Clavius, Christoph, Gnomonices libri octo, in quibus non solum horologiorum solariu[m], sed aliarum quo[quam] rerum, quae ex gnomonis umbra cognosci possunt, descriptiones geometricè demonstrantur

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        <div xml:id="echoid-div105" type="section" level="1" n="33">
          <p>
            <s xml:id="echoid-s2125" xml:space="preserve">
              <pb o="31" file="0051" n="51" rhead="LIBER PRIMVS."/>
            los æquales illis, quos ordinatim applicatæ cum diametro Paraboles conſtituunt. </s>
            <s xml:id="echoid-s2126" xml:space="preserve">Multo magis conue-
              <lb/>
            niet hæc ratio conis Scalcnis, cum triangulum per axem ad coni baſim rectum eſt, quia tunc, ex propoſ. </s>
            <s xml:id="echoid-s2127" xml:space="preserve">7.
              <lb/>
            </s>
            <s xml:id="echoid-s2128" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2129" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2130" xml:space="preserve">Apoll. </s>
            <s xml:id="echoid-s2131" xml:space="preserve">ordinatim applicatæ ſunt ad diametrum Paraboles perpendiculares, quemadmodum in
              <lb/>
            cono recto, ita vt E H, ſit quoque axis Parabolæ.</s>
            <s xml:id="echoid-s2132" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2133" xml:space="preserve">PRO hyperbolis verò oppoſitis demonſtranda ſunt duo alia lemmata, quæ omni cono tam recto,
              <lb/>
            quàm ſcaleno conucniunt; </s>
            <s xml:id="echoid-s2134" xml:space="preserve">quorum primum hoc eſt.</s>
            <s xml:id="echoid-s2135" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div109" type="section" level="1" n="34">
          <head xml:id="echoid-head37" xml:space="preserve">LEMMA PRIMVM.</head>
          <p>
            <s xml:id="echoid-s2136" xml:space="preserve">DATO cono, & </s>
            <s xml:id="echoid-s2137" xml:space="preserve">diametro tranſuerſa Hyperbolarum oppoſitarum, inuenire
              <lb/>
              <note position="left" xlink:label="note-0051-01" xlink:href="note-0051-01a" xml:space="preserve">10</note>
              <note position="right" xlink:label="note-0051-02" xlink:href="note-0051-02a" xml:space="preserve">Inuentio late-
                <lb/>
              ris recti hyper-
                <lb/>
              bolarũ oppoſi-
                <lb/>
              tarum, quatũ
                <lb/>
              diameter tranſ-
                <lb/>
              uerſa in cono
                <lb/>
              data ſit.</note>
            latus rectum Hyperboles.</s>
            <s xml:id="echoid-s2138" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2139" xml:space="preserve">SIT datus conus A B C, in quo triangulum per axem A B C, producatur {q́ue} conus vnà
              <lb/>
            cum triangulo per axem ad verticem A, vt fiant duo coni A B C, A D E, ad verticem
              <lb/>
            A, coniuncti. </s>
            <s xml:id="echoid-s2140" xml:space="preserve">Secetur quoque vtraque ſuperficies conica plano non per verticem facien
              <lb/>
            te ſectiones F G H, I K L, quæ hyperbolæ ſunt oppoſitæ, ex propoſ. </s>
            <s xml:id="echoid-s2141" xml:space="preserve">14. </s>
            <s xml:id="echoid-s2142" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2143" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2144" xml:space="preserve">Apallonij,
              <lb/>
            quarum diameter tranſuerſa communis F I, & </s>
            <s xml:id="echoid-s2145" xml:space="preserve">la-
              <lb/>
              <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a" number="34">
                <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0051-01"/>
              </figure>
            tera recta æqualia. </s>
