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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        <div xml:id="echoid-div398" type="section" level="1" n="135">
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            <s xml:id="echoid-s7119" xml:space="preserve">
              <pb o="118" file="0138" n="138" rhead="GNOMONICES"/>
            linquunt arcus ſemidiurnos quadrantis Eclipticæ borealis inter principium ♈, & </s>
            <s xml:id="echoid-s7120" xml:space="preserve">principium ♋, inter-
              <lb/>
            poſiti; </s>
            <s xml:id="echoid-s7121" xml:space="preserve">quirurſum æquales ſunt arcubus ſemidiurnis reliqui quadrantis borealis inter principium ♎, & </s>
            <s xml:id="echoid-s7122" xml:space="preserve">
              <lb/>
            principium ♋, interiecti.</s>
            <s xml:id="echoid-s7123" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7124" xml:space="preserve">IDEM prorſus dicendum eſt de latitudine ortiua, occiduave: </s>
            <s xml:id="echoid-s7125" xml:space="preserve">quia ſi ſupputentur pro vno quadran
              <lb/>
            tc
              <unsure/>
            Eclipticę ſiue auſtralis, ſiue borealis, omnes amplitudines ortiuę occiduęve, habebuntur etiam latitu-
              <lb/>
            dines ortiuæ, ſiue occiduę pro omnibus punctis Eclipticę, vt ex demonſtr atione patet: </s>
            <s xml:id="echoid-s7126" xml:space="preserve">quia nimirum de-
              <lb/>
            clinationes punctorum vnius quadrantis Eclipticæ, ex quibus latitudines ortiuę & </s>
            <s xml:id="echoid-s7127" xml:space="preserve">occiduę ſupputantur,
              <lb/>
            æquales ſunt declinationibus reliquorum punctorum Eclipticæ, vt alibi ostenſum eſt à nobis.</s>
            <s xml:id="echoid-s7128" xml:space="preserve"/>
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        <div xml:id="echoid-div401" type="section" level="1" n="136">
          <head xml:id="echoid-head139" xml:space="preserve">PROBLEMA 14. PROPOSITIO 35.</head>
          <note position="left" xml:space="preserve">10</note>
          <p>
            <s xml:id="echoid-s7129" xml:space="preserve">CREPVSCVLORVM magnitudines in quocunque parallelo
              <lb/>
            ad datam poli altitudinem inuenire.</s>
            <s xml:id="echoid-s7130" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7131" xml:space="preserve">QVAMVIS problema hoc per triangula ſphærica poſſit abſolui, vt in ſcholio ſequenti de-
              <lb/>
            monſtrabimus, multo tamen breuius, & </s>
            <s xml:id="echoid-s7132" xml:space="preserve">facilius idem per triangula plana rectilinea expediemus.
              <lb/>
            </s>
            <s xml:id="echoid-s7133" xml:space="preserve">Sit enim Meridianus Analemmatis A B C D, cuius centrum E; </s>
            <s xml:id="echoid-s7134" xml:space="preserve">diameter Horizontis A C; </s>
            <s xml:id="echoid-s7135" xml:space="preserve">Ver-
              <lb/>
            ticalis B D; </s>
            <s xml:id="echoid-s7136" xml:space="preserve">axis mundi F G; </s>
            <s xml:id="echoid-s7137" xml:space="preserve">diameter Aequatoris H I; </s>
            <s xml:id="echoid-s7138" xml:space="preserve">diameter paralleli Solis K L, ſiue borealis,
              <lb/>
            ſiue auſtralis, circa quam ſemicirculus paralleli deſcribatur K P L, cuius centrum erit punctum
              <lb/>
              <note position="left" xlink:label="note-0138-02" xlink:href="note-0138-02a" xml:space="preserve">20</note>
            M, vbi axis, & </s>
            <s xml:id="echoid-s7139" xml:space="preserve">paralleli diameter ſe interſecant; </s>
            <s xml:id="echoid-s7140" xml:space="preserve">quandoquidem recta E M, perpendicularis ad
              <lb/>
            K L, ipſam K L, bifariam diuidit. </s>
            <s xml:id="echoid-s7141" xml:space="preserve">Ex puncto autem S, vbi diameter paralleli diametrum Hori-
              <lb/>
              <note position="left" xlink:label="note-0138-03" xlink:href="note-0138-03a" xml:space="preserve">3. tertij.</note>
            zontis ſecat, erigatur ad K L, perpendicularis S Q, quæ communis ſectio erit Horizontis, & </s>
            <s xml:id="echoid-s7142" xml:space="preserve">paral-
              <lb/>
            leli. </s>
            <s xml:id="echoid-s7143" xml:space="preserve">Quoniam enim tam Horizon, quàm parallelus ad Meridianum rectus eſt, erit eorum com-
