Clavius, Christoph, Geometria practica

Table of Notes

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            <s xml:id="echoid-s15205" xml:space="preserve">
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            AHK, demittaturque perpendicularis HI. </s>
            <s xml:id="echoid-s15206" xml:space="preserve">Quoniam igitur ponitur arcus BD,
              <lb/>
            ad rectam AD, vt AD, hoc eſt, vt A B, ad AF; </s>
            <s xml:id="echoid-s15207" xml:space="preserve">eſt que vt A B, ſemidiameter ad ſe-
              <lb/>
            midiametrum A F, ita arcus B D, ad arcum F G; </s>
            <s xml:id="echoid-s15208" xml:space="preserve">(Cum enim ſit, vt lib. </s>
            <s xml:id="echoid-s15209" xml:space="preserve">4. </s>
            <s xml:id="echoid-s15210" xml:space="preserve">capit. </s>
            <s xml:id="echoid-s15211" xml:space="preserve">7.
              <lb/>
            </s>
            <s xml:id="echoid-s15212" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s15213" xml:space="preserve">1. </s>
            <s xml:id="echoid-s15214" xml:space="preserve">demonſtrauimus, diameter ad diametrum, vt circumferentia ad circũ-
              <lb/>
              <note symbol="a" position="right" xlink:label="note-355-01" xlink:href="note-355-01a" xml:space="preserve">15. quinti.</note>
            ferentiam; </s>
            <s xml:id="echoid-s15215" xml:space="preserve"> erit quo que ſemidiameter AB, ad ſemidiametrum AF, vt eadem
              <note symbol="b" position="right" xlink:label="note-355-02" xlink:href="note-355-02a" xml:space="preserve">15. quinti.</note>
            cumferentia ad eandem circumferentiam: </s>
            <s xml:id="echoid-s15216" xml:space="preserve"> ac proinde etiam, vt quarta pars circumferentiæ ad quartam partem circumferentiæ, hoc eſt, vt arcus BD, ad ar-
              <lb/>
              <note symbol="c" position="right" xlink:label="note-355-03" xlink:href="note-355-03a" xml:space="preserve">11. quinti.</note>
            cum F G.) </s>
            <s xml:id="echoid-s15217" xml:space="preserve"> Erit quoque arcus B D, ad rectam A D, vtidem arcus B D, ad
              <note symbol="d" position="right" xlink:label="note-355-04" xlink:href="note-355-04a" xml:space="preserve">9. quinti.</note>
            F G; </s>
            <s xml:id="echoid-s15218" xml:space="preserve"> ac proptera æquales erunt recta A D, & </s>
            <s xml:id="echoid-s15219" xml:space="preserve">arcus F G. </s>
            <s xml:id="echoid-s15220" xml:space="preserve">Quia verò ex præce- denti coroll. </s>
            <s xml:id="echoid-s15221" xml:space="preserve">eſt, vt arcus B D, ad arcum BK, ita recta AD, ad rectam HI, & </s>
            <s xml:id="echoid-s15222" xml:space="preserve">vt ar-
              <lb/>
              <note symbol="e" position="right" xlink:label="note-355-05" xlink:href="note-355-05a" xml:space="preserve">ſchol. 33.
