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
51 39
52 40
53 41
54 42
55 43
56 44
57 45
58 46
59 47
60 48
61 49
62 50
63 51
64 52
65 53
66 54
67 55
68 56
69 57
70 58
71 59
72 60
73 61
74 62
75 63
76 64
77 65
78 66
79 67
80 70
< >
page |< < (44) of 445 > >|
IO. BAPT. BENED.
    <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-div134" type="math:theorem" level="3" n="67">
              <pb o="44" rhead="IO. BAPT. BENED." n="56" file="0056" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0056"/>
            </div>
            <div xml:id="echoid-div136" type="math:theorem" level="3" n="68">
              <head xml:id="echoid-head84" xml:space="preserve">THEOREMA
                <num value="68">LXVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s582" xml:space="preserve">CVR numero per numerum diuiſo,
                  <reg norm="productoque" type="simple">productoq́;</reg>
                duorum numerorum per pro-
                  <lb/>
                ueniens multiplicato, quod vltimò productum eſt, diuiſi numeri ſemper qua
                  <lb/>
                dratum exiſtat.</s>
              </p>
              <p>
                <s xml:id="echoid-s583" xml:space="preserve">Exempli gratia, ſi diuidamus .10. per .2. proueniens erit .5. quo producto ex duo
                  <lb/>
                bus numeris multiplicato, nempe .20. habe
                  <lb/>
                bimus .100. quadratum numeri diuiſi.</s>
              </p>
              <figure position="here">
                <image file="0056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0056-01"/>
              </figure>
              <p>
                <s xml:id="echoid-s584" xml:space="preserve">Cuius gratia duo numeri ſint
                  <var>.a.</var>
                et
                  <var>.e.</var>
                por
                  <lb/>
                  <var>.a.</var>
                per
                  <var>.e.</var>
                diuiſo detur
                  <var>.u.</var>
                tum
                  <var>.o.</var>
                produ-
                  <lb/>
                ctum
                  <var>.a.</var>
                in
                  <var>.e.</var>
                eſſe conſtituatur, quo per
                  <var>.u.</var>
                  <lb/>
                multiplicato dabitur
                  <var>.x.</var>
                quadratum
                  <var>.a.</var>
                pro-
                  <lb/>
                ptereà quòd
                  <var>.a.</var>
                medium eſt proportionale
                  <lb/>
                inter
                  <var>.o.</var>
                et
                  <var>.u.</var>
                ex .35. theoremate. </s>
                <s xml:id="echoid-s585" xml:space="preserve">itaque
                  <lb/>
                ex .16. ſexti aut .20. ſeptimi, propoſiti veri-
                  <lb/>
                tas eluceſcet.</s>
              </p>
            </div>
            <div xml:id="echoid-div137" type="math:theorem" level="3" n="69">
              <head xml:id="echoid-head85" xml:space="preserve">THEOREMA
                <num value="69">LXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s586" xml:space="preserve">CVR numero aliquo per duos alios multiplicato & diuiſo, ſi per horum duo-
                  <lb/>
                rum productum, ſumma duorum primorum productorum diuiſa fuerit, vl-
                  <lb/>
                timum proueniens, ſummæ duorum primorum prouenientium æquale ſit.</s>
              </p>
              <p>
                <s xml:id="echoid-s587" xml:space="preserve">Exempli gratia, proponitur numerus .24. per .8. et .6. multiplicandus & diuiden
                  <lb/>
                dus ſumma productorum crit .336. prouenientium autem .7. ſi igitur ſummam .336.
                  <lb/>
                productorum per productum duorum ſecundorum numerorum nempe .48. diuiſe-
                  <lb/>
                rimus, proueniens pariter erit .7.</s>
              </p>
              <p>
                <s xml:id="echoid-s588" xml:space="preserve">In cuius
                  <reg norm="gratiam" type="context">gratiã</reg>
                primus numerus ſignificetur linea
                  <var>.q.b.</var>
                multiplicandus & diuiden-
                  <lb/>
                dus numeris deſignatis per
                  <var>.k.m.</var>
                et
                  <var>.y.m.</var>
                productorum ſumma ſit
                  <var>.k.z.</var>
                prouenien-
                  <lb/>
                tium autem
                  <var>.a.e</var>
                : et
                  <var>.a.o.</var>
                ex
                  <var>.k.m.</var>
                et
                  <var>.o.e.</var>
                ex
                  <var>.y.m</var>
                : tum productum
                  <var>.k.m.</var>
                in
                  <var>.m.y.</var>
                ſit
                  <var>.f.
