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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            <div xml:id="echoid-div730" type="section" level="3" n="41">
              <div xml:id="echoid-div730" type="letter" level="4" n="1">
                <pb o="378" rhead="IO. BAPT. BENED." n="390" file="0390" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0390"/>
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              <div xml:id="echoid-div732" type="letter" level="4" n="2">
                <head xml:id="echoid-head556" style="it" xml:space="preserve">De incommenſur abilitate, in longitudine perpendicu-
                  <lb/>
                laris trianguli æquilateri cum eiuſdem latere.</head>
                <head xml:id="echoid-head557" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s4473" xml:space="preserve">ID quod à me poſtulas eſt omnino impoſſibile, velles enim duos numeros inueni
                    <lb/>
                  re inter ſe ita ſe habentes, vt ſe habent perpendicularis in triangulo æquilatero
                    <lb/>
                  cum vno eius laterum, quod vero hoc fieri non poſſit, conſidera in figura præcedenti
                    <lb/>
                  triangulum æquilaterum
                    <var>.d.l.q.</var>
                  cuius perpendicularis ſit
                    <var>.d.o.</var>
                  quæ diuidit
                    <var>.l.q.</var>
                  per
                    <lb/>
                  æqualia in
                    <var>.o.</var>
                  vnde ex .4. ſecundi Euclidis, quadratum
                    <var>.l.q.</var>
                  (ideſt
                    <var>.d.q.</var>
                  ) quadruplum
                    <lb/>
                  erit quadrato
                    <var>.o.q.</var>
                  & ex penultima primi ęquale quadratis
                    <var>.d.o.</var>
                  et
                    <var>.o.q.</var>
                  </s>
                  <s xml:id="echoid-s4474" xml:space="preserve">quare erit ſeſ-
                    <lb/>
                  quitertium quadrato ipſius
                    <var>.d.o.</var>
                  & ita quadratum
                    <var>.d.o.</var>
                  erit triplum quadrato ipſius
                    <var>.
                      <lb/>
                    o.q.</var>
                  hæe autem proportiones non ſunt vt numeri quadrati ad numerum quadratum
                    <lb/>
                  quod ſi ita fuiſſent, ſequeretur ternarium numerum eſſe quadratum ex .22. octaui.
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4475" xml:space="preserve">Cum igitur non ſint vt numeri quadrati ad numerum quadratum, ſequitur ex ſepti-
                    <lb/>
                  ma decimi
                    <var>.d.o.</var>
                  eſſe incommenſurabilem ipſi
                    <var>.l.q.</var>
                  ſeu
                    <var>.d.q.</var>
                  in longitudine.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4476" xml:space="preserve">Vel dicamus ita, proportio quadrati ipſius
                    <var>.l.q.</var>
                  ad quadratum ipſius
                    <var>.o.d.</var>
                  eſt in ge
                    <lb/>
                  nere ſuperparticulari, cum ſit ſeſquitertia, vnde quadratum ipſius
                    <var>.d.o.</var>
                  numeris da-
                    <lb/>
                  ri non poteſt, eo quod ſi dabilis fuiſſet, ſequeretur, quod inter quadratum ipſius. l
                    <unsure/>
                    <var>.
                      <lb/>
                    q.</var>
                  & ipſius
                    <var>.d.o.</var>
                  eſſet aliquis numerus medius proportionalis ex .16. octaui, vnde ex
                    <lb/>
                  octaua eiuſdem vnitas diuiſibilis eſſet, quod fieri non poteſt.</s>
                </p>
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                  <image file="0390-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0390-01"/>
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              </div>
              <div xml:id="echoid-div733" type="letter" level="4" n="3">
                <head xml:id="echoid-head558" style="it" xml:space="preserve">De triangulo & Pentagono æquilatero</head>
                <head xml:id="echoid-head559" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s4477" xml:space="preserve">MOdum quem conſideraui circa triangulum æquilaterum & pentagonum, vt
                    <lb/>
                  tibi ſignificaui ita ſe habet.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4478" xml:space="preserve">Probandum primò eſt pentagonum altiorem eſſe triangulo ſibi iſoperimetro.
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4479" xml:space="preserve">Iam tibi notam puto proportionem lateris trianguli ad latus pentagoni eſſe vt .5.
                    <lb/>
                  ad .3.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4480" xml:space="preserve">Sit igitur pentagonus
                    <var>.b.d.m.g.v.</var>
                  cuius periferia diſtenta ſit
                    <var>.K.z.</var>
                  baſis autem
                    <var>.m.
                      <lb/>
                    g.</var>
                  bifariam diuiſa ſit in puncto
                    <var>.a.</var>
                    <reg norm="ductaque" type="simple">ductaq́;</reg>
                    <var>.a.b</var>
                  :
                    <var>b.g.</var>
                  et
                    <var>.b.m.</var>
                  clarum erit
                    <var>.a.b.</var>
                  perdicu-
                    <lb/>
                  larem eſſe ad
                    <var>.m.g.</var>
                  ex .8. primi Eucli. cum
                    <var>.b.m.</var>
                  et
                    <var>.b.g.</var>
                  (baſes triangulorum
                    <var>.b.d.m.</var>
                  </s>
                </p>
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