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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[31.] PROBLEMA 2. PROPOSITIO 8.
Page: 46 (26)
[32.] SCHOLIVM.
Page: 49 (29)
[33.] LEMMA.
Page: 49 (29)
[34.] LEMMA PRIMVM.
Page: 51 (31)
[35.] LEMMA II.
Page: 51 (31)
[36.] LEMMA PRIMVM.
Page: 53 (33)
[37.] LEMMA II.
Page: 53 (33)
[38.] THEOREMA 7. PROPOSITIO 9.
Page: 57 (37)
[39.] SCHOLIVM.
Page: 58 (38)
[40.] THEOREMA 8. PROPOSITIQ 10.
Page: 59 (39)
[41.] COROLLARIVM.
Page: 60 (40)
[42.] SCHOLIVM.
Page: 60 (40)
[43.] THEOREMA 9. PROPOSITIO 11.
Page: 62 (42)
[44.] SCHOLIVM.
Page: 63 (43)
[45.] COROLLARIVM PRIMVM.
Page: 63 (43)
[46.] COROLLARIVM II.
Page: 63 (43)
[47.] THEOREMA 10. PROPOSITIO 12.
Page: 63 (43)
[48.] COROLLARIVM.
Page: 64 (44)
[49.] THEOREMA 11. PROPOSITIO 13.
Page: 64 (44)
[50.] THEOREMA 12. PROPOSITIO 14.
Page: 65 (45)
[51.] SCHOLIVM.
Page: 66 (46)
[52.] THEOREMA 13. PROPOSITIO 15.
Page: 66 (46)
[53.] LEMMA.
Page: 69 (49)
[54.] COROLLARIVM.
Page: 69 (49)
[55.] THEOREMA 14. PROPOSITIO 16.
Page: 69 (49)
[56.] COROLLARIVM.
Page: 70 (50)
[57.] THEOREMA 15. PROPOSITIO 17.
Page: 70 (50)
[58.] LEMMA.
Page: 72 (52)
[59.] SCHOLIVM.
Page: 72 (52)
[60.] THEOREMA 16. PROPOSITIO 18.
Page: 72 (52)
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          <p>
            <s xml:id="echoid-s941" xml:space="preserve">
              <pb o="13" file="0033" n="33" rhead="LIBER PRIMVS."/>
            ipſorum ducuntur: </s>
            <s xml:id="echoid-s942" xml:space="preserve">atque eodem modo paralleli ſingulorum graduum Eclipticæ inu ſtigari poſ-
              <lb/>
            ſunt; </s>
            <s xml:id="echoid-s943" xml:space="preserve">ſi nimirum circulus M P N Q, in ſingulos gra dus diſtribuatur, & </s>
            <s xml:id="echoid-s944" xml:space="preserve">reliqua fiant, quæ prius.
              <lb/>
            </s>
            <s xml:id="echoid-s945" xml:space="preserve">Nam in vniuerſu@ rectæ, quæ ipſi P Q, parallelæ ſunt, abſcindunt ex Meridiano arcus declinatio-
              <lb/>
            num eorum arcuum Eclipticæ, qui arcubus circuli M P N Q, ſi miles ſunt, ſicut & </s>
            <s xml:id="echoid-s946" xml:space="preserve">duodecim ſi-
              <lb/>
            gna Zodiaci duodecim arcubus Q X, X Y, &</s>
            <s xml:id="echoid-s947" xml:space="preserve">c. </s>
            <s xml:id="echoid-s948" xml:space="preserve">ſimilia ſunt. </s>
            <s xml:id="echoid-s949" xml:space="preserve">Quod quidem hac fere ratione cum
              <lb/>
            Petro Nonio lib. </s>
            <s xml:id="echoid-s950" xml:space="preserve">2. </s>
            <s xml:id="echoid-s951" xml:space="preserve">de arte nauigandi demonſtrabimus.</s>
            <s xml:id="echoid-s952" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s953" xml:space="preserve">INTELLIGATVR circa E M, deſcriptus ſemicirculus Eclipticæ A M B, & </s>
            <s xml:id="echoid-s954" xml:space="preserve">circa E H, ſe-
              <lb/>
              <note position="right" xlink:label="note-0033-01" xlink:href="note-0033-01a" xml:space="preserve">Demon ſtrati@
                <lb/>
              deſcriptionis
                <lb/>
              Analemmatis.
