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-div176" type="section" level="1" n="52">
          <pb o="49" file="0069" n="69" rhead="LIBER PRIMVS."/>
        </div>
        <div xml:id="echoid-div180" type="section" level="1" n="53">
          <head xml:id="echoid-head56" xml:space="preserve">LEMMA.</head>
          <p>
            <s xml:id="echoid-s3206" xml:space="preserve">SI duos circulos inæquales tangant duæ lineæ rectæ diametris æquidiſtantes,
              <lb/>
            coniunganturq́; </s>
            <s xml:id="echoid-s3207" xml:space="preserve">puncta contactuum, & </s>
            <s xml:id="echoid-s3208" xml:space="preserve">centra duabus rectis lineis, quibus per duo
              <lb/>
            puncta ſemidiametrorum æqualiter à centris remota parallelæ agantur ſecantes
              <lb/>
            circulorum peripherias; </s>
            <s xml:id="echoid-s3209" xml:space="preserve">erunt rectę inter lineas tangentes, & </s>
            <s xml:id="echoid-s3210" xml:space="preserve">peripherias interce-
              <lb/>
            ptæ, inæquales, minorq́; </s>
            <s xml:id="echoid-s3211" xml:space="preserve">ea, quæ extra maioreni circulum exiſtit.</s>
            <s xml:id="echoid-s3212" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3213" xml:space="preserve">SIT circulus a Y Z, maior circulo H B C, & </s>
            <s xml:id="echoid-s3214" xml:space="preserve">vtrumque
              <unsure/>
            tangant rectæ α δ, H λ,
              <lb/>
            æquidiſtantes diametris Y Z, B C, connectantur {q́ue} puncta contactuum α, H, & </s>
            <s xml:id="echoid-s3215" xml:space="preserve">centra
              <lb/>
              <note position="left" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">10</note>
            β, μ, rectis α β, H μ; </s>
            <s xml:id="echoid-s3216" xml:space="preserve">ſumptis autem rectis β γ, μ ξ, æqualibus ex ſemidiametris β Z, μ C,
              <lb/>
            agantur per γ, ξ, rectis α β, H μ, parallelæ γ δ, ξ λ, ſecantes peripherias in ε, P. </s>
            <s xml:id="echoid-s3217" xml:space="preserve">Dico
              <lb/>
            rectam δ ε, minorem eſſe, quàm
              <lb/>
              <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a" number="51">
                <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0069-01"/>
              </figure>
            λ P. </s>
            <s xml:id="echoid-s3218" xml:space="preserve">Cum enim circulus α Y Z,
              <lb/>
            maior ſit circulo H B C, erit & </s>
            <s xml:id="echoid-s3219" xml:space="preserve">
              <lb/>
            ſemidiameter α β, ſemidiame-
              <lb/>
            tro H μ, maior. </s>
            <s xml:id="echoid-s3220" xml:space="preserve">Abſciſſa ergo
              <lb/>
            recta α A, quæ ipſi H μ, ſit æqua-
              <lb/>
            lis, deſcribatur ad interuallum
              <lb/>
              <note position="left" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">10</note>
            A α, ex A, circulus α E, qui æ-
              <lb/>
            qualis erit circulo H B C, propter
              <lb/>
            æqualitatem ſemidiametrorum
              <lb/>
            α A, H μ, tanget {q́ue} circulũ α Y Z,
              <lb/>
            in α. </s>
            <s xml:id="echoid-s3221" xml:space="preserve">Et quoniam ducta ex A,
              <lb/>
