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
181 169
182 170
183 171
184 172
185 173
186 174
187 175
188 176
189 177
190 178
191 179
192 180
193 181
194 182
195 183
196 184
197 185
198 186
199 187
200 188
201 189
202 190
203 191
204 192
205 193
206 194
207 195
208 196
209 197
210 198
< >
page |< < (38) 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-div120" type="math:theorem" level="3" n="59">
              <pb o="38" rhead="IO. BAPT. BENED." n="50" file="0050" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0050"/>
              <p>
                <s xml:id="echoid-s521" xml:space="preserve">Hoc vt demonſtremus, primus nu-
                  <lb/>
                  <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a" number="67">
                    <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0050-01"/>
                  </figure>
                merus linea
                  <var>.a.b.</var>
                ſignificetur, quam di-
                  <lb/>
                uiſam cogitemus in puncto
                  <var>.c.</var>
                in partes
                  <lb/>
                quæſitas, ex quo præſupponitur duas li-
                  <lb/>
                neas
                  <var>.a.c.</var>
                et
                  <var>.c.b.</var>
                duo quadrata eſſe, quæ
                  <lb/>
                in altera figura ſignificetur per
                  <var>.d.</var>
                et
                  <var>.e.</var>
                  <lb/>
                productum autem radicum cognitum
                  <var>.
                    <lb/>
                  f.</var>
                quandoquidem datum eſt, cuius qua-
                  <lb/>
                dratum æquale erit producto quadra-
                  <lb/>
                torum
                  <var>.d.e.</var>
                adinuicem, nempe
                  <var>.b.c.</var>
                in
                  <var>.a.c.</var>
                ex .19. theoremate huius. </s>
                <s xml:id="echoid-s522" xml:space="preserve">Quod verbi
                  <lb/>
                gratia ſit
                  <var>.x.</var>
                  <reg norm="itaque" type="simple">itaq;</reg>
                cognitum, quo facto, doctrinam .45. theorematis libri huius ſecuti,
                  <lb/>
                propoſitum conſequemur.</s>
              </p>
            </div>
            <div xml:id="echoid-div122" type="math:theorem" level="3" n="60">
              <head xml:id="echoid-head76" xml:space="preserve">THEOREMA
                <num value="60">LX</num>
              .</head>
              <p>
                <s xml:id="echoid-s523" xml:space="preserve">CVR productum differentiæ duarum radicum in ſummam ipſarum, ſemper
                  <lb/>
                differentia ſit quadratorum ipſarum radicum.</s>
              </p>
              <p>
                <s xml:id="echoid-s524" xml:space="preserve">
                  <reg norm="Exempli" type="context">Exẽpli</reg>
                gratia, quoslibet duos numeros pro radicibus ſumpſerimus, vt potè .3. et
                  <num value="5">.
                    <lb/>
                  5.</num>
                quorum differentia eſt .2. certè ſi differentiam hanc per ſummam radicum ſcili-
                  <lb/>
                cet .8. multiplicauerimus, dabitur numerus .16. quod productum differentia eſt
                  <lb/>
                ſuorum quadratorum, nempeinter .9. et .25.</s>
              </p>
              <p>
                <s xml:id="echoid-s525" xml:space="preserve">Hoc vt ſpeculemur, duæ radices in linea
                  <var>.n.i.</var>
                ſignificentur, quarum vna ſit
                  <var>.n.c.</var>
                &
                  <lb/>
                altera
                  <var>.c.i.</var>
                ipſarum autem differentia
                  <var>.n.t.</var>
                ex quo
                  <var>.t.
                    <lb/>
                  c.</var>
                æqualis erit
                  <var>.c.i</var>
                . </s>
                <s xml:id="echoid-s526" xml:space="preserve">Tum cogitato toto quadrato
                  <var>.d.i.</var>
                  <lb/>
                  <figure xlink:label="fig-0050-02" xlink:href="fig-0050-02a" number="68">
                    <image file="0050-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0050-02"/>
                  </figure>
                cum diametro
                  <var>.d.i.</var>
                  <reg norm="ductaque" type="simple">ductaq́</reg>
                parallela lateri
                  <var>.n.d.</var>
                à
                  <lb/>
                puncto
                  <var>.c.</var>
                & altera à puncto
                  <var>.t.</var>
                & à puncto
                  <var>.o.</var>
                tertia
                  <lb/>
                ipſi
                  <var>.n.i.</var>
                & à puncto
                  <var>.a.</var>
                quarta
                  <var>.x.a.e.</var>
                parallela ipſi
                  <var>.
                    <lb/>
                  o.</var>
                inueniemus
                  <var>.b.n.</var>
                productum eſſe differentiæ
                  <var>.n.
                    <lb/>
                  t.</var>
                in ſumma radicum
                  <var>.n.i.</var>
                & cum
                  <var>.d.o.</var>
                et
                  <var>.a.o.</var>
                ſint
                  <lb/>
                quadrata radicum prædictarum: </s>
                <s xml:id="echoid-s527" xml:space="preserve">b.e. æquale erit
                  <var>.
                    <lb/>
                  n.u.</var>
                cum vtrunque horum productorum æquale ſit
                  <var>.
                    <lb/>
                  x.u.</var>
                ex quo gnomon
                  <var>.e.d.u.</var>
                æqualis erit producto
                  <var>.
                    <lb/>
                  b.n.</var>
                quod ſcire cupiebamus.</s>
              </p>
            </div>
            <div xml:id="echoid-div124" type="math:theorem" level="3" n="61">
              <head xml:id="echoid-head77" xml:space="preserve">THEOREMA
                <num value="61">LXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s528" xml:space="preserve">CVR propoſitum aliquem numerum diuiſuri in duas eiuſmodi partes, vt diffe-
                  <lb/>
                rentia radicum quadratarum æqualis ſit alteri numero propoſito, cuius ta-
                  <lb/>
                men quadratum dimidij primi quadratum non excedat. </s>
                <s xml:id="echoid-s529" xml:space="preserve">Rectè ſecundum numerum
                  <lb/>
                in ſeipſum multiplicant, productum verò ex primo numero detrahunt,
                  <reg norm="rurſusque" type="simple">rurſusq́;</reg>
                di
                  <lb/>
                midium reſidui quadrant, & quadratum hoc ex quadrato dimidij primi ſubtrahunt,
                  <lb/>
                atque ita radice quadrata reſidui, dimidio primi coniuncta, pars maior datur, qua
                  <lb/>
                ex ipſo dimidio detracta, pars minor relinquitur.</s>
              </p>
              <p>
                <s xml:id="echoid-s530" xml:space="preserve">Exempli gratia, propoſito numero .20. ita ut propoſitum eſt, diuidendo, nem-
                  <lb/>
                pe vt differentia radicum quadratarum dictarum partium æqualis ſit binario, bina-
                  <lb/>
                rium hocin ſeipſum multiplicabimus, cuius quadratum .4. è primo numero .20. de­ </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum