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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          <p>
            <s xml:id="echoid-s5101" xml:space="preserve">
              <pb o="92" file="0112" n="112" rhead="GNOMONICES"/>
            lis ponitur arcui E L; </s>
            <s xml:id="echoid-s5102" xml:space="preserve">crit quoque G O, ipſi k T, æqualis. </s>
            <s xml:id="echoid-s5103" xml:space="preserve">Cum ergo G O, K T, ſint etiam paral-
              <lb/>
              <note position="left" xlink:label="note-0112-01" xlink:href="note-0112-01a" xml:space="preserve">28. primi.</note>
            lelæ, propterea quòd anguli O G T, k T G, recti ſint, ex conſtructione; </s>
            <s xml:id="echoid-s5104" xml:space="preserve">erunt & </s>
            <s xml:id="echoid-s5105" xml:space="preserve">G T, O K, æqua-
              <lb/>
              <note position="left" xlink:label="note-0112-02" xlink:href="note-0112-02a" xml:space="preserve">33. primi.</note>
            les, & </s>
            <s xml:id="echoid-s5106" xml:space="preserve">parallelæ. </s>
            <s xml:id="echoid-s5107" xml:space="preserve">Quare cum angulus A G O, rectus ſit, erit & </s>
            <s xml:id="echoid-s5108" xml:space="preserve">angulus G O k, rectus. </s>
            <s xml:id="echoid-s5109" xml:space="preserve">Quoniam igi
              <lb/>
              <note position="left" xlink:label="note-0112-03" xlink:href="note-0112-03a" xml:space="preserve">29. primi.</note>
              <figure xlink:label="fig-0112-01" xlink:href="fig-0112-01a" number="75">
                <image file="0112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0112-01"/>
              </figure>
            tur recta G O, ad rectas O L, O k, perpendicula-
              <lb/>
            ris eſt, erit eadem G O, ad planum per O L, O k, du-
              <lb/>
              <note position="left" xlink:label="note-0112-04" xlink:href="note-0112-04a" xml:space="preserve">4. vndec.</note>
              <figure xlink:label="fig-0112-02" xlink:href="fig-0112-02a" number="76">
                <image file="0112-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0112-02"/>
              </figure>
            ctum perpendicularis: </s>
            <s xml:id="echoid-s5110" xml:space="preserve">ac propterea & </s>
            <s xml:id="echoid-s5111" xml:space="preserve">planum cir-
              <lb/>
            culi D E B F, per G O, ductum, ad idem planum
              <lb/>
              <note position="left" xlink:label="note-0112-05" xlink:href="note-0112-05a" xml:space="preserve">38. vndec.</note>
            per O L, O K, ductum erit rectum. </s>
            <s xml:id="echoid-s5112" xml:space="preserve">Quare perpen-
              <lb/>
            dicularis ex k, in planum D E B F, demiſſa cidet
              <lb/>
            in rectam L N, communem ſectionem plani DE-
              <lb/>
              <note position="left" xlink:label="note-0112-06" xlink:href="note-0112-06a" xml:space="preserve">38. vndec.</note>
              <note position="left" xlink:label="note-0112-07" xlink:href="note-0112-07a" xml:space="preserve">10</note>
            B F, & </s>
            <s xml:id="echoid-s5113" xml:space="preserve">eius, quod per O L, O k, ducitur. </s>
            <s xml:id="echoid-s5114" xml:space="preserve">Quòd au-
              <lb/>
            tem in Q, cadat, ita demonſtrabitur. </s>
            <s xml:id="echoid-s5115" xml:space="preserve">Cadat, ſi he-
              <lb/>
            ri poteſt, in aliud punctum, vt in R. </s>
            <s xml:id="echoid-s5116" xml:space="preserve">Quoniam igi
              <lb/>
            tur L S, M P, parallelæ ſunt, erit vt L M, ad M G,
              <lb/>
              <note position="left" xlink:label="note-0112-08" xlink:href="note-0112-08a" xml:space="preserve">@. ſe@t@.</note>
            ita S P, ad P G; </s>
            <s xml:id="echoid-s5117" xml:space="preserve">& </s>
            <s xml:id="echoid-s5118" xml:space="preserve">componendo, vt L G, ad M G, ita
              <lb/>
            S G, ad P G: </s>
            <s xml:id="echoid-s5119" xml:space="preserve">Sed L G, M G, ipſis E G, H G, æqua-
              <lb/>
            les ſunt, per definitionem circuli; </s>
            <s xml:id="echoid-s5120" xml:space="preserve">& </s>
            <s xml:id="echoid-s5121" xml:space="preserve">S G, P G, ipſis
              <lb/>
              <note position="left" xlink:label="note-0112-09" xlink:href="note-0112-09a" xml:space="preserve">34. primi.</note>
            L O, Q O, æquales, ob rectangula S O, P O. </s>
            <s xml:id="echoid-s5122" xml:space="preserve">Ergo
              <lb/>
            erit quoque, vt E G, ad H G, ita L O, ad Q O; </s>
            <s xml:id="echoid-s5123" xml:space="preserve">& </s>
