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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            <div xml:id="echoid-div223" type="math:theorem" level="3" n="117">
              <p>
                <s xml:id="echoid-s1058" xml:space="preserve">
                  <pb o="80" rhead="IO. BAPT. BENED." n="92" file="0092" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0092"/>
                coniunctis denotetur, et
                  <var>.b.e.</var>
                ſit tertia pars ipſius
                  <var>.a.b.</var>
                prioris numeri im aginati, et. b
                  <lb/>
                c. tertia pars ipſius,
                  <var>b.d.</var>
                ſecundi numeri propoſiti, vnde coniunctum vnius harum ter
                  <lb/>
                tiarum
                  <reg norm="partium" type="context">partiũ</reg>
                  <reg norm="cum" type="context">cũ</reg>
                alia ſit
                  <var>.e.c.</var>
                quod quidem
                  <var>.e.c.</var>
                eſſe tertiam partem ſummæ duorum
                  <lb/>
                primorum ideſt
                  <var>.a.d.</var>
                aſſero. </s>
                <s xml:id="echoid-s1059" xml:space="preserve">Iam manifeſtum eſt ipſius
                  <var>.d.b.</var>
                ad
                  <var>.b.c.</var>
                eſſe quemadmo
                  <lb/>
                dum ipſius
                  <var>.a.b.</var>
                ad
                  <var>.b.e.</var>
                vnde viciſſim ipſius
                  <var>.d.b.</var>
                ad
                  <var>.b.a.</var>
                erit quemadmodum ipſius
                  <var>.b.
                    <lb/>
                  c.</var>
                ad
                  <var>.b.e.</var>
                & coniunctim ipſius
                  <var>.d.a.</var>
                ad
                  <var>.a.b.</var>
                quemadmodum ipſius
                  <var>.c.e.</var>
                ad
                  <var>.e.b.</var>
                & viciſ-
                  <lb/>
                ſim ipſius
                  <var>.d.a.</var>
                ad
                  <var>.c.e.</var>
                quemadmodum ipſius
                  <var>.b.a.</var>
                ad
                  <var>.b.e.</var>
                ſed proportio ipſius
                  <var>.b.a.</var>
                ad
                  <var>.
                    <lb/>
                  b.e.</var>
                eſt tripla, ergo ea quæ eſt ipſius
                  <var>.a.d.</var>
                ad
                  <var>.e.c.</var>
                erit quoque tripla; </s>
                <s xml:id="echoid-s1060" xml:space="preserve">vnde ſumendo
                  <var>.e.
                    <lb/>
                  c.</var>
                pro tertia parte ipſius
                  <var>.a.d.</var>
                & ab ipſa
                  <var>.e.c.</var>
                ſubtrahendo tertiam partem ipſius
                  <var>.a.b.</var>
                  <lb/>
                tertia pars ipſius
                  <var>.b.d.</var>
                remanebit
                  <var>.b.c</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s1061" xml:space="preserve">Aut alio hoc modo, ſupponendo
                  <var>.e.c.</var>
                tertiam partem ipſius
                  <var>.a.d.</var>
                et
                  <var>.e.b.</var>
                ipſius
                  <var>.a.
                    <lb/>
                  b.</var>
                exiſter. </s>
                <s xml:id="echoid-s1062" xml:space="preserve">Dico
                  <var>.b.c.</var>
                tertiam partem ipſius
                  <var>.b.d.</var>
                futuram:</s>
                <s xml:id="echoid-s1063" xml:space="preserve">quia ſi totius
                  <var>.a.d.</var>
                ad totum
                  <lb/>
                  <var>e.c.</var>
                ita ſe habet, quemadmodum
                  <var>.a.b.</var>
                à toto
                  <var>.a.d.</var>
                diffecti atque diuulſi ad
                  <var>.e.b.</var>
                à toto
                  <var>.
