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 contents

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[4.28.] Occultam fuiße grauisſimo Stagirit & canſam ſcintilla-tionis ſtellarum. CAP. XXVIII.
[4.29.] Daricontinuum infinitum motum ſuper rectam at que finitam lineam. CAP. XXIX.
[4.30.] Non eſſe ſolis calorem à motu localι ipſius corporis ſolaris, ut Ariſtoteli placuit. CAP. XXX.
[4.31.] Vnde caloris ſolis prode at incrementum & state, et byeme decrementum. CAP. XXXI.
[4.32.] Nullum corpus ſenſus expers à ſono offendi, præterquam Aristoteles crediderit. CAP. XXXII.
[4.33.] Pytagoreorum opinionem de ſonitu corporum cælestium non fuiſſe ab Aristotele ſublatam. CAP. XXXIII.
[4.34.] Deraro et denſo nonnulla, minus diligenter à Peripateticis perpenſa. CAP. XXXIIII.
[4.35.] Motum rectum curuo poſſe comparari etiam diſentiente Ariſtotele. CAP. XXXV.
[4.36.] Minus ſufficienter exploſam fuiſſe ab Ariſtotele opinionem cre-dentium plures mundos exiſtere. CAP. XXXVI.
[4.37.] Anrectè loquutus ſit Phyloſopbus de extenſione luminis per uacuum. CAP. XXXVII.
[4.38.] An rectè phyloſophiœ penus Ariſtoteles ſenſerit de loco im-pellendo à pyramide. CAP. XXXVIII.
[4.39.] Examinatur quam ualida ſit ratio Aristotelis de inalterabilitate Cœli. CAP. XXXIX.
[5.] IN QVINTVM EVCLIDIS LIBRVM
[Item 5.1.]
[5.1.1.] Horum autem primum est.
[5.1.2.] SECVNDVM.
[5.1.3.] TERTIVM. Quę est εuclidis ſeptima propoſitio.
[5.1.4.] QVARTVM. εuclidis uerò nona propoſitio.
[5.1.5.] QVINTVM. Euclidis uerò octaua propoſitio.
[5.1.6.] SEXTVM. εuclidis uerò decima propoſitio.
[5.1.7.] SEPTIMVM. Euclidis uerò undecima propoſitio.
[5.1.8.] OCTAVVM. εuclidis uerò duodecima propoſitio.
[5.1.9.] NONVM. Euclidis uero tertiadecima propoſitio.
[5.1.10.] DECIMVM.
[5.1.11.] VNDECIMVM.
[5.1.12.] DVODECIMVM.
[Item 5.2.]
[5.2.1.] THEOR.I. II. ET III.
[5.2.2.] THEOREM. IIII.
[5.2.3.] THEOR.V. ET VI.
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              <p>
                <s xml:id="echoid-s605" xml:space="preserve">
                  <pb o="46" rhead="IO. BAPT. BENED." n="58" file="0058" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0058"/>
                  <var>g.m.</var>
                  <reg norm="cogiteturque" type="simple">cogiteturq́;</reg>
                rectangulum
                  <var>.y.x.</var>
                & rectangulum
                  <var>.k.x</var>
                . </s>
                <s xml:id="echoid-s606" xml:space="preserve">Itaque dabitur eadem pro
                  <lb/>
                portio
                  <var>.k.m.</var>
                ad
                  <var>.m.x.</var>
                nempe
                  <var>.k.x.</var>
                rectanguli ad
                  <var>.m.g.</var>
                quæ eſt
                  <var>.b.a.</var>
                ad
                  <var>.o.e.</var>
                et
                  <var>.y.x.</var>
                ad
                  <var>.m.
