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-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div579" type="section" level="3" n="19">
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                <p>
                  <s xml:id="echoid-s3715" xml:space="preserve">
                    <pb o="301" rhead="EPISTOL AE" n="313" file="0313" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0313"/>
                  litatis à linea
                    <var>.d.o</var>
                  . </s>
                  <s xml:id="echoid-s3716" xml:space="preserve">Accipiemus igitur
                    <var>.t.i.</var>
                  pro ſinu arcus
                    <var>.t.u.</var>
                  qui eſt graduum .3. hoc
                    <lb/>
                  eſt ſexageſima pars ſemicirculi graduum .180. quapropter
                    <var>.t.i.</var>
                  erit partium .5233. ta-
                    <lb/>
                  lium qualium
                    <var>.o.u.</var>
                  eſt .100000. cuius
                    <var>.t.i.</var>
                  quadratum demptum cum fuerit à quadra-
                    <lb/>
                  to ſemidiametri
                    <var>.o.t.</var>
                  relinquet nobis quadratum ipſius
                    <var>.o.i.</var>
                  quæ quidem
                    <var>.o.i.</var>
                  vt radix
                    <lb/>
                  quadrata, erit partium .99862. talium qualium ſemidiameter eſt .100000. vnde
                    <var>.i.u.</var>
                    <lb/>
                  reſiduum diametri, remanebit partium .138. </s>
                  <s xml:id="echoid-s3717" xml:space="preserve">Vel ſic, cum cognitus ſit nobis arcus
                    <var>.
                      <lb/>
                    t.u.</var>
                  illicò cognoſcemus ſinum arcus
                    <var>.p.t.</var>
                    <reg norm="complementum" type="context">complementũ</reg>
                  vnius quartæ, qui ſinus æ qua-
                    <lb/>
                  lis erit ferè arcui
                    <var>.o.i.</var>
                  partium .99862. vnde
                    <var>.i.u.</var>
                  erit, vt dictum eſt, partium .138. quę
                    <lb/>
                  quidem
                    <var>.i.u.</var>
                  æqualis eſt ferè ſinui arcus
                    <var>.u.e.</var>
                  & ita etiam
                    <var>.u.e.</var>
                  </s>
                  <s xml:id="echoid-s3718" xml:space="preserve">quare ſi diuiſa fuerit to-
                    <lb/>
                  ta
                    <var>.a.u.</var>
                    <reg norm="partium" type="context">partiũ</reg>
                  .200000. per .138. proueniet nobis .1449. & ſic angulus
                    <var>.t.d.u.</var>
                  erit vna
                    <lb/>
                  partium .1449. anguli
                    <var>.a.d.u</var>
                  . </s>
                  <s xml:id="echoid-s3719" xml:space="preserve">Confideremus igitur quomodo fieri poteſt, vt oculo
                    <lb/>
                  compræhendatur hæc tam parua particula luminis lunaris.</s>
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            <div xml:id="echoid-div584" type="section" level="3" n="20">
              <div xml:id="echoid-div584" type="letter" level="4" n="1">
                <head xml:id="echoid-head448" xml:space="preserve">SOLVTIONES ALIQVAE.</head>
                <head xml:id="echoid-head449" style="it" xml:space="preserve">Paulo Aemilio Raifestaim.</head>
                <p>
                  <s xml:id="echoid-s3720" xml:space="preserve">Poſt eas literas quas proximè ad te dedi, Franciſcus Monardus mihi retulit tuas
                    <lb/>
                  nonnullas dubitationes circa noſtrum Theorema Arithmeticum .116. quarum
                    <lb/>
                  prima eſt, quod ſi numerus
                    <var>.a.</var>
                    <reg norm="cogitatus" type="simple">cogitatꝰ</reg>
                  , eſſet æqualis .4. </s>
                  <s xml:id="echoid-s3721" xml:space="preserve">tunc ipſe non eſſet multiplex
                    <lb/>
                  ipſi .4. de quo tamen nullam mentionem feci. </s>
                  <s xml:id="echoid-s3722" xml:space="preserve">Idem etiam inquis, ſi
                    <var>.a.</var>
                  fuiſſet .5. 6. 7.
                    <lb/>
                  nec non .1. 2. et .3. </s>
                  <s xml:id="echoid-s3723" xml:space="preserve">Cui reſpondi, quod
                    <reg norm="quanuis" type="context">quãuis</reg>
                  nullam fecerim mentionem de æqua
                    <lb/>
                  litate ipſius
                    <var>.a.</var>
                  cum .4. nihiltamen refert, </s>
                  <s xml:id="echoid-s3724" xml:space="preserve">propterea quod quando ita fuiſſet, nihi-
                    <lb/>
                  lominus eaſdem conditiones ſubiret,
                    <reg norm="quemadodum" type="context">quemadodũ</reg>
                  ſi fuiſſet duplus, triplus, aut qua
                    <lb/>
                  druplus. eo quod à genere multiplici, æqualitas, formam diuerſam non induat. </s>
                  <s xml:id="echoid-s3725" xml:space="preserve">Qua
                    <lb/>
                  re idem eueniet ſi
                    <var>.a.</var>
                  fuerit .4. 5. 6. 7. vt ſi eſſet .8. 9. 10. et .11. & ſic de cæteris, excepto
                    <lb/>
                  quod in proprijs multiplicibus, vel in ſuperantibus ipſis multiplicibus
                    <var>.a.</var>
                  menſurare
                    <lb/>
                  tur ab ipſo .4. plus quam ſemel. </s>
                  <s xml:id="echoid-s3726" xml:space="preserve">Quod autem dicis. de .1. 2. et .3. nihil eſt, quia,
                    <lb/>
                  vt in ſecunda ſumma, hoc eſt in tertio termino maximo, reliquus tertius terminus,
                    <lb/>
                  ideſt .9. non compræhendetur, ita nobis indicabit primum numerum ſumptum mi
                    <lb/>
                  norem eſſe quaternario. </s>
                  <s xml:id="echoid-s3727" xml:space="preserve">Quæ omnia, exipſa noſtra thęoria ibidem expreſſa ma-
                    <lb/>
                  nifeſtantur. </s>
                  <s xml:id="echoid-s3728" xml:space="preserve">Quid autem circa hoc Frater Lucas dicat, neſcio, quia ipſius opus
                    <lb/>
                  ad manus meas nunquam peruenit, ſatis enim mihi fuit, in Tartalea hanc praxim
                    <lb/>
                  vidiſſe, ratio vero nullibi à me reperta fuit. </s>
                  <s xml:id="echoid-s3729" xml:space="preserve">Tartalea enim multos citat authores,
                    <lb/>
                  quorum ſcripta ego nunquam vidi, vt Leonardi Piſani, Proſdocimi, Petri Borghi,
                    <lb/>
                  Fratris Lucæ, Ioannis Sfortunati,
                    <reg norm="cæterorumque" type="simple">cæterorumq́;</reg>
                  ſimilium.</s>
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