                <unsure/>
              </note>
            micirculus Aequatoris A H B, & </s>
            <s xml:id="echoid-s955" xml:space="preserve">vtriuſque ſectio communis ſit recta A B; </s>
            <s xml:id="echoid-s956" xml:space="preserve">ſitq́; </s>
            <s xml:id="echoid-s957" xml:space="preserve">A, principium ♈,
              <lb/>
            & </s>
            <s xml:id="echoid-s958" xml:space="preserve">B, principium ♎. </s>
            <s xml:id="echoid-s959" xml:space="preserve">Et quoniam M, eſt principium ♋, vel ♑, cum H M, portio Meridiani
              <lb/>
            circuli ſit maxima declinatio ſolis; </s>
            <s xml:id="echoid-s960" xml:space="preserve">diſtat autem vtrumque horũ ab æquinoctialibus punctis qua-
              <lb/>
              <note position="left" xlink:label="note-0033-02" xlink:href="note-0033-02a" xml:space="preserve">10</note>
            drante integro; </s>
            <s xml:id="echoid-s961" xml:space="preserve">erunt arcus A M, B M, quadrantes, atque adeo anguli A E M, B E M, recti. </s>
            <s xml:id="echoid-s962" xml:space="preserve">Secet
              <lb/>
            iam recta X R, in plano Meridiani per arcum H M, & </s>
            <s xml:id="echoid-s963" xml:space="preserve">rectas E H, E M, M O, ducto rectã M O,
              <lb/>
            in puncto φ, & </s>
            <s xml:id="echoid-s964" xml:space="preserve">rectam E M, in puncto C. </s>
            <s xml:id="echoid-s965" xml:space="preserve">Intelligatur quoque per rectam X R, planũ duci Aequa-
              <lb/>
            tori A H B, parallelum occurrens rectæ E M in C, (quoniam enim circulus M P N Q, cum in
              <lb/>
            Analemmate iaceat in plano Meridiani, ad Aequatorem rectus eſt, eſtq́; </s>
            <s xml:id="echoid-s966" xml:space="preserve">Q H P E, communis ſe-
              <lb/>
            ctio Aequatoris & </s>
            <s xml:id="echoid-s967" xml:space="preserve">eiuſdem pla
              <lb/>
            ni Meridiani, & </s>
            <s xml:id="echoid-s968" xml:space="preserve">recta X R, di-
              <lb/>
            ctæ ſectioni Q H P E, parallela,
              <lb/>
              <figure xlink:label="fig-0033-01" xlink:href="fig-0033-01a" number="11">
                <image file="0033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0033-01"/>
              </figure>
            poterit per ipſam X R, duci
              <lb/>
            planũ Aequatori æquidiſtans.)
              <lb/>
            </s>
            <s xml:id="echoid-s969" xml:space="preserve">
              <note position="left" xlink:label="note-0033-03" xlink:href="note-0033-03a" xml:space="preserve">20</note>
            faciensq́; </s>
            <s xml:id="echoid-s970" xml:space="preserve">in Ecliptica quidem
              <lb/>
            cõmunem ſectionem D K, re-
              <lb/>
            ctam; </s>
            <s xml:id="echoid-s971" xml:space="preserve">In Sphæra autem circu-
              <lb/>
              <note position="right" xlink:label="note-0033-04" xlink:href="note-0033-04a" xml:space="preserve">3. vndec.</note>
            lum D γ k, ex propos. </s>
            <s xml:id="echoid-s972" xml:space="preserve">1. </s>
            <s xml:id="echoid-s973" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s974" xml:space="preserve">1.
