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
91 79
92 80
93 81
94 82
95 89
96 84
97 85
98 96
99 87
100 88
101 89
102 90
103 91
104 92
105 93
106 94
107 95
108 96
109 97
110 98
111 99
112 100
113 101
114 102
115 103
116 104
117 105
118 106
119 107
120 108
< >
page |< < (60) 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-div179" type="math:theorem" level="3" n="91">
              <p>
                <s xml:id="echoid-s788" xml:space="preserve">
                  <pb o="60" rhead="IO. BAPT. BENED." n="72" file="0072" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0072"/>
                per duab. vnitatibus ſuperficialibus creſcere,
                  <reg norm="quarum" type="context">quarũ</reg>
                  <reg norm="ſingularum" type="context">ſingularũ</reg>
                radix æqualis eſt
                  <var>.g.</var>
                ne
                  <lb/>
                ceſſariò ſequitur gnomonem
                  <var>.b.a.g.</var>
                duabus partibus aut vnitatibus gnomonem
                  <var>.d.
                    <lb/>
                  o.p.</var>
                ſuperare, ita vt gnomon
                  <var>.b.a.g.</var>
                ſeptem vnitatibus, aut partibus ſuperficialibus
                  <lb/>
                quadratis conſtet. </s>
                <s xml:id="echoid-s789" xml:space="preserve">Quare eadem ratione gnomon
                  <var>.f.s.h.</var>
                conſtabit nouem ſimilibus.
                  <lb/>
                </s>
                <s xml:id="echoid-s790" xml:space="preserve">Itaque æqualis erit quadrato
                  <var>.q.o</var>
                . </s>
                <s xml:id="echoid-s791" xml:space="preserve">Quamobrem verum eſt, quòd quadrato
                  <var>.q.o.</var>
                  <lb/>
                coniuncto quadrato
                  <var>.q.a.</var>
                proueniet quadratum
                  <var>.q.s.</var>
                cuius radix ita differet à
                  <var>.q.g.</var>
                vt
                  <var>.
                    <lb/>
                  q.g.</var>
                à
                  <var>.q.p</var>
                : ex quo tres radices arithmeticè inter ſe continuæ proportionales erunt.
                  <lb/>
                </s>
                <s xml:id="echoid-s792" xml:space="preserve">Idipſum dico ſi
                  <var>.q.p.</var>
                fuerit .6. et
                  <var>.q.g</var>
                : 8: </s>
                <s xml:id="echoid-s793" xml:space="preserve">tunc enim ſingulæ partes
                  <var>.q.k.p.g.h.</var>
                æquipol
                  <lb/>
                lebunt duabus vnitatibus, quæ cogitabuntur
                  <lb/>
                  <figure xlink:label="fig-0072-01" xlink:href="fig-0072-01a" number="101">
                    <image file="0072-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0072-01"/>
                  </figure>
                in ſummam collectæ, ut cum patribus
                  <var>.q.k.p.
                    <lb/>
                  g.h.</var>
                integris contemplari liceat. </s>
                <s xml:id="echoid-s794" xml:space="preserve">Idem acci-
                  <lb/>
                det fi
                  <var>.q.p.</var>
                erit .9. et
                  <var>.q.g.</var>
                12. fingulæ enim par-
                  <lb/>
                tes
                  <var>.q.K.p.g.h.</var>
                tripartitæ erunt. </s>
                <s xml:id="echoid-s795" xml:space="preserve">Idcircò dixi
                  <lb/>
                gnomonem
                  <var>.f.s.h.</var>
                tam amplum cogitari de-
                  <lb/>
                bere, quam gnomon
                  <var>.b.a.g.</var>
                nempè ut
                  <var>.h.</var>
                æqua
                  <lb/>
                lis ſit
                  <var>.g</var>
                . </s>
                <s xml:id="echoid-s796" xml:space="preserve">Idem occurret ſi
                  <var>.q.g.</var>
                erit .12. et
                  <var>.q.p.</var>
                  <lb/>
                quinque, quod cum fuerit patebitex præce-
                  <lb/>
                dentis theorematis ſpeculatione, gnomonem
                  <lb/>
                  <var>f.s.h</var>
                : 25. vnitatibus conſtare, cogitatum am-
                  <lb/>
                plitudinis ſimplicis vnitatis denominatæ in
                  <var>.q.
                    <lb/>
                  p.</var>
                aut
                  <var>.q.g.</var>
                non amplitudinis gnomonis
                  <var>.b.a.g.</var>
                  <lb/>
                qui ſeptem vnitatibus latus eſſet. </s>
                <s xml:id="echoid-s797" xml:space="preserve">Cum igitur
                  <var>.
