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-div194" type="math:theorem" level="3" n="103">
              <p>
                <s xml:id="echoid-s885" xml:space="preserve">
                  <pb o="66" rhead="IO. BAPT. BENED." n="78" file="0078" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0078"/>
                affirmabant. </s>
                <s xml:id="echoid-s886" xml:space="preserve">Quæ ſanè regula, non ſemper, etſi interdum vera ſit.</s>
              </p>
              <p>
                <s xml:id="echoid-s887" xml:space="preserve">Sumebant hi exemplum progreſſionis, quæ ab vnitate incohata creſcit per bina
                  <lb/>
                rium, in qua per accidens euenit vt numerus dimidium vltimi termini proximè ſe-
                  <lb/>
                quens, nempe è duabus partibus vltimi termini maior, æqualis ſit numero termino
                  <lb/>
                rum, qui per ſe vnus è producentibus, ex ijs que .94. theoremate diximus, eſſe debet;
                  <lb/>
                </s>
                <s xml:id="echoid-s888" xml:space="preserve">alter vero producens, qui per ſe dimidium ſummæ primi & vltimi eſſe debet, per
                  <lb/>
                accidens pars maior eſt duarum vltimi termini, & alteri producenti æqualis.</s>
              </p>
              <p>
                <s xml:id="echoid-s889" xml:space="preserve">Aut alio modo ratiocinemur, dicentes, in huiuſmodi progreſſione dimidium
                  <lb/>
                ſummæ vltimi termini cum primo, ſemper medium proportionale eſt inter eam
                  <lb/>
                ſummam & dimidium numeri terminorum, etenim huiuſmodi ſumma numero ter-
                  <lb/>
                minorum ſemper dupla eſt, prout .94. theoremate tradimus. </s>
                <s xml:id="echoid-s890" xml:space="preserve">Itaque ex .20. ſeptimi,
                  <lb/>
                quadratum partis maioris, producto ſummæ dictæ in numerum dimidij
                  <reg norm="terminorum" type="context">terminorũ</reg>
                  <lb/>
                æquale erit, quod productum per ſe ſummæ progreſſionis eſt æquale. </s>
                <s xml:id="echoid-s891" xml:space="preserve">At in cæte-
                  <lb/>
                ris eiuſmodi progreſſionibus fallit regula, vt ex ſupradictis facilè demonſtratur.</s>
              </p>
            </div>
            <div xml:id="echoid-div195" type="math:theorem" level="3" n="104">
              <head xml:id="echoid-head121" xml:space="preserve">THEOREMA
                <num value="104">CIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s892" xml:space="preserve">PErmultis terminis ad libitum propoſitis, diſpoſitis nihilominus progreſſio-
                  <lb/>
                ne, aut proportionalitate geometrica continua, ſi minimus ex maximo & exfe-
                  <lb/>
                quenti minimum detrahatur, reſiduum maximi, eam proportionem ad fum-
                  <lb/>
                mam reliquorum omnium terminorum retinebit, quam reſiduum ſecundi ad pri-
                  <lb/>
                mum.</s>
              </p>
              <p>
                <s xml:id="echoid-s893" xml:space="preserve">Proponuntur, exempli gratia, quatuor termini .3. 12. 48. 192. continui geome-
                  <lb/>
                tricè proportionales, ſi primum, hoc eſt minimum, ex ſecundo, & maximo detra
                  <lb/>
                has, exſecundo ſupererit .9. ex maximo .189. quod ſi minimum per reſiduum maxi
                  <lb/>
                mi multiplicaueris, hoc eſt .189. orietur .567. tum ſi huiuſmodi productum per .9.