              <lb/>
            </s>
            <s xml:id="echoid-s975" xml:space="preserve">Theod. </s>
            <s xml:id="echoid-s976" xml:space="preserve">tranſeuntem per pun-
              <lb/>
            ctum γ, in quo recta X R, arcũ
              <lb/>
            Meridiani H M, ſecat. </s>
            <s xml:id="echoid-s977" xml:space="preserve">Quo-
              <lb/>
            niam igitur eſt, vt M C, ad C E,
              <lb/>
              <note position="right" xlink:label="note-0033-05" xlink:href="note-0033-05a" xml:space="preserve">2. ſexti.</note>
            ita M φ, ad φ O, erit & </s>
            <s xml:id="echoid-s978" xml:space="preserve">compo
              <lb/>
            nendo, vt M E, ad CE, ita M O,
              <lb/>
              <note position="left" xlink:label="note-0033-06" xlink:href="note-0033-06a" xml:space="preserve">30</note>
            ad φ O; </s>
            <s xml:id="echoid-s979" xml:space="preserve">& </s>
            <s xml:id="echoid-s980" xml:space="preserve">permutando, vt M E, ſemidiameter Eclipticæ ad M O, ſemidiametrũ circuli M P Q,
              <lb/>
            ita C E, ad φ O: </s>
            <s xml:id="echoid-s981" xml:space="preserve">Eſt autem C E, æqualis ſinui arcus D A, hoc eſt, rectæ D F, ex D, ad A B, ad rectos
              <lb/>
              <note position="right" xlink:label="note-0033-07" xlink:href="note-0033-07a" xml:space="preserve">34. primi.</note>
            angulos ductæ (ſunt enim A B, D K, communes ſectiones planorum parallelorũ, nempe Aequa-
              <lb/>
              <note position="right" xlink:label="note-0033-08" xlink:href="note-0033-08a" xml:space="preserve">16. undec.</note>
            toris A H B, & </s>
            <s xml:id="echoid-s982" xml:space="preserve">circuli D γ K, factæ ab Eclipticæ plano A M B, parallelæ nec non & </s>
            <s xml:id="echoid-s983" xml:space="preserve">C E, D F, pa-
              <lb/>
              <note position="right" xlink:label="note-0033-09" xlink:href="note-0033-09a" xml:space="preserve">28. primi.</note>
            rallelæ) & </s>
            <s xml:id="echoid-s984" xml:space="preserve">φ O, eadem ratione ęqualis ſinui arcus Q X, hoc eſt, rectæ X ω, quæ ad Q E, perpendi-
              <lb/>
            cularis eſt. </s>
            <s xml:id="echoid-s985" xml:space="preserve">Igitur ſemidiametri M E, M O, eandem habent proportionem, quam ſinus D F, X ω,
              <lb/>
            ac propterea arcus A D, Q X, ſimiles ſunt, vt mox lemmate ſequenti demonſtrabimus. </s>
            <s xml:id="echoid-s986" xml:space="preserve">Oſtenden
              <lb/>
            dum ergo eſt, arcum H γ, quem aufert parallela X R, ex Meridiano, æqualem eſſe arcui declina-
              <lb/>
            tionis, quam habet Eclipticæ arcus A D, quem arcui Q X, circuli M P Q, ſimilem iam demonſtra
              <lb/>
            uimus. </s>
            <s xml:id="echoid-s987" xml:space="preserve">quod quidem facile præſtabimus hoc modo. </s>
            <s xml:id="echoid-s988" xml:space="preserve">Deſcripto per polos mundi, hoc eſt, per po-
              <lb/>
              <note position="left" xlink:label="note-0033-10" xlink:href="note-0033-10a" xml:space="preserve">40</note>
            los parallelorum A H B, D γ K, & </s>
            <s xml:id="echoid-s989" xml:space="preserve">per D, punctum circulo maximo D G, erit arcus D G, arcus de-
              <lb/>
            clinationis puncti D, cum intercipiatur inter ipſum punctum, & </s>
            <s xml:id="echoid-s990" xml:space="preserve">Aequatorem. </s>
            <s xml:id="echoid-s991" xml:space="preserve">Cum ergo arcus
              <lb/>
            circulorum maximorum, qui per polos parallelorum deſcribuntur, inter ipſos parallelos interce-
              <lb/>
            pti, ex propoſitione 15. </s>
            <s xml:id="echoid-s992" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s993" xml:space="preserve">2. </s>
            <s xml:id="echoid-s994" xml:space="preserve">Theodoſii, æquales ſint; </s>
            <s xml:id="echoid-s995" xml:space="preserve">Sint autem arcus H γ, D G, circulorum ma-
              <lb/>
            ximorum per polos parallelorum A H B, D γ K, deſcriptorum, intercipianturq́; </s>
            <s xml:id="echoid-s996" xml:space="preserve">inter ipſos paral-
              <lb/>
            lelos, æqualis erit arcus H γ, arcui D G. </s>
            <s xml:id="echoid-s997" xml:space="preserve">Aufert igitur in Analemmate parallela X R, arcum H γ,
              <lb/>
            æqualem arcui declinationis illius arcus Eclipticæ, qui arcui Q X, ſimilis eſt, qualis eſtarcus A D.
