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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[2.9.] CAP. IX.
[2.10.] CAP.X.
[2.11.] CAP. XI. ALITER IDEM.
[2.12.] JACOBO SOLDATO MEDIOLANENSI Serenißimi Ducis Sabaudiæ Architecto peritißimo. CAP. VII.
[2.13.] AD EVNDEM IACOBVM. CAP. XIII.
[2.14.] CAP. XIIII.
[2.15.] CAP. XV.
[3.] DE MECHANICIS.
[3.1.] De differentia ſitus brachiorum libra. CAP.I.
[3.2.] De proportione ponderis extremitatis brachij libr & in diuerſo ſitu ab orizontali. CAP. II.
[3.3.] Quòd quantit as cuiuſlibet ponderis, aut uirtus mouens re-ſpectu alterius quantitatis cognoſcatur beneficio perpendicularium ductarum à centro libr & ad line am inclinationis. CAP. III.
[3.4.] Quemadmodum exſupradictis cauſis omnes staterarum & uectium cauſæ dependeant. CAP. IIII.
[3.5.] De quibuſdam rebus animaduerſione dignis. CAP.V.
[3.6.] De ratione cuiuſdam uis adauctæ. CAP. VI.
[3.7.] De quibuſdam erroribus Nicolai Tartaleæ circa pondera corporum & eorum motus, quorum aliqui deſumpti fuerunt à fordano ſcriptore quodam antiquo. CAP. VII.
[3.8.] CAP. VIII.
[3.9.] Quòdſummaratione ſtateræper æqualia interualla ſint diuiſæ. CAP. IX.
[3.10.] Quòd line a circularis non habe at concauum cum con-uexo coniunctum, & quod Aristo. cir caproportio nes motuum aberrauerit. CAP.X.
[3.11.] Quod Aristo. in prima mechanicarum quæstionum eius quod inquir it, uer am cauſam non attulerit. CAP. XI.
[3.12.] De uer a cauſa ſecundæ, & tertiæ quæstionis mechanicæ ab Ariſtotele nonperſpecta. CAP. XII.
[3.13.] Quòd Ariſtotelisratio in 6. quæſtione poſit a non ſit admittenda. CAP. XIII.
[3.14.] Quòdrationes ab Ariſtotele de octaua quæstione confictæ ſufficient es non ſint. CAP. XIIII.
[3.15.] Quod Aristotelis ratio none queſtionis admittendanon ſit. CAP. XV.
[3.16.] Quod Aristotelis rationes de decima queſtione ſint reijciende. CAP. XVI.
[3.17.] De uer a cauſa .12. questionis mechanice. CAP. XVII.
[3.18.] De decimatertia questione. CAP. XVIII.
[3.19.] De decimaquart a queſtione. CAP. XIX.
[3.20.] De uer a r atione .17. queſtionis. CAP. XX.
[3.21.] De uera & intrinſeca cauſa trocble arum. CAP. XXI.
[3.22.] Depropria cauſa .24. quæſtionis. CAP. XXII.
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            <div xml:id="echoid-div94" type="math:theorem" level="3" n="44">
              <p>
                <s xml:id="echoid-s389" xml:space="preserve">
                  <pb o="29" rhead="THEOR. ARITH." n="41" file="0041" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0041"/>
                to maiore
                  <var>.c.t.</var>
                extractum quare reſiduum qua-
                  <lb/>
                  <figure xlink:label="fig-0041-01" xlink:href="fig-0041-01a" number="56">
                    <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0041-01"/>
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                drati
                  <var>.c.p.</var>
                cognitum erit, quam quantitatem co-
                  <lb/>
                gnitam, cum ſit ſecundo loco data, cogitemus
                  <lb/>
                detrahi è toto quadrato cognito
                  <var>.q.e.</var>
                ex quo
                  <lb/>
                ſumma duorum ſupplementorum
                  <var>.q.o.</var>
                et
                  <var>.o.e.</var>
                  <lb/>
                cognoſcetur, vnà cum quadratis
                  <var>.u.n.</var>
                et
                  <var>.p.a.</var>
                du
                  <lb/>
                plo ſcilicet
                  <var>.q.a.</var>
                quo diuiſo per duplum
                  <var>.q.h.</var>
                aut
                  <lb/>
                ſimplex
                  <var>.q.a.</var>
                per
                  <var>.q.h.</var>
                ſimplicem, dabitur
                  <var>.a.h.</var>
                  <lb/>
                nempe
                  <var>.p.h.</var>
                minor numerus quæſitus.</s>
              </p>
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            <div xml:id="echoid-div96" type="math:theorem" level="3" n="45">
              <head xml:id="echoid-head61" xml:space="preserve">THEOREMA
                <num value="45">XLV</num>
              .</head>
              <p>
                <s xml:id="echoid-s390" xml:space="preserve">CVR volentes diuidere numerum propoſitum in duas eiuſmodi partes, vt pro
                  <lb/>
                ductum vnius in alteram, alteri numero propoſito æquetur, rectè dimidium
                  <lb/>
                primi dati numeri in ſeipſum multiplicant, ex quo quadrato ſecundum datum nu-
                  <lb/>
                merum detrahunt,
                  <reg norm="reſiduique" type="simple">reſiduiq́;</reg>
                radicem ſumunt, qua coniuncta vni dimidio primi nu-
                  <lb/>
                meri, pars maior datur, ex altero verò dimidio detracta, minorem manifeſtabit.</s>
              </p>
              <p>
                <s xml:id="echoid-s391" xml:space="preserve">Exempli gratia, ſi numerus partiendus eſſet .34. alter verò numerus eſſet .64. cui
                  <lb/>
                productum vnius partis in alteram æquale eſſe deberet. </s>
                <s xml:id="echoid-s392" xml:space="preserve">Dimidium primi numeri, in
                  <lb/>
                ſeipſum multiplicaremus, cuius quadratum eſſet .289. de quo detracto ſecundo nu-
                  <lb/>
                mero nempe .64. remaneret .225. cuius quadrata radix nempe .15. coniuncta .17.