                <lb/>
              ſexti.</note>
            cus BD, ad arcum BK, ita eſt arcus FG, ad arcum FH, quod arcus B D, B K, ar- cubus FG, FH, ſimiles ſint; </s>
            <s xml:id="echoid-s15223" xml:space="preserve"> erit quoque recta AD, ad rectam HI, vt arcus F
              <note symbol="f" position="right" xlink:label="note-355-06" xlink:href="note-355-06a" xml:space="preserve">11. quinti.</note>
            ad arcum FH. </s>
            <s xml:id="echoid-s15224" xml:space="preserve">Cum ergo oſtenſa ſit recta A D, arcui F G, æqualis: </s>
            <s xml:id="echoid-s15225" xml:space="preserve">erit
              <note symbol="g" position="right" xlink:label="note-355-07" xlink:href="note-355-07a" xml:space="preserve">14. quinti.</note>
            que recta HI, arcui F H, ęqualis quod eſt abſurdum. </s>
            <s xml:id="echoid-s15226" xml:space="preserve">Eſt enim recta H I, minor
              <lb/>
              <note symbol="h" position="right" xlink:label="note-355-08" xlink:href="note-355-08a" xml:space="preserve">3. tertij.</note>
            arcu F H, cum ea ſit ſemiſsis chordæ ſubten dentis arcum duplum arcus F H: </s>
            <s xml:id="echoid-s15227" xml:space="preserve">
              <note symbol="i" position="right" xlink:label="note-355-09" xlink:href="note-355-09a" xml:space="preserve">ſchol. 27.
                <lb/>
              tertij.</note>
            (Nam recta A F, ſecat eam chordam bifariam; </s>
            <s xml:id="echoid-s15228" xml:space="preserve"> ac proinde & </s>
            <s xml:id="echoid-s15229" xml:space="preserve">arcum) chorda autem ſemper ſuo arcu minor ſit. </s>
            <s xml:id="echoid-s15230" xml:space="preserve">Non ergo eſt arcus B D, ad ſemidiametrum
              <lb/>
            AD, vt AD, ad rectam maiorem baſe AE, Quadratricis.</s>
            <s xml:id="echoid-s15231" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15232" xml:space="preserve">
              <emph style="sc">Sit</emph>
            deinde, ſi fieri poteſt, vt arcus BD, ad AD, ita A D, ad A I, min orem baſe
              <lb/>
            AE. </s>
            <s xml:id="echoid-s15233" xml:space="preserve">Deſcripto igitur ex centro A, per I, Quadrante IL, erigatur ex I, ad AE, per-
              <lb/>
            pendicularis I H, ſecans Quadratricem in H, puncto, per quod ſemidia-
              <lb/>
            meter ducatur AK, ſecans arcum IL, in M. </s>
            <s xml:id="echoid-s15234" xml:space="preserve">Oſtendemus ergo, vt prius, arcum
              <lb/>
            IL, rectæ AD, æqualem eſſe. </s>
            <s xml:id="echoid-s15235" xml:space="preserve">Item ita eſſe arcum BD, ad arcum BK, hoc eſt, arcum
              <lb/>
            I L, ad arcum I M, vt eſt recta A D; </s>
            <s xml:id="echoid-s15236" xml:space="preserve">ad rectam H I. </s>
            <s xml:id="echoid-s15237" xml:space="preserve">Quare cum arcus
              <lb/>
              <note symbol="k" position="right" xlink:label="note-355-10" xlink:href="note-355-10a" xml:space="preserve">14. quinti.</note>
            IL, oſtenſus ſit æqualis rectæ A D, erit quoq; </s>
            <s xml:id="echoid-s15238" xml:space="preserve">arcus I M, æqualis rectæ HI. </s>
            <s xml:id="echoid-s15239" xml:space="preserve">
              <note symbol="l" position="right" xlink:label="note-355-11" xlink:href="note-355-11a" xml:space="preserve">2. coroll. 36.
                <lb/>
              tertij.</note>
            eſt abſurdum. </s>
            <s xml:id="echoid-s15240" xml:space="preserve">Eſt enimrecta H I, maior arcu I M. </s>
            <s xml:id="echoid-s15241" xml:space="preserve">Nam ſi ex H, duceretur ver-
              <lb/>
            ſus G, alia recta tangens circulum IL, ſicut H I, eundẽ tangit in I, eſſent hę
              <note symbol="m" position="right" xlink:label="note-355-12" xlink:href="note-355-12a" xml:space="preserve">ſchol. 27.
