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-div113" type="math:theorem" level="3" n="55">
              <pb o="35" rhead="THEOREM. ARIT." n="47" file="0047" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0047"/>
              <p>
                <s xml:id="echoid-s467" xml:space="preserve">Sumantur enimtres numeri continui proportionales, cuiuſcunque denique pro
                  <lb/>
                portionalitatis, qui in ſummam colligantur, ac poſtmodum, regula de trib. dica-
                  <lb/>
                mus. </s>
                <s xml:id="echoid-s468" xml:space="preserve">Si ſumma hæc primo numero propoſito in tres partes diuidendo reſpondet,
                  <lb/>
                cuireſpondebit vna ex tribus partibus huiuſcę
                  <reg norm="summæ" type="context">sũmæ</reg>
                ? </s>
                <s xml:id="echoid-s469" xml:space="preserve">idem dereliquis duabus pa
                  <unsure/>
                rti
                  <lb/>
                bus dico.</s>
              </p>
              <p>
                <s xml:id="echoid-s470" xml:space="preserve">Exempli gratia, ſi proponatur numerus .57. diuidendus in tres continuas partes
                  <lb/>
                proportionales proportione ſeſquialtera, tres numeros in eiuſmodi proportio-
                  <lb/>
                nalitate diſtinctos ſumemus, vt potè .4. 6. 9. qui in ſummam collecti dabunt
                  <reg norm="ſum- mam" type="context">ſum-
                    <lb/>
                  mã</reg>
                .19.
                  <reg norm="dicemusque" type="simple">dicemusq́;</reg>
                ſi .19. dant .4. quid
                  <reg norm="dabunt" type="context">dabũt</reg>
                .57? </s>
                <s xml:id="echoid-s471" xml:space="preserve">vnde proueniens vnius partis erit
                  <num value="12">.
                    <lb/>
                  12</num>
                . </s>
                <s xml:id="echoid-s472" xml:space="preserve">Tum ſi dicamus, ſi .19. dat .6. quid dabit .57? </s>
                <s xml:id="echoid-s473" xml:space="preserve">nempe dabit .18. </s>
                <s xml:id="echoid-s474" xml:space="preserve">Poſtremò, ſi
                  <num value="19">.
                    <lb/>
                  19.</num>
                dat .9. quid dabit .57? </s>
                <s xml:id="echoid-s475" xml:space="preserve">nempe .26. atque ita dabitur .18. cuius quadratum æqua-
                  <lb/>
                bitur producto reliquarum duarum partium inter ſe.</s>
              </p>
              <p>
                <s xml:id="echoid-s476" xml:space="preserve">Quod vt ſciamus, numerus propoſitus in tres quaſlibet partes diuidendus ſi-
                  <lb/>
                gnificetur linea
                  <var>.a.d.</var>
                tres autem numeri dictæ proportionalitatis, lineis
                  <var>.e.f</var>
                :
                  <var>f.g.</var>
                  <lb/>
                et
                  <var>.g.h.</var>
                directè inter ſe coniunctis denotentur. </s>
                <s xml:id="echoid-s477" xml:space="preserve">Cogitemus pariter lineam
                  <var>.d.a.</var>
                in
                  <lb/>
                tres partes diuiſam
                  <var>.a.b</var>
                :
                  <var>b.c.</var>
                et
                  <var>.c.d.</var>
                eadem cum cæteris proportionalitate, </s>
                <s xml:id="echoid-s478" xml:space="preserve">tunc ea-
                  <lb/>
                dem erit proportio
                  <var>.a.d.</var>
                ad quamlibet ſuarum partium, quæ eſt
                  <var>.e.h.</var>
                ad reſponden
                  <lb/>
                tem ipſius in
                  <var>.a.d</var>
                : Verbi gratia reſpondentem
                  <var>.a.b.</var>
                ipſi
                  <var>.e.f.</var>
                et
                  <var>.b.c</var>
                :
                  <var>f.g.</var>
                et
                  <var>.c.d</var>
                :
                  <var>g.h</var>
                . </s>
                <s xml:id="echoid-s479" xml:space="preserve">Di
                  <lb/>
                co enim quòd ita ſe habebit
                  <var>.a.d.</var>
                ad
                  <var>.c.d.</var>
                ſicut
                  <var>.e.h.</var>
                ad
                  <var>.g.h</var>
                . </s>
                <s xml:id="echoid-s480" xml:space="preserve">Nam cum ſic ſe habeat
                  <var>.a.
