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-div708" type="section" level="3" n="36">
              <div xml:id="echoid-div710" type="letter" level="4" n="2">
                <p>
                  <s xml:id="echoid-s4369" xml:space="preserve">
                    <pb o="366" rhead="IO. BAPT. BENED." n="378" file="0378" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0378"/>
                  proportionalis inter
                    <var>.g.</var>
                  et
                    <var>.h</var>
                  . </s>
                  <s xml:id="echoid-s4370" xml:space="preserve">quare
                    <var>.g.</var>
                  et
                    <var>.h.</var>
                  non erunt minimi in ea proportione, quia
                    <lb/>
                  vnitas diuiſibilis eſſet ſi
                    <var>.g.h.</var>
                  minimi fuiſſent, quod non conceditur, ſint igitur mini
                    <lb/>
                  mi in dicta proportione
                    <var>.a.</var>
                  et
                    <var>.b.</var>
                  quorum differentia erit vnitas, vt ſcis,
                    <reg norm="ſitque" type="simple">ſitq́;</reg>
                    <var>.c.</var>
                  quadra
                    <lb/>
                  tum ipſius
                    <var>.g.</var>
                  et
                    <var>.d.</var>
                  quadratum ipſius
                    <var>.K</var>
                  . </s>
                  <s xml:id="echoid-s4371" xml:space="preserve">tunc clarum erit ex .11. octaui, quod propor-
                    <lb/>
                  tio ipſius c. ad
                    <var>.d.</var>
                  eadem erit quæ
                    <var>.g.</var>
                  ad
                    <var>.h.</var>
                  hoc eſt vt ipſius
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  vnde ſi vnus termi.
                    <lb/>
                  norum
                    <var>.a.</var>
                  vel
                    <var>.b.</var>
                  eſſet quadratus, reliquus etiam quadratus eſſet ex .22. octaui, & ex
                    <lb/>
                  16. eiuſdem, inter
                    <var>.a.</var>
                  et
                    <var>.b.</var>
                  reperiretur aliquis medius numerus proportionalis, quod
                    <lb/>
                  fieri non poteſt ex hypotheſi, cum inter
                    <var>.a.</var>
                  et
                    <var>.b.</var>
                  nullus ſit numerus, quia differunt in
                    <lb/>
                  ter ſe per vnitatem tantummodo. </s>
                  <s xml:id="echoid-s4372" xml:space="preserve">Nunc autem cum nullus numerorum
                    <var>.a.</var>
                  vel
                    <var>.b.</var>
                  qua
                    <lb/>
                  dratus ſit, ponatur quod
                    <var>.f.</var>
                  quadratus ſit ipſius
                    <var>.b.</var>
                  et
                    <var>.e.</var>
                  ſit productum ipſius
                    <var>.a.</var>
                  in
                    <var>.b.</var>
                  vn
                    <lb/>
                  de ex .18. ſeptimi, proportio ipſius
                    <var>.e.</var>
                  ad
                    <var>.f.</var>
                  erit vt. ipſius
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  hoc eſt vt ipſius
                    <var>.c.</var>
                  ad
                    <lb/>
                  d. quapropter
                    <var>.e.</var>
                  erit quadratus ex .22. octaui, cuius latus tetragonicum eſſet
                    <reg norm="medium" type="context">mediũ</reg>
                    <lb/>
                  proportionale inter
                    <var>.a.</var>
                  et
                    <var>.b.</var>
                  ex .20. ſeptimi, quod eſt impoſſibile, vt iam dixi, cum
                    <var>.a.</var>
                    <lb/>
                  et
                    <var>.b.</var>
                  ſint inui cem conſequentes, vnus poſt alium immediatè.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4373" xml:space="preserve">Superius enim dixi hunc modum eſſe vniuerſalem,
                    <lb/>
