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-div219" type="math:theorem" level="3" n="115">
              <pb o="78" rhead="IO. BAPT. BENED." n="90" file="0090" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0090"/>
              <p>
                <s xml:id="echoid-s1034" xml:space="preserve">Exempli gratia, diſtant loca .100. milliaribus à ſe inuicem; </s>
                <s xml:id="echoid-s1035" xml:space="preserve">vnus autem viator
                  <lb/>
                ſingulis diebus .15 milliaria, alter .10. conficit ſi ita que .15. cum .10.
                  <reg norm="coniungamus" type="context">coniũgamus</reg>
                ,
                  <lb/>
                ſumma erit .25. per
                  <reg norm="quam" type="context">quã</reg>
                diuiſis milliaribus .100. totius interualli proferetur .4. nume
                  <lb/>
                rus quæſitus dierum quo viatoribus iter agendum erit prius quam ſibi obuient.</s>
              </p>
              <p>
                <s xml:id="echoid-s1036" xml:space="preserve">In cuius ſpeculationis gratiam totum iter ſignificetur linea
                  <var>.a.u</var>
                : primi autem via-
                  <lb/>
                toris iter diurnum fit
                  <var>.a.e.</var>
                & alterius
                  <var>.u.o</var>
                : terminus verò
                  <var>.i.</var>
                ſit occurſus ita vt eodem
                  <lb/>
                tempore, alter ſpacium
                  <var>.a.i.</var>
                alter
                  <var>.u.i.</var>
                confecerit, ſpacij autem
                  <var>.a.e.</var>
                tempus
                  <lb/>
                per
                  <var>.b.</var>
                ſignificetur & tempus ſpacij
                  <var>.u.o.</var>
                per
                  <var>.c.</var>
                quæ tempora erunt inter ſe
                  <lb/>
                æqualia, porrò ſpacij
                  <var>.a.i.</var>
                tempus per
                  <var>.d.</var>
                & ſpacij
                  <var>.u.i.</var>
                per
                  <var>.f.</var>
                denotetur, æquali
                  <lb/>
                bus inquam, ex quo eadem proportio erit
                  <var>.a.e.</var>
                ad
                  <var>.a.i.</var>
                quæ
                  <var>.b.</var>
                ad
                  <var>.d.</var>
                et
                  <var>.o.u.</var>
                ad
                  <var>.u.i.</var>
                quæ
                  <lb/>
                c. ad
                  <var>.f.</var>
                vnde permutando eadem erit proportio itineris ipſius
                  <var>.b.</var>
                ad iter ipſius
                  <var>.c.</var>
                quæ
                  <lb/>
                itineris
                  <var>.d.</var>
                ad iter ipſius
                  <var>.f.</var>
                & componendo itinerum ipſius
                  <var>.b.c.</var>
                ad iter
                  <var>.c.</var>
                vt itinerum
                  <var>.
                    <lb/>
                  d.f.</var>
                ad iter
                  <var>.f.</var>
                & permutando itinerum
                  <lb/>
                  <var>b.c.</var>
                ad itinera
                  <var>.d.f.</var>
                vt itineris
                  <var>.c.</var>
                ad. iter
                  <lb/>
                  <figure xlink:label="fig-0090-01" xlink:href="fig-0090-01a" number="122">
                    <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0090-01"/>
                  </figure>
                ipſius
                  <var>.f.</var>
                meritò itaque quęritur ſi itine
                  <lb/>
                ra
                  <var>.b.c.</var>
                dat itinera
                  <var>.d.f.</var>
                quid dabit tem-
                  <lb/>
                pus
                  <var>.c.</var>
                nempe dabit tempus
                  <var>.f.</var>
                ſed
                  <var>.c.</var>
                  <lb/>
                ſignatum eſt pro vna die, </s>
                <s xml:id="echoid-s1037" xml:space="preserve">quare in pro
                  <lb/>
                poſito exemplo
                  <var>.f.</var>
                ſignificabit 4: dies.</s>
              </p>
            </div>
            <div xml:id="echoid-div221" type="math:theorem" level="3" n="116">
              <head xml:id="echoid-head134" xml:space="preserve">THEOREMA
                <num value="116">CXVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s1038" xml:space="preserve">ANtiquorum monumentis traditum motum reperimus diuinandi numeri quem
                  <lb/>
                quis mente conceperit, quo iubemus eum qui numerum cogitauerit, dimi-
                  <lb/>
                dium cogitari numeri addere cogitato, atque huic ſummæ, rurſus eiuſdem ſummę
                  <lb/>
                dimidium adiungere, tum quærimus, quoties noueratius totam eam ſummam ingre
                  <lb/>
                diatur patefactis fractis ſi qui occurrant.</s>
              </p>
              <p>
                <s xml:id="echoid-s1039" xml:space="preserve">Exempli gratia, ſi quis cogitaſſet numerum .12. iubebant huic dimidium addi,
                  <lb/>
                nempe .6. ex quo ſumma erat .18. iubebant, præterea dimidium huius ſummæ nem-
                  <lb/>
                pe .9. toti ſummæ adiungi, quæ fuiſſet .27. adhæc quærebant ſibi patefieri quoties
                  <num value="9">.
