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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[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)
[61.] THEOREMA 17. PROPOSITIO 19.
Page: 73 (53)
[62.] SCHOLIVM.
Page: 74 (54)
[63.] THEOREMA 18. PROPOSITIO 20.
Page: 75 (55)
[64.] SCHOLIVM.
Page: 75 (55)
[65.] Linea horæ 24. ab ortu vel occaſu. Vel horizontalis linea.
Page: 76 (56)
[66.] Linea horæ 12. ab ortu vel occaſu.
Page: 76 (56)
[67.] Linea horæ ſextæ à meridie vel media nocte.
Page: 77 (57)
[68.] Linea horæ 12. à meridie vel media nocte.
Page: 77 (57)
[69.] Linea horæ 23. ab ortu vel occaſu.
Page: 81 (61)
[70.] Linea horæ 22. ab ortu vel occaſu.
Page: 81 (61)
[71.] Linea horæ 21. ab ortu vel occaſu.
Page: 82 (62)
[72.] Linea horæ 20. ab ortu vel occaſu.
Page: 82 (62)
[73.] Linea horæ 19. ab ortu vel occaſu.
Page: 83 (63)
[74.] Linea horæ 18. ab ortu vel occaſu.
Page: 83 (63)
[75.] Linea horæ 17. ab ortu vel occaſu.
Page: 84 (64)
[76.] Linea horæ 16. ab ortu vel occaſu.
Page: 84 (64)
[77.] Linea horæ 15. ab ortu vel occaſu.
Page: 85 (65)
[78.] Linea horæ 14. ab ortu vel occaſu.
Page: 85 (65)
[79.] Linea horæ 13. ab ortu vel occaſu.
Page: 86 (66)
[80.] In horologio, quod circulo horæ 13. ab ortu vel occaſu æquidiſtat, lineæ quarumlibet duarum horarum huius tabu læ ſunt, ex ſcholio propoſ. 22. parallelæ.
Page: 86 (66)
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        <div xml:id="echoid-div491" type="section" level="1" n="139">
          <p style="it">
            <s xml:id="echoid-s8988" xml:space="preserve">
              <pb o="139" file="0159" n="159" rhead="LIBER PRIMVS."/>
            mediet ates harum rectarum compoſitarum æquales erunt. </s>
            <s xml:id="echoid-s8989" xml:space="preserve">Vt in duob{us} tropicis mediet as rectæ compo-
              <lb/>
            ſię ex ſinu altitudinis meridianæ ♋, & </s>
            <s xml:id="echoid-s8990" xml:space="preserve">ſinu depreſſionis meridianę ♋, vel ex ſinu altitudinis meridia-
              <lb/>
            næ ♑, & </s>
            <s xml:id="echoid-s8991" xml:space="preserve">ſinu depreſſionis ♑, erit 68151. </s>
            <s xml:id="echoid-s8992" xml:space="preserve">eadem permanens, & </s>
            <s xml:id="echoid-s8993" xml:space="preserve">vtilis ad omnium horarum altitudines,
              <lb/>
            Sole in principio ♋, vel ♑, exiſtente, inuestigandas.</s>
            <s xml:id="echoid-s8994" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8995" xml:space="preserve">RVRVS in omnibus parallelis ſemper vſurpatur idem ſinus totus 100000. </s>
            <s xml:id="echoid-s8996" xml:space="preserve">in Solis altitudini-
              <lb/>
            bus perquirendis.</s>
            <s xml:id="echoid-s8997" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8998" xml:space="preserve">POSTREMO recta λ N, hoc eſt, differentia inter ſinum altitudinis meridianæ, & </s>
            <s xml:id="echoid-s8999" xml:space="preserve">medietatem
              <lb/>
            prædictam K λ, nunquam mutatur in duobus parallelis oppoſitis: </s>
            <s xml:id="echoid-s9000" xml:space="preserve">quia eadem differentia est omnino in-
              <lb/>
            ter ſinum depreſſionis meridianæ, & </s>
