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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          <p style="it">
            <s xml:id="echoid-s7431" xml:space="preserve">
              <pb o="123" file="0143" n="143" rhead="LIBER PRIMVS."/>
            tudinis, & </s>
            <s xml:id="echoid-s7432" xml:space="preserve">ſinu grad. </s>
            <s xml:id="echoid-s7433" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7434" xml:space="preserve">compoſita, vt accidit in tertia figura huius propoſ. </s>
            <s xml:id="echoid-s7435" xml:space="preserve">Subtracto autem arcu ſe-
              <lb/>
            midiurno K Q, ex arcu K R, inuento, perſpicuum eſt, arcum crepuſculi Q R, relinqui. </s>
            <s xml:id="echoid-s7436" xml:space="preserve">Quòd ſi dicta me-
              <lb/>
            dietas K λ, æqualis fuerit dictæ rectę K O, erit arcus compoſitus ex arcu ſemidiurno, & </s>
            <s xml:id="echoid-s7437" xml:space="preserve">arcu crepuſcu-
              <lb/>
            li quadrans, vt in quarta figura huius propoſ. </s>
            <s xml:id="echoid-s7438" xml:space="preserve">apparet.</s>
            <s xml:id="echoid-s7439" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7440" xml:space="preserve">QVIN etiam beneficio altitudinis meridianæ, & </s>
            <s xml:id="echoid-s7441" xml:space="preserve">depreſſionis meridianæ, breui admodum calculo
              <lb/>
            arcus ſemidiurnos ſupputabimus, licet eoſdem alio modo in præcedenti propoſ. </s>
            <s xml:id="echoid-s7442" xml:space="preserve">inuenerimus. </s>
            <s xml:id="echoid-s7443" xml:space="preserve">Quoniam
              <lb/>
            enim in omnibus figuris huius propoſ. </s>
            <s xml:id="echoid-s7444" xml:space="preserve">& </s>
            <s xml:id="echoid-s7445" xml:space="preserve">etiam in priori, quę in ſcholio ponitur, eſt vt K λ, medietas re-
              <lb/>
              <note position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve">2. vel 4. ſexti</note>
              <note position="right" xlink:label="note-0143-02" xlink:href="note-0143-02a" xml:space="preserve">Arcus ſemidiur
                <lb/>
              nus qua ratio-
                <lb/>
              ne ſupputandus
                <lb/>
              aliter quàm ſ@-
                <lb/>
              pra.</note>
            ctæ compoſitæ ex ſinu altitudinis meridianæ, & </s>
            <s xml:id="echoid-s7446" xml:space="preserve">ſinu depreſſionis meridianæ ad λ N, differentiam inter
              <lb/>
            prædictam medietatem, & </s>
            <s xml:id="echoid-s7447" xml:space="preserve">ſinum altitudinis meridianæ, ita K M, ſinus totus ad M S: </s>
            <s xml:id="echoid-s7448" xml:space="preserve">Si fiat vt K λ,
              <lb/>
            medietas rectę ex ſinu altitudinis meridianæ, & </s>
            <s xml:id="echoid-s7449" xml:space="preserve">ſinu depreſſionis meridianæ compoſitæ, ad λ N, differen-
              <lb/>
              <note position="left" xlink:label="note-0143-03" xlink:href="note-0143-03a" xml:space="preserve">10</note>
            tiam inter dictam medietatem, & </s>
            <s xml:id="echoid-s7450" xml:space="preserve">ſinum meridianæ altitudinis, ita K M, ſinus totus ad aliud, inuenietur
              <lb/>
