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-div175" type="math:theorem" level="3" n="89">
              <pb o="58" rhead="IO. BAPT. BENED." n="70" file="0070" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0070"/>
              <p>
                <s xml:id="echoid-s763" xml:space="preserve">Exempli gratia, caſu ſeſe offerunt hi quatuor numeri .8. 5. 3. 2. multiplicato .8.
                  <lb/>
                per .5. & hoc .40. per .3. rurſus hoc .120. per .2. vltimum productum eſſet .240. æqua
                  <lb/>
                le producto .15. (quod ex .5. in .3. oritur) in productum .16. quod ex .8. in .2. pro-
                  <lb/>
                fertur.</s>
              </p>
              <p>
                <s xml:id="echoid-s764" xml:space="preserve">Cuius ſpeculationis gratia, cogitemus quatuor numeros quatuor lineis
                  <var>.a.e.i.o.</var>
                  <lb/>
                ſignifi cari, productum autem
                  <var>.e.</var>
                in
                  <var>.i.</var>
                eſſe
                  <var>.m.f.</var>
                et
                  <var>.r.s.</var>
                ſimiliter & productum
                  <var>.a.</var>
                in
                  <var>.o.</var>
                eſ-
                  <lb/>
                ſe
                  <var>.m.z</var>
                : et
                  <var>.z.f.</var>
                productum eſſe
                  <var>.m.f.</var>
                in
                  <var>.m.z.</var>
                cui productum
                  <var>.a.</var>
                in
                  <var>.e.</var>
                multiplicatum per
                  <lb/>
                i. & hoc tandem per
                  <var>.o.</var>
                æquari debet.</s>
              </p>
              <p>
                <s xml:id="echoid-s765" xml:space="preserve">Sit itaque
                  <var>.u.y.</var>
                productum
                  <var>.a.</var>
                in
                  <var>.e.</var>
                quod
                  <var>.u.y.</var>
                per
                  <var>.i.</var>
                multiplicatum proferat
                  <var>.u.s.</var>
                  <lb/>
                hocq́ue
                  <var>.u.s.</var>
                multiplicatum per
                  <var>.o</var>
                . </s>
                <s xml:id="echoid-s766" xml:space="preserve">Dico quod dabit numerum æqualem numero
                  <var>.f.z.</var>
                  <lb/>
                Quamobrem
                  <var>.r.s.</var>
                aut
                  <var>.m.f.</var>
                quod idem eſt, in figura præcedentis theore matis ſigni-
                  <lb/>
                ficetur linea
                  <var>.n.u.</var>
                & linea
                  <var>.r.u.</var>
                hu-
                  <lb/>
                ius, nempe
                  <var>.a.</var>
                ſignificetur per
                  <var>.u.t.</var>
                  <lb/>
                  <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a" number="98">
                    <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0070-01"/>
                  </figure>
                præcedentis, ex quo numerus pro
                  <lb/>
                ducti
                  <var>.u.s.</var>
                præſentis, in præcedenti
                  <lb/>
                ſignificabitur producto
                  <var>.n.t.</var>
                quod
                  <lb/>
                  <reg norm="productum" type="simple context">ꝓductũ</reg>
                  <var>.u.s.</var>
                  <reg norm="pręsens" type="context">pręsẽs</reg>
                  <reg norm="per" type="simple">ꝑ</reg>
                  <reg norm="præsens" type="context">præsẽs</reg>
                  <var>.o.</var>
                mul­
                  <lb/>
                tiplicatum, quod erat in præceden
                  <lb/>
                ti
                  <var>.u.c.</var>
                ſignificabitur per
                  <var>.d.u.</var>
                præce
                  <lb/>
                dentis, quod non modo ex multi-
                  <lb/>
                plicatione
                  <var>.n.t.</var>
                præcedentis, nempe
                  <var>.u.s.</var>
                præſentis. in
                  <var>.u.c.</var>
                præcedentis æquali
                  <var>.o.</var>
                præ-
                  <lb/>
                ſentis oritur, ſed etiam ex
                  <var>.c.t.</var>
                præcedentis æquali
