Clavius, Christoph, Geometria practica

Table of contents

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[191.] II. EX diametro aream circuli vera minorem inueſtigare.
[192.] III. EX circumferentia aream circuli vera maiorem colligere.
[193.] IV. EX circumferentia aream circuli vera minorem concludere.
[194.] DE AREA SEGMENTORVM CIRCVLI. Capvt VIII.
[196.] II.
[197.] III.
[198.] IV.
[200.] VI.
[201.] FINIS LIBRI QVARTI.
[202.] GEOMETRIÆ PRACTICÆ LIBER QVINTVS.
[203.] AREAS Solidorum, corporumue perſcrutans.
[204.] DE AREA PARALLELEPIP EDO-rum, Priſmatum, & Cylindrorum. Capvt I.
[205.] DE AREA PYRAMIDVM & Conorum. Capvt II.
[206.] DL AREA FRVSTI PYRA-midis, & Coni. Capvt III.
[207.] SCHOLIVM.
[208.] DE AREA QVINQVE COR-porum regularium. Capvt IV.
[209.] Capvt V.
[210.] PROPOSITIO I.
[211.] COROLLARIVM.
[212.] PROPOSITIO II.
[213.] COROLLARIVM.
[214.] PROPOSITIO III.
[215.] COROLLARIVM.
[216.] PROPOSITIO IV.
[217.] PROPOSITIO V.
[218.] PROPOSITIO VI.
[219.] PROPOSITIO VII.
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        <div xml:id="echoid-div617" type="section" level="1" n="218">
          <pb o="222" file="252" n="252" rhead="GEOMETR. PRACT."/>
          <p>
            <s xml:id="echoid-s10001" xml:space="preserve">
              <emph style="sc">Iam</emph>
            verè ſi circumferentia circuli maximi alicuius ſphærę ſit 1. </s>
            <s xml:id="echoid-s10002" xml:space="preserve">diuidatur que
              <lb/>
            per 3 {10/71}. </s>
            <s xml:id="echoid-s10003" xml:space="preserve">producetur diameter {71/223}. </s>
            <s xml:id="echoid-s10004" xml:space="preserve">maior quam vera, ex coroll. </s>
            <s xml:id="echoid-s10005" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s10006" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10007" xml:space="preserve">de
              <lb/>
            dimenſione circuli. </s>
            <s xml:id="echoid-s10008" xml:space="preserve">Si igitur eius ſemiſsis {71/446}. </s>
            <s xml:id="echoid-s10009" xml:space="preserve">in {1/2}. </s>
            <s xml:id="echoid-s10010" xml:space="preserve">ſemiſſem circumferentiæ
              <lb/>
            ducatur, procreabitur area circuli maximi {71/892}. </s>
            <s xml:id="echoid-s10011" xml:space="preserve">maior, quẽ vera: </s>
            <s xml:id="echoid-s10012" xml:space="preserve">quæ rurſus du-
              <lb/>
            cta in {2/3}. </s>
            <s xml:id="echoid-s10013" xml:space="preserve">diametri inuentę, (quæ etiam maior eſt, quam vera) nimirum in
              <lb/>
            {142/669}. </s>
            <s xml:id="echoid-s10014" xml:space="preserve">gignet, vtinſra in regula 2. </s>
            <s xml:id="echoid-s10015" xml:space="preserve">oſtendam, ſoliditatem ſphæræ {10082/596748}. </s>
            <s xml:id="echoid-s10016" xml:space="preserve">hoc eſt
              <lb/>
            {5041/298374}. </s>
            <s xml:id="echoid-s10017" xml:space="preserve">maiorem tamen, quam veram. </s>
            <s xml:id="echoid-s10018" xml:space="preserve"> Maior ergo erit proportio cubi
              <note symbol="a" position="left" xlink:label="note-252-01" xlink:href="note-252-01a" xml:space="preserve">8. quinti.</note>
            circumferentia 1. </s>
            <s xml:id="echoid-s10019" xml:space="preserve">deſcripti, qui cubus eſt 1. </s>
            <s xml:id="echoid-s10020" xml:space="preserve">ad ſphęram, quam ad {5041/298374}. </s>
            <s xml:id="echoid-s10021" xml:space="preserve">Cum
              <lb/>
            ergo ſit 1. </s>
            <s xml:id="echoid-s10022" xml:space="preserve">ad {5041/298374}. </s>
            <s xml:id="echoid-s10023" xml:space="preserve">vt 298374. </s>
            <s xml:id="echoid-s10024" xml:space="preserve">ad 5041. </s>
            <s xml:id="echoid-s10025" xml:space="preserve">(Quoniam enim ex propoſitione
              <lb/>
            2. </s>
            <s xml:id="echoid-s10026" xml:space="preserve">Minutiarum ad finem libr. </s>
            <s xml:id="echoid-s10027" xml:space="preserve">9. </s>
            <s xml:id="echoid-s10028" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s10029" xml:space="preserve">eadem eſt proportio Numeratoris
              <lb/>
            5041. </s>
            <s xml:id="echoid-s10030" xml:space="preserve">ad denominatorem 298374. </s>
            <s xml:id="echoid-s10031" xml:space="preserve">quæ minutię {5041/298374}. </s>
            <s xml:id="echoid-s10032" xml:space="preserve">ad ſuum integrum 1.
