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-div221" type="math:theorem" level="3" n="116">
              <p>
                <s xml:id="echoid-s1047" xml:space="preserve">
                  <pb o="79" rhead="THEOR. ARITH." n="91" file="0091" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0091"/>
                quoque ſumma par nunquam exiſtet, cuius medietatem aliquod medium ſemper
                  <lb/>
                ingredietur, & hanc ob cauſam poſterior ſumma cum fracto ſemper erit, & nume-
                  <lb/>
                rum deſumptum maiorem eſſe multiplici ad quatuor per duo ſignificabit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1048" xml:space="preserve">At verò ſi inter impares reponatur, aut eorum erit qui ſuperant multiplicem
                  <lb/>
                ipſius quatuor per vnum, ſeu per tria, quod hinc innoteſcet, nempe, quia ſi eorum
                  <lb/>
                erit qui dictum multiplicem per vnum tantum vincunt, ſua medietate ipſi numero
                  <lb/>
                addita, & præter hanc medietatem medio etiam integro adiuncto, tota hæc prior
                  <lb/>
                ſumma in numerum parem ſemper euadet, vnde in poſteriori ſumma nullus nume-
                  <lb/>
                rus fractus conſpicietur, & hanc ob
                  <reg norm="causam" type="context">causã</reg>
                multiplici ipſius .4. vnitas ſemper addetur.</s>
              </p>
              <p>
                <s xml:id="echoid-s1049" xml:space="preserve">Sed ſi numerus deſumptus, in ſerie eorum, qui multiplicem ipſius .4. pertria ſu-
                  <lb/>
                perant, collocabitur, hinc compræhendetur, quia primæ ſummæ numerus cum
                  <lb/>
                media vnitate ſemper impar erit, vnde ſecunda ſumma præter integras cum me-
                  <lb/>
                dia vnitate nobis ſemper occur ret.</s>
              </p>
              <p>
                <s xml:id="echoid-s1050" xml:space="preserve">Quod autem nobis prodere faciamus an in prima diuiſione, & ſecunda numerus
                  <lb/>
                aliquis fractus conſiſtat, eò tantum nobis inſeruit, quò deueniamus in cognitionem
                  <lb/>
                an numerus animo conceptus multiplicem ipſius .4. per vnum, per duo, aut tria ſupe
                  <lb/>
                ret. </s>
                <s xml:id="echoid-s1051" xml:space="preserve">Quòd etiam medias eas vnitates ad integros reducere faciamus, eò tantum re
                  <lb/>
                fertur, vt minori labore eum, qui numerum imaginatione compræhendit, onere-
                  <lb/>
                mus, quia reuera numerus impar nunquam mente concipi poteſt, quin aliquis fra-
                  <lb/>
                ctus in prima diuiſione, aut in ſecunda ſequatur: </s>
                <s xml:id="echoid-s1052" xml:space="preserve">vnde à numeris imparibus, qui mul
                  <lb/>
                tiplicem ipſius .4. unitatis tantum exceſſu
                  <reg norm="ſuperant" type="context">ſuperãt</reg>
                , poſterior ſumma
                  <reg norm="cum" type="context">cũ</reg>
                quarta parte
                  <lb/>
                vnitatis, præter integros numeros, & ab imparibus qui dictum multiplicem ipſius
                  <num value="4">.
                    <lb/>
                  4.</num>
                per tria vincunt, cum tribus quartis vnius integri præter integras vnitates ; </s>
                <s xml:id="echoid-s1053" xml:space="preserve">& à
                  <lb/>
                numeris paribus, qui multiplicem ipſius .4. per duo cum medietate vnitatis præter
                  <lb/>
                integros ſemper procedit. </s>
                <s xml:id="echoid-s1054" xml:space="preserve">Ita cum is qui numerum ſecum conſiderat, ſi in nume-
                  <lb/>
                  <figure xlink:label="fig-0091-01" xlink:href="fig-0091-01a" number="123">
                    <image file="0091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0091-01"/>
                  </figure>
                ris fractis verſatus eſſet, qui eum in-
                  <lb/>
                terrogat prudenter ſe gereret, ſi ſibi
                  <lb/>
                declarari curaret, quis nam ex fractis
                  <lb/>
                ſu per integros
                  <reg norm="ſecundæ" type="context">ſecũdæ</reg>
                  <reg norm="summæ" type="context">sũmæ</reg>
                remane
                  <lb/>
                ret, quia
                  <reg norm="per" type="simple">ꝑ</reg>
                quot quarta integros
                  <reg norm="ſecun- dæ" type="context">ſecũ-
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                  dæ</reg>
                ſummæ ſuperaret, per
                  <reg norm="totidem" type="context">totidẽ</reg>
                inte
                  <lb/>
                gros numerus mente conceptus multiplicem ipſius .4. ſuperaret.</s>
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            <div xml:id="echoid-div223" type="math:theorem" level="3" n="117">
              <head xml:id="echoid-head135" xml:space="preserve">THEOREMA
                <num value="117">CXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1055" xml:space="preserve">VNDE fiat, vt ſi ali quis quemuis numerum animo compræhendat, eique
                  <lb/>
                numero alium etiam quemlibet numerum propoſitum addat, & à tertia par
                  <lb/>
                te huius ſummæ tertiam partem numeri imaginati detrah et, reſiduum ſecundi nu-
                  <lb/>
                meri adiuncti, ideſt propoſiti, tertia pars erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1056" xml:space="preserve">Vt exempli gratia, ſi aliquis de numero denario cogitaſſet,
                  <reg norm="huicque" type="simple">huicq́;</reg>
                .24. adderet,
                  <lb/>
                vnde triginta quatuor efficerent, detra hendo nunc tertiam partem numeri de na-
                  <lb/>
                rij cogitatione concepti, ideſt .3. cum tertia parte vnius, à tertia parte huius ſum mæ
                  <lb/>
                ideſt ab vndecim & vna tertia parte remanerent .8. ideſt tertia pars numeri additi.
                  <lb/>
                </s>
                <s xml:id="echoid-s1057" xml:space="preserve">Id quod mihi inter iocos in honeſtorum hominum cætu in mentem venit.</s>
              </p>
              <p>
                <s xml:id="echoid-s1058" xml:space="preserve">Pro cuius ratione, prior numerus ima
                  <lb/>
                  <figure xlink:label="fig-0091-02" xlink:href="fig-0091-02a" number="124">
                    <image file="0091-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0091-02"/>
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                ginatus mediante linea
                  <var>.a.b.</var>
                et is, qui ad-
                  <lb/>
                ditus eſt
                  <reg norm="intercedente" type="context">intercedẽte</reg>
                linea
                  <var>.b.d.</var>
                è directo </s>
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