            ſinus rectus M S, cuius arcus P Q, in ſignis borealibus additus quadranti, in auſtralibus verò à qua-
              <lb/>
            drante ſubtractus, dabit arcum ſemidiurnum K Q.</s>
            <s xml:id="echoid-s7451" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7452" xml:space="preserve">IDEM hac ratione conſequemur. </s>
            <s xml:id="echoid-s7453" xml:space="preserve">Quoniam in eiſdem figuris eſt, vt K λ, medietas prædicta ad
              <lb/>
              <note position="right" xlink:label="note-0143-04" xlink:href="note-0143-04a" xml:space="preserve">Arcus ſemidiur
                <lb/>
              nus quomodo
                <lb/>
              aliter inueſtige-
                <lb/>
              tur.</note>
            K N, ſinum altitudinis meridianæ, ita K M, ſinus totus ad K S, ſinum verſum arcus ſemidiurni
              <lb/>
            K Q: </s>
            <s xml:id="echoid-s7454" xml:space="preserve">Si fiat vt K λ, medietas rectæ compoſitæ ex ſinu altitudinis meridianæ, & </s>
            <s xml:id="echoid-s7455" xml:space="preserve">ſinu depreſſionis meri-
              <lb/>
            dianę ad K N, ſinum altitudinis meridianæ, ita K M, ſinus totus ad aliud, obtinebimus K S, ſinum ver-
              <lb/>
            ſum arcus ſemidiurni K Q. </s>
            <s xml:id="echoid-s7456" xml:space="preserve">Ex quo ſinu verſo ita adipiſcemur arcum ſemidiurnum K Q. </s>
            <s xml:id="echoid-s7457" xml:space="preserve">In ſignis borea-
              <lb/>
            libus dematur ex ſinu verſo inuento ſinus totus, & </s>
            <s xml:id="echoid-s7458" xml:space="preserve">reſidui ſinus recti arcus P Q, ad quadrantem adij-
              <lb/>
            ciatur; </s>
            <s xml:id="echoid-s7459" xml:space="preserve">In ſignis verò auſtralib{us} contra, ſin{us} verſ{us} inuentus ex ſinu toto auferatur, & </s>
            <s xml:id="echoid-s7460" xml:space="preserve">reſidui ſinus
              <lb/>
              <note position="left" xlink:label="note-0143-05" xlink:href="note-0143-05a" xml:space="preserve">20</note>
            recti arcus P Q, ex quadrante rurſus ſubtrahatur. </s>
            <s xml:id="echoid-s7461" xml:space="preserve">Exurget enim ſemper ſiue ex illa additione, ſiue ex
              <lb/>
            hac ſubtractione arcus ſemidiurnus.</s>
            <s xml:id="echoid-s7462" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7463" xml:space="preserve">HAEC autem omnia intelligenda ſunt, quando complementum altitudinis poli maius eſt declina-
              <lb/>
              <note position="right" xlink:label="note-0143-06" xlink:href="note-0143-06a" xml:space="preserve">Arcus diurnus
                <lb/>
              quoniodo ſe ha
                <lb/>
              beat, quando cõ
                <lb/>
              plementum al-
                <lb/>
              titudinis poli
                <lb/>
              maius non fue-
                <lb/>
              rit declinatione
                <lb/>
              propoſiti paral-
                <lb/>
              leli.</note>
            tione paralleli illius, cuius arcus ſemidiurnus inueſtigatur. </s>
            <s xml:id="echoid-s7464" xml:space="preserve">Nam quando complementum illud non eſt ma-
              <lb/>
            ius, tanget vel parallelus Horizontem, at que ita eius arcus diurnus, ſi borealis eſt, continebit horas 24.