              <lb/>
            </s>
            <s xml:id="echoid-s10033" xml:space="preserve">erit conuertendo, vt 298374. </s>
            <s xml:id="echoid-s10034" xml:space="preserve">ad 5041. </s>
            <s xml:id="echoid-s10035" xml:space="preserve">ita 1. </s>
            <s xml:id="echoid-s10036" xml:space="preserve">ad {5042/298374}.) </s>
            <s xml:id="echoid-s10037" xml:space="preserve">maior erit proportio
              <lb/>
            cubi ex circumferentia 1. </s>
            <s xml:id="echoid-s10038" xml:space="preserve">deſcripti, qui cubus eſt 1. </s>
            <s xml:id="echoid-s10039" xml:space="preserve">ad ſphæram, qaam 298374. </s>
            <s xml:id="echoid-s10040" xml:space="preserve">
              <lb/>
            ad 5041. </s>
            <s xml:id="echoid-s10041" xml:space="preserve">Cum igitur ita ſe habeat cubus ex circumferentia maximi circuli
              <lb/>
            in ſphæra ad ſphæram qualibet deſcriptus, vt cubus circumferentiæ 1. </s>
            <s xml:id="echoid-s10042" xml:space="preserve">ad ſuam
              <lb/>
            ſphæram, vtinitio huius propoſ. </s>
            <s xml:id="echoid-s10043" xml:space="preserve">oſtendimus; </s>
            <s xml:id="echoid-s10044" xml:space="preserve">Conſtatid, quod primo loco
              <lb/>
            proponitur.</s>
            <s xml:id="echoid-s10045" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10046" xml:space="preserve">
              <emph style="sc">Rvrsvs</emph>
            ſi circumferentia 1. </s>
            <s xml:id="echoid-s10047" xml:space="preserve">diuidatur per 3 {1/7}. </s>
            <s xml:id="echoid-s10048" xml:space="preserve">producetur diameter {7/22}.
              <lb/>
            </s>
            <s xml:id="echoid-s10049" xml:space="preserve">minor quam vera, ex coroll. </s>
            <s xml:id="echoid-s10050" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s10051" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10052" xml:space="preserve">de dimenſione circuli. </s>
            <s xml:id="echoid-s10053" xml:space="preserve">Si igitur eius ſe-
              <lb/>
            miſsis {7/44}. </s>
            <s xml:id="echoid-s10054" xml:space="preserve">ducatur in {1/2}. </s>
            <s xml:id="echoid-s10055" xml:space="preserve">ſemiſſem circumferentiæ, procreabitur area circuli ma-
              <lb/>
            ximi {7/88}: </s>
            <s xml:id="echoid-s10056" xml:space="preserve">minor, quam vera: </s>
            <s xml:id="echoid-s10057" xml:space="preserve">quę rurſus ducta in {2/3}. </s>
            <s xml:id="echoid-s10058" xml:space="preserve">diametri inuentę (quæ etiam
              <lb/>
            minor eſt, quam vera) nimirum in {14/66}. </s>
            <s xml:id="echoid-s10059" xml:space="preserve">hoc eſt in {7/33}. </s>
            <s xml:id="echoid-s10060" xml:space="preserve">producet, vt infra in regula
              <lb/>
            2. </s>
            <s xml:id="echoid-s10061" xml:space="preserve">docebo, ſoliditatem ſphærę {49/2904}. </s>
            <s xml:id="echoid-s10062" xml:space="preserve">minoremtamen, quam veram. </s>
            <s xml:id="echoid-s10063" xml:space="preserve"> Minor
              <note symbol="b" position="left" xlink:label="note-252-02" xlink:href="note-252-02a" xml:space="preserve">@. quinti.</note>
            go erit proportio cubi ex circumferentia 1. </s>
            <s xml:id="echoid-s10064" xml:space="preserve">deſcripti, qui cubus eſt 1. </s>
            <s xml:id="echoid-s10065" xml:space="preserve">ad ſphæ-
              <lb/>
            ram, quam ad {49/2904}. </s>
            <s xml:id="echoid-s10066" xml:space="preserve">Cum ergo ſit 1. </s>
            <s xml:id="echoid-s10067" xml:space="preserve">ad {49/2904}. </s>
            <s xml:id="echoid-s10068" xml:space="preserve">vt 2904. </s>
            <s xml:id="echoid-s10069" xml:space="preserve">ad 49. </s>
            <s xml:id="echoid-s10070" xml:space="preserve">(Quoniam enim ex
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s10071" xml:space="preserve">2. </s>
            <s xml:id="echoid-s10072" xml:space="preserve">Minutiarum ad finem libr. </s>
            <s xml:id="echoid-s10073" xml:space="preserve">9. </s>
            <s xml:id="echoid-s10074" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s10075" xml:space="preserve">eadem proportio eſt Numerato-
