Clavius, Christoph, Geometria practica

Table of contents

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[171.] DE AREA RECTANGVLORVM Capvt I.
[172.] DE AREA TRIANGVLORVM Capvt II.
[173.] DE AREA QVADRILATERORVM non rectangulorum. Capvt III.
[174.] DE AREA MVLTIL ATERARVM figurarum irregularium. Capvt IV.
[175.] DE AREA MVLTILATERA-rum figurarum regularium. Capvt V.
[176.] De dimenſione circuli ex Archimede. Capvt VI.
[177.] PROPOSITIO I.
[178.] SCHOLIVM.
[179.] PROPOSITIO II.
[180.] COROLLARIVM.
[181.] PROPOSITIO III.
[182.] DE AREA CIRCVLI, INVENTIONE-que circumferentiæ ex diametro, & diametri ex circumfetentia. Capvt VII.
[184.] II.
[185.] III.
[186.] IIII.
[187.] PROPOSITIO I.
[188.] PROPOSITIO II.
[189.] PROPOSITIO III.
[190.] I. EX diametro aream circuli vera maiorem inueſtigare.
[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.
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          <pb o="193" file="223" n="223" rhead="LIBER QVARTVS."/>
          <p>
            <s xml:id="echoid-s8392" xml:space="preserve">
              <emph style="sc">Deniqve</emph>
            quia quatuor lineæ EI, EB, EL, EM, proportionales ſunt, quod
              <lb/>
              <note symbol="a" position="right" xlink:label="note-223-01" xlink:href="note-223-01a" xml:space="preserve">16. ſexti.</note>
            tam priores duę, quam poſteriores duę habeant proportionem duplam; </s>
            <s xml:id="echoid-s8393" xml:space="preserve"> erit rectangulum ſub EI, diametro, & </s>
            <s xml:id="echoid-s8394" xml:space="preserve">EM, quarta parte perip heriæ comprehen-
              <lb/>
            ſum, rectangulo ſub EB, ſemidiametro, & </s>
            <s xml:id="echoid-s8395" xml:space="preserve">EL, ſemiperipheria comprehenſo
              <lb/>
            æquale. </s>
            <s xml:id="echoid-s8396" xml:space="preserve">Cum ergo hoc in prima parte oſtenſum ſit æquale circulo, erit quo que
              <lb/>
            illud eidem circulo æquale. </s>
            <s xml:id="echoid-s8397" xml:space="preserve">quod eſt tertium.</s>
            <s xml:id="echoid-s8398" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8399" xml:space="preserve">
              <emph style="sc">Vt</emph>
            fra ctiones interdum vitentur, ducenda erit tota diameter in totam cir-
              <lb/>
              <note position="right" xlink:label="note-223-02" xlink:href="note-223-02a" xml:space="preserve">Area ſemi-
                <lb/>
              circuli, Qua-
                <lb/>
              drantis, octa-
                <lb/>
              uæpartis, & c.</note>
            cumferentiam. </s>
            <s xml:id="echoid-s8400" xml:space="preserve">Quarta enim pars numeri producti area erit circuli: </s>
            <s xml:id="echoid-s8401" xml:space="preserve">propterea
              <lb/>
            quod numerus productus quadruplus eſt numeri producti ex ſemidiametro in
              <lb/>
            ſemicircumferentiam, vt liquet.</s>
            <s xml:id="echoid-s8402" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8403" xml:space="preserve">
              <emph style="sc">Seqvitvr</emph>
            ex prima parte, aream ſemicirculi produci ex ſemidiametro in
              <lb/>
            quartam partem circumferentię: </s>
            <s xml:id="echoid-s8404" xml:space="preserve">quia nimirum producitur ſemiſsis eius, quod
              <lb/>
            fit ex ſemidiametro in ſemiſſem peripheriæ. </s>
            <s xml:id="echoid-s8405" xml:space="preserve">item aream Quadrantis procreari
              <lb/>
            ex ſemidiametro in octauam partem circumferentię: </s>
            <s xml:id="echoid-s8406" xml:space="preserve">Et aream octauę partis ex
              <lb/>
            ſemidiametro in ſextamdecimam partem circumferentię: </s>
            <s xml:id="echoid-s8407" xml:space="preserve">Et aream ſextę de-
              <lb/>
            cimę partis ex ſemidiametro in {1/32}. </s>
            <s xml:id="echoid-s8408" xml:space="preserve">circumferentię, & </s>
            <s xml:id="echoid-s8409" xml:space="preserve">ſic deinceps: </s>
            <s xml:id="echoid-s8410" xml:space="preserve">quia ſem-
              <lb/>
            per producitur ſemiſsis pręcedentis producti; </s>
            <s xml:id="echoid-s8411" xml:space="preserve">quemadmodum & </s>
            <s xml:id="echoid-s8412" xml:space="preserve">partes cir-
              <lb/>
            culi ſemiſſes ſunt pręcedentium partium: </s>
            <s xml:id="echoid-s8413" xml:space="preserve">nimirum Quadrans ſemiſsis eſt ſe-
              <lb/>
            micirculi; </s>
            <s xml:id="echoid-s8414" xml:space="preserve">& </s>
            <s xml:id="echoid-s8415" xml:space="preserve">octaua pars ſemiſsis Quadrantis; </s>
            <s xml:id="echoid-s8416" xml:space="preserve">& </s>
            <s xml:id="echoid-s8417" xml:space="preserve">ſextadecima pars ſemiſsis o-
              <lb/>