            ad γ δ, perpendicularis A D,
              <lb/>
            ipſi β γ, parallela eſt, parallelogrammum erit A γ; </s>
            <s xml:id="echoid-s3222" xml:space="preserve">ac propterea recta A D, rectæ β γ, hoc
              <lb/>
              <note position="right" xlink:label="note-0069-03" xlink:href="note-0069-03a" xml:space="preserve">28. primi.</note>
            eſt, rectæ μ ξ, æqualis erit. </s>
            <s xml:id="echoid-s3223" xml:space="preserve">Cum ergo μ ξ, minor ſit ſemidiametro μ C, vel H μ, hoc eſt,
              <lb/>
              <note position="left" xlink:label="note-0069-04" xlink:href="note-0069-04a" xml:space="preserve">30</note>
              <note position="right" xlink:label="note-0069-05" xlink:href="note-0069-05a" xml:space="preserve">34. primi.</note>
            quàm α A, quæ æqualis eſt ipſi H μ, erit quoque A D, minor, quàm α A, ac idcirco
              <lb/>
            punctum D, intra circulum α E, exiſtet. </s>
            <s xml:id="echoid-s3224" xml:space="preserve">Quare circunferentia α E, rectam D δ, ſeca-
              <lb/>
            bit infra punctum ε, nempe in E. </s>
            <s xml:id="echoid-s3225" xml:space="preserve">Quia ver ò ductis rectis A E, μ P, quadratum ex A E,
              <lb/>
            quadratis ex A D, D E, & </s>
            <s xml:id="echoid-s3226" xml:space="preserve">quadratum ex μ P, quadratis ex μ ξ, ξ P, æquale eſt; </s>
            <s xml:id="echoid-s3227" xml:space="preserve">ſunt {q́ue}
              <lb/>
              <note position="right" xlink:label="note-0069-06" xlink:href="note-0069-06a" xml:space="preserve">47. primi.</note>
            quadrata ex A E, μ P, inter ſe æqualia; </s>
            <s xml:id="echoid-s3228" xml:space="preserve">erunt quadrata ex A D, D E, quadratis ex μ ξ,
              <lb/>
            ξ P, æqualia. </s>
            <s xml:id="echoid-s3229" xml:space="preserve">Ablatis ergo æqualibus quadratis rectarum A D, μ ξ, reliqua quadrata ex
              <lb/>
            D E, ξ P, æqualia erunt, ac propterea & </s>
            <s xml:id="echoid-s3230" xml:space="preserve">rectæ ipſæ æquales. </s>
            <s xml:id="echoid-s3231" xml:space="preserve">Cum ergo & </s>
            <s xml:id="echoid-s3232" xml:space="preserve">totæ D δ, ξ λ,
              <lb/>
            æquales ſint, quòd D δ, ipſi A α, & </s>
            <s xml:id="echoid-s3233" xml:space="preserve">ξ λ, ipſi μH, æqualis ſit; </s>
            <s xml:id="echoid-s3234" xml:space="preserve">erunt quoque reliquæ δ E,
              <lb/>
              <note position="right" xlink:label="note-0069-07" xlink:href="note-0069-07a" xml:space="preserve">34. primi.</note>
            λ P, æquales. </s>
            <s xml:id="echoid-s3235" xml:space="preserve">Eſt autem δ ε, minor quàm δ E. </s>
            <s xml:id="echoid-s3236" xml:space="preserve">Igitur δ ε, minor quoque erit, quàm λ P,
              <lb/>
              <note position="left" xlink:label="note-0069-08" xlink:href="note-0069-08a" xml:space="preserve">40</note>
            quod erat demonſtrandum.</s>
            <s xml:id="echoid-s3237" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s3238" xml:space="preserve">EX his manifeſtum eſt, in figura ſuperiori rectam δ ε, minorem eſſe recta λ P, vt
              <lb/>
              <note position="right" xlink:label="note-0069-09" xlink:href="note-0069-09a" xml:space="preserve">Lin@æ horarũ
                <lb/>
              12. & 24. ab or.
