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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              <p>
                <pb o="101" rhead="THEOREM. ARIT." n="113" file="0113" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0113"/>
                <s xml:id="echoid-s1314" xml:space="preserve">25. ad radicem cubam .10000. quæ quidem proportiones æquales inuicem ſunt, cu
                  <lb/>
                tam vna, quàm alia, ſit tertia pars totius.</s>
              </p>
              <p>
                <s xml:id="echoid-s1315" xml:space="preserve">Pro cuius ratione cogitem is
                  <var>.a.b.</var>
                eſſe aliquod totum, quod multiplicare cupimus
                  <lb/>
                per duas tertias, quod
                  <reg norm="quidem" type="context">quidẽ</reg>
                nihil aliud eſt, quàm accipere duas tertias partes vnius
                  <lb/>
                totius ſuperficialis, imaginemur igitur hoc totum
                  <var>.a.b.</var>
                lineare diuiſum eſſe in tertias
                  <lb/>
                partes mediantibus
                  <var>.e.</var>
                et
                  <var>.d</var>
                . </s>
                <s xml:id="echoid-s1316" xml:space="preserve">& tunc multiplicando ipſum per 2. tertias lineares produ-
                  <lb/>
                ctum erit
                  <var>.a.c.</var>
                ſex vnitatum ſuperficialium, quod quidem productum poſteà diuiſum
                  <lb/>
                per .3. dabit
                  <var>.d.c.</var>
                hoc eſt duas tertias ſuperficiales (quæ eſt tertia pars ipſius
                  <var>.a.c.</var>
                ) &
                  <lb/>
                ęquales numero
                  <var>.c.b.</var>
                duabus vnitatibus linearibus, ideſt duabus tertijs ipſius
                  <var>.a.b</var>
                . </s>
                <s xml:id="echoid-s1317" xml:space="preserve">No
                  <lb/>
                tandum etiam eſt, quòd cum ferè omnia reducantur ad regulam de tribus, proptereà
                  <lb/>
                etiam multiplicatio alicuius quantitatis per aliam quantitatem, nihil aliud eſt quàm
                  <lb/>
                quædam operatio ipſius regulæ de tribus, vt eyempli gratia volo multiplicare .25.
                  <lb/>
                per 20. hoc nihil aliud eſt niſi quærere alium numerum ita proportionatum ad .25.
                  <lb/>
                vt 20. ſe habetad vnum, vnde multiplicando .25. cum .20. & productum diuidendo
                  <lb/>
                per vnum exregula de tribus, prouentus eſt idem numerus ipſius producti, & propte
                  <lb/>
                rea cum volumus multiplicare aliquem numerum per fractos hoc nihil aliud eſt
                  <lb/>
                quàm quærere aliquem numerum ita proportionatum ad ipſum numerum datum,
                  <lb/>
                vt ſe habet numerator ad denominatorem, exempli gratia ſi .24. aliquis voluerit mul
                  <lb/>
                tiplicare per duo tertia hoc idem eſt vt ſi quæreret numerum ad quem .24. ita ſe
                  <lb/>
                habeat, vt .3. ad .2. & idem dico de proportionibus, hoc eſt quod aliud non eſt mulri-
                  <lb/>
                plicare aliquam proportionem per fractos, quàm aliam proportionem quærere ad
                  <lb/>
                  <reg norm="quam" type="context">quã</reg>
                data ſe habeat, vt denominator ſe
                  <reg norm="hent" type="context">hẽt</reg>
                ad
                  <reg norm="numeratorem" type="context">numeratorẽ</reg>
                ; </s>
                <s xml:id="echoid-s1318" xml:space="preserve">& hoc exregula de tribus
                  <lb/>
                perficitur,
                  <reg norm="conſtituendo" type="context context">cõſtituẽdo</reg>
                  <reg norm="denominatorem" type="context">denominatorẽ</reg>
                in primo loco, quilocus eſt diuiſoris, numerato
                  <lb/>
                  <reg norm="rem" type="context">rẽ</reg>
                verò in
                  <reg norm="ſecundo" type="context">ſecũdo</reg>
                loco,
                  <reg norm="multiplicando" type="context">multiplicãdo</reg>
                poſteà pro
                  <lb/>
                portionem per
                  <reg norm="numeratorem" type="context">numeratorẽ</reg>
                , &
                  <reg norm="productum" type="context">productũ</reg>
                  <reg norm="diuidem" type="context">diuidẽ</reg>
                  <lb/>
                  <figure xlink:label="fig-0113-01" xlink:href="fig-0113-01a" number="156">
                    <image file="0113-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0113-01"/>
                  </figure>
                do per denominatorem, prouentus demum erit
                  <lb/>
                proportio, ad quam data ſe habebit, vt denomi-
                  <lb/>
                nator ſe
                  <reg norm="hent" type="context">hẽt</reg>
                ad numeratorem ex ratione ipſius re
                  <lb/>
                gulę de tribus. </s>
                <s xml:id="echoid-s1319" xml:space="preserve">Ratio verò methodi
                  <reg norm="diuidendi" type="context">diuidẽdi</reg>
                  <reg norm="vnam" type="context">vnã</reg>
                  <lb/>
                datam
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                per fractos, ex ſe ſatis patet,
                  <lb/>
                cum idem ſit modus diuidendi quemhbet nume
                  <lb/>
                rum integrum per fractos. </s>
                <s xml:id="echoid-s1320" xml:space="preserve">Quare, quæ vnius,
                  <lb/>
                & alterius eſt ratio.</s>
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            <div xml:id="echoid-div291" type="math:theorem" level="3" n="153">
              <head xml:id="echoid-head172" xml:space="preserve">THEOREMA
                <num value="153">CLIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1321" xml:space="preserve">NIcolaus Tartalea in .3. lib. quintæ partis numerorum ſoluit .24. quæſitum ſi-
                  <lb/>
                bi propoſitum à Hieronymo Cardano, via particulari & non generali. </s>
                <s xml:id="echoid-s1322" xml:space="preserve">Quæ-
                  <lb/>
                ſitum autem tale eſt quamlibet propoſitam rectam lineam in duas partes ita diuide
                  <lb/>
                re via Euclidis, ut cubus totius lineæ ad cubos partium ſe habeat in proportione
                  <lb/>
                tripla.</s>
              </p>
              <p>
                <s xml:id="echoid-s1323" xml:space="preserve">Tartalea igitur inquit quòd vt ſatisfiat ſpeculatiuis ingenijs ſoluendum ſit huiuſ-
                  <lb/>
                modi quæſitum, ſecando lineam propoſitam
                  <var>.a.b.</var>
                in tres æquales partes, quarum vna
                  <lb/>
                fit
                  <var>.c.b.</var>
                vnde problema ſolutum erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1324" xml:space="preserve">Verum dicit, ſed hæc non eſt methodus generalis, proptereà, quod cum tale
                  <lb/>
                problema alterius fuiſlet proportionis quam triplæ, talis methodus nihil valeret.</s>
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