            <s xml:id="echoid-s2146" xml:space="preserve">Vtriuſque ergo lat{us} rectum
              <lb/>
            ita inueniem{us}. </s>
            <s xml:id="echoid-s2147" xml:space="preserve">Per A, ducatur A M, ipſi F I,
              <lb/>
            parallela ſecans B C, in M; </s>
            <s xml:id="echoid-s2148" xml:space="preserve">fiat{q́ue} vt C M, altera
              <lb/>
              <note position="left" xlink:label="note-0051-03" xlink:href="note-0051-03a" xml:space="preserve">20</note>
            parsbaſis, ad A M, ita A M, ad M N. </s>
            <s xml:id="echoid-s2149" xml:space="preserve">Rurſ{us}
              <lb/>
            fiat, vt M N, ad B M, alteram baſis partem, ita
              <lb/>
              <note position="right" xlink:label="note-0051-04" xlink:href="note-0051-04a" xml:space="preserve">11. ſexti.</note>
              <note position="right" xlink:label="note-0051-05" xlink:href="note-0051-05a" xml:space="preserve">12. ſexti.</note>
            F I, tranſuerſa diameter ad F O. </s>
            <s xml:id="echoid-s2150" xml:space="preserve">Dico F O, eſſe la-
              <lb/>
            tusrectum vtriuſque Hyperboles; </s>
            <s xml:id="echoid-s2151" xml:space="preserve">hoc eſt, eſſe re-
              <lb/>
            ctam, iuxta quam poſſunt ordinatim applicatæ ad
              <lb/>
            diametrum vtriuſque hyperboles. </s>
            <s xml:id="echoid-s2152" xml:space="preserve">Sit enim re-
              <lb/>
            ctangulum B C, contentum ſub baſis partibus B M,
              <lb/>
            M C; </s>
            <s xml:id="echoid-s2153" xml:space="preserve">& </s>
            <s xml:id="echoid-s2154" xml:space="preserve">ad M C, applicetur rectangulum C N, ſub
              <lb/>
              <note position="left" xlink:label="note-0051-06" xlink:href="note-0051-06a" xml:space="preserve">30</note>
              <note position="right" xlink:label="note-0051-07" xlink:href="note-0051-07a" xml:space="preserve">17. ſexti.</note>
            M C, M N, contentum, quod æquale erit quadrato
              <lb/>
            rectæ A M, propterea quòdtres rectæ M C, A M,
              <lb/>
            M N, continuè proportionales ſunt ex conſtructio-
              <lb/>
            ne: </s>
            <s xml:id="echoid-s2155" xml:space="preserve">erit{q́ue} B M N, vna linearecta, quòd duo an-
              <lb/>
              <note position="right" xlink:label="note-0051-08" xlink:href="note-0051-08a" xml:space="preserve">14. primi.</note>
            guli ad M, recti ſint. </s>
            <s xml:id="echoid-s2156" xml:space="preserve">Quoniam igitur eſt, vt
              <lb/>
            M N, ad B M, ita F I, ad F O; </s>
            <s xml:id="echoid-s2157" xml:space="preserve">Vt autem M N, ad B M, ita eſt rectangulum C N, hoc est,
              <lb/>
              <note position="right" xlink:label="note-0051-09" xlink:href="note-0051-09a" xml:space="preserve">1. ſexti.</note>
            quadratum ex A M, ad rectangulum B C, ſub baſis partibus B M, M C, contentum; </s>
            <s xml:id="echoid-s2158" xml:space="preserve">erit
              <lb/>
            quoque vt quadratum ex A M, adrectangulum ſub B M, M C, ita tranſuerſa diameter
              <lb/>
            F I, ad rectam F O. </s>
            <s xml:id="echoid-s2159" xml:space="preserve">Eſt igitur F O, latus rectum hyperboles, ex propoſ. </s>
            <s xml:id="echoid-s2160" xml:space="preserve">12. </s>
            <s xml:id="echoid-s2161" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2162" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2163" xml:space="preserve">Apollo-
              <lb/>
              <note position="left" xlink:label="note-0051-10" xlink:href="note-0051-10a" xml:space="preserve">40</note>
            nij, hoc eſt, Recta, iuxta quam poſſunt or dinatim applicatæ, &</s>
