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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            <div xml:id="echoid-div98" type="math:theorem" level="3" n="46">
              <p>
                <s xml:id="echoid-s396" xml:space="preserve">
                  <pb o="30" rhead="IO. BAPT. BENED." n="42" file="0042" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0042"/>
                rimus, ſi ſumma vnius dictorum prouenientium cum vnitate dat primum numerum,
                  <lb/>
                quid ipſa eadem vnitas dabit? </s>
                <s xml:id="echoid-s397" xml:space="preserve">ex quo propoſitum oriatur.</s>
              </p>
              <p>
                <s xml:id="echoid-s398" xml:space="preserve">Exempli gratia, proponuntur tres numeri, primus .20. ſecundus .34. tertius .8.
                  <lb/>
                Iam quærimus diuidere primum .20. in duas partes quæ mutuò diuiſæ prębeant duo
                  <lb/>
                prouenientia, quorum ſumma tanta ſit vt per eam diuiſo .34. proueniat numerus
                  <lb/>
                æqualis tertio numero .8. </s>
                <s xml:id="echoid-s399" xml:space="preserve">Quod vt præſtemus iubet regula ſecundum .34. per
                  <reg norm="tertium" type="context">tertiũ</reg>
                  <num value="8">.
                    <lb/>
                  8.</num>
                diuidi, vnde proueniet .4. cum vna quarta parte, quod proueniens erit ſumma pro
                  <lb/>
                uenientium ex diuiſione duarum partium quæſitarum, quæ ſi diſtinguere volueri-
                  <lb/>
                mus, præcedentis theorematis methodum ſequemur, vnitate ſuperficiali pro ſecun
                  <lb/>
                do numero propoſito ſumpta, ac ſi diceremus, diuidatur .4. cum vna quarta parte
                  <lb/>
                in duas eiuſmodi partes, vt productum vnius in alteram ſit vnitas ſuperficialis, cer-
                  <lb/>
                tè fractis integris cum quarta parte coniungendis, darentur vnitatis decemſeptem
                  <lb/>
                quartæ lineares, verum cum neceſſe ſit, ex præcedenti theoremate, dimidium in
                  <lb/>
                ſeipſum multiplicare,
                  <reg norm="eſſetque" type="simple">eſſetq́;</reg>
                dimidium .8. quartarum partium cum octaua, com-
                  <lb/>
                modius totum conſtituetur .34. octauarum, quarum dimidium, nempe decemſep-
                  <lb/>
                tem octauæ, in ſeipſum multiplicatum erunt .289. ſexageſimæ quartæ vnius integri
                  <lb/>
                ſuperficialis, quandoquidem
                  <reg norm="integrum" type="context">integrũ</reg>
                ſuperficiale, cuius vnitas linearis in .8. partes
                  <lb/>
                diuiditur eſt .64. vt ex primo theoremate huius libri depræhendi poteſt. </s>
                <s xml:id="echoid-s400" xml:space="preserve">Nunc vni-
                  <lb/>
                tate hac ſuperficiali, nempe .64. ex .289. detracta, ſupererit .225. cuius radix qua-
                  <lb/>
                drata, ſcilicet .15. coniuncta dimidio dictorum prouenientium, nempe .17. dabit
                  <lb/>
                maius proueniens .32.
                  <reg norm="detractaque" type="simple">detractaq́;</reg>
                ex altero dimidio, dabit proueniens minus .2. hoc
                  <lb/>
                eſt pro maiore proueniente .32. octauas, & pro minore duas, quatuor ſcilicet inte-
                  <lb/>
                gros pro maiore, & quartam partem vnius integri pro minore. </s>
                <s xml:id="echoid-s401" xml:space="preserve">Nunc ſi ex regula
                  <lb/>
                de tribus dixerimus, ſi .4. iuncta vni, nempe .5. dant .20. primum numerum, quid
                  <lb/>
                dabunt .4. integra (proueniens inquam maius)
                  <reg norm="dabunt" type="context">dabũt</reg>
                certè .16. partem maiorem.