              <lb/>
            </s>
            <s xml:id="echoid-s998" xml:space="preserve">Idemq́ue dicendum eſt de reliquis parallelis Y S, Z T, & </s>
            <s xml:id="echoid-s999" xml:space="preserve">α V. </s>
            <s xml:id="echoid-s1000" xml:space="preserve">Conſtat ergo arcus H γ, H β, H δ, & </s>
            <s xml:id="echoid-s1001" xml:space="preserve">
              <lb/>
            H ε. </s>
            <s xml:id="echoid-s1002" xml:space="preserve">æquales eſſe declinationibus reliquorum ſignorum Zodiaci inter ♋ & </s>
            <s xml:id="echoid-s1003" xml:space="preserve">♑, quandoqui-
              <lb/>
            dem arcus ſignorũ in Ecliptica ſimiles ſunt arcubus Q X, X Y, &</s>
            <s xml:id="echoid-s1004" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1005" xml:space="preserve">in circulo M P N Q. </s>
            <s xml:id="echoid-s1006" xml:space="preserve">tam enim
              <lb/>
              <note position="left" xlink:label="note-0033-11" xlink:href="note-0033-11a" xml:space="preserve">50</note>
            hi, quàm illi, duodecimæ partes ſunt ſuorum circulorum. </s>
            <s xml:id="echoid-s1007" xml:space="preserve">Quoniam verò ſectiones parallelorũ
              <lb/>
            per ſignorum initia ductorum factæ à Meridiani plano parallelæ ſunt, liquido conſtat, parallelas
              <lb/>
            illas per puncta M, β, γ, H, &</s>
            <s xml:id="echoid-s1008" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1009" xml:space="preserve">ductas, eſſe diametros parallelorum, cum auferant ex circulo A B
              <lb/>
            C D, arcus ſimiles illis, quos ex Meridiano ab@cindunt re uera diametri dictorum parallelorum,
              <lb/>
            vt ante dictum eſt. </s>
            <s xml:id="echoid-s1010" xml:space="preserve">Quòd ſi circulus A B C D, æqualis eſſet Meridiano in Sphæra, tranſirent om-
              <lb/>
            nino per illas parallelas paralleli per initia ſignorum ducti. </s>
            <s xml:id="echoid-s1011" xml:space="preserve">Idem prorſus demonſtrabimus, ſi pro
              <lb/>
            Meridiano circulus A B C D, intelligatur quicunque alius circulus maximus per polos mundi
              <lb/>
            ductus, qualis eſt Colurus ſolſtitiorum, vt ſupra in definitione Analemmatis diximus. </s>
            <s xml:id="echoid-s1012" xml:space="preserve">Analemma
              <lb/>
            ergo ad quamcunque poli altitudinem deſcripſimus. </s>
            <s xml:id="echoid-s1013" xml:space="preserve">Quod erat faciendum.</s>
            <s xml:id="echoid-s1014" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div46" type="section" level="1" n="16">
          <head xml:id="echoid-head19" xml:space="preserve">LEMMA.</head>
          <p>
            <s xml:id="echoid-s1015" xml:space="preserve">QVAM proportionem habent ſinus toti, hoc eſt, ſemidiametri </s>
          </p>
        </div>
      </text>
    </echo>
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