                  <lb/>
                dimidio .34. proferet .32. maiorem partem,
                  <reg norm="detractoque" type="simple">detractoq́;</reg>
                ex .17. ſupereſſet .2. pars
                  <lb/>
                inquam minor.</s>
              </p>
              <p>
                <s xml:id="echoid-s393" xml:space="preserve">Cuius ſpeculationis cauſa, primus numerus propoſitus ſignificetur linea
                  <var>.a.d.</var>
                cu-
                  <lb/>
                ius dimidium
                  <var>.c.d.</var>
                cognitum erit, vnà etiam eius quadratum
                  <var>.c.f.</var>
                quo diuiſo per dia
                  <lb/>
                metrum
                  <var>.e.d.</var>
                ſupponantur partes ignotæ
                  <lb/>
                  <figure xlink:label="fig-0041-02" xlink:href="fig-0041-02a" number="57">
                    <image file="0041-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0041-02"/>
                  </figure>
                ipſius
                  <var>.a.d.</var>
                eſſe
                  <var>.a.b.</var>
                et
                  <var>.b.d.</var>
                & à puncto
                  <var>.b.</var>
                  <lb/>
                duci lineam
                  <var>.b.h.g.</var>
                parallelam
                  <var>.d.f.</var>
                et
                  <var>.m.
                    <lb/>
                  h.k.</var>
                parallelam
                  <var>.d.a.</var>
                extructa figura ſimi
                  <lb/>
                li figuræ quintæ ſecundi Eucli. </s>
                <s xml:id="echoid-s394" xml:space="preserve">quare da
                  <lb/>
                bitur
                  <reg norm="gnomon" type="context">gnomõ</reg>
                  <var>.l.d.g.</var>
                æqualis producto
                  <var>.b.
                    <lb/>
                  k.</var>
                & proinde cognitus, quo detracto è
                  <lb/>
                quadrato,
                  <var>c.f.</var>
                remanebit quadratum
                  <var>.g.l.</var>
                  <lb/>
                cuius radice æquali
                  <var>.c.b.</var>
                coniuncta
                  <var>.a.c.</var>
                  <lb/>
                & detracta ex
                  <var>.c.d.</var>
                partes
                  <var>.a.b.</var>
                et
                  <var>.b.d.</var>
                quæſitæ dabuntur.</s>
              </p>
            </div>
            <div xml:id="echoid-div98" type="math:theorem" level="3" n="46">
              <head xml:id="echoid-head62" xml:space="preserve">THEOREMA
                <num value="46">XLVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s395" xml:space="preserve">CVR propoſitis tribus numeris, quorum prior in duas eiuſmodi partes diui-
                  <lb/>
                dendus ſit, ut mutuò diuiſæ, & per ſummam prouenientium diuiſo ſecundo
                  <lb/>
                numero, proueniens vltimum ſit æquale tertio numerorum propoſitorum. </s>
                <s xml:id="echoid-s396" xml:space="preserve">Conſul
                  <lb/>
                tiſsimum ſit ſecundum numerum per tertium diuidere, ex quo proueniens ſit ſum-
                  <lb/>
                ma prouenientium è duabus partibus mutuò diuiſis, quam ſummam ſi quis velit di-
                  <lb/>
                ſtinguere, rectè poſſit medio operationis
                  <reg norm="pręcedentis" type="context">pręcedẽtis</reg>
                theorematis
                  <reg norm="sumpta" type="context">sũpta</reg>
                vnitate ſuper
                  <lb/>
                ficiali pro ſecundo numero diſtinctis poſtmodum prouenientibus, rectè meo iudi-
                  <lb/>
                cio operabimur per
                  <reg norm="regulam" type="context">regulã</reg>
                de tribus (quod fuit ab antiquis prætermiſſum) Si dixe- </s>
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