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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            <s xml:id="echoid-s1842" xml:space="preserve">
              <pb o="27" file="0047" n="47" rhead="LIBER PRIMVS."/>
            baſim B C, ſiue infra in Ellipſi, vt ex tribus propoſitionibus proximè dictis conſtat. </s>
            <s xml:id="echoid-s1843" xml:space="preserve">Sumantur
              <lb/>
            in diametro E D, quotcunque partes ſiue æquales, ſiue inæquales E k, k L, & </s>
            <s xml:id="echoid-s1844" xml:space="preserve">per puncta k, L, agan-
              <lb/>
            tur baſi B C, parallelæ F H, G I; </s>
            <s xml:id="echoid-s1845" xml:space="preserve">eruntq́ue tam partes C H, H I, quàm E k, k L, (ſumendo in Elli-
              <lb/>
              <note position="right" xlink:label="note-0047-01" xlink:href="note-0047-01a" xml:space="preserve">2. ſexti.</note>
            pſi punctum E, in B C, baſi trianguli) partibus B F, F G, proportionales: </s>
            <s xml:id="echoid-s1846" xml:space="preserve">Immo in parabola æqua-
              <lb/>
            les ſunt partes C H, H I, partibus E k, k L, propter parallelogramma C k, k I.</s>
            <s xml:id="echoid-s1847" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">34. primi.</note>
          <p>
            <s xml:id="echoid-s1848" xml:space="preserve">QVOD ſi conus rectus fuerit, vt ſunt omnes illi, quibus in deſcriptionibus horologiorum
              <lb/>
            vtimur, (omnes enim hi recti ſunt, cum eorum axes ſint partes axis mundi, qui ad parallelos pri-
              <lb/>
            mi motus, nempead baſes conorum, per propoſ. </s>
            <s xml:id="echoid-s1849" xml:space="preserve">10. </s>
            <s xml:id="echoid-s1850" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s1851" xml:space="preserve">1. </s>
            <s xml:id="echoid-s1852" xml:space="preserve">Theod. </s>
            <s xml:id="echoid-s1853" xml:space="preserve">rectus eſt) commodiſſime ita
              <lb/>
            agemus. </s>
            <s xml:id="echoid-s1854" xml:space="preserve">Sumantur in recta B D, quotcunque partes ſiue æquales, ſiue inæquales B F, F G, & </s>
            <s xml:id="echoid-s1855" xml:space="preserve">his
              <lb/>
            æquales in alio latere C H, H I, ſingulæ ſingulis, iunganturq́; </s>
            <s xml:id="echoid-s1856" xml:space="preserve">rectæ F H, G I, ſecantes diametrum
              <lb/>
              <note position="left" xlink:label="note-0047-03" xlink:href="note-0047-03a" xml:space="preserve">10</note>
            D E, in K, L. </s>
            <s xml:id="echoid-s1857" xml:space="preserve">Nam hę lineę, cum ſecent latera A B, A C, proportionaliter, parallelæ erunt, propor
              <lb/>
              <note position="right" xlink:label="note-0047-04" xlink:href="note-0047-04a" xml:space="preserve">2. ſexti.</note>
            tionalesq́; </s>
            <s xml:id="echoid-s1858" xml:space="preserve">propterea erunt partes E K, K L, partibus B F, F G. </s>
            <s xml:id="echoid-s1859" xml:space="preserve">Exponatur deinde ſeorſum baſis
              <lb/>
            B E C, trianguli A B C, & </s>
            <s xml:id="echoid-s1860" xml:space="preserve">ex puncto E, quod inſtar ſit omnium punctorum E, K, L, perpendicula-
              <lb/>
            ris educatur E M, atque in rectam B C, ex puncto E, in vtramque partem transferantur partes
              <lb/>
            K F, K H, & </s>
            <s xml:id="echoid-s1861" xml:space="preserve">L G, L I; </s>
            <s xml:id="echoid-s1862" xml:space="preserve">ita vt EF, E G, partibus K F, L G, & </s>
            <s xml:id="echoid-s1863" xml:space="preserve">E H, E I, partibus K H, L I, ſint equales:
              <lb/>
            </s>
            <s xml:id="echoid-s1864" xml:space="preserve">quæ quidem ex parte C, in parabola omnes in punctum C, cadent, propterea quòd E C, K H, L I,
              <lb/>
              <note position="right" xlink:label="note-0047-05" xlink:href="note-0047-05a" xml:space="preserve">34. primi.</note>
            ęquales ſint. </s>
            <s xml:id="echoid-s1865" xml:space="preserve">In Hyperbola autem ſemper minores fient, quàm E C, & </s>
            <s xml:id="echoid-s1866" xml:space="preserve">in Ellipſi maiores, vt patet.
