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-s3357" xml:space="preserve">
              <pb o="51" file="0071" n="71" rhead="LIBER PRIMVS."/>
            culorum A C, B D, G H. </s>
            <s xml:id="echoid-s3358" xml:space="preserve">Deſcribatur enim maximus parallelorum L M, qui per puncta E, F, tran
              <lb/>
            ſibit, cum per præcedentem propoſ. </s>
            <s xml:id="echoid-s3359" xml:space="preserve">E F, ſit communis ſectio trium circulorum A C, B D, L M,
              <lb/>
            propterea quòd A C, BD, tangunt parallelum B C,
              <lb/>
              <figure xlink:label="fig-0071-01" xlink:href="fig-0071-01a" number="53">
                <image file="0071-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0071-01"/>
              </figure>
            in punctis oppoſitis, & </s>
            <s xml:id="echoid-s3360" xml:space="preserve">L M, eſt parallelorum ma-
              <lb/>
            ximus. </s>
            <s xml:id="echoid-s3361" xml:space="preserve">Deſcripto autem per polos parallelorum,
              <lb/>
            & </s>
            <s xml:id="echoid-s3362" xml:space="preserve">per contactum B, circulo maximo G L H M,
              <lb/>
            tranſibit hic idem per polos quoque circuli B D,
              <lb/>
            per propoſ. </s>
            <s xml:id="echoid-s3363" xml:space="preserve">5. </s>
            <s xml:id="echoid-s3364" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3365" xml:space="preserve">2. </s>
            <s xml:id="echoid-s3366" xml:space="preserve">Theodoſii. </s>
            <s xml:id="echoid-s3367" xml:space="preserve">Quare ſecabit
              <lb/>
            ſegmenta B E F, L E F, per propoſ. </s>
            <s xml:id="echoid-s3368" xml:space="preserve">9. </s>
            <s xml:id="echoid-s3369" xml:space="preserve">eiuſdem,
              <lb/>
            bifariam. </s>
            <s xml:id="echoid-s3370" xml:space="preserve">Cum ergo L E F, ſemicirculus ſit, quòd
              <lb/>
              <note position="left" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">10</note>
            maximi circuli ſe bifariam ſecent, per propoſ. </s>
            <s xml:id="echoid-s3371" xml:space="preserve">11.
              <lb/>
            </s>
            <s xml:id="echoid-s3372" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3373" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3374" xml:space="preserve">Theodoſii, erunt arcus L E, L F, quadran-
              <lb/>
            res. </s>
            <s xml:id="echoid-s3375" xml:space="preserve">Quoniam verò circuli maximi G L H M,
              <lb/>
            G E H F, per G, H, polos parallelorum B C,
              <lb/>
            L M, deſcripti ſunt, erunt per propoſ. </s>
            <s xml:id="echoid-s3376" xml:space="preserve">10. </s>
            <s xml:id="echoid-s3377" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3378" xml:space="preserve">2. </s>
            <s xml:id="echoid-s3379" xml:space="preserve">
              <lb/>
            Theodoſii, arcus inter ipſos intercepti ſimiles: </s>
            <s xml:id="echoid-s3380" xml:space="preserve">ſunt
              <lb/>
            autem arcus B I, B k, paralleli B C, inter ipſos inter-
              <lb/>
            cepti, ex hypotheſi, quadrantes. </s>
            <s xml:id="echoid-s3381" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s3382" xml:space="preserve">arcus pa-
              <lb/>
            ralleli L M, intercepti inter eoſdem, quadrantes e-
              <lb/>
            runt: </s>
            <s xml:id="echoid-s3383" xml:space="preserve">ac proinde, cum L E, L F, oſtenſi ſint quadrantes, tranſibit circulus G H, per puncta E, F,
              <lb/>
              <note position="left" xlink:label="note-0071-02" xlink:href="note-0071-02a" xml:space="preserve">20</note>
            atque adeo vtrunque circulum A C, B D, per rectam E F, ſecabit. </s>
            <s xml:id="echoid-s3384" xml:space="preserve">Quare recta E F, communis
              <lb/>
            eſt ſectio trium circulorum A C, B D, G H. </s>
            <s xml:id="echoid-s3385" xml:space="preserve">Quod oſtendendum erat.</s>
            <s xml:id="echoid-s3386" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3387" xml:space="preserve">TANGANT deinde in eadem Sphæra A B C D, eundem parallelum B C, in punctis non
              <lb/>
            oppoſitis E, F, duo circuli maximi E G, F H, quorum communis ſectio ſit recta I K, quæ diameter
              <lb/>
            erit ipſorum, cum per propoſ. </s>
            <s xml:id="echoid-s3388" xml:space="preserve">11. </s>
            <s xml:id="echoid-s3389" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3390" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3391" xml:space="preserve">Theod, ſe
              <lb/>
              <figure xlink:label="fig-0071-02" xlink:href="fig-0071-02a" number="54">
                <image file="0071-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0071-02"/>
              </figure>
            mutuo bifariam ſecent: </s>
            <s xml:id="echoid-s3392" xml:space="preserve">Secet autem eundem pa-
              <lb/>
            rallelum B C, alius circulus maximus L M, per pa-
              <lb/>
            ralleli polos L, M, & </s>
            <s xml:id="echoid-s3393" xml:space="preserve">per axem L M, incedens, in
              <lb/>
            punctis O, P, æqualiter diſtantibus à punctis E, F,
              <lb/>
            ita vt arcus O E, O F, & </s>
            <s xml:id="echoid-s3394" xml:space="preserve">P B E, P C F, æquales ſint.
