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-div115" type="math:theorem" level="3" n="56">
              <p>
                <s xml:id="echoid-s491" xml:space="preserve">
                  <pb o="36" rhead="IO. BAPT. BENED." n="48" file="0048" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0048"/>
                tum .12. cum dimidio erit .155. quod æquale erit ſummæ quadratorum duarum par
                  <lb/>
                tium, nempe .100. cum .55.</s>
              </p>
              <p>
                <s xml:id="echoid-s492" xml:space="preserve">Hoc vt
                  <reg norm="demonſtremus" type="context">demõſtremus</reg>
                , numerus diuiſibilis propoſitus ſignificetur linea
                  <var>.a.d.</var>
                & ſum
                  <lb/>
                ma radicum, noſtro modo ſumptarum, linea
                  <var>.e.h.</var>
                quarum prima & maior ſit
                  <var>.e.f.</var>
                ſe-
                  <lb/>
                cunda
                  <var>.f.g.</var>
                tertia
                  <var>.g.h.</var>
                cogitemus etiam lineam
                  <var>.a.d.</var>
                ea ratione diuiſam eſſe qua
                  <var>.e.h.</var>
                  <lb/>
                patebit cnim ex modo præcedentis theorematis vnamquanque partium
                  <var>.a.d.</var>
                ita ſe
                  <lb/>
                habituram ad ſuum totum ſicut ſe habent ſingulæ
                  <var>.e.h.</var>
                ad ſuum. </s>
                <s xml:id="echoid-s493" xml:space="preserve">Quod ideo dico, vt
                  <lb/>
                intelligamus rectè nos dicere. </s>
                <s xml:id="echoid-s494" xml:space="preserve">Si
                  <var>.e.h.</var>
                dat
                  <var>.a.d.</var>
                ergo
                  <var>.e.f.</var>
                dabit
                  <var>.a.b.</var>
                  <reg norm="atque" type="simple">atq;</reg>
                ita de cæteris.
                  <lb/>
                </s>
                <s xml:id="echoid-s495" xml:space="preserve">Quare permutando ſic ſe habebit
                  <var>.a.b.</var>
                ad
                  <var>.b.c.</var>
                ſicut
                  <var>.e.f.</var>
                ad
                  <var>.f.g.</var>
                idem dico de reliquis.
                  <lb/>
                </s>
                <s xml:id="echoid-s496" xml:space="preserve">Igitur ex .18. ſexti aut .11. octaui, eadem erit proportio quadrati
                  <var>.a.b.</var>
                ad
                  <reg norm="quadratum" type="context">quadratũ</reg>
                  <var>.
                    <lb/>
                  b.c.</var>
                quæ quadrati
                  <var>.e.f.</var>
                ad quadratum
                  <var>.f.g.</var>
                tota enim ſunt æqualia, cum eorum partes
                  <lb/>
                ſimiles inter ſe ſunt æquales. </s>
                <s xml:id="echoid-s497" xml:space="preserve">Idem dico de proportione qu@drati
                  <var>.a.b.</var>
                nempe ita
                  <lb/>
                ſe habere ad
                  <var>.c.d.</var>
                ſicut quadratum
                  <var>.e.f.</var>
                ad quadratum
                  <var>.g.h.</var>
                ex quo ex .24. quinti pro-
                  <lb/>
                portio quadrati
                  <var>.a.b.</var>
                ad ſummam quadratorum duarum partium
                  <var>.b.c.</var>
                et
                  <var>.c.d.</var>
                ſic ſe ha
                  <lb/>
                bebit ut quadrati
                  <var>.e.f.</var>
                ad ſummam quadra-
                  <lb/>
                torum
                  <var>.f.g.</var>
                et
                  <var>.g.h</var>
                . </s>
                <s xml:id="echoid-s498" xml:space="preserve">At quadratum
                  <var>.e.f.</var>
                æquale
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0048-01a" xlink:href="fig-0048-01"/>
                eſt ſummæ quadratorum
                  <var>.f.g.</var>
                et
                  <var>.g.h.</var>
                igitur
                  <lb/>
                ſic etiam ſe habebit quadratum
                  <var>.a.b.</var>
                nempe
                  <lb/>
                æquale quadratis
                  <var>.b.c.</var>
                et
                  <var>.c.g</var>
                . </s>
                <s xml:id="echoid-s499" xml:space="preserve">Idipſum de cæ
                  <lb/>