                    <lb/>
                  b.</var>
                ad
                  <var>.b.c.</var>
                ſicut
                  <var>.e.f.</var>
                ad
                  <var>.f.g.</var>
                ex præſuppoſito, permutando ſic ſe habebit
                  <var>.a.b.</var>
                ad
                  <var>.e.f.</var>
                ſi-
                  <lb/>
                cut
                  <var>.b.c.</var>
                ad
                  <var>.f.g.</var>
                & eadem ratione ſic ſe habe-
                  <lb/>
                bit
                  <var>.c.d.</var>
                ad
                  <var>.g.h.</var>
                ſicut
                  <var>.b.c.</var>
                ad
                  <var>.f.g.</var>
                &
                  <reg norm="conſequen- ter" type="context">cõſequen-
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0047-01a" xlink:href="fig-0047-01"/>
                  ter</reg>
                ſicut
                  <var>.a.b.</var>
                ad
                  <var>.e.f.</var>
                ex quo ex .13. quinti ſic
                  <lb/>
                ſe habebit tota
                  <var>.a.d.</var>
                ad totam
                  <var>.e.h.</var>
                ſicut
                  <var>.c.d.</var>
                  <lb/>
                ad
                  <var>.g.h.</var>
                aut
                  <var>.b.c.</var>
                ad
                  <var>.f.g.</var>
                aut
                  <var>.a.b.</var>
                ad
                  <var>.e.f.</var>
                per-
                  <lb/>
                mutando itaque propoſitum manifeſtum erit, ipſum autem productum
                  <var>.a.b.</var>
                in
                  <var>.c.b.</var>
                  <lb/>
                æquale erit quadrato
                  <var>.b.c.</var>
                ex .15. fexti aut .20. ſeptimi.</s>
              </p>
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                <figure xlink:label="fig-0047-01" xlink:href="fig-0047-01a">
                  <image file="0047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0047-01"/>
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            <div xml:id="echoid-div115" type="math:theorem" level="3" n="56">
              <head xml:id="echoid-head72" xml:space="preserve">THEOREMA
                <num value="56">LVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s481" xml:space="preserve">
                  <emph style="sc">VEteres</emph>
                aliud quoque problema indeterminatum propoſuerunt, quod ex
                  <lb/>
                more ratione à me definietur, eſt autem eiuſmodi.</s>
              </p>
              <p>
                <s xml:id="echoid-s482" xml:space="preserve">Quomodo propoſitus numerus in tres eiuſmodi partes diuidatur, vt
                  <reg norm="quadratum" type="context">quadratũ</reg>
                  <lb/>
                vnius æquale fit fummæ quadratorum reliquarum duarum partium.</s>
              </p>
              <p>
                <s xml:id="echoid-s483" xml:space="preserve">Hoc vt efficiamus tria quadrata ſeparata ſumamus,
                  <reg norm="quorum" type="context">quorũ</reg>
                  <reg norm="vnum" type="context">vnũ</reg>
                æquale ſit reliquis
                  <lb/>
                duobus; </s>
                <s xml:id="echoid-s484" xml:space="preserve">
                  <reg norm="eorum" type="context">eorũ</reg>
                  <reg norm="autem" type="context">autẽ</reg>
                radices in ſummam ſimul colligantur, tum regulam de tribus ſe
                  <lb/>
                quemur, ratione præcedenti theoremate demonſtrata, & rectè vt infra docebimus,
                  <lb/>
                quod autem dico de quadratis, etiam de cubis, & quibuſuis dignitatibus aſſero.</s>
              </p>
              <p>
                <s xml:id="echoid-s485" xml:space="preserve">Exempli gratia, ſi numerus diuiſibilis proponatur .30. in tres eiuſmodi partes di
                  <lb/>
                uidendus, vt quadratum vnius æquale ſit ſummæ quadratorum reliquarum duarum
                  <lb/>
                partium, in primis radices trium quadratorum ſumemus, ſic quomodocunque ſe
                  <lb/>
                habentes, vt maius ipſorum æquale ſit ſummæ reliquorum duorum, verbi gratia .25.
                  <lb/>
                16. et .9. nempe .5. 4. et .3. quæ ſi colligantur in ſummam efficiunt .12. </s>
                <s xml:id="echoid-s486" xml:space="preserve">Tum ex regu-
                  <lb/>
                la de tribus dicemus, ſi .12. reſpondet .30: </s>
                <s xml:id="echoid-s487" xml:space="preserve">cui, 5. radix maior reſpondebit? </s>
                <s xml:id="echoid-s488" xml:space="preserve">nem-
                  <lb/>
                pe .12. cum dimidio.</s>
              </p>
              <p>
                <s xml:id="echoid-s489" xml:space="preserve">Deinde ſi dixerimus ſi .12. valet .30. quid valebit .4. radix media? </s>
                <s xml:id="echoid-s490" xml:space="preserve">nempe vale-
                  <lb/>
                bit .10. tertia autem minor .7. cum dimidio. </s>
                <s xml:id="echoid-s491" xml:space="preserve">Itaquetota ſumma erit .30. & quadra- </s>
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