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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THEOREM. ARIT.
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              <p>
                <s xml:id="echoid-s1370" xml:space="preserve">
                  <pb o="107" rhead="THEOREM. ARIT." n="119" file="0119" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0119"/>
                tio erit eius differentiæ, quæ eſt inter primam & fecundam ſummam, ad differen-
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                tiam quæ eſt inter primas earum partes, quæ illius differentiæ, quæ eſt inter ſecun-
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                dam & tertiam ſummam, ad differentiam, quæ eft inter primas illarum partes, ſed
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                harum .4. differentiarum, tres nobis cognitæ ſunt, ideft .12. 2. et .9. ergo ex regula de
                  <lb/>
                tribus ab Eucli. in .20. ſeptì
                  <unsure/>
                mi ſpeculata inueniebatur quarta differentia, quæ eft .1.
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                cum dimidio.</s>
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              <p>
                <s xml:id="echoid-s1371" xml:space="preserve">A compofitis ſummis idem etiam proueniet, ſed non vt ex proprijs caufis, & per
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                ſe, ſedper accidens. </s>
                <s xml:id="echoid-s1372" xml:space="preserve">Nam quamuis eadem differentia fit inter 71. et .59. quæ in-
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                ter .60. et .48. &
                  <reg norm="eadem" type="context">eadẽ</reg>
                inter .59. et .50. quæ inter .48. et .39. </s>
                <s xml:id="echoid-s1373" xml:space="preserve">Nihilominus non eft
                  <reg norm="eadem" type="context">eadẽ</reg>
                  <lb/>
                proportio (propriè) ipſius .71. ad .59. quæ ipſius .60. ad .48. nec ea quæ ipſius .59. ad
                  <num value="50">.
                    <lb/>
                  50.</num>
                eft quæ ipſius .48. ad .39: </s>
                <s xml:id="echoid-s1374" xml:space="preserve">Vnde non erit eadem proportio ipſius .71. ad .59. quæ
                  <lb/>
                ipfius .10. ad .8. ne@ea quæ eft ipfius .59. ad .50. quæ ipſius .8. ad .6. cum dimidio. </s>
                <s xml:id="echoid-s1375" xml:space="preserve">Sed
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                minores illis. </s>
                <s xml:id="echoid-s1376" xml:space="preserve">Nam ex æqualibus additamentis diminuuntur proportiones maio-
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                ris inęqualitatis.</s>
              </p>
              <p>
                <s xml:id="echoid-s1377" xml:space="preserve">A fimplicibus igitur ſummis pendet ratio huiuſmodi effectus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1378" xml:space="preserve">Si vero prima pars fecundæ poſitionis effet .4. tunc ſecunda eius pars effet .8. & ter-
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                tia .12. quarum ſumma effet .24. (harum fimplicium partium ſeilicet) & minor vera
                  <lb/>
                (39.) per .15. & differens à ſumma primarum. (60.) per .36. & differentia primarum
                  <lb/>
                partium effet .6. differentia vero primæpartis ſecundæ poſitionis, a prima parte quę
                  <lb/>
                fita effet .2. cum dimidio. </s>
                <s xml:id="echoid-s1379" xml:space="preserve">Vnde in huiuſmodi exemplo videre eft quare colligan-
                  <lb/>
                tur errores inuicem, quando alter eorum eccedit, reliquus vero deficit à numero pro
                  <lb/>
                pofito. </s>
                <s xml:id="echoid-s1380" xml:space="preserve">Quod quidem ob aliam caufam non fit, nifi vt cognoſcatur differentia .36.
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                differentia ſcilicet ſimplicium ſummarum ipſarum poſitionum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1381" xml:space="preserve">Secundus autem modus ab antiquis magis exercitatus eſt, quod multiplicabant
                  <lb/>
                diametraliter errores cum primis partibus, hoc eſt primum errorem cum prima par
                  <lb/>
                te, hoc eſt cum numero ſecundæ poſitionis, ſecundum vero errorem cum prima
                  <lb/>
                parte, hoc eſt cum numero primæ poſitionis, differentiam poſteà vel aggregatum
                  <lb/>
                horum duorum productorum diuidebant per differentiam vel aggregatum dicto-
                  <lb/>
                rum errorum, proueniens poſteà erat prima pars quæſita numeri propoſiti. </s>
                <s xml:id="echoid-s1382" xml:space="preserve">Vn-
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                de oriebantur tria producta, quorum
                  <reg norm="tertium" type="context">tertiũ</reg>
                , hoc eſt differentia, ſeu aggregatum il-
                  <lb/>
                lorum conſtituebatur ex differentia feuaggregato errorum, & ex numero quæ-
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                fito.</s>
              </p>
              <p>
                <s xml:id="echoid-s1383" xml:space="preserve">Vtin præfenti exemplo, primus error eſt .21. qui multiplicatus cum prima par-
                  <lb/>
                te ſecundæ poſitionis, quæ eſt .8. producit .168.
                  <reg norm="ſecundus" type="context">ſecũdus</reg>
                verò error eſt .9. qui multi-
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                plicatus cum prima parte primę poſitionis producit .90. differentia autem horum
                  <lb/>
                productorum eſt .78. quæ diuifa per differentiam errorum, quæ eſt 12. dabit .6.
                  <reg norm="cum" type="context">cũ</reg>
                di
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                midio, pro prima parte quæſita dati numeri diuiſibilis, qui erat .50.</s>
              </p>
              <p>
                <s xml:id="echoid-s1384" xml:space="preserve">Hæc omnia rectè ſe habent. </s>
                <s xml:id="echoid-s1385" xml:space="preserve">Sed, vt ſupra dixi diuiſor non eft per ſe differentia
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                errorum, neque etiam differentia per ſe ſummarum compoſitarum, fed bene fim-
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                plicium.</s>
              </p>
              <p>
                <s xml:id="echoid-s1386" xml:space="preserve">Pro cuius rei ſpeculatione, accipiendæ ſunt ſummæ ſimplices, quarum differen-
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                tiæ per ſe vtiles ſunt in huiuſmodi operatione; </s>
                <s xml:id="echoid-s1387" xml:space="preserve">& quia etiam rationes veritatis ex
                  <lb/>
                iſtis, & non ex illis fluunt; </s>
                <s xml:id="echoid-s1388" xml:space="preserve">quamuis tam vnæ, quam aliæ ſint eædem in quantitate,
                  <lb/>
                ideſt æquales.</s>
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