              <lb/>
            </s>
            <s xml:id="echoid-s1867" xml:space="preserve">In omnibus tamen erunt partes B F, FG, in primis figuris, (voco primas figuras, ipſos conos, ſecun-
              <lb/>
            das autem, eas, in quibus ſeorſum expoſuimus baſim B E C.) </s>
            <s xml:id="echoid-s1868" xml:space="preserve">partibus B F, F G, in ſecundis, nec
              <lb/>
            non & </s>
            <s xml:id="echoid-s1869" xml:space="preserve">C H, H I, in primis, partibus C H, H I, in ſecundis proportionales. </s>
            <s xml:id="echoid-s1870" xml:space="preserve">Ducta enim G N, in
              <lb/>
              <note position="left" xlink:label="note-0047-06" xlink:href="note-0047-06a" xml:space="preserve">20</note>
            primis figuris, parallela ipſi D E, erit vt G B, ad B N, ita G F, ad F O; </s>
            <s xml:id="echoid-s1871" xml:space="preserve">(cum triangula G B N, GFO,
              <lb/>
              <note position="right" xlink:label="note-0047-07" xlink:href="note-0047-07a" xml:space="preserve">4. ſexti.</note>
            ſimilia ſint, ex corolla. </s>
            <s xml:id="echoid-s1872" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s1873" xml:space="preserve">4 lib. </s>
            <s xml:id="echoid-s1874" xml:space="preserve">6. </s>
            <s xml:id="echoid-s1875" xml:space="preserve">Euclidis) & </s>
            <s xml:id="echoid-s1876" xml:space="preserve">permutando, vt G B, ad G F, ita B N, ad F O.
              <lb/>
            </s>
            <s xml:id="echoid-s1877" xml:space="preserve">Cum ergo B N, in primis figuris, equalis ſit ipſi B G, in ſecundis; </s>
            <s xml:id="echoid-s1878" xml:space="preserve">& </s>
            <s xml:id="echoid-s1879" xml:space="preserve">F O, in primis, ipſi F G, in ſe-
              <lb/>
            cundis; </s>
            <s xml:id="echoid-s1880" xml:space="preserve">(propterea quòd N E, ipſi G L, in primis, hoc eſt, ipſi G E, in ſecundis, ſit æqualis; </s>
            <s xml:id="echoid-s1881" xml:space="preserve">& </s>
            <s xml:id="echoid-s1882" xml:space="preserve">B N,
              <lb/>
            idcirco ipſi B G, & </s>
            <s xml:id="echoid-s1883" xml:space="preserve">F O, ipſi F G. </s>
            <s xml:id="echoid-s1884" xml:space="preserve">Poſitæ enim ſunt E B, E F, E G, in ſecundis figuris ipſis E B, K F,
              <lb/>
            L G, in primis, æquales.) </s>
            <s xml:id="echoid-s1885" xml:space="preserve">erit quoque vt G B, ad G F, in primis, ita B G, ad F G, in ſecundis; </s>
            <s xml:id="echoid-s1886" xml:space="preserve">& </s>
            <s xml:id="echoid-s1887" xml:space="preserve">di-
              <lb/>
            uidendo, vt F B, ad G F, in primis, ita B F, ad F G, in ſecundis. </s>
            <s xml:id="echoid-s1888" xml:space="preserve">Idemq́; </s>
            <s xml:id="echoid-s1889" xml:space="preserve">oſtendemus de C H, H I,
              <lb/>
            ſi ex I, ducatur in primis figuris ipſi D E, parallela. </s>
            <s xml:id="echoid-s1890" xml:space="preserve">Vnde ſi B F, F G, ęquales fuerint in primis fi-
              <lb/>
            guris, erunt & </s>
            <s xml:id="echoid-s1891" xml:space="preserve">E K, K L, in primis, nec non & </s>
            <s xml:id="echoid-s1892" xml:space="preserve">B F, F G, & </s>
