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
41 29
42 30
43 31
44 32
45 33
46 34
47 35
48 36
49 37
50 38
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
< >
page |< < (33) of 445 > >|
THEOR. ARITH.
    <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-div106" type="math:theorem" level="3" n="50">
              <p>
                <s xml:id="echoid-s444" xml:space="preserve">
                  <pb o="33" rhead="THEOR. ARITH." n="45" file="0045" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0045"/>
                numero, verbi gratia .92. præcepit regula detrahi primum numerum ex ſecundo,
                  <lb/>
                nempe .20. ex .92. cuius reſiduum, ſcilicet .72. conſeruetur, tum detrahi iubet bi
                  <lb/>
                narium ex primo, ſic in propoſito exemplo remanebunt .18. huius autem .18. dimi
                  <lb/>
                dium in ſeipſum multiplicari iubet, quod cum ſit .9. datur numerus .81. ex quo .81.
                  <lb/>
                primum numerum conſeruatum, nempe .72. vult regula detrahi, ſic remanebit .9.
                  <lb/>
                tum huius .9. quadrata radix detrahenda eſt ex dimidio ipſius .18. quod fuit ante qua
                  <lb/>
                dratum, ſic ſupererit .6. hoc eſt .9. excepta radice quadrata, qui .6. erit minor pars
                  <lb/>
                quæſita, maior verò .14. quarum productum .84. coniunctum cum partium differen
                  <lb/>
                tia præbet exactè .92.</s>
              </p>
              <p>
                <s xml:id="echoid-s445" xml:space="preserve">Cuius rei hæc eſt ſpeculatio. </s>
                <s xml:id="echoid-s446" xml:space="preserve">Primus numerus minor, qui proponitur diuiſibilis
                  <lb/>
                ſignificetur linea
                  <var>.q.g.</var>
                maior vero linea
                  <var>.x.</var>
                tum cogitemus
                  <var>.q.g.</var>
                diuiſam, cuius maior
                  <lb/>
                pars ſit
                  <var>.q.o.</var>
                minor
                  <var>.o.g.</var>
                differentia
                  <var>.q.p.</var>
                ex quo
                  <var>.p.o.</var>
                æqualis erit
                  <var>.o.g.</var>
                ſit autem produ-
                  <lb/>
                ctum
                  <var>.b.o</var>
                . </s>
                <s xml:id="echoid-s447" xml:space="preserve">Oportet igitur, ut
                  <var>.b.o.</var>
                ſimul cum differentia
                  <var>.q.p.</var>
                æquale ſit numero
                  <var>.x.</var>
                ſe-
                  <lb/>
                cundò propoſito, qui notus eſt, </s>
                <s xml:id="echoid-s448" xml:space="preserve">quare etiam ſumma producti
                  <var>.b.o.</var>
                cum differentia
                  <lb/>
                  <var>q.p.</var>
                cognita erit, ex qua detracto primo numero
                  <var>.q.g.</var>
                reſiduum cognitum erit, nunc
                  <lb/>
                igitur quodnam erit hoc reſiduum? </s>
                <s xml:id="echoid-s449" xml:space="preserve">attendamus qua ratione ex ſumma
                  <var>.b.o.</var>
                et
                  <var>.q.p.</var>
                  <lb/>
                detrahenda ſit
                  <var>.q.g</var>
                . </s>
                <s xml:id="echoid-s450" xml:space="preserve">In primis ſi ſubtraxerimus ex dicta ſumma
                  <var>.q.p.</var>
                quę pars eſt
                  <var>.q.g.</var>
                  <lb/>
                ſupererit detrahenda
                  <var>.p.g.</var>
                ex
                  <var>.b.o.</var>
                pars inquam ipſius
                  <var>.q.g.</var>
                quod fiet quotieſcunque
                  <lb/>
                cogitauerimus
                  <var>.q.o.</var>
                duabus vnitatibus diminutam, et per
                  <var>.o.g.</var>
                multiplicatam, ſit au-
                  <lb/>
                tem productum
                  <var>.b.e.</var>
                nam cum
                  <var>.o.g.</var>
                toties
                  <var>.b.o.</var>
