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-s4915" xml:space="preserve">
              <pb o="89" file="0109" n="109" rhead="LIBER PRIMVS."/>
            nationem muri à Verticali propriè dicto, & </s>
            <s xml:id="echoid-s4916" xml:space="preserve">à meridie in ortum, vel occaſum, prout vmbra nobis ad mu-
              <lb/>
            rum conuerſis ad dextram, vel ad ſiniſtram ceciderit, vt proxime dictum eſt. </s>
            <s xml:id="echoid-s4917" xml:space="preserve">Quod intelligendum eſt
              <lb/>
            de muro ad meridiem vergente. </s>
            <s xml:id="echoid-s4918" xml:space="preserve">Nam murus ad Boream ſpectans in meridie non illuminatur à Sole, niſi
              <lb/>
            Sol borealior ipſo muro fuerit, quod in Zona torrida contingere poteſt. </s>
            <s xml:id="echoid-s4919" xml:space="preserve">Quod cum acciderit, & </s>
            <s xml:id="echoid-s4920" xml:space="preserve">vmbra
              <lb/>
            nobis ad murum conuerſis ceciderit ad ſiniſtram, declinabit murus à Septentrione in ortum, ſi verò ad
              <lb/>
            dextram, à Septentrione in occaſum.</s>
            <s xml:id="echoid-s4921" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s4922" xml:space="preserve">HAEC omnia accommodari poſſunt etiam planis inclinatis ad Horizontem, ſi ſupra lineam, quæ
              <lb/>
              <note position="right" xlink:label="note-0109-01" xlink:href="note-0109-01a" xml:space="preserve">Declinatio
                <unsure/>
              pla-
                <lb/>
              norum, quæ ad
                <lb/>
              Horizontẽ ſunt
                <lb/>
              inclinata, qua
                <lb/>
              tatione repert@a-
                <lb/>
              tur.</note>
            in eiuſinodi planis Horizonti ducitur parallela, ſtatuatur planum rectum ad Horizontem, obſeruatio{q́ue}
              <lb/>
            fiat in facie huius plani, quæ cum plano inclinato angulum obtuſum constituit.
              <lb/>
            </s>
            <s xml:id="echoid-s4923" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div308" type="section" level="1" n="116">
          <head xml:id="echoid-head119" xml:space="preserve">THEOREMA 21. PROPOSITIO 24.</head>
          <p>
            <s xml:id="echoid-s4924" xml:space="preserve">SI à circunferentia circuli maximi in ſphęra ſuper alium circulum
              <lb/>
              <note position="right" xlink:label="note-0109-03" xlink:href="note-0109-03a" xml:space="preserve">Perpendicula-
                <lb/>
              res cadentes à
                <lb/>
              circunferentia
                <lb/>
              maximi circuli
                <lb/>
              in ſphæra in pla
                <lb/>
              num alterius
                <lb/>
              circuli maximi,
                <lb/>
              ad quem ille in
                <lb/>
              clinatus eſt, fa-
                <lb/>
              ciunt in poſte-
                <lb/>
              riori circulo ma
                <lb/>
              ximo Ellipſina
                <unsure/>
              .</note>
            maximum inclinati perpendiculares ad eiuſdem circuli maximi planũ
              <lb/>
            ducantur, cadent omnes in lineam, quæ Ellipſis appellatur; </s>
            <s xml:id="echoid-s4925" xml:space="preserve">cuius qui-
              <lb/>
            dem diameter maior eadem eſt, quæ communis ſectio ipſorum circulo-
              <lb/>
            rum, nempe eorum diameter, minor verò determinatur interuallo per-
              <lb/>
              <note position="left" xlink:label="note-0109-04" xlink:href="note-0109-04a" xml:space="preserve">20</note>
            pendicularium cadentium ab extremitate alterius diametri circuli incli
              <lb/>
            nati, quæ priorem diametrum, hoc eſt, communem ſectionem, ad re-
              <lb/>
            ctos angulos diuidit.</s>
            <s xml:id="echoid-s4926" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4927" xml:space="preserve">SIT in ſphæra circulus maximus A B C D, cuius centrum E, ad circulum maximum AFCG,
              <lb/>
            inclinatus, & </s>
            <s xml:id="echoid-s4928" xml:space="preserve">circulus A F C G, ſecet circulum A B C D, in centro E, vt ſit diameter A C, commu
              <lb/>
            nis ſectio circulorum A B C D, A F C G, ducaturq́;
              <lb/>
            </s>
            <s xml:id="echoid-s4929" xml:space="preserve">
              <figure xlink:label="fig-0109-01" xlink:href="fig-0109-01a" number="71">
