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>
            <s xml:id="echoid-s6451" xml:space="preserve">
              <pb o="107" file="0127" n="127" rhead="LIBER PRIMVS."/>
            lij ſequentis. </s>
            <s xml:id="echoid-s6452" xml:space="preserve">Sed tamen præſtat inuenire declinationes parallelorum Borealium. </s>
            <s xml:id="echoid-s6453" xml:space="preserve">Nam horũ de-
              <lb/>
            clinationes æquales ſunt declinationibus reſpondentium parallelorum auſtralium. </s>
            <s xml:id="echoid-s6454" xml:space="preserve">Vt declinatio
              <lb/>
            paralleli borealis, cuius arcus diurnus continet horas 14. </s>
            <s xml:id="echoid-s6455" xml:space="preserve">æqualis eſt declinationi paralleli auſtra-
              <lb/>
            lis, cuius arcus diurnus horas 10. </s>
            <s xml:id="echoid-s6456" xml:space="preserve">complectitur, nempe qui cum illo horas 24. </s>
            <s xml:id="echoid-s6457" xml:space="preserve">conſtituit. </s>
            <s xml:id="echoid-s6458" xml:space="preserve">Nam ar-
              <lb/>
            cus nocturnus E D F, paralleli borealis, cuius diameter B D, declinationemq́; </s>
            <s xml:id="echoid-s6459" xml:space="preserve">habens D M, æqua-
              <lb/>
            lis eſt, per propoſ. </s>
            <s xml:id="echoid-s6460" xml:space="preserve">19. </s>
            <s xml:id="echoid-s6461" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s6462" xml:space="preserve">2. </s>
            <s xml:id="echoid-s6463" xml:space="preserve">Theodoſii, arcui diurno paralleli auſtralis, qui priori ęqnalis eſt, & </s>
            <s xml:id="echoid-s6464" xml:space="preserve">op-
              <lb/>
            poſitus, hoc eſt, qui declinationem habet æqualem. </s>
            <s xml:id="echoid-s6465" xml:space="preserve">Vnde inuenta declinatione paralleli borealis,
              <lb/>
            habetur etiam declinatio paralleli auſtralis oppoſiti, & </s>
            <s xml:id="echoid-s6466" xml:space="preserve">æqualis, cuius arcus diurnus cum arcu
              <lb/>
            diurno illius complet horas 24.</s>
            <s xml:id="echoid-s6467" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6468" xml:space="preserve">QVONIAM verò modus proxime deſcriptus requirit accuratam diuiſionẽ circuli ABCD,
              <lb/>
              <note position="left" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">10</note>
            in horas, ſemiſſes horarũ, & </s>
            <s xml:id="echoid-s6469" xml:space="preserve">quadrantes, poterimus ex ſinubus idem conſequi ſine vlla diuiſione
              <lb/>
            circuli, hac arte. </s>
            <s xml:id="echoid-s6470" xml:space="preserve">Sit Analemma A B C D, cuius centrum E; </s>
            <s xml:id="echoid-s6471" xml:space="preserve">Horizon B D; </s>
            <s xml:id="echoid-s6472" xml:space="preserve">Aequator A C; </s>
            <s xml:id="echoid-s6473" xml:space="preserve">paralle-
              <lb/>
              <note position="right" xlink:label="note-0127-02" xlink:href="note-0127-02a" xml:space="preserve">Declinatio pa-
                <lb/>
              ralleli, cuius ar-
                <lb/>
              cus diutnus ſit
                <lb/>
              notus, qua arte
                <lb/>
              per ſinus per-
                <lb/>
              quiratur.</note>
            lus ſiue boreus, ſiue auſtrinus H I; </s>
            <s xml:id="echoid-s6474" xml:space="preserve">axis mundi F G, ſecans parallelum in K. </s>
            <s xml:id="echoid-s6475" xml:space="preserve">Deſcribatur ex K, circa
              <lb/>
            H I, ſen@icirculus paralleli H M I, cui occurrat axis productus in M. </s>
            <s xml:id="echoid-s6476" xml:space="preserve">Ducatur quoque ex L, ad H I,
              <lb/>
            perpendicularis L N, quæ erit communis ſectio paralleli H M I, & </s>
            <s xml:id="echoid-s6477" xml:space="preserve">Horizontis; </s>
            <s xml:id="echoid-s6478" xml:space="preserve">(Quoniã enim tam
              <lb/>
            Horizon, quàm parallelus rectus eſt ad Meridianum, erit quoque cõmunis eorum ſectio ad eun-
              <lb/>
              <note position="right" xlink:label="note-0127-03" xlink:href="note-0127-03a" xml:space="preserve">19. vndec.</note>
            dem perpendicularis in puncto L,
              <lb/>
              <figure xlink:label="fig-0127-01" xlink:href="fig-0127-01a" number="93">
                <image file="0127-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0127-01"/>
              </figure>
            vbi ſe mutuo ſecant, ac proinde & </s>
            <s xml:id="echoid-s6479" xml:space="preserve">ad
              <lb/>
            rectam H I, per defin. </s>
            <s xml:id="echoid-s6480" xml:space="preserve">3. </s>
            <s xml:id="echoid-s6481" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s6482" xml:space="preserve">11. </s>
            <s xml:id="echoid-s6483" xml:space="preserve">Eucl.
