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-s2308" xml:space="preserve">
              <pb o="34" file="0054" n="54" rhead="GNOMONICES"/>
            nor quàm E F. </s>
            <s xml:id="echoid-s2309" xml:space="preserve">Quare & </s>
            <s xml:id="echoid-s2310" xml:space="preserve">A C, dimidia ipſius A B, minor erit quàm D F, dimidia ipſius
              <lb/>
            E F. </s>
            <s xml:id="echoid-s2311" xml:space="preserve">Deſcripto igitur circa D F, ſemicirculo, accommodetur in eorecta F I, æqualis re-
              <lb/>
              <note position="left" xlink:label="note-0054-01" xlink:href="note-0054-01a" xml:space="preserve">1. quarti.</note>
            ctæ A C, quæ minor eſt ostenſa, quàm D F, ſubtendatur{q́ue} recta D I, quæ minor quoque
              <lb/>
              <note position="left" xlink:label="note-0054-02" xlink:href="note-0054-02a" xml:space="preserve">15. tertij.</note>
            erit, quàm D F. </s>
            <s xml:id="echoid-s2312" xml:space="preserve">Abſcindantur vtrinque ex D, rectæ D N, D O, ipſi D I, æquales. </s>
            <s xml:id="echoid-s2313" xml:space="preserve">Dico
              <lb/>
            t
              <unsure/>
            am rectangulum ſub E N, N F, ad rectam E F, applicatum, deficiens{q́ue} quadrato ex N F,
              <lb/>
            quàm rectangulum ſub F O, O E, ad eandem rectam E F, applicatum, deficiens{q́ue} qua-
              <lb/>
            drato ex O E, æquale eße quadrato ex A C, hoc eſt, quartæ parti rectanguli ſub E F, E I.
              <lb/>
            </s>
            <s xml:id="echoid-s2314" xml:space="preserve">Deſcripto enim ex D F, quadrato D H, perficia-
              <lb/>
              <note position="left" xlink:label="note-0054-03" xlink:href="note-0054-03a" xml:space="preserve">10</note>
              <figure xlink:label="fig-0054-01" xlink:href="fig-0054-01a" number="38">
                <image file="0054-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0054-01"/>
              </figure>
            tur figura, vt vides. </s>
            <s xml:id="echoid-s2315" xml:space="preserve">Quoniam igitur parallelo-
              <lb/>
              <note position="left" xlink:label="note-0054-04" xlink:href="note-0054-04a" xml:space="preserve">34. ſexti.</note>
            gramma D H, G L, L F, circa eandem diame-
              <lb/>
            trum exiſtentia, ſimilia ſunt, eſt{q́ue} D H, quadra-
              <lb/>
            tum, erunt quoque G L, L F, quadrata. </s>
            <s xml:id="echoid-s2316" xml:space="preserve">Et quia
              <lb/>
            quadratum D H, æquale est quadratis ex F I,
              <lb/>
              <note position="left" xlink:label="note-0054-05" xlink:href="note-0054-05a" xml:space="preserve">47. primi.</note>
            D I, hoc eſt, quadrato ex A C, vna cum quadra-
              <lb/>
            to G L; </s>
            <s xml:id="echoid-s2317" xml:space="preserve">(Eſt enim angul{us} D I F, rect{us}, & </s>
            <s xml:id="echoid-s2318" xml:space="preserve">rectæ
              <lb/>
              <note position="left" xlink:label="note-0054-06" xlink:href="note-0054-06a" xml:space="preserve">31. tertij.</note>
            F I, D I, æquales fuerunt rectis A C, D N, vel
              <lb/>
            K L.) </s>
            <s xml:id="echoid-s2319" xml:space="preserve">erit gnomon K N H, quadrato ex A C, æqualis. </s>
            <s xml:id="echoid-s2320" xml:space="preserve">Cum ergo gnomon K N H, æqua-
              <lb/>
