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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          <div xml:id="echoid-div7" type="chapter" level="2" n="1">
            <div xml:id="echoid-div36" type="math:theorem" level="3" n="15">
              <pb o="12" rhead="IO. BAPT. BENED." n="24" file="0024" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0024"/>
            </div>
            <div xml:id="echoid-div38" type="math:theorem" level="3" n="16">
              <head xml:id="echoid-head32" xml:space="preserve">THEOREMA
                <num value="16">XVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s179" xml:space="preserve">INuenire autem cupienti cuius numeri, duæ tertiæ, ſint quatuor quintę partes, mul­
                  <lb/>
                tiplicandę eſſent duæ tertiæ per denominantem communem, & productum diui-
                  <lb/>
                dendum per quatuor quintas ipſius de-
                  <lb/>
                nominantis. </s>
                <s xml:id="echoid-s180" xml:space="preserve">Ac ſi quis diceret ſi
                  <var>.e.</var>
                dat
                  <var>.
                    <lb/>
                    <figure xlink:label="fig-0024-01" xlink:href="fig-0024-01a" number="23">
                      <image file="0024-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0024-01"/>
                    </figure>
                  o.</var>
                quid dabit
                  <var>.a</var>
                ? </s>
                <s xml:id="echoid-s181" xml:space="preserve">nempe dabit
                  <var>.u.</var>
                nam in
                  <lb/>
                propoſito exemplo, terminus
                  <var>.a.</var>
                loco
                  <var>.e.</var>
                  <lb/>
                duos ſortietur denominantes, cognitum
                  <lb/>
                videlicet
                  <var>.o.</var>
                et
                  <var>.u.</var>
                incognitum quod po-
                  <lb/>
                ſtea cognitum oritur ex regula de tribus, vt dictum eſt.</s>
              </p>
            </div>
            <div xml:id="echoid-div40" type="math:theorem" level="3" n="17">
              <head xml:id="echoid-head33" xml:space="preserve">THEOREMA
                <num value="17">XVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s182" xml:space="preserve">QVA ratione cognoſci poterit proportionem quantitatis cenſicæ cenſicæ ad
                  <lb/>
                ſimilem quantitatem quadruplam eſſe ad eam, quæ eſt ſuarum radicum; </s>
                <s xml:id="echoid-s183" xml:space="preserve">pro-
                  <lb/>
                portionem
                  <reg norm="autem" type="context">autẽ</reg>
                primarum relatarum eſſe quintuplam,
                  <reg norm="atque" type="simple">atq;</reg>
                ita deinceps?</s>
              </p>
              <p>
                <s xml:id="echoid-s184" xml:space="preserve">Cuiusrei gratia,
                  <reg norm="ſciendus" type="context">ſciẽdus</reg>
                eſt modus
                  <reg norm="productionis" type="simple">ꝓductionis</reg>
                  <reg norm="harum" type="context">harũ</reg>
                  <reg norm="dignitatum" type="context">dignitatũ</reg>
                qui oritur ex produ-
                  <lb/>
                ctione primæ radicis in ſeipſam, prout qui
                  <reg norm="cubum" type="context">cubũ</reg>
                requirit, ducat radicé in ſuo quadra-
                  <lb/>
                to, & orietur cubus, hæc poſtea ducta in cubum,
                  <reg norm="quantitatem" type="context">quantitatẽ</reg>
                cenſicam
                  <reg norm="cenſicam" type="context">cenſicã</reg>
                , et in
                  <lb/>
                hanc, prædictam radicem, dabit quantitatem primam relatam. </s>
                <s xml:id="echoid-s185" xml:space="preserve">Quod vbi ſciueri-
                  <lb/>
                mus, meminiſſe oportet Euclidem decimaoctaua ſexti aut .11. octaui docere, pro-
                  <lb/>
                portionem quadrati ad
                  <reg norm="quadratum" type="context">quadratũ</reg>
                , duplam eſſe proportioni ſuarum radicum, & .36.