                  hoc eſt quod hac methodo poſſumus in cognitionem
                    <lb/>
                  vcnire, quod non ſolum in duas æquales partes diui-
                    <lb/>
                    <figure xlink:label="fig-0378-01" xlink:href="fig-0378-01a" number="418">
                      <image file="0378-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0378-01"/>
                    </figure>
                  di non poſſit, ſed nec in tres, nec quatuor nec quot vo
                    <lb/>
                  lueris. </s>
                  <s xml:id="echoid-s4374" xml:space="preserve">Primum enim quod non in tres diuidatur à te
                    <lb/>
                  ipſo cognoſces ope
                    <reg norm="cuborum" type="context">cuborũ</reg>
                  vice
                    <reg norm="quadratorum" type="context">quadratorũ</reg>
                  , opevero
                    <lb/>
                    <reg norm="cenſuum" type="context">cenſuũ</reg>
                    <reg norm="cenſuum" type="context context">cẽſuũ</reg>
                  , ve
                    <unsure/>
                  l qui cognouerit eam
                    <reg norm="proportionem" type="context">proportionẽ</reg>
                    <lb/>
                  eſſe indiuiſibilem per æqualia, illicò etiam cognoſcet
                    <lb/>
                  indiuiſibilem eſſe per quatuor partes, ope verò pri-
                    <lb/>
                  morum relatorum, cognoſcet non eſſe diuiſibilem per
                    <lb/>
                    <reg norm="quinque" type="simple">quinq;</reg>
                  partes, & ſic de cęteris, ſed mediantibus ijs
                    <lb/>
                  quas ſcripſi de iſtis dignitatibus in libro
                    <reg norm="Thęorematum" type="context">Thęorematũ</reg>
                    <lb/>
                  arithmeticorum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4375" xml:space="preserve">Id autem quod Illuſtriſſimus Daniel Barbarus ſcri
                    <lb/>
                  bit in quinta parte ſuæ perſpectiuæ, ſi ſupra aliquo im
                    <lb/>
                  mobili, atque magno pariete facere volueris, te opor
                    <lb/>
                  tebit hoc ex reflexione radij ſolaris à ſpeculo plano
                    <lb/>
                  perficere.</s>
                </p>
              </div>
            </div>
            <div xml:id="echoid-div713" type="section" level="3" n="37">
              <div xml:id="echoid-div713" type="letter" level="4" n="1">
                <head xml:id="echoid-head540" xml:space="preserve">DE INVENTIONE DIAMETRI
                  <lb/>
                circuli circunſcribentis triangulum.</head>
                <head xml:id="echoid-head541" style="it" xml:space="preserve">Francbino Triuultio.</head>
                <p>
                  <s xml:id="echoid-s4376" xml:space="preserve">
                    <emph style="sc">QVod</emph>
                  mihi nunc proponis eſt triangulum, cuius baſis cum angulo ſibi op
                    <lb/>
                  poſito dantur. </s>
                  <s xml:id="echoid-s4377" xml:space="preserve">
                    <reg norm="Vellesque" type="simple">Vellesq́;</reg>
                  diametrum circuli apti eum triangulum circnn-
                    <lb/>
                  ſcribere inuenire in diſcreto.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4378" xml:space="preserve">Sit igitur triangulum
                    <var>.a.b.g.</var>
                  cuius baſis
                    <var>.b.g.</var>
                  ſimul cum angulo
                    <var>.a.</var>
                  ei op-
                    <lb/>
                  poſito data ſit in numeris. </s>
                  <s xml:id="echoid-s4379" xml:space="preserve">Imaginetur ergo circulas circunſeribens ipſum triangu-
                    <lb/>
                  lum
                    <var>.b.p.g.q.</var>
                  cuius diameter ſit
                    <var>.q.p.</var>
                  perpendicularis eius baſi
                    <var>.b.g.</var>
                  vnde
                    <var>.b.g.</var>
                  diuiſa
                    <lb/>
                  erit per æqualia ab ipſo diametro in puncto
                    <var>.m.</var>
                  per tertiam tertij, protrahatur etiam </s>
                </p>
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