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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THEOREM. ARIT.
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              <p>
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                eſſe gnomoni
                  <var>.e.c.u.</var>
                  <reg norm="itemque" type="simple">itemq́;</reg>
                gnomonem
                  <var>.b.f.d.</var>
                æqualem gnomoni
                  <var>.b.o.d.</var>
                at hic gno-
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                mon
                  <var>.b.o.d.</var>
                ex præſuppoſito, maior eſt gnomone
                  <var>.e.o.u.</var>
                duabus vnitatibus
                  <var>.b.</var>
                et
                  <var>.d.</var>
                  <lb/>
                Itaque etiam gnomon
                  <var>.b.f.d.</var>
                duabus vnitatibus gnomonem
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                ſuperabit. </s>
                <s xml:id="echoid-s777" xml:space="preserve">Qua-
                  <lb/>
                re
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                erit impar immediatè ſequens ternarium, qui coniunctus quadrato
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                  <lb/>
                quadratum ſubſequens componet. </s>
                <s xml:id="echoid-s778" xml:space="preserve">Eadem ratione probabitur de quadrato
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                ſe
                  <lb/>
                quenti
                  <var>.o.f.</var>
                & gnomone
                  <var>.i.n.a.</var>
                cum hic ordo ſpeculationis ſit vniuerſalis. </s>
                <s xml:id="echoid-s779" xml:space="preserve">In
                  <lb/>
                quo cernitur quemlibet gnomonem ſibi
                  <reg norm="contiguum" type="context">contiguũ</reg>
                inferiorem ſemper duabus vni-
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                tat ibus excedere, cumque quadrata non niſi gnomonibus ſibi inuicem ſuccedant.
                  <lb/>
                </s>
                <s xml:id="echoid-s780" xml:space="preserve">Sed
                  <reg norm="cum" type="context">cũ</reg>
                primus
                  <var>.e.c.u.</var>
                diſpar fuerit,
                  <reg norm="proculdubio" type="simple">ꝓculdubio</reg>
                  <reg norm="etiam" type="context">etiã</reg>
                  <reg norm="neceſſarioque" type="simple">neceſſarioq́;</reg>
                cæteri diſpares
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                .
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                <s xml:id="echoid-s781" xml:space="preserve">Ex qua ſpeculatione, oritur regula ab antiquis tradita
                  <lb/>
                inueniendi vltimi numeri diſparis
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                ad
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                    <anchor type="figure" xlink:label="fig-0071-01a" xlink:href="fig-0071-01"/>
                  ſitionem</reg>
                alicuius quadrati. </s>
                <s xml:id="echoid-s782" xml:space="preserve">Vt ſi quis ſeire deſideret nu-
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                merum vltimum diſparem, quo mediante quadratum
                  <var>.
                    <lb/>
                  o.n.</var>
                conſtitutum fuit, quod aliud non eſt quam ſcire
                  <lb/>
                quantus ſit numerus vltimi gnomonis
                  <var>.i.n.a.</var>
                æqualis gno
                  <lb/>
                moni
                  <var>.i.o.a</var>
                . </s>
                <s xml:id="echoid-s783" xml:space="preserve">Itaque vt ſciamus hunc gnomonem
                  <var>.i.o.a.</var>
                  <lb/>
                patet duplicandam eſſe radicem
                  <var>.o.e.b.i.</var>
                  <reg norm="dabiturque" type="simple punctuation">dabiturq́,</reg>
                  <var>.o.e.
                    <lb/>
                  b.i.</var>
                et
                  <var>.o.u.d.a.</var>
                vbi bis reperitur
                  <var>.o.</var>
                nos autem tantummo
                  <lb/>
                do quærimus ſcire gnomonem .i.b.e.o.u.d.a. </s>
                <s xml:id="echoid-s784" xml:space="preserve">Itaque
                  <lb/>
                minor eſt vnitate duplo radicis, cum unitas
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                bis repe-
                  <lb/>
                tatur, quæ tamen in gnomone ſemel tantum ſumebatur.</s>
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              <head xml:id="echoid-head108" xml:space="preserve">THEOREMA
                <num value="91">XCI</num>
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              <p>
                <s xml:id="echoid-s785" xml:space="preserve">CVR ſumma quadratorum, quorum radices ſunt in proportione ſeſquitertia
                  <lb/>
                nempe .4. ad .3. quadrata ſit.</s>
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              <p>
                <s xml:id="echoid-s786" xml:space="preserve">Exempli gratia, ſumemus quadratum .3. ſcilicet 9. quod in ſummam cum qua-
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                drato .4. colligemus, nempè .16.
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                quadratum .25. & ita quadratum .6. hoc eſt
                  <num value="36">.
                    <lb/>
                  36.</num>
                collectum cum quadrato .8. nempè .64. efficiet quadratum .100. ita etiam qua-
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                dratum .9. hoceſt .81. coniunctum quadrato .12. nempè .144. producet quadra-
                  <lb/>
                tum .225.</s>
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              <p>
                <s xml:id="echoid-s787" xml:space="preserve">In cuius gratiam ſint duo quadrata ſubſcripta
                  <var>.q.o.</var>
                et
                  <var>.q.a.</var>
                quorum radices ſint
                  <var>.q.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0071-02a" xlink:href="fig-0071-02"/>
                g. et
                  <var>.q.p.</var>
                hoc eſt
                  <var>.q.g.</var>
                quatuor vnitatum, et
                  <var>.q.
                    <lb/>
                  p.</var>
                trium, ex quo
                  <var>.q.a.</var>
                erit .16. vnitatum et
                  <var>.q.o.</var>
                  <lb/>
                nouem. </s>
                <s xml:id="echoid-s788" xml:space="preserve">Ad hæc cogitemus applicari quadra-
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                to
                  <var>.q.a.</var>
                gnomonem
                  <var>.f.s.h.</var>
                tam amplum ſiue la-
                  <lb/>
                tum
                  <reg norm="quam" type="context">quã</reg>
                gnomon
                  <var>.b.a.g.</var>
                nempè vt
                  <var>.h.</var>
                ſit æqua
                  <lb/>
                lis .g: g. verò differentia ſit qua
                  <var>.q.g.</var>
                maior eſt
                  <var>.
                    <lb/>
                  q.p.</var>
                  <reg norm="huncque" type="simple">huncq́;</reg>
                gnomonem
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                dico ęqualem eſ
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                ſe quadrato
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                nam ex preſuppoſito
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                terra
                  <lb/>
                dicem
                  <var>.q.p.</var>
                ingreditur, & quater
                  <var>.q.g.</var>
                ex quo,
                  <lb/>
                tres partes
                  <var>.q.k.p.</var>
                inter ſe æquales ſunt vnde
                  <lb/>
                etiam quadratum
                  <var>.q.o.</var>
                nouem partibus ſuper-
                  <lb/>
                ficialibus quadratis conſtabit, quarum ſingula
                  <lb/>
                rum radix æqualis erit
                  <var>.g.</var>
                cumque præcedenti
                  <lb/>
                theoremate didicerimus quemlibet gnomo-
                  <lb/>
                nem quadrati immediatè ſequentis æquę amplitudinis cum gnomone præcedentis, </s>
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