Clavius, Christoph, Geometria practica

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          <p>
            <s xml:id="echoid-s7362" xml:space="preserve">
              <pb o="176" file="206" n="206" rhead="GEOMETR. PRACT."/>
            in hexagono continet rectum cum parte tertia, hoc eſt, grad. </s>
            <s xml:id="echoid-s7363" xml:space="preserve">120. </s>
            <s xml:id="echoid-s7364" xml:space="preserve">Et quoniam
              <lb/>
            ſemidiameter ſecat angulum figuræ bifariam, vt conſtat ex demonſtratione
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s7365" xml:space="preserve">12. </s>
            <s xml:id="echoid-s7366" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7367" xml:space="preserve">4. </s>
            <s xml:id="echoid-s7368" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s7369" xml:space="preserve">fit vt ſi ſemiſsis lateris, in quod perpendicularis cadit,
              <lb/>
            ponatur ſinus totus, perpendicularis ſit Tangens ſemiſsis anguli figuræ, & </s>
            <s xml:id="echoid-s7370" xml:space="preserve">ſemi-
              <lb/>
            diameter, Secans. </s>
            <s xml:id="echoid-s7371" xml:space="preserve">Si fiat ergo,
              <lb/>
              <note style="it" position="right" xlink:label="note-206-01" xlink:href="note-206-01a" xml:space="preserve">
                <lb/>
              Vt lat{us} 100000, ſin{us} \\ tot{us}. # ad ſemiſſem \\ lateris: # ita Tangens ſemiſſis an- \\ guli, vel ſecans. # ad aliud,
                <lb/>
              </note>
            prodibit tam perpendicularis, quàm ſemidiameter in partibus lateris figuræ.
              <lb/>
            </s>
            <s xml:id="echoid-s7372" xml:space="preserve">Verbi gratia, in Hexagono regulari, cuius vnum latus ſit 12. </s>
            <s xml:id="echoid-s7373" xml:space="preserve">ſi fiat, vt 100000. </s>
            <s xml:id="echoid-s7374" xml:space="preserve">ſi-
              <lb/>
            nus totus ad 6. </s>
            <s xml:id="echoid-s7375" xml:space="preserve">ſemiſſem lateris: </s>
            <s xml:id="echoid-s7376" xml:space="preserve">ita 173205. </s>
            <s xml:id="echoid-s7377" xml:space="preserve">tangens ſemiſsis anguli Hexagoni
              <lb/>
            ad aliud, reperietur perpendicularis 10 {39230/100000}. </s>
            <s xml:id="echoid-s7378" xml:space="preserve">vel 10 {3923/10000}. </s>
            <s xml:id="echoid-s7379" xml:space="preserve">Item ſi fiat, vt
              <lb/>
            100000. </s>
            <s xml:id="echoid-s7380" xml:space="preserve">ſinus totus ad 6. </s>
            <s xml:id="echoid-s7381" xml:space="preserve">ſemiſſem lateris; </s>
            <s xml:id="echoid-s7382" xml:space="preserve">ita 200000. </s>
            <s xml:id="echoid-s7383" xml:space="preserve">Secans ſemiſsis anguli
              <lb/>
            liexagoni ad aliud, exibit ſemidiameter figuræ 12.</s>
            <s xml:id="echoid-s7384" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7385" xml:space="preserve">
              <emph style="sc">Itaqve</emph>
            ſi perpendicularis 10 {3923/10000}. </s>
            <s xml:id="echoid-s7386" xml:space="preserve">ducatur in 36. </s>
            <s xml:id="echoid-s7387" xml:space="preserve">ſemiſſem ambitus He-
              <lb/>
            xagoni ex tribus lateribus conflatam, producetur area Hexagoni 374 {1228/10000}. </s>
            <s xml:id="echoid-s7388" xml:space="preserve">vel
              <lb/>
            374 {307/2500}. </s>
            <s xml:id="echoid-s7389" xml:space="preserve">Eodemque modo procedendum eſt in aliis figuris regularibus, in
              <lb/>
            quibus angulorum ſemiſſes in tabula Tangentium, ac ſecantium accipi poſ-
              <lb/>
            ſunt.</s>
            <s xml:id="echoid-s7390" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7391" xml:space="preserve">3. </s>
            <s xml:id="echoid-s7392" xml:space="preserve">
              <emph style="sc">Qvoniam</emph>
            vero circa quamlibet figuram regularem circulus deſcribi
              <lb/>
            poteſt, vt ex lib. </s>
            <s xml:id="echoid-s7393" xml:space="preserve">4. </s>
            <s xml:id="echoid-s7394" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s7395" xml:space="preserve">conſtet, proponemus hic plurimarum figurarum regu-
              <lb/>
            larium latera in partibus diametri circuli ambientis 20000000. </s>
            <s xml:id="echoid-s7396" xml:space="preserve">vel ſemidiame-
              <lb/>
            tri, ſiue ſinus totius 10000000. </s>
            <s xml:id="echoid-s7397" xml:space="preserve">ex probatis auctoribus, vt earum areæ magis ex-
              <lb/>
              <note position="left" xlink:label="note-206-02" xlink:href="note-206-02a" xml:space="preserve">Quare ex la-
                <lb/>
              tere figuræ re-
                <lb/>
              gularis dato
                <lb/>
              non ſemper
                <lb/>
              poſſit inueniri
                <lb/>
              area, niſi figu-
                <lb/>
              raipſa deſcri-
                <lb/>
              pta ſit.</note>
            quiſitè inueniri poſsint per regulam Num. </s>
            <s xml:id="echoid-s7398" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7399" xml:space="preserve">propoſitam. </s>
            <s xml:id="echoid-s7400" xml:space="preserve">Nam ſi ex ſolo vno
