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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            <div xml:id="echoid-div211" type="math:theorem" level="3" n="112">
              <p>
                <s xml:id="echoid-s974" xml:space="preserve">
                  <pb o="73" rhead="THEOREM. ARITH." n="85" file="0085" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0085"/>
                nis, ex quo
                  <var>.m.k.</var>
                prolixior erit
                  <var>.e.d.</var>
                ex præſup poſito. </s>
                <s xml:id="echoid-s975" xml:space="preserve">Poſtmodum
                  <var>.m.e.</var>
                et
                  <var>.k.d.</var>
                dua-
                  <lb/>
                bus lineis rectis coniungantur, quæ productæ concurrentin puncto
                  <var>.b.</var>
                ducatur pari-
                  <lb/>
                ter
                  <var>.e.g.</var>
                à puncto
                  <var>.e.</var>
                parallela
                  <var>.b.k.</var>
                et
                  <var>.m.a</var>
                :
                  <var>e.h.</var>
                et
                  <var>.b.q.</var>
                parallelæ
                  <var>.f.n.</var>
                ex quo
                  <var>.f.m.</var>
                æqua-
                  <lb/>
                lis erit
                  <var>.n.a.</var>
                et
                  <var>.m.h</var>
                :
                  <var>a.e.</var>
                et
                  <var>.h.q</var>
                :
                  <var>e.o.</var>
                et
                  <var>.g.k</var>
                :
                  <var>e.d.</var>
                et
                  <var>.f.q</var>
                :
                  <var>n.o.</var>
                ex .34. primi Eucli. </s>
                <s xml:id="echoid-s976" xml:space="preserve">vnde pro
                  <lb/>
                portio
                  <var>.m.h.</var>
                ad
                  <var>.h.q.</var>
                erit vt
                  <var>.m.g.</var>
                ad
                  <var>.g.k.</var>
                quandoquidem vtraque æqualis eſt propor-
                  <lb/>
                tioni
                  <var>.m.e.</var>
                ad
                  <var>.e.b.</var>
                ex .2. ſexti, ſed cum
                  <var>.m.k.</var>
                et
                  <var>.g.k.</var>
                notæ ſint, pariter cognoſcetur
                  <var>.m.
                    <lb/>
                  g.</var>
                ſecundum reſiduum, cum etiam notæ ſint
                  <var>.n.e.</var>
                et
                  <var>.n.a</var>
                . </s>
                <s xml:id="echoid-s977" xml:space="preserve">Itaque cognoſcemus
                  <var>.a.e.</var>
                hoc
                  <lb/>
                eſt
                  <var>.m.h.</var>
                cognitis verò
                  <var>.m.g</var>
                :
                  <var>g.k.</var>
                et
                  <var>.m.h.</var>
                ex .15. ſexti aut .20. ſeptimi cognoſcetur
                  <var>.h.
                    <lb/>
                  q.</var>
                erit igitur
                  <var>.a.e.</var>
                aut quod idem eſt
                  <var>.m.</var>
                hprimum reſiduum, et
                  <var>.m.g.</var>
                ſecundum, et
                  <var>.h.
                    <lb/>
                  q.</var>
                aut
                  <var>.e.o.</var>
                proueniens, et
                  <var>.n.o.</var>
                et
                  <var>.f.q.</var>
                itinera vtriuſque viatoris inter ſe æqualia.
                  <lb/>
                </s>
                <s xml:id="echoid-s978" xml:space="preserve">Nec verò prætermittenda eſt ſpeculatio vltimæ rationis inueniendæ quantitatis
                  <lb/>
                diei, quæ conſtat ope diuiſionis producti
                  <var>.m.h.</var>
                in .24. per
                  <var>.m.g</var>
                . </s>
                <s xml:id="echoid-s979" xml:space="preserve">Ea autem eiuſmodi
                  <lb/>
                eſt. </s>
                <s xml:id="echoid-s980" xml:space="preserve">Probatum fuit ſic ſe habere
                  <var>.m.h.</var>
                ad
                  <var>.h.q.</var>
                ut
                  <var>.m.g.</var>
                ad
                  <var>.g.k</var>
                . </s>
                <s xml:id="echoid-s981" xml:space="preserve">Itaque componendo
                  <lb/>
                ſic ſe habebit
                  <var>.m.q.</var>
                ad
                  <var>.h.q.</var>
                vt
                  <var>.m.k.</var>
                ad
                  <var>.g.k.</var>
                & permutando
                  <var>.m.q.</var>
                ad
                  <var>.m.k.</var>
                vt
                  <var>.h.q.</var>
                ad
                  <var>.g.
