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
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
81 71
82 70
83 71
84 72
85 73
86 74
87 75
88 76
89 77
90 78
< >
page |< < (56) 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-div167" type="math:theorem" level="3" n="85">
              <p>
                <s xml:id="echoid-s740" xml:space="preserve">
                  <pb o="56" rhead="IO. BAPT. BENED." n="68" file="0068" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0068"/>
                poſitorum .3. et .2. multiplicato .3. per .10.
                  <reg norm="ſummam" type="context">ſummã</reg>
                .6. cum .4. dantur .30. quod pro-
                  <lb/>
                ductum æquale erit producto .2. per .15. nempe per ſummam 9. et .6.</s>
              </p>
              <p>
                <s xml:id="echoid-s741" xml:space="preserve">Quod vt cognoſcamus, tres quan
                  <lb/>
                  <figure xlink:label="fig-0068-01" xlink:href="fig-0068-01a" number="94">
                    <image file="0068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0068-01"/>
                  </figure>
                titates continuæ proportionales ſint
                  <lb/>
                  <var>b.a.p.</var>
                proportione
                  <var>.d.q.</var>
                productum
                  <lb/>
                autem
                  <var>.d.</var>
                in ſummam
                  <var>.a.</var>
                cum
                  <var>.p.</var>
                ſit
                  <var>.f.t.</var>
                  <lb/>
                & productum
                  <var>.q.</var>
                in ſummam
                  <var>.b.a.</var>
                ſit
                  <var>.
                    <lb/>
                  K.h.</var>
                et
                  <var>.K.n.</var>
                ſit æqualis
                  <var>.b.</var>
                et
                  <var>.n.o.</var>
                æqua
                  <lb/>
                lis
                  <var>.a.</var>
                & ita etiam
                  <var>.o.u.</var>
                eidem
                  <var>.a.</var>
                et
                  <var>.u.t.</var>
                  <lb/>
                æqualis
                  <var>.p.</var>
                et
                  <var>.h.o.</var>
                ipſi
                  <var>.q.</var>
                et
                  <var>.f.o.</var>
                ipſi
                  <var>.d</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s742" xml:space="preserve">quare ita ſe habebit
                  <var>.K.n.</var>
                ad
                  <var>.n.o.</var>
                ſicut
                  <lb/>
                  <var>o.u.</var>
                ad
                  <var>.u.t.</var>
                & componendo
                  <var>.K.o.</var>
                ad
                  <var>.
                    <lb/>
                  n.o.</var>
                vt
                  <var>.o.t.</var>
                ad
                  <var>.u.t.</var>
                & permutando
                  <var>.K.
                    <lb/>
                  o.</var>
                ad
                  <var>.o.t.</var>
                vt
                  <var>.n.o.</var>
                hoc eſt
                  <var>.o.u.</var>
                ad
                  <var>.u.t.</var>
                &
                  <lb/>
                pariter
                  <var>.f.o.</var>
                ad
                  <var>.o.h.</var>
                vt
                  <var>.o.u.</var>
                ad
                  <var>.u.t</var>
                . </s>
                <s xml:id="echoid-s743" xml:space="preserve">Ita-
                  <lb/>
                que ſicut
                  <var>.k.o.</var>
                ad
                  <var>.o.t.</var>
                ex quo ex .15. ſexti aut .20. ſeptimi
                  <var>.K.h.</var>
                æqualis erit
                  <var>.f.t</var>
                .</s>
              </p>
            </div>
            <div xml:id="echoid-div169" type="math:theorem" level="3" n="86">
              <head xml:id="echoid-head103" xml:space="preserve">THEOREMA
                <num value="86">LXXXVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s744" xml:space="preserve">CVR multiplicatis ſingulis tribus quantitatibus continuis proportionalibus in
                  <lb/>
                reliquas duas, ſex producta æqualia ſint producto dupli ſummæ ipſarum trium
                  <lb/>
                in mediam proportionalem.</s>
              </p>
              <p>
                <s xml:id="echoid-s745" xml:space="preserve">Exempli gratia, proponuntur hitres termini continui proportionales .9. 6. 4. pro
                  <lb/>
                ductum .9. in .6. erit .54. at .9. in .4. erit .36. et .6. in .9: 54. et .6. in .4: 24. et .4. in .9: 36. et
                  <num value="4">.
                    <lb/>
                  4.</num>
                in .6: 24. quæ producta ſimul collecta efficiunt numerum .228 ſed
                  <reg norm="tantum" type="context">tantũ</reg>
                eſt pro-
                  <lb/>
                ductum dupli ſummæ trium terminorum in ſecundum nempe .38 in .6.</s>
              </p>
              <p>
                <s xml:id="echoid-s746" xml:space="preserve">Cuius
                  <reg norm="intelligentiæ" type="context">intelligẽtiæ</reg>
                cauſa, tres termini
                  <reg norm="continui" type="context">cõtinui</reg>
                proportionales ſignificentur linea
                  <var>.
                    <lb/>
                  b.e.</var>
                nempe
                  <var>.b.d</var>
                :
                  <var>d.c</var>
                :
                  <var>c.e.</var>
                cuius duplum ſit
                  <var>.u.e.</var>
                et
                  <var>.b.f.</var>
                æqualis ſit
                  <var>.b.d.</var>
                et
                  <var>.f.n</var>
                :
                  <var>d.c.</var>
                et
                  <var>.n.u</var>
                :
                  <lb/>
                c. e productum verò
                  <var>.u.e.</var>
                in
                  <var>.d.c.</var>
                ſit
                  <var>.u.s.</var>
                cui dico æqualem eſſe ſummam productorum
                  <lb/>
                ſingulorum trium terminorum in reliquos duos. </s>
                <s xml:id="echoid-s747" xml:space="preserve">Quamobrem ducantur perpendi-
                  <lb/>
                culares
                  <var>.c.g</var>
                :
                  <var>d.o</var>
                :
                  <var>b.i</var>
                :
                  <var>f.a.</var>
                et
                  <var>.n.p.</var>
                inter
                  <var>.u.e.</var>
                et
                  <var>.q.s.</var>
                ex quo pro producto
                  <var>.c.e.</var>
                in
                  <var>.c.d.</var>
                ha-
                  <lb/>
                bebimus rectangulum
                  <var>.c.s.</var>
                & rectan-
                  <lb/>
                  <figure xlink:label="fig-0068-02" xlink:href="fig-0068-02a" number="95">
                    <image file="0068-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0068-02"/>
                  </figure>
                gulum
                  <var>.d.g.</var>
                pro producto
                  <var>.c.e.</var>
                in
                  <var>.d.b.</var>
                  <lb/>
                ex .16. ſexti aut .20. ſeptimi itemq́ue
                  <lb/>
                rectangulum
                  <var>.q.n.</var>
                pro producto
                  <var>.d.c.</var>
                  <lb/>
                in
                  <var>.c.e.</var>
                & rectangulum
                  <var>.b.o.</var>
                ex
                  <var>.d.c.</var>
                in
                  <var>.
                    <lb/>
                  b.d.</var>
                & rectangulum
                  <var>.b.a.</var>
                ex
                  <var>.b.d.</var>
                in
                  <var>.d.
                    <lb/>
                  c.</var>
                et
                  <var>.p.f.</var>
                ex
                  <var>.d.b.</var>
                in
                  <var>.c.e.</var>
                ex .16. aut .20.
                  <lb/>
                prędictas. </s>
                <s xml:id="echoid-s748" xml:space="preserve">Quare ſex producta æquantur inter ſe,
                  <reg norm="replentque" type="simple">replentq́</reg>
                productum
                  <var>.u.s.</var>
                ex quo
                  <lb/>
                verum eſt propoſitum.</s>
              </p>
            </div>
            <div xml:id="echoid-div171" type="math:theorem" level="3" n="87">
              <head xml:id="echoid-head104" xml:space="preserve">THEOREMA
                <num value="87">LXXXVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s749" xml:space="preserve">QVA ratione cognoſci poſſit
                  <reg norm="verum" type="context">verũ</reg>
                eſſe proportionem ſummæ quatuor quan-
                  <lb/>
                titatum continuarum proportionalium ad ſummam ſecundæ & tertiæ, ean-
                  <lb/>
                dem eſſe, quæ ſummæ primæ & tertiæ ad ſecundam ſimplicem.</s>
              </p>
              <p>
                <s xml:id="echoid-s750" xml:space="preserve">Exempli gratia, ſi inue nirentur hæ quatuor quantitates continuæ proportiona-
                  <lb/>
                es .16. 8. 4. 2. earum ſumma erit .30. ſunima verò ſecundæ & tertiæ .12. tum ſumma </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum