Clavius, Christoph, Geometria practica

Table of contents

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[211.] COROLLARIVM.
Page: 249 (219)
[212.] PROPOSITIO II.
Page: 249 (219)
[213.] COROLLARIVM.
Page: 249 (219)
[214.] PROPOSITIO III.
Page: 250 (220)
[215.] COROLLARIVM.
Page: 250 (220)
[216.] PROPOSITIO IV.
Page: 251 (221)
[217.] PROPOSITIO V.
Page: 251 (221)
[218.] PROPOSITIO VI.
Page: 251 (221)
[219.] PROPOSITIO VII.
Page: 252 (222)
[220.] I.
Page: 253 (223)
[221.] II.
Page: 253 (223)
[222.] ALITER.
Page: 255 (225)
[223.] ALITER.
Page: 255 (225)
[224.] I.
Page: 256 (226)
[225.] II.
Page: 257 (227)
[226.] III.
Page: 257 (227)
[227.] IIII.
Page: 257 (227)
[228.] I.
Page: 258 (228)
[229.] II.
Page: 258 (228)
[230.] III.
Page: 258 (228)
[231.] IIII.
Page: 258 (228)
[232.] DE AREA SEGMENTO-rum ſphæræ. Capvt VI.
Page: 259 (229)
[233.] ALITER.
Page: 261 (231)
[234.] DE AREA SPHÆROIDIS, EIVSDEM-que portionum. Capvt VII.
Page: 262 (232)
[235.] DE AREA CONOIDIS parabolici. Capvt VIII.
Page: 262 (232)
[236.] DE AREA CONOIDIS Hyperbolici. Capvt IX.
Page: 263 (233)
[237.] DE AREA DOLIORVM. Capvt X.
Page: 263 (233)
[238.] DE AREA CORPORVM. omnino irregularium. Capvt XI.
Page: 264 (234)
[239.] DE SVPERFICIE CONVEXA coni & cylindri recti. Capvt XII.
Page: 265 (235)
[240.] FINIS LIBRI QVINTI.
Page: 265 (235)
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          <p>
            <s xml:id="echoid-s10384" xml:space="preserve">
              <pb o="226" file="256" n="256" rhead="GEOMETR. PRACT."/>
            priori: </s>
            <s xml:id="echoid-s10385" xml:space="preserve"> æqualia erunt ipſa parallelepipeda: </s>
            <s xml:id="echoid-s10386" xml:space="preserve">Sed prius eſt, per 2. </s>
            <s xml:id="echoid-s10387" xml:space="preserve">partem
              <note symbol="a" position="left" xlink:label="note-256-01" xlink:href="note-256-01a" xml:space="preserve">34. vndec.</note>
            2. </s>
            <s xml:id="echoid-s10388" xml:space="preserve">regulæ, æquale ſphærę. </s>
            <s xml:id="echoid-s10389" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s10390" xml:space="preserve">poſterius: </s>
            <s xml:id="echoid-s10391" xml:space="preserve">atque idcirco ſphæra gignetur ex
              <lb/>
            {1/2}. </s>
            <s xml:id="echoid-s10392" xml:space="preserve">areæ circuli maximi in {4/5}. </s>
            <s xml:id="echoid-s10393" xml:space="preserve">diametri. </s>
            <s xml:id="echoid-s10394" xml:space="preserve">quod eſt quintum.</s>
            <s xml:id="echoid-s10395" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10396" xml:space="preserve">
              <emph style="sc">Item</emph>
            intelligantur duo parallelepipeda, quorum vnius baſis æqualis ſit {2/3}.
              <lb/>
            </s>
            <s xml:id="echoid-s10397" xml:space="preserve">
              <note position="left" xlink:label="note-256-02" xlink:href="note-256-02a" xml:space="preserve">Demonſtratio
                <lb/>
              ſextæpartis.</note>
            areæ circuli maximi in ſphæra, & </s>
            <s xml:id="echoid-s10398" xml:space="preserve">altitudo diametro: </s>
            <s xml:id="echoid-s10399" xml:space="preserve">alterius verò baſis æqualis
              <lb/>
            ſit {1/3}. </s>
            <s xml:id="echoid-s10400" xml:space="preserve">areæ maximi circuli, & </s>
            <s xml:id="echoid-s10401" xml:space="preserve">altitudo duplæ diametro. </s>
            <s xml:id="echoid-s10402" xml:space="preserve">Et quia horum paralle-
              <lb/>
            lepipedorum baſes cum altitudinibus reciprocantur: </s>
            <s xml:id="echoid-s10403" xml:space="preserve">quod tam baſis in priori
              <lb/>
            ſit dupla baſis in poſteriori, quam altitudo in poſteriori altitudinis in priori:
              <lb/>
            </s>
            <s xml:id="echoid-s10404" xml:space="preserve"> eruntipſa parallelepipeda æqualia: </s>
            <s xml:id="echoid-s10405" xml:space="preserve">Sed prius per 3. </s>
            <s xml:id="echoid-s10406" xml:space="preserve">partem huius 2. </s>
            <s xml:id="echoid-s10407" xml:space="preserve">regulę,
              <note symbol="b" position="left" xlink:label="note-256-03" xlink:href="note-256-03a" xml:space="preserve">34. vndec.</note>
            quale eſt ipſi ſphærę. </s>
            <s xml:id="echoid-s10408" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s10409" xml:space="preserve">poſterius: </s>
            <s xml:id="echoid-s10410" xml:space="preserve">Ac proinde ſphæra ex dupla dia-
              <lb/>
            metro in {1/3}. </s>
            <s xml:id="echoid-s10411" xml:space="preserve">areæ circuli maximi procreabitur. </s>
            <s xml:id="echoid-s10412" xml:space="preserve">quod ſexto loco eſt propo-
              <lb/>
            ſitum.</s>
            <s xml:id="echoid-s10413" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10414" xml:space="preserve">
              <emph style="sc">Intelligantvr</emph>
            quoque duo parallelepipeda, quorum vnius baſis con-
              <lb/>
              <note position="left" xlink:label="note-256-04" xlink:href="note-256-04a" xml:space="preserve">Demonſtratio
                <lb/>
              ſeptimæ partis.</note>
            tineat {1/3}. </s>
            <s xml:id="echoid-s10415" xml:space="preserve">ſuperficiei ſphærę, & </s>
            <s xml:id="echoid-s10416" xml:space="preserve">altitudo {1/3}. </s>
            <s xml:id="echoid-s10417" xml:space="preserve">diametri: </s>
            <s xml:id="echoid-s10418" xml:space="preserve">alterius verò baſis compre-
              <lb/>
            hendat {1/6}. </s>
            <s xml:id="echoid-s10419" xml:space="preserve">ſuperficiei, & </s>
            <s xml:id="echoid-s10420" xml:space="preserve">altitudo æqualis ſit diametro. </s>
            <s xml:id="echoid-s10421" xml:space="preserve">Et quoniam baſes cum
              <lb/>
            altitudinibus ſunt reciprocę, quod ita ſit {1/3}. </s>
            <s xml:id="echoid-s10422" xml:space="preserve">ſuperficiei baſis videlicet prioris
              <lb/>
            parallelepipedi, ad {1/6}. </s>
            <s xml:id="echoid-s10423" xml:space="preserve">ſuperficiei, id eſt, ad baſem poſterioris, vt altitudo poſte-
              <lb/>
            rioris, nempe diameter, ad prioris altitudinem, nimirum ad {1/2}. </s>
            <s xml:id="echoid-s10424" xml:space="preserve">diametri, cum v-
              <lb/>
            traque proportio ſit dupla: </s>
            <s xml:id="echoid-s10425" xml:space="preserve"> ipſa parallelepipeda æqualia erunt: </s>
            <s xml:id="echoid-s10426" xml:space="preserve">Sed prius
              <note symbol="c" position="left" xlink:label="note-256-05" xlink:href="note-256-05a" xml:space="preserve">34. vndec.</note>
            ſphærę, per 1. </s>
            <s xml:id="echoid-s10427" xml:space="preserve">partem huius 2. </s>
            <s xml:id="echoid-s10428" xml:space="preserve">regulæ æquale eſt. </s>
            <s xml:id="echoid-s10429" xml:space="preserve">Igitur, & </s>
            <s xml:id="echoid-s10430" xml:space="preserve">poſterius: </s>
            <s xml:id="echoid-s10431" xml:space="preserve">hoc eſt,
              <lb/>
            ſphærę ſoliditas producetur ex diametro in ſextam partem ſuperficiei, quod eſt
              <lb/>
            ſeptimum.</s>
            <s xml:id="echoid-s10432" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10433" xml:space="preserve">
              <emph style="sc">Deniqve</emph>
            concipiantur duo parallelepipeda, quorum vnius baſis ſit {1/3}.