            <s xml:id="echoid-s1893" xml:space="preserve">C H, H I, in ſecundis, æquales, vt ex fi-
              <lb/>
            guris apparet. </s>
            <s xml:id="echoid-s1894" xml:space="preserve">Sumpſimus enim facilitatis gratia partes B F, F G, in primis figuris æquales.</s>
            <s xml:id="echoid-s1895" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">30</note>
          <p>
            <s xml:id="echoid-s1896" xml:space="preserve">POST Hæc circa diametros B C, F H, G I, ſemicirculi deſcribantur ſecantes rectam E M, in
              <lb/>
            punctis M, P, Q. </s>
            <s xml:id="echoid-s1897" xml:space="preserve">Habebuntur autem ſemidiametri, ſi axis coni in primis figuris ducatur ſecans ba-
              <lb/>
            ſim trianguli bifariam. </s>
            <s xml:id="echoid-s1898" xml:space="preserve">Hic enim diuidet etiam omnes diametros F H, G I, & </s>
            <s xml:id="echoid-s1899" xml:space="preserve">reliquas, bifariam,
              <lb/>
            vt in ſcholio propoſ. </s>
            <s xml:id="echoid-s1900" xml:space="preserve">4. </s>
            <s xml:id="echoid-s1901" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s1902" xml:space="preserve">6. </s>
            <s xml:id="echoid-s1903" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s1904" xml:space="preserve">oſtẽdimus. </s>
            <s xml:id="echoid-s1905" xml:space="preserve">Quare ſi in primis figuris accipiamus diſtantias in-
              <lb/>
            ter axem coni, & </s>
            <s xml:id="echoid-s1906" xml:space="preserve">puncta E, K, L, easq́; </s>
            <s xml:id="echoid-s1907" xml:space="preserve">transferamus in ſecundas figuras à puncto E, in lineã
              <unsure/>
            B E C,
              <lb/>
            vel ad partes B, vel ad partes C, prout primæ figuræ indicant, habebimus centra, &</s>
            <s xml:id="echoid-s1908" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1909" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1910" xml:space="preserve">POSTREMO diameter ſectionis conicæ D E, ſeorſum diuidatur, vt in cono, hoc eſt, E K,
              <lb/>
            K L, ęquales ſint partibus E K, KL, in cono, ſingulę ſingulis: </s>
            <s xml:id="echoid-s1911" xml:space="preserve">Et per E, K, L, ad D E, perpendicula-
              <lb/>
            res educantur; </s>
            <s xml:id="echoid-s1912" xml:space="preserve">quod quidem facile fiet, & </s>
            <s xml:id="echoid-s1913" xml:space="preserve">breuiſſimè, (præſertim quando plurima puncta fuerint
              <lb/>
            ſumpta in diametro D E,) ſi per E, perpendicularem eduxeris, à cuius duobus punctis ipſi D E, pa-
              <lb/>
              <note position="left" xlink:label="note-0047-09" xlink:href="note-0047-09a" xml:space="preserve">40</note>
            rallelæ erigantur, diuidanturq́ue, vt D E. </s>
            <s xml:id="echoid-s1914" xml:space="preserve">Nam rectæ puncta diuiſionum coniungẽtes erunt ad DE,
              <lb/>
            perpendiculares in punctis K, L, propterea quòd hac ratione ad rectas E K, E L, parallelogramma
              <lb/>
              <note position="right" xlink:label="note-0047-10" xlink:href="note-0047-10a" xml:space="preserve">29. primi.</note>
            ſint conſtituta, quæ rectangula ſunt, obangulum rectum ad E, conſtitutum, vt manifeſtum eſt.
              <lb/>
            </s>
            <s xml:id="echoid-s1915" xml:space="preserve">Quod ſi ordinatim applicatæ ad D E, diametrum ſectionis non ſint ad ipſam perpendiculares, (vt
              <lb/>
            fit in conis ſcalenis, cum triangulum per axem non eſt rectum ad baſim coni, vt conſtat ex propoſ. </s>
            <s xml:id="echoid-s1916" xml:space="preserve">
              <lb/>
            7. </s>
            <s xml:id="echoid-s1917" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s1918" xml:space="preserve">1. </s>
            <s xml:id="echoid-s1919" xml:space="preserve">Apollonii) ducendę erunt per puncta E, K, L, in tertiis figuris, lineę parallelę facientes an-
              <lb/>
            gulos ad diametrum D E, ęquales illis, quos ordinatim applicatę in primis figuris faciunt.</s>
            <s xml:id="echoid-s1920" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1921" xml:space="preserve">POST hęc ex perpendicularibus, parallelisve per puncta E, K, L, ductis, in tertiis figuris, ad
              <lb/>
            vtramque partem punctorum E, K, L, abſcindantur rectę E M, k P, L Q, rectis E M, E P, E Q, in
              <lb/>
            ſecundis figuris, ęquales, nimirum k P, ęqualis illi, quę inter diametrum F H, & </s>
            <s xml:id="echoid-s1922" xml:space="preserve">eius ſemicirculũ
              <lb/>
              <note position="left" xlink:label="note-0047-11" xlink:href="note-0047-11a" xml:space="preserve">50</note>
            intercipitur, qualis eſt E P, in ſecundis figuris; </s>
            <s xml:id="echoid-s1923" xml:space="preserve">& </s>
            <s xml:id="echoid-s1924" xml:space="preserve">L Q, equalis ipſi E Q, inter diametrum G I,
              <lb/>
            eiusq́ ſemicirculum poſitam, & </s>
            <s xml:id="echoid-s1925" xml:space="preserve">ſic de cęteris, obſeruando diligenter, quę puncta diametri D E,
              <lb/>
            quibus diametris ſemicirculorum reſpondeant. </s>
            <s xml:id="echoid-s1926" xml:space="preserve">Iam ſi puncta D, Q, P, &</s>
            <s xml:id="echoid-s1927" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1928" xml:space="preserve">appoſitè linea qua-
              <lb/>
            dam flexa coniunxeris, deſcripta erit ſectio conica propoſita, nempe Parabole, Hyperbole, vel El-
              <lb/>
            lipſis, vt mox demonſtrabimus. </s>
            <s xml:id="echoid-s1929" xml:space="preserve">Ex his manifeſtum eſt, quò crebriora fuerint puncta in diame-
              <lb/>
            tro D E, eò aptius ſectionem conicam deſcribi, vt vides factum eſſe in Hyperbola, & </s>
            <s xml:id="echoid-s1930" xml:space="preserve">Ellipſi vtra-
              <lb/>
            que; </s>
            <s xml:id="echoid-s1931" xml:space="preserve">ſumptum enim eſt in his ſectionibus aliud punctum præter K, L. </s>
            <s xml:id="echoid-s1932" xml:space="preserve">Quod ſi augere inſtituas
              <lb/>
            Parabolẽ, & </s>
            <s xml:id="echoid-s1933" xml:space="preserve">Hyperbolem, augendi erunt coni, & </s>
            <s xml:id="echoid-s1934" xml:space="preserve">puncta infra baſim B C, ſumenda ad ęqualitatẽ
              <unsure/>
              <lb/>
            punctorum F, G, &</s>
            <s xml:id="echoid-s1935" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1936" xml:space="preserve">vt figurę indicant. </s>
            <s xml:id="echoid-s1937" xml:space="preserve">Quod idem dicendum eſt de Ellipſi, cuius diameter ſecet
              <lb/>
            alterum latus trianguli per axem infra baſim, vt in ſecunda Ellipſi; </s>
            <s xml:id="echoid-s1938" xml:space="preserve">alias enim pars tantum Ellipſis
              <lb/>
            deſcriberetur M P Q D Q P M. </s>
            <s xml:id="echoid-s1939" xml:space="preserve">In priori porrò Ellipſi, cuius diameter baſim trianguli non </s>
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