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. ARITH.
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              <pb o="11" rhead="THEOREM. ARITH." n="23" file="0023" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0023"/>
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            <div xml:id="echoid-div32" type="math:theorem" level="3" n="13">
              <head xml:id="echoid-head29" xml:space="preserve">THEOREMA.
                <num value="13">XIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s168" xml:space="preserve">
                  <emph style="sc">CVr</emph>
                diuidentibus numerum diuiſibilem per proueniens, oritur numerus diui-
                  <lb/>
                dens?</s>
              </p>
              <p>
                <s xml:id="echoid-s169" xml:space="preserve">Sit ſubſcriptus rectangulus
                  <var>.o.e.</var>
                numerus diuiſi
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0023-01a" xlink:href="fig-0023-01"/>
                bilis, qui producitur, tam ex
                  <var>.a.o.</var>
                in
                  <var>.a.e.</var>
                quám ex
                  <var>.a.
                    <lb/>
                  e.</var>
                in
                  <var>.a.o</var>
                . </s>
                <s xml:id="echoid-s170" xml:space="preserve">quare ſi
                  <var>.a.o.</var>
                diuidens fuerit
                  <var>.a.e.</var>
                proue-
                  <lb/>
                niens erit, ſi veró
                  <var>.a.e.</var>
                diuidens extiterit,
                  <var>a.o.</var>
                pro-
                  <lb/>
                ueniens erit futurum.</s>
              </p>
              <div xml:id="echoid-div32" type="float" level="4" n="1">
                <figure xlink:label="fig-0023-01" xlink:href="fig-0023-01a">
                  <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div34" type="math:theorem" level="3" n="14">
              <head xml:id="echoid-head30" xml:space="preserve">THEOREMA.
                <num value="14">XIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s171" xml:space="preserve">HOcipſum, alia
                  <reg norm="quoque" type="simple">quoq;</reg>
                uia licebit ſpeculari.</s>
              </p>
              <p>
                <s xml:id="echoid-s172" xml:space="preserve">Sit linea
                  <var>.a.</var>
                  <reg norm="denotans" type="context">denotãs</reg>
                numerum diuiſibilem, et
                  <var>.o.</var>
                primi prouenientis linea
                  <var>.e.</var>
                pri
                  <lb/>
                mi diuidentis
                  <var>.u.</var>
                ſecundi prouenientis ideſt cum
                  <var>.o.</var>
                pro diuidente ſumetur. </s>
                <s xml:id="echoid-s173" xml:space="preserve">Iam ex
                  <lb/>
                indicata definitione diuiſionis nono theoremate huius libri, dabitur proportio
                  <var>.a.</var>
                  <lb/>
                ad
                  <var>.o.</var>
                prout datur
                  <var>.e.</var>
                ad vnitatem ſignificatam li-
                  <lb/>
                nea
                  <var>.i.</var>
                & permutatim
                  <var>.a.</var>
                ad
                  <var>.e.</var>
                ſicut
                  <var>.o.</var>
                ad
                  <var>.i.</var>
                ſed
                  <var>.a.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0023-02a" xlink:href="fig-0023-02"/>
                ad
                  <var>.u.</var>
                ſic ſe habet prout
                  <var>.o.</var>
                ad
                  <var>.i.</var>
                ex eadem definitio-
                  <lb/>
                ne diuiſionis,
                  <reg norm="itaque" type="simple">itaq;</reg>
                ſic ſe habebit
                  <var>.a.</var>
                ad
                  <var>.u.</var>
                ſicut
                  <var>.a.</var>
                ad
                  <var>.
                    <lb/>
                  e.</var>
                vnde
                  <var>.u.</var>
                æqualis erit
                  <var>.e.</var>
                ex .9. quinti.</s>
              </p>
              <div xml:id="echoid-div34" type="float" level="4" n="1">
                <figure xlink:label="fig-0023-02" xlink:href="fig-0023-02a">
                  <image file="0023-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div36" type="math:theorem" level="3" n="15">
              <head xml:id="echoid-head31" xml:space="preserve">THEOREMA.