              <lb/>
            </s>
            <s xml:id="echoid-s6484" xml:space="preserve">Ergo L N, perpendicularis ad H I,
              <lb/>
              <note position="left" xlink:label="note-0127-04" xlink:href="note-0127-04a" xml:space="preserve">20</note>
            cõmunis ſectio erit Horizontis & </s>
            <s xml:id="echoid-s6485" xml:space="preserve">pa
              <lb/>
            ralleli) atque adeo H N, erit arcus ſe-
              <lb/>
            midiurnus dicti paralleli, qui poni-
              <lb/>
            tur notus. </s>
            <s xml:id="echoid-s6486" xml:space="preserve">Tandem iungatur recta
              <lb/>
            E H. </s>
            <s xml:id="echoid-s6487" xml:space="preserve">Quoniam igitur in triangulo
              <lb/>
            E
              <emph style="sc">K</emph>
            L, ſi recta E L, ponatur ſinus to-
              <lb/>
            tus, recta K L, ſinus eſt anguli K E L,
              <lb/>
            altitudinis poli, & </s>
            <s xml:id="echoid-s6488" xml:space="preserve">E K, ſinus anguli
              <lb/>
            E L k, complementi altitudinis poli;
              <lb/>
            </s>
            <s xml:id="echoid-s6489" xml:space="preserve">ſi fiat, vt K L, ſinus altitudinis poli,
              <lb/>
              <note position="left" xlink:label="note-0127-05" xlink:href="note-0127-05a" xml:space="preserve">30</note>
            ad E K, ſinum complementi altitudi-
              <lb/>
            nis poli, ita K L, quatenus ſinus re-
              <lb/>
            ctus arcus M N, quo arcus ſemidiur-
              <lb/>
            nus H N, notus, & </s>
            <s xml:id="echoid-s6490" xml:space="preserve">quadrans H M,
              <lb/>
            inter ſe differunt, ad aliud, nota fiet
              <lb/>
            recta E k, in partibus ſinus totius
              <lb/>
            K H, reſpectu cuius K L, ſinus rectus
              <lb/>
            eſt arcus M N. </s>
            <s xml:id="echoid-s6491" xml:space="preserve">Quia igitur duo qua-
              <lb/>
            drata rectarum E K, k H, quadrato rectæ E H, æqualia ſunt; </s>
            <s xml:id="echoid-s6492" xml:space="preserve">ſi ſimul addantur quadratum ſinus to-
              <lb/>
              <note position="right" xlink:label="note-0127-06" xlink:href="note-0127-06a" xml:space="preserve">47. primi.</note>
            tius K H, & </s>
            <s xml:id="echoid-s6493" xml:space="preserve">quadratum rectæ E K, notæ iam factæ in partibus ſinus totius K H, notum erit qua-
              <lb/>
              <note position="left" xlink:label="note-0127-07" xlink:href="note-0127-07a" xml:space="preserve">40</note>
            dratum rectæ E H, cuius radix quadrata dabit rectam E H, notam in ijſdem partibus ſinus to-
              <lb/>
            tius k H. </s>
            <s xml:id="echoid-s6494" xml:space="preserve">Itaque ſi fiat, vt E H, quatenus nota in partibus ſinus totius K H, ad E K, quatenus
              <lb/>
            nota eſt in ijſdem partibus, ita E H, quatenus ſinus totus, ad aliud, nota fiet E K, in partibus
              <lb/>
            ſinus totius E H. </s>
            <s xml:id="echoid-s6495" xml:space="preserve">Quare & </s>
            <s xml:id="echoid-s6496" xml:space="preserve">A H, arcus reſpondens ſinui E k, ex tabula ſinuum cognitus
              <lb/>
            erit, nempe declinatio paralleli, cuius arcus diurnus, ac proinde & </s>
            <s xml:id="echoid-s6497" xml:space="preserve">ſemidiurnus H N, no-
              <lb/>
            tus ponitur.</s>
            <s xml:id="echoid-s6498" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6499" xml:space="preserve">EXEMPLVM. </s>
            <s xml:id="echoid-s6500" xml:space="preserve">Sit datus arcus diurnus horarum 14. </s>
            <s xml:id="echoid-s6501" xml:space="preserve">vel 10. </s>
            <s xml:id="echoid-s6502" xml:space="preserve">atque adeo ſemidiurnus arcus
              <lb/>
              <note position="right" xlink:label="note-0127-08" xlink:href="note-0127-08a" xml:space="preserve">Exemplu@.</note>
            H N, horarum 7. </s>
            <s xml:id="echoid-s6503" xml:space="preserve">vel 5. </s>
            <s xml:id="echoid-s6504" xml:space="preserve">hoc eſt, grad. </s>
            <s xml:id="echoid-s6505" xml:space="preserve">105. </s>
            <s xml:id="echoid-s6506" xml:space="preserve">vel 75. </s>
            <s xml:id="echoid-s6507" xml:space="preserve">ex quo fiet arcus M N, horæ vnius, nempe grad.
              <lb/>
            </s>
            <s xml:id="echoid-s6508" xml:space="preserve">15. </s>
            <s xml:id="echoid-s6509" xml:space="preserve">cum quadrans H M, contineat horas 6. </s>
            <s xml:id="echoid-s6510" xml:space="preserve">hoc eſt, grad. </s>
            <s xml:id="echoid-s6511" xml:space="preserve">90. </s>
            <s xml:id="echoid-s6512" xml:space="preserve">Si igitur fiat, vt k L, ſinus altitudinis
              <lb/>
            poli 66913. </s>
            <s xml:id="echoid-s6513" xml:space="preserve">ad E K, ſinum complementi altitudinis poli 74314. </s>
            <s xml:id="echoid-s6514" xml:space="preserve">ita K L, ſinus arcus M N, quo
              <lb/>
              <note position="left" xlink:label="note-0127-09" xlink:href="note-0127-09a" xml:space="preserve">50</note>
            differt arcus ſemidiurnus à quadrãte, nempe in dato exemplo, ſinus arcus grad. </s>
            <s xml:id="echoid-s6515" xml:space="preserve">15. </s>
            <s xml:id="echoid-s6516" xml:space="preserve">qui eſt 25881.
              <lb/>
            </s>
            <s xml:id="echoid-s6517" xml:space="preserve">ad aliud, inuenietur E K, partium ferè 28744, qualium K L, eſt 25881. </s>
            <s xml:id="echoid-s6518" xml:space="preserve">Quia verò K H, ſinus totus
              <lb/>
            earundem partium eſt 100000. </s>
            <s xml:id="echoid-s6519" xml:space="preserve">ſi quadrati numeri ſinuum E K, KH, ſimul addantur, fiet quadra-
              <lb/>
            tum rectę E H, partium ferè 10826217536. </s>
            <s xml:id="echoid-s6520" xml:space="preserve">atque adeò recta E H, erit quaſi partium 104049. </s>
            <s xml:id="echoid-s6521" xml:space="preserve">
              <lb/>
            reſpectu ſinus totius K H. </s>
            <s xml:id="echoid-s6522" xml:space="preserve">Igitur ſi fiat, vt E H, 104049. </s>
            <s xml:id="echoid-s6523" xml:space="preserve">ad E K, 28744. </s>
            <s xml:id="echoid-s6524" xml:space="preserve">ita E H, ſinus to-
              <lb/>
            tus 100000. </s>
            <s xml:id="echoid-s6525" xml:space="preserve">ad aliud, reperietur E K, ſinus declinationis paralleli H I, reſpectu ſinus totius
              <lb/>
            E H, partium ferè 27625. </s>
            <s xml:id="echoid-s6526" xml:space="preserve">cuius arcus eſt grad. </s>
            <s xml:id="echoid-s6527" xml:space="preserve">16. </s>
            <s xml:id="echoid-s6528" xml:space="preserve">Min. </s>
            <s xml:id="echoid-s6529" xml:space="preserve">2. </s>
            <s xml:id="echoid-s6530" xml:space="preserve">Tanta eſt declinatio paralle-
              <lb/>
              <note position="right" xlink:label="note-0127-10" xlink:href="note-0127-10a" xml:space="preserve">Declinatio pa-
                <lb/>
              ralleli, cuius ar-
                <lb/>
              cus. diurnus da-
                <lb/>
              tus ſit, quo pa-
                <lb/>
              cto aliter per ſi-
                <lb/>
              nus ſit inuefti-
                <lb/>
              gandus.</note>
            li
              <emph style="sc">H</emph>
            I, borealis quidem, ſi dies maior eſt 12. </s>
            <s xml:id="echoid-s6531" xml:space="preserve">horis, auſtralis autem, ſi minor. </s>
            <s xml:id="echoid-s6532" xml:space="preserve">Eadem ratio eſt
              <lb/>
            in cæteris.</s>
            <s xml:id="echoid-s6533" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6534" xml:space="preserve">FACILIVS idem obtinebimus per triangula ſphęrica hoc modo. </s>
            <s xml:id="echoid-s6535" xml:space="preserve">Sit
              <emph style="sc">H</emph>
            orizon A B C D;
              <lb/>
            </s>
            <s xml:id="echoid-s6536" xml:space="preserve">Meridianus A C; </s>
            <s xml:id="echoid-s6537" xml:space="preserve">Aequator B D, Meridianum ſecans in E; </s>
            <s xml:id="echoid-s6538" xml:space="preserve">parallelus ſiue borealis, ſiue </s>
          </p>
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