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. ARITH.
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              <p>
                <s xml:id="echoid-s703" xml:space="preserve">
                  <pb o="53" rhead="THEOREM. ARITH." n="65" file="0065" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0065"/>
                proportione diuidentium, quamuis ex aduerſo.</s>
              </p>
              <p>
                <s xml:id="echoid-s704" xml:space="preserve">Cuius ratio ex .15. ſexti aut .20. ſeptimi dependet. </s>
                <s xml:id="echoid-s705" xml:space="preserve">prout in ſubſcripto ordine fa-
                  <lb/>
                cillimè deprehendi poteſt.</s>
              </p>
            </div>
            <div xml:id="echoid-div159" type="math:theorem" level="3" n="81">
              <head xml:id="echoid-head98" xml:space="preserve">THEOREMA
                <num value="81">LXXXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s706" xml:space="preserve">CVR quantitate in tres continuas partes proportionales ſecta, & per ſingulas
                  <lb/>
                ipſarum diuiſa, ſumma trium prouenientium quadrato medij prouenientis
                  <lb/>
                æqualis eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s707" xml:space="preserve">Exempli gratia, proponitur .14. diuidendus in tres continuas partes proportio-
                  <lb/>
                nales, nempe .8. 4. 2.
                  <reg norm="ipſeque" type="simple">ipſeq́;</reg>
                numerus .14. per ſingulas diuiditur, ex quo tria proue-
                  <lb/>
                nientia oriuntur, nempe ex prima parte .8.
                  <reg norm="proueniens" type="context">proueniẽs</reg>
                erit .1. cum tribus quartis par
                  <lb/>
                tibus ex ſecunda .4. datur proueniens .3. cum dimidio vnius, & ex tertia .2. proue-
                  <lb/>
                nient .7. integri, qui in ſummam collecti dant .12. integros & vnam quartam par-
                  <lb/>
                tem tantumdem, videlicet quantum quadratum prouenientis medij, nempe .3.
                  <lb/>
                cum dimidio.</s>
              </p>
              <p>
                <s xml:id="echoid-s708" xml:space="preserve">Cuius ſpeculationis gratia, totalis numerus ſignificetur linea
                  <var>.n.c.</var>
                qui in tres par-
                  <lb/>
                tes diuidatur
                  <var>.n.a</var>
                :
                  <var>a.e.</var>
                et
                  <var>.e.c.</var>
                quæ ſint continuæ proportionales, quarum ſingulis,
                  <lb/>
                numerum
                  <var>.n.c.</var>
                diuiſum eſſe cogitemus, proueniens autem ex diuiſione
                  <var>.n.c.</var>
                per
                  <var>.n.
                    <lb/>
                  a.</var>
                ſit
                  <var>.i.d.</var>
                quod verò prouenit ex diuiſione
                  <var>.n.c.</var>
                per
                  <var>.a.e.</var>
                ſit
                  <var>.d.u.</var>
                proueniens quoque ex
                  <lb/>
                diuiſione
                  <var>.n.c.</var>
                per
                  <var>.e.c.</var>
                ſit
                  <var>.u.o.</var>
                quorum ſumma ſit
                  <var>.i.o.</var>
                quæ aſſeritur eſſe numeri æqua-
                  <lb/>
                lis numero quadrati
                  <var>.d.u</var>
                . </s>
                <s xml:id="echoid-s709" xml:space="preserve">Quod hac ratione probabo, producatur linea
                  <var>.i.o.</var>
                donec
                  <var>.
                    <lb/>
                  o.p.</var>
                æqualis ſit
                  <var>.o.u.</var>
                  <reg norm="erigaturque" type="simple">erigaturq́;</reg>
                  <var>.o.m.</var>
                æqualis
                  <var>.d.i.</var>
                perpendiculariter
                  <var>.o.p.</var>
                in puncto
                  <var>.o.</var>
                  <lb/>
                quæ producatur donec
                  <var>.o.q.</var>
                vnitati ſit æqualis,
                  <reg norm="terminenturque" type="simple">terminenturq́;</reg>
                duo rectangula
                  <var>.m.p.</var>
                  <lb/>
                et
                  <var>.q.i.</var>
                ex quo habebimus rectangulum, aut productum
                  <var>.m.p.</var>
                æquale quadrato
                  <var>.d.u.</var>
                  <lb/>
                ex .16 ſexti aut .20. ſeptimi, quandoquidem tria prouenientia
                  <var>.o.u</var>
                :
                  <var>u.d.</var>
                et
                  <var>.d.i.</var>
                ex
                  <lb/>
                pręcedenti theoremate ſunt inter ſe continua proportionalia, proportionalitate qua
                  <lb/>
                partes
                  <var>.n.c</var>
                . </s>
                <s xml:id="echoid-s710" xml:space="preserve">Iam verò ſi probauero
                  <var>.q.i.</var>
                productum, producto
                  <var>.m.p.</var>
                æquale eſſe, pro-
                  <lb/>
                poſitum quoque probatum erit. </s>
                <s xml:id="echoid-s711" xml:space="preserve">Numerus enim producti
                  <var>.q.i.</var>
                æqualis eſt numero.
