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-div181" type="math:theorem" level="3" n="92">
              <p>
                <s xml:id="echoid-s808" xml:space="preserve">
                  <var>
                    <pb o="61" rhead="THEOR. ARITH." n="73" file="0073" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0073"/>
                  n.k.</var>
                ipſius quadratum numerorum integrorum cognoſcetur, cui addito gnomone
                  <var type="gnomon">.
                    <lb/>
                  n.o.K.</var>
                cognoſcemus numerum
                  <var>.u.i.</var>
                quæſitum.</s>
              </p>
              <p>
                <s xml:id="echoid-s809" xml:space="preserve">Sed cum nobis hæc via, tenenda propoſitum non fuit, hoc eſt primo loco inue
                  <lb/>
                niendi quadrati minoris
                  <var>.n.K.</var>
                ideo ſupereſt probandum gnomonem
                  <var>.t.o.c.</var>
                vnitati
                  <reg norm="ae- qualem" type="simple">ę-
                    <lb/>
                  qualem</reg>
                eſſe, nempe quadratulo
                  <var>.m.a.</var>
                quod patebit, ſi conſideremus nos ſumpſiſſe
                  <lb/>
                rectangulum
                  <var>.r.c.</var>
                pro dimidio gnomonis
                  <var>.n.o.K</var>
                . </s>
                <s xml:id="echoid-s810" xml:space="preserve">etenim ſi ſupplemento etiam
                  <var>.n.r.</var>
                qua
                  <lb/>
                dratulum æquale
                  <var>.m.a.</var>
                adderetur, pateret gnomonem
                  <var>.n.a.K.</var>
                cum dicto quadratulo
                  <lb/>
                collectum, æqualem eſſe gnomoni
                  <var>.n.o.K</var>
                : cum duo ſupplementa
                  <var>.m.t.</var>
                et
                  <var>.m.c.</var>
                inter ſe
                  <lb/>
                fint æqualia. </s>
                <s xml:id="echoid-s811" xml:space="preserve">Quamobrem inuento quadrato
                  <var>.t.c.</var>
                ex dimidio gnomonis cognito,
                  <lb/>
                additur vnitas, gnomon ſcilicet
                  <var>.t.o.c.</var>
                ex quo cognoſcitur numerus
                  <var>.u.i.</var>
                quæſitus.
                  <lb/>
                </s>
                <s xml:id="echoid-s812" xml:space="preserve">Quod autem quadratum
                  <var>.g.p.</var>
                numeris integris conſtet, hac ratione probatur viſum
                  <lb/>
                enim fuit ſupra quadratum
                  <var>.n.K.</var>
                verè quadratum eſſe, & numeris integris conſtare,
                  <lb/>
                pariter etiam
                  <var>.t.c.</var>
                  <reg norm="ſeque" type="simple">ſeq́;</reg>
                mutuo conſequi (nam
                  <var>.K.c.</var>
                eſt vnitas linearis) ex quo gnomon
                  <lb/>
                  <var>n.a.K.</var>
                numero diſpari conſtabit, ex ijs quæ .90. theoremate probata fuerunt. </s>
                <s xml:id="echoid-s813" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                  <lb/>
                ex eodem theoremate neceſſe eſt gnomonem
                  <var>.t.d.c.</var>
                etiam numero diſpari conſtare,
                  <lb/>
                ita vt à numero
                  <var>.n.a.K.</var>
                non niſi duabus vnitatibus differat, nempe vt
                  <var>.c.p.</var>
                ſit vnitas li-
                  <lb/>
                nearis, ſed ita reuera eſt, numerus enim
                  <var>.u.d.i.</var>
                ex præſuppoſito par eſt, </s>
                <s xml:id="echoid-s814" xml:space="preserve">quare nume
                  <lb/>
                rus
                  <var>.t.d.c.</var>
                diſpar erit, cum alterum vnitate ſuperet, videlicet gnomone
                  <var>.t.o.c.</var>
                vnita
                  <lb/>
                ri æquali, tum
                  <var>.n.a.K.</var>
                minor eſt
                  <var>.n.o.K.</var>
                ex eodem gnomone
                  <var>.t.o.c.</var>
                unitati æquali. </s>
