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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IO. BAPT. BENED.
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            <div xml:id="echoid-div109" type="math:theorem" level="3" n="52">
              <p>
                <pb o="34" rhead="IO. BAPT. BENED." n="46" file="0046" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0046"/>
                <s xml:id="echoid-s455" xml:space="preserve">11. dabuntur .110. quo producto multiplicato cum .12. dabuntur .1320. hoc pro
                  <lb/>
                ueniens per primum nempe .10. diuiſum dabit .132. numerum æqualem producto
                  <lb/>
                ſecundi in tertium numerorum propoſitorum, ſcilicet .132.</s>
              </p>
              <p>
                <s xml:id="echoid-s456" xml:space="preserve">Hoc vt ſpeculemur, primus numerus ſignificetur line
                  <var>a.o.u.</var>
                ſecundus
                  <var>.e.o.</var>
                tertius
                  <var>.
                    <lb/>
                  e.a.</var>
                productum verò
                  <var>.o.u.</var>
                in
                  <var>.o.e.</var>
                ſit
                  <var>.o.i.</var>
                ipſius ve
                  <lb/>
                  <var>.o.i.</var>
                per
                  <var>.e.a.</var>
                  <reg norm="productum" type="context">productũ</reg>
                  <reg norm="corporeum" type="context">corporeũ</reg>
                ſit
                  <var>.i.c.</var>
                tum
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/>
                  <reg norm="productum" type="context">productũ</reg>
                  <var>.e.o.</var>
                in
                  <var>.e.a.</var>
                ſit
                  <var>.e.c</var>
                . </s>
                <s xml:id="echoid-s457" xml:space="preserve">Dico
                  <reg norm="nunc" type="context">nũc</reg>
                quod di-
                  <lb/>
                uiſo numero corporeo
                  <var>.i.c.</var>
                per
                  <reg norm="primum" type="context">primũ</reg>
                  <var>.o.u.</var>
                  <reg norm="proue" type="simple">ꝓue</reg>
                  <lb/>
                niens æquale erit numero producti
                  <var>.e.c</var>
                . </s>
                <s xml:id="echoid-s458" xml:space="preserve">Qua-
                  <lb/>
                re in primis cogitandum eſt, quod cum produ-
                  <lb/>
                ctum
                  <var>.i.c.</var>
                ortum fuerit ex multiplicatione
                  <var>.o.i.</var>
                  <lb/>
                in
                  <var>.e.a</var>
                : dictum
                  <var>.o.i.</var>
                toties ingredietur
                  <var>.i.c.</var>
                quo-
                  <lb/>
                ties vnitas reperitur in
                  <var>.e.a.</var>
                eadem ratione, to-
                  <lb/>
                ties
                  <var>.e.c.</var>
                in
                  <var>.i.c.</var>
                quot vnitates erunt in
                  <var>.o.u</var>
                . </s>
                <s xml:id="echoid-s459" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                  <lb/>
                ſequitur quòd diuiſo
                  <var>.i.c.</var>
                per
                  <var>o.u.</var>
                proueniens ſit
                  <lb/>
                  <var>e.c.</var>
                corporeum, æquale nihilominus producto
                  <var>.e.c.</var>
                ſuperficiali.</s>
              </p>
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                <figure xlink:label="fig-0046-01" xlink:href="fig-0046-01a">
                  <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0046-01"/>
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            <div xml:id="echoid-div111" type="math:theorem" level="3" n="53">
              <head xml:id="echoid-head69" xml:space="preserve">THEOREMA
                <num value="53">LIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s460" xml:space="preserve">CVR diuidens propoſitum numerum in tres partes ſic ſe habentes vt produ-
                  <lb/>
                ctum primi in ſecundam, in tertia
                  <reg norm="multiplicatum" type="context">multiplicatũ</reg>
                , præbeat numerum alteri nu-
                  <lb/>
                mero propoſito æqualem. </s>
                <s xml:id="echoid-s461" xml:space="preserve">Rectè ſecundum numerum per quemcunque alium mino
                  <lb/>
                rem primo diuidit, qui diuidens vna erit ex tribus partibus quæſitis, proueniens
                  <lb/>
                autem erit productum vnius in alteram reliquarum duarum, quarum ſumma cogni
                  <lb/>
                ta erit, detracto numero diuidente ex primo dato, quam quidem ſi diſtinguere
                  <lb/>
                quis voluerit, vtetur theoremate .45.</s>
              </p>
              <p>
                <s xml:id="echoid-s462" xml:space="preserve">Exempli gratia, proponitur numerus .20. in tres partes diuidendus, quæ ſic ſe
                  <lb/>
                habeant, ut productum primæ in ſecundam in tertia multiplicatum det .90. itaque
                  <lb/>
                ſumenda erit pro prima vna pars ipſius .20. quæcunque illa ſit, verbi gratia .2. qua
                  <lb/>
                ſecundus numerus, nempe .90. diuidatur, dabitur igitur .45. quod erit productum
                  <lb/>
                cæterarum partium inter ſe, quarum ſumma eſt .18. quam ſummam ſi diſtinguere
                  <lb/>
                volueris in cęteris duabus partibus ſeparatis, vteris .45. theoremate, vt quàm citiſ-
                  <lb/>
                ſimè quod cupis exequaris, erunt autem partes .3. et .15.</s>
              </p>
              <p>
                <s xml:id="echoid-s463" xml:space="preserve">In cuius ſpeculationis gratiam nihil aliud occurrit, quàm quod præcedenti theo-
                  <lb/>
                remate, & ſuperiore .45. allatum eſt.</s>
              </p>
            </div>
            <div xml:id="echoid-div112" type="math:theorem" level="3" n="54">
              <head xml:id="echoid-head70" xml:space="preserve">THEOREMA
                <num value="54">LIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s464" xml:space="preserve">
                  <emph style="sc">DIvidere</emph>
                numerum in .3. eiuſmodi partes, vt quadratum vnius ſit æquale
                  <lb/>
                producto reliquarum duarum inter ſe, idem omnino eſt cum 51. theoremate.
                  <lb/>
                </s>
                <s xml:id="echoid-s465" xml:space="preserve">Nam qui ſumet quamlibet partem propoſiti numeri, quæ tertia parte maior tamen
                  <lb/>
                non ſit,
                  <reg norm="reſiduumque" type="simple">reſiduumq́</reg>
                in duas tales partes diuiſerit, vt prima ſumpta, media proportio
                  <lb/>
                nalis ſit ex probatione .51. theoremate allata, propoſitum conſequetur.</s>
              </p>
            </div>
            <div xml:id="echoid-div113" type="math:theorem" level="3" n="55">
              <head xml:id="echoid-head71" xml:space="preserve">THEOREMA
                <num value="55">LV</num>
              .</head>
              <p>
                <s xml:id="echoid-s466" xml:space="preserve">ID ipſum alia ratione ab ea diuerſa
                  <reg norm="quam" type="context">quã</reg>
                .51. theoremate adduximus,
                  <reg norm="profici" type="simple">ꝓfici</reg>
                poteſt.</s>
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