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 |< < (31) 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-div98" type="math:theorem" level="3" n="46">
              <p>
                <s xml:id="echoid-s407" xml:space="preserve">
                  <pb o="31" rhead="THEOREM. ARITH." n="43" file="0043" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0043"/>
                ad vnitatem
                  <var>.o.i.</var>
                  <reg norm="permutandoque" type="simple">permutandoq́;</reg>
                  <var>.e.a.</var>
                ad
                  <var>.a.u.</var>
                ſicut
                  <var>.t.n.</var>
                ad
                  <var>.n.i.</var>
                & componendo
                  <var>.e.a.u.</var>
                  <lb/>
                ad
                  <var>a.u.</var>
                ſicut
                  <var>.t.n.i.</var>
                ad
                  <var>.n.i</var>
                : & euerſim
                  <var>.e.a.u.</var>
                ad
                  <var>.e.a.</var>
                vt
                  <var>.t.n.i.</var>
                ad
                  <var>.t.n</var>
                . </s>
                <s xml:id="echoid-s408" xml:space="preserve">Quare, ex .20. ſepti
                  <lb/>
                mi, recte vtimur regula de tribus. </s>
                <s xml:id="echoid-s409" xml:space="preserve">Idem & de altera parte dico, quamuis qui vnam
                  <lb/>
                teneat, alteram quo que habiturus ſit. </s>
                <s xml:id="echoid-s410" xml:space="preserve">Non mirum tamen ſi huiuſmodi problema
                  <lb/>
                ab antiquis definitum non fuerit, qui hanc vltimam partem non cognouerunt.</s>
              </p>
            </div>
            <div xml:id="echoid-div100" type="math:theorem" level="3" n="47">
              <head xml:id="echoid-head63" xml:space="preserve">THEOREMA
                <num value="47">XLVII</num>
              .</head>
              <p>
                <s xml:id="echoid-s411" xml:space="preserve">CVR duobus numeris mutuó diuiſis, ſi per ſummam prouenientium, produ-
                  <lb/>
                ctum vnius in alterum multiplicetur, vltimum productum, ſummæ quadra-
                  <lb/>
                tn
                  <gap extent="2"/>
                m duorum numerorum æquale futurum ſit.</s>
              </p>
              <p>
                <s xml:id="echoid-s412" xml:space="preserve">Exempli gratia, propoſitis .16. et .4. mutuò diuiſis, ſumma prouenientium erit
                  <num value="4">.
                    <lb/>
                  4.</num>
                integrorum cum quarta parte, qua ſumma multiplicata cum producto
                  <reg norm="primorum" type="context">primorũ</reg>
                  <lb/>
                numerorum, nempe .64. dabuntur .272. integri ſuperficiales, qui ſummæ quadra-
                  <lb/>
                torum duorum numerorum æquantur.</s>
              </p>
              <p>
                <s xml:id="echoid-s413" xml:space="preserve">Hoc vt conſideremus, duo numeri partibus
                  <var>.a.e.</var>
                et
                  <var>.e.i.</var>
                in linea
                  <var>.a.i.</var>
                ſignificentur,
                  <lb/>
                quorum productum ſit
                  <var>.e.d.</var>
                &
                  <reg norm="quadratum" type="context">quadratũ</reg>
                ipſius
                  <var>.a.e.</var>
                ſit
                  <var>.e.p</var>
                : ipſius verò
                  <var>.e.i.</var>
                ſit
                  <var>.e.q.</var>
                pro-
                  <lb/>
                ueniens
                  <reg norm="autem" type="wordlist">aũt</reg>
                ex diuiſione
                  <var>.e.i.</var>
                per
                  <var>.a.e.</var>
                ſit
                  <var>.o.u.</var>
                proueniens
                  <reg norm="autem" type="wordlist">aũt</reg>
                  <var>.a.e.</var>
                per
                  <var>.e.i.</var>
                ſit
                  <var>.o.t.</var>
                quo-
                  <lb/>
                rum ſumma ſit
                  <var>.o.u.t.</var>
                tum productum
                  <var>.e.d</var>
                : linea
                  <var>.u.n.</var>
                ſignificetur ad angulum
                  <reg norm="rectum" type="context">rectũ</reg>
                  <lb/>
                coniuncta in puncto
                  <var>.u.</var>
                extremo ipſius
                  <var>.o.u.t.</var>
                productum
                  <reg norm="autem" type="wordlist">aũt</reg>
                  <var>.u.o.t.</var>
                in
                  <var>.u.n.</var>
                ſit
                  <var>.n.t</var>
                . </s>
                <s xml:id="echoid-s414" xml:space="preserve">Iam
                  <lb/>
                probandum nobis eſt
                  <var>.n.t.</var>
                æqualem eſſe ſummæ duorum quadratorum
                  <var>.q.e.p</var>
                . </s>
                <s xml:id="echoid-s415" xml:space="preserve">Quod
                  <lb/>
                ſingillatim probo, & aſſero productum
                  <var>.o.n.</var>
                æquale eſſe quadrato
                  <var>.q.e.</var>
                &
                  <reg norm="productum" type="context">productũ</reg>
                  <var>.