                  <var>.m.z.</var>
                præſentis in
                  <var>.n.u.</var>
                præceden
                  <lb/>
                tis æquali
                  <var>.m.f.</var>
                præſentis. </s>
                <s xml:id="echoid-s767" xml:space="preserve">Itaque verum eſt propoſitum.</s>
              </p>
            </div>
            <div xml:id="echoid-div177" type="math:theorem" level="3" n="90">
              <head xml:id="echoid-head107" xml:space="preserve">THEOREMA
                <num value="90">XC</num>
              .</head>
              <p>
                <s xml:id="echoid-s768" xml:space="preserve">CVR quibuſlibet & quantiſuis numeris in ſummam collectis, ſi ab vnitate in ſe-
                  <lb/>
                cunda ſpecie progreſſionis arithmeticę imparium numerorum progreſſi fue-
                  <lb/>
                rimus, eiuſmodi ſumma ſemper eſt quadratus numerus.</s>
              </p>
              <p>
                <s xml:id="echoid-s769" xml:space="preserve">Exempli gratia, ſi horum quatuor diſparium numerorum
                  <reg norm="ſummam" type="context">ſummã</reg>
                , in dicta pro-
                  <lb/>
                greſſione arithmetica quis ſumat, principio ab vnitate ſumpto, nempe .1. 3. 5. 7. ſum-
                  <lb/>
                ma erit .16. numerus quadratus inquam. </s>
                <s xml:id="echoid-s770" xml:space="preserve">Idem de cæteris.</s>
              </p>
              <p>
                <s xml:id="echoid-s771" xml:space="preserve">Quamobrem animaduertendum eſt, vnitatem, tam ſumi pro ſui ipſius radicem,
                  <lb/>
                quam pro quadrato, cubo, cenſo cenſi, primo relato, & alia quauis dignitate.
                  <lb/>
                </s>
                <s xml:id="echoid-s772" xml:space="preserve">Nunc autem pro quadrato ſumamus per
                  <var>.o.</var>
                ſignificato,
                  <reg norm="cogitemusque" type="simple">cogitemusq́</reg>
                quadratum
                  <var>.o.</var>
                  <lb/>
                includi quadrato vnitatem ſequenti, quod, vt patet, eſt quatuor vnitatum, ac pro-
                  <lb/>
                priè primum quadratum numerorum, ex quo etiam nomen accepit, vnde ex ſimi-
                  <lb/>
                litudine quam cætera quadrata cum hoc primo retinent, ex quaternario denomina-
                  <lb/>
                tionem acceperunt. </s>
                <s xml:id="echoid-s773" xml:space="preserve">
                  <reg norm="Hocitaque" type="simple">Hocitaq;</reg>
                ſit
                  <var>.o.u.c.e.</var>
                ita ex communi ſcientia quadrato
                  <var>.o.</var>
                iun-
                  <lb/>
                gitur gnomon
                  <var>.e.c.u.</var>
                conſtans tribus vnitatibus, quare primus gnomon, numero im-
                  <lb/>
                pari conſtat. </s>
                <s xml:id="echoid-s774" xml:space="preserve">Scimus etiam ex additione numeri binarij ad imparem, numeris di-
                  <lb/>
                ſparibus ſummam excreſcere, cum propius accedere
                  <reg norm="quam" type="context">quã</reg>
                binario nequeant, ex quo
                  <lb/>
                medio binario, ſibi inuicem ſuccedunt. </s>
                <s xml:id="echoid-s775" xml:space="preserve">Dico igitur quòd quinario ternarium ſub
                  <lb/>
                ſequente, coniuncto quadrato
                  <var>.o.u.c.e.</var>
                profertur quadratum, quod in numeris, bi-
                  <lb/>
                narij quadratum ſequitur,
                  <reg norm="eritque" type="simple">eritq́;</reg>
                ternarij,
                  <reg norm="quodque" type="simple">quodq́;</reg>
                ſignificetur per
                  <var>.o.f.</var>
                patet enim pri
                  <lb/>
                mo non differre ab
                  <var>.o.c.</var>
                præter quam gnomone
                  <var>.b.f.d.</var>
                qui coniungitur quadrato
                  <var>.o.
                    <lb/>
                  c.</var>
                quique duabus vnitatibus maior eſt
                  <var>.e.c.u</var>
                . </s>
                <s xml:id="echoid-s776" xml:space="preserve">
                  <reg norm="Iam" type="context">Iã</reg>
                ſcimus gnomonem
                  <var>.e.o.u.</var>
                æqualem </s>
              </p>
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