            <s xml:id="echoid-s9001" xml:space="preserve">medietatem alteram θ λ, vt ex figuris manifeſtum eſt. </s>
            <s xml:id="echoid-s9002" xml:space="preserve">Conſtat autẽ
              <lb/>
            ex demonſtratis, ſinum depreſſionis meridianę paralleli borealis æqualem eſſe ſinui altitudinis meridianæ
              <lb/>
              <note position="left" xlink:label="note-0159-01" xlink:href="note-0159-01a" xml:space="preserve">10</note>
            paralleli auſtralis oppoſiti: </s>
            <s xml:id="echoid-s9003" xml:space="preserve">Vnde ſemper eadem differentia erit inter medietatem prædictam, & </s>
            <s xml:id="echoid-s9004" xml:space="preserve">ſinum
              <lb/>
            altitudinis meridianæ tam paralleli borealis, quàm oppoſiti auſtralis. </s>
            <s xml:id="echoid-s9005" xml:space="preserve">Vt in duobus tropicis erit huiuſmo
              <lb/>
            di differentia hic numerus ferè 26681. </s>
            <s xml:id="echoid-s9006" xml:space="preserve">qui nunquam mutãdus erit, donec omnes altitudines inuentæ ſint
              <lb/>
            in duobus tropicis, & </s>
            <s xml:id="echoid-s9007" xml:space="preserve">ad quem numerum modo adijcienda eſt recta T λ, inuenta, modo ex eodem detra-
              <lb/>
              <note position="right" xlink:label="note-0159-02" xlink:href="note-0159-02a" xml:space="preserve">
                <gap/>
              </note>
            henda in ſignis borealibus, vel certè in australibus ſignis ipſemet numerus 26681. </s>
            <s xml:id="echoid-s9008" xml:space="preserve">ſubducendus eſt ex re-
              <lb/>
            cta T λ, inuenta, vt habeatur ſinus T N, altitudinis Solis, vt ex ſuperioribus patet.</s>
            <s xml:id="echoid-s9009" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9010" xml:space="preserve">QVOD ſi per alterum modum altitudines Solis inueſtigare placuerit in duobus oppoſitis parallelis,
              <lb/>
              <note position="right" xlink:label="note-0159-03" xlink:href="note-0159-03a" xml:space="preserve">Qui numeri ij-
                <lb/>
              dẽ permaneãt, ſi
                <lb/>
              altitudines So-
                <lb/>
              lis indagen tur
                <lb/>
              per ſecundum
                <lb/>
              modũ pro ſingu
                <lb/>
              lis horis cuiuſ-
                <lb/>
              uis paralleli.</note>
            permanebit quidem in vno eodem parallelo & </s>
            <s xml:id="echoid-s9011" xml:space="preserve">ſinus verſus arcus ſemidiurni, et ſinus rectus altitudinis
              <lb/>
            meridianæ ſmeper idẽ ſed ad altitudi@es inquir endas in altero parallelo, qui ei opponitur, proprius ſinus
              <lb/>
            tam verſus arcus ſemidiurni, quàm rectus altitudinis meridianę accipiendus crit. </s>
            <s xml:id="echoid-s9012" xml:space="preserve">Solum id commodi ha-
              <lb/>
              <note position="left" xlink:label="note-0159-04" xlink:href="note-0159-04a" xml:space="preserve">20</note>
            bebimus, quòd detr acto ſinu verſo arcus ſemidiurni vnius paralleli ex tota diametro, hoc eſt, ex 200000.
