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. BAPT. BENED.
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                    <pb o="420" rhead="IO. BAPT. BENED." n="432" file="0432" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0432"/>
                  à centro circuli, ipſum triangulum circunſcribentis, terminatur, & à baſi, vt in tertio
                    <lb/>
                  propoſito decimæſeptimæ quartidecimi Eucli. probatur, ex quo ſequitur proportio­
                    <lb/>
                  nem huiuſmodi perpendicularis ad axem Tetraedri, hoc eſt ad
                    <var>.a.c.</var>
                  ſeſquioctauam
                    <lb/>
                  eſſe in potentia, ex penultima primi Eucli. </s>
                  <s xml:id="echoid-s5086" xml:space="preserve">Sed cum
                    <var>.d.c.</var>
                  tertia pars ſit ipſius
                    <var>.d.a.</var>
                  vt
                    <lb/>
                  etiam ex .2. propoſito, ſeu corollario decimæſeptimæ .14. lib. diſcurrere licet, cum ex
                    <lb/>
                  dicto corollario
                    <var>.d.c.</var>
                  ſit ſexta pars ipſius
                    <var>.a.b</var>
                  . </s>
                  <s xml:id="echoid-s5087" xml:space="preserve">Quare
                    <var>.d.c.</var>
                  quarta pars erit ipſius
                    <var>.a.c.</var>
                  vn
                    <lb/>
                  de
                    <var>.a.c.</var>
                  ſeſquitertia erit ipſi
                    <var>.a.d.</var>
                  in longitudine,
                    <reg norm="ideoque" type="simple">ideoq́;</reg>
                  quadratum ipſius
                    <var>.a.d.</var>
                  ad qua-
                    <lb/>
                  dratum ipſius
                    <var>.a.c.</var>
                  erit vt .9. ad .16: </s>
                  <s xml:id="echoid-s5088" xml:space="preserve">& ita duplum quadrati ipſius
                    <var>.a.d.</var>
                  hoc eſt quadra-
                    <lb/>
                  tum ipſius
                    <var>.b.f.</var>
                  ad quadratum ipſius
                    <var>.a.c.</var>
                  erit, vt .18. ad .16. hoc eſt ſeſquioctauum, er-
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                  go
                    <var>.b.f.</var>
                  æqualis erit dictæ perpendiculari, ex .9. quinti.</s>
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                  <s xml:id="echoid-s5089" xml:space="preserve">Cubus poſtea ipſius
                    <var>.b.e.</var>
                  erit partium .1539838576570176.</s>
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                <p>
                  <s xml:id="echoid-s5090" xml:space="preserve">Pro Octaedro deinde, accipies productum diametri in ſemidiametrum, quod
                    <lb/>
                  productum, æquale erit quadrato diuidenti per æqualia Octaedron, hocigitur pro-
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                  ductum, multiplicando per .100000. ſemidiametrum ſphæræ, tibi dabit columnam
                    <lb/>
                  quadrilateram cuius tertia pars, erit partium .666666666666666. cuius duplum
                    <lb/>
                  erit ipſum Octaedron partium .1333333333333.</s>
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                  <s xml:id="echoid-s5091" xml:space="preserve">Pro Icoſaedro autem, oportet prius quantitatem perpendicularis inuenire, quæ
                    <lb/>
                  perpendicularis, per æqualia diuidit baſim ipſius Icoſaedri, quæ vt radix quadrata
                    <lb/>
                  trium quartarum quadrati lateris ipſius baſis, erit partium .91055. talium, qualium
                    <lb/>
                  dictum latus erit partium .105142. cuius medietas eſt .52571. quæ medietas ſi mul-
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                  tiplicata fuerit cum dicta perpendiculari, dabit totam baſim ſuperficialem, hoc eſt
                    <lb/>
                  ſuperficiem vnius trianguli æquilateris partium ſuperficialium .4786852405. quo
                    <lb/>
                  facto, accipe quadratum duarum tertiarum ipſius, hic ſupra dictæ perpendicularis,