                    <lb/>
                  s.t.</var>
                quadrato
                  <var>.e.p</var>
                . </s>
                <s xml:id="echoid-s416" xml:space="preserve">Nam ex .35. theoremate patet numerum
                  <var>.e.i.</var>
                medium eſſe
                  <reg norm="pro- portionalem" type="context">pro-
                    <lb/>
                  portionalẽ</reg>
                inter
                  <var>.e.d.</var>
                et
                  <var>.o.u</var>
                : cum numerus
                  <var>.e.i.</var>
                ex præſuppoſito ab
                  <var>.e.a.</var>
                multiplicetur
                  <lb/>
                & diuidatur, cuius multiplicationis produ-
                  <lb/>
                ctum eſt
                  <var>.d.e</var>
                : nempe
                  <var>.u.n.</var>
                & proueniens ex
                  <lb/>
                  <figure xlink:label="fig-0043-01" xlink:href="fig-0043-01a" number="59">
                    <image file="0043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0043-01"/>
                  </figure>
                diuiſione eſt
                  <var>.o.u</var>
                : </s>
                <s xml:id="echoid-s417" xml:space="preserve">quare ex dicto theorema-
                  <lb/>
                te
                  <var>.e.i.</var>
                media proportionalis eſt inter
                  <var>.u.n.</var>
                et
                  <var>.
                    <lb/>
                  u.o</var>
                . </s>
                <s xml:id="echoid-s418" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                productum
                  <var>.o.n.</var>
                æquale eſt qua-
                  <lb/>
                drato
                  <var>.e.q.</var>
                ex .16. ſexti vel .20. ſeptimi. </s>
                <s xml:id="echoid-s419" xml:space="preserve">Idem
                  <lb/>
                dico de producto
                  <var>.s.t.</var>
                  <reg norm="nempe" type="context">nẽpe</reg>
                æquale eſſe qua-
                  <lb/>
                drato
                  <var>.e.p.</var>
                quandoquidem numerus
                  <var>.a.e.</var>
                ab
                  <lb/>
                  <var>e.i.</var>
                multiplicatur ac diuiditur, cuius multi-
                  <lb/>
                plicationis productum eſt
                  <var>.d.e.</var>
                nempe
                  <var>o.s.</var>
                &
                  <lb/>
                proueniens ex diuiſione
                  <var>.o.t</var>
                : </s>
                <s xml:id="echoid-s420" xml:space="preserve">inter quæ ex .
                  <lb/>
                35. theoremate
                  <var>.a.e.</var>
                media proportionalis
                  <lb/>
                eſt. </s>
                <s xml:id="echoid-s421" xml:space="preserve">Quare ex allatis propoſitionibus
                  <reg norm="productum" type="context">productũ</reg>
                  <var>.s.t.</var>
                æquale eſt quadrato
                  <var>.e.p.</var>
                ſed
                  <reg norm="totum" type="context">totũ</reg>
                  <lb/>
                productum
                  <var>.n.t.</var>
                ſumma eſt duorum productorum
                  <var>.o.n.</var>
                et
                  <var>.s.t.</var>
                ex prima ſecundi Eucli.
                  <lb/>
                </s>
                <s xml:id="echoid-s422" xml:space="preserve">Itaque verum eſſe quod dictum eſt, conſequitur.</s>
              </p>
            </div>
            <div xml:id="echoid-div102" type="math:theorem" level="3" n="48">
              <head xml:id="echoid-head64" xml:space="preserve">THEOREMA
                <num value="48">XLVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s423" xml:space="preserve">CVR ſi quis maiorem duorum numerorum ſola vnitate inter ſe differentium,
                  <lb/>
                per minorem diuidat,
                  <reg norm="maioremque" type="simple">maioremq́;</reg>
                per proueniens multiplicet, productum,
                  <lb/>
                  <reg norm="summæ" type="context">sũmæ</reg>
                ipſius maioris cum eodem proueniente æquale erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s424" xml:space="preserve">Exempli gratia .10 per .9. diuiſo, datur vnum cum nona parte, quo multiplica-
                  <lb/>
                to per proueniens, ipſo nempe .10: </s>
                <s xml:id="echoid-s425" xml:space="preserve">datur productum .11. cum nona parte, tantum ſci­ </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum