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

Table of contents

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[101.] In horologio, quod circulo horæ 7. à meridie vel media nocte æquidiſtat, lineæ quarumlibet duarum horarum \\ huius tabulæ ſunt parallelæ, vt ex ſcholio propoſ. 22. liquido conſtat.
Page: 94 (74)
[102.] Linea horæ 5. à meridie vel media nocte.
Page: 94 (74)
[103.] Linea horæ quartæ à meridie vel media nocte.
Page: 95 (75)
[104.] In horologio, quod circulo horæ quartæ à merdie vel media nocte æquidiſtat, lineæ quarumlibet duarum hora- \\ rum huius tabulæ ſunt parallelæ, vt ex ſcholio propoſ. 22. manifeſte colligitur.
Page: 95 (75)
[105.] Linea horæ tertiæ à meridie vel media nocte.
Page: 95 (75)
[106.] Linea horæ ſ cundæ à meridie vel media nocte.
Page: 96 (76)
[107.] Linea horæ primæ a meridie vel media nocte.
Page: 96 (76)
[108.] THEOREMA 19. PROPOSITIO 21.
Page: 97 (77)
[109.] COROLLARIVM.
Page: 97 (77)
[110.] SCHOLIVM.
Page: 97 (77)
[111.] THEOREMA 20. PROPOSITIO 22.
Page: 97 (77)
[112.] COROLLARIVM.
Page: 97 (77)
[113.] SCHOLIVM.
Page: 98 (78)
[114.] PROBLEMA 3. PROPOSITIO 23.
Page: 99 (79)
[115.] SCHOLIVM.
Page: 102 (82)
[116.] THEOREMA 21. PROPOSITIO 24.
Page: 109 (89)
[117.] SCHOLIVM.
Page: 110 (90)
[118.] PROBLEMA 4. PROPOSITIO 25.
Page: 110 (90)
[119.] COROLLARIVM.
Page: 111 (91)
[120.] PROBLEMA 5. PROPOSITIO 26.
Page: 111 (91)
[121.] COROLLARIVM.
Page: 112 (92)
[122.] PROBLEMA 6. PROPOSITIO 27.
Page: 113 (93)
[123.] PROBLEMA. 7. PROPOSITIO 28.
Page: 113 (93)
[124.] SCHOLIVM I.
Page: 115 (95)
[125.] COROLLARIVM.
Page: 115 (95)
[126.] SCHOLIVM II.
Page: 116 (96)
[127.] PROBLEMA 8. PROPOSITIO 29.
Page: 119 (99)
[128.] PROBLEMA. 9. PROPOSITIO 30.
Page: 121 (101)
[129.] PROBLEMA 10. PROPOSITIO 31.
Page: 123 (103)
[130.] PROBLEMA 11. PROPOSITIO 32.
Page: 125 (105)
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          <p>
            <s xml:id="echoid-s4986" xml:space="preserve">
              <pb o="90" file="0110" n="110" rhead="GNOMONICES"/>
            M L, ita rectangulum ſub C E, E A, ad rectangulum ſub C M, M A. </s>
            <s xml:id="echoid-s4987" xml:space="preserve">Sed ex eadem propoſ. </s>
            <s xml:id="echoid-s4988" xml:space="preserve">21.
              <lb/>
            </s>
            <s xml:id="echoid-s4989" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4990" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4991" xml:space="preserve">Apollonij, ſi circa diametrum maiorem AC, & </s>
            <s xml:id="echoid-s4992" xml:space="preserve">minorem H I, (Eſt enim A C, maior quàm
              <lb/>
            H I, cum A C, diameter circuli A F C G, ęqualis ſit diametro F G, eiuſdem circuli) ellipſis deſcri-
              <lb/>
            batur, & </s>
            <s xml:id="echoid-s4993" xml:space="preserve">ex quouis puncto ipſi H E, parallela du-
              <lb/>
              <figure xlink:label="fig-0110-01" xlink:href="fig-0110-01a" number="72">
                <image file="0110-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0110-01"/>
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            catur, hoc eſt, ordinatim applicata ad diametrũ
              <lb/>
            A C, quadratum ex H E, ad quadratum illius pa-
              <lb/>
            rallelæ eſt, vt rectangulum ſub C E, E A, ad re-
              <lb/>
            ctangulum ſub partibus diametri A C, quas pa-
              <lb/>
            rallela illa facit. </s>
            <s xml:id="echoid-s4994" xml:space="preserve">Igitur punctum L, in illam El-
              <lb/>
            lipſim cadet, cuius maior diameter A C, & </s>
            <s xml:id="echoid-s4995" xml:space="preserve">mi-
              <lb/>
              <note position="left" xlink:label="note-0110-01" xlink:href="note-0110-01a" xml:space="preserve">10</note>
            nor HI; </s>
            <s xml:id="echoid-s4996" xml:space="preserve">quandoquidem eſt, vt quadratum ex
              <lb/>
            H E, ad quadratum ex L M, ita rectangulum ſub
              <lb/>
            C E, E A, ad rectangulum ſub C M, M A; </s>
            <s xml:id="echoid-s4997" xml:space="preserve">alias
              <lb/>
            pars foret ęqualis toti. </s>
            <s xml:id="echoid-s4998" xml:space="preserve">Si enim illa Ellipſis non