                  <lb/>
                </s>
                <s xml:id="echoid-s712" xml:space="preserve">ſummæ
                  <var>.i.o</var>
                . </s>
                <s xml:id="echoid-s713" xml:space="preserve">Habemus autem ex definitione diuiſionis ita ſe habere
                  <var>.n.c.</var>
                ad
                  <var>.i.d.</var>
                ſicut
                  <var>.
                    <lb/>
                  n.a.</var>
                ad
                  <var>.o.q</var>
                . </s>
                <s xml:id="echoid-s714" xml:space="preserve">Itaque permutando ſic ſe habebit
                  <var>.n.c.</var>
                ad
                  <var>.n.a.</var>
                ſicut
                  <var>.d.i.</var>
                hoc eſt
                  <var>.m.o.</var>
                ad
                  <var>.
                    <lb/>
                  o.q.</var>
                ſed ſicut ſe habet
                  <var>.n.c.</var>
                ad
                  <var>.n.a.</var>
                ita pariter ſe habet
                  <var>.i.o.</var>
                ad
                  <var>.o.u.</var>
                hoc eſt ad
                  <var>.o.p</var>
                . </s>
                <s xml:id="echoid-s715" xml:space="preserve">Ita-
                  <lb/>
                que
                  <var>.i.o.</var>
                ad
                  <var>.o.p.</var>
                ſic ſe habebit ſicut
                  <var>.m.o.</var>
                ad
                  <var>.o.q.</var>
                ex quo ex .15. ſexti aut .20. ſeptimi
                  <var>.
                    <lb/>
                  q.i.</var>
                æqualis erit
                  <var>.m.p.</var>
                & conſequenter quadrato
                  <var>.d.u</var>
                . </s>
                <s xml:id="echoid-s716" xml:space="preserve">Vt autem lector minori labo-
                  <lb/>
                re cognoſcere queat
                  <var>.i.o.</var>
                ad
                  <var>.o.u.</var>
                ſic ſe habere, vt
                  <var>.n.c.</var>
                ad
                  <var>.n.a.</var>
                ſciendum eſt quòd, ſic
                  <lb/>
                ſe habet
                  <var>.i.d.</var>
                ad
                  <var>.d.u.</var>
                ut
                  <var>.c.e.</var>
                ad
                  <var>.e.a.</var>
                ex quo componendo ſic ſe habebit
                  <var>.i.u.</var>
                ad
                  <var>.d.u.</var>
                ſi-
                  <lb/>
                cut
                  <var>.c.a.</var>
                ad
                  <var>.a.e.</var>
                & permutando ita
                  <var>.i.u.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0065-01a" xlink:href="fig-0065-01"/>
                ad
                  <var>.c.a.</var>
                vt
                  <var>.d.u.</var>
                ad
                  <var>.e.a.</var>
                ſed cum ex
                  <reg norm="præ- cedenti" type="context">præ-
                    <lb/>
                  cedẽti</reg>
                theoremate ſic ſe habeat
                  <var>.d.u.</var>
                  <lb/>
                ad
                  <var>.u.o.</var>
                ſicut
                  <var>.e.a.</var>
                ad
                  <var>.a.n.</var>
                permutando
                  <lb/>
                ſic ſe habebit
                  <var>.d.u.</var>
                ad
                  <var>.a.e.</var>
                ſicut
                  <var>.u.o.</var>
                ad
                  <lb/>
                  <var>a.n.</var>
                ex quo ex .11. quinti ſic ſe habe-
                  <lb/>
                bit
                  <var>.i.u.</var>
                ad
                  <var>.c.a.</var>
                prout
                  <var>.o.u.</var>
                ad
                  <var>.a.n.</var>
                per-
                  <lb/>
                mutandoq́ue
                  <var>.i.u.</var>
                ad
                  <var>.u.o.</var>
                vt
                  <var>.c.a.</var>
                ad
                  <var>.a.n.</var>
                & componendo, ita
                  <var>.i.o.</var>
                ad
                  <var>.u.o.</var>
                ſicut
                  <var>.c.n.</var>
                  <lb/>
                ad
                  <var>.a.n</var>
                .</s>
              </p>
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                <figure xlink:label="fig-0065-01" xlink:href="fig-0065-01a">
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