                <s xml:id="echoid-s815" xml:space="preserve">Ita
                  <lb/>
                que
                  <var>.n.a.K.</var>
                minor erit
                  <var>.u.d.i.</var>
                per vnitatem, & minor
                  <var>.t.d.c.</var>
                per duas unitates, ex quo ſe-
                  <lb/>
                quitur
                  <var>.g.p.</var>
                eſſe quadratum
                  <reg norm="integrorum" type="context">integrorũ</reg>
                ex dicto theoremate ac con ſequens quadrato
                  <lb/>
                  <var>t.c</var>
                . </s>
                <s xml:id="echoid-s816" xml:space="preserve">quare
                  <var>.c.p.</var>
                vnitas erit, & radices
                  <var>.q.K.</var>
                et
                  <var>.q.p.</var>
                horum quadratorum numero bina-
                  <lb/>
                rio inter ſe different. </s>
                <s xml:id="echoid-s817" xml:space="preserve">Vnà etiam ſcienda eſt cauſa, cur numerus propoſitus neceſſa
                  <lb/>
                  <figure xlink:label="fig-0073-01" xlink:href="fig-0073-01a" number="102">
                    <image file="0073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0073-01"/>
                  </figure>
                riò binario maior eſſe debeat. </s>
                <s xml:id="echoid-s818" xml:space="preserve">Etenim
                  <reg norm="cum" type="context">cũ</reg>
                ipſe
                  <lb/>
                ſit futurus gnomon
                  <var>.n.o.K.</var>
                nec poſſit minor eſſe
                  <lb/>
                numero ternario, vt patet ex .90. theoremate,
                  <lb/>
                idcirco ſequitur neceſſariò maiorem eſſe bina-
                  <lb/>
                rio debere. </s>
                <s xml:id="echoid-s819" xml:space="preserve">Quòd ſi diſpar numerus propone-
                  <lb/>
                retur, nec forma operis nec ſpeculationis
                  <reg norm="mutan- da" type="context">mutã-
                    <lb/>
                  da</reg>
                eſſet. </s>
                <s xml:id="echoid-s820" xml:space="preserve">Non erit tamen neceſſarium vt ipſa
                  <lb/>
                quadrata
                  <var>.n.K.</var>
                et
                  <var>.g.p.</var>
                numeris integris conſta-
                  <lb/>
                rent. </s>
                <s xml:id="echoid-s821" xml:space="preserve">Sæpius enim fractis
                  <reg norm="componerentur" type="context">cõponerentur</reg>
                , quod
                  <lb/>
                ex .90. theoremate facile erit ſpeculari nihilo-
                  <lb/>
                minus fractis integris,
                  <reg norm="ipſisque" type="simple">ipſisq́;</reg>
                collectis cum ſuis
                  <lb/>
                fractis ſummæ eſſent quadratæ.</s>
              </p>
            </div>
            <div xml:id="echoid-div183" type="math:theorem" level="3" n="93">
              <head xml:id="echoid-head110" xml:space="preserve">THEOREMA
                <num value="93">XCIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s822" xml:space="preserve">CVR propoſitis duobus numeris altero pari, altero verò diſpari, duplo primi
                  <lb/>
                minore per vnitatem, ſi alium inuenire numerum voluerimus, cui alterum iſto
                  <lb/>
                rum coniunctum proferat quadratum, & altero detracto, quadratum ſuperſit. </s>
                <s xml:id="echoid-s823" xml:space="preserve">Re-
                  <lb/>
                ctè datos numeros in ſummam colligemus, quam ſummam in duas quam maximas
                  <lb/>
                poterimus partes diuidemus, quarum vna pari, altera diſpari conſtet, tum vtran-
                  <lb/>
                que in ſeipſam multiplicabimus, & quadrato minori, duorum numerorum propo-
                  <lb/>
                ſitorum quemuis ademus, ex quo cupimus nobis quadratum minus ſupereſſe, & pro
                  <lb/>
                ueniet nobis numerum quæſitum.</s>
              </p>
              <p>
                <s xml:id="echoid-s824" xml:space="preserve">Exempli gtatia, proponuntur numeri .11. et .6. quorum alter alicui numero ad- </s>
              </p>
            </div>
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