              <lb/>
            latere in aliqua menſura cognito aream inueſtigare velimus, quemadmodum in
              <lb/>
            Hexagono factum eſt, occurrẽt multæ difficultates acmagnę, propterea quod
              <lb/>
            ſemidiametri, perpendiculareſque ex ſinubus, Tangentibus, ac ſecantibus erui
              <lb/>
            non poſſunt, niſi quando ſemiſsis anguli figuræ comprehendit pręcisè gradus,
              <lb/>
            vel gradus cum minutis, vel gradus cumminutis & </s>
            <s xml:id="echoid-s7401" xml:space="preserve">ſecundis: </s>
            <s xml:id="echoid-s7402" xml:space="preserve">(quamuis ſi ſe-
              <lb/>
            cunda adſint, neceſſe ſit partem proportionalem adhibere) ſicut in Hexago-
              <lb/>
            no paulo ante factum eſt, cuius anguli ſemiſsis continet grad. </s>
            <s xml:id="echoid-s7403" xml:space="preserve">60. </s>
            <s xml:id="echoid-s7404" xml:space="preserve">quod in
              <lb/>
            quam plurimis figuris non contingit. </s>
            <s xml:id="echoid-s7405" xml:space="preserve">Nam, verbi gratia, in Heptagono omnes
              <lb/>
            7. </s>
            <s xml:id="echoid-s7406" xml:space="preserve">anguli ex ſcholio propoſ. </s>
            <s xml:id="echoid-s7407" xml:space="preserve">32. </s>
            <s xml:id="echoid-s7408" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7409" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7410" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s7411" xml:space="preserve">æquiualent 10. </s>
            <s xml:id="echoid-s7412" xml:space="preserve">rectis, ideo que v-
              <lb/>
            nus angulus complectitur vnum rectum cum {3/7}. </s>
            <s xml:id="echoid-s7413" xml:space="preserve">hoc eſt, gradus 128. </s>
            <s xml:id="echoid-s7414" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7415" xml:space="preserve">34.
              <lb/>
            </s>
            <s xml:id="echoid-s7416" xml:space="preserve">Sec. </s>
            <s xml:id="echoid-s7417" xml:space="preserve">17. </s>
            <s xml:id="echoid-s7418" xml:space="preserve">Ter. </s>
            <s xml:id="echoid-s7419" xml:space="preserve">8. </s>
            <s xml:id="echoid-s7420" xml:space="preserve">Quar. </s>
            <s xml:id="echoid-s7421" xml:space="preserve">34. </s>
            <s xml:id="echoid-s7422" xml:space="preserve">&</s>
            <s xml:id="echoid-s7423" xml:space="preserve">c. </s>
            <s xml:id="echoid-s7424" xml:space="preserve">atque eius ſemiſsis grad. </s>
            <s xml:id="echoid-s7425" xml:space="preserve">64. </s>
            <s xml:id="echoid-s7426" xml:space="preserve">min. </s>
            <s xml:id="echoid-s7427" xml:space="preserve">17. </s>
            <s xml:id="echoid-s7428" xml:space="preserve">Secun. </s>
            <s xml:id="echoid-s7429" xml:space="preserve">8. </s>
            <s xml:id="echoid-s7430" xml:space="preserve">
              <lb/>
            Ter. </s>
            <s xml:id="echoid-s7431" xml:space="preserve">34. </s>
            <s xml:id="echoid-s7432" xml:space="preserve">Quar. </s>
            <s xml:id="echoid-s7433" xml:space="preserve">17. </s>
            <s xml:id="echoid-s7434" xml:space="preserve">&</s>
            <s xml:id="echoid-s7435" xml:space="preserve">c. </s>
            <s xml:id="echoid-s7436" xml:space="preserve">ex qua ſemiſſe nequit Tangens pro perpendiculari, ne-
              <lb/>
            que ſecans pro ſemidiametro per tabulas Tangentium, atque ſecantium excer-
              <lb/>
            pi poteſt. </s>
            <s xml:id="echoid-s7437" xml:space="preserve">Idemque in aliis figuris innumeris accidere comperies. </s>
            <s xml:id="echoid-s7438" xml:space="preserve">Quod ſi
              <lb/>
            ſcientia inuenta eſſet conſtruendi omnes figuras regulares, ſuperari aliquo mo-
              <lb/>
            do poſſet hæc diffi cultas: </s>
            <s xml:id="echoid-s7439" xml:space="preserve">propterea quod cognito latere vno in quantiſcun-
              <lb/>
            que partibus, cognoſci quoque poſſet tam perpendicularis, quam ſemidiame-
              <lb/>
            ter in iiſdem partibus, beneficio inſtrumenti partium, vt lib. </s>
            <s xml:id="echoid-s7440" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7441" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s7442" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7443" xml:space="preserve">ad finem
              <lb/>
            Num. </s>
            <s xml:id="echoid-s7444" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7445" xml:space="preserve">tradidimus. </s>
            <s xml:id="echoid-s7446" xml:space="preserve">Sed quia pauciſsimas figuras æquilateras deſcribere no-
              <lb/>
            uimus intra circulum, areas plurimarum ignorari ncceſſe eſt. </s>
            <s xml:id="echoid-s7447" xml:space="preserve">Hanc ob cau-
              <lb/>
            ſam nonnulli Geometræ, inter quos ſtrenuam operam nauauit Ludolphus,
              <lb/>
            à Collen, ingenti labore latera figurarum, ſiue deſcriptæ eæ ſint, ſiue non, ex-
              <lb/>
            plorarunt: </s>
            <s xml:id="echoid-s7448" xml:space="preserve">quamuis earum anguli præter gradus, ac minuta comprehendant</s>
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