              <lb/>
            </s>
            <s xml:id="echoid-s7465" xml:space="preserve">vel ſi auſiralis, hor. </s>
            <s xml:id="echoid-s7466" xml:space="preserve">0. </s>
            <s xml:id="echoid-s7467" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s7468" xml:space="preserve">0. </s>
            <s xml:id="echoid-s7469" xml:space="preserve">Vel totus parallelus borealis ſupra Horizontem extat, & </s>
            <s xml:id="echoid-s7470" xml:space="preserve">auſtralis infra,
              <lb/>
            atque ita nullus erit arcus diurnus, ſed continua dies in parallelo boreali, & </s>
            <s xml:id="echoid-s7471" xml:space="preserve">continua nox in auſtrali erit. </s>
            <s xml:id="echoid-s7472" xml:space="preserve">
              <lb/>
            Quæ omnia in ſecunda figura ſcholij huius propoſ. </s>
            <s xml:id="echoid-s7473" xml:space="preserve">apparent, vbi complementum altitudinis poli C I, mi-
              <lb/>
            nus eſt declinatione I L, paralleli borealis, & </s>
            <s xml:id="echoid-s7474" xml:space="preserve">rurſus idem complementum A H, minus declinatione
              <lb/>
            H K, paralleli auſtralis. </s>
            <s xml:id="echoid-s7475" xml:space="preserve">Quòd ſi complementum illud declinationi foret æquale, tangeret parallelus bo-
              <lb/>
              <note position="left" xlink:label="note-0143-07" xlink:href="note-0143-07a" xml:space="preserve">30</note>
            realis Horizontem in C, & </s>
            <s xml:id="echoid-s7476" xml:space="preserve">australis in A, vt manifeſtum eſt.</s>
            <s xml:id="echoid-s7477" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7478" xml:space="preserve">VT autem omnibus numeris abſoluta ſit demonſtratio huius propoſ. </s>
            <s xml:id="echoid-s7479" xml:space="preserve">trademus etiam inuentionem
              <lb/>
            crepuſculorum in ſphærarecta, vbi multo facilius inueniuntur. </s>
            <s xml:id="echoid-s7480" xml:space="preserve">Sit Meridianus Analemmatis A B C D;
              <lb/>
            </s>
            <s xml:id="echoid-s7481" xml:space="preserve">Horizontis recti diameter A C, per polos A, C, tranſiens; </s>
            <s xml:id="echoid-s7482" xml:space="preserve">Paral-
              <lb/>
              <figure xlink:label="fig-0143-01" xlink:href="fig-0143-01a" number="103">
                <image file="0143-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0143-01"/>
              </figure>
            leli Horizontis, in quo principium, vel terminus crepuſculi ponitur,
              <lb/>
              <note position="right" xlink:label="note-0143-08" xlink:href="note-0143-08a" xml:space="preserve">Crepuſculum
                <lb/>
              in ſphæra recta,
                <lb/>
              quo pacto in-
                <lb/>
              dagandum.</note>
            diameter F G; </s>
            <s xml:id="echoid-s7483" xml:space="preserve">Aequatoris diameter B D, ſecans F G, in H; </s>
            <s xml:id="echoid-s7484" xml:space="preserve">diame
              <lb/>
            ter paralleli Solis c
              <unsure/>
            uiuſcunque I K, ſecans A C, F G, in L, & </s>
            <s xml:id="echoid-s7485" xml:space="preserve">M.
              <lb/>
            </s>
            <s xml:id="echoid-s7486" xml:space="preserve">Ex L, centro paralleli deſcribatur ſemicirculus ipſius I N K, exten
              <lb/>
            danturq́, A C, F G, vſque ad N, & </s>
            <s xml:id="echoid-s7487" xml:space="preserve">O.</s>
            <s xml:id="echoid-s7488" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7489" xml:space="preserve">ERIT igitur A F, vel C G, arcus crepuſculi, Sole in Aequa-
              <lb/>
              <note position="left" xlink:label="note-0143-09" xlink:href="note-0143-09a" xml:space="preserve">40</note>
            tore exiſtente, cuius ſinus eſt recta E H, reſpondens arcui grad. </s>
            <s xml:id="echoid-s7490" xml:space="preserve">18.