              <lb/>
            munis ſectio ad eundem perpendicularis; </s>
            <s xml:id="echoid-s7144" xml:space="preserve">atque adeo, per defin. </s>
            <s xml:id="echoid-s7145" xml:space="preserve">3. </s>
            <s xml:id="echoid-s7146" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7147" xml:space="preserve">11. </s>
            <s xml:id="echoid-s7148" xml:space="preserve">Euclidis, & </s>
            <s xml:id="echoid-s7149" xml:space="preserve">ad diametrũ
              <lb/>
              <note position="left" xlink:label="note-0138-04" xlink:href="note-0138-04a" xml:space="preserve">19. vndec.</note>
            paralleli K L. </s>
            <s xml:id="echoid-s7150" xml:space="preserve">Cum ergo communis illa ſectio per punctum S, vbi ſe mutuo ſecant Horizon, & </s>
            <s xml:id="echoid-s7151" xml:space="preserve">
              <lb/>
            parallelus in Meridiano, tranſeat, erit S Q, perpendicularis exiſtens ad k L, communis ſectio Ho-
              <lb/>
            rizontis, & </s>
            <s xml:id="echoid-s7152" xml:space="preserve">paralleli. </s>
            <s xml:id="echoid-s7153" xml:space="preserve">Igitur K Q, arcus erit ſemidiurnus paralleli, cum intercipiatur inter Hori-
              <lb/>
            zontem, qui per punctum Q, tranſit, & </s>
            <s xml:id="echoid-s7154" xml:space="preserve">Meridianum, qui in k, parallelum ſecat. </s>
            <s xml:id="echoid-s7155" xml:space="preserve">Ducatur quoque
              <lb/>
            diameter paralleli Horizontis X Y, infra Horizontem, in quo exiſtit Sol, cum crepuſculum matu-
              <lb/>
              <note position="left" xlink:label="note-0138-05" xlink:href="note-0138-05a" xml:space="preserve">30</note>
            tinum incipit, vel veſpertinum deſinit, ita vt tam arcus A X, quàm C Y, comprehendat grad. </s>
            <s xml:id="echoid-s7156" xml:space="preserve">18.
              <lb/>
            </s>
            <s xml:id="echoid-s7157" xml:space="preserve">Tot enim gradibus ab Horizonte abeſt Sol in initio matutini crepuſculi, vel in fine veſpertini,
              <lb/>
            vt communis eſt opinio, licet aliqui auctores minorem faciant hanc diſtantiam, & </s>
            <s xml:id="echoid-s7158" xml:space="preserve">alij maiorem,
              <lb/>
            vt refert Petrus Nonius in parte ſecunda libri de crepuſculis propoſ. </s>
            <s xml:id="echoid-s7159" xml:space="preserve">1. </s>
            <s xml:id="echoid-s7160" xml:space="preserve">Secet autem diameter X Y,
              <lb/>
            paralleli Solis diametrum K L, in T, puncto, à quo ad K L, educatur perpendicularis T R, quæ
              <lb/>
            communis ſectio erit paralleli Solis, & </s>
            <s xml:id="echoid-s7161" xml:space="preserve">paralleli Horizontis, in quo principium eſt, vel finis crepu-
              <lb/>
            ſculi. </s>
            <s xml:id="echoid-s7162" xml:space="preserve">Quoniam enim tam parallelus Solis, quà parallelus Horizontis ad Meridianum rectus
              <lb/>
            eſt, erit & </s>
            <s xml:id="echoid-s7163" xml:space="preserve">eorum ſectio communis ad eundem, atque adeò per definitionem 3. </s>
            <s xml:id="echoid-s7164" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7165" xml:space="preserve">11. </s>
            <s xml:id="echoid-s7166" xml:space="preserve">Euclidis,
              <lb/>
              <note position="left" xlink:label="note-0138-06" xlink:href="note-0138-06a" xml:space="preserve">19. vndec.</note>
            & </s>
            <s xml:id="echoid-s7167" xml:space="preserve">ad rectam K L, in Meridiani plano exiſtentem perpendicularis, qualis eſt recta T R. </s>
            <s xml:id="echoid-s7168" xml:space="preserve">Erit ergo
              <lb/>
            longitudo crepuſculi arcus Q R, quandoquidem Sol in puncto R, exiſtit, quando occupat Hori-
              <lb/>
              <note position="left" xlink:label="note-0138-07" xlink:href="note-0138-07a" xml:space="preserve">40</note>
            zontis parallelum X Y, in quo principium eſt crepuſculi matutini, vel finis veſpertini; </s>
            <s xml:id="echoid-s7169" xml:space="preserve">& </s>
            <s xml:id="echoid-s7170" xml:space="preserve">in pun-
              <lb/>
            cto Q, oritur, vel occidit. </s>
            <s xml:id="echoid-s7171" xml:space="preserve">Cum enim T R, ſit communis ſectio paralleli Solis, & </s>
