Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div536" type="section" level="1" n="193">
          <pb o="199" file="229" n="229" rhead="LIBER QVARTVS."/>
          <p>
            <s xml:id="echoid-s8795" xml:space="preserve">
              <emph style="sc">Inventa</emph>
            circumferentia ex diametro, vel diametro ex circumferentia, re-
              <lb/>
            perietur area circuli, vt Num. </s>
            <s xml:id="echoid-s8796" xml:space="preserve">1. </s>
            <s xml:id="echoid-s8797" xml:space="preserve">tra ditum eſt: </s>
            <s xml:id="echoid-s8798" xml:space="preserve">ſi nimirum ſemidiameter in ſemicir-
              <lb/>
            cumferentiam ducatur: </s>
            <s xml:id="echoid-s8799" xml:space="preserve">vel tota circumferentia in ſemiſſem ſemidiametri: </s>
            <s xml:id="echoid-s8800" xml:space="preserve">vel
              <lb/>
            deniq; </s>
            <s xml:id="echoid-s8801" xml:space="preserve">tota diameter in quartam partem circumferentiæ. </s>
            <s xml:id="echoid-s8802" xml:space="preserve">quæ quidem area mi-
              <lb/>
            nus à vera diſtabit, quam illa, quæ ex proportione Archimedis inuenitur. </s>
            <s xml:id="echoid-s8803" xml:space="preserve">Sed
              <lb/>
            quia diffi cilius eſt per magnos numeros calculum inſtituere, quam per minores,
              <lb/>
            vſus artificum obtinuit, vt proportio Archimedis ad calculum ad hibeatur. </s>
            <s xml:id="echoid-s8804" xml:space="preserve">Quã-
              <lb/>
            do tamẽ deſideratur accuratior calculus, vtendum erit poſteriori hac propor-
              <lb/>
            tione Ludolphi, præſertimin maioribus circulis.</s>
            <s xml:id="echoid-s8805" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div540" type="section" level="1" n="194">
          <head xml:id="echoid-head206" xml:space="preserve">DE AREA SEGMENTORVM CIRCVLI.</head>
          <head xml:id="echoid-head207" xml:space="preserve">
            <emph style="sc">Capvt</emph>
          VIII.</head>
          <handwritten number="68"/>
          <p>
            <s xml:id="echoid-s8806" xml:space="preserve">1. </s>
            <s xml:id="echoid-s8807" xml:space="preserve">
              <emph style="sc">Sit</emph>
            primum propoſitus ſector circuli ABCD, comprehenſus duabus ſe-
              <lb/>
            midiametris AB, AD, &</s>
            <s xml:id="echoid-s8808" xml:space="preserve">arcu BCD. </s>
            <s xml:id="echoid-s8809" xml:space="preserve">Huius aream ita explorabimus. </s>
            <s xml:id="echoid-s8810" xml:space="preserve">Si tam
              <lb/>
            ſemidiameter AB, nota ſit, nimirum palmorum 7. </s>
            <s xml:id="echoid-s8811" xml:space="preserve">quam arcus B C D, palmorum
              <lb/>
            videlicet 3 {2/3}. </s>
            <s xml:id="echoid-s8812" xml:space="preserve">ducatur ſemidiameter 7. </s>
            <s xml:id="echoid-s8813" xml:space="preserve">in {11/6}. </s>
            <s xml:id="echoid-s8814" xml:space="preserve">id eſt, in ſemiſſem arcus, Produ-
              <lb/>
            ctus enim numerus 12 {5/6}. </s>
            <s xml:id="echoid-s8815" xml:space="preserve">palm. </s>
            <s xml:id="echoid-s8816" xml:space="preserve">erit area ſectoris ABCD, vt demonſtrabimus. </s>
            <s xml:id="echoid-s8817" xml:space="preserve">Si
              <lb/>
            autem neque ſemidiameter AB, ne que perip heria BCD, data ſit, menſuranda erit
              <lb/>
            ſemidiameter aliqua menſura nota, & </s>
            <s xml:id="echoid-s8818" xml:space="preserve">ſecundum
              <lb/>
              <figure xlink:label="fig-229-01" xlink:href="fig-229-01a" number="144">
                <image file="229-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/229-01"/>
              </figure>
            eandem menſuram inuenienda circumſerẽtia cir-
              <lb/>
            culi per regulas antecedentis capit. </s>
            <s xml:id="echoid-s8819" xml:space="preserve">necnonrecta
              <lb/>
            BD. </s>
            <s xml:id="echoid-s8820" xml:space="preserve">Deinde ſiat, vt AB, nota in aſſumpta menſu-
              <lb/>
            ra ad ſinum totum 100000. </s>
            <s xml:id="echoid-s8821" xml:space="preserve">ita BD, nota in eadem
              <lb/>
            menſura aſſumpta ad aliud. </s>
            <s xml:id="echoid-s8822" xml:space="preserve">Numerus enim pro-
              <lb/>
            creatus dabit rectam B D, cognitam in partibus ſi-
              <lb/>
            nus totius. </s>
            <s xml:id="echoid-s8823" xml:space="preserve">Huius autem medietas ſinus erit ſemiſsis arcus B D: </s>