                    <lb/>
                  g.</var>
                quæ
                  <var>.b.a.</var>
                ad
                  <var>.a.o.</var>
                ſed ex prima ſexti aut .18. vel .19. ſeptimi, ſic ſe habet rectangu-
                  <lb/>
                lum
                  <var>.k.y.</var>
                ad
                  <var>.x.y.</var>
                ſicut
                  <var>.k.m.</var>
                ad
                  <var>.m.x.</var>
                </s>
                <s xml:id="echoid-s607" xml:space="preserve">quare ſicut
                  <var>.b.a.</var>
                ad
                  <var>.o.e.</var>
                ex .11. quinti, & eiuſdem
                  <lb/>
                rectanguli
                  <var>.k.y.</var>
                ad rectangulum
                  <var>.k.x.</var>
                ſicut
                  <var>.y.m.</var>
                ad
                  <var>.x.m.</var>
                nempe
                  <var>.b.a.</var>
                ad
                  <var>.a.o</var>
                . </s>
                <s xml:id="echoid-s608" xml:space="preserve">Quare
                  <lb/>
                ex communi ſcientia, ſic ſe habebit duplum rectanguli
                  <var>.k.y.</var>
                ad ſummam
                  <var>.y.x.</var>
                cum
                  <var>.
                    <lb/>
                  k.x.</var>
                rectangulorum, ſicut duplum
                  <var>.b.a.</var>
                ad ſummam
                  <var>.a.o.e.</var>
                et proportio ſummæ re-
                  <lb/>
                ctangulorum
                  <var>.y.x.</var>
                et
                  <var>.k.x.</var>
                duplo
                  <var>.g.m.</var>
                ſicut duplum
                  <var>.b.a.</var>
                ad
                  <var>.a.o.e</var>
                . </s>
                <s xml:id="echoid-s609" xml:space="preserve">Igitur ſumma duo-
                  <lb/>
                rum rectangulorum
                  <var>.y.x.</var>
                et
                  <var>.x.k.</var>
                media proportionalis erit inter duplum rectanguli
                  <var>.
                    <lb/>
                  k.y.</var>
                & duplum vnitatis ſuperſicialis
                  <var>.g.m</var>
                . </s>
                <s xml:id="echoid-s610" xml:space="preserve">Nunc terminetur rectangulum
                  <var>.a.r.</var>
                ex quo
                  <lb/>
                dabitur eadem proportio dupli
                  <var>.a.s.</var>
                ad
                  <var>.a.r.</var>
                ſicut dupli
                  <var>.b.a.</var>
                ad
                  <var>.a.e.</var>
                ex propoſitioni-
                  <lb/>
                bus notatis, ſexti aut ſeptimi. </s>
                <s xml:id="echoid-s611" xml:space="preserve">Quare etiam ſicut dupli rectanguli
                  <var>.k.y.</var>
                ad
                  <reg norm="ſummam" type="context">ſummã</reg>
                  <lb/>
                rectangulorum
                  <var>.y.x.</var>
                et
                  <var>.k.x</var>
                . </s>
                <s xml:id="echoid-s612" xml:space="preserve">Iam verò ſi conſtituatur
                  <var>.e.c.</var>
                pro vnitate lineari ipſius
                  <var>.
                    <lb/>
                  e.r.</var>
                certi erimus numerum
                  <var>.a.c.</var>
                æqualem eſſe
                  <var>.a.e.</var>
                & proportionem
                  <var>.r.e.</var>
                ad
                  <var>.e.c.</var>
                hoc
                  <lb/>
                eſt
                  <var>.a.r.</var>
                ad
                  <var>.a.c.</var>
                eandem quæ
                  <var>.y.x.</var>
                et
                  <var>.x.k.</var>
                rectangulorum ad
                  <var>.m.g.</var>
                ex prædictis rationi-
                  <lb/>
                bus, & ex hypotheſi, nempe quòd
                  <var>.
                    <lb/>
                  e.r.</var>
                æqualis ſit numero
                  <var>.k.m.y.</var>
                  <lb/>
                  <figure xlink:label="fig-0058-01" xlink:href="fig-0058-01a" number="79">
                    <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0058-01"/>
                  </figure>
                hoc eſt rectangulorum
                  <var>.y.x.</var>
                et
                  <var>.x.