                    <lb/>
                  e.c.</var>
                diſractum, ergo ex
                  <ref id="ref-0011">.19. lib. quinti Eu-
                    <lb/>
                    <figure xlink:label="fig-0092-01" xlink:href="fig-0092-01a" number="125">
                      <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0092-01"/>
                    </figure>
                  clid.</ref>
                reſidui
                  <var>.b.d.</var>
                totius
                  <var>.a.d.</var>
                ad reſiduum
                  <var>.b.c.</var>
                  <lb/>
                totius
                  <var>.e.c.</var>
                erit, vt totius
                  <var>.a.d.</var>
                ad
                  <reg norm="totum" type="context">totũ</reg>
                  <var>.e.c.</var>
                at-
                  <lb/>
                que hic quidem modus rem
                  <reg norm="propoſitam" type="context">propoſitã</reg>
                ſpe-
                  <lb/>
                culandi mihi aptior & commodior eſſe videtur.</s>
              </p>
            </div>
            <div xml:id="echoid-div226" type="math:theorem" level="3" n="118">
              <head xml:id="echoid-head136" xml:space="preserve">THEOREMA
                <num value="118">CXVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1064" xml:space="preserve">PErmulta ac varia problemata inuenerunt antiqui, longioribus verò vijs reſolu-
                  <lb/>
                ta, proptereà quòd
                  <reg norm="non" type="context">nõ</reg>
                ſemper nobis ſuccurrit breuiſſima in vnaquaque re ex-
                  <lb/>
                plicatio. </s>
                <s xml:id="echoid-s1065" xml:space="preserve">Vt exempli gratia, proponitur numerus .50. diuidendus in tres tales par-
                  <lb/>
                tes, quod ſecunda dupla ſit primę, & adhuc eam ſuperet tribus vnitatibus, tertia ve
                  <lb/>
                rò æqualis ſit aggregato primæ cum ſecunda, & amplius ipſum aggregatum ſuperet
                  <lb/>
                quinque vnitatibus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1066" xml:space="preserve">Ad hoc autem quæſitum ſoluendum antiqui vtebantur regula falſi, quod reuera
                  <lb/>
                breuiori modo poteſt ſolui, videlicet detra hendo illud ſecundum exceſſum, quin-
                  <lb/>
                que ſcilicet ex .50. ita vt nobis .45. remaneret, cui medietati hoc eſt .22. cum dimidia
                  <lb/>
                vnitate, ſi addiderimus illud quinque habebimus .27. cum dimidia vnitate pro ter-
                  <lb/>
                tia parte quæſita ipſius numeri .50. deinde ſi ab eodem numero .22. cum dimidia
                  <lb/>
                vnitate detractum fuerit illud .3. primus exceſſus datus, remanebit .19. cum dimi-
                  <lb/>
                dia vnitate, cuius tertia pars, hoc eſt .6. cum dimidia vnitate, prima pars, ex tri-
                  <lb/>
                bus quæſita erit, quæ quidem ſi detraxerimus ex .19. cum dimidia vnitate, reli-
                  <lb/>
                quum erit .13. cui
                  <reg norm="cum" type="context">cũ</reg>
                additus fuerit primus exceſſus ideſt .3. </s>
                <s xml:id="echoid-s1067" xml:space="preserve">Iam propoſitum re-
                  <lb/>
                ſultabit nobis .16. pro ſecunda parte quæſita.</s>
              </p>
              <p>
                <s xml:id="echoid-s1068" xml:space="preserve">Ratio verò huiuſmodi operationis talis eſt, ſit verbi gratia totalis numerus pro-
                  <lb/>
                poſitus ſignificatus per lineam
                  <var>.a.b.</var>
                cuius ſecundæ partis numerus datus ſignificetur
                  <lb/>
                per lineam
                  <var>.g.</var>
                & numerus tertiæ partis propoſitus per lineam
                  <var>.h</var>
                . </s>
                <s xml:id="echoid-s1069" xml:space="preserve">Nunc dempta
                  <var>.h.</var>
                ex
                  <lb/>
                  <var>a.b.</var>
                nobis cognita, remanebit
                  <var>.f.a.</var>
                qua
                  <reg norm="quidem" type="context">quidẽ</reg>
                per æqualia imaginatione diuiſa in pun
                  <lb/>
                cto
                  <var>.e.</var>
                & ipſi
                  <var>.e.f.</var>
                addita
                  <var>.f.b.</var>
                tota
                  <var>.e.b.</var>
                nobis cognita erit, quæ quidem tertia pars
                  <lb/>
                quæſita ipſius
                  <var>.a.b.</var>
                erit, proptereà quòd
                  <var>.a.e.</var>
                (quæ æqualis eſt ipſi
                  <var>.e.f.</var>
                ) erit ſumma
                  <lb/>
                primæ, & ſecundæ partis. </s>
                <s xml:id="echoid-s1070" xml:space="preserve">Detrahatur poſteà. g. ex
                  <var>.e.a.</var>
                & remanebit
                  <var>.d.a.</var>
                cuius ter
                  <lb/>
                tia pars ſit
                  <var>.a.c.</var>
                quæ quidem prima pars quæſita erit, & nunc cognita, & ita
                  <var>.c.d.</var>
                  <lb/>
                cognita, cui cum addita fuerit
                  <var>.d.e.</var>
                habebimus ſecundam partem quæſitam, quæ
                  <lb/>
                compo- </s>
              </p>
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