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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                <pb o="382" rhead="IO. BAPT. BENED." n="394" file="0394" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0394"/>
                <p>
                  <s xml:id="echoid-s4523" xml:space="preserve">In eo quod à me petis, mittendo te ad Eutotium, tibi non ſatisfacerem, cum Eu-
                    <lb/>
                  totius citet ſextum librum Pergei, quem nunquam vidimus,
                    <reg norm="ſupponatque" type="simple">ſupponatq́;</reg>
                  ea, quæ nec
                    <lb/>
                  ipſe nec alius vnquam quod ſcimus probauit.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4524" xml:space="preserve">Deſideras enim demonſtrationem illius quod Archimedes dicit inter primam,
                    <lb/>
                  & ſecundam propoſitionem ſecundi libri, vbi tractat de centris grauium, propte-
                    <lb/>
                  rea quod illud ſupponit pro manifeſto.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4525" xml:space="preserve">Sit enim figura hic ſubſcripta, ferè ſimilis parabolæ poſitæ in .2. propoſitione di
                    <lb/>
                  cti libri, vt in impreſſione Baſileenſi habetur,
                    <reg norm="ſintque" type="simple">ſintq́;</reg>
                  diuiſæ duæ
                    <var>.a.b.</var>
                  et
                    <var>.b.c.</var>
                  per æqua
                    <lb/>
                  lia à punctis
                    <var>.x.</var>
                  et
                    <var>.u.</var>
                    <reg norm="protractisque" type="simple">protractisq́;</reg>
                    <var>.f.x.</var>
                  et
                    <var>.u.i.</var>
                  ad
                    <var>.b.d.</var>
                  quæ inuicem etiam erunt parallelę
                    <lb/>
                  ex .30. primi Eucli. </s>
                  <s xml:id="echoid-s4526" xml:space="preserve">vnde ipſæ etiam, diametri erunt ipſarum portionum: </s>
                  <s xml:id="echoid-s4527" xml:space="preserve">vt ex eo col
                    <lb/>
                  ligere eſt, quod in .49. primi lib. Pergei probatur. </s>
                  <s xml:id="echoid-s4528" xml:space="preserve">Imaginando poſtea ad puncta
                    <var>.b.
                      <lb/>
                    f.</var>
                  er
                    <unsure/>
                    <var>.i.</var>
                  tres contingentes, manifeſtum erit punctum
                    <var>.b.</var>
                  illud eſſe quod terminat alti-
                    <lb/>
                  tudinem huiuſmodi portionis, et
                    <var>.f.</var>
                  et
                    <var>.i.</var>
                  terminantia altitudines partialium, ex .5. ſe­
                    <lb/>
                  cundi ipſius Pergei, eo quod dictæ contingentes paralellæ erunt ipſis baſibus, vnde
                    <lb/>
                  trianguli inſcripti, eaſdem habebunt altitudines, quas portiones ipſæ, quod erit ex
                    <lb/>
                  mente Archimedis. </s>
                  <s xml:id="echoid-s4529" xml:space="preserve">Et ſic deinceps poteris multiplicare angulos ſiguræ rectilineæ
                    <lb/>
                  in parabola, quæ deſignata erit vt deſiderat Archimedes, qui quidem dicit, quod
                    <lb/>
                  protractæ cum fuerint aliæ deinceps poſt
                    <var>.f.i.</var>
                  ipſæ inuicem ęquidiſtantes
                    <reg norm="erunt" type="context">erũt</reg>
                  , diuiſę-
                    <lb/>
                  q́ue peræqualia ab
                    <var>.d.b.</var>
                  quod
                    <reg norm="quanuis" type="context">quãuis</reg>
                    <reg norm="verum" type="context">verũ</reg>
                  ſit,
                    <reg norm="tantum" type="wordlist/context">tñ</reg>
                  ab Eutotio non ſatis
                    <reg norm="demonſtratum" type="context context">demõſtratũ</reg>
                    <lb/>
                  eſt, cum ſupponat
                    <var>.a.f.b.</var>
                  æqualem eſſe ipſi
                    <var>.b.i.c.</var>
                  probare volens eius diametros æqua
                    <lb/>
                  les eſſe abſque aliqua citata ratione, quæ quidem ratio eſſet conuerſum .4. propoſi-
                    <lb/>
                  tionis libri de conoidalibus. </s>
                  <s xml:id="echoid-s4530" xml:space="preserve">Sed oporteret nos
                    <reg norm="etiam" type="context">etiã</reg>
                  videre .6. librum ipſius Pergei,
                    <lb/>
                  & propterea tibi non ſatisfacerem.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4531" xml:space="preserve">Eſto igitur, ut inuenta ſit linea
                    <var>.K.</var>
                  cuius productum in
                    <var>.u.i.</var>
                  æquale ſit qua drato ip
                    <lb/>
                  ſius
                    <var>.u.c.</var>
                  inuenta etiam ſit linea
                    <var>.h.</var>
                  cuius productum cum
                    <var>.f.x.</var>
                  æquale ſit quadrato ip-
                    <lb/>
                  ſius
                    <var>.a.x.</var>
                  vnde ex conuerſo .49. primi ipſius Pergei, proportio ipſius
                    <var>.K.</var>
                  ad
                    <var>.b.c.</var>
                  erit ut
                    <lb/>
                  ipſius
                    <var>.b.c.</var>
                  ad
                    <var>.b.d.</var>
                  & ipſius
                    <var>.h.</var>
                  ad
                    <var>.a.b.</var>
                  vt ipſius
                    <var>.a.b.</var>
                  ad
                    <var>.b.d</var>
                  . </s>
                  <s xml:id="echoid-s4532" xml:space="preserve">Erit igitur ex .16. ſexti Eucl.