              <lb/>
            tranſit per punctum L, tranſeat ſi fieri poteſt, per
              <lb/>
            N. </s>
            <s xml:id="echoid-s4999" xml:space="preserve">Erit igitur per propoſ. </s>
            <s xml:id="echoid-s5000" xml:space="preserve">21. </s>
            <s xml:id="echoid-s5001" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5002" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5003" xml:space="preserve">Apollonij,
              <lb/>
            vt rectangulum CE, E A, ad rectangulum ſub C M, M A, hoc eſt, vt quadratum ex H E, ad qua-
              <lb/>
            dratum ex L M, ita quadaatum ex H E, ad quadratum ex N M. </s>
            <s xml:id="echoid-s5004" xml:space="preserve">Aequalia ſunt igitur quadrata ex
              <lb/>
              <note position="left" xlink:label="note-0110-02" xlink:href="note-0110-02a" xml:space="preserve">9. quinti.</note>
            L M, & </s>
            <s xml:id="echoid-s5005" xml:space="preserve">N M, atque adeò & </s>
            <s xml:id="echoid-s5006" xml:space="preserve">rectę L M, N M, ęquales, totum & </s>
            <s xml:id="echoid-s5007" xml:space="preserve">pars. </s>
            <s xml:id="echoid-s5008" xml:space="preserve">Quod eſt abſurdum.
              <lb/>
            </s>
            <s xml:id="echoid-s5009" xml:space="preserve">Tranſit ergo Ellipſis illa per punctum L, ac proinde punctum L, in Ellipſim cadit, cuius maior
              <lb/>
              <note position="left" xlink:label="note-0110-03" xlink:href="note-0110-03a" xml:space="preserve">20</note>
            diameter A C, & </s>
            <s xml:id="echoid-s5010" xml:space="preserve">minor HI. </s>
            <s xml:id="echoid-s5011" xml:space="preserve">Eodem modo oſtendemus & </s>
            <s xml:id="echoid-s5012" xml:space="preserve">alia puncta, in quæ à circunferentia
              <lb/>
            circuli. </s>
            <s xml:id="echoid-s5013" xml:space="preserve">A B C D, perpendiculares cadunt, in eadem Ellipſi eſſe. </s>
            <s xml:id="echoid-s5014" xml:space="preserve">Quocirca ſi à circunferentia cir-
              <lb/>
            culi maximi in ſphæra, &</s>
            <s xml:id="echoid-s5015" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5016" xml:space="preserve">Quod erat demon ſtrandum.</s>
            <s xml:id="echoid-s5017" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div311" type="section" level="1" n="117">
          <head xml:id="echoid-head120" style="it" xml:space="preserve">SCHOLIVM.</head>
          <p style="it">
            <s xml:id="echoid-s5018" xml:space="preserve">HOC theorema proponitur à Federico Commandino vniuer ſalius in libello de horologiorum deſcri-
              <lb/>
            ptione; </s>
            <s xml:id="echoid-s5019" xml:space="preserve">adeo vt etiamſi planum, in quo circulus A F C G, non ſecet circulum inclinatum A B C D, per
              <lb/>
            centrum, vel nullo modo, & </s>
            <s xml:id="echoid-s5020" xml:space="preserve">ſiue A B C D, ſit maximus circulus in ſphæra, ſiue quicunque, tamen per-
              <lb/>
            pendiculares ductæ à circunferentia circuli A B C D, ad planum A F C G, cadant in Ellipſim. </s>
            <s xml:id="echoid-s5021" xml:space="preserve">Nam ſi
              <lb/>
              <note position="left" xlink:label="note-0110-04" xlink:href="note-0110-04a" xml:space="preserve">30</note>
            planum, in quo circulus A F C G, non ſecet circulum A B C D, per centrum, vel nullo modo, ita propo-
              <lb/>
            ſitum colligit. </s>
            <s xml:id="echoid-s5022" xml:space="preserve">Ducto alio plano ipſi A F C G, æquidiſtante, quod circulum A B C D, ſecet in centro E, ſi-
              <lb/>
            militer demonſtrabitur, vt prius, perpendiculares à circuli A B C D, circunferentia ad planum illud de-
              <lb/>
            miſſas in Ellipſim cadere: </s>
            <s xml:id="echoid-s5023" xml:space="preserve">quæ quidem lineæ, cum vlterius productæ ad planum A F C G, quod illi æqui-
              <lb/>
            diſtat, eandem poſitionem habeant, cadent & </s>
            <s xml:id="echoid-s5024" xml:space="preserve">eoloco in Ellipſim, cuius maior diameter æqualis erit dia-
              <lb/>
            metro A C, circuli A B C D, minor vero æqualis interuallo H I, perpendicularium B H, D I, quæ ab ex-
              <lb/>
            tremitatibus alteri{us} diametri B D, ducuntur.</s>
            <s xml:id="echoid-s5025" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s5026" xml:space="preserve">NOS autem propoſuimus theorema de circulis maximis in ſphæra duntaxat, quia in his ſolis appare
              <lb/>
            bit eius vſus in noſtra hac Gnomonica.</s>
            <s xml:id="echoid-s5027" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">40</note>
        </div>
        <div xml:id="echoid-div313" type="section" level="1" n="118">
          <head xml:id="echoid-head121" xml:space="preserve">PROBLEMA 4. PROPOSITIO 25.</head>
          <p>
            <s xml:id="echoid-s5028" xml:space="preserve">IN circunferentia circuli maximi in ſphęra ad alium circulum ma-
              <lb/>
            ximum inclinati ſumptis duobus punctis extremis diametri commu-
              <lb/>
            nem eorum ſectionem ad rectos angulos ſecantis, quo loco perpendi-
              <lb/>
            culares ab his ductæ ad alium circulum cadant, ſi nota fuerit inclina-
              <lb/>
            tio, inueſtigare.</s>
            <s xml:id="echoid-s5029" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">50</note>
          <p>
            <s xml:id="echoid-s5030" xml:space="preserve">SIT in ſphęra circulus maximus A B C D, ad circulum maximum A F C E, inclinatus, ſitq́;
              <lb/>
            </s>
            <s xml:id="echoid-s5031" xml:space="preserve">eorum ſectio communis diameter A C, ad quam in plano circuli A B C D, per centrum G, alia
              <lb/>
            diameter ducatur perpendicularis B D. </s>
            <s xml:id="echoid-s5032" xml:space="preserve">Oportet igitur inueſtigare, quo loco perpendiculares à
              <lb/>
            punctis D, B, in planum circuli A F C E, demiſſę cadant. </s>
            <s xml:id="echoid-s5033" xml:space="preserve">In plano circuli A F C E, ducatur alia
              <lb/>
              <note position="left" xlink:label="note-0110-07" xlink:href="note-0110-07a" xml:space="preserve">Inuentio pun-
                <lb/>
              ctorum, in quæ
                <lb/>
              cadunt perpen-
                <lb/>
              diculares ab ex-
                <lb/>
              tremitatibus,
                <lb/>
              diametri circu-
                <lb/>
              li ad alium cir-
                <lb/>
              culum inclina-
                <lb/>
              ti
                <unsure/>
              .</note>
            diameter E F, ad A C, perpendicularis, ſitq́; </s>
            <s xml:id="echoid-s5034" xml:space="preserve">angulus inclinationis, quę nota ponitur E G H, ita vt
              <lb/>
            arcus E H, æqualis ſitarcui inclinationis circuli A B C D, ad circulum A F C E; </s>
            <s xml:id="echoid-s5035" xml:space="preserve">& </s>
            <s xml:id="echoid-s5036" xml:space="preserve">ab H, ducatur
              <lb/>
            HI, ad EF, perpendicularis. </s>
            <s xml:id="echoid-s5037" xml:space="preserve">Dico perpendicularem à D, ad planum circuli A F C E, demiſſam
              <lb/>
            cadere in punctum I. </s>
            <s xml:id="echoid-s5038" xml:space="preserve">Ducto enim per E G, D G, plano faciente in ſphæra ſemicirculum E D F, ex
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s5039" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5040" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5041" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5042" xml:space="preserve">Theodoſii, erit hicad circulos A F C E, A B C D, rectus. </s>
            <s xml:id="echoid-s5043" xml:space="preserve">(Nam cum C G, per-
              <lb/>
            pendicularis ſit ad E G, D G, erit eadem quoque ad planum per E G, D G, ductum, id eſt, ad ſe-
              <lb/>
              <note position="left" xlink:label="note-0110-08" xlink:href="note-0110-08a" xml:space="preserve">4. vndec.</note>
            micirculum E D F, recta, atque adeo & </s>
            <s xml:id="echoid-s5044" xml:space="preserve">plana circulorum A F C E, A B C D, per C G, ducta </s>
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