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-div24" type="math:theorem" level="3" n="10">
              <pb o="9" rhead="THEOR. ARITH." n="21" file="0021" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0021"/>
              <p>
                <s xml:id="echoid-s128" xml:space="preserve">Hinc manifeſte patet quamlibet
                  <reg norm="diuiſionem" type="context">diuiſionẽ</reg>
                aut partitionem oriri ex regula de tri-
                  <lb/>
                bus, quandoquidem ſinguli diuidentes æquipollent vni integro, & loco illius ſu-
                  <lb/>
                muntur. </s>
                <s xml:id="echoid-s129" xml:space="preserve">Perinde enim eſt diuidere centum per viginti, ac
                  <reg norm="regulam" type="context">regulã</reg>
                obſeruare de tri-
                  <lb/>
                bus
                  <reg norm="dicentes" type="context">dicẽtes</reg>
                , ſi viginti æquipollent vni, quibus
                  <reg norm="ęquiualebunt" type="context">ęquiualebũt</reg>
                  <reg norm="centum" type="context">cẽtum</reg>
                ? </s>
                <s xml:id="echoid-s130" xml:space="preserve">Hoc autem ex ſub
                  <lb/>
                ſequenti figura facile deprehendetur, in qua linea
                  <var>.a.b.</var>
                ſignificat viginti, et
                  <var>.a.o.</var>
                  <reg norm="vni- tatem" type="context">vni-
                    <lb/>
                  tatẽ</reg>
                  <reg norm="linearem" type="context">linearẽ</reg>
                , et
                  <var>.a.c.</var>
                vnitates lineares
                  <reg norm="centum" type="context">centũ</reg>
                : </s>
                <s xml:id="echoid-s131" xml:space="preserve">o.c. verò centum vnitates ſuperficiales,
                  <lb/>
                et
                  <var>.a.d.</var>
                  <reg norm="quinque" type="simple">quinq;</reg>
                vnitates lineares, et
                  <var>.d.b.</var>
                centum vnitates ſuperficiales, ex quo manife-
                  <lb/>
                ftè deprehenditur quòd quemadmodum multiplicare, nihil aliud eſt, quam inueni
                  <lb/>
                re
                  <reg norm="productum" type="context">productũ</reg>
                ex duobus lateribus propoſitis, it a partiri nihil aliud eſt, quam da-
                  <lb/>
                to vno latere inuenire aliud latus producti propoſiti.</s>
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              <figure position="here" number="16">
                <image file="0021-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0021-01"/>
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              <figure position="here" number="17">
                <image file="0021-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0021-02"/>
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              <p>
                <s xml:id="echoid-s132" xml:space="preserve">Nam
                  <reg norm="quotieſcunque" type="simple">quotieſcunq;</reg>
                  <reg norm="ratiocinantes" type="context">ratiocinãtes</reg>
                dicimus tantundem numeri, immediate produci
                  <lb/>
                mus ſuperficiem,
                  <reg norm="mediante" type="context">mediãte</reg>
                vnitate in huiuſmodi numero, qui numerus
                  <reg norm="antequam" type="context">antequã</reg>
                pro-
                  <lb/>
                ducatur in vnitatem, mente concipiendus eſt tanqua m linearis, tanquam linea in-
                  <lb/>
                quam diuiſa in totidem particulas lineares, ſingulas continuas & æquales vnitati
                  <lb/>
                propoſitæ. </s>
                <s xml:id="echoid-s133" xml:space="preserve">
                  <reg norm="Cum" type="context">Cũ</reg>
                verò productus fuerit numerus in vnitate ſuperficialis, erit ac ſi tot eſ-
                  <lb/>
                ſent vnitates quadratæ, quod ſi ita non eſſet, nulla mentio facienda eſſet quo-
                  <lb/>
                rumuis
                  <reg norm="fractorum" type="context">fractorũ</reg>
                . </s>
                <s xml:id="echoid-s134" xml:space="preserve">Ex
                  <reg norm="eadem" type="context">eadẽ</reg>
                regula de tribus reduci poteſtad praxim
                  <reg norm="tertium" type="context">tertiũ</reg>
                theorema.</s>
              </p>
              <p>
                <s xml:id="echoid-s135" xml:space="preserve">Quare cupientes ſcire quæ ſint illæ partes, quæ ſunt tres quartę, ipſarum quin-
                  <lb/>
                que ſextarum, dicemus ſi quatuor dant tria, quid dabunt
                  <reg norm="quinque" type="simple">quinq;</reg>
                ſextæ? </s>
                <s xml:id="echoid-s136" xml:space="preserve">dabunt .15.
