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-s3163" xml:space="preserve">
              <pb o="48" file="0068" n="68" rhead="GNOMONICES"/>
            mutuo ſecent, ita palam fiet. </s>
            <s xml:id="echoid-s3164" xml:space="preserve">Si enim non ſe interſecent, cum ſint in eodem plano horologii, pa-
              <lb/>
            rallelæ erunt. </s>
            <s xml:id="echoid-s3165" xml:space="preserve">Cum igitur vtraque H I, P Q, ipſi V X, parallela ſit, & </s>
            <s xml:id="echoid-s3166" xml:space="preserve">inter ſe parallelæ erunt H I,
              <lb/>
              <note position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">9. vndec.</note>
            P Q. </s>
            <s xml:id="echoid-s3167" xml:space="preserve">Eſt autem & </s>
            <s xml:id="echoid-s3168" xml:space="preserve">H K, ipſi P Q, parallela, vt ſupra demonſtrauimus. </s>
            <s xml:id="echoid-s3169" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s3170" xml:space="preserve">H I, H k, inter
              <lb/>
              <figure xlink:label="fig-0068-01" xlink:href="fig-0068-01a" number="50">
                <image file="0068-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0068-01"/>
              </figure>
              <note position="left" xlink:label="note-0068-02" xlink:href="note-0068-02a" xml:space="preserve">10</note>
              <note position="left" xlink:label="note-0068-03" xlink:href="note-0068-03a" xml:space="preserve">20</note>
              <note position="left" xlink:label="note-0068-04" xlink:href="note-0068-04a" xml:space="preserve">30</note>
              <note position="left" xlink:label="note-0068-05" xlink:href="note-0068-05a" xml:space="preserve">9. vndec.</note>
            ſe parallelæ erunt. </s>
            <s xml:id="echoid-s3171" xml:space="preserve">Quod eſt abſurdũ, cùm conueniant in H Secant ergo ſe mutuo P Q, V X. </s>
            <s xml:id="echoid-s3172" xml:space="preserve">Qua-
              <lb/>
            re V X, vtramque H λ, P Q, ſecans, per punctum λ, vbi ſe mutuo ſecant H λ, Q P, tranſibit. </s>
            <s xml:id="echoid-s3173" xml:space="preserve">Nam
              <lb/>
            ſi verbi gratia, V X, ſecet rectam H λ, citra punctum λ, vel vltra, ſi producatur vlterius, ſtatim ſe-
              <lb/>
            cabit planum circuli B C, in eodem puncto, in quo rectam H λ, ſecat, ac proinde ſecare non po-
              <lb/>
            terit amplius rectam Q P, in eodem plano circuli B C, exiſtentem. </s>
            <s xml:id="echoid-s3174" xml:space="preserve">Pari ratione, ſi V X, ſecet P Q,
              <lb/>
              <note position="left" xlink:label="note-0068-06" xlink:href="note-0068-06a" xml:space="preserve">40</note>
            citra, vel vltra punctum λ, ſecabit quoque producta ſtatim planum circuli B C, in eodem puncto,
              <lb/>
            in quo rectam P Q, ſecat, ac propterea ſecare non poterit rectam H λ, in eodem ſemper plano
              <lb/>
            circuli B C, exiſtentem. </s>
            <s xml:id="echoid-s3175" xml:space="preserve">Quare V X, niſi per punctum λ, tranſeat, vtramque H λ, Q P, non ſeca-
              <lb/>
            bit, Quod eſt abſurdum. </s>
            <s xml:id="echoid-s3176" xml:space="preserve">oſtenſum eſt enim eam ſecare vtramque. </s>
            <s xml:id="echoid-s3177" xml:space="preserve">Eodem modo in δ, ſe mutuo
              <lb/>
              <note position="left" xlink:label="note-0068-07" xlink:href="note-0068-07a" xml:space="preserve">16. vndec.</note>
            ſecabunt tres rectæ α δ, θ ε, V X. </s>
            <s xml:id="echoid-s3178" xml:space="preserve">Quoniam igitur rectæ μ β, ξ γ, parallelæ ſunt, vtpote commu-
              <lb/>
            nes ſectiones planorum parallelorum F G, M N, factæ à plano circuli D B C E, nec non & </s>
            <s xml:id="echoid-s3179" xml:space="preserve">rectæ
              <lb/>
            μ ξ, β γ, nempe communes ſectiones planorum parallelorum B C, Y Z, factæ à plano eodem
              <lb/>
              <note position="left" xlink:label="note-0068-08" xlink:href="note-0068-08a" xml:space="preserve">34. primi.</note>
