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="113" rhead="THEOREM. ARIT." n="125" file="0125" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0125"/>
              <p>
                <s xml:id="echoid-s1429" xml:space="preserve">PRoponitur etiam quoddam vas, cuius pes ſit quarta pars totius vaſis cum oper
                  <lb/>
                culo, pars autem media ſine operculo, ſit quinta pars ipſius pedis, operculum
                  <lb/>
                verò .18. libras pendeat. </s>
                <s xml:id="echoid-s1430" xml:space="preserve">quæritur nunc quantitas dicti pedis.</s>
              </p>
              <p>
                <s xml:id="echoid-s1431" xml:space="preserve">Ex methodo enim antiquorum inuentus eſt pes .4. cum .14. decimisnonis ta-
                  <lb/>
                lium partium, ſeu librarum, qualium operculus eſt .18. </s>
                <s xml:id="echoid-s1432" xml:space="preserve">Videamus igitur & nos ex
                  <lb/>
                noſtra figura, quo pacto hoc reſpondeat veritati.</s>
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              <p>
                <s xml:id="echoid-s1433" xml:space="preserve">Inuenta enim ſunt tria producta,
                  <reg norm="iam" type="context">iã</reg>
                orta ex dicta methodo .10. 100. 90. quæ ſuis
                  <lb/>
                locis notaui, vt in ſigura
                  <var>.H.</var>
                ſubſcripſi etiam duas illorum poſitiones .5. et .10. cum
                  <lb/>
                ſua differentia .5. & cum productum .10. oriretur ab vno latere .10. reliquum erat
                  <num value="1">.
                    <lb/>
                  1.</num>
                quod ſuo loco notaui, ſimiliter quia .100. productum, pro vno eius laterum erat
                  <num value="5">.
                    <lb/>
                  5.</num>
                reliquum autem .20. ſuo loco poſui, & quia differentia inter .20. et .1. duo latera,
                  <lb/>
                quę eſt .19. æqualis eſt ei, quæ inter duo conſequentia duarum poſitionum, etiam
                  <lb/>
                ſuo loco ipſam conſtitui, ſed quia hæc differentia eſt vnum laterum producti .90. er
                  <lb/>
                go reliquum latus
                  <reg norm="quæſitum" type="context">quæſitũ</reg>
                erit .4. cum .14. decimisnonis, rectè igitur operatur.
                  <lb/>
                </s>
                <s xml:id="echoid-s1434" xml:space="preserve">ſed cum eadem proportio ſit inter differentiam .5. ſuperiorem, et .19. inferiorem,
                  <lb/>
                quæ eſt vnius
                  <reg norm="antecedentis" type="context">antecedẽtis</reg>
                ad ſuum
                  <lb/>
                conſequens, </s>
                <s xml:id="echoid-s1435" xml:space="preserve">quare .10.
                  <reg norm="antecedens" type="context">antecedẽs</reg>
                  <lb/>
                habebit pro ſuo conſequenti .38.
                  <lb/>
                  <figure xlink:label="fig-0125-01" xlink:href="fig-0125-01a" number="170">
                    <image file="0125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0125-01"/>
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                et .5. habebit .19. et .4. cum .14. de-
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                cimisnonis habebit .18. rectè
                  <reg norm="igitur" type="simple">igit̃</reg>
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                dictum fuiſſet ſi .19. prouenit .à .5.
                  <lb/>
                à quo proueniet .18?</s>
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              <p>
                <s xml:id="echoid-s1436" xml:space="preserve">Huiuſmodi autem rei ratio ita
                  <lb/>
                ſe
                  <reg norm="hent" type="context">hẽt</reg>
                , eſto linea
                  <var>.a.e.u.</var>
                cuius pars
                  <lb/>
                a. ſit quarta reliquarum
                  <var>.e.u.</var>
                iuncta
                  <lb/>
                rum, ſed
                  <var>.e.</var>
                ſit quinta ipſius
                  <var>.a</var>
                . </s>
                <s xml:id="echoid-s1437" xml:space="preserve">Tunc
                  <lb/>
                clarum erit quod
                  <var>.e.</var>
                erit vigeſima
                  <lb/>
                dictarum
                  <var>.e.u.</var>
                </s>
                <s xml:id="echoid-s1438" xml:space="preserve">quare erit decima-
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                nona ipſius
                  <var>.u.</var>
                ſed
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                u.
                  <reg norm="sumpta" type="context">sũpta</reg>
                ſit vt
                  <num value="18">.
                    <lb/>
                  18.</num>
                rectè igitur dici poteſt, ſi .u: ut
                  <num value="19">.
                    <lb/>
                  19.</num>
                prouenit ab
                  <var>.a.</var>
                ut quinque, à
                  <lb/>
                quot ipſius
                  <var>.a.</var>
                proueniet
                  <var>.u.</var>
                ut .18.</s>
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              <p>
                <s xml:id="echoid-s1439" xml:space="preserve">Quis enim non uidet quod diui
                  <lb/>
                ſa cum fuerit
                  <var>.u.</var>
                in partes .19. quod
                  <lb/>
                quinque illarum æquabuntur ipſi
                  <var>.
                    <lb/>
                  a.</var>
                cum quælibet fuerit æqualis
                  <var>.e.</var>
                  <lb/>
                quintæ parti ipſius
                  <var>.a</var>
                .</s>
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              <p>
                <s xml:id="echoid-s1440" xml:space="preserve">HAc igitur mea numerorum diſpoſitione mediante reperiuntur ipſi numeri in
                  <lb/>
                feriores naturaliter conſequentes, correſpondentesq́ue ipſis ſuperioribus an
                  <lb/>
                tecedentibus; </s>
                <s xml:id="echoid-s1441" xml:space="preserve">quamuis multoties
                  <reg norm="contingere" type="context">cõtingere</reg>
                poſſit, ut generationes ſeu com-
                  <lb/>
                poſitiones ipſorum ignorentur: </s>
                <s xml:id="echoid-s1442" xml:space="preserve">& quia tam à differentijs errorum, quam ab illis,
                  <lb/>
                quę ſunt inter ueros conſequentes numeros ( propter eorum æqualitatem ) elicitur
                  <lb/>
                ipſa ueritas, proptereà rectè antiqui illis vſi ſunt, quamuis ſint potius ſenſum
                  <lb/>
                ſequuti, uel experientiam, quam rationem: </s>
                <s xml:id="echoid-s1443" xml:space="preserve">quæ quidem ratio pendet ab ipſis na-
                  <lb/>
                turalibus numeris conſequentibus ( ut ſupra uidimus ) etſi incognitis ut plurimum,
                  <lb/>
                quod ſi ipſos inuenire primò nobis datum fuiſſet, unica
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                poſitio ſuffice- </s>
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