                  <lb/>
                vigeſimas quartas, quæ quindecim ſunt tres quartæ ipſius .20. viginti
                  <reg norm="autem" type="context">autẽ</reg>
                  <reg norm="quinque" type="simple">quinq;</reg>
                ſex
                  <lb/>
                tæ vigintiquatuor, quandoquidem nos numerum quęrimus, cui ita proportionentur
                  <lb/>
                  <reg norm="quinque" type="simple">quinq;</reg>
                ſextæ alterius numeri, ſicut quatuor ad tria, vnde ſic ſe habent .20. ad .15. ſi-
                  <lb/>
                cut .4. ad .3. ipſe autem .20.
                  <reg norm="quinque" type="simple">quinq;</reg>
                ſextę partes ſunt vigintiquatuor, vt per ſe
                  <reg norm="notum" type="context">notũ</reg>
                eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s137" xml:space="preserve">Ex eadem regula de tribus, huiuſmodi quęſito reſponderi poteſt, ſi conſtituamus
                  <lb/>
                prædictas
                  <reg norm="quinque" type="simple">quinq;</reg>
                ſextas eſſe numerum, cuius tres quartæ quęrantur, dicentes, ſi vnus
                  <lb/>
                integer dat tres quartas, quid dabunt
                  <reg norm="quinque" type="simple">quinq;</reg>
                ſextæ? </s>
                <s xml:id="echoid-s138" xml:space="preserve">quare ſequentes regulam de
                  <lb/>
                tribus, dabuntur quindecim vigeſimæ quartæ. </s>
                <s xml:id="echoid-s139" xml:space="preserve">Valet eadem regula de tribus; </s>
                <s xml:id="echoid-s140" xml:space="preserve">vt quis
                  <lb/>
                ſcire poſſit, quæ pars aut partes numeri propoſiti ſit aliquis numerus.</s>
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              <p>
                <s xml:id="echoid-s141" xml:space="preserve">Exempli gratia, ſcire cupienti, quæ pars aut partes ipſius vigintiquatuor ſint ſex-
                  <lb/>
                decim, conſtituentur .24. tanquam vnum totum, cuius pars aut partes ſint ſexdecim,
                  <lb/>
                dicemus igitur ſi .24. dant ſexdecim, quid dabit vnum? </s>
                <s xml:id="echoid-s142" xml:space="preserve">ſexdecim videlicet vigeſi-
                  <lb/>
                masquartas, quæ cum ad primos numeros reductæ fuerint, erunt duæ tertiæ.
                  <lb/>
                </s>
                <s xml:id="echoid-s143" xml:space="preserve">Eadem ratione qui ſcire uellet, quæ partes aut pars eſſent tres quartæ, octo no-
                  <lb/>
                narum, diceret, ſi octo nonæ danttres quartas, quid dabit vnum? </s>
                <s xml:id="echoid-s144" xml:space="preserve">prouenient .27.
                  <lb/>
                trigeſimęſecundæ.</s>
              </p>
              <p>
                <s xml:id="echoid-s145" xml:space="preserve">Subſeruit pariter ad
                  <reg norm="ſciendum" type="context">ſciendũ</reg>
                  <reg norm="naturam" type="context">naturã</reg>
                  <reg norm="partium" type="context">partiũ</reg>
                numeri propoſiti. </s>
                <s xml:id="echoid-s146" xml:space="preserve">Exempli cauſa, ſi quis
                  <lb/>
                quærat, cuius numeri, duodecim ſint duæ tertiæ partes. </s>
                <s xml:id="echoid-s147" xml:space="preserve">Dicet ſi duo dant tria, quid </s>
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