              <lb/>
            </s>
            <s xml:id="echoid-s3395" xml:space="preserve">
              <note position="left" xlink:label="note-0071-03" xlink:href="note-0071-03a" xml:space="preserve">30</note>
            Dico circulum L M, ſecare vtrumque maximum
              <lb/>
            E G, F H, per rectam I k, hoc eſt, tranſire per
              <lb/>
            puncta I, K, ita vt recta I K, ſit communis ſectio
              <lb/>
            trium circulorum maximorum E G, F H, L M.
              <lb/>
            </s>
            <s xml:id="echoid-s3396" xml:space="preserve">Cum enim circulus L M, ſecet parallelum B C,
              <lb/>
            per polos, ſecabit ipſum, per propoſ. </s>
            <s xml:id="echoid-s3397" xml:space="preserve">15. </s>
            <s xml:id="echoid-s3398" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3399" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3400" xml:space="preserve">
              <lb/>
            Theodoſii, bifariam. </s>
            <s xml:id="echoid-s3401" xml:space="preserve">Sectio igitur communis O P,
              <lb/>
            diameter erit paralleli B C, tranſiens per centrum
              <lb/>
            Q, in quod axis L M, cadit, per propoſ. </s>
            <s xml:id="echoid-s3402" xml:space="preserve">10. </s>
            <s xml:id="echoid-s3403" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3404" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3405" xml:space="preserve">
              <lb/>
            Theodoſii. </s>
            <s xml:id="echoid-s3406" xml:space="preserve">Sit quoque R, centrum ſphęrę, per quod
              <lb/>
              <note position="left" xlink:label="note-0071-04" xlink:href="note-0071-04a" xml:space="preserve">40</note>
            & </s>
            <s xml:id="echoid-s3407" xml:space="preserve">axis L M, & </s>
            <s xml:id="echoid-s3408" xml:space="preserve">I K, diameter circulorum maximorũ
              <lb/>
            tranſit. </s>
            <s xml:id="echoid-s3409" xml:space="preserve">Et quia circuli in Sphęra ſe mutuo tangere dicuntur, cum communis ſectio planorum, in
              <lb/>
            quibus circuli exiſtunt, vtrum que circulum tangit, ex defin. </s>
            <s xml:id="echoid-s3410" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3411" xml:space="preserve">2. </s>
            <s xml:id="echoid-s3412" xml:space="preserve">Theodofii: </s>
            <s xml:id="echoid-s3413" xml:space="preserve">Sint communes ſe-
              <lb/>
            ctiones circulorum E G, B C, & </s>
            <s xml:id="echoid-s3414" xml:space="preserve">F H, B C, rectę E S, F S, tangentes ipſos circulos. </s>
            <s xml:id="echoid-s3415" xml:space="preserve">Ductis ergo è
              <lb/>
            centro Q, ſemidiametris Q E, Q F, erunt anguli S E Q, S F Q, recti, & </s>
            <s xml:id="echoid-s3416" xml:space="preserve">idcirco, ducta recta E F,
              <lb/>
              <note position="right" xlink:label="note-0071-05" xlink:href="note-0071-05a" xml:space="preserve">18. tertij.</note>
            anguli S E F, S F E, rectis minores. </s>
            <s xml:id="echoid-s3417" xml:space="preserve">Quare rectæ E S, F S, in eodem plano paralleli B C, quem tan
              <lb/>
            gunt, exiſtentes conuenient in aliquo puncto, vtpote in S, per 11. </s>
            <s xml:id="echoid-s3418" xml:space="preserve">pronunciatum lib. </s>
            <s xml:id="echoid-s3419" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3420" xml:space="preserve">Euclidis.
