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. ARIT.
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            <div xml:id="echoid-div217" type="math:theorem" level="3" n="114">
              <p>
                <s xml:id="echoid-s1023" xml:space="preserve">
                  <pb o="77" rhead="THEOREM. ARIT." n="89" file="0089" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0089"/>
                eo qui per
                  <var>.b.d.</var>
                ad tempus quo
                  <var>.q.p.</var>
                ſolum, qui per
                  <var>.q.p.</var>
                mouetur (mo-
                  <lb/>
                tus enim continui regulares & vniformes conſtituuntur) eadem ratione ita-
                  <lb/>
                que ea erit proportio
                  <var>.q.k.</var>
                ad
                  <var>.b.c.</var>
                quæ
                  <var>.q.g.</var>
                ad
                  <var>.q.p.</var>
                & cum probatum
                  <lb/>
                fuerit ita ſe habere
                  <var>.k.n.</var>
                ad
                  <var>.c.o.</var>
                vt
                  <var>.q.k.</var>
                ad
                  <var>.b.c.</var>
                itaque ſic ſe habebit
                  <var>.k.n.</var>
                ad
                  <var>.c.
                    <lb/>
                  o.</var>
                ut
                  <var>.q.g.</var>
                ad
                  <var>.q.p.</var>
                probatum etiam fuit ita ſe habere
                  <var>.q.k.</var>
                ad
                  <var>.k.n.</var>
                vt
                  <var>.b.c.</var>
                ad
                  <var>.c.o.</var>
                ex quo
                  <lb/>
                componendo ſic ſe habebit
                  <var>.q.n.</var>
                ad
                  <var>.n.k.</var>
                vt
                  <var>.b.o.</var>
                ad
                  <var>.o.c.</var>
                & permutando ita
                  <var>.q.n.</var>
                ad
                  <var>.b.
                    <lb/>
                  o.</var>
                vt
                  <var>.k.n.</var>
                ad
                  <var>.c.o.</var>
                hoc eſt
                  <var>.q.g.</var>
                ad
                  <var>.q.p.</var>
                nempe vt tempus lenti ad tempus velocis itine-
                  <lb/>
                rantis, & componendo ita
                  <var>.q.n.</var>
                cum
                  <var>.o.b.</var>
                hoc eſt
                  <var>.b.d.</var>
                ad
                  <var>.b.o.</var>
                vt ſumma
                  <reg norm="dierum" type="context">dierũ</reg>
                vnius &
                  <lb/>
                alterius viatoris ad
                  <reg norm="minorem" type="context">minorẽ</reg>
                  <reg norm="numerum" type="context">numerũ</reg>
                  <reg norm="dierum" type="context">dierũ</reg>
                velocioris. </s>
                <s xml:id="echoid-s1024" xml:space="preserve">Breuiter
                  <reg norm="itaque" type="simple">itaq;</reg>
                obtineremus in
                  <lb/>
                  <reg norm="tentum" type="context">tentũ</reg>
                  <reg norm="quando" type="wordlist">qñ</reg>
                diceremus ſi ſumma dierum, quibus iter agitur à viatoribus talis eſt (20) re-
                  <lb/>
                ſpectu numeri dierum velocioris(9) qualis & cui
                  <reg norm="reſpondebit" type="context">reſpõdebit</reg>
                totum ſpacium
                  <var>.b.d</var>
                ? </s>
                <s xml:id="echoid-s1025" xml:space="preserve">vn-
                  <lb/>
                de dabitur ſpacium
                  <var>.b.o.</var>
                vnde reliqua omnia nobis cognita emergent.</s>
              </p>
              <p>
                <s xml:id="echoid-s1026" xml:space="preserve">Cum autem antiquorum regula iubeat numerum dierum vnius, cum numero die-
                  <lb/>
                rum alterius multiplicari, ac poſtmodum diuidi productum per ſummam omnium
                  <lb/>
                dierum, rectèid quidem fit. </s>
                <s xml:id="echoid-s1027" xml:space="preserve">Nam cum ſic ſe habeat
                  <var>.b.d.</var>
                ad
                  <var>.b.o.</var>
                vt ſumma omnium
                  <lb/>
                dierum ad minorem quantitatem dierum velocioris ſcilicet. </s>
                <s xml:id="echoid-s1028" xml:space="preserve">Ideo temporis propor
                  <lb/>
                tio à mobili per
                  <var>.b.d.</var>
                abſumpti ad tempus mobilis per
                  <var>.b.o.</var>
