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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            <s xml:id="echoid-s7514" xml:space="preserve">
              <pb o="124" file="0144" n="144" rhead="GNOMONICES"/>
            cus Q R, crepuſculinota. </s>
            <s xml:id="echoid-s7515" xml:space="preserve">Eadem autem S T, ablata ex L S, ſinu verſo arcus ſeminocturni paralleli bo-
              <lb/>
            realis oppoſiti, qui ſinui verſo K S, arcus ſemidiurni æqualis eſt in parallelo auſtrali, vt diximus, relinquet
              <lb/>
              <figure xlink:label="fig-0144-01" xlink:href="fig-0144-01a" number="104">
                <image file="0144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0144-01"/>
              </figure>
              <note position="left" xlink:label="note-0144-01" xlink:href="note-0144-01a" xml:space="preserve">10</note>
            T L, ſinum verſum arcus L R, qui ablatus ex arcu ſeminocturno L Q, paralleli borealis, quem arcui ſe-
              <lb/>
            midiurno k Q, paralleli auſtralis æqualem eſſe tradidimus, relinquet arcum crepuſculi Q R. </s>
            <s xml:id="echoid-s7516" xml:space="preserve">Itaque vt
              <lb/>
            vides, vnica multiplicatione, & </s>
            <s xml:id="echoid-s7517" xml:space="preserve">diuiſione inuenitur recta S T, in vtroque parallelo, beneficio cuius cre-
              <lb/>
              <note position="left" xlink:label="note-0144-02" xlink:href="note-0144-02a" xml:space="preserve">20</note>
            puſculum vtrumque elicitur, vt expoſuimus. </s>
            <s xml:id="echoid-s7518" xml:space="preserve">Malumus autem reperire prædictam rectam S T, in paral-
              <lb/>
            lelo auſtrali, quàm in boreali; </s>
            <s xml:id="echoid-s7519" xml:space="preserve">quia & </s>
            <s xml:id="echoid-s7520" xml:space="preserve">ſinus rectus k N, altitudinis meridianæ paralleli auſtralis minor
              <lb/>
            eſt ſinu recto altitudinis meridianæ paralleli borealis, & </s>
            <s xml:id="echoid-s7521" xml:space="preserve">ſinus verſus K S, arcus ſemidiurni paralleli au-
              <lb/>
            stralis ſinu verſo arcus ſemidiurni paralleli ſeptentrionalis. </s>
            <s xml:id="echoid-s7522" xml:space="preserve">Vnde facilior efficitur multiplicatio, atque
              <lb/>
            diuiſio per numeros minores.</s>
            <s xml:id="echoid-s7523" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">Crepuſculum
            <lb/>
          pontinuum in
            <lb/>
          parallelo borea
            <lb/>
          li quando.</note>
          <p style="it">
            <s xml:id="echoid-s7524" xml:space="preserve">ADVERTENDVM autem eſt hic, quando recta S T, in parallelo auſtrali inuenta fuerit ma-
              <lb/>
            ior ſinu verſo K S, arcus ſemidiurni, (quod quidem accidit, cum ſinus altitudinis meridianæ K N, minor
              <lb/>
            eſt ſinu grad. </s>
            <s xml:id="echoid-s7525" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7526" xml:space="preserve">vt in ſecunda figura contingit) perpetuum eſſe crepuſculum in parallelo boreali oppoſi-
              <lb/>
            to: </s>
            <s xml:id="echoid-s7527" xml:space="preserve">quia tunc recta S T, in parallelo boreali auferri nequit ex L S, ſinu verſo arcus ſeminocturni, qui ſi-
              <lb/>
            nui verſo K S, arcus ſeminocturni paralleli auſtralis æqualis eſt, atqu@ adeo minor quoque quàm recta
              <lb/>
              <note position="left" xlink:label="note-0144-04" xlink:href="note-0144-04a" xml:space="preserve">30</note>