            <s xml:id="echoid-s7172" xml:space="preserve">Horizontis pa-
              <lb/>
            ralleli X Y, ſecabunt ſeſe mutuo hi paralleli in R, atque adeò Sol in puncto R, exiſtens vtrumque
              <lb/>
            parallelum poſſidebit. </s>
            <s xml:id="echoid-s7173" xml:space="preserve">Eodem pacto in puncto Q, exiſtens Horizontem tenebit, cum vt paulo
              <lb/>
            ante diximus, S Q, ſit communis ſectio Horizontis, & </s>
            <s xml:id="echoid-s7174" xml:space="preserve">paralleli Solis. </s>
            <s xml:id="echoid-s7175" xml:space="preserve">Demittatur tandem ex K,
              <lb/>
            ad A C, perpendicularis K N, ſecans X Y, in O, eritq́; </s>
            <s xml:id="echoid-s7176" xml:space="preserve">K N, ſinus altitudinis meridianæ Solis in pa
              <lb/>
            rallelo K L, exiſtentis, hoc eſt, ſinus arcus K A; </s>
            <s xml:id="echoid-s7177" xml:space="preserve">& </s>
            <s xml:id="echoid-s7178" xml:space="preserve">recta K O, compoſita ex K N, ſinu altitudinis
              <lb/>
            meridianæ, & </s>
            <s xml:id="echoid-s7179" xml:space="preserve">N O, ſinu diſtantię Solis ſub Horizonte in principio crepuſculi matutini, uel fine
              <lb/>
            veſpertini. </s>
            <s xml:id="echoid-s7180" xml:space="preserve">Sic etiam erit K S, ſinus verſus arcus ſemidiurni K Q; </s>
            <s xml:id="echoid-s7181" xml:space="preserve">& </s>
            <s xml:id="echoid-s7182" xml:space="preserve">K T, ſinus verſus arcus K R,
              <lb/>
            ex arcu ſemidiurno k Q, & </s>
            <s xml:id="echoid-s7183" xml:space="preserve">arcu crepuſculi Q R, quem videlicer Sol in ſuo parallelo percurrit,
              <lb/>
              <note position="left" xlink:label="note-0138-08" xlink:href="note-0138-08a" xml:space="preserve">Crepuſeuli ma
                <lb/>
              gnitudo, quo
                <lb/>
              modo inueſtige
                <lb/>
              tur, Sole exi-
                <lb/>
              ſtente in quo-
                <lb/>
              cunque paralle
                <lb/>
              10.</note>
              <note position="left" xlink:label="note-0138-09" xlink:href="note-0138-09a" xml:space="preserve">50</note>
            dum crepuſculum durat, compoſitus.</s>
            <s xml:id="echoid-s7184" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7185" xml:space="preserve">ITAQVE quoniam in triangulo K O T, recta S N, baſi O T, parallela eſt, erit vt k N, ſinus
              <lb/>
            altitudinis Meridianæ ad K O, rectam compoſitam ex ſinu altitudinis meridianæ, & </s>
            <s xml:id="echoid-s7186" xml:space="preserve">ex N O, ſinu
              <lb/>
            diſtantiæ Solis ſub Horizonte in principio crepuſculi matutini, vel fine veſpertini, ita K S, ſinus
              <lb/>
            verſus ſemidiurni arcus ad K T, ſinum verſum arcus compoſiti ex arcu ſemidiurno, & </s>
            <s xml:id="echoid-s7187" xml:space="preserve">arcu crepu-
              <lb/>
              <note position="left" xlink:label="note-0138-10" xlink:href="note-0138-10a" xml:space="preserve">2. ſexti.</note>
            ſculi. </s>
            <s xml:id="echoid-s7188" xml:space="preserve">Quamobrem ſi fiat, vt ſinus altitudinis meridianæ (quo pacto autem meridiana altitudo in-
              <lb/>
            ueniatur ex cognita declinatione Solis, in ſequenti ſcholio exponemus) ad rectam compoſitam
              <lb/>
            ex ſinu altitudinis meridianę, & </s>
            <s xml:id="echoid-s7189" xml:space="preserve">ſinu grad. </s>
            <s xml:id="echoid-s7190" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7191" xml:space="preserve">(tot enim gradibus ſecundum communem ſenten-
              <lb/>
            tiam Sol infra Horizontem deprimitur in Verticali per centrum Solis tunc tranſeuntis) ita ſinus
              <lb/>
            verſus arcus ſemidiurni ad aliud, habebitur ſinus verſus arcus compoſiti ex arcu ſemidiurno, & </s>
            <s xml:id="echoid-s7192" xml:space="preserve">
              <lb/>
            arcu crepuſculi. </s>
            <s xml:id="echoid-s7193" xml:space="preserve">Ex hoc ergo ſinu verſo ita inueniemus quantitatem crepuſculi. </s>
            <s xml:id="echoid-s7194" xml:space="preserve">Si ſinus verſus </s>
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