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-div271" type="math:theorem" level="3" n="142">
              <pb o="96" rhead="IO. BAPT. BENED." n="108" file="0108" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0108"/>
              <p>
                <s xml:id="echoid-s1246" xml:space="preserve">Vnde cum aliquis diceret priori modo, dolium habeo vrnarum .400. vini, & per
                  <lb/>
                vices .25. extraxi & impleui ipſum, vt dictum eſt. </s>
                <s xml:id="echoid-s1247" xml:space="preserve">Nunc verò velim ſcire proportio-
                  <lb/>
                nem vini ad a quam hac vltima vice. </s>
                <s xml:id="echoid-s1248" xml:space="preserve">Nunc igitur ſi procedemus iuxta doctrinam
                  <lb/>
                primi exempli huius theorematis, obtinebimus quod quærebamus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1249" xml:space="preserve">Sed ſi diceret iuxta Tartaleæ quæſitum, hoc eſt dolium habeo, quod ignoro quot
                  <lb/>
                  <reg norm="nam" type="context">nã</reg>
                urnas contineat, volo tamen per .25. vices extrahere, & implere vt
                  <reg norm="ſupradictum" type="context">ſupradictũ</reg>
                  <lb/>
                eſt, ita vt vltima vice proportio vini ad aquam ſit ſeſquialtera. </s>
                <s xml:id="echoid-s1250" xml:space="preserve">Tunc ſi iuxta mo-
                  <lb/>
                dum ſecundi exempli huius theorematis procedemus habebimus quod cupimus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1251" xml:space="preserve">Alio etiam modo aliquis quærere poſſet, hoc eſt, habeo
                  <reg norm="dolium" type="context">doliũ</reg>
                quod capit .400.
                  <lb/>
                vrnas. </s>
                <s xml:id="echoid-s1252" xml:space="preserve">Habeo etiam vas trium vrnarum, quo mediante me oportet extrahere, &
                  <lb/>
                implere. </s>
                <s xml:id="echoid-s1253" xml:space="preserve">Velim tamen ſcire quoties me hoc facere oporteat, ita vt poſtrema vi-
                  <lb/>
                ce vinum ſe habeat ad aquam in proportione ſeſquialtera, vnde multoties accidet
                  <lb/>
                vltimam extractionem, & impletionem mutilatam, ſeu imperfectam, euadere.</s>
              </p>
              <p>
                <s xml:id="echoid-s1254" xml:space="preserve">Exempli gratia, ſi proportio vini ad aquam in vltima miſtione deberet eſſe vt
                  <var>.n.
                    <lb/>
                  u.</var>
                ad
                  <var>.n.a.</var>
                ita vt extrema vice fuiſſet
                  <var>.t.m.</var>
                quæ quidem
                  <var>.t.m.</var>
                excederet terminum per
                  <var>.
                    <lb/>
                  n.m.</var>
                quæ
                  <var>.n.m.</var>
                reuera eſſet nobis cognita, eò quòd ex priori modo hic ſupra dicto
                  <lb/>
                proportio
                  <var>.a.m.</var>
                ad
                  <var>.m.u.</var>
                nobis in-
                  <lb/>
                noteſceret, & proportio
                  <var>.n.a.</var>
                ad
                  <var>.