            <s xml:id="echoid-s2164" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2165" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">Quomodo
            <lb/>
          quarta pats ſub
            <lb/>
          diametro trãſ-
            <lb/>
          uerſa hyperbo-
            <lb/>
          les, & latere re-
            <lb/>
          cto cõprehenſi
            <lb/>
          applicetur ad
            <lb/>
          diametrũ trã
            <unsure/>
          ſ-
            <lb/>
          uerſam ex vtra-
            <lb/>
          que parte, ita vt
            <lb/>
          excedat figura
            <lb/>
          quadraata.</note>
        </div>
        <div xml:id="echoid-div112" type="section" level="1" n="35">
          <head xml:id="echoid-head38" xml:space="preserve">LEMMA II.</head>
          <p>
            <s xml:id="echoid-s2166" xml:space="preserve">QVARTAM partem rectanguli ſub diametro tranſuerſa Hyperboles, & </s>
            <s xml:id="echoid-s2167" xml:space="preserve">late-
              <lb/>
            re recto comprehenſi ad tranſuerſam diametrum ex vtraque parte applicare, ita vt
              <lb/>
            excedatfigura quadrata.</s>
            <s xml:id="echoid-s2168" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2169" xml:space="preserve">POSITA eadem figura, reperiatur inter tranſuer ſam diametrum F I, & </s>
            <s xml:id="echoid-s2170" xml:space="preserve">latus re-
              <lb/>
            ctum F O, media proportionalis A B, quæ bifariam ſecetur in C. </s>
            <s xml:id="echoid-s2171" xml:space="preserve">Erit igitur quadratum
              <lb/>
              <note position="right" xlink:label="note-0051-12" xlink:href="note-0051-12a" xml:space="preserve">13. ſexti.</note>
              <note position="left" xlink:label="note-0051-13" xlink:href="note-0051-13a" xml:space="preserve">50</note>
            ex A B, æquale rectangulo ſub F I, F O; </s>
            <s xml:id="echoid-s2172" xml:space="preserve">at que adeo quadratum ex A C, quod ex ſcholio
              <lb/>
              <note position="right" xlink:label="note-0051-14" xlink:href="note-0051-14a" xml:space="preserve">17. ſexti.</note>
            propoſ. </s>
            <s xml:id="echoid-s2173" xml:space="preserve">4. </s>
            <s xml:id="echoid-s2174" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2175" xml:space="preserve">2. </s>
            <s xml:id="echoid-s2176" xml:space="preserve">Euclidis, quarta pars eſt quadrati ex A B, quartæ parti rectanguli ſub F I,
              <lb/>
            F O, æquale erit. </s>
            <s xml:id="echoid-s2177" xml:space="preserve">Huic igitur quadrato ex A C, applicabim{us} ad diametrum tranſuer ſam
              <lb/>
            F I, ex vtraque parte, æquale rectangulum excedens figura quadrata, hoc modo. </s>
            <s xml:id="echoid-s2178" xml:space="preserve">Diuiſa
              <lb/>
            recta F I, bifariam in D, fiat angulus rectus H K L, & </s>
            <s xml:id="echoid-s2179" xml:space="preserve">recta H K, rectæ A C, & </s>
            <s xml:id="echoid-s2180" xml:space="preserve">recta
              <lb/>
            K L, rectæ D I, æqualis, connectatur{q́ue} recta H L, quæ maior erit, quàm recta K L, hoc
              <lb/>
              <note position="right" xlink:label="note-0051-15" xlink:href="note-0051-15a" xml:space="preserve">19. primi.</note>
            eſt, quàm D I, propterea quòd H L, maiori angulo opponatur, quàm K L. </s>
            <s xml:id="echoid-s2181" xml:space="preserve">Producta recta
              <lb/>
            F I, in vtramque partem, abſcindantur vtrinque ex D, rectæ D Q, D R, ipſi H L, æqua-
              <lb/>
            les. </s>
            <s xml:id="echoid-s2182" xml:space="preserve">Dico tam rectangulum ſub F Q, Q I, applicatum ad F I, excedens{q́ue} quadrato </s>
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