                  <lb/>
                </s>
                <s xml:id="echoid-s402" xml:space="preserve">Tum ſi dixerimus, ſi quarta pars coniuncta vnitati dat .20: </s>
                <s xml:id="echoid-s403" xml:space="preserve">quid dabit quarta illa
                  <lb/>
                pars (hoc eſt proueniens minus) dabit
                  <reg norm="profectò" type="simple">ꝓfectò</reg>
                quatuor ſcilicet
                  <reg norm="minorem" type="context">minorẽ</reg>
                partem, quod
                  <lb/>
                ab antiquis certè ignoratum fuit, qui, inuentis prouenientibus quieuerunt, ne-
                  <lb/>
                ſcientes ijs vti ad inueniendas duas primi numeri partes.</s>
              </p>
              <p>
                <s xml:id="echoid-s404" xml:space="preserve">Cuius ſpeculationis gratia, demus primum numerum ſignificari linea
                  <var>.e.u.</var>
                cuius
                  <lb/>
                partes
                  <var>.e.a.</var>
                &
                  <var>a.u.</var>
                ſint quæ quæruntur, alter verò numerus ſignificetur linea
                  <var>.b.
                    <lb/>
                  d.</var>
                tertius linea
                  <var>.g.f.</var>
                proueniens
                  <reg norm="autem" type="wordlist">aũt</reg>
                diuiſionis
                  <var>.e.a.</var>
                per
                  <var>.a.u.</var>
                ſit
                  <var>.n.t.</var>
                diuiſionis
                  <reg norm="autem" type="wordlist">aũt</reg>
                  <var>.a.u.</var>
                  <lb/>
                per
                  <var>.a.e.</var>
                ſit
                  <var>.t.o.</var>
                ſumma erit
                  <var>.n.t.o.</var>
                vnitas verò
                  <var>.n.i.</var>
                et
                  <var>.o.i</var>
                . </s>
                <s xml:id="echoid-s405" xml:space="preserve">Iam ſi numerus
                  <var>.f.g.</var>
                tertiò
                  <lb/>
                propoſitus ex diuiſione ſecundi per
                  <var>.o.t.n.</var>
                proferri debet. </s>
                <s xml:id="echoid-s406" xml:space="preserve">Ex .13. theoremate patet,
                  <lb/>
                quòd ſi
                  <var>.b.d.</var>
                per
                  <var>.g.f.</var>
                diuiſerimus, proferetur
                  <var>.o.t.n.</var>
                qui cum fuerit inuentus,
                  <reg norm="ſummam" type="context">ſummã</reg>
                  <lb/>
                eſſe oportet
                  <reg norm="duorum" type="context">duorũ</reg>
                  <reg norm="prouenientium" type="context">prouenientiũ</reg>
                , ex diuiſione mutua
                  <reg norm="duorum" type="context">duorũ</reg>
                numerorum, nempe
                  <var>.
                    <lb/>
                  a.e.</var>
                per
                  <var>.a.u.</var>
                et
                  <var>.a.u.</var>
                per
                  <var>.a.e.</var>
                deinde manifeſtum eſt ex .24. aut .25. theoremate
                  <reg norm="eorum" type="context">eorũ</reg>
                  <lb/>
                productum (multiplicatis prouenientibus adinuicem) vnitatem ſuperficialem futu
                  <lb/>
                ram eſſe. </s>
                <s xml:id="echoid-s407" xml:space="preserve">Hactenus igitur, totum
                  <var>.o.n.</var>
                ex doctrina præcedentis theorematis diui-
                  <lb/>
                ditur in puncto
                  <var>.t.</var>
                ita vt productum
                  <var>.o.t.</var>
                in
                  <var>.t.n.</var>
                  <lb/>
                ſolam vnitatem ſuperficialem
                  <reg norm="contineat" type="context">cõtineat</reg>
                , quo
                  <lb/>
                  <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a" number="58">
                    <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0042-01"/>
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                facto, ſi, vt antedictum eſt, cogitauerimus
                  <var>.n.
                    <lb/>
                  t.</var>
                  <reg norm="proueniens" type="context">proueniẽs</reg>
                eſſe ex diuiſione
                  <var>.e.a.</var>
                per
                  <var>.a.u.</var>
                et
                  <var>.
                    <lb/>
                  t.o.</var>
                proueniens ex diuiſione
                  <var>.a.u.</var>
                per
                  <var>.a.e.</var>
                pa-
                  <lb/>
                tebit ex definitione diuiſionis, quod eadem
                  <lb/>
                erit proportio
                  <var>.a.e.</var>
                ad
                  <var>.n.t.</var>
                quæ eſt
                  <var>.a.u.</var>
                ad vni-
                  <lb/>
                tatem
                  <var>.n.i.</var>
                et
                  <var>.a.u.</var>
                ad
                  <var>.o.t.</var>
                eadem quæ eſt
                  <var>.e.a.</var>
                </s>
              </p>
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