                    <lb/>
                  9.</num>
                ſummam prædictam ingrederetur, & ſi in prima aut ſecunda diuiſione aut
                  <reg norm="etiam" type="context">etiã</reg>
                  <lb/>
                vtraque, fracti reperirentur, ac quoties nouem vltimam ſummam ingrediebatur,
                  <lb/>
                toties .4. multiplicabant. </s>
                <s xml:id="echoid-s1040" xml:space="preserve">Quod ſi in prima diuiſione fracti erant, vltimo produ-
                  <lb/>
                cto addebant vnitatem; </s>
                <s xml:id="echoid-s1041" xml:space="preserve">ſin verò in ſecunda, binarium adiungebant, ex quo exa-
                  <lb/>
                ctus numerus quæſitus proferebatur.</s>
              </p>
              <p>
                <s xml:id="echoid-s1042" xml:space="preserve">Pro cuius rei ratione ſit
                  <var>.a.</var>
                numerus cogitatione compræhenſus et
                  <var>.e.</var>
                ipſius
                  <var>.a.</var>
                cum
                  <lb/>
                eiuſdem medietate ſumma et
                  <var>.i.</var>
                ipſius
                  <var>.e.</var>
                cum eiuſdem medietate itidem ſumma, vn
                  <lb/>
                de
                  <var>.i.e.a.</var>
                tres numeri continui proportionales, in ſeſquialtera proportione euadent.
                  <lb/>
                </s>
                <s xml:id="echoid-s1043" xml:space="preserve">Sumantur nunc tres numeri .4. 6. 9. in eadem proportionalitate. </s>
                <s xml:id="echoid-s1044" xml:space="preserve">Vnde ratione ęqua
                  <lb/>
                litatis proportionum ita ſe habebit
                  <var>.i.</var>
                ad
                  <var>.a.</var>
                  <reg norm="quemadmodum" type="wordlist">quẽadmodum</reg>
                .9. ad .4. & permutando
                  <var>.i.</var>
                  <lb/>
                ad .9. quemadmodum
                  <var>.a.</var>
                ad .4. & ob id .4. toties ingredietur
                  <var>.a.</var>
                quoties .9. ipſam
                  <var>.i</var>
                . </s>
                <s xml:id="echoid-s1045" xml:space="preserve">Sed
                  <lb/>
                quia ſępe contingit, vt in ſecunda diuiſione, aut in ambabus etiam diuiſionibus re
                  <lb/>
                periantur numeri fracti, anima duertendum eſt numerum animo compræhenſum
                  <var>.a.</var>
                  <lb/>
                ſcilicet aut parem aut imparem ſemper futurum. </s>
                <s xml:id="echoid-s1046" xml:space="preserve">Si par eſt, aut multiplex erit ad
                  <num value="4">.
                    <lb/>
                  4.</num>
                aut non. </s>
                <s xml:id="echoid-s1047" xml:space="preserve">Si priori modo ſe habebit in duabus diuiſionibus, nullus numerus fra-
                  <lb/>
                ctus admittetur, ſed ſi ad .4. multiplex non erit, à multiplicibus per duo ſemper dif
                  <lb/>
                feret, & ſi per medium diuidatur, eiuſdem medietas impar ſemper erit, vnde prior </s>
              </p>
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