                <image file="0109-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0109-01"/>
              </figure>
            in circulo A B C D, alia diameter B D, ſecans A C,
              <lb/>
            ad rectos angulos: </s>
            <s xml:id="echoid-s4930" xml:space="preserve">ducatur quoque in plano circu-
              <lb/>
            li A F C G, alia diameter F G, ad eandem A G, per-
              <lb/>
              <note position="left" xlink:label="note-0109-05" xlink:href="note-0109-05a" xml:space="preserve">30</note>
            pendicularis. </s>
            <s xml:id="echoid-s4931" xml:space="preserve">Quoniam igitur recta C E, rectis
              <lb/>
            B E, F E, ſeſe in E, ſecantibus inſiſtit ad rectos an-
              <lb/>
            gulos, erit eadem C E, ad planum per B E, F E, du-
              <lb/>
              <note position="right" xlink:label="note-0109-06" xlink:href="note-0109-06a" xml:space="preserve">4. vndec.</note>
            ctum ad angulos rectos. </s>
            <s xml:id="echoid-s4932" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s4933" xml:space="preserve">plana circulo-
              <lb/>
            rum A F C G, A B C D, per C E, tranſeuntia ad idẽ
              <lb/>
              <note position="right" xlink:label="note-0109-07" xlink:href="note-0109-07a" xml:space="preserve">18. vndec.</note>
            planum per B E, F E, ductum erunt recta, eritq́ue
              <lb/>
            F G, communis ſectio plani per B E, F E, ducti, & </s>
            <s xml:id="echoid-s4934" xml:space="preserve">
              <lb/>
            plani circuli A F C G, quod ad illud rectum eſt
              <lb/>
            oſtenſum; </s>
            <s xml:id="echoid-s4935" xml:space="preserve">atque adeò ſi ex puncto B, quod in pla-
              <lb/>
            no per B E, F E, ducto exiſtit, linea perpendicularis ducatur ad planum A F C G, nempe B H,
              <lb/>
              <note position="left" xlink:label="note-0109-08" xlink:href="note-0109-08a" xml:space="preserve">40</note>
            ipſa in F E, communem ſectionem plani per B E, F E, ducti, & </s>
            <s xml:id="echoid-s4936" xml:space="preserve">plani A F C G, cadet. </s>
            <s xml:id="echoid-s4937" xml:space="preserve">Cadat ergo
              <lb/>
              <note position="right" xlink:label="note-0109-09" xlink:href="note-0109-09a" xml:space="preserve">38. vndec.</note>
            in H. </s>
            <s xml:id="echoid-s4938" xml:space="preserve">Eodem modo perpendicularis ex D, ad idem planum A F C G, ducta in rectam F G, cadet,
              <lb/>
            vt in I: </s>
            <s xml:id="echoid-s4939" xml:space="preserve">eruntq́; </s>
            <s xml:id="echoid-s4940" xml:space="preserve">rectæ E H, E I, inter ſe æquales. </s>
            <s xml:id="echoid-s4941" xml:space="preserve">Cum enim in triangulis E B H, E D I, anguli ad
              <lb/>
            H, I, recti ſint, ex defin. </s>
            <s xml:id="echoid-s4942" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4943" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4944" xml:space="preserve">11. </s>
            <s xml:id="echoid-s4945" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4946" xml:space="preserve">& </s>
            <s xml:id="echoid-s4947" xml:space="preserve">anguli ad verticẽ E, ęquales, Item & </s>
            <s xml:id="echoid-s4948" xml:space="preserve">latera E B, E D, ęqua-
              <lb/>
              <note position="right" xlink:label="note-0109-10" xlink:href="note-0109-10a" xml:space="preserve">15. primi.</note>
            lia, erunt quoque latc
              <unsure/>
            ra E H, E I, ęqualia. </s>
            <s xml:id="echoid-s4949" xml:space="preserve">Sumpto autem quouis alio puncto K, in circunferentia
              <lb/>
              <note position="right" xlink:label="note-0109-11" xlink:href="note-0109-11a" xml:space="preserve">26. primi.</note>
            circuli A B C D, ab eodem ad idem planum A F C G, perpendicularis demittatur K L. </s>
            <s xml:id="echoid-s4950" xml:space="preserve">Dico pun
              <lb/>
            ctum L, cadere in Ellipſim, cuius quidem diameter maior eſt A C, circulorum diameter, & </s>
            <s xml:id="echoid-s4951" xml:space="preserve">mi-
              <lb/>
            nor recta H I. </s>
            <s xml:id="echoid-s4952" xml:space="preserve">Ducta enim in plano A F C G, ex L, ad A C, perpendiculari L M, cum & </s>
            <s xml:id="echoid-s4953" xml:space="preserve">H E, ad
              <lb/>
            eandẽ A C, ſit perpendicularis, erunt H E, L M, inter ſe parallelæ: </s>