                    <lb/>
                  m</var>
                . </s>
                <s xml:id="echoid-s589" xml:space="preserve">Dico quòd ſi
                  <var>.k.z.</var>
                per
                  <var>.f.m.</var>
                diuiſerimus proueni et
                  <var>.a.e</var>
                . </s>
                <s xml:id="echoid-s590" xml:space="preserve">Quod cum ſic fuerit, erit
                  <lb/>
                quoque verum quòd diuiſa
                  <var>.k.z.</var>
                per
                  <var>.a.e.</var>
                proueniet
                  <var>.f.m.</var>
                numerus ſcilicet æqualis
                  <lb/>
                numero
                  <var>.f.m.</var>
                ex .13. theoremate huius. </s>
                <s xml:id="echoid-s591" xml:space="preserve">Itaque quotieſcunque probauero quòd di-
                  <lb/>
                uiſa
                  <var>.k.z.</var>
                per
                  <var>.a.e.</var>
                proueniat numerus æqualis ipſi
                  <var>.f.m.</var>
                propoſitum verum eſſe con
                  <lb/>
                ſequetur. ex .13. theoremate. </s>
                <s xml:id="echoid-s592" xml:space="preserve">Quòd ſi proueniens ex diuiſione
                  <var>.k.z.</var>
                per
                  <var>.a.e.</var>
                æqua
                  <lb/>
                le fuerit
                  <var>.f.m.</var>
                patet ex .7. quinti quòd
                  <reg norm="eadem" type="context">eadẽ</reg>
                erit proportio numeri
                  <var>.k.m.y.</var>
                ad ipſum
                  <lb/>
                proueniens, quæ ad numerum
                  <var>.f.m</var>
                . </s>
                <s xml:id="echoid-s593" xml:space="preserve">Cogitemus
                  <reg norm="itaque" type="simple">itaq;</reg>
                  <var>.k.u.</var>
                æqualem
                  <var>.a.e.</var>
                ſuper quam
                  <lb/>
                mente concipiamus rectangulum
                  <var>.u.p.</var>
                æqualem
                  <var>.k.z.</var>
                ex quo eadem erit proportio
                  <var>.
                    <lb/>
                  k.p.</var>
                ad
                  <var>.k.y.</var>
                quæ
                  <var>.g.k.</var>
                ad
                  <var>.k.u.</var>
                ex .15. ſexti, aut, 20. ſeptimi, numerus autem
                  <var>.k.p.</var>
                erit
                  <lb/>
                proueniens, quod probandum eſt æquale eſſe
                  <var>.f.m</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s594" xml:space="preserve">Probabitur autem ſic, ex .9. quinti, nempe demonſtrato quòd numerus
                  <var>.k.p.</var>
                ean
                  <lb/>
                dem proportionem habeat ad numerum
                  <var>.k.y.</var>
                quam habet numerus
                  <var>.f.m.</var>
                ad eundem
                  <lb/>
                  <var>k.y</var>
                . </s>
                <s xml:id="echoid-s595" xml:space="preserve">Sed probatum eſt ſic ſe habere
                  <var>.k.g.</var>
                ad
                  <var>.k.u.</var>
                ſicut
                  <var>.k.p.</var>
                ad
                  <var>.k.y.</var>
                ſufficiet igitur pro-
                  <lb/>
                bare ſic ſe habere
                  <var>.k.g.</var>
                ad
                  <var>.k.u.</var>
                ſicut
                  <var>.f.m.</var>
                ad
                  <var>.k.y</var>
                . </s>
                <s xml:id="echoid-s596" xml:space="preserve">Sed
                  <var>.k.g.</var>
                dicitur æqualis eſſe
                  <var>.q.b</var>
                : et
                  <var>.k.</var>
                  <lb/>
                u; </s>
                <s xml:id="echoid-s597" xml:space="preserve">a.e. ſatis erit igitur probare ita ſe habere
                  <var>.q.b.</var>
                ad
                  <var>.a.e.</var>
                ſicut
                  <var>.f.m.</var>
                ad
                  <var>.k.y</var>
                . </s>
                <s xml:id="echoid-s598" xml:space="preserve">Scimus au-
                  <lb/>
                tem quòd eadem eſt proportio
                  <var>.q.b.</var>
                ad
                  <var>.a.o.</var>
                quæ
                  <var>.m.k.</var>
                ad vnitatem, quæ ſit
                  <var>.x.</var>
                & quod
                  <lb/>
                proportio
                  <var>.o.e.</var>
                ad
                  <var>.q.b.</var>
                eadem eſt, quæ
                  <var>.x.</var>
                ad
                  <var>.m.y.</var>
                ex definitione diuiſionis. </s>
                <s xml:id="echoid-s599" xml:space="preserve">Quare
                  <lb/>
                ex æqualitate proportionum eadem erit proportio
                  <var>.k.m.</var>
                ad
                  <var>.m.y.</var>
                quæ
                  <var>.e.o.</var>
                ad
                  <var>.o.a.</var>
                & </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>