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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[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)
[131.] SCHOLIVM.
Page: 125 (105)
[132.] PROBLEMA 12. PROPOSITIO 33.
Page: 126 (106)
[133.] SCHOLIVM.
Page: 129 (109)
[134.] PROBLEMA 13. PROPOSITIO 34.
Page: 134 (114)
[135.] SCHOLIVM.
Page: 137 (117)
[136.] PROBLEMA 14. PROPOSITIO 35.
Page: 138 (118)
[137.] SCHOLIVM.
Page: 140 (120)
[138.] PROBLEMA 15. PROPOSITIO 36.
Page: 144 (124)
[139.] SCHOLIVM.
Page: 158 (138)
[140.] FINIS PRIMI LIBRI.
Page: 160 (140)
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          <p style="it">
            <s xml:id="echoid-s5646" xml:space="preserve">
              <pb o="99" file="0119" n="119" rhead="LIBER PRIMVS."/>
            fiet, & </s>
            <s xml:id="echoid-s5647" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5648" xml:space="preserve">Eadem ratione ex E P, & </s>
            <s xml:id="echoid-s5649" xml:space="preserve">E T, notis nota fiet P T. </s>
            <s xml:id="echoid-s5650" xml:space="preserve">Igitur vt antea, iterum notus erit angu-
              <lb/>
            lus P R T, altitudinis poli, & </s>
            <s xml:id="echoid-s5651" xml:space="preserve">c.</s>
            <s xml:id="echoid-s5652" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div347" type="section" level="1" n="127">
          <head xml:id="echoid-head130" xml:space="preserve">PROBLEMA 8. PROPOSITIO 29.</head>
          <p>
            <s xml:id="echoid-s5653" xml:space="preserve">DATO plano vel ad Meridianum, & </s>
            <s xml:id="echoid-s5654" xml:space="preserve">Horizontem, vel ad Meri-
              <lb/>
            dianum tantum, vel ad Horizontem tantum inclinato, quanta ſit poli
              <lb/>
            altitudo ſupra ipſum, deprehendere.</s>
            <s xml:id="echoid-s5655" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">10</note>
          <p>
            <s xml:id="echoid-s5656" xml:space="preserve">SIT planum circuli A B C D, cuius centrum E, & </s>
            <s xml:id="echoid-s5657" xml:space="preserve">ad Meridianum, & </s>
            <s xml:id="echoid-s5658" xml:space="preserve">ad Horizontem, vel ad
              <lb/>
              <note position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">Altitudo poli
                <lb/>
              ſupra planum
                <lb/>
              inclinatum ad
                <lb/>
              Meridianum,
                <lb/>
              & Horizontem,
                <lb/>
              vel ad Meridia-
                <lb/>
              num t
                <unsure/>
              antum,
                <lb/>
              quomodo inue
                <lb/>
              miatur.</note>
            Meridianum tantum inclinatum, & </s>
            <s xml:id="echoid-s5659" xml:space="preserve">communis ipſius, ac Meridiani ſectio B D. </s>
            <s xml:id="echoid-s5660" xml:space="preserve">Inuento autem,
              <lb/>
            ex coroll. </s>
            <s xml:id="echoid-s5661" xml:space="preserve">propoſitionis præcedentis, arcu Meridiani inter planum inclinatum, & </s>
            <s xml:id="echoid-s5662" xml:space="preserve">polum mun-
              <lb/>
              <figure xlink:label="fig-0119-01" xlink:href="fig-0119-01a" number="84">
                <image file="0119-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0119-01"/>
              </figure>
            di arcticum, ſumatur illi ęqualis D F. </s>
            <s xml:id="echoid-s5663" xml:space="preserve">Inueniatur
              <lb/>
            quoque per coroll. </s>
            <s xml:id="echoid-s5664" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s5665" xml:space="preserve">25. </s>
            <s xml:id="echoid-s5666" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s5667" xml:space="preserve">minor
              <lb/>
            diameter Ellipſis, quam perpendiculares ex cir-
              <lb/>
            cunferentia Meridiani in planum inclinatum de-
              <lb/>
            miſſæ faciunt, quæ ſit G H, ſecans maiorem B D,
              <lb/>
            ad angulos rectos, & </s>
            <s xml:id="echoid-s5668" xml:space="preserve">circa G H, circulus deſcri-
              <lb/>
            batur, cuius circunferentiam ſecet recta ducta E F,
              <lb/>
              <note position="left" xlink:label="note-0119-03" xlink:href="note-0119-03a" xml:space="preserve">20</note>
            in I. </s>
            <s xml:id="echoid-s5669" xml:space="preserve">Deinde per F, agatur minori diametro paral-
              <lb/>
            lela F K; </s>
            <s xml:id="echoid-s5670" xml:space="preserve">per I, autem maiori diametro parallela
              <lb/>
            I K, ſecans priorem in K, puncto, per quod dia-
              <lb/>
            meter ducatur A C, ad quam ex K, perpendicula-
              <lb/>
            ris erigatur K L, ſecans circulum A B C D, in L.