                    <lb/>
                  q.p.</var>
                quinque vnitatibus linearibus conſtet ſcimus
                  <var>.q.o</var>
                : 25. ſuperficialibus conſtare,
                  <lb/>
                collecto itaque in ſummam quadrato
                  <var>.q.o.</var>
                cum quadrato
                  <var>.q.a.</var>
                cognoſcetur quadra-
                  <lb/>
                tum
                  <var>.q.s.</var>
                vnà etiam eius radix. </s>
                <s xml:id="echoid-s798" xml:space="preserve">Eadem ratione, alia multa quadrata ſimilia contem-
                  <lb/>
                plari licebit.</s>
              </p>
            </div>
            <div xml:id="echoid-div181" type="math:theorem" level="3" n="92">
              <head xml:id="echoid-head109" xml:space="preserve">THEOREMA
                <num value="92">XCII</num>
              .</head>
              <p>
                <s xml:id="echoid-s799" xml:space="preserve">CVR propoſito numero pari maiori binario, qui detrahi & in ſummam colli-
                  <lb/>
                gi debeat ex altero numero quærendo, vt tam reſiduum quam ſumma ſint
                  <lb/>
                quadrata numerorum integrornm. </s>
                <s xml:id="echoid-s800" xml:space="preserve">Rectè dimidium propoſiti numeri in ſeipſum
                  <lb/>
                multiplicamus, & quadrato huic addimus vnitatem,
                  <reg norm="eritque" type="simple">eritq́;</reg>
                numerus quæfitus.</s>
              </p>
              <p>
                <s xml:id="echoid-s801" xml:space="preserve">Exempli gratia proponitur .12. numerus detrahendus, & coniungendus nume-
                  <lb/>
                ro inueſtigando, ut reſiduum detractionis, & ſumma ſint quadrati numeri. </s>
                <s xml:id="echoid-s802" xml:space="preserve">Addi-
                  <lb/>
                ta vnitate ipſi .36. quadrato dimidij, dabitur .37. numerus quæſitus.</s>
              </p>
              <p>
                <s xml:id="echoid-s803" xml:space="preserve">Cuius ſpeculationis gratia, ſubſcripta quatuor quadrata cogitemus
                  <var>.g.p</var>
                :
                  <var>u.i</var>
                :
                  <var>t.c</var>
                :
                  <var>n.
                    <lb/>
                  K.</var>
                  <reg norm="cogitemusque" type="simple">cogitemusq́;</reg>
                quadratum
                  <var>.g.p.</var>
                eſſe quadratum ſummæ,
                  <var>K.n.</var>
                verò reſidui ſubtractio-
                  <lb/>
                nis:
                  <var>u.i.</var>
                  <reg norm="autem" type="wordlist">aũt</reg>
                numerum
                  <reg norm="inueſtigandum" type="context context">inueſtigãdũ</reg>
                , ex quo gnomon
                  <var>.u.d.i.</var>
                cognoſcetur ita etiam et
                  <var>.n.
                    <lb/>
                  o.K.</var>
                qui inter ſe ſunt æquales. </s>
                <s xml:id="echoid-s804" xml:space="preserve">Iam certi erimus
                  <var>.e.i.</var>
                eſſe plus quam dimidium gno-
                  <lb/>
                monis
                  <var>.n.o.K</var>
                . </s>
                <s xml:id="echoid-s805" xml:space="preserve">Itaque cogitemus rectangulum
                  <var>.r.c.</var>
                exactum
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                eſſe gnomonis
                  <var>.
                    <lb/>
                  n.o.K.</var>
                ex unitatibus ſuperficialibus quarum una erit
                  <var>.m.a</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s806" xml:space="preserve">Cuius numeri quadratum ſit
                  <var>.t.c.</var>
                vnde etiam cognitum & cum
                  <var>.K.c.</var>
                ex communi
                  <lb/>
                ſcientia ſit vnitas linearis, </s>
                <s xml:id="echoid-s807" xml:space="preserve">propterea quod
                  <var>.m.a.</var>
                eſt ſuperficialis hoc eſt quadrata,
                  <lb/>
                quæ detracta ex
                  <var>.q.c.</var>
                dimidio gnomonis
                  <var>.n.o.K.</var>
                (quamuis lineari) ſupererit
                  <var>.K.q.</var>
                co
                  <lb/>
                gnita, numerorum integrorum (nota
                  <var>q.K.i.</var>
                ſemper minor erit duabus vnitatibus li-
                  <lb/>
                nearibus & maior vna ex dictis vnitatibus, ut ex te ipſo contemplari potes) </s>
                <s xml:id="echoid-s808" xml:space="preserve">quare
                  <var>. </var>
                </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum