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]

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THEOREM. ARIT.
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            <div xml:id="echoid-div281" type="math:theorem" level="3" n="147">
              <p>
                <s xml:id="echoid-s1291" xml:space="preserve">
                  <pb o="99" rhead="THEOREM. ARIT." n="111" file="0111" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0111"/>
                ni
                  <var>.o.</var>
                ad
                  <var>.c</var>
                . </s>
                <s xml:id="echoid-s1292" xml:space="preserve">Idem dico de reliquis proportionibus ſuperparticularibus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1293" xml:space="preserve">Sed ſi data proportio numerorum fuerit ex ſuper partientibus, vt exempli gra-
                  <lb/>
                tia de quinque ad tria, efficiemus, vt
                  <var>.a.</var>
                et
                  <var>.e.</var>
                ſint prima relata ipſius
                  <var>.o.</var>
                et
                  <var>.c.</var>
                vnde
                  <lb/>
                proportio
                  <var>.a.</var>
                ad
                  <var>.e.</var>
                ita ſe habe-
                  <lb/>
                bit ad proportionem
                  <var>.o.</var>
                ad
                  <var>.c.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0111-01a" xlink:href="fig-0111-01"/>
                vt quinque ad
                  <reg norm="vnum" type="context">vnũ</reg>
                & propor-
                  <lb/>
                tio
                  <var>.i.</var>
                ad
                  <var>.c.</var>
                ut tria ad
                  <reg norm="vnum" type="context">vnũ</reg>
                . </s>
                <s xml:id="echoid-s1294" xml:space="preserve">Qua-
                  <lb/>
                re proportio
                  <var>.a.</var>
                ad
                  <var>.e.</var>
                ad pro-
                  <lb/>
                portionem
                  <var>.i.</var>
                ad
                  <var>.c.</var>
                ſe habebit,
                  <lb/>
                vt quinque ad tria, & ſic de reliquis.</s>
              </p>
              <div xml:id="echoid-div281" type="float" level="4" n="1">
                <figure xlink:label="fig-0111-01" xlink:href="fig-0111-01a">
                  <image file="0111-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0111-01"/>
                </figure>
              </div>
              <p>
                <s xml:id="echoid-s1295" xml:space="preserve">Pro alijs, eundem ordinem ſeruando, obtinebimus quod volumus.</s>
              </p>
            </div>
            <div xml:id="echoid-div283" type="math:theorem" level="3" n="148">
              <head xml:id="echoid-head167" xml:space="preserve">THEOREMA
                <num value="148">CXLVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1296" xml:space="preserve">QVamuis in .16. ſexti et .20. ſeptimi manifeſtè pateat ratio, quare rectè fiatac
                  <lb/>
                cipiendam radicem quadratam illius producti, quod fit ex duobus datis
                  <lb/>
                terminis, vt medium proportionale geometricè inter ipſos habeamus: </s>
                <s xml:id="echoid-s1297" xml:space="preserve">nihilomi-
                  <lb/>
                nus, quia per aliam methodum hoc idem ſcire poſſumus, inconueniens non erit a-
                  <lb/>
                liquid circa hoc dicere.</s>
              </p>
              <p>
                <s xml:id="echoid-s1298" xml:space="preserve">Cogitemus igitur exempli gratia, tres numeros continuè proportionales geo-
                  <lb/>
                metricè
                  <var>.a.b</var>
                :
                  <var>c.d.</var>
                et
                  <var>.e.f.</var>
                quorum
                  <var>.a.b.</var>
                et
                  <var>.e.f.</var>
                tantummodo nobis cogniti ſint, imagine-
                  <lb/>
                mur etiam
                  <var>.g.a.</var>
                eſſe productum quod fit ex
                  <var>.a.b.</var>
                in
                  <var>.e.f.</var>
                et
                  <var>.d.k.</var>
                quadratum
                  <var>.c.d.</var>
                et
                  <var>.a.h.</var>
                  <lb/>
                id quod fit ex
                  <var>.a.b.</var>
                vnde eandem proportionem habebimus
                  <var>.a.h.</var>
                ad
                  <var>.a.g.</var>
                quæ eſt
                  <var>.h.b.</var>
                  <lb/>
                ad
                  <var>.b.g.</var>
                ex prima .6. aut .18. vel .19. ſepti-
                  <lb/>
                mi, ſed per .11. octaui ita eſt quadrati
                  <var>.a.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0111-02a" xlink:href="fig-0111-02"/>
                h. ad quadratum
                  <var>.k.d.</var>
                vt
                  <var>.a.b.</var>
                ad
                  <var>.e.f.</var>
                hoc
                  <lb/>
                eſt vt
                  <var>.h.b.</var>
                ad
                  <var>.b.g.</var>
                ergo per .11. quinti ita
                  <lb/>
                erit
                  <var>.a.h.</var>
                ad
                  <var>.a.g.</var>
                vt ad
                  <var>.k.d.</var>
                vnde
                  <var>.a.g.</var>
                æqua
                  <lb/>
                le erit
                  <var>.k.d.</var>
                per .9. quinti. </s>
                <s xml:id="echoid-s1299" xml:space="preserve">Rectè ergo erit
                  <lb/>
                accipere radicem quadratam
                  <var>.a.g.</var>
                pro
                  <var>.c.
                    <lb/>
                  d.</var>
                quod etiam eſt diuidere vnam datam
                  <lb/>
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                per æqualia, hoc eſt in duas
                  <lb/>
                æquales partes, non dubito quin poſſer aliquis dicere non oportere vti poſteriori-
                  <lb/>
                bus Theorematibus ad demonſtrandum priora illis, ſed hoc .148. dictum ſit luden
                  <lb/>
                di loco.</s>
              </p>
              <div xml:id="echoid-div283" type="float" level="4" n="1">
                <figure xlink:label="fig-0111-02" xlink:href="fig-0111-02a">
                  <image file="0111-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0111-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div285" type="math:theorem" level="3" n="149">
              <head xml:id="echoid-head168" xml:space="preserve">THEOREMA
                <num value="149">CXLIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s1300" xml:space="preserve">
                  <emph style="sc">Vnde</emph>
                fiat
                  <reg norm="quod" type="simple">ꝙ</reg>
                ſi quis inuenire voluerit ſecundum terminum ex quatuor nume
                  <lb/>
                ris continuè, & geometricè proportionalibus, quorum duo extremi tantum-
                  <lb/>
                modo nobis cogniti ſint, rectè factum ſit quadrare primum eorum, & hoc quadra-
                  <lb/>
                tum poſteà per alium terminum cognitum multiplicare, cuius producti demum ac-
                  <lb/>
                cipere radicem cubam pro ſecundo termino quæſito, hocloco videbimus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1301" xml:space="preserve">Imaginemur quatuor terminos continuè proportionales, vt dictum eſt, eſſe.</s>
              </p>
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