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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                <p>
                  <s xml:id="echoid-s5071" xml:space="preserve">
                    <pb o="419" rhead="EPISTOLAE." n="431" file="0431" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0431"/>
                  gi, methodo etiam qua vtebar dum in iſtisrebus me aliquo modo exercebam.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5072" xml:space="preserve">Quotieſcunque igitur ſcire volueris quantitatem corpulentiæ
                    <reg norm="cuiuſque" type="simple">cuiuſq;</reg>
                    <reg norm="quinque" type="simple">quinq;</reg>
                  cor-
                    <lb/>
                  porum regularium ab vna
                    <reg norm="eademque" type="simple">eademq́;</reg>
                  ſphæra terminatorum ſeu
                    <reg norm="circunſcriptibilium" type="context">circunſcriptibiliũ</reg>
                  cu-
                    <lb/>
                  rabis primum, cognoſcere quantitatem lateris
                    <reg norm="cuiusque" type="simple">cuiusq́;</reg>
                  eorum, talium partium, qua-
                    <lb/>
                  lium ſemidiameter dictæ ſphæræ ſit .100000. extabulis ſinuum Nicolai Copernici.
                    <lb/>
                  </s>
                  <s xml:id="echoid-s5073" xml:space="preserve">Propone igitur tibiante oculos figuram ſemicircularem vltimæ propoſitionis .13.
                    <lb/>
                  lib. Eucli. & inuenies
                    <var>.c.d.</var>
                  tertiam partem ſemidiametri
                    <var>.d.b.</var>
                  eſſe partium .33333. æ-
                    <lb/>
                  qualem ſinui arcus
                    <var>.f.e.</var>
                  graduum .19. mi .28. qui quidem arcus
                    <reg norm="demptus" type="context">dẽptus</reg>
                    <reg norm="cum" type="context">cũ</reg>
                  fuerit à tota
                    <lb/>
                  quarta
                    <var>.b.f.</var>
                  remanebitarcus
                    <var>.e.b.</var>
                  gra .70. mi .32. cuius corda erit latus exaedri, quod
                    <unsure/>
                    <lb/>
                  latus ita cognoſces, ſumendo ſcilicet ſinum medietatis
                    <var>.b.e.</var>
                  hoc eſt ſinum gra .35. mi
                    <num value="16">.
                      <lb/>
                    16.</num>
                  qui erit partium .57738. cuius duplum erit partium .115476. pro latere cubi.</s>
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                <p>
                  <s xml:id="echoid-s5074" xml:space="preserve">Dempto poſtea quadrato lateris exaedri, & quadrato totius diametri
                    <var>.a.b.</var>
                  reſi-
                    <lb/>
                  dui radix quadrata, erit
                    <var>.a.e.</var>
                  latus Tetraedri. </s>
                  <s xml:id="echoid-s5075" xml:space="preserve">Vel ſi duplicaueris ſinum dimidij ar-
                    <lb/>
                  cus
                    <var>.a.e.</var>
                  qui quidem arcus, componitur ex quarta
                    <var>.a.f.</var>
                  & ex arcu
                    <var>.f.e.</var>
                  iam inuento, ſiue,
                    <lb/>
                  vt reſiduus totius dimidij circuli, dempto
                    <var>.b.e.</var>
                  iam ſupra inuento, habebimus idem
                    <lb/>
                  latus
                    <var>.a.e.</var>
                  partium .163294.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5076" xml:space="preserve">Pro latere verò Octaedri accipere potes radicem quadratam dupli quadrati ip-
                    <lb/>
                  ſius
                    <var>.d.b.</var>
                  & habebis
                    <var>.f.b.</var>
                  latus quæſitum. </s>
                  <s xml:id="echoid-s5077" xml:space="preserve">Vel ſi malis accipe duplum ſinus medietatis
                    <lb/>
                  arcus
                    <var>.b.f.</var>
                  quod duplum erit
                    <var>.f.b.</var>
                  partium .14142.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5078" xml:space="preserve">Pro latere verò Duodecaedri, diuide latus Exaedri ex methodo .11. ſecundi
                    <lb/>
                  Eucli. cuius maior pars erit latus quæſitum, partium .71368.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5079" xml:space="preserve">Sed pro latere Icoſaedri, te primum oportebit inuenire quantitatem anguli
                    <var>g.d.
