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]

Page concordance

< >
Scan Original
71 59
72 60
73 61
74 62
75 63
76 64
77 65
78 66
79 67
80 70
81 71
82 70
83 71
84 72
85 73
86 74
87 75
88 76
89 77
90 78
91 79
92 80
93 81
94 82
95 89
96 84
97 85
98 96
99 87
100 88
< >
page |< < (30) of 445 > >|
    <echo version="1.0">
      <text type="book" xml:lang="la">
        <div xml:id="echoid-div7" type="body" level="1" n="1">
          <div xml:id="echoid-div7" type="chapter" level="2" n="1">
            <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"/>
                  </figure>
                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>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum