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 style="it">
            <s xml:id="echoid-s2182" xml:space="preserve">
              <pb o="32" file="0052" n="52" rhead="GNOMONICES"/>
            QI, quàm rectangulum ſub I R, R F, ad candem I F, applicatum, cxcedens{q́ue} quadrato
              <unsure/>
              <lb/>
            ex R F, æquale eſſe quadrato ex A C, hoceſt, quartæ partirectanguli ſub F I, F O. </s>
            <s xml:id="echoid-s2183" xml:space="preserve">De-
              <lb/>
            ſcripto enim ex D I, quadrato D E, ducatur per Q, ipſi I E, parallela P N, occurrensrectæ
              <lb/>
            G E, productæ in P, & </s>
            <s xml:id="echoid-s2184" xml:space="preserve">diametro G I, productæ in N, perficiatur{q́ue} figura, vt vides. </s>
            <s xml:id="echoid-s2185" xml:space="preserve">Quo-
              <lb/>
            niam igitur pallelogramma D E, M P, N I, circa eandem diametrum exiſtentia ſimilia
              <lb/>
              <note position="left" xlink:label="note-0052-01" xlink:href="note-0052-01a" xml:space="preserve">24. ſexti.</note>
            ſunt, eſt{q́ue} D E, quadratum; </s>
            <s xml:id="echoid-s2186" xml:space="preserve">erunt quo-
              <lb/>
              <note position="left" xlink:label="note-0052-02" xlink:href="note-0052-02a" xml:space="preserve">47. primi.</note>
              <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a" number="35">
                <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0052-01"/>
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            que M P, N I, quadrata. </s>
            <s xml:id="echoid-s2187" xml:space="preserve">Et quoniam
              <lb/>
            quadratum ex H L, æquale eſt quadratis
              <lb/>
            ex H K, K L; </s>
            <s xml:id="echoid-s2188" xml:space="preserve">eſt autem recta H L, rectæ
              <lb/>
              <note position="left" xlink:label="note-0052-03" xlink:href="note-0052-03a" xml:space="preserve">10</note>
            D Q, ſeu M N, & </s>
            <s xml:id="echoid-s2189" xml:space="preserve">recta H K, rectæ AC,
              <lb/>
            & </s>
            <s xml:id="echoid-s2190" xml:space="preserve">recta K L, rectæ D I, æqualis, ex con-
              <lb/>
            ſtructione; </s>
            <s xml:id="echoid-s2191" xml:space="preserve">Erit quoque quadratũ M P,
              <lb/>
            ex D Q, ſeu M N, deſoriptum, æquale
              <lb/>
            quadrato D E, ex D I, deſcripto, vnà cũ
              <lb/>
            quadrato ex A C. </s>
            <s xml:id="echoid-s2192" xml:space="preserve">Quare ablato commu-
              <lb/>
            ni quadrato D E, erit@ reliqu{us} gnomon
              <lb/>
            D N E, æqualis reliquo quadrato ex AC.
              <lb/>
            </s>
            <s xml:id="echoid-s2193" xml:space="preserve">
              <note position="left" xlink:label="note-0052-04" xlink:href="note-0052-04a" xml:space="preserve">20</note>
            Cum ergo gnomon D N E, æqualis ſit re-
              <lb/>
            ctangulo F N, (Nam cum F M, ipſi M I,
              <lb/>
              <note position="left" xlink:label="note-0052-05" xlink:href="note-0052-05a" xml:space="preserve">36. primi.</note>
            hoc eſt, ipſi I P, æquale ſit, addito communi M Q, æquale erit F N, gnomoni D N E,) eris
              <unsure/>
              <lb/>
              <note position="left" xlink:label="note-0052-06" xlink:href="note-0052-06a" xml:space="preserve">43. primi.</note>
            quoque rectangulum F N, contcntum ſuh F Q, Q I, (quòdrecta Q I, rectæ Q N, æqua-
              <lb/>
            lis ſit, ob quadratum I N, %%%% æquale quadrato ex A C, hoc eſt, quartæ parti quadrati ex AB,
              <lb/>
            hoc eſt, rectanguli ſub F I, F O, comprehenſi. </s>
            <s xml:id="echoid-s2194" xml:space="preserve">Applicatum eſt ergo ad F I, diametrũ tranſ-
              <lb/>
            uerſam rectangulum ſub F Q, Q I, æquale quartæ parti rectanguli ſub F I, F O, exce-
              <lb/>
            dens quadrato rectæ Q I. </s>
            <s xml:id="echoid-s2195" xml:space="preserve">Eodem modo demonſtr abitur rectangulum ſub I R, R F, ap-
