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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THEOREM. ARIT.
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            <div xml:id="echoid-div177" type="math:theorem" level="3" n="90">
              <p>
                <s xml:id="echoid-s776" xml:space="preserve">
                  <pb o="59" rhead="THEOREM. ARIT." n="71" file="0071" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0071"/>
                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-
                  <lb/>
                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
                  <var>.e.c.u.</var>
                ſuperabit. </s>
                <s xml:id="echoid-s777" xml:space="preserve">Qua-
                  <lb/>
                re
                  <var>.b.f.d.</var>
                erit impar immediatè ſequens ternarium, qui coniunctus quadrato
                  <var>.o.c.</var>
                  <lb/>
                quadratum ſubſequens componet. </s>
                <s xml:id="echoid-s778" xml:space="preserve">Eadem ratione probabitur de quadrato
                  <var>.o.n.</var>
                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-
                  <lb/>
                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
                  <reg norm="erunt" type="context">erũt</reg>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s781" xml:space="preserve">Ex qua ſpeculatione, oritur regula ab antiquis tradita
                  <lb/>
                inueniendi vltimi numeri diſparis
                  <reg norm="concurrentis" type="context">cõcurrentis</reg>
                ad
                  <reg norm="compo­ ſitionem" type="context">cõpo­
                    <lb/>
                    <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-
                  <lb/>
                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
                  <var>.o.</var>
                bis repe-
                  <lb/>
                tatur, quæ tamen in gnomone ſemel tantum ſumebatur.</s>
              </p>
              <div xml:id="echoid-div177" type="float" level="4" n="1">
                <figure xlink:label="fig-0071-01" xlink:href="fig-0071-01a">
                  <image file="0071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0071-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div179" type="math:theorem" level="3" n="91">
              <head xml:id="echoid-head108" xml:space="preserve">THEOREMA
                <num value="91">XCI</num>
              .</head>
              <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>
              </p>
              <p>
                <s xml:id="echoid-s786" xml:space="preserve">Exempli gratia, ſumemus quadratum .3. ſcilicet 9. quod in ſummam cum qua-
                  <lb/>
                drato .4. colligemus, nempè .16.
                  <reg norm="eritque" type="simple">eritq́;</reg>
                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-
                  <lb/>
                dratum .9. hoceſt .81. coniunctum quadrato .12. nempè .144. producet quadra-
                  <lb/>
                tum .225.</s>
              </p>
              <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-
                  <lb/>
                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
                  <var>.f.s.h.</var>
                dico ęqualem eſ
                  <lb/>
                ſe quadrato
                  <var>.q.o.</var>
                nam ex preſuppoſito
                  <var>.g.</var>
                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>
              </p>
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