                    <lb/>
                  k</var>
                . </s>
                <s xml:id="echoid-s982" xml:space="preserve">Sed cum ſic ſe habeat
                  <var>.m.h.</var>
                ad
                  <var>.h.q.</var>
                vt
                  <var>.m.g.</var>
                ad
                  <var>.g.k.</var>
                permutando ſic ſe habebit
                  <var>.m.
                    <lb/>
                  h.</var>
                ad
                  <var>.m.g.</var>
                vt
                  <var>.h.q.</var>
                ad
                  <var>.g.k.</var>
                itaque
                  <lb/>
                ex .11. quinti ita
                  <var>.m.h.</var>
                ad
                  <var>.m.g.</var>
                vt
                  <var>.
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0085-01a" xlink:href="fig-0085-01"/>
                  m.q.</var>
                ad
                  <var>.m.k.</var>
                ex quo permutando
                  <lb/>
                  <var>m.h.</var>
                ad
                  <var>.m.q.</var>
                vt
                  <var>.m.g.</var>
                ad
                  <var>.m.k.</var>
                ſed
                  <lb/>
                  <reg norm="cum" type="context">cũ</reg>
                  <var>.m.k.</var>
                ſit motus toti diei reſpon
                  <lb/>
                dens, ſecurè dicere poterimus, ſi
                  <lb/>
                  <var>m.g.</var>
                talis eſt reſpectu horarum
                  <num value="24">.
                    <lb/>
                  24.</num>
                ſignificatarum per
                  <var>.m.k.</var>
                qualis
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0085-02a" xlink:href="fig-0085-02"/>
                erit
                  <var>.m.h.</var>
                & quo
                  <lb/>
                tæ parti dieire-
                  <lb/>
                ſpondens: </s>
                <s xml:id="echoid-s983" xml:space="preserve">quæ
                  <lb/>
                  <reg norm="poſtmodum" type="context">poſtmodũ</reg>
                erit
                  <var>.
                    <lb/>
                  m.q.</var>
                quæ, vt
                  <reg norm="di- ctum" type="context">di-
                    <lb/>
                  ctũ</reg>
                fuit, talis eſt
                  <lb/>
                reſpectu
                  <var>.m.k.</var>
                  <lb/>
                qualis
                  <var>.m.h.</var>
                re-
                  <lb/>
                ſpectu
                  <var>.m.g</var>
                . </s>
                <s xml:id="echoid-s984" xml:space="preserve">Reli
                  <lb/>
                quę duæ ſpecula
                  <lb/>
                tiones priorum
                  <lb/>
                  <reg norm="modorum" type="context">modorũ</reg>
                , vna &
                  <lb/>
                eadem eſt,
                  <reg norm="facilisque" type="simple">facilisq́;</reg>
                per ſe mediocriter intelligenti. </s>
                <s xml:id="echoid-s985" xml:space="preserve">Eodem modo reliquæ omnes
                  <lb/>
                progreſſiones ſecundi viatoris
                  <reg norm="cum" type="context">cũ</reg>
                rectangulo primi conferri ex hoc theoremate
                  <lb/>
                poterunt.</s>
              </p>
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                <figure xlink:label="fig-0085-01" xlink:href="fig-0085-01a">
                  <image file="0085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0085-01"/>
                </figure>
                <figure xlink:label="fig-0085-02" xlink:href="fig-0085-02a">
                  <image file="0085-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0085-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div214" type="math:theorem" level="3" n="113">
              <head xml:id="echoid-head130" xml:space="preserve">THEOREMA
                <num value="113">CXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s986" xml:space="preserve">PRoponitur & aliud, primum ſcilicet viatorem iter incipere diebus aliquot an-
                  <lb/>
                tè ſecundum, primum tamen lentius, quàm ſecundum ambulare, & utrunque
                  <lb/>
                eorum certa quædam milliaria conficere. </s>
                <s xml:id="echoid-s987" xml:space="preserve">Iam ſiſcire voluerimus in quot diebus
                  <lb/>
                ſeſe conſequentur, uulgaris regula iubet, inſpici quot milliaria primus ſolus iter a-
                  <lb/>
                gens confecerit, tum animaduerti differentiam diurnam motus vnius ab altero,
                  <reg norm="atque" type="simple">atq;</reg>
                  <lb/>
                milliarium numerum primi viatoris ſoli abundantis per hanc
                  <reg norm="differentiam" type="context">differentiã</reg>
                diuidi, pro
                  <lb/>
                ueniens autem erit numerus dierum quæſitus.</s>
              </p>
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