            <s xml:id="echoid-s5124" xml:space="preserve">
              <lb/>
            permutando, vt E G, ad L O, ita H G, ad Q O; </s>
            <s xml:id="echoid-s5125" xml:space="preserve">atq;
              <lb/>
            </s>
            <s xml:id="echoid-s5126" xml:space="preserve">
              <note position="left" xlink:label="note-0112-10" xlink:href="note-0112-10a" xml:space="preserve">20</note>
            adeò vt quadratum ex E G, ad quadratum ex L O, ita quadratum ex H G, ad quadratum ex Q O:
              <lb/>
            </s>
            <s xml:id="echoid-s5127" xml:space="preserve">
              <note position="left" xlink:label="note-0112-11" xlink:href="note-0112-11a" xml:space="preserve">@@. ſex@@.</note>
            Sed eſt, ex propoſ. </s>
            <s xml:id="echoid-s5128" xml:space="preserve">21. </s>
            <s xml:id="echoid-s5129" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5130" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5131" xml:space="preserve">Apollonij, vt quadratum ex E G, ad quadratum ex L O, ita rectangu-
              <lb/>
            gulum ſub B G, GD, ad rectangulum ſub B O, O D, quòd E G, L O, ſint ordinatim ductæ ad B D,
              <lb/>
            diametrum circuli D E B F, nempe perpendiculares. </s>
            <s xml:id="echoid-s5132" xml:space="preserve">Erit ergo quoque vt rectangulum ſub B G,
              <lb/>
            G D, ad rectangulum ſub B O, O D, ita quadratum ex H G, ad quadratum ex Q O. </s>
            <s xml:id="echoid-s5133" xml:space="preserve">Et quoniam, cũ
              <lb/>
            Ellipſis diametrorum D B, HI, ponatur tranſire per R, (eò quod in R, dicatur cadere perpendicu
              <lb/>
            laris ex K, demiſſa,) eſt quoque, per eandem propoſitionem 21. </s>
            <s xml:id="echoid-s5134" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5135" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5136" xml:space="preserve">Apollonii, vt rectangulum
              <lb/>
            ſub B G, G D, ad rectangulum ſub B O, O D, ita quadratum ex H G, ad quadratum ex R O, quòd
              <lb/>
            H G, R O, ſint ordinatim applicatę ad diametrum BD. </s>
            <s xml:id="echoid-s5137" xml:space="preserve">Erit igitur vt quadratum ex H G, ad qua-
              <lb/>
            dratum ex Q O, ita idem quadratum ex H G, ad quadratum ex R O. </s>
            <s xml:id="echoid-s5138" xml:space="preserve">Quare quadrata ex Q O,
              <lb/>
              <note position="left" xlink:label="note-0112-12" xlink:href="note-0112-12a" xml:space="preserve">9. quar@@.</note>
              <note position="left" xlink:label="note-0112-13" xlink:href="note-0112-13a" xml:space="preserve">30</note>
            R O, ęqualia ſunt, ac propterea & </s>
            <s xml:id="echoid-s5139" xml:space="preserve">lineę Q O, R O, ęquales, totum & </s>
            <s xml:id="echoid-s5140" xml:space="preserve">pars. </s>
            <s xml:id="echoid-s5141" xml:space="preserve">Quod eſt abſurdum.
              <lb/>
            </s>
            <s xml:id="echoid-s5142" xml:space="preserve">Perpendicularis ergo à k, demiſſa non cadit in aliud punctum, quàm in Q. </s>
            <s xml:id="echoid-s5143" xml:space="preserve">Eodem modo ſum-
              <lb/>
            ptis alijs punctis in circunferentia circuli A B C D, inueniemus, quo loco perpendicutares ab ip-
              <lb/>
            ſis ductę in planum circuli D E B F, cadant. </s>
            <s xml:id="echoid-s5144" xml:space="preserve">Quapropter in circunferentia circuli maximi in ſphę
              <lb/>
            ra ad alium circulum, &</s>
            <s xml:id="echoid-s5145" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5146" xml:space="preserve">Quod faciendum erat.</s>
            <s xml:id="echoid-s5147" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div319" type="section" level="1" n="121">
          <head xml:id="echoid-head124" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s5148" xml:space="preserve">EX his manifeſtè patet modus deſcribendæ Ellipſis, cuius diametri datæ ſint. </s>
            <s xml:id="echoid-s5149" xml:space="preserve">Si enim duę diametri
              <lb/>
            D B, H I, ita aptentur, vt ſeſe bifariam in G, & </s>
            <s xml:id="echoid-s5150" xml:space="preserve">ad angulos rectos ſecent; </s>
            <s xml:id="echoid-s5151" xml:space="preserve">& </s>
            <s xml:id="echoid-s5152" xml:space="preserve">ex centro G, & </s>
            <s xml:id="echoid-s5153" xml:space="preserve">interuallis
              <lb/>
              <note position="left" xlink:label="note-0112-14" xlink:href="note-0112-14a" xml:space="preserve">Quo mode de-
                <lb/>
              ſcribenda fit El