            <s xml:id="echoid-s8824" xml:space="preserve">ac proinde ex
              <lb/>
            tabula ſinuũ ſemiſsis BC, in gradib. </s>
            <s xml:id="echoid-s8825" xml:space="preserve">nota erit, ideoq; </s>
            <s xml:id="echoid-s8826" xml:space="preserve">totus arcus BD, nõ ignora-
              <lb/>
            bitur. </s>
            <s xml:id="echoid-s8827" xml:space="preserve">Et quiatota circuli circumferentia nota facta eſt in aſſumpta menſura: </s>
            <s xml:id="echoid-s8828" xml:space="preserve">ſi
              <lb/>
            fiat vt grad. </s>
            <s xml:id="echoid-s8829" xml:space="preserve">360. </s>
            <s xml:id="echoid-s8830" xml:space="preserve">ad totam circumferentia in aſſumpta menſura cognitam, ita ar-
              <lb/>
            cus BD, in gradibus cognitus ad aliud, cognoſcetur idem arcus B D, in menſura
              <lb/>
            aſſumpta. </s>
            <s xml:id="echoid-s8831" xml:space="preserve">Quare, vt prius, area ſectoris A B C D, reperietur. </s>
            <s xml:id="echoid-s8832" xml:space="preserve">Poſlent quoque
              <lb/>
            gradus in arcu BD, contenti inueſtigari beneficio quadrantis alicuius in gradus
              <lb/>
            diuiſi, adhibita doctrina cap. </s>
            <s xml:id="echoid-s8833" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8834" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s8835" xml:space="preserve">1. </s>
            <s xml:id="echoid-s8836" xml:space="preserve">Nume. </s>
            <s xml:id="echoid-s8837" xml:space="preserve">10. </s>
            <s xml:id="echoid-s8838" xml:space="preserve">tradita, vt minuta etiam cogno-
              <lb/>
            ſcantur, quando in arcu BD, vltra gradus aliqua particula ſupereſt.</s>
            <s xml:id="echoid-s8839" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8840" xml:space="preserve">
              <emph style="sc">Aream</emph>
            porro ſectoris produci ex ſemidiametro in ſemiſſem arcus ſectoris,
              <lb/>
              <handwritten xlink:label="hd-229-2" xlink:href="hd-229-2a" number="69"/>
            ſic demonſtro. </s>
            <s xml:id="echoid-s8841" xml:space="preserve">Sit quadrans B E, & </s>
            <s xml:id="echoid-s8842" xml:space="preserve">ſemicirculus BEF. </s>
            <s xml:id="echoid-s8843" xml:space="preserve"> Et quoniam eſt, vt
              <note symbol="a" position="right" xlink:label="note-229-01" xlink:href="note-229-01a" xml:space="preserve">33. ſexti.</note>
            cus B D, ad quadrantem BE, ita ſector ABCD, ad ſectorem ABDE: </s>
            <s xml:id="echoid-s8844" xml:space="preserve">erit quo que
              <lb/>
            ex ſcholio propoſ. </s>
            <s xml:id="echoid-s8845" xml:space="preserve">22. </s>
            <s xml:id="echoid-s8846" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s8847" xml:space="preserve">5. </s>
            <s xml:id="echoid-s8848" xml:space="preserve">Eucli. </s>
            <s xml:id="echoid-s8849" xml:space="preserve">vt arcus BD, ad quadruplum quadrantis BE,
              <lb/>
            hoc eſt, ad totam circumferentiam, ita ſector A B C D, ad quadruplum ſectoris
              <lb/>
            A B D E, hoc eſt, ad totum circulum. </s>
            <s xml:id="echoid-s8850" xml:space="preserve"> Vt autem arcus BD, ad totam
              <note symbol="b" position="right" xlink:label="note-229-02" xlink:href="note-229-02a" xml:space="preserve">15. quinti.</note>
            ferentiam, ita eſt BC, ſemiſsis arcus BD, ad BEF, ſemiſſem totius circumferentiæ.
              <lb/>
            </s>
            <s xml:id="echoid-s8851" xml:space="preserve">Igitur erit quo que vt B C, ad B E F, ita ſector A B C D, ad totum circulum. </s>
            <s xml:id="echoid-s8852" xml:space="preserve">
              <note symbol="c" position="right" xlink:label="note-229-03" xlink:href="note-229-03a" xml:space="preserve">1. ſexti.</note>
            Sed vt B C, ad B E F, ita eſt rectangulum ſub A B, B C, ad rectangulum ſub
              <lb/>
            AB, BEF. </s>
            <s xml:id="echoid-s8853" xml:space="preserve">Ergo erit quoq; </s>
            <s xml:id="echoid-s8854" xml:space="preserve">ſector ABCD, ad totũ circulum, vtrectangulum ſub
              <lb/>
            AB, BC, ad rectangulum ſub AB, BEF. </s>
            <s xml:id="echoid-s8855" xml:space="preserve">Cum ergo vt cap. </s>
            <s xml:id="echoid-s8856" xml:space="preserve">7. </s>
            <s xml:id="echoid-s8857" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s8858" xml:space="preserve">1. </s>
            <s xml:id="echoid-s8859" xml:space="preserve"/>
          </p>
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