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="105" rhead="THEOREM. ARITH." n="117" file="0117" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0117"/>
                potius veras rationes
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                fundamenta huiuſmodi operationis oftendere, fu-
                  <lb/>
                mendo eadem exempla propoſita abipſis practicis, & maximè à Nicolao Tartalea
                  <lb/>
                viro accuratiffimo, qui vbicunque potuit ſpeculatus eſt cauiſas
                  <reg norm="ipſarum" type="context">ipſarũ</reg>
                operationum,
                  <lb/>
                etſi de huiuſmodi falſi regula circa finem cap .8. lib. 17. promittat poſtea loqui, nub-
                  <lb/>
                libi tamen loquutus eft. </s>
                <s xml:id="echoid-s1357" xml:space="preserve">Monendum etiam cenſeo, me nihil de rationibus regulæ
                  <lb/>
                falſi ſimplicis dicturum, cum ex ſeipſis ſatis appareant, quod non ita eſt de poſitio-
                  <lb/>
                nibus duplis. </s>
                <s xml:id="echoid-s1358" xml:space="preserve">Incipiam ergo à primo problemate lib. 17. ipſius Tartaleæ, quo
                  <reg norm="etiam" type="context">etiã</reg>
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                ipſe vtitur pro exemplo docendi gratia, ipſam regulam duplæ poſitionis, quod qui
                  <lb/>
                dem problema aliter à me
                  <reg norm="ſolutum" type="context">ſolutũ</reg>
                fuit in .118. </s>
                <s xml:id="echoid-s1359" xml:space="preserve">Theoremate huius mei lib. quod ſimi
                  <lb/>
                liter ob hanc demum occaſionem mihi oblatam, alia etiam via, ſpeculatus ſumidem
                  <lb/>
                poſſe fieri, quæ quidem via ſeu methodus generalis erit, & ita ſe habet.</s>
              </p>
              <p>
                <s xml:id="echoid-s1360" xml:space="preserve">Accipio enim propoſitum numerum diuiſibilem, à quo detraho ſummam
                  <lb/>
                datorum numerorum, primo duplicato, eo quòd tam in ſecunda quam in
                  <lb/>
                tertia parte reperitur, vt in propofito exemplo, datus numerus eft, 50. à
                  <lb/>
                quo detraho ſummam dictorum numerorum, quæ eſt .11. nam tres, & tres, &
                  <lb/>
                quinque ſunt vndecim, eo quòd primus ingreditur in ſecunda, & in tertia parte,
                  <lb/>
                dempto igitur hoc numero .11. ex .50. remanet .39. qui quidem numerus intelligen-
                  <lb/>
                dus eſt pro ſumma trium partium ſimplicium adhuc incognitarum, à quo extrahen
                  <lb/>
                da eſt prima, eo modo quo nunc proponam exregula de tribus, hoc eſt aggregan
                  <lb/>
                do dictas partes ſimplices ſine aliqua additione vtcunque volueris (ſed commodius
                  <lb/>
                erit in minimis numeris) iuxta propoſitum, quod quidem propoſitum eſt, vt ſecun
                  <lb/>
                da pars dupla ſit primæ, tertia verò æqualis fit primæ & ſecundæ, quæ partes in di-
                  <lb/>
                ctis minimis numeris, ita diſpoſitæ erunt .1. 2. 3. quarum ſumma erit .6. </s>
                <s xml:id="echoid-s1361" xml:space="preserve">Nunc ſi ex
                  <lb/>
                regula de tribus dixerimus, cum hæc ſumma proueniat nobis ab vno, à quo proue-
                  <lb/>
                niet .39. et veniet nobis .6. cum dimidio pro prima parte quæfita in propoſito nume-
                  <lb/>
                ro .39. cum ergo habuerimus primam
                  <reg norm="partem" type="context">partẽ</reg>
                , reliquas poſteà illicò cognoſcemus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1362" xml:space="preserve">Huiuſmodi verò operationis ratio ex ſe manifeſta patet, eo quòd proportio ſum
                  <lb/>
                mæ partium in minimis numeris ad primam eorum partem eadem eſſe debet, quæ
                  <lb/>
                ipſius .39. ad primam partem quæſitam huiuſmodi aggregati partium
                  <reg norm="ſimplicium" type="context">ſimpliciũ</reg>
                , ſed
                  <lb/>
                quia nemo adhuc, quod ſciam, ſatis animaduertit rationem modorum, qui ab anti-
                  <lb/>
                quis obſeruati ſunt, qui quidem modi duo ſunt circa hoc Helcataym duplæ falſæ
                  <lb/>
                pofitionis, igitur non prætermittam aliquid de hacreſpeculari, & primo de pri-
                  <lb/>
                mo modo.</s>
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              <p>
                <s xml:id="echoid-s1363" xml:space="preserve">In primis igitur
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                eft,
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                  <figure xlink:label="fig-0117-01" xlink:href="fig-0117-01a" number="161">
                    <image file="0117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0117-01"/>
                    <caption xml:id="echoid-caption1" xml:space="preserve">Compositorum</caption>
                  </figure>
                veritas ita inueniri poterit eo-
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                rum modo, me diantibus ſimpli­
                  <lb/>
                cibus partibus, vt
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                  <reg norm="median- tibus" type="simple">median-
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                  tibꝰ</reg>
                  <reg norm="compoſitis" type="context">cõpoſitis</reg>
                , ut in pręſenti
                  <reg norm="exem" type="context">exẽ</reg>
                  <lb/>
                plo pro primis pofitionibus ac-
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                ceperunt .10. et .8. pro ſecundis
                  <lb/>
                verò compoſitis
                  <reg norm="cum" type="context">cũ</reg>
                numero .3.
                  <lb/>
                  <reg norm="inuenerunt" type="context">inuenerũt</reg>
                .23. et .19. pro tertijs
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                  <reg norm="autem" type="wordlist">aũt</reg>
                  <reg norm="compoſitis" type="context">cõpoſitis</reg>
                  <reg norm="cum" type="context">cũ</reg>
                  <reg norm="quinque" type="simple">quinq;</reg>
                , notaue
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                runt .38. et .32. vnde prima ſum
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                marefultauit .71. ſecunda verò
                  <lb/>
                59. ita
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                  <reg norm="primus" type="simple">primꝰ</reg>
                error remanebat
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                21.
                  <reg norm="ſecundus" type="context simple">ſecũdꝰ</reg>
                  <reg norm="autem" type="wordlist">aũt</reg>
                .9. vt in figura
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                .</s>
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