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-div179" type="math:theorem" level="3" n="91">
              <p>
                <s xml:id="echoid-s788" xml:space="preserve">
                  <pb o="60" rhead="IO. BAPT. BENED." n="72" file="0072" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0072"/>
                per duab. vnitatibus ſuperficialibus creſcere,
                  <reg norm="quarum" type="context">quarũ</reg>
                  <reg norm="ſingularum" type="context">ſingularũ</reg>
                radix æqualis eſt
                  <var>.g.</var>
                ne
                  <lb/>
                ceſſariò ſequitur gnomonem
                  <var>.b.a.g.</var>
                duabus partibus aut vnitatibus gnomonem
                  <var>.d.
                    <lb/>
                  o.p.</var>
                ſuperare, ita vt gnomon
                  <var>.b.a.g.</var>
                ſeptem vnitatibus, aut partibus ſuperficialibus
                  <lb/>
                quadratis conſtet. </s>
                <s xml:id="echoid-s789" xml:space="preserve">Quare eadem ratione gnomon
                  <var>.f.s.h.</var>
                conſtabit nouem ſimilibus.
                  <lb/>
                </s>
                <s xml:id="echoid-s790" xml:space="preserve">Itaque æqualis erit quadrato
                  <var>.q.o</var>
                . </s>
                <s xml:id="echoid-s791" xml:space="preserve">Quamobrem verum eſt, quòd quadrato
                  <var>.q.o.</var>
                  <lb/>
                coniuncto quadrato
                  <var>.q.a.</var>
                proueniet quadratum
                  <var>.q.s.</var>
                cuius radix ita differet à
                  <var>.q.g.</var>
                vt
                  <var>.
                    <lb/>
                  q.g.</var>
                à
                  <var>.q.p</var>
                : ex quo tres radices arithmeticè inter ſe continuæ proportionales erunt.
                  <lb/>
                </s>
                <s xml:id="echoid-s792" xml:space="preserve">Idipſum dico ſi
                  <var>.q.p.</var>
                fuerit .6. et
                  <var>.q.g</var>
                : 8: </s>
                <s xml:id="echoid-s793" xml:space="preserve">tunc enim ſingulæ partes
                  <var>.q.k.p.g.h.</var>
                æquipol
                  <lb/>
                lebunt duabus vnitatibus, quæ cogitabuntur
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0072-01a" xlink:href="fig-0072-01"/>
                in ſummam collectæ, ut cum patribus
                  <var>.q.k.p.
                    <lb/>
                  g.h.</var>
                integris contemplari liceat. </s>
                <s xml:id="echoid-s794" xml:space="preserve">Idem acci-
                  <lb/>
                det fi
                  <var>.q.p.</var>
                erit .9. et
                  <var>.q.g.</var>
                12. fingulæ enim par-
                  <lb/>
                tes
                  <var>.q.K.p.g.h.</var>
                tripartitæ erunt. </s>
                <s xml:id="echoid-s795" xml:space="preserve">Idcircò dixi
                  <lb/>
                gnomonem
                  <var>.f.s.h.</var>
                tam amplum cogitari de-
                  <lb/>
                bere, quam gnomon
                  <var>.b.a.g.</var>
                nempè ut
                  <var>.h.</var>
                æqua
                  <lb/>
                lis ſit
                  <var>.g</var>
                . </s>
                <s xml:id="echoid-s796" xml:space="preserve">Idem occurret ſi
                  <var>.q.g.</var>
                erit .12. et
                  <var>.q.p.</var>
                  <lb/>
                quinque, quod cum fuerit patebitex præce-
                  <lb/>
                dentis theorematis ſpeculatione, gnomonem
                  <lb/>
                  <var>f.s.h</var>
                : 25. vnitatibus conſtare, cogitatum am-
                  <lb/>
                plitudinis ſimplicis vnitatis denominatæ in
                  <var>.q.
                    <lb/>
                  p.</var>
                aut
                  <var>.q.g.</var>
                non amplitudinis gnomonis
                  <var>.b.a.g.</var>
                  <lb/>
                qui ſeptem vnitatibus latus eſſet. </s>
                <s xml:id="echoid-s797" xml:space="preserve">Cum igitur
                  <var>.
