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
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
81 71
82 70
83 71
84 72
85 73
86 74
87 75
88 76
89 77
90 78
< >
page |< < (52) 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-div154" type="math:theorem" level="3" n="78">
              <pb o="52" rhead="IO. BAPT. BENED." n="64" file="0064" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0064"/>
              <p>
                <s xml:id="echoid-s688" xml:space="preserve">Sint exempli gratia .4. quantitates
                  <var>.a.b</var>
                :
                  <var>c.d</var>
                :
                  <var>e.f</var>
                : et
                  <var>.g.h</var>
                : inuicem proportionales in
                  <lb/>
                proportionalitate arithmetica. </s>
                <s xml:id="echoid-s689" xml:space="preserve">Hoc eſt vt quæ proportio (licet impropriè dicta)
                  <lb/>
                eſt ipſius
                  <var>.a.b.</var>
                ad
                  <var>.c.d.</var>
                  <reg norm="eadem" type="context">eadẽ</reg>
                ſit ipſius
                  <var>.e.f.</var>
                ad
                  <var>.g.h</var>
                . </s>
                <s xml:id="echoid-s690" xml:space="preserve">Tunc permutando dico eandem pro
                  <lb/>
                portionem fore ipſius
                  <var>.a.b.</var>
                ad
                  <var>.e.f.</var>
                quæ ipſius
                  <var>.c.d.</var>
                ad
                  <var>.g.h</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s691" xml:space="preserve">Nam, ex hypotheſi, differentia qua
                  <var>.a.b.</var>
                ſuperat
                  <var>.c.d.</var>
                (quæ ſit
                  <var>.m.b.</var>
                ) æqualis eſt
                  <lb/>
                differentiæ qua
                  <var>.e.f.</var>
                ſuperat
                  <var>.g.h.</var>
                (quæ ſit
                  <var>.i.f.</var>
                ) vnde
                  <var>.a.m.</var>
                reſiduum ex
                  <var>.a.b.</var>
                æquale erit
                  <lb/>
                  <var>c.d.</var>
                & reſiduum
                  <var>.e.i.</var>
                æquale
                  <var>.g.h</var>
                . </s>
                <s xml:id="echoid-s692" xml:space="preserve">Sit igitur exempli gratia
                  <var>.c.d.</var>
                maior
                  <var>.g.h.</var>
                per
                  <var>.c.n.</var>
                  <lb/>
                vnde
                  <var>.n.d.</var>
                æqualis erit
                  <var>.g.h.</var>
                </s>
                <s xml:id="echoid-s693" xml:space="preserve">quare
                  <var>.a.m.</var>
                maior erit
                  <var>.e.i.</var>
                per
                  <var>.a.K.</var>
                æqualem
                  <var>.c.n.</var>
                ex com-
                  <lb/>
                muni ſcientia. </s>
                <s xml:id="echoid-s694" xml:space="preserve">Vnde
                  <var>.K.m.</var>
                æqualis erit
                  <var>.n.d.</var>
                hoc eſt ipſi
                  <var>.g.h.</var>
                hoc eſt ipſi
                  <var>e.i</var>
                . </s>
                <s xml:id="echoid-s695" xml:space="preserve">Quare ex
                  <lb/>
                communi conceptu
                  <var>.b.K.</var>
                æqualis erit ipſi
                  <var>.f.e.</var>
                ſed
                  <var>.n.d.</var>
                æqualis eſt
                  <var>.g.h.</var>
                vt dictum eſt.
                  <lb/>
                </s>
                <s xml:id="echoid-s696" xml:space="preserve">Cum ergo
                  <var>.b.K.</var>
                æqualis ſit
                  <var>.e.f.</var>
                et
                  <var>.d.n.</var>
                ipſi
                  <var>.g.h.</var>
                et
                  <var>.a.b.</var>
                maior ſit ipſa
                  <var>.K.b.</var>
                per
                  <var>.a.K.</var>
                æqua-
                  <lb/>
                lem ipſi
                  <var>.c.n.</var>
                per quam
                  <var>c.n</var>
                :
                  <var>d.c.</var>
                maior eſt ipſa
                  <var>.d.n.</var>
                ſequitur verum eſſe
                  <reg norm="propoſitum" type="context">propoſitũ</reg>
                hoc
                  <lb/>
                eſt, quod eadem proportio ſit ipſius
                  <var>.a.b.</var>
                ad
                  <var>.e.f.</var>
                quæ
                  <var>.c.d.</var>
                ad
                  <var>.g.h.</var>
                arithmetice ſcilicet.</s>
              </p>
              <figure position="here" number="87">
                <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-01"/>
              </figure>
            </div>
            <div xml:id="echoid-div155" type="math:theorem" level="3" n="79">
              <head xml:id="echoid-head96" xml:space="preserve">THEOREMA
                <num value="79">LXXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s697" xml:space="preserve">CVR prouenientia duorum numerorum diuidentium eiuſdem numeri diuiſi-
                  <lb/>
                bilis, geometricè
                  <reg norm="eandem" type="context">eandẽ</reg>
                inter ſe
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                ſeruant,
                  <reg norm="quam" type="context">quã</reg>
                ipſimet
                  <reg norm="diuidentes" type="context">diuidẽtes</reg>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s698" xml:space="preserve">Exempli gratia ſi per ſenarium & octonarium numerus vigintiquatuor diuida-
                  <lb/>
                tur, prouenientia erunt .4. et .3. eadem proportione, qua diuidentes.</s>
              </p>
              <p>
                <s xml:id="echoid-s699" xml:space="preserve">Cuius eſt ratio numerus diuiſibilis ſignificetur rectangulis
                  <var>.u.x.</var>
                et
                  <var>.n.e.</var>
                diuidentes
                  <lb/>
                autem ſint
                  <var>.u.o.</var>
                et
                  <var>.e.o.</var>
                </s>
                <s xml:id="echoid-s700" xml:space="preserve">quare ex ijs, quæ .10.
                  <lb/>
                  <figure xlink:label="fig-0064-02" xlink:href="fig-0064-02a" number="88">
                    <image file="0064-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-02"/>
                  </figure>
                theoremate dicta fuerunt
                  <var>.u.x.</var>
                per
                  <var>.u.o.</var>
                diui-
                  <lb/>
                ſo dabit
                  <var>.x.o.</var>
                & diuiſo
                  <var>.n.e.</var>
                per
                  <var>.e.o.</var>
                dabit
                  <var>.o.
                    <lb/>
                  n</var>
                . </s>
                <s xml:id="echoid-s701" xml:space="preserve">Dicimus itaque
                  <reg norm="eandem" type="context">eandẽ</reg>
                eſſe
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                  <lb/>
                  <var>o.x.</var>
                ad
                  <var>.o.n.</var>
                quæ
                  <var>.e.o.</var>
                ad
                  <var>.o.u.</var>
                quod patet ſub
                  <lb/>
                ſcriptam figuram conſiderantibus, in qua,
                  <lb/>
                ex .15. ſexti aut .20. ſeptimi, eadem propor-
                  <lb/>
                tio cernitur
                  <var>.o.x.</var>
                ad
                  <var>.o.n.</var>
                quæ
                  <var>.o.e.</var>
                ad
                  <var>.o.u</var>
                .</s>
              </p>
            </div>
            <div xml:id="echoid-div157" type="math:theorem" level="3" n="80">
              <head xml:id="echoid-head97" xml:space="preserve">THEOREMA
                <num value="80">LXXX</num>
              .</head>
              <p>
                <s xml:id="echoid-s702" xml:space="preserve">CVR quauis quantitate, tribus
                  <lb/>
                  <figure xlink:label="fig-0064-03" xlink:href="fig-0064-03a" number="89">
                    <image file="0064-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-03"/>
                  </figure>
                aut quatuor aut etiam pro libi-
                  <lb/>
                to pluribus diuidentibus numeris di-
                  <lb/>
                uifa, prouenientia eandem prorſus
                  <lb/>
                inter ſe proportionem ſeruabunt,
                  <lb/>
                quam ipſi diuidentes habere compe
                  <lb/>
                riuntur.</s>
              </p>
              <p>
                <s xml:id="echoid-s703" xml:space="preserve">Exempli gratia, proponitur nu-
                  <lb/>
                merus .60. quinque numeris diuiden
                  <lb/>
                dus, vtpotè .30. 20. 15. 12. 10. pro-
                  <lb/>
                uenientia erunt .2. 3. 4. 5. 6. eadem </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>