                    <lb/>
                  k</var>
                . </s>
                <s xml:id="echoid-s613" xml:space="preserve">Quamobrem
                  <var>.a.r.</var>
                ex communi
                  <lb/>
                ſcientia
                  <reg norm="medium" type="context">mediũ</reg>
                proportionale erit
                  <lb/>
                inter duplum
                  <var>.a.s.</var>
                & duplum
                  <var>.a.c.</var>
                  <reg norm="ea­ demque" type="context simple">ea­
                    <lb/>
                  dẽq́;</reg>
                  <reg norm="proportio" type="simple">ꝓportio</reg>
                dupli prędicti
                  <var>.a.s.</var>
                ad
                  <lb/>
                duplum
                  <var>.a.c.</var>
                ex æqualitate propor-
                  <lb/>
                tionum ſimul collectarum, eadem
                  <lb/>
                erit qùæ proportio dupli rectangu-
                  <lb/>
                li
                  <var>.k.y.</var>
                ad duplum
                  <var>.m.g.</var>
                hoc eſt
                  <var>.a.s.</var>
                  <lb/>
                ſimplicis ad ſimplicem
                  <var>.a.c.</var>
                quæ ſim
                  <lb/>
                plicis rectanguli
                  <var>.k.y.</var>
                ad ſimplicem
                  <lb/>
                vnitatem
                  <var>.g.m.</var>
                ſic enim ſe habet ſim
                  <lb/>
                plex ad ſimplex, ſicut duplum ad
                  <lb/>
                duplum. </s>
                <s xml:id="echoid-s614" xml:space="preserve">Sed pariter ita ſe habet
                  <var>.a.s.</var>
                ad
                  <var>.a.</var>
                c
                  <unsure/>
                . cogitato
                  <var>.a.c.</var>
                tamquam proueniente
                  <lb/>
                ex diuiſione
                  <var>.a.s.</var>
                per rectangulum
                  <var>.k.y.</var>
                vt conſtitutum eſt, ſicut
                  <var>.k.y.</var>
                ad
                  <var>.m.g.</var>
                ex defi-
                  <lb/>
                nitione diuiſionis vt iam dictum eſt, </s>
                <s xml:id="echoid-s615" xml:space="preserve">quare numerus
                  <var>.a.c.</var>
                æqualis erit numero
                  <var>.a.o.e</var>
                .</s>
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            <div xml:id="echoid-div140" type="math:theorem" level="3" n="71">
              <head xml:id="echoid-head87" xml:space="preserve">THEOREMA
                <num value="71">LXXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s616" xml:space="preserve">CVR propoſitis .4. numeris, duobus nempe diuidentibus ac duobus diuiden-
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                dis, ſi
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                diuiſi fuerint,
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                  <reg norm="prouenientia" type="context">proueniẽtia</reg>
                  <reg norm="inuicem" type="context">inuicẽ</reg>
                multiplicata
                  <reg norm="quenuis" type="context">quẽuis</reg>
                nu
                  <lb/>
                merum producant, qui ſeruetur, ſi deinde ijdem numeri verſa vice mutuo diuiſi fue
                  <lb/>
                rint, & inter ſe multiplicata prouenientia,
                  <reg norm="productum" type="context">productũ</reg>
                hoc, primo ſeruato numero
                  <lb/>
                æquale erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s617" xml:space="preserve">Exempli gratia propoſitis his .4. numeris .20. 30. 5. 10. duo autem .20. ſcilicet
                  <lb/>
                et .30. ſint numeri diuidendi, porrò .5. et .10. numeri diuidentes,
                  <reg norm="nempe" type="context">nẽpe</reg>
                vt primo .20
                  <lb/>
                per .5. diuidatur, tum .30. per .10. producetur .4. et .3. qui ſimul multiplicati
                  <reg norm="proferent" type="context">proferẽt</reg>
                  <num value="12">.
                    <lb/>
                  12.</num>
                tum .20. per .10. d iuiſo et .30. per .5. prouenientia erunt .2. 6. quæ inter ſe multi-
                  <lb/>
                plicata producent etiam .12.</s>
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