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
31 19
32 20
33 21
34 22
35 23
36 24
37 25
38 26
39 27
40 28
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
< >
page |< < (41) of 445 > >|
THEOREM. 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-div129" type="math:theorem" level="3" n="63">
              <p>
                <s xml:id="echoid-s548" xml:space="preserve">
                  <pb o="41" rhead="THEOREM. ARITH." n="53" file="0053" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0053"/>
                mentum eſt quadrati
                  <var>.q.d.</var>
                totalis. </s>
                <s xml:id="echoid-s549" xml:space="preserve">Quare duplicato
                  <var>.a.i.</var>
                & coniuncto
                  <var>.b.</var>
                cognoſci-
                  <lb/>
                mustotum
                  <var>.q.d.</var>
                & conſequenter
                  <var>.a.d.</var>
                ſuam radicem, hoc eſt ſummam duarum radi
                  <lb/>
                cum
                  <var>.a.g.</var>
                et
                  <var>.g.d.</var>
                quæ medio
                  <var>.a.i.</var>
                cognito, & quadrageſimoquinto theoremate ſingu-
                  <lb/>
                læ cognoſcuntur.</s>
              </p>
            </div>
            <div xml:id="echoid-div130" type="math:theorem" level="3" n="64">
              <head xml:id="echoid-head80" xml:space="preserve">THEOREMA
                <num value="64">LXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s550" xml:space="preserve">CVR propoſitum aliquem num erum in duas eiuſmodi partes diuiſuri, vt ſum-
                  <lb/>
                ma radicum dictarum partium æqualis ſit alteri numero propoſito. </s>
                <s xml:id="echoid-s551" xml:space="preserve">Rectè ſe-
                  <lb/>
                cundum numerum in ſeipſum multiplicant, ex quo quadrato, primum datum nu-
                  <lb/>
                merum detrahunt,
                  <reg norm="rurſusque" type="simple">rurſusq́;</reg>
                reſiduum in ſeipſum multiplicant, & ex eo quadrato
                  <lb/>
                quartam partem deſumunt,
                  <reg norm="quam" type="context">quã</reg>
                ex quadrato dimidij primi numeri detrahunt, radi-
                  <lb/>
                cemq́ue qua dratam reſidui cum iunxerint, & ex dimidio primi numeri detraxerint,
                  <lb/>
                partes quæſitæ proferuntur.</s>
              </p>
              <p>
                <s xml:id="echoid-s552" xml:space="preserve">Exempli gratia, ſi proponeretur primus numerus .20. diuidendus et .6. ſecundus
                  <lb/>
                pro ſumma radicum, hunc ſecundum .6. in ſeipſum multiplicabimus,
                  <reg norm="dabiturque" type="simple">dabiturq́;</reg>
                nu-
                  <lb/>
                merus .36. ex quo quadrato primus numerus detrahetur,
                  <reg norm="ſupereritque" type="simple">ſupereritq́;</reg>
                numerus .16.
                  <lb/>
                qui quadratus dabit .256. cuius numeri quarta pars ſumetur, nempe .64. quæ ex qua
                  <lb/>
                drato dimidij primi numeri detrahetur, nempe .100.
                  <reg norm="ſupereritque" type="simple">ſupereritq́;</reg>
                .36. cuius radix qua
                  <lb/>
                drata .6. coniuncta & detracta ex .10. dabit .16. partem maiorem et .4. minorem.</s>
              </p>
              <p>
                <s xml:id="echoid-s553" xml:space="preserve">Cuius rei hæc ſpeculatio, primus numerus diuiſibilis ſignificetur linea
                  <var>.a.b.</var>
                diui-
                  <lb/>
                ſa in puncto
                  <var>.e.</var>
                in partes adhuc incognitas, et
                  <var>.a.c.</var>
                ſit productum
                  <var>.a.e.</var>
                in
                  <var>.e.b.</var>
                item
                  <var>.q.
                    <lb/>
                  p.</var>
                ſecundum numerum ſignificet, æqualem ſummæ radicum, quæ puncto
                  <var>.n.</var>
                diſtin-
                  <lb/>
                guantur. </s>
                <s xml:id="echoid-s554" xml:space="preserve">Poſtmodum totum quadratum
                  <var>.p.d.</var>
                erigatur (quod nobis eſt cognitum),
                  <lb/>
                in duo quadrata diuiſum
                  <var>.o.p.</var>
                et
                  <var>.o.d.</var>
                quorum ſumma
                  <var>.a.b.</var>
                cum detur, cognita rema-
                  <lb/>
                net ſumma
                  <reg norm="duorum" type="context">duorũ</reg>
                  <reg norm="ſupplementorum" type="context">ſupplementorũ</reg>
                  <var>.o.u.</var>
                et
                  <var>.o.q.</var>
                qua quadrata
                  <reg norm="cum" type="context">cũ</reg>
                fuerit dabit quadru
                  <lb/>
                  <reg norm="plum" type="context">plũ</reg>
                quadrati
                  <reg norm="ſupplementi" type="context">ſupplemẽti</reg>
                  <var>.o.q.</var>
                  <reg norm="nempe" type="context">nẽpe</reg>
                  <reg norm="quadruplum" type="context">quadruplũ</reg>
                producti
                  <var>.a.c.</var>
                etenim
                  <var>.a.c.</var>
                ex .19. theo
                  <lb/>
                remate huius libri quadratum eft ipſius
                  <var>.q.o.</var>
                  <reg norm="ſicque" type="simple">ſicq́;</reg>
                poterant etiam veteres quadrare
                  <lb/>
                dimidium differentiæ
                  <var>.a.b.</var>
                ab
                  <var>.p.d.</var>
                nempe quadrato tantummodo ſupplemento
                  <var>.q.
                    <lb/>
                  o</var>
                . </s>
                <s xml:id="echoid-s555" xml:space="preserve">Tunc habito
                  <var>.a.c.</var>
                eius ope tanquam producti
                  <var>.a.e.</var>
                in
                  <var>.e.b.</var>
                ex .45. theoremate ſingu
                  <lb/>
                læ partes cognoſcentur.</s>
              </p>
              <p>
                <s xml:id="echoid-s556" xml:space="preserve">Quod alia etiam ratione præſtari poterat, nempe cognito ſupplemento
                  <var>.
                    <lb/>
                  q.o.</var>
                diſtinguendæ radices
                  <var>q.n.</var>
                et
                  <var>.n.p.</var>
                ex .45. theoremate, quibus cognitis, eorum
                  <lb/>
                etiam quadrata cognoſcuntur.</s>
              </p>
              <figure position="here">
                <image file="0053-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0053-01"/>
              </figure>
              <figure position="here">
                <image file="0053-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0053-02"/>
              </figure>
            </div>
          </div>
        </div>
      </text>
    </echo>