                <num value="15">XV</num>
              .</head>
              <p>
                <s xml:id="echoid-s174" xml:space="preserve">VNde prouenit, vt qui velit cognoſcere cuius numeri quatuor quintæ par-
                  <lb/>
                tes, ſint duæ tertię, aut quid ſimile,
                  <reg norm="conſultiſſime" type="context">cõſultiſſime</reg>
                faciat, ſi ad unam
                  <reg norm="eandemque" type="simple">eandemq;</reg>
                  <lb/>
                denominationem reduxerit.</s>
              </p>
              <p>
                <s xml:id="echoid-s175" xml:space="preserve">Prout in propoſito exemplo,
                  <reg norm="cum" type="context">cũ</reg>
                  <reg norm="denominans" type="context">denominãs</reg>
                  <reg norm="communis" type="context">cõmunis</reg>
                ſit quindecim, cuius duæ ter
                  <lb/>
                tiæ ſunt
                  <reg norm="decem" type="context">decẽ</reg>
                , & quatuor quintæ duodecim,
                  <reg norm="communis" type="context">cõmunis</reg>
                  <reg norm="autem" type="context">autẽ</reg>
                denominans .15. multipli
                  <lb/>
                candus ſit per quatuor quintas, ſcilicet duodecim, & productum diuidendum per
                  <lb/>
                duas tertias, hoc eſt decem, ex quo oriantur decemocto quęſitus numerus?</s>
              </p>
              <p>
                <s xml:id="echoid-s176" xml:space="preserve">Quod ad
                  <reg norm="reductionem" type="context">reductionẽ</reg>
                  <reg norm="numeratorum" type="context">numeratorũ</reg>
                ad vnam & eandem denominationem attinet,
                  <lb/>
                ea de cauſa fit quo uti poſſimus regula de tribus, quæ tribus tantummodo notis ter-
                  <lb/>
                minis indiget, quo quartus à prędictis dependens, inueniri poſſit, quandoquidem
                  <lb/>
                bini illi reſpectus, tribus terminis comprehendi
                  <reg norm="poſsunt" type="context">poſsũt</reg>
                . </s>
                <s xml:id="echoid-s177" xml:space="preserve">At quod ad multiplicatio-
                  <lb/>
                nem ſpectat denominantis
                  <reg norm="communis" type="context">cõmunis</reg>
                  <reg norm="cum" type="context">cũ</reg>
                numerante denominantis in cogniti & diui-
                  <lb/>
                ſionem producti per numerantem
                  <reg norm="cognitum" type="context">cognitũ</reg>
                illę nihil aliud ſunt, quam
                  <reg norm="quartum" type="context">quartũ</reg>
                  <reg norm="terminum" type="context">terminũ</reg>
                  <lb/>
                inuenire, ita proportionatum tertio, vt ſecundus primo.</s>
              </p>
              <p>
                <s xml:id="echoid-s178" xml:space="preserve">Excmpli gratia, ſit
                  <var>.a.</var>
                  <reg norm="denotans" type="context">denotãs</reg>
                nume-
                  <lb/>
                rantem denominantis cogniti, qui ſigni
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0023-03a" xlink:href="fig-0023-03"/>
                ficetur linea
                  <var>.o.</var>
                et
                  <var>.e.</var>
                ſit denominantis in-
                  <lb/>
                cogniti numerans, denotati linea
                  <var>.u.</var>
                imò
                  <lb/>
                verò & cogniti
                  <var>.o.</var>
                nempe quatuor
                  <lb/>
                quintæ, Iam ſi
                  <var>.o.</var>
                cum
                  <var>.e.</var>
                multiplicemus, & productum per
                  <var>.a.</var>
                diuidemus dabitur
                  <var>.u.</var>
                  <lb/>
                ſic ſe habens ad
                  <var>.e.</var>
                ſicut
                  <var>.o.</var>
                ad
                  <var>.a.</var>
                ex .20. ſeptimi.</s>
              </p>
              <div xml:id="echoid-div36" type="float" level="4" n="1">
                <figure xlink:label="fig-0023-03" xlink:href="fig-0023-03a">
                  <image file="0023-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0023-03"/>
                </figure>
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