              <lb/>
            ris 49. </s>
            <s xml:id="echoid-s10076" xml:space="preserve">ad denominatorem 2904. </s>
            <s xml:id="echoid-s10077" xml:space="preserve">quæ minutię {49/2904}. </s>
            <s xml:id="echoid-s10078" xml:space="preserve">ad ſuum integrum 1. </s>
            <s xml:id="echoid-s10079" xml:space="preserve">erit
              <lb/>
            conuertendo, vt 2904. </s>
            <s xml:id="echoid-s10080" xml:space="preserve">ad 49. </s>
            <s xml:id="echoid-s10081" xml:space="preserve">ita 1. </s>
            <s xml:id="echoid-s10082" xml:space="preserve">ad {49/2904}.) </s>
            <s xml:id="echoid-s10083" xml:space="preserve">minor erit proportio cubi deſcripti
              <lb/>
            ex circumferentia 1. </s>
            <s xml:id="echoid-s10084" xml:space="preserve">qui cubus eſt 1. </s>
            <s xml:id="echoid-s10085" xml:space="preserve">ad ſphæram, quam 2904. </s>
            <s xml:id="echoid-s10086" xml:space="preserve">ad 49. </s>
            <s xml:id="echoid-s10087" xml:space="preserve">Cum igi-
              <lb/>
            tur ita ſe habeat cubus ex circumferentia maximi circuli ſphærę cuiuslibet de-
              <lb/>
            ſcriptus, ad ſphæram, vt cubus circumferentiæ 1. </s>
            <s xml:id="echoid-s10088" xml:space="preserve">ad ſuam ſphęram, vtinitio hu-
              <lb/>
            ius propoſitionis demonſtrauimus, patet etiam id, quod ſecundo loco propo-
              <lb/>
            ſitum erat.</s>
            <s xml:id="echoid-s10089" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div621" type="section" level="1" n="219">
          <head xml:id="echoid-head236" xml:space="preserve">PROPOSITIO VII.</head>
          <p>
            <s xml:id="echoid-s10090" xml:space="preserve">CVBVS diametri ſphæræ ad ſphæram, maiorem proportionem habet,
              <lb/>
            quam 21. </s>
            <s xml:id="echoid-s10091" xml:space="preserve">ad 11. </s>
            <s xml:id="echoid-s10092" xml:space="preserve">minorem verò, quam 426. </s>
            <s xml:id="echoid-s10093" xml:space="preserve">ad 223.</s>
            <s xml:id="echoid-s10094" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10095" xml:space="preserve">
              <emph style="sc">Cvm</emph>
            enim ſit, vt cubus diametri cuiuslibet ſphærę ad cubum diametri al-
              <lb/>
            terius ſphærę, ita ſphæra ad ſphęram ; </s>
            <s xml:id="echoid-s10096" xml:space="preserve">quod vtra que proportio ſit trip
              <note symbol="c" position="left" xlink:label="note-252-03" xlink:href="note-252-03a" xml:space="preserve">33. vndec. &
                <lb/>
              18. duodec.</note>
            proportionis diametrorum: </s>
            <s xml:id="echoid-s10097" xml:space="preserve">erit permutando, vt cubus diametri cuiuslibet
              <lb/>
            ſphærę ad ipſam ſphęram, ita cubus diametri alterius ſphærę ad ipſam ſphęram.</s>
            <s xml:id="echoid-s10098" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10099" xml:space="preserve">
              <emph style="sc">Qvod</emph>
            ſi diameter alicuius ſphærę ponatur 1. </s>
            <s xml:id="echoid-s10100" xml:space="preserve">multipliceturq; </s>
            <s xml:id="echoid-s10101" xml:space="preserve">per 3 {1/7}. </s>
            <s xml:id="echoid-s10102" xml:space="preserve">
              <note symbol="d" position="left" xlink:label="note-252-04" xlink:href="note-252-04a" xml:space="preserve">corol 2. de
                <lb/>
              Dimenſ. cir-
                <lb/>
              culi.</note>
            ueniet circuli maximi circumferentia {@@/7}. </s>
            <s xml:id="echoid-s10103" xml:space="preserve">maior quam vera. </s>
            <s xml:id="echoid-s10104" xml:space="preserve">Eius ergo ſemiſ@is
              <lb/>
            {11/7}. </s>
            <s xml:id="echoid-s10105" xml:space="preserve">in {1/2}. </s>
            <s xml:id="echoid-s10106" xml:space="preserve">ſemiſſem diametri ducta efficiet {11/@@}. </s>
            <s xml:id="echoid-s10107" xml:space="preserve">aream ipſius maximi circuli vera </s>
          </p>
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