              <note position="left" xlink:label="note-0054-07" xlink:href="note-0054-07a" xml:space="preserve">20</note>
            lis quoque ſit rectangulo E L, (Nam cum E K, ipſi K F, hoc eſt, ipſi N H, æquale ſit; </s>
            <s xml:id="echoid-s2321" xml:space="preserve">addi-
              <lb/>
              <note position="left" xlink:label="note-0054-08" xlink:href="note-0054-08a" xml:space="preserve">36. primi.</note>
            to communi D L, fit totum E L, toti gnomoni K N H, æquale) erit quoque rectangulum
              <lb/>
            E L, contentum ſub E N, N F; </s>
            <s xml:id="echoid-s2322" xml:space="preserve">(quòd recta N F, rectæ N L, æqualis ſit, ob quadratum
              <lb/>
            L F.) </s>
            <s xml:id="echoid-s2323" xml:space="preserve">æquale quadrato ex A C. </s>
            <s xml:id="echoid-s2324" xml:space="preserve">Applicatum eſt ergo ad E F, diametrum tranſuerſam
              <lb/>
            rectangulum ſub E N, N F, æquale quartæ parti rectanguli ſub E F, E I, deficiens{q́ue} qua-
              <lb/>
            drato rectæ N F. </s>
            <s xml:id="echoid-s2325" xml:space="preserve">Eodem modo demonſtrabitur rectangulum ſub F O, O E, applicatum ad
              <lb/>
            E F, deficiens{q́ue} quadrato ex E O, æquale eße quartæ parti rectanguli ſub E F, E I. </s>
            <s xml:id="echoid-s2326" xml:space="preserve">Quod
              <lb/>
            eſt propoſitum.</s>
            <s xml:id="echoid-s2327" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">Alia deſcriptio
            <lb/>
          Ellipſis in pla-
            <lb/>
          no.</note>
          <p style="it">
            <s xml:id="echoid-s2328" xml:space="preserve">HIS præmiſſis ſit E F, axis tranſuerſus Ellipſis E F, & </s>
            <s xml:id="echoid-s2329" xml:space="preserve">lat{us} rectum E I, datum ex lemmate 1.
              <lb/>
            </s>
            <s xml:id="echoid-s2330" xml:space="preserve">
              <note position="left" xlink:label="note-0054-10" xlink:href="note-0054-10a" xml:space="preserve">30</note>
            Applicetur per 2. </s>
            <s xml:id="echoid-s2331" xml:space="preserve">lemma, ad E F, ex vtraque parte rectangulum tam ſub F O, O E, quàm ſub E N,
              <lb/>
            N F, quartæ parti rectanguli ſub E F, E I, æquale, quorum illud quidem deficiat quadrato ex E O, hoc
              <lb/>
            vero, quadrato ex F N. </s>
            <s xml:id="echoid-s2332" xml:space="preserve">Et diuiſa N O, bifariam in A, ſumantur inter A, & </s>
            <s xml:id="echoid-s2333" xml:space="preserve">N, quotlibet puncta vt-
              <lb/>
            cunque B, C, D. </s>
            <s xml:id="echoid-s2334" xml:space="preserve">Deinde ad interuallum E A, vel F A, ex punctis O, & </s>
            <s xml:id="echoid-s2335" xml:space="preserve">N, deſcribantur quatuor ar-
              <lb/>
            cus ſemutuo ſecantes hinc inde in G. </s>
            <s xml:id="echoid-s2336" xml:space="preserve">Item ex eiſdem punctis O, & </s>
            <s xml:id="echoid-s2337" xml:space="preserve">N, ad interuallum E B, quatuor ar-
              <lb/>
            cus deſcribantur, quos in puncto H, ſecent alij quatuor arcus ex eiſdem punctis ad interuallum F B, de-
              <lb/>
            ſcripti. </s>
            <s xml:id="echoid-s2338" xml:space="preserve">Eodem modo ad interualla E C, F C, ex eiſdem punctis O, & </s>
            <s xml:id="echoid-s2339" xml:space="preserve">N, arcus deſcripti ſemutuo ſecent