                <lb/>
              lipſis, cuius dia-
                <lb/>
              @@etri datæ ſint@</note>
              <figure xlink:label="fig-0112-03" xlink:href="fig-0112-03a" number="77">
                <image file="0112-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0112-03"/>
              </figure>
            G D, G H, circuli deſcribantur: </s>
            <s xml:id="echoid-s5154" xml:space="preserve">producta autem H I,
              <lb/>
              <note position="left" xlink:label="note-0112-15" xlink:href="note-0112-15a" xml:space="preserve">40</note>
            vtrinque, ſumantur arcus ęquales E L, L A, A B, B C,
              <lb/>
            quotcunque, & </s>
            <s xml:id="echoid-s5155" xml:space="preserve">his æquales F N, N k, k P, P R; </s>
            <s xml:id="echoid-s5156" xml:space="preserve">idemq́
              <lb/>
            @@
              <unsure/>
            at in altero ſemicirculo E B F; </s>
            <s xml:id="echoid-s5157" xml:space="preserve">iunganturq́ue puncta
              <lb/>
            L, A, B, &</s>
            <s xml:id="echoid-s5158" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5159" xml:space="preserve">cum centro G, rectis ſecantibus circulum
              <lb/>
            H M I, in partes, quæ ſimiles erunt partibus E L, L A,
              <lb/>
            &</s>
            <s xml:id="echoid-s5160" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5161" xml:space="preserve">exijs, quę in commentarijs in ſphæram ad finem
              <lb/>
            capitis 1. </s>
            <s xml:id="echoid-s5162" xml:space="preserve">ſcripſimus. </s>
            <s xml:id="echoid-s5163" xml:space="preserve">Deinde bina quælibet puncta cir
              <lb/>
            culi maioris à D, vel B, hinc inde remota æqualiter
              <lb/>
            connectantur lineis rectis; </s>
            <s xml:id="echoid-s5164" xml:space="preserve">itemq́; </s>
            <s xml:id="echoid-s5165" xml:space="preserve">bina quælibet pun-
              <lb/>
            cta circuli minoris ab H, vel I, hinc inde ęqualiter quo-
              <lb/>
            que remota alijs lineis rectis; </s>
            <s xml:id="echoid-s5166" xml:space="preserve">ac poſtremo puncta, vbi
              <lb/>
            coierint quęque duæ lineæ, quę per diuiſiones ſibi re-
              <lb/>
              <note position="left" xlink:label="note-0112-16" xlink:href="note-0112-16a" xml:space="preserve">50</note>
            ſpondentes tranſeunt, notentur, cadẽt ea omnia in El-
              <lb/>
            lipſim, cuius diametri D B, H I, vt demonſtratum eſt.
              <lb/>
            </s>
            <s xml:id="echoid-s5167" xml:space="preserve">Nam in ſuperiori figura oſtendimus punctum Q, vbi
              <lb/>
            coeuntrectæ L N, P M, quarum illa minori diametro
              <lb/>
            H I, hæc verò maiori D B, parallela eſt, cadere in El-
              <lb/>
            lipſim, quæ quidem parallelæ ducuntur per puncta L,
              <lb/>
            M, interſe reſpondentia, hoc eſt, auferentia arcus ſimiles E L, H M. </s>
            <s xml:id="echoid-s5168" xml:space="preserve">Cum ergo hic idem fiat, propterea
              <lb/>
            quòd lineę omnes puncta correſpondentia connectentes parallelæ ſunt diametris H I, D B, ex ijs, quę in
              <lb/>
            ſcholio propoſ. </s>
            <s xml:id="echoid-s5169" xml:space="preserve">27. </s>
            <s xml:id="echoid-s5170" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5171" xml:space="preserve">3. </s>
            <s xml:id="echoid-s5172" xml:space="preserve">Euclidis demonſtrauimus, perſpicuum eſt, omnia illa puncta in Ellipſim cadere. </s>
            <s xml:id="echoid-s5173" xml:space="preserve">
              <lb/>
            Idẽ fiet, etiã ſi arcus E L, L A, &</s>
            <s xml:id="echoid-s5174" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5175" xml:space="preserve">nõ fint æquales, dummodo per puncta reſpondentia L, M, &</s>
            <s xml:id="echoid-s5176" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5177" xml:space="preserve">ducant@r
              <lb/>
            parallelæ diametris Ellipſis, &</s>
            <s xml:id="echoid-s5178" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5179" xml:space="preserve">Quare ſi lineam appoſitè, congruenterq́; </s>
            <s xml:id="echoid-s5180" xml:space="preserve">eiu@modi puncta
              <unsure/>
            coniungentem
              <lb/>
            duxerimus, Ellipſis deſcripta erit. </s>
            <s xml:id="echoid-s5181" xml:space="preserve">Quod erat faciendum.</s>
            <s xml:id="echoid-s5182" xml:space="preserve"/>
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