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]

Page concordance

< >
Scan Original
51 39
52 40
53 41
54 42
55 43
56 44
57 45
58 46
59 47
60 48
61 49
62 50
63 51
64 52
65 53
66 54
67 55
68 56
69 57
70 58
71 59
72 60
73 61
74 62
75 63
76 64
77 65
78 66
79 67
80 70
< >
page |< < (66) of 445 > >|
    <echo version="1.0">
      <text type="book" xml:lang="la">
        <div xml:id="echoid-div7" type="body" level="1" n="1">
          <div xml:id="echoid-div7" type="chapter" level="2" n="1">
            <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>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum