Clavius, Christoph, Geometria practica

Table of contents

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[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)
[241.] GEOMETRIÆ PRACTICÆ LIBER SEXTVS.
Page: 266 (236)
[242.] THOREMA 1. PROPOSITIO 1.
Page: 267 (237)
[243.] PROBLEMA 1. PROPOSITIO 2.
Page: 269 (239)
[244.] PROBL. 2. PROPOS. 3.
Page: 276 (246)
[245.] ALITER.
Page: 276 (246)
[246.] ALITER.
Page: 277 (247)
[247.] PROBL. 3. PROPOS. 4.
Page: 278 (248)
[248.] SCHOLIVM.
Page: 282 (252)
[249.] PROBLEMA 4. PROPOSITIO 5.
Page: 283 (253)
[250.] ALITER.
Page: 284 (254)
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          <p>
            <s xml:id="echoid-s10107" xml:space="preserve">
              <pb o="223" file="253" n="253" rhead="LIBER QVINTVS."/>
            iorem: </s>
            <s xml:id="echoid-s10108" xml:space="preserve">ac proinde ſi hęc area maior, quam vera, ducatur in {2/3}. </s>
            <s xml:id="echoid-s10109" xml:space="preserve">diametri, gigne-
              <lb/>
            tur, vt infra in regula 2. </s>
            <s xml:id="echoid-s10110" xml:space="preserve">dicetur, ſoliditas ſphærę {22/42}. </s>
            <s xml:id="echoid-s10111" xml:space="preserve">hoc eſt {11/21}. </s>
            <s xml:id="echoid-s10112" xml:space="preserve">maior quam ve-
              <lb/>
              <note symbol="a" position="right" xlink:label="note-253-01" xlink:href="note-253-01a" xml:space="preserve">8. quinti.</note>
            ra. </s>
            <s xml:id="echoid-s10113" xml:space="preserve"> Igitur cubus diametri 1. </s>
            <s xml:id="echoid-s10114" xml:space="preserve">qui eſt 1. </s>
            <s xml:id="echoid-s10115" xml:space="preserve">ad ſphæram, proportionem habebit maio- rem, quam ad {11/2@}. </s>
            <s xml:id="echoid-s10116" xml:space="preserve">Cum ergo ſit 1. </s>
            <s xml:id="echoid-s10117" xml:space="preserve">ad {11/21}. </s>
            <s xml:id="echoid-s10118" xml:space="preserve">vt 21. </s>
            <s xml:id="echoid-s10119" xml:space="preserve">ad 11. </s>
            <s xml:id="echoid-s10120" xml:space="preserve">(Quia enim ex propoſ. </s>
            <s xml:id="echoid-s10121" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10122" xml:space="preserve">Mi-
              <lb/>
            nutiarum ad finem lib. </s>
            <s xml:id="echoid-s10123" xml:space="preserve">9. </s>
            <s xml:id="echoid-s10124" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s10125" xml:space="preserve">eadem eſt proportio Numeratoris 11. </s>
            <s xml:id="echoid-s10126" xml:space="preserve">ad Deno-
              <lb/>
            minatorem 21. </s>
            <s xml:id="echoid-s10127" xml:space="preserve">quæ Minutiæ {11/21}. </s>
            <s xml:id="echoid-s10128" xml:space="preserve">ad ſuum integrum 1. </s>
            <s xml:id="echoid-s10129" xml:space="preserve">erit conuertendo vt 21. </s>
            <s xml:id="echoid-s10130" xml:space="preserve">ad
              <lb/>
            11. </s>
            <s xml:id="echoid-s10131" xml:space="preserve">ita 1. </s>
            <s xml:id="echoid-s10132" xml:space="preserve">ad {11/21}.) </s>
            <s xml:id="echoid-s10133" xml:space="preserve">maior erit proportio cubi 1. </s>
            <s xml:id="echoid-s10134" xml:space="preserve">ex diametro 1. </s>
            <s xml:id="echoid-s10135" xml:space="preserve">deſcripti ad ſphæram,
              <lb/>
            quam 21. </s>
            <s xml:id="echoid-s10136" xml:space="preserve">ad 11. </s>
            <s xml:id="echoid-s10137" xml:space="preserve">Et quia, vt initio huius propoſitionis oſtendimus, ita eſt cubus
              <lb/>
            diametri cuiuſuis alterius ſphæræ ad ipſam ſphærã, vt cubus diametri 1. </s>
            <s xml:id="echoid-s10138" xml:space="preserve">ad ſuam
              <lb/>
            ſphæram, verum eſt, quod primo loco eſt propoſitum.</s>
            <s xml:id="echoid-s10139" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10140" xml:space="preserve">
              <emph style="sc">Item</emph>
            ſi diameter 1. </s>