              <lb/>
            </s>
            <s xml:id="echoid-s7491" xml:space="preserve">Igitur ſi ſumatur arcus grad. </s>
            <s xml:id="echoid-s7492" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7493" xml:space="preserve">habebitur arcus crepuſculi in ſphæ-
              <lb/>
            rarecta, Sole in æquinoctijs exiſtente.</s>
            <s xml:id="echoid-s7494" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7495" xml:space="preserve">SOLE verò in quouis parallelo exiſtente, vt in parallelo INK,
              <lb/>
            erit arcus O N, longitudo crepuſculi. </s>
            <s xml:id="echoid-s7496" xml:space="preserve">Si igitur fiat, vt I L, quatenus
              <lb/>
            pars eſt ſinus totius B E, maximi circuli, hoc eſt, quatenus ſinus eſt complementi declinationis, (eſt
              <lb/>
            enim arcus I C, vel I A, complementum declinationis) ad L M, ſinum grad. </s>
            <s xml:id="echoid-s7497" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7498" xml:space="preserve">eiuſdem circuli maximi,
              <lb/>
            ita I L, quatenus ſinus totus in parallelo I N K, ad aliud, c
              <unsure/>
            ognita erit L M, in partibus ſinus totius I L;
              <lb/>
            </s>
            <s xml:id="echoid-s7499" xml:space="preserve">atque adeò @ius arcus N O, qui crepuſculum metitur, notus erit. </s>
            <s xml:id="echoid-s7500" xml:space="preserve">Atque hoc verum eſt, ſiue I K, ſit dia-
              <lb/>
            meter paralleli auſtralis, ſiue borealis, vt ex figura manifeſtum eſt.</s>
            <s xml:id="echoid-s7501" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">50</note>
          <p style="it">
            <s xml:id="echoid-s7502" xml:space="preserve">NON eſt autem prætereundum, vno fere nos labore peruestigare poſſe crepuſcula duorum paralle-
              <lb/>
              <note position="right" xlink:label="note-0143-11" xlink:href="note-0143-11a" xml:space="preserve">Crepuſcula duo
                <lb/>
              rum parallclo-
                <lb/>
              rum oppoſito-
                <lb/>
              rum, qua ratio-
                <lb/>
              ne vnico labore
                <unsure/>
                <lb/>
              inueniantur.</note>
            lorum oppoſitorum, ſi primo modo vtamur. </s>
            <s xml:id="echoid-s7503" xml:space="preserve">Quoniam enim arcus ſemidiurnus cuiuſuis paralleli æqualis
              <lb/>
            eſt ar cui ſeminocturno, vt in ſcholio antecedentis propoſ. </s>
            <s xml:id="echoid-s7504" xml:space="preserve">oſtendimus, erit ſinus verſus K S, arcus ſemi-
              <lb/>
            diurni paralleli australis æqualis ſinui verſo L S, arcus ſeminocturni paralleli borealis oppoſiti, vt in
              <lb/>
            ſequẽti prima figura apparet. </s>
            <s xml:id="echoid-s7505" xml:space="preserve">Est autẽ & </s>
            <s xml:id="echoid-s7506" xml:space="preserve">S T, portio diametri paralleli auſtralis portioni S T, diametri
              <lb/>
              <note position="right" xlink:label="note-0143-12" xlink:href="note-0143-12a" xml:space="preserve">34. primi.</note>
            paralleli borealis æqualis. </s>
            <s xml:id="echoid-s7507" xml:space="preserve">Igitur ſi fiat, vt K N, ſinus altitudinis meridianæ paralleli auſtralis ad
              <lb/>
            N O, ſinum grad. </s>
            <s xml:id="echoid-s7508" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7509" xml:space="preserve">ita K S, ſinus verſus arcus ſemidiurni paralleli auſtralis, ad aliud, nota erit re-
              <lb/>
            cta S T; </s>
            <s xml:id="echoid-s7510" xml:space="preserve">cum eandem proportionem habeat K N, ad N O, quam k S, ad S T. </s>
            <s xml:id="echoid-s7511" xml:space="preserve">Hæc autem S T, nota,
              <lb/>
              <note position="right" xlink:label="note-0143-13" xlink:href="note-0143-13a" xml:space="preserve">2. ſ@nti.</note>
            (quam fortaſſis facilius inuenies ratione ad finem huius propoſ. </s>
            <s xml:id="echoid-s7512" xml:space="preserve">præſcripta) ſi adijciatur ad ſinum ver-
              <lb/>
            ſum K S, arcus ſemidiurni paralleli auſtralis, conflabitur K T, ſinus verſus arcus K R, compoſiti ex ar-
              <lb/>
            @u ſemidiurno K Q, & </s>
            <s xml:id="echoid-s7513" xml:space="preserve">arcu crepuſculi Q R. </s>
            <s xml:id="echoid-s7514" xml:space="preserve">Dempto ergo ſemidiurno arcu K Q, remanebit </s>
          </p>
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