            S T. </s>
            <s xml:id="echoid-s7528" xml:space="preserve">Quare nunquam illa nocte Sol ad parallelum Horizontis, cuius diameter X Y, perueniet, & </s>
            <s xml:id="echoid-s7529" xml:space="preserve">c.</s>
            <s xml:id="echoid-s7530" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s7531" xml:space="preserve">DENIQVE vt nihil prætermittamus, quod ad rem, in qua verſamur, pertineat, placet quoque
              <lb/>
              <note position="left" xlink:label="note-0144-05" xlink:href="note-0144-05a" xml:space="preserve">Crepuſcula per
                <lb/>
              triangula ſphæ-
                <lb/>
              rica, quo modo
                <lb/>
              ſupputanda.</note>
            viam præſcribere, qua per triangula ſphęrica poſſint crepuſcula ſupputari. </s>
            <s xml:id="echoid-s7532" xml:space="preserve">Sit ergo, vt in tertia figura,
              <lb/>
            Horizon A B C D; </s>
            <s xml:id="echoid-s7533" xml:space="preserve">Meridianus B D; </s>
            <s xml:id="echoid-s7534" xml:space="preserve">Aequator A F C; </s>
            <s xml:id="echoid-s7535" xml:space="preserve">Verticalis proprie dictus A E C; </s>
            <s xml:id="echoid-s7536" xml:space="preserve">parallelus
              <lb/>
            ſiue borealis, ſiue auſtralis G I H; </s>
            <s xml:id="echoid-s7537" xml:space="preserve">parallelus Horizontis infra Horizontem K P, in quo Sol exiſtit in
              <lb/>
            principio matutini crepuſculi, vel in fine veſpertini, ita vt diſtet ab Horizonte grad. </s>
            <s xml:id="echoid-s7538" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7539" xml:space="preserve">ſecundum magis
              <lb/>
            receptam ſententiam, ſecet{q́ue} parallelum Solis in puncto K; </s>
            <s xml:id="echoid-s7540" xml:space="preserve">eritq́
              <unsure/>
            , K G, arcus crepuſculi, quem nimirum
              <lb/>
            Sol in ſuo parallelo percurrit à principio crepuſculi vſque ad finem, & </s>
            <s xml:id="echoid-s7541" xml:space="preserve">G I, arcus ſemidiurnus. </s>
            <s xml:id="echoid-s7542" xml:space="preserve">Ducatur
              <lb/>
            ex M, polo arctico per G, K, duo circuli maximi M G, M K, ſecantes Aequatorem in O, & </s>
            <s xml:id="echoid-s7543" xml:space="preserve">N. </s>
            <s xml:id="echoid-s7544" xml:space="preserve">Et quia
              <lb/>
            per propoſ. </s>
            <s xml:id="echoid-s7545" xml:space="preserve">10. </s>
            <s xml:id="echoid-s7546" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s7547" xml:space="preserve">2. </s>
            <s xml:id="echoid-s7548" xml:space="preserve">Theod. </s>
            <s xml:id="echoid-s7549" xml:space="preserve">arcus F O, arcui I G, & </s>
            <s xml:id="echoid-s7550" xml:space="preserve">arcus O N, arcui G K, ſimilis eſt, erit F O, ar-
              <lb/>
            cus ſemidiurnus paralleli G H, & </s>
            <s xml:id="echoid-s7551" xml:space="preserve">O N, arcus crepuſculi, quem ita notum efficiemus. </s>
            <s xml:id="echoid-s7552" xml:space="preserve">Ducto per verticem
              <lb/>
              <note position="left" xlink:label="note-0144-06" xlink:href="note-0144-06a" xml:space="preserve">40</note>
            E, & </s>
            <s xml:id="echoid-s7553" xml:space="preserve">per K, locum Solis in principio crepuſculi matutini, vel in fine veſpertini, circulo maximo E K, qui
              <lb/>
            ſecet Horizontem in L, erit arcus K L, grad. </s>