                <lb/>
              tertij.</note>
            tangentes æquales, arcuſq; </s>
            <s xml:id="echoid-s15242" xml:space="preserve">inter eas interceptus ſecaretur bifariam in M, pro- pterea quod angulus ab eis comprehenſus bifariam diuideretur à recta AH, ac
              <note symbol="n" position="right" xlink:label="note-355-13" xlink:href="note-355-13a" xml:space="preserve">4. vel 8.
                <lb/>
              primi.</note>
            proinde & </s>
            <s xml:id="echoid-s15243" xml:space="preserve">angulus in centro A, ſi ad alterum punctum conta ctus recta adiun-
              <lb/>
            geretur: </s>
            <s xml:id="echoid-s15244" xml:space="preserve"> ideoque arcus, quibus inſiſtunt, æquales forent. </s>
            <s xml:id="echoid-s15245" xml:space="preserve">Igitur cum, vt lib.</s>
            <s xml:id="echoid-s15246" xml:space="preserve">
              <note symbol="o" position="right" xlink:label="note-355-14" xlink:href="note-355-14a" xml:space="preserve">26. tertij.</note>
            8. </s>
            <s xml:id="echoid-s15247" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s15248" xml:space="preserve">1. </s>
            <s xml:id="echoid-s15249" xml:space="preserve">probabimus cum Archimede, duæ illætangentes ſimul maiores
              <lb/>
            ſint arcu ab eis comprehenſo, erit & </s>
            <s xml:id="echoid-s15250" xml:space="preserve">earum ſemiſsis HI, maior ſemiſſe IM, illius
              <lb/>
            arcus. </s>
            <s xml:id="echoid-s15251" xml:space="preserve">Non eſt ergo arcus BD, ad ſemidiametrum AD, vt AD, ad rectam minorẽ
              <lb/>
            baſe AE, Quadratricis; </s>
            <s xml:id="echoid-s15252" xml:space="preserve">Sed neque vt AD, ad maiorem, ſicut oſtenſum eſt. </s>
            <s xml:id="echoid-s15253" xml:space="preserve">Igitur
              <lb/>
            vt AD, ad ipſam baſem AE. </s>
            <s xml:id="echoid-s15254" xml:space="preserve">quod demonſtrandum erat.</s>
            <s xml:id="echoid-s15255" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div929" type="section" level="1" n="324">
          <head xml:id="echoid-head351" xml:space="preserve">COROLLARIVM I.</head>
          <note position="right" xml:space="preserve">Rectam cir-
            <lb/>
          cunferentiæ
            <lb/>
          circuli æqua-
            <lb/>
          lem reperire.</note>
          <p>
            <s xml:id="echoid-s15256" xml:space="preserve">HINC facilè rectam reperiemus arcui Quadrantis, ex quo Quadratrix
              <lb/>
            deſcripta eſt, ac proinde & </s>
            <s xml:id="echoid-s15257" xml:space="preserve">ſemicircumferentiæ, immo & </s>
            <s xml:id="echoid-s15258" xml:space="preserve">toti circũ-
              <lb/>
            ferentiæ æqualem.</s>
            <s xml:id="echoid-s15259" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15260" xml:space="preserve">
              <emph style="sc">Qvoniam</emph>
            eſt arcus B D, ad ſemidiametrum A D, vt A D, ad ba-
              <lb/>
            ſem Quadratricis A E; </s>
            <s xml:id="echoid-s15261" xml:space="preserve">erit conuertendo quoque A E, ad A D, vt A D, ad
              <lb/>
              <note symbol="p" position="right" xlink:label="note-355-16" xlink:href="note-355-16a" xml:space="preserve">11. quinti.</note>
            arcum B D. </s>
            <s xml:id="echoid-s15262" xml:space="preserve">Si igitur duabus rectis A E, A D, inueniatur tertia proportionalis; </s>
            <s xml:id="echoid-s15263" xml:space="preserve"> erit AD, ad eam tertiam, vt ad arcum BD, cum vtraq; </s>
            <s xml:id="echoid-s15264" xml:space="preserve">proportio ſit eadem, </s>
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