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. AR IT.
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            <div xml:id="echoid-div124" type="math:theorem" level="3" n="61">
              <p>
                <s xml:id="echoid-s530" xml:space="preserve">
                  <pb o="39" rhead="THEOREM. AR IT." n="51" file="0051" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0051"/>
                trahemus,
                  <reg norm="ſupereritque" type="simple">ſupereritq́;</reg>
                numerus .16. cuius dimidium ſcilicet .8. in ſeipſum multipli-
                  <lb/>
                cabimus,
                  <reg norm="dabiturque" type="simple">dabiturq́;</reg>
                numerus .64. qui cum ex quadrato dimidij primi detractus fue-
                  <lb/>
                rit, nempe ex .100. & reſiduo .36. radix quadrata nempe .6. coniuncta denario, di-
                  <lb/>
                midio primi, dabit .16. partem maiorem, & ex denario detracta, partem minorem.</s>
              </p>
              <p>
                <s xml:id="echoid-s531" xml:space="preserve">Cuius ſpeculationis cauſa, primus numerus
                  <lb/>
                propoſitus ſigniſicetur linea
                  <var>.x.y.</var>
                pro voto diui-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/>
                ſa in puncto
                  <var>.c.</var>
                et
                  <var>.x.t.</var>
                productum ſit ipſius
                  <var>.x.
                    <lb/>
                  c.</var>
                in
                  <var>.c.y.</var>
                pariter etiam
                  <var>.q.p.</var>
                ſit ſumma radicum
                  <lb/>
                quadratarum, nempe
                  <var>.q.g.</var>
                ipſius
                  <var>.t.c.</var>
                et
                  <var>.g.p.</var>
                ip-
                  <lb/>
                ſius
                  <var>.c.y</var>
                . </s>
                <s xml:id="echoid-s532" xml:space="preserve">Tum ſuper
                  <var>.q.p.</var>
                extruatur & diuidatur
                  <lb/>
                quadratum
                  <var>.q.u.</var>
                ea ratione qua .41. theoremate
                  <lb/>
                aut .29. diuiſimus, in quo ſanè quadrato, quadra
                  <lb/>
                tum ipſius
                  <var>.q.i.</var>
                cernemus datæ differentiæ, & in
                  <lb/>
                eo collocata quadrata
                  <var>.x.c.</var>
                et
                  <var>.c.y.</var>
                ita etiam &
                  <lb/>
                rationem, qua cognoſcimus productum
                  <var>.g.r.</var>
                (vſi
                  <lb/>
                modo .29. theorematis) cuius quidem
                  <var>.g.r.</var>
                qua-
                  <lb/>
                dratum, ex .19. theoremate æquale erit produ-
                  <lb/>
                cto
                  <var>.x.t.</var>
                ideo etiam
                  <reg norm="cognitum" type="context">cognitũ</reg>
                , ac proinde cum no
                  <lb/>
                uerimus
                  <var>.x.y.</var>
                ſi rationem ſequemur .45. theore
                  <lb/>
                mate cognoſcemus non ſolum ratione .41. theoremate allata hocrectè perfici, ſed
                  <lb/>
                hac etiam alia ratione.</s>
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                <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a">
                  <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div126" type="math:theorem" level="3" n="62">
              <head xml:id="echoid-head78" xml:space="preserve">THEOREMA
                <num value="62">LXII</num>
              .</head>
              <p>
                <s xml:id="echoid-s533" xml:space="preserve">CVR propoſitum numerum diuiſuri in duas eiuſmodi partes, vt differentia
                  <lb/>
                  <reg norm="ſuarum" type="context">ſuarũ</reg>
                  <reg norm="radicum" type="context">radicũ</reg>
                  <reg norm="quadratarum" type="context">quadratarũ</reg>
                æqualis ſit alteri numero propoſito. </s>
                <s xml:id="echoid-s534" xml:space="preserve">Cuius
                  <reg norm="tamen" type="wordlist">tamẽ</reg>
                  <reg norm="qua- dratum" type="context">qua-
                    <lb/>
                  dratũ</reg>
                maius non ſit quadrato medietatis ipſius primi propoſiti numeri. </s>
                <s xml:id="echoid-s535" xml:space="preserve">Rectè
                  <reg norm="etiam" type="context">etiã</reg>
                  <lb/>
                  <reg norm="quadratum" type="context">quadratũ</reg>
                dimidij ſecundi numeri ex dimidio primi
                  <reg norm="detrahunt" type="context">detrahũt</reg>
                ,
                  <reg norm="reſiduique" type="simple">reſiduiq́;</reg>
                radicem per
                  <lb/>
                ſecundum multiplicant, & productum ex dimidio primi detrahunt, vt reſiduum
                  <lb/>
                pars quæſita minor ſit, & illud alterum totius reſiduum, pars maior.</s>
              </p>
              <p>
                <s xml:id="echoid-s536" xml:space="preserve">Exempli gratia, ſi numerus .50. in
                  <lb/>
                prædictas duas partes diuidendus pro-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0051-02a" xlink:href="fig-0051-02"/>
                poneretur, & alter etiam .6. quadratum
                  <lb/>
                dimidij ſecundi numeri eſſet .9. eo detra
                  <lb/>
                cto ex dimidio primi, remaneret .16. cu
                  <lb/>
                ius radix .4. ſcilicet per totum ſecundum
                  <lb/>
                nempe .6. multiplicata, proferet .24.
                  <lb/>
                quo producto ex dimidio primi detra-
                  <lb/>
                cto, nempe .25. dabitur .1. pars minor,
                  <lb/>
                maior
                  <reg norm="autem" type="context">autẽ</reg>
                erit
                  <reg norm="reſidum" type="context">reſidũ</reg>
                .50. hoc eſt .49.
                  <lb/>
                radices autem erunt .1. et .7. differentes
                  <lb/>
                inter ſe, numero ſenario.</s>
              </p>
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                <figure xlink:label="fig-0051-02" xlink:href="fig-0051-02a">
                  <image file="0051-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0051-02"/>
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              <p>
                <s xml:id="echoid-s537" xml:space="preserve">Hocvt ſciamus, duo numeri lineis
                  <reg norm="ſi- gnificentur" type="context">ſi-
                    <lb/>
                  gnificẽtur</reg>
                , primus linea .b:
                  <reg norm="ſecundus" type="context">ſecũdus</reg>
                linea
                  <var>.
                    <lb/>
                  c.</var>
                duæ autem partes
                  <var>.b.</var>
                duobus quadra-
                  <lb/>
                tis
                  <var>.q.i.</var>
                et
                  <var>.i.d.</var>
                notentur, eorum verò radi-
                  <lb/>
                ces lineis
                  <var>.a.g.</var>
                et
                  <var>.g.d.</var>
                differentia porrò ip
                  <lb/>
                ſi
                  <var>.c.</var>
                æqualis & co gnita ſit
                  <var>.a.h.</var>
                ex quo
                  <var>.h.</var>
                </s>
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