                    <lb/>
                    <reg norm="ipſumque" type="simple">ipſumq́;</reg>
                  deme ex quadrato ſemidiametri ſphæræ, hoc eſt, ex quadrato
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                  .100000
                    <lb/>
                  radix poſtea quadrata reſidui, erit partium .79468. & hæc erit perpendicularis à cen
                    <lb/>
                  tro ſphærę ad vnam baſim ipſius Icoſaedri, quam volueris, quam perpendicularem
                    <lb/>
                  ſi multiplicaueris cum quantitate ſuperficiali, hic ſuperius reperta, vnius baſis, con-
                    <lb/>
                  ſequeris columnam trilateram partium .380401586920540. cuius tertia pars, erit
                    <lb/>
                  partium .126800528973513. pro vna ex .20. </s>
                  <s xml:id="echoid-s5092" xml:space="preserve">Pyramidibus ipſum corpus compo-
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                  nentibus. </s>
                  <s xml:id="echoid-s5093" xml:space="preserve">Breuius tamen hoc efficiens, ſi multiplicaueris baſim dictam, cum tertia
                    <lb/>
                  parte ipſius perpendicularis, hanc poſtea pyramidem multiplicando per .20. habebis
                    <lb/>
                  totam corpulentiam ipſius Icoſaedri partium .2536010579470260.</s>
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                  <s xml:id="echoid-s5094" xml:space="preserve">Pro Duodecaedro demum, accipe ſinum gra .36. qui
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                  ſunt pro dimidio quin
                    <lb/>
                  tæ partis totius gyri circularis,
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                    <reg norm="quidem" type="context">quidẽ</reg>
                  ſinus, erit partium .58778. cuius quadratum
                    <lb/>
                  ſi
                    <reg norm="dem" type="context">dẽ</reg>
                  pſeris ex quadrato
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                  .100000. ſemidiametri circuli
                    <reg norm="circunſcribentis" type="context">circũſcribentis</reg>
                    <reg norm="aliquem" type="context">aliquẽ</reg>
                    <reg norm="pen- tago" type="context">pẽ-
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                    tago</reg>
                  num æquilaterum, & æquiangulum, </s>
                  <s xml:id="echoid-s5095" xml:space="preserve">tunc radix reſidui, erit perpendicularis du-
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                  cta à centro dicti circuli ad medium vnius lateris ipſius pentagoni, quæ perp endicu
                    <lb/>
                  laris, erit partium .80902. talium qualium medietas lateris dicti fuerit .58778.
                    <lb/>
                  Nunc verò dicendo ſi .58778. dat .80902. quid nobis dabit .35684? </s>
                  <s xml:id="echoid-s5096" xml:space="preserve">medietas lateris
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                  ipſius Duodecaedri, vnde da bit .49116. pro perpendiculari, à centro ipſius penta-
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                  goni, ad latus ipſius Duodecaedri, quæ multiplicata cum me dietate ſupradicta ip-
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                  ſius lat eris, hoc eſt cum .35684. producet vnum ex quinque triangulis componenti-
                    <lb/>
                  bus vn um pentagonum, ſeu vnam baſim ipſius Duodecaedri, quod quidem triangu
                    <lb/>
                  lum, erit partium .1752655344. ſu perficialium, quas ſi per quinque multiplicaueris
                    <lb/>
                  habeb is vnam baſim pentagonam dicti corporis partium .8763276720. </s>
                  <s xml:id="echoid-s5097" xml:space="preserve">Dicendum
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                  poſtea eſt, ſi ad .80901. conuenit ſemidiameter circularis partium .100000. quid
                    <reg norm="con" type="context">cõ</reg>
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                  ueniet partibus .49116. dabit .60711. pro tali ſemidiametro circulari, cuius quadra- </s>
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