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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                <pb o="249" rhead="EPISTOLAE." n="261" file="0261" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0261"/>
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                  <s xml:id="echoid-s3139" xml:space="preserve">Verum nolo te in ea, quæfalſa eſt, opinione conſiſtere, nonidem, & cum octona-
                    <lb/>
                  rio, ſenario, vel quinario, aut quouis alio numero poſſe efficere, cum eademmet ra
                    <lb/>
                  tio, quæ in ſeptenario, aut nouenario,
                    <reg norm="ent" type="context">ẽt</reg>
                  in cæteris perhibeatur. </s>
                  <s xml:id="echoid-s3140" xml:space="preserve">Ponamus
                    <reg norm="exemplum" type="context">exemplũ</reg>
                    <lb/>
                  hos tres or dinum numeros velle ſupputare, quorum primus ſit .679. ſecundus .846.
                    <lb/>
                  & tertius .935. & illorum
                    <reg norm="ſummam" type="context">ſummã</reg>
                  .2460. nunc maiorem numerum primi ordinis ab
                    <lb/>
                  octonario menſi, proijciendo, remanebit .7. deinde maiorem numerum demendo à
                    <lb/>
                  ſecundo or dine, reſiduum erit .6. ac ſi idem in tertio ordine fecerimus, erit nobis re-
                    <lb/>
                  liquum .7. </s>
                  <s xml:id="echoid-s3141" xml:space="preserve">Demum tria hæc reſidua in vnum collecta .20. efficient, à quibus ſi nume
                    <lb/>
                  rum maiorem ab octonario menſum dempſeris, ſupererunt .4. & totidem à ſumma
                    <num value="2460">.
                      <lb/>
                    2460.</num>
                  remanebunt, reiecto maiori numero ab octonario menſo. </s>
                  <s xml:id="echoid-s3142" xml:space="preserve">Atque idem me-
                    <lb/>
                  dio quouis alio numero, euenire poteſt.</s>
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                <p>
                  <s xml:id="echoid-s3143" xml:space="preserve">Cuius ratio tam perſe clara atque euidens eſt, quod ſi ſummam trium
                    <reg norm="reliquorum" type="context">reliquorũ</reg>
                  ,
                    <lb/>
                  quæ eſt .20. à ſumma .2460. ſubduxeris, remanebunt .2440. pro ſumma trium nume
                    <lb/>
                  rorum dictorum trium ordinum ab octonario menſorum, cui numero addito .16. pro
                    <lb/>
                  maiori numero ſummę
                    <reg norm="reliquorum" type="context">reliquorũ</reg>
                  , qui ab octonario menſus ſit, ſupererunt .4. </s>
                  <s xml:id="echoid-s3144" xml:space="preserve">At ſi per
                    <lb/>
                    <reg norm="ſenarium" type="context">ſenariũ</reg>
                    <reg norm="experimentum" type="context context">experimẽtũ</reg>
                  feceris, remanebit
                    <var>.o.</var>
                  & ſic de reliquis per ordinem
                    <reg norm="procedendo" type="context">procedẽdo</reg>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3145" xml:space="preserve">Verum poſſes ſciſcitari, quare velocius, exceſſus ordinum, potius per
                    <reg norm="noue- narium" type="context">noue-
                      <lb/>
                    nariũ</reg>
                  , quam per cæteros numeros, prout
                    <reg norm="docent" type="context">docẽt</reg>
                  practici, inueniri queat, videlicet ag
                    <lb/>
                  gregando prius duas figuras numerorum primæ ſummæ, deinde alias duas. </s>
                  <s xml:id="echoid-s3146" xml:space="preserve">Exem-
                    <lb/>
                  plum ſit primus ordo .679. colligendo .6. et .7. faciunt 13. & cum hæc ſumma ſit dua
                    <lb/>
                  rum figurarum, ſupputantur & ipſæ, è quibus prodeunt .4. & conſimilis erit proba-
                    <lb/>
                  tio numeri .67. facta per .9. quod idem eſt, ac ſi quis diuidat .67. per .9. ex quo reli-
                    <lb/>
                  qui erunt ſemper .4.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3147" xml:space="preserve">At quo ratio huiuſce perſpicuè dignoſci poſſit, in primis ſciendum eſt, cuique
                    <lb/>
                  ex ſe cognitum, atque exploratum eſſe, denarium numerum vnitate nouenarium ſu
                    <lb/>
                  perare, & ex hoc ſequitur, ſex denarios continere in ſe ſex nouenarios, & ſex vni-
                    <lb/>
                  tates.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3148" xml:space="preserve">At ſex vnitates, vna cum .7. faciunt .13. & quia in .13. eſt denarius, igitur in illo erit
                    <lb/>
                  vnitas ſupra .9. </s>
                  <s xml:id="echoid-s3149" xml:space="preserve">Quæ vnitas addita ternario, præbet nobis ſuperfluum, per quod .67.
                    <lb/>
                  ſuperat .54. iunctum cum .9. ſcilicet ſummam .63.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3150" xml:space="preserve">Idem dicinon poteſt de octonario, ſeptenario, vel ſenario, & de reliquis, quo-
                    <lb/>
                  niam numerus denariorum, in cæteris minoribus nouenario non præbet illico nu-
                    <lb/>
                  merum exceſſus maioris numeri, qui à numero probationis menſus eſt. </s>
                  <s xml:id="echoid-s3151" xml:space="preserve">Et quod di
                    <lb/>
                  co de probatione aggregationis, idem intelligo de alijs operationibus, vt puta ſub-
                    <lb/>
                  tractionis, multiplicationis, & partitionis ſeu diuiſionis.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3152" xml:space="preserve">Vnde autem oriatur, vt in partitionis probatione opus ſit probationem euentus
                    <lb/>
                  cum diuiſionis probatione multiplicare, & productum cum fractionis probatione
                    <lb/>
                  ſupputare, ſeu aggregare, tibi non erit ignotum, quoties animaduerteris, quod
                    <lb/>
                  productum ipſius euentus cum diuiſore, adiunctum fractioni, perpetuo ſe æquat nu
                    <lb/>
                  mero diuiſibili. </s>
                  <s xml:id="echoid-s3153" xml:space="preserve">Et quoniam numeri probationum ſunt partes, quæ remanent ex
                    <lb/>
                  ipſis totis, detractis maioribus numeris ab eo dimenſis, quo pro communi men-
                    <lb/>
                  ſura vtimur (prout .7. vel .9. aut alium numerum, quem voluerimus) par eſt vt ex ip-
                    <lb/>
                  ſarum remanentibus partibus, velut ex ipſis totis idem fiat.</s>
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