                  <lb/>
                vndecimi aut .11. octaui, cubi ad
                  <reg norm="cubum" type="context">cubũ</reg>
                triplam eſſe, ego verò nunc aſſero, cenſici cen
                  <lb/>
                ſici ad radicum proportionem quadruplam eſſe, primi verò relati ad primum re-
                  <lb/>
                latum quintuplam
                  <reg norm="atque" type="simple">atq;</reg>
                ita gradatim.</s>
              </p>
              <p>
                <s xml:id="echoid-s186" xml:space="preserve">Cuius ſpeculationis gratia, detur linea
                  <var>.d.</var>
                quæ cubum maiorem ſignificet. et
                  <var>.b.</var>
                  <lb/>
                minorem
                  <var>.c.</var>
                verò ſit radixipſius
                  <var>.d.</var>
                et
                  <var>.e.</var>
                ipſius
                  <var>.b.</var>
                ita ordinate adinuicem, vt in ſub-
                  <lb/>
                ſcripta figura cernitur. </s>
                <s xml:id="echoid-s187" xml:space="preserve">Iam
                  <var>.c.</var>
                cum
                  <var>.d.</var>
                producatur
                  <reg norm="proueniatque" type="simple">proueniatq́;</reg>
                  <var>.q.</var>
                cenſicum cenſi-
                  <lb/>
                cum, tum producatur
                  <var>.e.</var>
                cum
                  <var>.b.</var>
                et dabitur
                  <var>.p.</var>
                alterum cenſicum cenſicum. </s>
                <s xml:id="echoid-s188" xml:space="preserve">Dico
                  <lb/>
                igitur proportionem
                  <var>.q.</var>
                ad
                  <var>.p.</var>
                quadruplam eſſe proportioni
                  <var>.c.</var>
                ad
                  <var>.e.</var>
                hac de
                  <lb/>
                cauſa quòd proportio
                  <var>.q.</var>
                ad
                  <var>.p.</var>
                compo-
                  <lb/>
                natur ex proportione
                  <var>.d.</var>
                ad
                  <var>.b.</var>
                et
                  <var>.c.</var>
                ad
                  <var>.e.</var>
                  <lb/>
                  <figure xlink:label="fig-0024-02" xlink:href="fig-0024-02a" number="24">
                    <image file="0024-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0024-02"/>
                  </figure>
                prout facile ex .24. ſexti, aut quinta octaui
                  <lb/>
                depręhenditur. </s>
                <s xml:id="echoid-s189" xml:space="preserve">Quare
                  <reg norm="cum" type="context">cũ</reg>
                proportio
                  <var>.d.</var>
                ad
                  <var>.
                    <lb/>
                  b.</var>
                proportioni
                  <var>.c.</var>
                ad
                  <var>.e.</var>
                tripla ſit, patet pro-
                  <lb/>
                portionem
                  <var>.q.</var>
                ad
                  <var>.p.</var>
                quadruplam eſſe pro-
                  <lb/>
                portioni
                  <var>.c.</var>
                ad
                  <var>.e</var>
                . </s>
                <s xml:id="echoid-s190" xml:space="preserve">Idem de cæteris dignitati
                  <lb/>
                bus dico, ſumptis ſemper
                  <var>.d</var>
                et
                  <var>.b.</var>
                pro duo-
                  <lb/>
                bus cenſibus cenſuum, aut duobus primis relatis, aut alio quouis axiomate.</s>
              </p>
            </div>
            <div xml:id="echoid-div42" type="math:theorem" level="3" n="18">
              <head xml:id="echoid-head34" xml:space="preserve">THEOREMA.
                <num value="18">XVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s191" xml:space="preserve">CVR diuidentibus nobis dignitatem, per dignitatem, radix prouenientis: </s>
                <s xml:id="echoid-s192" xml:space="preserve">pro
                  <lb/>
                ueniens ſit diuiſionis vnius radicis per alteram?</s>
              </p>
              <p>
                <s xml:id="echoid-s193" xml:space="preserve">Sint exempli gratia duę lineæ
                  <var>.b.q.</var>
                et
                  <var>.f.g.</var>
                quæ ſignificent duas radices cuiuſuis
                  <lb/>
                dignitatis; </s>
                <s xml:id="echoid-s194" xml:space="preserve">
                  <reg norm="demusque" type="simple">demusq́;</reg>
                eſſe radices duorum quadratorum,
                  <reg norm="quadratumque" type="simple">quadratumq́;</reg>
                ipſius
                  <var>b.q.</var>
                  <lb/>
                per quadratum ipſius
                  <var>.f.g.</var>
                diuidatur; </s>
                <s xml:id="echoid-s195" xml:space="preserve">quadrataq́ue radix prouenientis ſit
                  <var>.d.q.</var>
                  <lb/>
                vnitas verò linearis ſit
                  <var>.i.g</var>
                . </s>
                <s xml:id="echoid-s196" xml:space="preserve">Dico ipſam
                  <var>.d.q.</var>
                eſſe proueniens ex diuiſione
                  <var>.b.q.</var>
                  <lb/>
                per
                  <var>.f.g</var>
                . </s>
                <s xml:id="echoid-s197" xml:space="preserve">Patet enim ex definitione diuiſionis nono theoremate tradita quadra- </s>
              </p>
            </div>
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