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-div274" type="math:theorem" level="3" n="143">
              <p>
                <s xml:id="echoid-s1262" xml:space="preserve">
                  <pb o="97" rhead="THEOREM. ARIT." n="109" file="0109" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0109"/>
                nem
                  <var>.c.d.</var>
                ad
                  <var>.d.e.</var>
                ſi
                  <var>.c.d.</var>
                accipiemus, vt medium inter
                  <var>.a.d.</var>
                et
                  <var>.d.e.</var>
                cognoſcemus etiam
                  <lb/>
                proportionem
                  <var>.a.d.</var>
                ad
                  <var>.d.e.</var>
                </s>
                <s xml:id="echoid-s1263" xml:space="preserve">quare etiam eam quæ
                  <var>.a.e.</var>
                ad
                  <var>.d.e.</var>
                collocando poſteà.
                  <lb/>
                  <var>d.e.</var>
                inter
                  <var>.e.f.</var>
                et
                  <var>.a.e.</var>
                innoteſcet ea, quæ eſt
                  <var>.a.e.</var>
                ad
                  <var>.e.f.</var>
                & ita gradatim accedenrus ad
                  <lb/>
                perfectam cognitionem proportionis totius
                  <var>.a.l.</var>
                ad
                  <var>.k.l</var>
                . </s>
                <s xml:id="echoid-s1264" xml:space="preserve">Nunc autem mediante
                  <var>.k.l.</var>
                  <lb/>
                cognoſcemus proportionem totius
                  <var>.a.l.</var>
                ad
                  <var>.i.k.</var>
                & hac mediante, cam cognoſcemus,
                  <lb/>
                quæ totius
                  <var>.a.l.</var>
                ad
                  <var>.g.h.</var>
                & hac mediante eam quæ totius
                  <var>.a.l.</var>
                ad
                  <var>.f.g.</var>
                & ſic gradatim, co
                  <lb/>
                gnita nobis erit proportio totius
                  <lb/>
                lineæ
                  <var>.a.l.</var>
                ad ſuam partem
                  <var>.a.c.</var>
                be-
                  <lb/>
                  <figure xlink:label="fig-0109-01" xlink:href="fig-0109-01a" number="150">
                    <image file="0109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0109-01"/>
                  </figure>
                neficio poſteà totius lineæ
                  <var>.a.l.</var>
                co
                  <lb/>
                gnoſcemus proportionem
                  <var>a.c.</var>
                ad
                  <lb/>
                  <var>a.b.</var>
                & ſic aliarum reſpectu lineæ
                  <var>.a.b.</var>
                vt quærebatur, quæ quidem propoſitio, etſi car
                  <lb/>
                danica uocetur leuiſſima tamen eſt.</s>
              </p>
            </div>
            <div xml:id="echoid-div276" type="math:theorem" level="3" n="144">
              <head xml:id="echoid-head163" xml:space="preserve">THEOREMA
                <num value="144">CXLIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1265" xml:space="preserve">QVamuis multi de modo in ſumma colligendi, ſubtrahendi,
                  <reg norm="multiplicandi" type="context">multiplicãdi</reg>
                , & di
                  <lb/>
                uidendi proportiones ſcripſerint, nullus tamen (quod ſciam) perfectè, ac
                  <lb/>
                ſcientificè ſpeculatus eſt has operationes, quapropter hanc rem cum ſilentio tranſi
                  <lb/>
                re nolui, quin aliquid de ipſa conſcribam à ſumma dictarum proportionum in-
                  <lb/>
                cohando.</s>
              </p>
              <p>
                <s xml:id="echoid-s1266" xml:space="preserve">Quotieſcunque igitur volunt duas proportiones inuicem aggregare, ſimul ea-
                  <lb/>
                rum antecedentia multiplicant, & ſimiliter earum conſequentia. </s>
                <s xml:id="echoid-s1267" xml:space="preserve">Tunc proportio
                  <lb/>
                terminata ab illis productis euadit in ſummam illarum duarum propoſitarum
                  <lb/>
                proportionum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1268" xml:space="preserve">Vt exempli gratia, ſi voluerimus colligere proportionem ſeſquialteram cum ſeſ-
                  <lb/>
                quitertia, multiplicando .3. cum .4. antecedentia ſcilicet, pro ductum erit .12. poſteà
                  <lb/>
                multiplicando .2. cum .3. conſequentia, tunc productum erit .6. </s>
                <s xml:id="echoid-s1269" xml:space="preserve">Proportio igitur,
                  <lb/>
                quæ inter .12. et .6. reperitur. (quæ dupla eſt) eſt ſumma propoſitarum
                  <reg norm="proportionum" type="context">proportionũ</reg>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s1270" xml:space="preserve">Cuius rei ſpeculatio erit huiuſmodi ſint
                  <var>.x.</var>
                et
                  <var>.u.</var>
                  <lb/>
                duo antecedentia quarunruis proportionum
                  <var>.t.</var>
                  <lb/>
                  <figure xlink:label="fig-0109-02" xlink:href="fig-0109-02a" number="151">
                    <image file="0109-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0109-02"/>
                  </figure>
                verò et. n ſint eorum conſequentia, productum
                  <lb/>
                autem antecedentium ſit
                  <var>.a.g.</var>
                illud verò quod
                  <reg norm="con" type="context">cõ</reg>
                  <lb/>
                ſequentium ſit
                  <var>.d.a.</var>
                vnde proportio
                  <var>.a.g.</var>
                ad
                  <var>.a.d.</var>
                  <lb/>
                compoſita erit ex proportione
                  <var>.x.</var>
                ad
                  <var>.t.</var>
                & ex ea,
                  <lb/>
                quæ eſt
                  <var>.u.</var>
                ad
                  <var>.n.</var>
                per .24. ſexti vel quintam octaui.
                  <lb/>
                </s>
                <s xml:id="echoid-s1271" xml:space="preserve">Patet igitur ratio rectè faciendi, vt ſuprà dictum
                  <lb/>
                eſt.</s>
              </p>
            </div>
            <div xml:id="echoid-div278" type="math:theorem" level="3" n="145">
              <head xml:id="echoid-head164" xml:space="preserve">THEOREMA
                <num value="145">CXLV</num>
              .</head>
              <p>
                <s xml:id="echoid-s1272" xml:space="preserve">QVotieſcunque deinde detrahere volunt vnam proportionem ex altera mul-
                  <lb/>
                tiplicant antecedens vnius cum conſequenti alterius. </s>
                <s xml:id="echoid-s1273" xml:space="preserve">Tunc proportio, quę
                  <lb/>
                inter talia duo producta incluſa reperitur, eſt reſiduum, ſeu differentia illarum dua-
                  <lb/>
                rum proportionum datarum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1274" xml:space="preserve">Vt exempli gratia, ſi aliquis vellet ex proportione dupla detrahere ſeſquialte-
                  <lb/>
                ram, multiplicaret .2. antecedens duplæ cum .2. conſequenti ſeſquialteræ, quorum
                  <lb/>
                productum eſſet .4. pro antecedenti reſiduę proportionis. </s>
                <s xml:id="echoid-s1275" xml:space="preserve">Deinde multiplicaret .3
                  <lb/>
                antecedens ſeſquialteræ cum .1. conſequenti duplæ, & productum eſſet .3. pro
                  <reg norm="con- ſequenti" type="context">cõ- </reg>
                </s>
              </p>
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