                    <lb/>
                  quadratum
                    <var>.b.c.</var>
                  æquale producto ipſius
                    <var>.K.</var>
                  in
                    <var>.b.d.</var>
                  & quadratum
                    <var>.a.b.</var>
                  æquale produ-
                    <lb/>
                  cto ipſius
                    <var>.h.</var>
                  in
                    <var>.b.d.</var>
                  & ex prima ſexti, ita erit ipſius
                    <var>.K.</var>
                  ad
                    <var>.h.</var>
                  vt producti quod fit ex
                    <var>.K.</var>
                    <lb/>
                  in
                    <var>.b.d.</var>
                  ad productum ipſius
                    <var>.h.</var>
                  in
                    <var>.b.d.</var>
                  hoc eſt vt quadrati ipſius
                    <var>.b.c.</var>
                  ad quadratum ip
                    <lb/>
                  ſius
                    <var>.b.a.</var>
                  ex .16. et .11. quinti, hoc eſt vt quadrati ipſius
                    <var>.u.c.</var>
                  ad quadratum ipſius
                    <var>.a.x.</var>
                    <lb/>
                  hoc eſt ut productum ipſius
                    <var>.k.</var>
                  in
                    <var>.u.i.</var>
                  ad productnm ipſius
                    <var>.h.</var>
                  in
                    <var>.x.f</var>
                  . </s>
                  <s xml:id="echoid-s4533" xml:space="preserve">Nunc ſi ipſius
                    <var>.k.</var>
                    <lb/>
                  ad
                    <var>.h.</var>
                  c
                    <unsure/>
                  ſt vt producti ipſius
                    <var>.K.</var>
                  in
                    <var>.u.i.</var>
                  ad productum ipſius
                    <var>.h.</var>
                  in
                    <var>.f.x.</var>
                  ergo ex .24. ſexti,
                    <lb/>
                  & communi conceptu, proportio ipſius
                    <var>.k.</var>
                  ad
                    <var>.h.</var>
                  compoſita erit ex ea quæ ipſius
                    <var>.u.i.</var>
                    <lb/>
                  ad
                    <var>.f.x.</var>
                  & ex ea quæ ipſius
                    <var>.k.</var>
                  ad
                    <var>.h</var>
                  . </s>
                  <s xml:id="echoid-s4534" xml:space="preserve">Cum ergo dempta fuerit proportio ipſius
                    <var>.k.</var>
                  ad
                    <var>.h.</var>
                    <lb/>
                  (vt ſimplex) à proportione ipſius
                    <var>.k.</var>
                  ad
                    <var>.h.</var>
                  (vt compoſita) reliquum nihil erit. </s>
                  <s xml:id="echoid-s4535" xml:space="preserve">Qua-
                    <lb/>
                  re
                    <var>.f.x.</var>
                  æqualis erit ipſi
                    <var>.u.i</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4536" xml:space="preserve">Sed quod
                    <var>.f.m.</var>
                  æqualis ſit ipſi
                    <var>.m.i</var>
                  . </s>
                  <s xml:id="echoid-s4537" xml:space="preserve">Videto in Eutotio, quia hoc ſatis ſui natura
                    <lb/>
                  facile eſt.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4538" xml:space="preserve">Sed accipe alium modum breuiorem ad probandum
                    <var>.f.x.</var>
                  eſſe æqualem ipſi
                    <var>.u.i</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4539" xml:space="preserve">Finge lineam
                    <var>.e.b.g.</var>
                  conting entem in puncto
                    <var>.b.</var>
                  prolungatisq́ue diametris
                    <var>f.
                      <lb/>
                    x.</var>
                  et
                    <var>.u.i.</var>
                  vſque ad contingentem ipſam, habebis
                    <var>.f.e.</var>
                  æqualem ipſi
                    <var>.f.x.</var>
                  et
                    <var>.g.i.</var>
                  ipſi
                    <var>.u.i.</var>
                    <lb/>
                  Ex .35. primi Pergei, producta poſtea
                    <var>.x.u.</var>
                  habeb is ex .2. ſexti Eucli
                    <var>.x.u.</var>
                  parallelam
                    <lb/>
                  ipſi
                    <var>.a.c.</var>
                  ſed
                    <var>.e.g.</var>
                  parallela eſt ipſimet
                    <var>.a.c.</var>
                  ex quinta ſecundi ipſius Pergei, </s>
                  <s xml:id="echoid-s4540" xml:space="preserve">quare ex .30
                    <lb/>
                  primi Euclid
                    <var>.e.g.</var>
                  parallela erit ipſi
                    <var>.u.x.</var>
                  & ex .34. eiuſdem æqualis erit
                    <var>.e.x.</var>
                  ipſi
                    <var>.u.g.</var>
                    <lb/>
                  vnde
                    <var>.f.x.</var>
                  etiam æqualis erit
                    <var>.u.i.</var>
                  ex communi conceptu.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4541" xml:space="preserve">Sed ne quid deſideres probabo
                    <var>.f.m.</var>
                  æqualem eſſe ipſi
                    <var>.m.i</var>
                  . </s>
                  <s xml:id="echoid-s4542" xml:space="preserve">Iam igitur ſcis quod </s>
                </p>
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