                eadem erit, quæ ſummæ
                  <lb/>
                omnium dierum ad namerum dierum velocioris. </s>
                <s xml:id="echoid-s1029" xml:space="preserve">Quarerectè dicemus, ſi eiuſmodi
                  <lb/>
                ſumma talem reſpectum habet ad minorem numerum dierum, quem numerum re-
                  <lb/>
                ſpiciet dies ipſius
                  <var>.b.d</var>
                ? </s>
                <s xml:id="echoid-s1030" xml:space="preserve">ex quo proferentur dies reſpondentes ipſi
                  <var>.b.o.</var>
                cætera iam
                  <lb/>
                dicta fuerunt.</s>
              </p>
              <p>
                <s xml:id="echoid-s1031" xml:space="preserve">Huiuſmodi verò ſpeculationis am-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0089-01a" xlink:href="fig-0089-01"/>
                plitudo ad pauciſſima verba reduci
                  <lb/>
                poteft, in cuius
                  <reg norm="gratiam" type="context">gratiã</reg>
                ſit ſubſcripta
                  <lb/>
                figura pars
                  <reg norm="inquam" type="context">inquã</reg>
                pręcedentis, in qua
                  <lb/>
                  <reg norm="conſtituamus" type="context simple">cõſtituamꝰ</reg>
                  <var>.o.n.</var>
                  <reg norm="locum" type="context">locũ</reg>
                  <reg norm="eum" type="context">eũ</reg>
                eſſe quo ſibi
                  <lb/>
                viatores obuient, ex quo ſpacium
                  <var>.q.
                    <lb/>
                  n.</var>
                à ſuo viatore conficietur, eo ipſo
                  <lb/>
                tempore, quo à ſuo ſpacium
                  <var>.b.o.</var>
                ita
                  <lb/>
                que eadem erit proportio
                  <var>.q.n.</var>
                ad
                  <var>.b.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0089-02a" xlink:href="fig-0089-02"/>
                o. quæ
                  <var>.q.g.</var>
                ad
                  <var>.b.d.</var>
                eadem erit
                  <reg norm="inquam" type="context">inquã</reg>
                  <lb/>
                proportio
                  <var>.d.o.</var>
                ad
                  <var>.o.b.</var>
                quæ numeri
                  <lb/>
                dierum eius, qui à
                  <var>.b.</var>
                pergit in
                  <var>.d.</var>
                ad
                  <lb/>
                numerum dierum alterius qui à
                  <var>.q.</var>
                in
                  <lb/>
                p. proficiſcitur, & componendo
                  <reg norm="eadem" type="context">eadẽ</reg>
                  <lb/>
                erit proportio
                  <var>.d.b.</var>
                ad
                  <var>.b.o.</var>
                quæ ſum-
                  <lb/>
                mæ dierum ad minorem numerum ipſorum, & eadem quæ dierum
                  <var>.b.d.</var>
                ad dies
                  <lb/>
                ipſius
                  <var>.b.o</var>
                .</s>
              </p>
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                <figure xlink:label="fig-0089-01" xlink:href="fig-0089-01a">
                  <image file="0089-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0089-01"/>
                </figure>
                <figure xlink:label="fig-0089-02" xlink:href="fig-0089-02a">
                  <image file="0089-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0089-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div219" type="math:theorem" level="3" n="115">
              <head xml:id="echoid-head133" xml:space="preserve">THEOREMA
                <num value="115">CXV</num>
              .</head>
              <p>
                <s xml:id="echoid-s1032" xml:space="preserve">CIRCA hæc ipſa itinera aliud quæritur peruenuſtè, in quo quæſito illud con­
                  <lb/>
                ſtituitur cognitum eſſe, nempe interuallum inter duo diuerſa loca, è quibus
                  <lb/>
                duo viatores eodem inſtanti vt ſibi occurrant proficiſcuntur,
                  <reg norm="certaque" type="simple">certaq́;</reg>
                milliaria ſin-
                  <lb/>
                gulis diebus conficiant, ita tamen, ut unus ordinatè plura altero ambulet, quæritur
                  <lb/>
                deinde quoto die ſibi occurrent. </s>
                <s xml:id="echoid-s1033" xml:space="preserve">Hoc autem fit diuiſo toto interuallo locorum per
                  <lb/>
                ſummam milliariorum quam vterque quotidie abſoluit.</s>
              </p>
            </div>
          </div>
        </div>
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