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-div13" type="math:theorem" level="3" n="4">
              <pb o="4" rhead="IO. BAPT. BENED." n="16" file="0016" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0016"/>
            </div>
            <div xml:id="echoid-div14" type="math:theorem" level="3" n="5">
              <head xml:id="echoid-head21" xml:space="preserve">THEOREMA
                <num value="5">V</num>
              .</head>
              <p>
                <s xml:id="echoid-s66" xml:space="preserve">
                  <emph style="sc">ALia</emph>
                quoque via prædicti effe
                  <lb/>
                  <figure xlink:label="fig-0016-01" xlink:href="fig-0016-01a" number="6">
                    <image file="0016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0016-01"/>
                  </figure>
                ctus cauſa, ſpeculando inno-
                  <lb/>
                teſcere poteſt, cuius rei gratia for-
                  <lb/>
                metur ſequens figura
                  <var>.e.o.a.u.n.</var>
                  <lb/>
                eiuſmodi, vt
                  <var>a.e.</var>
                ſit numerus li-
                  <lb/>
                linearis integrorum, &
                  <var>o.e.</var>
                produ-
                  <lb/>
                ctum numerantis ipſorum
                  <reg norm="fractorum" type="context">fractorũ</reg>
                  <lb/>
                in integris, ex quo
                  <var>.a.o.</var>
                erunt duæ
                  <lb/>
                tertiæ, verbigratia,
                  <var>a.i.</var>
                aut
                  <var>a.u.</var>
                qua-
                  <lb/>
                rum
                  <reg norm="linearum" type="context">linearũ</reg>
                ſingulę ſtatuuntur æqua
                  <lb/>
                les vnitati lineari, ſuperficies autem
                  <lb/>
                parallelogramma
                  <var>.u.n.</var>
                conſtituatur
                  <lb/>
                æqualis magnitudinis ſuperficiei
                  <var>.o.
                    <lb/>
                  e.</var>
                ex quo
                  <var>.u.n.</var>
                erit nobis cognita ſu-
                  <lb/>
                perficies. </s>
                <s xml:id="echoid-s67" xml:space="preserve">Cognoſcetur pariter quan
                  <lb/>
                titas partium
                  <var>.a.u.</var>
                quam in propoſi-
                  <lb/>
                to exemplo diximus eſſe trium par-
                  <lb/>
                tium. </s>
                <s xml:id="echoid-s68" xml:space="preserve">ex regula igitur de tribus, di-
                  <lb/>
                cemus ſi
                  <var>.u.a.</var>
                dat
                  <var>.a.e.</var>
                ſine dubio
                  <var>.o.
                    <lb/>
                  a.</var>
                dabit
                  <var>.a.n.</var>
                numerum linearem.
                  <lb/>
                quæ regula ex 15. ſexti in continuis,
                  <lb/>
                & ex 20. ſeptimi in diſcretis, depro-
                  <lb/>
                mitur. </s>
                <s xml:id="echoid-s69" xml:space="preserve">rectè igitur
                  <reg norm="multiplicantur" type="context">multiplicãtur</reg>
                fra-
                  <lb/>
                cti numerantes cum integris, & productum diuiditur per
                  <reg norm="denominantem" type="context">denominantẽ</reg>
                fractorum.</s>
              </p>
            </div>
            <div xml:id="echoid-div16" type="math:theorem" level="3" n="6">
              <head xml:id="echoid-head22" xml:space="preserve">THEOREMA
                <num value="6">VI</num>
              .</head>
              <p>
                <s xml:id="echoid-s70" xml:space="preserve">
                  <emph style="sc">ITem</emph>
                & alia ſpeculatione cognoſci poteſt hoc rectè fieri, mul-
                  <lb/>
                tiplicantes enim has duas tertias per decem, debemus conſide-
                  <lb/>
                  <figure xlink:label="fig-0016-02" xlink:href="fig-0016-02a" number="7">
                    <image file="0016-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0016-02"/>
                  </figure>
                rare quantitatem duarum
                  <reg norm="tertiarum" type="context">tertiarũ</reg>
                decies produci, ex quo oriuntur
                  <lb/>
                20. tertia, quandoquidem ſingulæ vnitates, </s>
                <s xml:id="echoid-s71" xml:space="preserve">tunc pro duobus ter-
                  <lb/>
                tijs ſumuntur, ſed c
                  <unsure/>
                um quilibet integer tria fragmenta contineat,
                  <lb/>
                ideo ex ratione partiendi quoties ternarius ingrediatur viginti,
                  <lb/>
                ſtatim cognoſcemus quod optabamus.</s>
              </p>
              <p>
                <s xml:id="echoid-s72" xml:space="preserve">Id ipſum accideret ſi integri in eiuſmodi ſpecie fractorum diui-
                  <lb/>
                derentur. quo facto hi multiplicandi eſſent cum numerante propo
                  <lb/>
                ſito, &
                  <reg norm="partiendum" type="context">partiendũ</reg>
                productum per quadratum denominantis.</s>
              </p>
              <p>
                <s xml:id="echoid-s73" xml:space="preserve">Cuius rei hæc eſt ſpeculatio. </s>
                <s xml:id="echoid-s74" xml:space="preserve">Sit linea
                  <var>.a.e.</var>
                conſtans ex
                  <reg norm="quinque" type="simple">quinq;</reg>
                  <lb/>
                integris numeris, quorum
                  <reg norm="vnuſquiſque" type="simple">vnuſquiſq;</reg>
                æqualis ſit
                  <var>.a.u.</var>
                vel
                  <var>.a.i.</var>
                &
                  <var>.a.o.</var>
                  <lb/>
                ſint duo tertia vnitatis integræ linearis. </s>
                <s xml:id="echoid-s75" xml:space="preserve">cogitemus nunc hos
                  <reg norm="quinque" type="simple">quinq;</reg>
                  <lb/>
                integros diuidi in ſua
                  <reg norm="fragmenta" type="context">fragmẽta</reg>
                linearia, quę in propoſito exemplo
                  <lb/>
                erunt 15. multiplicatis iam 15. cum propoſitis, videlicet
                  <var>a.o.</var>
                orie-
                  <lb/>
                tur productum
                  <var>.o.e.</var>
                triginta fragmentorum ſuperficialium,
                  <reg norm="quorum" type="context">quorũ</reg>
                  <lb/>
                in ſingulos integros ſuperficiales
                  <reg norm="cadunt" type="context">cadũt</reg>
                  <reg norm="nouem" type="context">nouẽ</reg>
                in hoc
                  <reg norm="exemplo" type="context">exẽplo</reg>
                , & cum
                  <lb/>
                notauerimus quoties
                  <reg norm="nouem" type="context">nouẽ</reg>
                ingrediatur triginta, propoſitum con-
                  <lb/>
                ſequemur.</s>
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