              <lb/>
            plicatum ad F I, excedens{q́ue} quadrato ex R F, æquale eſſe quartæ parti rectanguli ſub F I,
              <lb/>
              <note position="left" xlink:label="note-0052-07" xlink:href="note-0052-07a" xml:space="preserve">30</note>
            F O. </s>
            <s xml:id="echoid-s2196" xml:space="preserve">Quod est propoſitum.</s>
            <s xml:id="echoid-s2197" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2198" xml:space="preserve">HIS præmiſſis, ſit F I, axis tranſuerſus duarum hyperbolarum oppoſitarum F G H, I K L, vt in
              <lb/>
              <note position="left" xlink:label="note-0052-08" xlink:href="note-0052-08a" xml:space="preserve">Alia deſcriptio
                <lb/>
              hyperbolarum
                <lb/>
              oppoſitarum@in
                <lb/>
              plano.</note>
            figur a primi lemmatis, & </s>
            <s xml:id="echoid-s2199" xml:space="preserve">latus rectum F O, datum ex eodem primo lemmate, applicetur per ſecun-
              <lb/>
            dum lemma ad F I, ex vtraque parte rectangulum ſub F Q, Q I, & </s>
            <s xml:id="echoid-s2200" xml:space="preserve">I R, R F, quartæ parti rectanguli
              <lb/>
            ſub F I, F O, æquale, excedens{q́ue} quadrato ex I Q, & </s>
            <s xml:id="echoid-s2201" xml:space="preserve">F R, & </s>
            <s xml:id="echoid-s2202" xml:space="preserve">infra R, ſumantur vtcunque puncta
              <lb/>
            quotlibet A, B, C, D. </s>
            <s xml:id="echoid-s2203" xml:space="preserve">Deinde ad interuallum I A, deſcribantur ex punctis Q, & </s>
            <s xml:id="echoid-s2204" xml:space="preserve">R, quatuor arcus,
              <lb/>
            quos inpuncto E, ſecent alij quatuor arcus ex eiſdem punctis Q, & </s>
            <s xml:id="echoid-s2205" xml:space="preserve">R, ad interuallum F A, deſcripti.
              <lb/>
            </s>
            <s xml:id="echoid-s2206" xml:space="preserve">Item ex eiſdem punctis Q, & </s>
            <s xml:id="echoid-s2207" xml:space="preserve">R, ad interuallum I B, quatuor arcus deſcribantur, quos in puncto G, in-
              <lb/>
            terſecent alij quatuor ex eiſdem punctis Q, & </s>
            <s xml:id="echoid-s2208" xml:space="preserve">R, deſcripti ad interuallum F B. </s>
            <s xml:id="echoid-s2209" xml:space="preserve">Eodem modo ad inter-
              <lb/>
              <note position="left" xlink:label="note-0052-09" xlink:href="note-0052-09a" xml:space="preserve">40</note>
            ualla I C, F C, ex punctis Q, & </s>
            <s xml:id="echoid-s2210" xml:space="preserve">R, arcus deſcripti ſe mutuo ſecent in H, & </s>
            <s xml:id="echoid-s2211" xml:space="preserve">ſic de cæte
              <unsure/>
            ris punctis, ſi
              <lb/>
            quaſint; </s>
            <s xml:id="echoid-s2212" xml:space="preserve">obſeruando ſemper, vt bini maiores arcus ex ſingulis quatuor, qui ex Q, & </s>
            <s xml:id="echoid-s2213" xml:space="preserve">R, deſcribendi
              <lb/>
            ſunt, deſcribantur ex Q, vltra punctum F, & </s>
            <s xml:id="echoid-s2214" xml:space="preserve">bini ex R, vltra punctum I, bini autem minores ex Q,
              <lb/>
            citra punctum I, & </s>
            <s xml:id="echoid-s2215" xml:space="preserve">bini ex R, citra punctum F. </s>
            <s xml:id="echoid-s2216" xml:space="preserve">Nam per puncta F, E, G, H, & </s>
            <s xml:id="echoid-s2217" xml:space="preserve">I, E, G, H, oppoſitæ
              <lb/>
            hyperbolæ deſcribendæ erunt. </s>
            <s xml:id="echoid-s2218" xml:space="preserve">Quoniam enim recta Q E, hoc est, I A, ſuperat rectam E R, hoc eſt,
              <lb/>
            F A, diametro tranſuerſa F I; </s>
            <s xml:id="echoid-s2219" xml:space="preserve">Item recta Q G, rectam G R, eadem diametro ſuperat, & </s>
            <s xml:id="echoid-s2220" xml:space="preserve">ſic de cæ-
              <lb/>
            teris, tranſibunt hyperbolæ oppoſitæ, quarum axis F I, & </s>
            <s xml:id="echoid-s2221" xml:space="preserve">vertices F, I, per puncta E, G, H, quan-