                teris dignitatibus dices,
                  <reg norm="vterisque" type="simple">vterisq́;</reg>
                .21. theoremate huius libri.</s>
              </p>
              <div xml:id="echoid-div115" type="float" level="4" n="1">
                <figure xlink:label="fig-0048-01" xlink:href="fig-0048-01a">
                  <image file="0048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0048-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div117" type="math:theorem" level="3" n="57">
              <head xml:id="echoid-head73" xml:space="preserve">THEOREMA
                <num value="57">LVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s500" xml:space="preserve">
                  <emph style="sc">SImile</emph>
                quoque problema ab antiquis indeterminatum proponitur, quod eiuſ-
                  <lb/>
                modi eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s501" xml:space="preserve">An numerus aliquis in tres eiuſmodi partes di@idi poſſit, vt quadratum vnius æ-
                  <lb/>
                quale ſit ſummæ quadratorum cæterarum duarum partium ſimul cum producto
                  <lb/>
                vnius in alteram.</s>
              </p>
              <p>
                <s xml:id="echoid-s502" xml:space="preserve">Exempli gratia, ſi proponatur numerus .50. vt iam dictum eſt diuidendus, repe
                  <lb/>
                riendus erit alius quilibet numerus, qui tamen ſumma ſit trium radicum ſic ſe ha-
                  <lb/>
                bentium, vt quadratum vnius æquale ſit ſummæ quadratorum duarum partium ſi-
                  <lb/>
                mul cum producto vnius in alteram, eum autem qui primò occurrit ſumamus, utpo
                  <lb/>
                tè .30. qui ſumma eſt numerorum .6. 10. 14. partium ſic ſe habentium, vt quadratum
                  <lb/>
                ipſius .14. æquale ſit ſummæ quadratorum cæterarum partium ſimul cum produ-
                  <lb/>
                cto vnius in alteram, agamusq́ue regula de tribus, ac dicamus, ſi .30. valet
                  <num value="50">.
                    <lb/>
                  50.</num>
                quid valebit .14. nempe .23. cum tertia parte. </s>
                <s xml:id="echoid-s503" xml:space="preserve">Idem efficiemus in cæte-
                  <lb/>
                ris partibus, quarum vna erit .16. cum duabus tertijs, altera verò .10. abſque @ractis,
                  <lb/>
                ex quo quadratum primæ erit .544. cum .4. nonis, ſecundæ .277. cum ſeptem nonis,
                  <lb/>
                tertiæ .100. & productum ſecundæ in tertiam .166. cum .6. nonis, quod productum,
                  <lb/>
                cum quadratis ſecundæ & tertiæ collectum erit .544. cum .4. nonis.</s>
              </p>
              <p>
                <s xml:id="echoid-s504" xml:space="preserve">Huius rei ſpeculatio eadem eſt, quę fuit præcedentis theorematis vſquequo no-
                  <lb/>
                ueris eandem proportionem eſſe quadrati
                  <var>.a.b.</var>
                ad ſummam quadratorum
                  <var>.b.c.</var>
                et
                  <var>.c.
                    <lb/>
                  d.</var>
                quæ quadrati
                  <var>.e.f.</var>
                ad ſummam quadratorum
                  <var>.f.g.</var>
                et
                  <var>.g.h</var>
                . </s>
                <s xml:id="echoid-s505" xml:space="preserve">Sed cum hic non demus
                  <lb/>
                quadratum
                  <var>.e.f.</var>
                æquale ſummæ quadratorum
                  <var>.f.g.</var>
                et
                  <var>.g.h.</var>
                fed maius ex producto
                  <var>.g.h.</var>
                  <lb/>
                in
                  <var>.f.g.</var>
                aut quod idem eſt, è contrario, ſubſequentes figuræ cogitandæ erunt, qua-
                  <lb/>
                rum
                  <var>.i.</var>
                ſit quadratum
                  <var>.a.b</var>
                : l. ſit quadratum
                  <var>.e.f</var>
                : x. quadratum
                  <var>.b.c</var>
                : y. quadratum
                  <var>.f.g</var>
                : p.
                  <lb/>
                quadratum
                  <var>.c.d</var>
                : q. quadratum
                  <var>.g.h</var>
                : k. ſit productum
                  <var>.b.c.</var>
                in
                  <var>.c.d</var>
                : m. ſit productum
                  <var>.f.</var>
                </s>
              </p>
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