            <s xml:id="echoid-s4954" xml:space="preserve">Sed & </s>
            <s xml:id="echoid-s4955" xml:space="preserve">B H, K L, ſunt inter ſe
              <lb/>
              <note position="right" xlink:label="note-0109-12" xlink:href="note-0109-12a" xml:space="preserve">28. primi.</note>
            parallelę, quòd ad idem planũ A F C G, ſint perpendiculares. </s>
            <s xml:id="echoid-s4956" xml:space="preserve">Ergo planũ per B H, H E, ductum
              <lb/>
              <note position="left" xlink:label="note-0109-13" xlink:href="note-0109-13a" xml:space="preserve">50</note>
              <note position="right" xlink:label="note-0109-14" xlink:href="note-0109-14a" xml:space="preserve">6. vndec.</note>
            plano per k L, L M, ducto parallelũ erit: </s>
            <s xml:id="echoid-s4957" xml:space="preserve">& </s>
            <s xml:id="echoid-s4958" xml:space="preserve">propterea ipſorũ planorũ, & </s>
            <s xml:id="echoid-s4959" xml:space="preserve">circuli A B C D, cõmu-
              <lb/>
              <note position="right" xlink:label="note-0109-15" xlink:href="note-0109-15a" xml:space="preserve">15. vndec.</note>
            nes ſectiones, nempe rectæ B E, k M, parallelæ erunt. </s>
            <s xml:id="echoid-s4960" xml:space="preserve">Itaq; </s>
            <s xml:id="echoid-s4961" xml:space="preserve">quoniã B E, E H, ſeſe in E, tangentes, re-
              <lb/>
              <note position="right" xlink:label="note-0109-16" xlink:href="note-0109-16a" xml:space="preserve">16. vndec.</note>
            ctis k M, M L, in M, ſeſe tangentibus ſunt parallelæ, non ſunt autem in eodem plano, ſed in planis
              <lb/>
            parallelis, vt dictũ eſt, erit angulus B E H, angulo K M L, æqualis: </s>
            <s xml:id="echoid-s4962" xml:space="preserve">Sunt autem & </s>
            <s xml:id="echoid-s4963" xml:space="preserve">H, L, æquales, vt-
              <lb/>
              <note position="right" xlink:label="note-0109-17" xlink:href="note-0109-17a" xml:space="preserve">10. vndec.</note>
            potè recti, ex defin. </s>
            <s xml:id="echoid-s4964" xml:space="preserve">3. </s>
            <s xml:id="echoid-s4965" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4966" xml:space="preserve">11. </s>
            <s xml:id="echoid-s4967" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4968" xml:space="preserve">Ergo & </s>
            <s xml:id="echoid-s4969" xml:space="preserve">reliquus E B H, reliquo M k L, ęqualis erit, ex coroll. </s>
            <s xml:id="echoid-s4970" xml:space="preserve">1.
              <lb/>
            </s>
            <s xml:id="echoid-s4971" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s4972" xml:space="preserve">32. </s>
            <s xml:id="echoid-s4973" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4974" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4975" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s4976" xml:space="preserve">& </s>
            <s xml:id="echoid-s4977" xml:space="preserve">triangulũ B E H, triangulo k M L, æquiangulũ. </s>
            <s xml:id="echoid-s4978" xml:space="preserve">Quare erit vt B E, ad EH,
              <lb/>
              <note position="right" xlink:label="note-0109-18" xlink:href="note-0109-18a" xml:space="preserve">4. ſexti.</note>
            ita K M, ad M L, permutandoq́;</s>
            <s xml:id="echoid-s4979" xml:space="preserve">, vt B E, ad K M, ita E H, ad M L. </s>
            <s xml:id="echoid-s4980" xml:space="preserve">Igitur vt quadratum ex B E,
              <lb/>
            ad quadratum ex k M, ita quadratum ex E H, ad quadratũ ex M L: </s>
            <s xml:id="echoid-s4981" xml:space="preserve">Vt autẽ quadratũ ex B E, ad qua-
              <lb/>
              <note position="right" xlink:label="note-0109-19" xlink:href="note-0109-19a" xml:space="preserve">22. ſexti.</note>
            dratum ex K M, ita eſt, ex propoſ. </s>
            <s xml:id="echoid-s4982" xml:space="preserve">21. </s>
            <s xml:id="echoid-s4983" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s4984" xml:space="preserve">1. </s>
            <s xml:id="echoid-s4985" xml:space="preserve">Apollonij, rectangulum ſub C E, E A, ad rectangulũ
              <lb/>
            ſub C M, M A, propterea quòd B E, k M, in circulo A B C D, ad diametrum A C, ſint ductæ or-
              <lb/>
            dinatim, nempe perpendiculares. </s>
            <s xml:id="echoid-s4986" xml:space="preserve">Eſt igitur quoque, vt quadratum ex E H, ad quadratum </s>
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