              <lb/>
            </s>
            <s xml:id="echoid-s9013" xml:space="preserve">ſtatim habeamus ſinum verſam arcus ſemidiurni paralleli oppoſiti: </s>
            <s xml:id="echoid-s9014" xml:space="preserve">quia ſinus verſus arcus ſemidiurni
              <lb/>
            vnius paralleli eſt æqualis ſinui verſo arcus ſeminocturni alterius paralleli oppoſiti: </s>
            <s xml:id="echoid-s9015" xml:space="preserve">Perſpicuum autem
              <lb/>
            eſt, ſinum verſum arcus ſeminocturni ex tota diametro ſubductum relinq@ere arcum verſum arcus ſemi-
              <lb/>
            diurni, & </s>
            <s xml:id="echoid-s9016" xml:space="preserve">contra. </s>
            <s xml:id="echoid-s9017" xml:space="preserve">
              <lb/>
              <note position="right" xlink:label="note-0159-05" xlink:href="note-0159-05a" xml:space="preserve">Qui numeri nũ
                <lb/>
              quam mutẽtur,
                <lb/>
              ſi per vltimam
                <lb/>
              viam, quam an-
                <lb/>
              te rationẽ trian
                <lb/>
              gulorũ ſphæri-
                <lb/>
              corum tradidi-
                <lb/>
              mus, @nueſtigen
                <lb/>
              tur altitudines
                <lb/>
              Solis pro ſingu
                <lb/>
              lis hotis duorũ
                <lb/>
              parallelorũ op-
                <lb/>
              poſitorum.</note>
            </s>
          </p>
          <p style="it">
            <s xml:id="echoid-s9018" xml:space="preserve">AT vero in vltimailla via, quam proxime ante rationem ex triangulis ſphæricis depromptam ſcri-
              <lb/>
            pſimus, habebimus in oppoſitis parallelis non ſolum eandem ſemper medietatem rectę compoſitæ ex ſinu
              <lb/>
            altitudinis meridianę, & </s>
            <s xml:id="echoid-s9019" xml:space="preserve">ſinu meridianę depreſſionis, atque ſinum totum, verum etiam eoſaem ſinus ver
              <lb/>
            ſos diſtantiarum Solis à meridie in horis ęqualiter à Meridiano diſtantibus, vt ſunt horæ prima, & </s>
            <s xml:id="echoid-s9020" xml:space="preserve">vnde-
              <lb/>
            cima. </s>
            <s xml:id="echoid-s9021" xml:space="preserve">Item ſecunda & </s>
            <s xml:id="echoid-s9022" xml:space="preserve">decima, & </s>
            <s xml:id="echoid-s9023" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9024" xml:space="preserve">tam in parallelo ♋, quàm in parallelo ♑. </s>
            <s xml:id="echoid-s9025" xml:space="preserve">Vnde in eiſdem horis ea-
              <lb/>
              <note position="left" xlink:label="note-0159-06" xlink:href="note-0159-06a" xml:space="preserve">30</note>
            dem recta K T, inuenictur, adeo vt in parallelo oppoſito non opus ſit inueſtigare rurſus rectam K T, pro
              <lb/>
            illa hora, pro qua inuenta eſt eadem K T, in altero parallelo, ſed eadem omnino aſſumenda, vt detraha-
              <lb/>
            tur à ſinu altitudinis meridianæ propoſiti paralleli, & </s>
            <s xml:id="echoid-s9026" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s9027" xml:space="preserve">
              <note position="right" xlink:label="note-0159-07" xlink:href="note-0159-07a" xml:space="preserve">Qui ſinus ijdẽ
                <lb/>
              ſemper maneãt,
                <lb/>
              ſi per triangula
                <lb/>
              ſphærica altitu-
                <lb/>
              dines Solis in-
                <lb/>
              quirantur pro
                <lb/>
              ſingulis horis
                <lb/>
              duorum paral-
                <lb/>
              lelorum oppoſi
                <lb/>
              torum,</note>
            </s>
          </p>
          <p style="it">
            <s xml:id="echoid-s9028" xml:space="preserve">DENIQVE ſi quis altitudines Solis in duobus oppoſitis parallelis maluerit per triangula ſphęri
              <lb/>
            ca indagare, negotium etiam perfacile reddetur, quia idem ſemper ſinus totus, idemq́, ſinus declinationis
              <lb/>
            vtriuſq; </s>
            <s xml:id="echoid-s9029" xml:space="preserve">paralleli in omniũ horarũ altitudinibus perueſtigandis ret inendus eſt, vt ex dictis liquido cõſtat.</s>
            <s xml:id="echoid-s9030" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9031" xml:space="preserve">CÆTERVM vbi tanta eſt poli altitudo, vt totus parallelus aliquis borealis ſupra Horizontem