              <lb/>
            </s>
            <s xml:id="echoid-s5671" xml:space="preserve">Dico arcum C L, æqualem eſſe arcui, qui altitudi-
              <lb/>
            nem poli ſupra planum A B C D, metitur. </s>
            <s xml:id="echoid-s5672" xml:space="preserve">Quo-
              <lb/>
            niam enim arcus D F, ęqualis eſt ar cui Meridiani
              <lb/>
            inter planum A B C D, & </s>
            <s xml:id="echoid-s5673" xml:space="preserve">polum mundi, erit & </s>
            <s xml:id="echoid-s5674" xml:space="preserve">
              <lb/>
            reliquus F O, reliquo in Meridiano à polo vſque
              <lb/>
              <note position="left" xlink:label="note-0119-04" xlink:href="note-0119-04a" xml:space="preserve">30</note>
            ad diametrum, quæ ipſam B D, ſecat ad angulos rectos, & </s>
            <s xml:id="echoid-s5675" xml:space="preserve">à qua perpendiculares cadunt in puncta
              <lb/>
            G, H, æqualis. </s>
            <s xml:id="echoid-s5676" xml:space="preserve">Quare per ea, quæ propoſ. </s>
            <s xml:id="echoid-s5677" xml:space="preserve">26. </s>
            <s xml:id="echoid-s5678" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s5679" xml:space="preserve">demonſtrata ſunt, cadet perpendicularis
              <lb/>
            ex polo in planum A B C D, demiſſa in punctum K, ellipſis diametrorum B D, G H. </s>
            <s xml:id="echoid-s5680" xml:space="preserve">Sicut enim in
              <lb/>
            figura illius propoſitionis ſe habent arcus D L, L E, quibus in circulo inclinato ęquales ſunt ar-
              <lb/>
            cus D K, K A, ita hic ſe habent arcus D F, F O, quibus in Meridiano ad circulum A B C D, incli-
              <lb/>
            nato reſpondent arcus ęquales à D, vſque ad polum arcticum, & </s>
            <s xml:id="echoid-s5681" xml:space="preserve">à polo vſque ad diametrum, quę
              <lb/>
            ipſam B D, ad angulos rectos ſecat. </s>
            <s xml:id="echoid-s5682" xml:space="preserve">Quare vt ibi demonſtratum eſt, perpendicularem ex K, demiſ-
              <lb/>
            ſam cadere in punctum Q, vbi ſe interſecant rectæ L Q, M Q, diametris H I, B D, ellipſis æqui
              <lb/>
            diſtantes, ita quoque hic oſtendetur, perpendicularem ex polo demiſſam cadere in punctum K,
              <lb/>
            vbi ſe interſecant rectæ F K, I K, diametris G H, B D, ellipſis æquidiſtantes. </s>
            <s xml:id="echoid-s5683" xml:space="preserve">Sit igitur perpendi-
              <lb/>
              <note position="left" xlink:label="note-0119-05" xlink:href="note-0119-05a" xml:space="preserve">40</note>
            cularis à polo cadens K M, & </s>
            <s xml:id="echoid-s5684" xml:space="preserve">polus M; </s>
            <s xml:id="echoid-s5685" xml:space="preserve">intelligaturq́; </s>
            <s xml:id="echoid-s5686" xml:space="preserve">circulus maximus A M C, duci per rectas
              <lb/>
            A E K C, K M, qui neceſſario ad planum A B C D, rectus erit; </s>
            <s xml:id="echoid-s5687" xml:space="preserve">ac propterea cum per polum mun
              <lb/>
              <note position="right" xlink:label="note-0119-06" xlink:href="note-0119-06a" xml:space="preserve">18. vndec.</note>
            di M, tranſeat, inſtar Meridiani erit ipſius plani inclinati, recta autẽ A C, linea erit meridiana, & </s>
            <s xml:id="echoid-s5688" xml:space="preserve">
              <lb/>
            arcus C M, altitudinẽ poli ſupra idem planum metietur. </s>
            <s xml:id="echoid-s5689" xml:space="preserve">Ducantur quoque rectæ E L, E M, C L,
              <lb/>
            C M. </s>
            <s xml:id="echoid-s5690" xml:space="preserve">Quoniam igitur tam quadratum ex E L, quadratis ex E K, K L, quam quadratum ex E M,
              <lb/>
              <note position="right" xlink:label="note-0119-07" xlink:href="note-0119-07a" xml:space="preserve">47. primi.</note>
            quadratis ex E K, K M, æquale eſt; </s>
            <s xml:id="echoid-s5691" xml:space="preserve">propterea quòd anguli E K L, E KM, recti ſunt, ex conſtru-
              <lb/>
            ctione, & </s>
            <s xml:id="echoid-s5692" xml:space="preserve">ex defin. </s>
            <s xml:id="echoid-s5693" xml:space="preserve">3. </s>
            <s xml:id="echoid-s5694" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5695" xml:space="preserve">11. </s>
            <s xml:id="echoid-s5696" xml:space="preserve">Euclidis: </s>
