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-div313" type="section" level="1" n="118">
          <p>
            <s xml:id="echoid-s5044" xml:space="preserve">
              <pb o="91" file="0111" n="111" rhead="LIBER PRIMVS."/>
            c
              <unsure/>
            undem ſemicirculũ E D F, recta erunt, atque idcirco & </s>
            <s xml:id="echoid-s5045" xml:space="preserve">viciſſim hic ſemicirculus ad illa plana re-
              <lb/>
            ctus erit.) </s>
            <s xml:id="echoid-s5046" xml:space="preserve">Quare D E, arcus erit inclinationis circuli A B C D, ad circulum A F C E, proptereaq́;
              <lb/>
            </s>
            <s xml:id="echoid-s5047" xml:space="preserve">arcui E H, æqualis: </s>
            <s xml:id="echoid-s5048" xml:space="preserve">cadetq́ue perpendicularis ex
              <lb/>
              <figure xlink:label="fig-0111-01" xlink:href="fig-0111-01a" number="73">
                <image file="0111-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0111-01"/>
              </figure>
            D, ad planum A F C E, demiſſa in E F, communem
              <lb/>
              <note position="right" xlink:label="note-0111-01" xlink:href="note-0111-01a" xml:space="preserve">38. vndec.</note>
            ſectionem planorum A F C E, E D F; </s>
            <s xml:id="echoid-s5049" xml:space="preserve">quam dico
              <lb/>
            in I, cadere. </s>
            <s xml:id="echoid-s5050" xml:space="preserve">Si enim alio cadat, vt in k, erunt duo
              <lb/>
            anguli H I G, I G H, trianguli G H I, duobus an-
              <lb/>
            gulis D K G, K G D, trianguli G D K, ęquales;
              <lb/>
            </s>
            <s xml:id="echoid-s5051" xml:space="preserve">(Nam I G H, K G D, ęquales ſunt, ob æqualitatem
              <lb/>
              <note position="right" xlink:label="note-0111-02" xlink:href="note-0111-02a" xml:space="preserve">27. tertij.</note>
            arcuum E H, E D, & </s>
            <s xml:id="echoid-s5052" xml:space="preserve">H I G, D K G, recti ſunt, ex
              <lb/>
              <note position="left" xlink:label="note-0111-03" xlink:href="note-0111-03a" xml:space="preserve">10</note>
            conſtructione, & </s>
            <s xml:id="echoid-s5053" xml:space="preserve">definitione 3. </s>
            <s xml:id="echoid-s5054" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5055" xml:space="preserve">11. </s>
            <s xml:id="echoid-s5056" xml:space="preserve">Euclidis)
              <lb/>
            ſuntautem & </s>
            <s xml:id="echoid-s5057" xml:space="preserve">latera G H, G D, ducta à centro ſphę-
              <lb/>
            rę ad eius ſuperficiem, ęquales. </s>
            <s xml:id="echoid-s5058" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s5059" xml:space="preserve">latera G I,
              <lb/>
              <note position="right" xlink:label="note-0111-04" xlink:href="note-0111-04a" xml:space="preserve">26. primi.</note>
            G K, æqualia erunt, pars, & </s>
            <s xml:id="echoid-s5060" xml:space="preserve">totum. </s>
            <s xml:id="echoid-s5061" xml:space="preserve">Quod eſt abſur-
              <lb/>
            dum. </s>
            <s xml:id="echoid-s5062" xml:space="preserve">Non ergo perpendicularis à puncto D, de-
              <lb/>
            miſſa ad planum A F C E, alio cadit, quàm in I. </s>
            <s xml:id="echoid-s5063" xml:space="preserve">Eo-
              <lb/>
            dem modo reperiemus punctum, in quod cadit
              <lb/>
            perpendicularis ex B, demiſſa. </s>
            <s xml:id="echoid-s5064" xml:space="preserve">Cadet autem ſem-
              <lb/>
            per in punctum, puta M, quod tantum à centro
              <lb/>
            G, abeſt, quantum I, ab eodem diſtat. </s>