              <lb/>
            extet, ſiue illum tangat, vt in prima hac figura ſubiecta, ſiue non, vt in ſecunda, nihilo ſecius per priorem
              <lb/>
            modum in hac propoſ. </s>
            <s xml:id="echoid-s9032" xml:space="preserve">traditum altitudo Solis ex hora cognita, & </s>
            <s xml:id="echoid-s9033" xml:space="preserve">viciſſim ex altitudine Solis hora inue-
              <lb/>
              <note position="right" xlink:label="note-0159-08" xlink:href="note-0159-08a" xml:space="preserve">Quando paral-
                <lb/>
              lelus borealis to
                <lb/>
              tus eſt ſupra Ho
                <lb/>
              rizontem, qua
                <lb/>
              ratio ne altitu-
                <lb/>
              do Solis ex no-
                <lb/>
              ta hora, & uiciſ-
                <lb/>
              ſim ex altitudi-
                <lb/>
              ne Solis cogni-
                <lb/>
              ta exploranda
                <lb/>
              ſit hora.</note>
            nietur, dummodo in vtr aque figura meridies intelligatur eſſe in K, vbi Sol in Meridiano exiſtens maxi-
              <lb/>
              <note position="left" xlink:label="note-0159-09" xlink:href="note-0159-09a" xml:space="preserve">40</note>
            mam habet altitudinẽ, quæ ex decli-
              <lb/>
              <figure xlink:label="fig-0159-01" xlink:href="fig-0159-01a" number="116">
                <image file="0159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0159-01"/>
              </figure>
            natione paralleli inueſtiganda eſt, vt
              <lb/>
            ſupra in ſcholio propoſ. </s>
            <s xml:id="echoid-s9034" xml:space="preserve">præcedentis
              <lb/>
            declarauimus. </s>
            <s xml:id="echoid-s9035" xml:space="preserve">Verum nulla hic erit
              <lb/>
            depreſſio Meridiana, ſed in priori fi-
              <lb/>
            gura quidẽ recta K λ, erit medietas
              <lb/>
            ſinus altitudinis meridianę; </s>
            <s xml:id="echoid-s9036" xml:space="preserve">In poſte
              <lb/>
            riori verò eadem K λ, medietas erit
              <lb/>
            rectę K θ, quæ differentia eſt inter
              <lb/>
            K N, ſinum maioris altitudinis me-
              <lb/>
              <note position="left" xlink:label="note-0159-10" xlink:href="note-0159-10a" xml:space="preserve">50</note>
            ridianę K A, & </s>
            <s xml:id="echoid-s9037" xml:space="preserve">θ N, ſinum minoris
              <lb/>
            altitudinis meridianę L C: </s>
            <s xml:id="echoid-s9038" xml:space="preserve">quę qui-
              <lb/>
            dem minor altitudo C L, habebitur,
              <lb/>
            ſi ex arcu I L, declinationis detraha
              <lb/>
            tur arcus I C, complementi altitudi
              <lb/>
            nis poli. </s>
            <s xml:id="echoid-s9039" xml:space="preserve">Quòd ſi declinatio ęqualis fuerit complemento altitudinis poli, tanget parallelus Horizontem,
              <lb/>
            vt in priori figura accidit. </s>
            <s xml:id="echoid-s9040" xml:space="preserve">Quòd autem K λ, ſit medietas dictarum rectarum, ita probabitur. </s>
            <s xml:id="echoid-s9041" xml:space="preserve">Quoniam in
              <lb/>
            priori figur a eſt, vt k M, ad M C, ita K λ, ad λ N; </s>
            <s xml:id="echoid-s9042" xml:space="preserve">In poſteriori verò vt K M, ad M L, ita K λ, ad λ θ Eſt
              <lb/>
              <note position="right" xlink:label="note-0159-11" xlink:href="note-0159-11a" xml:space="preserve">2. ſex@@.</note>
            autemtam K M, ipſi M C, quàm K M, ipſi M L, æqualis, quòd hæ rectę ſint ſemidiametri ipſius paralle-
              <lb/>
            li; </s>
            <s xml:id="echoid-s9043" xml:space="preserve">erit quoque K λ, ipſi λ N, in priori figura, & </s>
            <s xml:id="echoid-s9044" xml:space="preserve">ipſi λ θ, in poſteriori ęqualis.</s>
            <s xml:id="echoid-s9045" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9046" xml:space="preserve">ITAQVE quoniam in priori figura, vbi parallelus Horizontem tangit, eſt vt K M, ſinus totus ad
              <lb/>
              <note position="right" xlink:label="note-0159-12" xlink:href="note-0159-12a" xml:space="preserve">2. vel 4. ſexti</note>
            </s>
          </p>
        </div>
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    </echo>
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