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="106" rhead="IO. BAPT. BENED." n="118" file="0118" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0118"/>
              <p>
                <s xml:id="echoid-s1364" xml:space="preserve">SEDijdem errores proueniunt exſummis partium ſimplicium.</s>
              </p>
              <p>
                <s xml:id="echoid-s1365" xml:space="preserve">Vtexempli gratia, in figura
                  <var>.B.</var>
                ſumma propoſita partium ſimplicium eſt .39.
                  <lb/>
                vt diximus, eo quòd ab ipſo .50. detraxerimus .11. ſumma ſcilicet numerorum adij
                  <lb/>
                ciendorum ad efficiendas partes compofitas, ſumma poſteà fimplicium partium
                  <lb/>
                primæ poſitionis, erit .60. eo quòd prima pars erat .10. ſecunda autem ſimplex 20.
                  <lb/>
                tertia verò fimplex .30. iuxta ordinem propoſiti. </s>
                <s xml:id="echoid-s1366" xml:space="preserve">Summa deinde ſimplicium
                  <reg norm="partium" type="context">partiũ</reg>
                  <lb/>
                fecundæ poſitionis effet .48. quia prima eius pars erat .8. ſecunda verò ſimplex .16.
                  <lb/>
                tertia autem ſimplex .24. vnde prima ſumma excederet datam .39. per .21. differen-
                  <lb/>
                tiæ, ſecunda verò per .9. vt ſupra vidimus de ſummis compoſitis à dato .50. compo-
                  <lb/>
                fito, & hoc quidem mirandum non eft, quod ſcilicet tres ſummæ fimplicium par-
                  <lb/>
                tium ſintinuicem inæqua-
                  <lb/>
                les, ijſdem differentijs me-
                  <lb/>
                  <figure xlink:label="fig-0118-01" xlink:href="fig-0118-01a" number="162">
                    <image file="0118-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0118-01"/>
                    <caption xml:id="echoid-caption2" xml:space="preserve">Simpricium</caption>
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                diantibus, quibus
                  <reg norm="differunt" type="context">differũt</reg>
                  <lb/>
                dictæ tres ſummæ compofi
                  <lb/>
                tæ, cum ab vnaquaque
                  <reg norm="con" type="context">cõ</reg>
                  <lb/>
                  <reg norm="poſitarum" type="context">poſitarũ</reg>
                ablatus fit nume-
                  <lb/>
                rus .11. æqualiter, vnde ex
                  <lb/>
                neceſſitate, permutando,
                  <lb/>
                  <reg norm="earum" type="context">earũ</reg>
                differentiæ
                  <reg norm="relinquem" type="context">relinquẽ</reg>
                  <lb/>
                dæ erant æquales inuicem
                  <lb/>
                ex
                  <ref id="ref-0017">.78. theoremate hu-
                    <lb/>
                  ius noſtri lib.</ref>
                ſummæ enim
                  <lb/>
                compofitæ erant .71. 59. et
                  <lb/>
                50. fimplices verò .60. 48.
                  <lb/>
                et .39. differentes à primis
                  <lb/>
                per .11. vt dictum eft, qua
                  <lb/>
                re veritas ita manabit à compofitis, quemadmodum à fimplicibus, ſed à fimplici-
                  <lb/>
                bus per ſe, & a compofitis per accidens vtiam iam videbimus.</s>
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              <p>
                <s xml:id="echoid-s1367" xml:space="preserve">ANtiquorumigitur primus m odus vtitur regula detribus, hocordine, multi-
                  <lb/>
                plicando ſcilicet ſecundum errorem, qui eft .9. cum differentia primarum par
                  <lb/>
                tium pofitarum, quæ eft .2. & productum diuidendo per differentiam errorum, quæ
                  <lb/>
                eft .12. proueniens poftea quod eft .1. cum dimidio additur hoc loco primæ parti ſe-
                  <lb/>
                cundæ poſitionis.
                  <reg norm="&c." type="unresolved">&c.</reg>
                quòd benè ſe habet. </s>
                <s xml:id="echoid-s1368" xml:space="preserve">Vbi animaduertendum eſt, quod ille
                  <lb/>
                numerus .12. non eft accipiendus per ſe vt differentia errorum hoc eft .21. et .9. nifi
                  <lb/>
                peràccidens, fed benè perfe, vt
                  <reg norm="differentia" type="context">differẽtia</reg>
                inter .60. er .48. ſimplices ſummas, quem
                  <lb/>
                admodum .9. in hoc propoſito eft differentia per ſe inter .48. et .39 per accidens ve-
                  <lb/>
                ro inter .59. et .50.</s>
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              <p>
                <s xml:id="echoid-s1369" xml:space="preserve">Cognoſcendum igitur eft mediante .24. quinti Eucli. quod eadem proportio
                  <lb/>
                eft primæ ſummæ (ſimplicium dico) ad ſuam primam partem, quæ ſecundæ ſum-
                  <lb/>
                mæ ad ſuam, & tertiæ ſummæ ad fuam fimiliter (vbi rectè etiam feciffent hoc in lo-
                  <lb/>
                co antiqui ſi multiplicauiffent tertiam ſummam fim plicem cum prima parte prioris
                  <lb/>
                fummæ fimplicis, & productum diuififfent per primam ſummam, vnde prima pars
                  <lb/>
                quæſita tertiæ ſummæ orta fuiffet, abſque ullo negotio ipfius plus velminus) </s>
                <s xml:id="echoid-s1370" xml:space="preserve">Quare
                  <lb/>
                habebimus tres terminos antecedentes ab vna parte, & tres terminos conſequen-
                  <lb/>
                tesab alia parte continentes vnam
                  <reg norm="eandemque" type="simple">eandemq́;</reg>
                proportionem, vnde ex .19. quinti,
                  <lb/>
                vel .12. ſeptimi eorum differentiæ proportionales erunt, hoc eft,
                  <reg norm="quod" type="simple">ꝙ</reg>
                eadem propor­ </s>
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