                      <lb/>
                    a.</var>
                  hoc eſt ipſius arcus
                    <var>.b.n.</var>
                  qui tali angulo ſubiacet, quod cum pluribus modis inue-
                    <lb/>
                  niri poſſit, nihilominus, hunc ſeruabis, inuenies primò quantitatem
                    <var>.d.g.</var>
                  quæ eſt ra
                    <lb/>
                  dix quadrata ſummæ duorum quadratorum hoc eſt
                    <var>.d.a.</var>
                  et
                    <var>.a.g.</var>
                  quæ
                    <var>.a.g.</var>
                  æqualis eſt
                    <lb/>
                  diametro
                    <var>.a.b.</var>
                  vt ſcis, dices poſtea, ſi
                    <var>.d.g.</var>
                  correſpondet ipſi
                    <var>.g.a.</var>
                  cui correſpondet
                    <var>.d.
                      <lb/>
                    h.</var>
                  ſemidiametro ſphæræ? </s>
                  <s xml:id="echoid-s5080" xml:space="preserve">tibi veniet
                    <var>.h.k.</var>
                  ſinus arcus
                    <var>.a.h.</var>
                  hoc eſt
                    <var>.b.n.</var>
                  graduum .63 -
                    <lb/>
                  min .26. cuius medietas gra .31. mi .43. pro ſinu ſuo habet partes .52571. cuius ſinus du
                    <lb/>
                  plum eſt partium .105142. pro latere Icoſaedri.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5081" xml:space="preserve">Incipiendo nunc à Tetraedro, ſcire debes, quod pars
                    <var>.a.c.</var>
                  totius diametri
                    <var>.a.b.</var>
                  æ-
                    <lb/>
                  qualis eſt axi ipſius Tetraedri, quæ quidem
                    <var>.a.c.</var>
                  vt ſubſeſquialtera ipſius
                    <var>.a.b.</var>
                  erit par
                    <lb/>
                  tium .13333.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5082" xml:space="preserve">Quæres poſtea quantitatem ſuperficialem vnius faciei ipſius Tetraedri, hac me-
                    <lb/>
                  thodo, inueniendo primum radicem quadratam trium quartarum quadrati
                    <lb/>
                  ipſius
                    <var>.a.e.</var>
                  lateris Tetraedri, eo quod latus hoc, ſeſquitertium in potentia eſt ipſi per­
                    <lb/>
                  pendiculari terminatę ab vno angulorum trianguli æquilateris & à latere ei oppoſi-
                    <lb/>
                  to ex .11. tertijdecimi ipſius Eucli. quę quidem perpendicularis, erit
                    <reg norm="partium" type="context">partiũ</reg>
                  .141416.
                    <lb/>
                  & hæc multiplicata cum medietate lateris trianguli, hoc eſt cum .81647. tibi dabit
                    <lb/>
                  ſuperficiem quæſitam, hoc eſt baſim Tetraedri
                    <reg norm="partium" type="context">partiũ</reg>
                    <reg norm="ſuperficialium" type="context">ſuperficialiũ</reg>
                  .11546192152.
                    <lb/>
                    <reg norm="Hanc" type="context">Hãc</reg>
                  demum baſim multiplicando cum tertia parte axis Tetraedri habebis corpu-
                    <lb/>
                  lentiam totius Tetraedri, quæ erit .513158964003488.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5083" xml:space="preserve">Neque tibi hoc loco occultare volo quandam meam animaduerſionem, quæ eſt,
                    <lb/>
                  quod diameter ſeu perpendicularis (ſupradicta) faciei ipſius Tetraedri, ſemper æ-
                    <lb/>
                  qualis eſt lateri ipſius Octaedri circunſcriptibilis ab eadem ſphæra, hoc eſt ipſi
                    <var>.b.f.</var>
                    <lb/>
                  quapropter quotieſcunque ipſam perpendicularem habere voluerimus accipiendo
                    <lb/>
                    <var>b.f.</var>
                  habebimus intentum. </s>
                  <s xml:id="echoid-s5084" xml:space="preserve">Et quod hoc verum ſit poſſumus ita demonſtrare.</s>
                </p>
                <p>
                  <s xml:id="echoid-s5085" xml:space="preserve">Primum, notum nobis eſt, ipſam perpendicularem, triplam eſſe eius parti, quæ </s>
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