            <s xml:id="echoid-s10141" xml:space="preserve">ducatur in 3 {10/71}. </s>
            <s xml:id="echoid-s10142" xml:space="preserve">producetur circumferentia
              <note symbol="b" position="right" xlink:label="note-253-02" xlink:href="note-253-02a" xml:space="preserve">corol. 2. d@
                <lb/>
              Dimenſ. cir-
                <lb/>
              culi.</note>
            circuli {223/71}. </s>
            <s xml:id="echoid-s10143" xml:space="preserve">minor quam vera. </s>
            <s xml:id="echoid-s10144" xml:space="preserve">Eius ergo ſemiſsis {223/142}. </s>
            <s xml:id="echoid-s10145" xml:space="preserve">ducta in {1/2}. </s>
            <s xml:id="echoid-s10146" xml:space="preserve">ſemiſſem diame-
              <lb/>
            tri@. </s>
            <s xml:id="echoid-s10147" xml:space="preserve">faciet {223/284}. </s>
            <s xml:id="echoid-s10148" xml:space="preserve">aream circuli maximi vera minorem; </s>
            <s xml:id="echoid-s10149" xml:space="preserve">ideo que ſi ea ducatur in
              <lb/>
            {2/3}. </s>
            <s xml:id="echoid-s10150" xml:space="preserve">diametri 1. </s>
            <s xml:id="echoid-s10151" xml:space="preserve">pro creabitur, vt infra in regula 2. </s>
            <s xml:id="echoid-s10152" xml:space="preserve">dicetur, ſoliditas ſphæræ {449/852}. </s>
            <s xml:id="echoid-s10153" xml:space="preserve">hoc
              <lb/>
            eſt, {223/476}. </s>
            <s xml:id="echoid-s10154" xml:space="preserve">minor, quam vera. </s>
            <s xml:id="echoid-s10155" xml:space="preserve"> Igitur cubus 1 diametri 1. </s>
            <s xml:id="echoid-s10156" xml:space="preserve">ad ſphæram habebit pro- portionem minorem, quam ad {223/426}. </s>
            <s xml:id="echoid-s10157" xml:space="preserve">Cum ergo ſit 1. </s>
            <s xml:id="echoid-s10158" xml:space="preserve">ad {223/426}. </s>
            <s xml:id="echoid-s10159" xml:space="preserve">vt 426. </s>
            <s xml:id="echoid-s10160" xml:space="preserve">ad 223.
              <lb/>
            </s>
            <s xml:id="echoid-s10161" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-253-03" xlink:href="note-253-03a" xml:space="preserve">8. quinti.</note>
            (Quia enim ex propoſ. </s>
            <s xml:id="echoid-s10162" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10163" xml:space="preserve">Minutiarum ad ſinem lib. </s>
            <s xml:id="echoid-s10164" xml:space="preserve">9. </s>
            <s xml:id="echoid-s10165" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s10166" xml:space="preserve">eadem proportio
              <lb/>
            eſt Numeratoris 223. </s>
            <s xml:id="echoid-s10167" xml:space="preserve">ad denominatorem 426. </s>
            <s xml:id="echoid-s10168" xml:space="preserve">quæ Minutiæ {223/426}. </s>
            <s xml:id="echoid-s10169" xml:space="preserve">ad ſuum
              <lb/>
            integrum 1. </s>
            <s xml:id="echoid-s10170" xml:space="preserve">erit conuertendo, vt 426. </s>
            <s xml:id="echoid-s10171" xml:space="preserve">ad 223. </s>
            <s xml:id="echoid-s10172" xml:space="preserve">ita 1. </s>
            <s xml:id="echoid-s10173" xml:space="preserve">ad {223/4@6}.) </s>
            <s xml:id="echoid-s10174" xml:space="preserve">minor erit pro-
              <lb/>
            portio cubi diametri 1. </s>
            <s xml:id="echoid-s10175" xml:space="preserve">ad ſuam ſphæram, quam 426. </s>
            <s xml:id="echoid-s10176" xml:space="preserve">ad 223. </s>
            <s xml:id="echoid-s10177" xml:space="preserve">Quoniam ve-
              <lb/>
            rò, vt ad initium huius propoſitionis oſtendimus, ita eſt cubus diametri cuiusli-
              <lb/>
            bet ſphæræ alterius ad ipſam ſphæram, vt cubus diametr 1. </s>
            <s xml:id="echoid-s10178" xml:space="preserve">ad ſuam ſphæram, li-
              <lb/>
            quet etiam id, quod ſecundo loco propoſitum eſt.</s>