            <s xml:id="echoid-s5065" xml:space="preserve">Quoniam enim in triangulis D G I, B G M, anguli ad I, M,
              <lb/>
              <note position="left" xlink:label="note-0111-05" xlink:href="note-0111-05a" xml:space="preserve">20</note>
            ex defin. </s>
            <s xml:id="echoid-s5066" xml:space="preserve">3. </s>
            <s xml:id="echoid-s5067" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s5068" xml:space="preserve">11. </s>
            <s xml:id="echoid-s5069" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s5070" xml:space="preserve">recti ſunt, anguliq́; </s>
            <s xml:id="echoid-s5071" xml:space="preserve">ad verticem G, æquales; </s>
            <s xml:id="echoid-s5072" xml:space="preserve">Item & </s>
            <s xml:id="echoid-s5073" xml:space="preserve">latera D G, B G, æqua
              <lb/>
              <note position="right" xlink:label="note-0111-06" xlink:href="note-0111-06a" xml:space="preserve">15. primi.</note>
            lia, cum ſint ſpheræ ſemidiametri; </s>
            <s xml:id="echoid-s5074" xml:space="preserve">erunt & </s>
            <s xml:id="echoid-s5075" xml:space="preserve">latera G I, G M, inter ſe ęqualia. </s>
            <s xml:id="echoid-s5076" xml:space="preserve">In circunferentia igi-
              <lb/>
              <note position="right" xlink:label="note-0111-07" xlink:href="note-0111-07a" xml:space="preserve">26. primi.</note>
            tur circuli maximi in ſphæra, &</s>
            <s xml:id="echoid-s5077" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5078" xml:space="preserve">Quod faciendum erat.</s>
            <s xml:id="echoid-s5079" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div315" type="section" level="1" n="119">
          <head xml:id="echoid-head122" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s5080" xml:space="preserve">EX his eadem via inueniemus diametrum minorem ellipſis illius, in quam perpendiculares à circun-
              <lb/>
              <note position="right" xlink:label="note-0111-08" xlink:href="note-0111-08a" xml:space="preserve">Inuentio min@
                <lb/>
              tis diametri El-
                <lb/>
              lipſis, quæ fit à
                <lb/>
              perpendiculari-
                <lb/>
              bus cadenubus
                <lb/>
              à circunferẽria
                <lb/>
              circuli inclinati
                <lb/>
              ad alium circu-
                <lb/>
              lum.</note>
            ferentia circuli inclinati in alium circulum demiſſæ cadunt. </s>
            <s xml:id="echoid-s5081" xml:space="preserve">Nam recta I M, inter puncta I, M, in quæ
              <lb/>
            dictæ perpendiculares cadunt, minor diameter eſt, per antecedentem propoſitionem.</s>
            <s xml:id="echoid-s5082" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">30</note>
        </div>
        <div xml:id="echoid-div317" type="section" level="1" n="120">
          <head xml:id="echoid-head123" xml:space="preserve">PROBLEMA 5. PROPOSITIO 26.</head>
          <p>
            <s xml:id="echoid-s5083" xml:space="preserve">IN circunferentia circuli maximi in ſphæra ad alium circulum ma-
              <lb/>
            ximum inclinati ſumptis quibuslibet punctis, quo loco perpendicula-
              <lb/>
            res ab his ductæ in alium circulum cadant, ſi inclinatio fuerit nota,
              <lb/>
            inquirere.</s>
            <s xml:id="echoid-s5084" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5085" xml:space="preserve">SIT in ſphæra circulus maximus A B C D, ad maximum D E B F, inclinatus, & </s>
            <s xml:id="echoid-s5086" xml:space="preserve">nota inclina-
              <lb/>
            tio, ſitq́; </s>
            <s xml:id="echoid-s5087" xml:space="preserve">eorum fectio communis diameter D B, per centrum G, tranſiens, ad quam ad angulos
