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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THEOR. ARITH.
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            <div xml:id="echoid-div140" type="math:theorem" level="3" n="71">
              <pb o="47" rhead="THEOR. ARITH." n="59" file="0059" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0059"/>
              <p>
                <s xml:id="echoid-s618" xml:space="preserve">Cuius rationem ſi quæris, ſignificentur .4. numeri lineis,
                  <var>a.e.o.u.</var>
                  <reg norm="diuidaturque" type="simple">diuidaturq́;</reg>
                .2.
                  <lb/>
                per
                  <var>.o.</var>
                &
                  <reg norm="oriatur" type="simple">oriat̃</reg>
                . s. & per
                  <var>.u.</var>
                  <reg norm="oriatur" type="simple">oriat̃</reg>
                  <var>.y.</var>
                et
                  <var>.
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0059-01a" xlink:href="fig-0059-01"/>
                  e.</var>
                diuiſo per
                  <var>.o.</var>
                oriatur
                  <var>.z.</var>
                & per
                  <var>.u.</var>
                  <lb/>
                proueniat
                  <var>.f.</var>
                tum
                  <var>.n.</var>
                ſit productum
                  <var>.z.</var>
                  <lb/>
                in
                  <var>.y.</var>
                et
                  <var>.m.</var>
                productum
                  <var>.s.</var>
                in
                  <var>.f</var>
                . </s>
                <s xml:id="echoid-s619" xml:space="preserve">Dico
                  <lb/>
                n. futurum æquale
                  <var>.m</var>
                . </s>
                <s xml:id="echoid-s620" xml:space="preserve">Sit deinde
                  <var>.
                    <lb/>
                  x.</var>
                vnitas, quare ex definitione diui-
                  <lb/>
                ſionis eadem erit proportio
                  <var>.s.</var>
                ad
                  <var>.a.</var>
                  <lb/>
                et
                  <var>.z.</var>
                ad
                  <var>.e.</var>
                quæ
                  <var>.x.</var>
                ad
                  <var>.o</var>
                . </s>
                <s xml:id="echoid-s621" xml:space="preserve">Sed ita ſe ha-
                  <lb/>
                bet
                  <var>.a.</var>
                ad
                  <var>.y.</var>
                et
                  <var>.e.</var>
                ad
                  <var>.f.</var>
                ſicut
                  <var>.u.</var>
                ad
                  <var>.x.</var>
                ex
                  <lb/>
                quo ſic ſe habebit
                  <var>.s.</var>
                ad
                  <var>.a.</var>
                ſicut
                  <var>.z.</var>
                ad
                  <lb/>
                e. et
                  <var>.a.</var>
                ad. y, ſicut
                  <var>.e.</var>
                ad
                  <var>.f</var>
                . </s>
                <s xml:id="echoid-s622" xml:space="preserve">Itaque ex
                  <lb/>
                æqualitate proportionum ſic ſe ha-
                  <lb/>
                bebit s. ad
                  <var>.y.</var>
                ſicut
                  <var>.z.</var>
                ad
                  <var>.f</var>
                . </s>
                <s xml:id="echoid-s623" xml:space="preserve">Igitur ex
                  <lb/>
                15. ſexti aut .20. ſeptimi productum
                  <var>.
                    <lb/>
                  n.</var>
                producto
                  <var>.m.</var>
                æquale erit.</s>
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            <div xml:id="echoid-div142" type="math:theorem" level="3" n="72">
              <head xml:id="echoid-head88" xml:space="preserve">THEOREMA
                <num value="72">LXXII</num>
              .</head>
              <p>
                <s xml:id="echoid-s624" xml:space="preserve">ALIVD quoque problema à me inuentum eſt, nempe vt proponantur .4.
                  <lb/>
                numeri qualeſcunque tandem, quorum duo diuiſibiles ſint, tertius diuiſor
                  <lb/>
                vnius è duobus pro libito,
                  <reg norm="quæramusque" type="simple">quæramusq́;</reg>
                alterius diuidentem, qui ſic ſe habeat vt pro
                  <lb/>
                ductum duorum prouenientium quarto numero propoſito ſit æquale.</s>
              </p>
              <p>
                <s xml:id="echoid-s625" xml:space="preserve">Exempli gratia, proponuntur .4. numeri .20. 48. 5. 12. porrò .20. et .48. numeri
                  <lb/>
                ſint diuiſibiles et .5.
                  <reg norm="diuidens" type="context">diuidẽs</reg>
                vnius, ut potè .20. </s>
                <s xml:id="echoid-s626" xml:space="preserve">
                  <reg norm="Quærendus" type="context">Quærẽdus</reg>
                nunc erit diuidens alterius
                  <lb/>
                nempe .48. eiuſmodi vt productum prouenientium æquale ſit .12. </s>
                <s xml:id="echoid-s627" xml:space="preserve">Diuidam itaque
                  <num value="20">.
                    <lb/>
                  20.</num>
                per .5.
                  <reg norm="prouenietque" type="simple">prouenietq́;</reg>
                4. quem per .48. multiplicabo, nempe per alterum diuiſibi-
                  <lb/>
                lem,
                  <reg norm="ſicque" type="simple">ſicq́;</reg>
                proueniet .192. quod productum per quartum numerum nempe .12. diui-
                  <lb/>
                fum dabit .16. qui erit diuidens quæſitus, quo diuiſo .48. proueniet .3. ſecundum ſci
                  <lb/>
                licet proueniens, quo per alterum hoc eſt .4. multiplicato producetur quartus nu-
                  <lb/>
                merus .12.</s>
              </p>
              <p>
                <s xml:id="echoid-s628" xml:space="preserve">Quod vt ſciamus, primus nume-
                  <lb/>
                rus diuiſibilis ſignificetur
                  <reg norm="rectangulo" type="context">rectãgulo</reg>
                  <var>.
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0059-02a" xlink:href="fig-0059-02"/>
                  a.i.</var>
                ſecundus rectangulo
                  <var>.o.u.</var>
                primus
                  <lb/>
                diuidens latere
                  <var>.a.e.</var>
                quartum nume-
                  <lb/>
                rum rectangulo
                  <var>.i.o.</var>
                primum proue-
                  <lb/>
                niens latere
                  <var>.e.i.</var>
                ſecundus diuidens la
                  <lb/>
                tere
                  <var>.e.u.</var>
                (hic autem eſt quem quæri-
                  <lb/>
                mus) tum alterum proueniens ſigni
                  <lb/>
                ficetur latere
                  <var>.e.o</var>
                . </s>
                <s xml:id="echoid-s629" xml:space="preserve">Iam
                  <reg norm="eadem" type="context">eadẽ</reg>
                erit pro-
                  <lb/>
                portio
                  <var>.e.i.</var>
                ad
                  <var>.e.u.</var>
                quæ
                  <var>.o.i.</var>
                ad
                  <var>.o.u.</var>
                  <lb/>
                Sed cum cognitæ ſint tres quantita-
                  <lb/>
                tes
                  <var>.e.i</var>
                :
                  <var>i.o</var>
                : et
                  <var>.o.u.</var>
                quarta quoque. e
                  <unsure/>
                  <var>.u.</var>
                exregula de tribus immediatè cognoſcetur,
                  <lb/>
                cætera in ſubſcripta figura facillimè patebunt.</s>
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                <figure xlink:label="fig-0059-02" xlink:href="fig-0059-02a">
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