            <s xml:id="echoid-s10179" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10180" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10181" xml:space="preserve">
              <emph style="sc">His</emph>
            præmiſsis, ſequuntur regulę ad inueſtigandam tam ſuperficiem
              <lb/>
            conuexam cuiuslibet ſphæræ, quam eiuſdem ſoliditatem.</s>
            <s xml:id="echoid-s10182" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div625" type="section" level="1" n="220">
          <head xml:id="echoid-head237" xml:space="preserve">I.</head>
          <p>
            <s xml:id="echoid-s10183" xml:space="preserve">SVPERFICIEM conuexam propoſitæ ſphæræ adinuenire.</s>
            <s xml:id="echoid-s10184" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10185" xml:space="preserve">
              <emph style="sc">Area</emph>
            maximi circuli datæ ſphærę quadruplicetur. </s>
            <s xml:id="echoid-s10186" xml:space="preserve">Productus enim nu-
              <lb/>
              <note position="right" xlink:label="note-253-04" xlink:href="note-253-04a" xml:space="preserve">Superfici@s
                <lb/>
              conuexa ſpa-
                <lb/>
              ræ.</note>
            merus conuexam ſphærę ſuperficiem exhibebit: </s>
            <s xml:id="echoid-s10187" xml:space="preserve">propterea quod per propoſ.
              <lb/>
            </s>
            <s xml:id="echoid-s10188" xml:space="preserve">31. </s>
            <s xml:id="echoid-s10189" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s10190" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10191" xml:space="preserve">Archimedis de ſphæra, & </s>
            <s xml:id="echoid-s10192" xml:space="preserve">Cylindro, ſuperficies ſphærę quadrupla eſt
              <lb/>
            circuli maximi.</s>
            <s xml:id="echoid-s10193" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10194" xml:space="preserve">
              <emph style="sc">Eadem</emph>
            ſuperficies procreabitur, ſi diameter ſphęrę in circumferentiam
              <lb/>
            circuli maximi ducatur: </s>
            <s xml:id="echoid-s10195" xml:space="preserve">propterea quod per propoſ. </s>
            <s xml:id="echoid-s10196" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10197" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s10198" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10199" xml:space="preserve">huius cap. </s>
            <s xml:id="echoid-s10200" xml:space="preserve">re-
              <lb/>
            ctangulum ſub diametro, & </s>
            <s xml:id="echoid-s10201" xml:space="preserve">circumferentia maximi circuli comprehenſum ſu-
              <lb/>
            perficiei conuexę ſphęrę eſt ęquale.</s>
            <s xml:id="echoid-s10202" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div627" type="section" level="1" n="221">
          <head xml:id="echoid-head238" xml:space="preserve">II.</head>
          <p>
            <s xml:id="echoid-s10203" xml:space="preserve">SOLIDITATEM propoſitæ ſphæræ exquirere.</s>
            <s xml:id="echoid-s10204" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s10205" xml:space="preserve">1. </s>
            <s xml:id="echoid-s10206" xml:space="preserve">SPH AER AE ſolidit{as} producitur ex ei{us} ſemidiametro in tertiam partem
              <lb/>
              <note position="right" xlink:label="note-253-05" xlink:href="note-253-05a" xml:space="preserve">Solidit{as}
                <unsure/>
                <lb/>
              ſphæræ.</note>
            ſuperficiei conuex@. </s>
            <s xml:id="echoid-s10207" xml:space="preserve">Velex {@/4} toti{us} diametri in {2/3}. </s>
            <s xml:id="echoid-s10208" xml:space="preserve">couuexæ ſuperficiei.</s>
            <s xml:id="echoid-s10209" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s10210" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10211" xml:space="preserve">ITEM ex duab{us} tertiis par@ib{us} diametri in aream circuli maximi.</s>
            <s xml:id="echoid-s10212" xml:space="preserve"/>
          </p>
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