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]

Table of handwritten notes

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            <div xml:id="echoid-div558" type="section" level="3" n="16">
              <div xml:id="echoid-div558" type="letter" level="4" n="1">
                <p>
                  <s xml:id="echoid-s3593" xml:space="preserve">
                    <pb o="290" rhead="IO. BAPT. BENED." n="302" file="0302" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0302"/>
                  rit tempus .33. minutorum ex h oris .2. min .24. reliquum erit hora .1. min .51. vnde
                    <lb/>
                  proportio aquæ, quæ in vaſe reperitur, ad eam, quæ totum vas implet, erit vt .111.
                    <lb/>
                  ad .144. </s>
                  <s xml:id="echoid-s3594" xml:space="preserve">Quare nunc poſſumus rectè dicere ex regula de tribus ſi .111. indigent mi-
                    <lb/>
                  nuta .33. temporis, ergo .144. indigent min .43. horæ, in quo tempore implebitur to-
                    <lb/>
                  tum vas omnibus fiſtulis operantibus.</s>
                </p>
              </div>
              <div xml:id="echoid-div559" type="letter" level="4" n="2">
                <head xml:id="echoid-head426" style="it" xml:space="preserve">Aliæ circuli noua paßiones.</head>
                <head xml:id="echoid-head427" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s3595" xml:space="preserve">VTad aſcendendum ignis, & ad
                    <reg norm="deſcendendum" type="context">deſcendendũ</reg>
                  quicquid graue natum eſt, ita ad
                    <lb/>
                  ſpeculandum humanus intellectus. </s>
                  <s xml:id="echoid-s3596" xml:space="preserve">nec quieſcit, dum poteſt, eſt enim ver-
                    <lb/>
                  ſatile,
                    <reg norm="agitandoque" type="simple">agitandoq́;</reg>
                  ſeſe cauſis rerum immiſcere, & abditum aliquid rimari,
                    <lb/>
                  conatur, & eſt in nobis, quaſi Diogenes quidam in Dolio.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3597" xml:space="preserve">Tibi igitur mitto quod vltimò inueni, alias ſcilicet nouas circuli paſſiones,
                    <lb/>
                  quæ ita ſe
                    <reg norm="habent" type="context">habẽt</reg>
                  . </s>
                  <s xml:id="echoid-s3598" xml:space="preserve">Sit circulus
                    <var>.a.b.c.</var>
                  in quo ſit
                    <var>.a.d.</var>
                  latus quadrati inſcriptibilis in ipſo
                    <lb/>
                  circulo, ct
                    <var>.b.c.</var>
                  ſit diameter ad rectos cum
                    <var>.a.d.</var>
                  in puncto
                    <var>.e.</var>
                  quod medium erit inter
                    <lb/>
                  a. et
                    <var>.d.</var>
                  ex .3. tertij Eucli. ſit ſimiliter
                    <var>.a.f.</var>
                  contingens ipſum circulum in puncto
                    <var>.a.</var>
                  quæ
                    <lb/>
                  protracta ſit vſque ad punctum
                    <var>.f.</var>
                  interſectionis cum diametro protracto, quod ita
                    <lb/>
                  eueniet cum anguli
                    <var>.a.e.f.</var>
                  et
                    <var>.f.a.e.</var>
                  minores ſint duobus rectis, eo quod angulus
                    <var>.f.a.e.</var>
                    <lb/>
                  acutus ſit, cum
                    <var>.a.d.</var>
                  tranſeat inter centrum et
                    <var>.f</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3599" xml:space="preserve">Dico nunc quod productum diametri
                    <var>.b.c.</var>
                  in parte
                    <var>.c.e.</var>
                  ipſius, æqualis erit produ-
                    <lb/>
                  cto ipſius
                    <var>.c.f.</var>
                  in
                    <var>.a.d</var>
                  . </s>
                  <s xml:id="echoid-s3600" xml:space="preserve">Protrahatur imaginatione
                    <var>.b.a.</var>
                  et
                    <var>.a.c.</var>
                  </s>
                  <s xml:id="echoid-s3601" xml:space="preserve">vnde ex .26. tertij Euclid.
                    <lb/>
                  habebimus angulum
                    <var>.d.a.c.</var>
                  æqualem angulo
                    <var>.a.b.c</var>
                  . </s>
                  <s xml:id="echoid-s3602" xml:space="preserve">ſed ex .31. eiuſdem angulus
                    <var>.f.a.