            <s xml:id="echoid-s7554" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7555" xml:space="preserve">Quoniam igitur in triangulo ſphærico E K M, omnia
              <lb/>
            latera nota ſunt, (Est enim E M, arcus complementi altitudinis poli; </s>
            <s xml:id="echoid-s7556" xml:space="preserve">& </s>
            <s xml:id="echoid-s7557" xml:space="preserve">arcus E K, compoſitus ex qua-
              <lb/>
            drante E L, & </s>
            <s xml:id="echoid-s7558" xml:space="preserve">arcu L K, grad. </s>
            <s xml:id="echoid-s7559" xml:space="preserve">18. </s>
            <s xml:id="echoid-s7560" xml:space="preserve">ita vt totus arcus E K, contineat grad. </s>
            <s xml:id="echoid-s7561" xml:space="preserve">108. </s>
            <s xml:id="echoid-s7562" xml:space="preserve">Denique arcus M K, in
              <lb/>
            parallelo boreali complementum eſt declinationis paralleli propoſiti, in auſtrali vero compoſitus ex qua
              <lb/>
            drante M N, & </s>
            <s xml:id="echoid-s7563" xml:space="preserve">arcu N k, declinationis) cognitus erit, per propoſ. </s>
            <s xml:id="echoid-s7564" xml:space="preserve">vltimam lib. </s>
            <s xml:id="echoid-s7565" xml:space="preserve">4. </s>
            <s xml:id="echoid-s7566" xml:space="preserve">Ioan. </s>
            <s xml:id="echoid-s7567" xml:space="preserve">Regiom. </s>
            <s xml:id="echoid-s7568" xml:space="preserve">de
              <lb/>
            triangulis, vel per propoſ. </s>
            <s xml:id="echoid-s7569" xml:space="preserve">45. </s>
            <s xml:id="echoid-s7570" xml:space="preserve">noſtrorum triangulorum ſphæricorum, angulus E M K. </s>
            <s xml:id="echoid-s7571" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s7572" xml:space="preserve">eius
              <lb/>
            arcus F N; </s>
            <s xml:id="echoid-s7573" xml:space="preserve">à quo ſi detrabatur arcus ſemidiurnus F O, reliquus erit arcus crepuſculi N O. </s>
            <s xml:id="echoid-s7574" xml:space="preserve">Quod
              <lb/>
            eſt propoſitum.</s>
            <s xml:id="echoid-s7575" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">50</note>
        </div>
        <div xml:id="echoid-div426" type="section" level="1" n="138">
          <head xml:id="echoid-head141" xml:space="preserve">PROBLEMA 15. PROPOSITIO 36.</head>
          <p>
            <s xml:id="echoid-s7576" xml:space="preserve">EX cognita diei hora altitudinem Solis ſupra Horizontem; </s>
            <s xml:id="echoid-s7577" xml:space="preserve">Et con-
              <lb/>
            tra ex altitudine Solis nota horam diei cognoſcere.</s>
            <s xml:id="echoid-s7578" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7579" xml:space="preserve">QVONIAM in ſequentibus libris horologia ſumus deſcripturi, ex quibus horas diei cogno
              <lb/>
            ſcamus, rem gratam me facturum multis arbitror, ſi in vltima hac propol. </s>
            <s xml:id="echoid-s7580" xml:space="preserve">huius libri modum o-
              <lb/>
            ſtendam, quo ſine horologio ex ſola altitudine ſolis cognita eaſdem horas, & </s>
            <s xml:id="echoid-s7581" xml:space="preserve">viciſſim ex hora co-
              <lb/>
            gnita altitudinem Solis inuenire poſſimus: </s>
            <s xml:id="echoid-s7582" xml:space="preserve">Vel ob eam præcipue cauſam, quòd magno vſui hæc
              <lb/>
            res ſit tũ ad horologia deſcribenda, tum maxime ad quadrantes, Cylindros horarios, & </s>
            <s xml:id="echoid-s7583" xml:space="preserve">alia id </s>
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