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
161 149
162 150
163 151
164 152
165 153
166 154
167 155
168 156
169 157
170 158
171 159
172 160
173 161
174 162
175 163
176 164
177 165
178 166
179 167
180 168
181 169
182 170
183 171
184 172
185 173
186 174
187 175
188 176
189 177
190 178
< >
page |< < (89) 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-div250" type="math:theorem" level="3" n="131">
              <pb o="89" rhead="THEOR. ARITH." n="101" file="0101" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0101"/>
            </div>
            <div xml:id="echoid-div252" type="math:theorem" level="3" n="132">
              <head xml:id="echoid-head150" xml:space="preserve">THEOREMA
                <num value="132">CXXXII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1156" xml:space="preserve">SED quia aliquis poſſet in dubium reuocare, an poſſibile ſit inuenire tertium
                  <lb/>
                terminum rationalem, ſeu communicantem duobus datis terminis inter ſe com
                  <lb/>
                municantibus in tali proportionalitate, hoc eſt harmonica. </s>
                <s xml:id="echoid-s1157" xml:space="preserve">Vthoc oſtendatur.</s>
              </p>
              <p>
                <s xml:id="echoid-s1158" xml:space="preserve">Sint duo termini dati
                  <var>.a.o.</var>
                et
                  <var>.a.e.</var>
                inter ſe communicantes, tertius verò inuentus
                  <lb/>
                ſit
                  <var>.a.c.</var>
                qui maximus, primò, ſit in ea proportionalitate, quem dico communicantem
                  <lb/>
                eſſe cum primis datis.</s>
              </p>
              <p>
                <s xml:id="echoid-s1159" xml:space="preserve">Nam ex conditionibus huiuſmodi proportionalitatis, habebimus primum ean-
                  <lb/>
                dem proportionem eſſe
                  <var>.a.c.</var>
                ad
                  <var>.a.o.</var>
                quæ eſt
                  <var>.e.c.</var>
                ad
                  <var>.e.o.</var>
                vnde permutando ita erit
                  <var>.a.
                    <lb/>
                  c.</var>
                ad
                  <var>.e.c.</var>
                vt
                  <var>.a.o.</var>
                ad
                  <var>.o.e.</var>
                & quia ex .9. decimi Euclid
                  <var>.a.o.</var>
                communicat cum
                  <var>.o.e.</var>
                </s>
                <s xml:id="echoid-s1160" xml:space="preserve">quare
                  <lb/>
                ex .10. eiuſdem
                  <var>.a.c.</var>
                communicabit cum
                  <var>.e.c.</var>
                & per .9. cum
                  <var>.a.e.</var>
                et per .8. cum
                  <var>.a.o.</var>
                  <lb/>
                quod
                  <unsure/>
                eſt propoſitum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1161" xml:space="preserve">Sed ſi datus fuerit maximus
                  <var>.a.c.</var>
                cum medio
                  <var>.a.e.</var>
                interſe communicantes mini-
                  <lb/>
                mum verò
                  <var>.a.o.</var>
                probabo
                  <reg norm="communicantem" type="context">cõmunicantem</reg>
                cum illis eſſe. </s>
                <s xml:id="echoid-s1162" xml:space="preserve">Cogitemus ergo
                  <var>.c.f.</var>
                æqua-
                  <lb/>
                jem eſſe differentiæ
                  <var>.c.e.</var>
                cognitæ, vnde habebimus proportionem,
                  <var>a.c.</var>
                ad
                  <var>.c.f.</var>
                vt
                  <var>.a.o.</var>
                  <lb/>
                ad
                  <var>.o.e.</var>
                & componendo
                  <var>.a.f.</var>
                ad
                  <var>.f.c.</var>
                vt
                  <var>.a.e.</var>
                ad
                  <var>.e.o.</var>
                & quia (ex ſuppoſito).
                  <var>a.c.</var>
                commu-
                  <lb/>
                nicat cum
                  <var>.e.c.</var>
                hoc eſt cum
                  <var>.c.f.</var>
                </s>
                <s xml:id="echoid-s1163" xml:space="preserve">quare
                  <lb/>
                ex eadem .9. dicti decimi
                  <var>.a.f.</var>
                et
                  <var>.f.c.</var>
                  <reg norm="erunt" type="context">erũt</reg>
                  <lb/>
                  <figure xlink:label="fig-0101-01" xlink:href="fig-0101-01a" number="138">
                    <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-01"/>
                  </figure>
                inter ſe communicantes, & per .10.