            <s xml:id="echoid-s5697" xml:space="preserve">Sunt autem quadrata rectarum E L, E M, æqualium ex cen-
              <lb/>
            tro ſphæræad eius ſuperficiem ductarum ęqualia; </s>
            <s xml:id="echoid-s5698" xml:space="preserve">erunt & </s>
            <s xml:id="echoid-s5699" xml:space="preserve">quadrata ex E K, K L, quadratis ex E K,
              <lb/>
            K M, ęqualia. </s>
            <s xml:id="echoid-s5700" xml:space="preserve">Dempto ergo communi quadrato ex E K, æquale erit quadratum ex K L, quadrato
              <lb/>
            ex K M, & </s>
            <s xml:id="echoid-s5701" xml:space="preserve">recta K L, rectæ K M, æqualis. </s>
            <s xml:id="echoid-s5702" xml:space="preserve">Itaque cum latera K L, K C, lateribus K M, K C, ſint
              <lb/>
              <note position="left" xlink:label="note-0119-08" xlink:href="note-0119-08a" xml:space="preserve">50</note>
            æqualia, angulosq́; </s>
            <s xml:id="echoid-s5703" xml:space="preserve">æquales cõprehendant, vtpote rectos, æqualis erit baſis C L, baſi C M; </s>
            <s xml:id="echoid-s5704" xml:space="preserve">ac proin
              <lb/>
              <note position="right" xlink:label="note-0119-09" xlink:href="note-0119-09a" xml:space="preserve">4. primi.</note>
            de & </s>
            <s xml:id="echoid-s5705" xml:space="preserve">arcus C L, æqualis erit arcui C M, qui altitudinẽ poli ſupra planũ A B C D, metitur. </s>
            <s xml:id="echoid-s5706" xml:space="preserve">Quod
              <lb/>
              <note position="right" xlink:label="note-0119-10" xlink:href="note-0119-10a" xml:space="preserve">28. tertij.</note>
            eſt propoſitum: </s>
            <s xml:id="echoid-s5707" xml:space="preserve">Atque hoc modo Federicus Cõmandinus ferè propoſitum exequitur, quanquã
              <lb/>
            de plano ad Horizontem tantum inclinato nihil dicat. </s>
            <s xml:id="echoid-s5708" xml:space="preserve">Quod idẽ nos ex ſinubus ita abſoluemus.</s>
            <s xml:id="echoid-s5709" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">Altitudo poli
            <lb/>
          ſupra planum
            <lb/>
          inclinatum ad
            <lb/>
          Meridianum, &
            <lb/>
          Horizon@ẽ qua
            <lb/>
          uia per ſinu@ in
            <lb/>
          quiratur.</note>
          <p>
            <s xml:id="echoid-s5710" xml:space="preserve">SIT Horizon A B C D, Meridianus A C; </s>
            <s xml:id="echoid-s5711" xml:space="preserve">planum ad Meridianum & </s>
            <s xml:id="echoid-s5712" xml:space="preserve">ad Horizontem inclina
              <lb/>
            tum E F, ſec
              <unsure/>
            ans Meridianum in G, vbicunque hoc contingat; </s>
            <s xml:id="echoid-s5713" xml:space="preserve">Polus mundi H, per quem & </s>
            <s xml:id="echoid-s5714" xml:space="preserve">polũ
              <lb/>
            plani inclinati E F, circulus maximus deſcribatur B D, ſecans planum inclinatum in I, atque adeo
              <lb/>
            per propoſ. </s>
            <s xml:id="echoid-s5715" xml:space="preserve">15. </s>
            <s xml:id="echoid-s5716" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5717" xml:space="preserve">1. </s>
            <s xml:id="echoid-s5718" xml:space="preserve">Theodoſij, ad angulos rectos; </s>
            <s xml:id="echoid-s5719" xml:space="preserve">metieturq́; </s>
            <s xml:id="echoid-s5720" xml:space="preserve">propterea arcus HI, altitudinem
              <lb/>
            poli ſupra planum E F. </s>
            <s xml:id="echoid-s5721" xml:space="preserve">Quoniam igitur in triangulo ſphęrico G H I, cuius angulus I, rectus eſt,
              <lb/>
            vt ſinus arcus Meridiani G H, qui inter planum inclinatum, & </s>
            <s xml:id="echoid-s5722" xml:space="preserve">polum interijcitur, ad ſinum angu
              <lb/>
            li recti I, hoc eſt, ad ſinum totum, ita eſt, per propoſ. </s>
            <s xml:id="echoid-s5723" xml:space="preserve">16. </s>
            <s xml:id="echoid-s5724" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5725" xml:space="preserve">4. </s>
            <s xml:id="echoid-s5726" xml:space="preserve">Ioan. </s>
            <s xml:id="echoid-s5727" xml:space="preserve">Regiom. </s>
            <s xml:id="echoid-s5728" xml:space="preserve">de triangulis, vel </s>
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