              <lb/>
            doquidem, vt vult propoſitio 51. </s>
            <s xml:id="echoid-s2222" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2223" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2224" xml:space="preserve">Apollonij, ſi lineæ rectæ ex punctis Q, R, ad vnum idem{q́ue} pun-
              <lb/>
            ctum Hyperboles inclinentur, maior minorcm ſuperat ipſo axe F I. </s>
            <s xml:id="echoid-s2225" xml:space="preserve">Si enim hyperbole, cuius axis F I,
              <lb/>
            & </s>
            <s xml:id="echoid-s2226" xml:space="preserve">vertex F, non tranſit per punctum E, tranſeat, ſi fieri pote
              <unsure/>
            ſt, per K, ſecans rectam Q E, in K, ſiue in-
              <lb/>
              <note position="left" xlink:label="note-0052-10" xlink:href="note-0052-10a" xml:space="preserve">50</note>
            fra E, ſiue ſupra; </s>
            <s xml:id="echoid-s2227" xml:space="preserve">coniungstur{q́ue} recta R K. </s>
            <s xml:id="echoid-s2228" xml:space="preserve">Quoniam igitur Hyperbole tranſit per K, ſuperabit recta
              <lb/>
            Q K, rectam K R, axe F I, per propoſ. </s>
            <s xml:id="echoid-s2229" xml:space="preserve">51. </s>
            <s xml:id="echoid-s2230" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2231" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2232" xml:space="preserve">Apollonij: </s>
            <s xml:id="echoid-s2233" xml:space="preserve">Sed eodem axe F I, ſuperat ex conſtructio-
              <lb/>
            ne recta Q E, rectam E R. </s>
            <s xml:id="echoid-s2234" xml:space="preserve">Idem ergo eſt exceſſus inter rectas Q K, K R, qui inter rectas Q E, E R.
              <lb/>
            </s>
            <s xml:id="echoid-s2235" xml:space="preserve">Quare permutando, ex lemmate propoſ. </s>
            <s xml:id="echoid-s2236" xml:space="preserve">79. </s>
            <s xml:id="echoid-s2237" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2238" xml:space="preserve">10. </s>
            <s xml:id="echoid-s2239" xml:space="preserve">Euclidis, idem exccſſus erit inter rectas Q K, Q E,
              <lb/>
            qui inter rectas K R, E R. </s>
            <s xml:id="echoid-s2240" xml:space="preserve">Cum ergo exceſſus inter Q K, & </s>
            <s xml:id="echoid-s2241" xml:space="preserve">Q E:</s>
            <s xml:id="echoid-s2242" xml:space="preserve">, ſit recta E K, erit quoque eadem re-
              <lb/>
            cta E K, exceſſus inter K R, & </s>
            <s xml:id="echoid-s2243" xml:space="preserve">E R. </s>
            <s xml:id="echoid-s2244" xml:space="preserve">Quare recta E K, addita minori@earum, fiet aggregatum exhis
              <lb/>
            duabus reliquæ æquale, ac proinde duo later a trianguli E K R, reliquo lateri æqualia erunt, ſed & </s>
            <s xml:id="echoid-s2245" xml:space="preserve">maio-
              <lb/>
              <note position="left" xlink:label="note-0052-11" xlink:href="note-0052-11a" xml:space="preserve">20. primi.</note>
            ra ſunt. </s>
            <s xml:id="echoid-s2246" xml:space="preserve">Quod eſt abſurdum. </s>
            <s xml:id="echoid-s2247" xml:space="preserve">Non ergo dicta Hyperbole per punctum K, ſed per E, tranſibit. </s>
            <s xml:id="echoid-s2248" xml:space="preserve">Eodem{q́ue}
              <lb/>
            pacto oſtendemus eandem pe
              <unsure/>
            r reliqua puncta G, H, &</s>
            <s xml:id="echoid-s2249" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2250" xml:space="preserve">tranſire, quod est propoſitum.</s>
            <s xml:id="echoid-s2251" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2252" xml:space="preserve">MANIFESTVM autem eſt, deſcriptione
              <unsure/>
            m hanc ſolum conuenire conis rectis, vel etiam Sca-
              <lb/>
            lenis, in quibus triangula per axem ad baſes conorum recta ſunt; </s>
            <s xml:id="echoid-s2253" xml:space="preserve">quia in his dunt axat diamc
              <unsure/>
            ter </s>
          </p>
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