            circuli D B C E; </s>
            <s xml:id="echoid-s3180" xml:space="preserve">æquales erunt μ ξ, β γ. </s>
            <s xml:id="echoid-s3181" xml:space="preserve">Et quia H λ, μ ξ, necnon & </s>
            <s xml:id="echoid-s3182" xml:space="preserve">α δ, β γ, parallelæ etiam ſunt,
              <lb/>
              <note position="left" xlink:label="note-0068-09" xlink:href="note-0068-09a" xml:space="preserve">28. primi.</note>
            propterea quòd anguli ω H λ, β α δ, ad puncta contactuum H, α, recti ſunt, (ſunt enim H μ, α β,
              <lb/>
            diametri circulorum B C, Y Z, nempe communes ſectiones ipſorum, & </s>
            <s xml:id="echoid-s3183" xml:space="preserve">plani F G, quod eos per
              <lb/>
              <note position="left" xlink:label="note-0068-10" xlink:href="note-0068-10a" xml:space="preserve">18. tertij.</note>
              <note position="left" xlink:label="note-0068-11" xlink:href="note-0068-11a" xml:space="preserve">50</note>
            polos bifariam ſecat, ex propo@. </s>
            <s xml:id="echoid-s3184" xml:space="preserve">15. </s>
            <s xml:id="echoid-s3185" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3186" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3187" xml:space="preserve">Theod.) </s>
            <s xml:id="echoid-s3188" xml:space="preserve">necnon & </s>
            <s xml:id="echoid-s3189" xml:space="preserve">anguli H μ ξ, α β γ, ad centra μ, β.
              <lb/>
            </s>
            <s xml:id="echoid-s3190" xml:space="preserve">Cum enim per conſtructionem, recta B C, rectam H k, ad angulos rectos ſecet, rectus erit angu-
              <lb/>
            lus H μ ξ. </s>
            <s xml:id="echoid-s3191" xml:space="preserve">Cum igitur rectis H μ, μ ξ, parallelæ ſint α β, β γ, erit & </s>
            <s xml:id="echoid-s3192" xml:space="preserve">angulus α β γ, ipſi H μ ξ, æqualis,
              <lb/>
              <note position="left" xlink:label="note-0068-12" xlink:href="note-0068-12a" xml:space="preserve">16. vndec.</note>
            rectus: </s>
            <s xml:id="echoid-s3193" xml:space="preserve">ſunt autem & </s>
            <s xml:id="echoid-s3194" xml:space="preserve">rectæ H μ, λ ξ, nec non α β, δ γ, parallelæ, nimirum ſectiones communes
              <lb/>
              <note position="left" xlink:label="note-0068-13" xlink:href="note-0068-13a" xml:space="preserve">10. vndec.</note>
            planorum parallelorum F G, M N, factæ à plano circuli B C, ſi de prioribus, & </s>
            <s xml:id="echoid-s3195" xml:space="preserve">à plano circuli
              <lb/>
              <note position="left" xlink:label="note-0068-14" xlink:href="note-0068-14a" xml:space="preserve">16. vndec.</note>
            Y Z, ſi de poſterioribus loquamur; </s>
            <s xml:id="echoid-s3196" xml:space="preserve">fit vt recta δ ε, minor ſit, quàm recta λ P, vt mox demonſtra-
              <lb/>
            bimus ſequenti lemmate. </s>
            <s xml:id="echoid-s3197" xml:space="preserve">Quare propinquior eſt recta V X, producta hyperbolæ O ε θ, in pun-
              <lb/>
            cto δ, quàm in puncto P; </s>
            <s xml:id="echoid-s3198" xml:space="preserve">atque ita deinceps, ſi longius producantur linea horaria, & </s>
            <s xml:id="echoid-s3199" xml:space="preserve">hyperbo-
              <lb/>
            le. </s>
            <s xml:id="echoid-s3200" xml:space="preserve">Idemq́ue oſtendemus in hyperbola R S T. </s>
            <s xml:id="echoid-s3201" xml:space="preserve">Item de linea horaria, quam facit circulus tan-
              <lb/>
            gens conicas ſuperficies in recta K L. </s>
            <s xml:id="echoid-s3202" xml:space="preserve">Quamobrem ſi circulus cuiuſuis horæ à meridie, vel media
              <lb/>
            nocte plano horologii æquidiſtet, &</s>
            <s xml:id="echoid-s3203" xml:space="preserve">c. </s>
            <s xml:id="echoid-s3204" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s3205" xml:space="preserve"/>
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