                  <lb/>
                ( refiduum ſecundi ) diuiſeris, proueniet .63. quod proueniens æquale erit ſummæ
                  <lb/>
                reliquorum omnium terminorum, maximo excepto. </s>
                <s xml:id="echoid-s894" xml:space="preserve">Ex quo inferre licet ex .20. ſe
                  <lb/>
                ptimi eandem proportionem eſſe .189. ad .63. quæ .9. ad .3. aut ſi reſiduum ſecundi
                  <lb/>
                per ſummam dictorum terminorum multiplicaueris produceturidem .567. </s>
                <s xml:id="echoid-s895" xml:space="preserve">quare
                  <lb/>
                ex .20. ſeptimi & cætera.</s>
              </p>
              <p>
                <s xml:id="echoid-s896" xml:space="preserve">Quod vt
                  <reg norm="ſcientificè" type="context">ſciẽtificè</reg>
                poſſimus, & in vniuerſum ſpeculari. </s>
                <s xml:id="echoid-s897" xml:space="preserve">Quatuor termini propo-
                  <lb/>
                ſiti, quatuor ſubſcriptis lineis
                  <reg norm="ſignificentur" type="context">ſignificẽtur</reg>
                  <var>.b.i</var>
                :
                  <var>c.a</var>
                :
                  <var>f.r</var>
                :
                  <var>m.s.</var>
                (quod
                  <reg norm="autem" type="wordlist">aũt</reg>
                de his quatuor di
                  <lb/>
                co de
                  <reg norm="centummillibus" type="context">centũmillibus</reg>
                , & eo amplius dicere poſſum.) </s>
                <s xml:id="echoid-s898" xml:space="preserve">Nunc minimus terminus
                  <var>.m.s.</var>
                ex
                  <lb/>
                maximo
                  <var>.b.i.</var>
                detrahatur,
                  <reg norm="ſuperſitque" type="simple">ſuperſitq́;</reg>
                  <var>.n.i.</var>
                  <reg norm="idemque" type="simple">idemq́;</reg>
                  <var>.m.s.</var>
                ex ſecundo termino
                  <var>.f.r.</var>
                ſubtra-
                  <lb/>
                hatur,
                  <reg norm="ſuperſitque" type="simple">ſuperſitq́;</reg>
                  <var>.o.r</var>
                . </s>
                <s xml:id="echoid-s899" xml:space="preserve">Dico proportionem
                  <var>.n.i.</var>
                ad ſummam reliquorum omnium ter-
                  <lb/>
                minorum
                  <var>.c.a</var>
                :
                  <var>f.r</var>
                :
                  <var>m.s.</var>
                eandem effe, quæ
                  <var>.o.r.</var>
                ad
                  <var>.m.s</var>
                . </s>
                <s xml:id="echoid-s900" xml:space="preserve">Quamobrem ex tertio & quar-
                  <lb/>
                to ſecundus
                  <var>.f.r.</var>
                  <reg norm="detrahatur" type="simple">detrahat̃</reg>
                , ex
                  <reg norm="tertioque" type="simple">tertioq́;</reg>
                ſuperſit
                  <var>.t.a.</var>
                & ex quarto
                  <var>.e.i.</var>
                ita etiam tertius
                  <var>.
                    <lb/>
                  c.a.</var>
                ex quarto
                  <var>.b.i.</var>
                  <reg norm="ſuperſitque" type="simple">ſuperſitq́;</reg>
                  <var>.d.i.</var>
                ſanè
                  <lb/>
                  <figure xlink:label="fig-0078-01" xlink:href="fig-0078-01a" number="106">
                    <image file="0078-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0078-01"/>
                  </figure>
                ſic ſe habebit
                  <var>.c.a.</var>
                ad
                  <var>.f.r.</var>
                vt
                  <var>.c.t.</var>
                ad
                  <var>.f.o.</var>
                  <lb/>
                vt
                  <reg norm="quisque" type="simple">quisq;</reg>
                per ſe ſcire poteſt. </s>
                <s xml:id="echoid-s901" xml:space="preserve">Quare ex
                  <lb/>
                19. quinti ſic ſe habebit
                  <var>.a.t.</var>
                ad
                  <var>.r.o.</var>
                vt
                  <var>.
                    <lb/>
                  c.a.</var>
                ad
                  <var>.f.r.</var>
                & permutando ita
                  <var>.a.t.</var>
                ad
                  <var>.a.
                    <lb/>
                  c.</var>
                vt
                  <var>.o.r.</var>
                ad
                  <var>.r.f.</var>
                & ſeparando ſic
                  <var>.a.t.</var>
                ad
                  <var>.
                    <lb/>
                  a.c.</var>
                (hoc eſt
                  <var>.f.r.</var>
                ) vt
                  <var>.r.o.</var>
                ad
                  <var>.o.f.</var>
                vide-
                  <lb/>
                licet
                  <var>.m.s</var>
                . </s>
                <s xml:id="echoid-s902" xml:space="preserve">
                  <reg norm="Idem" type="context">Idẽ</reg>
                dico de
                  <var>.d.i.</var>
                ad
                  <var>.a.c.</var>
                nem-
                  <lb/>
                pe ſic ſe habebit
                  <var>.d.i.</var>
                ad
                  <var>.a.c.</var>
                vt
                  <var>.a.t.</var>
                ad
                  <var>. </var>
                </s>
              </p>
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