                <lb/>
              vel occ. non co-
                <lb/>
              cunt in horolo
                <lb/>
              gio Meridiano
                <lb/>
              cũ ſectionibus
                <lb/>
              conicis factis in
                <lb/>
              conicis ſuperfi-
                <lb/>
              ciebus, quarum
                <lb/>
              baſes ſunt pa-
                <lb/>
              ralleli ſemper
                <lb/>
              apparentium,
                <lb/>
              ſemperq́; laten-
                <lb/>
              tium maximi.</note>
            in demonſtratione aſſumebatur; </s>
            <s xml:id="echoid-s3239" xml:space="preserve">propterea quòd circulus Y Z, maior eſt circulo B C, & </s>
            <s xml:id="echoid-s3240" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s3241" xml:space="preserve">vt lemma proponit.</s>
            <s xml:id="echoid-s3242" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div183" type="section" level="1" n="54">
          <head xml:id="echoid-head57" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s3243" xml:space="preserve">SEQVITVR ex ha@ propoſ. </s>
            <s xml:id="echoid-s3244" xml:space="preserve">lineas horarum 12. </s>
            <s xml:id="echoid-s3245" xml:space="preserve">& </s>
            <s xml:id="echoid-s3246" xml:space="preserve">24. </s>
            <s xml:id="echoid-s3247" xml:space="preserve">ab ortu, vel occaſu in horologio Meridia-
              <lb/>
            no non conuenire cum Hyperbolis, quas planum horologii facit, per propoſ. </s>
            <s xml:id="echoid-s3248" xml:space="preserve">6. </s>
            <s xml:id="echoid-s3249" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s3250" xml:space="preserve">in conicis ſu-
              <lb/>
            perficiebus, quarum baſes ſunt parallelus eorum, qui ſemper apparent, maximus, & </s>
            <s xml:id="echoid-s3251" xml:space="preserve">maximus eorum, qui
              <lb/>
              <note position="left" xlink:label="note-0069-10" xlink:href="note-0069-10a" xml:space="preserve">50</note>
            ſub terra occultantur: </s>
            <s xml:id="echoid-s3252" xml:space="preserve">quia Meridianus, hoc eſt, circulus horæ 12. </s>
            <s xml:id="echoid-s3253" xml:space="preserve">à meridie, vel media nocte, ęquidi-
              <lb/>
            ſtat plano horologii, ſecatq́; </s>
            <s xml:id="echoid-s3254" xml:space="preserve">maximum parallelorum ſemper apparentium in punctis, in quibus eundem
              <lb/>
            tangit & </s>
            <s xml:id="echoid-s3255" xml:space="preserve">circulus horæ 12. </s>
            <s xml:id="echoid-s3256" xml:space="preserve">ab ortu vel occaſu, & </s>
            <s xml:id="echoid-s3257" xml:space="preserve">circulus horę 24. </s>
            <s xml:id="echoid-s3258" xml:space="preserve">ſiue Horizon, vt conſtat ex figura
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s3259" xml:space="preserve">9. </s>
            <s xml:id="echoid-s3260" xml:space="preserve">huius lib.</s>
            <s xml:id="echoid-s3261" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">Lineæ horarum
            <lb/>
          6. & 18. ab or.
            <lb/>
          vel occ. non
            <lb/>
          coeunt in horo
            <lb/>
          logio Polari cũ
            <lb/>
          ſectionibus co-
            <lb/>
          nicis factis in
            <lb/>
          conicis ſuperfi-
            <lb/>
          ciebus, quarum
            <lb/>
          baſes ſunt pa-
            <lb/>
          ralleli ſemper
            <lb/>
          apparentium,
            <lb/>
          ſemperq́; laten
            <lb/>
          tium maximi.</note>
          <p>
            <s xml:id="echoid-s3262" xml:space="preserve">EODEM modo lineæ horarum 6. </s>
            <s xml:id="echoid-s3263" xml:space="preserve">& </s>
            <s xml:id="echoid-s3264" xml:space="preserve">18. </s>
            <s xml:id="echoid-s3265" xml:space="preserve">ab ortu, vel occaſu non coibunt cum eiſdem hyperbolis
              <lb/>
            in horologio polari. </s>
            <s xml:id="echoid-s3266" xml:space="preserve">Aequidiſtat enim horologium polare circulo horæ 6. </s>
            <s xml:id="echoid-s3267" xml:space="preserve">à meridie, vel media nocte,
              <lb/>
            qui quidem ſecat parallelum dictum in punctis, in quibus eundem tangunt circuli horarum 6. </s>
            <s xml:id="echoid-s3268" xml:space="preserve">& </s>
            <s xml:id="echoid-s3269" xml:space="preserve">18. </s>
            <s xml:id="echoid-s3270" xml:space="preserve">ab
              <lb/>
            ortu vel occaſu, ut ex eadem figura propoſ. </s>
            <s xml:id="echoid-s3271" xml:space="preserve">9. </s>
            <s xml:id="echoid-s3272" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s3273" xml:space="preserve">patet.</s>
            <s xml:id="echoid-s3274" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div185" type="section" level="1" n="55">
          <head xml:id="echoid-head58" xml:space="preserve">THEOREMA 14. PROPOSITIO 16.</head>
          <p>
            <s xml:id="echoid-s3275" xml:space="preserve">SI in Sphæra duo circuli maximi tangant vnum, eundemque </s>
          </p>
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