                    <lb/>
                  q.p.</var>
                quinque vnitatibus linearibus conſtet ſcimus
                  <var>.q.o</var>
                : 25. ſuperficialibus conſtare,
                  <lb/>
                collecto itaque in ſummam quadrato
                  <var>.q.o.</var>
                cum quadrato
                  <var>.q.a.</var>
                cognoſcetur quadra-
                  <lb/>
                tum
                  <var>.q.s.</var>
                vnà etiam eius radix. </s>
                <s xml:id="echoid-s798" xml:space="preserve">Eadem ratione, alia multa quadrata ſimilia contem-
                  <lb/>
                plari licebit.</s>
              </p>
              <div xml:id="echoid-div179" type="float" level="4" n="1">
                <figure xlink:label="fig-0071-02" xlink:href="fig-0071-02a">
                  <image file="0071-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0071-02"/>
                </figure>
                <figure xlink:label="fig-0072-01" xlink:href="fig-0072-01a">
                  <image file="0072-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0072-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div181" type="math:theorem" level="3" n="92">
              <head xml:id="echoid-head109" xml:space="preserve">THEOREMA
                <num value="92">XCII</num>
              .</head>
              <p>
                <s xml:id="echoid-s799" xml:space="preserve">CVR propoſito numero pari maiori binario, qui detrahi & in ſummam colli-
                  <lb/>
                gi debeat ex altero numero quærendo, vt tam reſiduum quam ſumma ſint
                  <lb/>
                quadrata numerorum integrornm. </s>
                <s xml:id="echoid-s800" xml:space="preserve">Rectè dimidium propoſiti numeri in ſeipſum
                  <lb/>
                multiplicamus, & quadrato huic addimus vnitatem,
                  <reg norm="eritque" type="simple">eritq́;</reg>
                numerus quæfitus.</s>
              </p>
              <p>
                <s xml:id="echoid-s801" xml:space="preserve">Exempli gratia proponitur .12. numerus detrahendus, & coniungendus nume-
                  <lb/>
                ro inueſtigando, ut reſiduum detractionis, & ſumma ſint quadrati numeri. </s>
                <s xml:id="echoid-s802" xml:space="preserve">Addi-
                  <lb/>
                ta vnitate ipſi .36. quadrato dimidij, dabitur .37. numerus quæſitus.</s>
              </p>
              <p>
                <s xml:id="echoid-s803" xml:space="preserve">Cuius ſpeculationis gratia, ſubſcripta quatuor quadrata cogitemus
                  <var>.g.p</var>
                :
                  <var>u.i</var>
                :
                  <var>t.c</var>
                :
                  <var>n.
                    <lb/>
                  K.</var>
                  <reg norm="cogitemusque" type="simple">cogitemusq́;</reg>
                quadratum
                  <var>.g.p.</var>
                eſſe quadratum ſummæ,
                  <var>K.n.</var>
                verò reſidui ſubtractio-
                  <lb/>
                nis:
                  <var>u.i.</var>
                  <reg norm="autem" type="wordlist">aũt</reg>
                numerum
                  <reg norm="inueſtigandum" type="context context">inueſtigãdũ</reg>
                , ex quo gnomon
                  <var>.u.d.i.</var>
                cognoſcetur ita etiam et
                  <var>.n.
                    <lb/>
                  o.K.</var>
                qui inter ſe ſunt æquales. </s>
                <s xml:id="echoid-s804" xml:space="preserve">Iam certi erimus
                  <var>.e.i.</var>
                eſſe plus quam dimidium gno-
                  <lb/>
                monis
                  <var>.n.o.K</var>
                . </s>
                <s xml:id="echoid-s805" xml:space="preserve">Itaque cogitemus rectangulum
                  <var>.r.c.</var>
                exactum
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                eſſe gnomonis
                  <var>.
                    <lb/>
                  n.o.K.</var>
                ex unitatibus ſuperficialibus quarum una erit
                  <var>.m.a</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s806" xml:space="preserve">Cuius numeri quadratum ſit
                  <var>.t.c.</var>
                vnde etiam cognitum & cum
                  <var>.K.c.</var>
                ex communi
                  <lb/>
                ſcientia ſit vnitas linearis, </s>
                <s xml:id="echoid-s807" xml:space="preserve">propterea quod
                  <var>.m.a.</var>
                eſt ſuperficialis hoc eſt quadrata,
                  <lb/>
                quæ detracta ex
                  <var>.q.c.</var>
                dimidio gnomonis
                  <var>.n.o.K.</var>
                (quamuis lineari) ſupererit
                  <var>.K.q.</var>
                co
                  <lb/>
                gnita, numerorum integrorum (nota
                  <var>q.K.i.</var>
                ſemper minor erit duabus vnitatibus li-
                  <lb/>
                nearibus & maior vna ex dictis vnitatibus, ut ex te ipſo contemplari potes) </s>
                <s xml:id="echoid-s808" xml:space="preserve">quare
                  <var>. </var>
                </s>
              </p>
            </div>
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