              <lb/>
            </s>
            <s xml:id="echoid-s10434" xml:space="preserve">
              <note position="left" xlink:label="note-256-06" xlink:href="note-256-06a" xml:space="preserve">Demonſtratio
                <lb/>
              octauæ partis.</note>
            ſuperficiei ſphæræ, & </s>
            <s xml:id="echoid-s10435" xml:space="preserve">altitudo ſemidiameter: </s>
            <s xml:id="echoid-s10436" xml:space="preserve">alterius autem baſis ſit {1/2}. </s>
            <s xml:id="echoid-s10437" xml:space="preserve">ſuperfi-
              <lb/>
            ciei, & </s>
            <s xml:id="echoid-s10438" xml:space="preserve">altitudo {1/3}. </s>
            <s xml:id="echoid-s10439" xml:space="preserve">diametri. </s>
            <s xml:id="echoid-s10440" xml:space="preserve">Quia verò baſes, & </s>
            <s xml:id="echoid-s10441" xml:space="preserve">altitudines recipro cantur, quod
              <lb/>
            ita ſit {1/3}. </s>
            <s xml:id="echoid-s10442" xml:space="preserve">ſuperficiei ad {1/2}. </s>
            <s xml:id="echoid-s10443" xml:space="preserve">ſuperficiei, nimirũ baſis prioris parallelepipedi ad ba-
              <lb/>
            ſem poſterioris, vt {1/3}. </s>
            <s xml:id="echoid-s10444" xml:space="preserve">diametriad {1/2}. </s>
            <s xml:id="echoid-s10445" xml:space="preserve">diametri, altitudo videlicet poſterioris pa-
              <lb/>
            rallelepipedi ad altitudinem prioris; </s>
            <s xml:id="echoid-s10446" xml:space="preserve"> æqualia eruntipſa parallelepipeda. </s>
            <s xml:id="echoid-s10447" xml:space="preserve">
              <note symbol="d" position="left" xlink:label="note-256-07" xlink:href="note-256-07a" xml:space="preserve">34. vndec.</note>
            ergo, per 1. </s>
            <s xml:id="echoid-s10448" xml:space="preserve">partem huius 2. </s>
            <s xml:id="echoid-s10449" xml:space="preserve">regulæ, prius ſit ſphæræ æquale, eidem quo que po-
              <lb/>
            ſterius æquale erit: </s>
            <s xml:id="echoid-s10450" xml:space="preserve">Ac propterea ſphæræ ſoliditas pro ducetur ex tertia part@
              <lb/>
            diametri in ſemiſſem conuexæ ſuperficiei. </s>
            <s xml:id="echoid-s10451" xml:space="preserve">quod eſt o ctauum.</s>
            <s xml:id="echoid-s10452" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10453" xml:space="preserve">3. </s>
            <s xml:id="echoid-s10454" xml:space="preserve">
              <emph style="sc">Iam</emph>
            vero ex propoſ. </s>
            <s xml:id="echoid-s10455" xml:space="preserve">4. </s>
            <s xml:id="echoid-s10456" xml:space="preserve">& </s>
            <s xml:id="echoid-s10457" xml:space="preserve">5. </s>
            <s xml:id="echoid-s10458" xml:space="preserve">huius cap. </s>
            <s xml:id="echoid-s10459" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s10460" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10461" xml:space="preserve">colliguntur quatuor ſe-
              <lb/>
            quentes regulæ, per quas ſuperficies ſphæræ conuexa inuenitur tum maior
              <lb/>
            quam vera, tum minor, tam ex circumferentia, quam ex diametro circuli ma-
              <lb/>
            ximi.</s>
            <s xml:id="echoid-s10462" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div643" type="section" level="1" n="224">
          <head xml:id="echoid-head241" xml:space="preserve">I.</head>
          <p>
            <s xml:id="echoid-s10463" xml:space="preserve">EX circumferentia circuli in ſphæra maximi ſuperficiem conuexam
              <lb/>
            ſphęrę procreare vera maiorem.</s>
            <s xml:id="echoid-s10464" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10465" xml:space="preserve">
              <emph style="sc">Fiat</emph>
            vt 223. </s>
            <s xml:id="echoid-s10466" xml:space="preserve">ad 71. </s>
            <s xml:id="echoid-s10467" xml:space="preserve">ita quadratum ex circumferentia maximi circuli data de-
              <lb/>
              <note position="left" xlink:label="note-256-08" xlink:href="note-256-08a" xml:space="preserve">Superfici{es}
                <lb/>
              ſphæræmaior,
                <lb/>
              quam vera.</note>
            ſcriptum ad aliud, pro dibitque ſphæræ ſuperficies maior quam vera. </s>
            <s xml:id="echoid-s10468" xml:space="preserve">Cum e-
              <lb/>
            nim per propoſ. </s>
            <s xml:id="echoid-s10469" xml:space="preserve">4. </s>
            <s xml:id="echoid-s10470" xml:space="preserve">huius cap. </s>
            <s xml:id="echoid-s10471" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s10472" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10473" xml:space="preserve">maior ſit proportio qua drati circumfe-
              <lb/>
            rentiæ circuli maximi ad ſuperficiem ſphæræ, quam 223. </s>
            <s xml:id="echoid-s10474" xml:space="preserve">ad 71. </s>
            <s xml:id="echoid-s10475" xml:space="preserve">ſit autem qua-
              <lb/>
            dratum datæ circumferentiæ ad numerum procreatum, vt 223. </s>
            <s xml:id="echoid-s10476" xml:space="preserve">ad 71. </s>
            <s xml:id="echoid-s10477" xml:space="preserve">habebit
              <lb/>
            quo que quadratum circumferentiæ datæ ad ſuperficiem ſphæræ ver@m, </s>
          </p>
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