                ingrediatur, quot ſunt in
                  <var>.q.o.</var>
                vnitates
                  <lb/>
                ex prima ſexti aut .18. vel .19. ſeptimi,
                  <reg norm="detrahendaque" type="simple">detrahendaq́;</reg>
                ſit
                  <var>.p.g.</var>
                ex
                  <var>.b.o.</var>
                quæ
                  <var>.p.g.</var>
                dupla
                  <lb/>
                eſt
                  <var>.o.g.</var>
                patebit
                  <var>.o.c.</var>
                æqualem eſſe
                  <var>.p.g.</var>
                fu-
                  <lb/>
                pererit ita que
                  <var>.b.e.</var>
                productum
                  <var>.q.e.</var>
                in
                  <var>.e.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0045-01a" xlink:href="fig-0045-01"/>
                i. cognitum, erutis autem ex
                  <var>.q.g.</var>
                ijſdem
                  <lb/>
                duabus vnitatibus, remanebit
                  <var>.q.i.</var>
                nobis
                  <lb/>
                nota, ex quo
                  <var>.e.i.</var>
                æqualis erit
                  <var>.e.c</var>
                . </s>
                <s xml:id="echoid-s451" xml:space="preserve">Cum
                  <lb/>
                igitur productum
                  <var>.q.e.</var>
                in
                  <var>.e.i.</var>
                cognoſcamus
                  <lb/>
                ſimul cum
                  <var>.q.i</var>
                : Sivoluerimus partes
                  <var>.q.e.</var>
                  <lb/>
                et
                  <var>.e.i.</var>
                cognoſcere, vtemur .45. theorema-
                  <lb/>
                te huius libri, & propoſitum obtinebimus, nam cognoſcemus
                  <var>.e.i.</var>
                & ex conſequen-
                  <lb/>
                ti
                  <var>.o.g.</var>
                eius æqualem.</s>
              </p>
              <div xml:id="echoid-div106" type="float" level="4" n="1">
                <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a">
                  <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0045-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div108" type="math:theorem" level="3" n="51">
              <head xml:id="echoid-head67" xml:space="preserve">THEOREMA
                <num value="51">LI</num>
              .</head>
              <p>
                <s xml:id="echoid-s452" xml:space="preserve">
                  <emph style="sc">DIvidere</emph>
                numerum in duas eiuſmodi partes, quæ pro medio proportionali
                  <lb/>
                alterum numerum propoſitum recipiant, primi dimidio minorem, aliud ni
                  <lb/>
                hil eſt, quàm binas primi numeri partes inuenire, quæ inter ſe multiplicatæ quadra
                  <lb/>
                to ſecundi numeri numerum æqualem proferant, ex .16. ſexti aut .20. ſeptimi, quod
                  <lb/>
                tamen .45. theoremate fuit à nobis ſpeculatum.</s>
              </p>
            </div>
            <div xml:id="echoid-div109" type="math:theorem" level="3" n="52">
              <head xml:id="echoid-head68" xml:space="preserve">THEOREMA
                <num value="52">LII</num>
              .</head>
              <p>
                <s xml:id="echoid-s453" xml:space="preserve">CVR pro poſitis tribus numeris quibuſcunque, ſi productum primi in ſecun-
                  <lb/>
                dum per tertium multiplicetur, atque ſecundum hoc productum
                  <reg norm="corporeum" type="context">corporeũ</reg>
                ,
                  <lb/>
                per primum numerum diuidatur, proueniens erit numerus æqualis producto ſe-
                  <lb/>
                cundi in tertium.</s>
              </p>
              <p>
                <s xml:id="echoid-s454" xml:space="preserve">Exempli cauſa, proponantur hi tres numeri .10. 11. 12.
                  <reg norm="multiplicenturque" type="simple">multiplicenturq́;</reg>
                .10.
                  <reg norm="cum" type="context">cũ</reg>
                .</s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>