                  <var>a.e.</var>
                  <lb/>
                communicabit cum
                  <var>.o.e.</var>
                & per .9.
                  <var>a.e.</var>
                cò
                  <lb/>
                municabit cum
                  <var>.a.o.</var>
                vnde per .8.
                  <var>a.o.</var>
                communicabit cum
                  <var>.a.c.</var>
                ſimiliter.</s>
              </p>
            </div>
            <div xml:id="echoid-div254" type="math:theorem" level="3" n="133">
              <head xml:id="echoid-head151" xml:space="preserve">THEOREMA
                <num value="133">CXXXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1164" xml:space="preserve">SED ſi nobis duo extremi termini propoſiti fuerint, & medium inuenire deſide
                  <lb/>
                remus in dicta proportionalitate, ita faciendum erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1165" xml:space="preserve">Sint, exempli gratia, duo termini dati
                  <var>.q.b.</var>
                et
                  <var>.b.r.</var>
                minor
                  <var>.b.r.</var>
                ex maiori
                  <var>.b.q.</var>
                de-
                  <lb/>
                trahatur, reſiduum verò
                  <var>.q.x.</var>
                multiplicetur per
                  <var>.b.r.</var>
                productum poſteà diuidatur per
                  <lb/>
                  <var>q.r.</var>
                vnde proueniet nobis
                  <var>.x.l.</var>
                pro differentia minori, quæ addita cum
                  <var>.b.x.</var>
                minimo
                  <lb/>
                termino, dabit nobis
                  <var>.b.l.</var>
                mcdium terminum harmonicum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1166" xml:space="preserve">Pro cuius ratione cogitemus dictum medium terminum
                  <var>.b.l.</var>
                iam inuentum eſſe,
                  <lb/>
                vnde ita erit proportio
                  <var>.q.l.</var>
                ad
                  <var>.l.x.</var>
                vt
                  <var>.q.b.</var>
                ad
                  <var>.b.r.</var>
                ex forma huius proportionalitatis,
                  <lb/>
                </s>
                <s xml:id="echoid-s1167" xml:space="preserve">quare coniunctim ita erit
                  <var>.q.r.</var>
                ad
                  <var>.r.b.</var>
                vt
                  <lb/>
                  <var>q.x.</var>
                ad
                  <var>.x.l.</var>
                & proptereà ex .20. ſeptimi
                  <lb/>
                  <figure xlink:label="fig-0101-02" xlink:href="fig-0101-02a" number="139">
                    <image file="0101-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0101-02"/>
                  </figure>
                productum, quod fit ex
                  <var>.q.r.</var>
                in
                  <var>.x.l.</var>
                æqua-
                  <lb/>
                le erit producto
                  <var>.q.x.</var>
                in
                  <var>.b.r</var>
                . </s>
                <s xml:id="echoid-s1168" xml:space="preserve">Rectè igitur
                  <lb/>
                fit cum diuiditur hoc productum per
                  <var>.q.r.</var>
                vt proueniat nobis
                  <var>.x.l.</var>
                differentia minor.</s>
              </p>
            </div>
            <div xml:id="echoid-div256" type="math:theorem" level="3" n="134">
              <head xml:id="echoid-head152" xml:space="preserve">THEOREMA
                <num value="134">CXXXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1169" xml:space="preserve">POſſumus etiam harmonicè diuidere vnam datam proportionem abſque aliqua
                  <lb/>
                diuiſione productorum, ne nobis fractiones proueniant, hoc modo videlicet.
                  <lb/>
                </s>
                <s xml:id="echoid-s1170" xml:space="preserve">Nobis propoſitum ſit diuidere harmonicè ſeſquialteram
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                inuenian-
                  <lb/>
                tur primo minimi termini huius proportionis ut putà .3. et .2. quarum ſumma, hoc
                  <lb/>
                eſt quinque, multiplicetur per minorem ideſt .2. vnde proueniet nobis .10. qui qui-
                  <lb/>
                dem erit minor terminus trium quæſitorum, quorum maximus erit productum ſum­ </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum