c.e.</var>
                  in
                    <var>.e.a.</var>
                  eo quod
                    <lb/>
                  quadratum
                    <var>.e.u.</var>
                  æquale eſt quadrato
                    <lb/>
                    <figure xlink:label="fig-0277-01" xlink:href="fig-0277-01a" number="308">
                      <image file="0277-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0277-01"/>
                    </figure>
                    <var>a.u.</var>
                  & quadrato
                    <var>.a.e.</var>
                  ex penultima
                    <lb/>
                  primi Eucli. </s>
                  <s xml:id="echoid-s3317" xml:space="preserve">hoc eſt producto quod
                    <lb/>
                  fit ex
                    <var>.c.a.</var>
                  in
                    <var>.a.e.</var>
                  ex .34. tertij
                    <reg norm="eiuſdem" type="context">eiuſdẽ</reg>
                  ,
                    <lb/>
                  & quadrato
                    <var>.a.e.</var>
                  hoc eſt producto,
                    <lb/>
                  quod fit ex
                    <var>.c.e.</var>
                  in
                    <var>.e.a.</var>
                  ex .3. ſecundi
                    <lb/>
                  eiuſdem.</s>
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                <p>
                  <s xml:id="echoid-s3318" xml:space="preserve">Inuenta poſtea
                    <var>.e.u.</var>
                  ponamus eam
                    <lb/>
                  vnius circuli ſemidiametrum eſſe, cu
                    <lb/>
                  ius ſuperficialis quantitas etiam inue
                    <lb/>
                  niatur, vt ſupra dictum eſt, quæ qui
                    <lb/>
                    <reg norm="dem" type="context">dẽ</reg>
                  æqualis erit ſuperficiei portionis
                    <lb/>
                    <var>n.e.u.</var>
                  ex .40. primi li. </s>
                  <s xml:id="echoid-s3319" xml:space="preserve">Archimedis de
                    <lb/>
                  ſphæra, & cyllindro.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3320" xml:space="preserve">Hæc autem quantitas vltimo
                    <reg norm="inuem" type="context">inuẽ</reg>
                    <lb/>
                  ta multiplicetur cum tertia parte ſe-
                    <lb/>
                  midiametri datæ ſphæræ, & habebi-
                    <lb/>
                  mus ſoliditatem vnius coni æqualis
                    <lb/>
                  aggregato ſoliditatis portionis
                    <var>.n.e.
                      <lb/>
                    u.</var>
                  ſimul ſumptę
                    <unsure/>
                  ,
                    <reg norm="cum" type="context">cũ</reg>
                  ſoliditate vnius co
                    <lb/>
                  ni, cuius axis ſit
                    <var>.a.o.</var>
                    <reg norm="reſiduum" type="context">reſiduũ</reg>
                  ſemidia-
                    <lb/>
                  metri noſtræ ſphæræ dempta
                    <var>.a.e.</var>
                  ba­ </s>
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