              <lb/>
            in I; </s>
            <s xml:id="echoid-s2340" xml:space="preserve">& </s>
            <s xml:id="echoid-s2341" xml:space="preserve">ſic de cæteris punctis, ſi qua ſint; </s>
            <s xml:id="echoid-s2342" xml:space="preserve">obſeruando ſemper, vt bini maiorcs arcus ex ſingulis quatuor,
              <lb/>
            qui ex O, & </s>
            <s xml:id="echoid-s2343" xml:space="preserve">N, deſcribendi ſunt, deſcribantur ex O, vltra punctum A, & </s>
            <s xml:id="echoid-s2344" xml:space="preserve">bini ex N, vltra idem pun-
              <lb/>
            ctum A; </s>
            <s xml:id="echoid-s2345" xml:space="preserve">bini autem minores ex O, citra punctum A, & </s>
            <s xml:id="echoid-s2346" xml:space="preserve">bini ex N, citra idem punctum A. </s>
            <s xml:id="echoid-s2347" xml:space="preserve">Nam per
              <lb/>
              <note position="left" xlink:label="note-0054-11" xlink:href="note-0054-11a" xml:space="preserve">40</note>
            puncta E, G, H, I, F, Ellipſis erit deſcribenda. </s>
            <s xml:id="echoid-s2348" xml:space="preserve">Quoniam enimtam rectæ N G, O G, hoc eſt, E A, F A,
              <lb/>
            quàm rectæ N H, O H, id est, E B, F B, &</s>
            <s xml:id="echoid-s2349" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2350" xml:space="preserve">axi E F, æquales ſunt, tranſibit Ellipſis, cuius axis E F,
              <lb/>
            per puncta E, G, H, I, F; </s>
            <s xml:id="echoid-s2351" xml:space="preserve">quandoquidem, vt vult propoſitio 52. </s>
            <s xml:id="echoid-s2352" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2353" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2354" xml:space="preserve">Apollonij, lineæ rectæ ex punctis
              <lb/>
            N, O, ad vnum idem{q́ue} Ellipſis punctum inclinatæ æquales ſunt axi E F. </s>
            <s xml:id="echoid-s2355" xml:space="preserve">Sinamque dicta Ellipſis non
              <lb/>
            tranſit per punctum I, tranſeat, ſifieri poteſt, per K, ſecans rectam N I, in K, vel vltra, vel citra I,
              <lb/>
            iungatur{q́ue}, recta O K. </s>
            <s xml:id="echoid-s2356" xml:space="preserve">Quoniam igitur Ellipſis prædicta tranſit per K, erunt rectæ N K, O K, ſimul æqua-
              <lb/>
            les axi E F, ex propoſ. </s>
            <s xml:id="echoid-s2357" xml:space="preserve">52. </s>
            <s xml:id="echoid-s2358" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2359" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2360" xml:space="preserve">Apollonij: </s>
            <s xml:id="echoid-s2361" xml:space="preserve">Sed per conſtructionem & </s>
            <s xml:id="echoid-s2362" xml:space="preserve">rectæ N I, O I, eidem axi E F,
              <lb/>
            æquales ſunt. </s>
            <s xml:id="echoid-s2363" xml:space="preserve">Igitur N K, O K, rectis N I, O I, æquales erunt: </s>
            <s xml:id="echoid-s2364" xml:space="preserve">Sed & </s>
            <s xml:id="echoid-s2365" xml:space="preserve">inæquales ſunt; </s>
            <s xml:id="echoid-s2366" xml:space="preserve">(Nam cadente pun-
              <lb/>
            cto K, vltra I, erunt rectæ I K, K O, maiores recta I O; </s>
            <s xml:id="echoid-s2367" xml:space="preserve">addita ergo communi I N, erunt N K, O K,
              <lb/>
              <note position="left" xlink:label="note-0054-12" xlink:href="note-0054-12a" xml:space="preserve">20. primi.</note>
            maiores, quàm N I, O I: </s>
            <s xml:id="echoid-s2368" xml:space="preserve">Cadente verò puncto K, citra I, erunt I K, I O, maiores, quàm K O. </s>
            <s xml:id="echoid-s2369" xml:space="preserve">addi-
              <lb/>
              <note position="left" xlink:label="note-0054-13" xlink:href="note-0054-13a" xml:space="preserve">20. primi.</note>
              <note position="left" xlink:label="note-0054-14" xlink:href="note-0054-14a" xml:space="preserve">50</note>
            ta ergo communi K N, erunt N I, O I, maiores, quàm N k, O K) Quod est abſurdum. </s>
            <s xml:id="echoid-s2370" xml:space="preserve">Non ergo dicta
              <lb/>
            Ellipſis per aliud punctum, quàm per I, tranſibit. </s>
            <s xml:id="echoid-s2371" xml:space="preserve">Eodc
              <unsure/>
            m{q́ue} modo demonſtrabimus eandem per reliqua
              <lb/>
            puncta H, G, &</s>
            <s xml:id="echoid-s2372" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2373" xml:space="preserve">tranſire, quod eſt propoſitum.</s>
            <s xml:id="echoid-s2374" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2375" xml:space="preserve">PERSPICVVM etiam eſt, hanc deſcriptionem non conuenire conis ſcalenis, niſi cum triangula
              <lb/>
              <note position="left" xlink:label="note-0054-15" xlink:href="note-0054-15a" xml:space="preserve">Qua ratione
                <lb/>
              Parabola qua-
                <lb/>
              liſ
                <unsure/>
              unque in
                <lb/>
              plano deſcriba-
                <lb/>
              tur.</note>
            per axem ad baſes conorum recta ſunt. </s>
            <s xml:id="echoid-s2376" xml:space="preserve">Tunc enim ſolum diameter ellipſis ad angulos rectos ſecat ordi-
              <lb/>
            natim applicatas, vt ex propoſ. </s>
            <s xml:id="echoid-s2377" xml:space="preserve">7. </s>
            <s xml:id="echoid-s2378" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s2379" xml:space="preserve">1. </s>
            <s xml:id="echoid-s2380" xml:space="preserve">Apoll. </s>
            <s xml:id="echoid-s2381" xml:space="preserve">conſtat, atque adeo axis eſt.</s>
            <s xml:id="echoid-s2382" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2383" xml:space="preserve">QVOD ſi vt cunque Parabolã aliquam, Hyperbolã, vel etiã duas oppoſitas, aut Ellipſim deſcribere
              <unsure/>
              <lb/>
            velimus, nulla habita ratione conorũ, à quibus oriuntur, accipiemus pro parabola axem cuiuſcunque ma-
              <lb/>
            gnitudinis E H, vt in ſuperiori parabola, & </s>
            <s xml:id="echoid-s2384" xml:space="preserve">in eo quotcunque partes æquales vt libet, et per puncta termi
              <lb/>
            nantia primam partem, & </s>
            <s xml:id="echoid-s2385" xml:space="preserve">ſequẽtes tres, & </s>
            <s xml:id="echoid-s2386" xml:space="preserve">ſequentes quinque, & </s>
            <s xml:id="echoid-s2387" xml:space="preserve">ſeque
              <unsure/>
            ntes ſeptem, &</s>
            <s xml:id="echoid-s2388" xml:space="preserve">c. </s>
            <s xml:id="echoid-s2389" xml:space="preserve">ducemus lineas
              <lb/>
            inter ſe parallelas; </s>
            <s xml:id="echoid-s2390" xml:space="preserve">ſumpta autem ex prima, quantacunque linea vtrinque A D, accipiemus eius </s>
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