                      <lb/>
                    c.</var>
                  æqualis eſt angulo
                    <var>.b</var>
                  . </s>
                  <s xml:id="echoid-s3603" xml:space="preserve">quare æqualis erit angulo
                    <var>.d.a.c.</var>
                  & ita habebimus per .3. ſexti
                    <lb/>
                  eandem proportionem
                    <var>.f.c.</var>
                  ad
                    <var>.c.e.</var>
                  quæ
                    <var>.f.a.</var>
                  ad
                    <var>.a.e.</var>
                  ſed
                    <var>.a.f.</var>
                  eſt æqualis ſemidiametro
                    <lb/>
                  circuli propoſiti, </s>
                  <s xml:id="echoid-s3604" xml:space="preserve">propterea quod ſi producta fuerit à puncto
                    <var>.a.</var>
                  ad centrum
                    <var>.o.</var>
                  ſemi
                    <lb/>
                  diameter
                    <var>.a.o.</var>
                  hæc cum
                    <var>.o.e.</var>
                  faciet dimidium angulirecti, cum ex ſuppoſito
                    <var>.a.d.</var>
                  la-
                    <lb/>
                  tus ſit quadrati inſcriptibilis in ipſo circulo. </s>
                  <s xml:id="echoid-s3605" xml:space="preserve">& cum
                    <var>.a.f.</var>
                  rectum ex .17. tertij, vnde an
                    <lb/>
                  gulus
                    <var>.f.</var>
                  erit ſimiliter medietas recti ex .32. primi, </s>
                  <s xml:id="echoid-s3606" xml:space="preserve">quare ex .6. eiuſdem
                    <var>.a.f.</var>
                  æqualis
                    <lb/>
                  erit
                    <var>.a.o</var>
                  . </s>
                  <s xml:id="echoid-s3607" xml:space="preserve">Ergo cum proportio
                    <var>.f.c.</var>
                  ad
                    <var>.c.e.</var>
                  ſit. vt
                    <var>.f.a.</var>
                  ad
                    <var>.a.e.</var>
                  erit ſimiliter vt
                    <var>.b.c.</var>
                  ad
                    <var>.a.d.</var>
                    <lb/>
                  hoc eſt ut dupli ad duplum, vnde ex .15. ſexti
                    <lb/>
                  manifeſtum erit propoſitum, ex quo alia paſ-
                    <lb/>
                    <figure xlink:label="fig-0302-01" xlink:href="fig-0302-01a" number="324">
                      <image file="0302-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0302-01"/>
                    </figure>
                  ſio oritur, hoc eſt, quod productum
                    <var>.f.c.</var>
                  in
                    <var>.a.
                      <lb/>
                    d.</var>
                  æ quale ſit qua drato ipſius
                    <var>.a.c.</var>
                  ratio eſt, quia
                    <lb/>
                  quadratum
                    <var>.a.c.</var>
                  æ quale eſt producto
                    <var>.b.c.</var>
                  in
                    <var>.c.
                      <lb/>
                    e.</var>
                  eo quod
                    <var>.a.c.</var>
                  media proportionalis eſt inter
                    <var>.
                      <lb/>
                    b.c.</var>
                  et
                    <var>.c.e.</var>
                  ex ſimilitudine triangulorum
                    <var>.a.b.c.</var>
                    <lb/>
                  et
                    <var>.e.a.c.</var>
                  nam anguli
                    <var>.b.a.c.</var>
                  et
                    <var>.a.e.c.</var>
                  recti ſunt
                    <lb/>
                  et
                    <var>.c.</var>
                    <reg norm="communis" type="context">cõmunis</reg>
                  , vnde
                    <var>.b.</var>
                  erit æqualis
                    <var>.e.a.c.</var>
                  ex .32
                    <lb/>
                  primi, ſequitur etiam, quod
                    <var>.a.c.</var>
                  ſit media pro
                    <lb/>
                  portionalis inter
                    <var>.a.d.</var>
                  et
                    <var>.f.c.</var>
                  & hæc etiam erit
                    <lb/>
                  alia circuli paſſio, & quia
                    <var>.a.c.</var>
                  eſt latus octago-
                    <lb/>
                  ni igitur tale latus
                    <reg norm="medium" type="context">mediũ</reg>
                  proportionale erit
                    <lb/>
                  inter latus quadrati. et
                    <var>.f.c.</var>
                    <reg norm="eiuſdem" type="context">eiuſdẽ</reg>
                  circuli, quę
                    <lb/>
                  quidem
                    <var>.f.c.</var>
                  eſt una portio diametri quadrati circunſcriptibilis ipſum circulum inter
                    <lb/>
                  circulum & angulum ipſius quadrati.</s>
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