              <lb/>
            </s>
            <s xml:id="echoid-s3421" xml:space="preserve">Et quoniam K I, communis ſectio circulorum E G, F H, conuenit quoque cum vtraque E S, F S,
              <lb/>
            vt mox oſtendemus lemmate ſequenti; </s>
            <s xml:id="echoid-s3422" xml:space="preserve">fit vt KI, producta vtrique occurrat in S. </s>
            <s xml:id="echoid-s3423" xml:space="preserve">Nam ſi alteram ip
              <lb/>
            ſarũ ſecaret infra, aut ſupra S, non coiret cũ reliqua, vt patet. </s>
            <s xml:id="echoid-s3424" xml:space="preserve">Si enim K I, occurrat, verbi gratia, re-
              <lb/>
              <note position="left" xlink:label="note-0071-06" xlink:href="note-0071-06a" xml:space="preserve">50</note>
            ctæ E S, alibi, quàm in puncto S, ſecabit ea producta ſtatim planũ circuli B C, in eo puncto, in quo
              <lb/>
            rectam E S, ſecat, ac proinde nullo modo ſecabit rectam F S, in plano eodem circuli BC, exiſtentẽ.
              <lb/>
            </s>
            <s xml:id="echoid-s3425" xml:space="preserve">Et ſi K I, occurrat rectæ F S, alibi quàm in puncto S, oſtendemus eodẽ modo, ipſam ſecare nõ poſ-
              <lb/>
            ſe rectam E S. </s>
            <s xml:id="echoid-s3426" xml:space="preserve">Quamobrem recta K I, niſi per punctum S, tranſeat, non ſecabit vtramque E S, F S,
              <lb/>
            Quod eſt abſurdum. </s>
            <s xml:id="echoid-s3427" xml:space="preserve">Vtramque enim ſecat, vt in lemmate ſequenti oſtendemus. </s>
            <s xml:id="echoid-s3428" xml:space="preserve">Ducantur iam
              <lb/>
            rectæ E O, F O, O S, in plano circuli B C. </s>
            <s xml:id="echoid-s3429" xml:space="preserve">Quia igitur arcus E O, F O, ponuntur æquales, æqua-
              <lb/>
            les erunt & </s>
            <s xml:id="echoid-s3430" xml:space="preserve">rectæ E O, F O: </s>
            <s xml:id="echoid-s3431" xml:space="preserve">Sunt autem & </s>
            <s xml:id="echoid-s3432" xml:space="preserve">tangentes S E, S F, per 2. </s>
            <s xml:id="echoid-s3433" xml:space="preserve">coroll. </s>
            <s xml:id="echoid-s3434" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s3435" xml:space="preserve">36. </s>
            <s xml:id="echoid-s3436" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3437" xml:space="preserve">3. </s>
            <s xml:id="echoid-s3438" xml:space="preserve">Eu-
              <lb/>
              <note position="right" xlink:label="note-0071-07" xlink:href="note-0071-07a" xml:space="preserve">28. tertij.</note>
            clidis, æquales. </s>
            <s xml:id="echoid-s3439" xml:space="preserve">Igitur erunt duo latera E O, O S, trianguli E O S, duobus lateribus F O, O S, trian
              <lb/>
            guli F O S, æqualia, & </s>
            <s xml:id="echoid-s3440" xml:space="preserve">baſis E S, baſi F S; </s>
            <s xml:id="echoid-s3441" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s3442" xml:space="preserve">anguli E O S, F O S, æquales erunt. </s>
            <s xml:id="echoid-s3443" xml:space="preserve">Non
              <lb/>
              <note position="right" xlink:label="note-0071-08" xlink:href="note-0071-08a" xml:space="preserve">8. primi.</note>
            aliter oſtendemus angulos E O Q, F O Q, æquales eſſe; </s>
            <s xml:id="echoid-s3444" xml:space="preserve">propterea quòd latera E O, O Q, trian-
              <lb/>
            guli E O Q, lateribus F O, O Q, trianguli F O Q, æqualia ſunt, & </s>
            <s xml:id="echoid-s3445" xml:space="preserve">baſis E Q, baſi F Q. </s>
            <s xml:id="echoid-s3446" xml:space="preserve"/>
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