                    <lb/>
                  n.u.</var>
                nobis data eſt ſimul cum
                  <reg norm="quan­ titate" type="context">quã­
                    <lb/>
                    <figure xlink:label="fig-0108-01" xlink:href="fig-0108-01a" number="149">
                      <image file="0108-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0108-01"/>
                    </figure>
                  titate</reg>
                  <var>.a.u.</var>
                </s>
                <s xml:id="echoid-s1255" xml:space="preserve">quare quantitas
                  <var>.n.u.</var>
                &
                  <lb/>
                  <var>m.u.</var>
                nobis cognita, remanebit, et
                  <lb/>
                  <var>n.m.</var>
                eorum differentia ſimiliter, etiam, et
                  <var>.t.n.</var>
                reſiduum vaſis, quo metimur, vnde
                  <lb/>
                neceſſe erit, quo
                  <unsure/>
                d vltima vice vas contineret ſolum
                  <var>.t.n.</var>
                reliqua uerò per ſe patent.</s>
              </p>
            </div>
            <div xml:id="echoid-div274" type="math:theorem" level="3" n="143">
              <head xml:id="echoid-head162" xml:space="preserve">THEOREMA
                <num value="143">CXLIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1256" xml:space="preserve">HIeronymus Cardanus in
                  <ref id="ref-0015">lib. ſuæ arithmeticæ cap .66. quæſtione .56.</ref>
                quam Car­
                  <lb/>
                danicam vocat, ita inquit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1257" xml:space="preserve">Quidam perambulauit prima die certam quantitatem ſpatij, & ſecunda die,
                  <reg norm="tan­ tò" type="context">tã­
                    <lb/>
                  tò</reg>
                plus proportionaliter, quantò diameter eſt maior coſta, & tertia die tantò plus
                  <lb/>
                ſecunda, quantò proportionaliter portio lineæ diuiſæ ſecundum proportionem ha
                  <lb/>
                bentem medium, & duo extrema excedit minorem portionem, & quarta die in
                  <lb/>
                proportione ad tertiam vt ſecunda ad primam, & quinta die proportionaliter tan-
                  <lb/>
                tò plus quarta, quantò in tertia plus ſecunda, & ita alternatis vicibus in diebus no-
                  <lb/>
                uem peregit nouem milliaria. </s>
                <s xml:id="echoid-s1258" xml:space="preserve">Quæritur igitur quantum ambulauit die prima.</s>
              </p>
              <p>
                <s xml:id="echoid-s1259" xml:space="preserve">Hoc autem nihil aliud eſt, quàm ſi aliquis diceret, propono tibi, exempli gratia,
                  <lb/>
                lineam
                  <var>.a.l.</var>
                nouem partibus inuicem non æqualibus ita diuiſam
                  <var>.a.c</var>
                :
                  <var>c.d</var>
                :
                  <var>d.e</var>
                : & cæte-
                  <lb/>
                ris, quarum partium proportiones tibi etiam do, vt putà.
                  <var>a.c.</var>
                ad
                  <var>.c.d.</var>
                et
                  <var>.c.d.</var>
                ad
                  <var>.d.e.</var>
                et
                  <var>.
                    <lb/>
                  d.e.</var>
                ad
                  <var>.e.f.</var>
                & ſic de cæteris vſque ad poſtremam
                  <var>.k.l.</var>
                quæ quidem proportiones ſint
                  <lb/>
                etiam inuicem diſſimiles, ſeu inæquales, do tibi etiam
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                totius lineæ
                  <var>.a.l.</var>
                  <lb/>
                ad
                  <var>.a.b.</var>
                ſuam partem, quæ vt in propoſito exemplo nonupla eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s1260" xml:space="preserve">Quæro nunc quam proportionem habebit
                  <var>.a.c.</var>
                ad
                  <var>.a.b.</var>
                & ſic de cæteris partibus
                  <lb/>
                eiuſdem ad eandem
                  <var>.a.b</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s1261" xml:space="preserve">Quod quidem facillimum erit ſpeculari, nec non operari vnicuique, qui omnino
                  <lb/>
                practicæ numerorum ignarus non fuerit, dum ab ordine ſcientifico non diſcedat.</s>
              </p>
              <p>
                <s xml:id="echoid-s1262" xml:space="preserve">Cum enim cognoſcimus proportionem
                  <var>.a.c.</var>
                ad
                  <var>.c.d.</var>
                conſequenter cognoſcemus
                  <lb/>
                ctiam proportion em aggregati
                  <var>.a.c.d.</var>
                ad
                  <var>.c.d.</var>
                cum autem cognouerimus proportio- </s>
              </p>
            </div>
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