              <lb/>
              <note position="left" xlink:label="note-0111-10" xlink:href="note-0111-10a" xml:space="preserve">40</note>
            rectos ducatur in circulo quidem A B C D, diame-
              <lb/>
              <figure xlink:label="fig-0111-02" xlink:href="fig-0111-02a" number="74">
                <image file="0111-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0111-02"/>
              </figure>
            rer A C, in circulo verò D E B F, diameter E F, in
              <lb/>
            quam cadent perpendiculares ex A, C, in circulum
              <lb/>
            D E B F, demiſſæ, vt in propoſitione præcedenti eſt
              <lb/>
            oſtenſum. </s>
            <s xml:id="echoid-s5088" xml:space="preserve">Cadant ergo in H, I, vt ſit D B, diame-
              <lb/>
            ter maior, & </s>
            <s xml:id="echoid-s5089" xml:space="preserve">H I, minor eius Ellipſis, quam perpen
              <lb/>
            diculares à circunferentia circuli A B C D, in pla-
              <lb/>
            num circuli D E B F, demiſſę faciunt, vt demonſtra
              <lb/>
            tum eſt. </s>
            <s xml:id="echoid-s5090" xml:space="preserve">Sumatur autem quodcunque punctum K,
              <lb/>
            in circunferentia A B C D. </s>
            <s xml:id="echoid-s5091" xml:space="preserve">Oportet igitur inquire-
              <lb/>
              <note position="left" xlink:label="note-0111-11" xlink:href="note-0111-11a" xml:space="preserve">50</note>
            re, quo loco perpendicularis à K, in planum D E-
              <lb/>
            B F, deducta cadat. </s>
            <s xml:id="echoid-s5092" xml:space="preserve">Sumatur arcui A K, æqualis ar-
              <lb/>
            cus E L, & </s>
            <s xml:id="echoid-s5093" xml:space="preserve">ducatur recta G L, quæ circulum H I,
              <lb/>
            circa minorem Ellipſis diametrum H I, deſcriptũ
              <lb/>
              <note position="right" xlink:label="note-0111-12" xlink:href="note-0111-12a" xml:space="preserve">Inuenti@ pun-
                <lb/>
              ctorum, in quæ
                <lb/>
              cadunt perpen-
                <lb/>
              diculares à quo
                <lb/>
              cunque puncto
                <lb/>
              circuli inclinati
                <lb/>
              ad alium circu-
                <lb/>
              lum.</note>
            ſecet in M. </s>
            <s xml:id="echoid-s5094" xml:space="preserve">Deinde per L, ducatur L N, parallela
              <lb/>
            minori diametro H I, quæ ſecet D B, in O; </s>
            <s xml:id="echoid-s5095" xml:space="preserve">& </s>
            <s xml:id="echoid-s5096" xml:space="preserve">per
              <lb/>
            M, ducatur P M, parallela maiori Ellipſis diame-
              <lb/>
            tro D B, ſecans L N, in Q. </s>
            <s xml:id="echoid-s5097" xml:space="preserve">Dico perpendicularem à K, in planum D E B F, demiſſam cadere in
              <lb/>
            punctum Q. </s>
            <s xml:id="echoid-s5098" xml:space="preserve">Quòd enim cadat in lineam L N, ita oſtendetur. </s>
            <s xml:id="echoid-s5099" xml:space="preserve">Ducta recta K O, erit hæc ipſi
              <lb/>
            A G, parallela. </s>
            <s xml:id="echoid-s5100" xml:space="preserve">Ductis enim L S, k T, ad E G, A G, perpendicularibus, cum G O, æqualis ſit ipſi
              <lb/>
              <note position="right" xlink:label="note-0111-13" xlink:href="note-0111-13a" xml:space="preserve">34. primi.</note>
            L S, ſinui recto arcus E L; </s>
            <s xml:id="echoid-s5101" xml:space="preserve">ſit autem L S, ſinus æqualis ipſi k T, ſinui recto arcus A K, qui </s>
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