            ctauę partis, & </s>
            <s xml:id="echoid-s8418" xml:space="preserve">c.</s>
            <s xml:id="echoid-s8419" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8420" xml:space="preserve">1. </s>
            <s xml:id="echoid-s8421" xml:space="preserve">
              <emph style="sc">Igitvr</emph>
            vt area circuli reperiatur, neceſſe eſt tam eius diametrum, quam
              <lb/>
            circumferentiam eſſe cognitam. </s>
            <s xml:id="echoid-s8422" xml:space="preserve">Quare trademus hic regulas nonnullas, per
              <lb/>
            quas ex data diametro circumferentia tum maior, tum minor, quam vera, ex
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s8423" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8424" xml:space="preserve">de Dimenſione circuli inueniatur. </s>
            <s xml:id="echoid-s8425" xml:space="preserve">Deinde alias regulas pręſcri-
              <lb/>
            bam, per quas area circulitum maior, tum minor, quam vera, vel ex ſola diame-
              <lb/>
            tro, vel ex ſola circumferentia cognita eruatur.</s>
            <s xml:id="echoid-s8426" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div512" type="section" level="1" n="183">
          <head xml:id="echoid-head191" xml:space="preserve">I.</head>
          <p>
            <s xml:id="echoid-s8427" xml:space="preserve">Ex data diametro circuli circumferentiam vera maiorem reperire.</s>
            <s xml:id="echoid-s8428" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8429" xml:space="preserve">FIAT vt 7 ad 22 ita data diameter, verbigratia, 28. </s>
            <s xml:id="echoid-s8430" xml:space="preserve">ad aliud, procreabitur que cir-
              <lb/>
            cumferentia 88. </s>
            <s xml:id="echoid-s8431" xml:space="preserve">maior quam vera. </s>
            <s xml:id="echoid-s8432" xml:space="preserve"> Quoniam enim proportio circumferentiæ
              <note symbol="b" position="right" xlink:label="note-223-03" xlink:href="note-223-03a" xml:space="preserve">2. de Dimeſ.
                <lb/>
              circuli:</note>
            diametrum minor eſt, quam tripla ſeſquiſeptima; </s>
            <s xml:id="echoid-s8433" xml:space="preserve">proportio autem 88. </s>
            <s xml:id="echoid-s8434" xml:space="preserve">ad 28.
              <lb/>
            </s>
            <s xml:id="echoid-s8435" xml:space="preserve">eſt tripla ſeſquiſeptima, eadem videlicet, quæ 22. </s>
            <s xml:id="echoid-s8436" xml:space="preserve">ad 7. </s>
            <s xml:id="echoid-s8437" xml:space="preserve"> erit numerus 88. </s>
            <s xml:id="echoid-s8438" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-223-04" xlink:href="note-223-04a" xml:space="preserve">10. quinti.</note>
            ior, quam circumferentia circuli, cuius diameter eſt 28.</s>
            <s xml:id="echoid-s8439" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div514" type="section" level="1" n="184">
          <head xml:id="echoid-head192" xml:space="preserve">II.</head>
          <p>
            <s xml:id="echoid-s8440" xml:space="preserve">EX data diametro circuli circumferentiam vera minorem elicere.</s>
            <s xml:id="echoid-s8441" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8442" xml:space="preserve">FIAT vt 71. </s>
            <s xml:id="echoid-s8443" xml:space="preserve">ad 223. </s>
            <s xml:id="echoid-s8444" xml:space="preserve">ita data diameter 28. </s>
            <s xml:id="echoid-s8445" xml:space="preserve">ad aliud, produc{et}urque circumferentia
              <lb/>
            87 {67/71}. </s>
            <s xml:id="echoid-s8446" xml:space="preserve">minor quam vera. </s>
            <s xml:id="echoid-s8447" xml:space="preserve"> Quoniam enim proportio circumferentię ad
              <note symbol="d" position="right" xlink:label="note-223-05" xlink:href="note-223-05a" xml:space="preserve">2. de Dimeſ.
                <lb/>
              circuli.</note>
            trum maior eſt, quam tripla ſuperdecupartiens ſeptuageſimas primas; </s>
            <s xml:id="echoid-s8448" xml:space="preserve">propor-
              <lb/>
            tio autem 87 {67/71}. </s>
            <s xml:id="echoid-s8449" xml:space="preserve">ad 28. </s>
            <s xml:id="echoid-s8450" xml:space="preserve">eſt tripla ſuperdecupartiens ſeptuageſimas primas, ni-
              <lb/>
            mirum eadem, quæ 223. </s>
            <s xml:id="echoid-s8451" xml:space="preserve">ad 28. </s>
            <s xml:id="echoid-s8452" xml:space="preserve"> erit circumferentia circuli, cuius diameter 28.</s>
            <s xml:id="echoid-s8453" xml:space="preserve">
              <note symbol="e" position="right" xlink:label="note-223-06" xlink:href="note-223-06a" xml:space="preserve">10. quinti.</note>
            maior, quam 87 {@/71}.</s>
            <s xml:id="echoid-s8454" xml:space="preserve"/>
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