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 style="it">
            <s xml:id="echoid-s3879" xml:space="preserve">
              <pb o="60" file="0080" n="80" rhead="GNOMONICES"/>
            eodem puncto F. </s>
            <s xml:id="echoid-s3880" xml:space="preserve">Si enim puncta cont actuum C, D, per diametrum ſunt oppoſita, ita vt arcus C G, D G,
              <lb/>
            ſint quadrantes, perſpicuum eſt ex ijs, quæ proximè demonſtrata ſunt, tres hos circulos ſe mutuo interſe-
              <lb/>
            care in Aequatore in vno eodemq́, puncto. </s>
            <s xml:id="echoid-s3881" xml:space="preserve">Si verò puncta contactuum C, D, non ſunt oppoſita, deſcri-
              <lb/>
            bantur per polum E, & </s>
            <s xml:id="echoid-s3882" xml:space="preserve">per contactus C, D, circuli maximi E C, E D. </s>
            <s xml:id="echoid-s3883" xml:space="preserve">Item per puncta C, G, arcus
              <lb/>
              <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a" number="61">
                <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0080-01"/>
              </figure>
            circuli maximi C H G, & </s>
            <s xml:id="echoid-s3884" xml:space="preserve">per puncta G, D, arcus
              <lb/>
            maximi circuli G I D, deſcribatur, ducantur{q́ue} chor
              <lb/>
            dę C G, G D. </s>
            <s xml:id="echoid-s3885" xml:space="preserve">Quoniam igitur per defin. </s>
            <s xml:id="echoid-s3886" xml:space="preserve">poli à Theo-
              <lb/>
            doſio traditam, rectæ ex polo E, ad puncta C, D,
              <lb/>
            cadentes æquales ſunt, erũt & </s>
            <s xml:id="echoid-s3887" xml:space="preserve">arcus E C, E D, æqua-
              <lb/>
              <note position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve">28. tertij.</note>
            les. </s>
            <s xml:id="echoid-s3888" xml:space="preserve">Rurſus, quia arcus C G, G D, paralleli C D,
              <lb/>
              <note position="left" xlink:label="note-0080-02" xlink:href="note-0080-02a" xml:space="preserve">10</note>
            æquales ſunt, erunt & </s>
            <s xml:id="echoid-s3889" xml:space="preserve">rectę C G, G D, æquales. </s>
            <s xml:id="echoid-s3890" xml:space="preserve">Igi-
              <lb/>
              <note position="left" xlink:label="note-0080-03" xlink:href="note-0080-03a" xml:space="preserve">29. tertij.</note>
            tur & </s>
            <s xml:id="echoid-s3891" xml:space="preserve">arcus maximorum circulorum C H G, G I D,
              <lb/>
              <note position="left" xlink:label="note-0080-04" xlink:href="note-0080-04a" xml:space="preserve">28. tertij.</note>
              <note position="left" xlink:label="note-0080-05" xlink:href="note-0080-05a" xml:space="preserve">Duo circuli ho
                <lb/>
              rarum ab or.
                <lb/>
              vel ò
                <unsure/>
              cc. tangen-
                <lb/>
              tes maximum
                <lb/>
              parallelorũ ſem
                <lb/>
              per apparentiũ
                <lb/>
              in duob
                <unsure/>
              us pun-
                <lb/>
              ctis quibuſcun-
                <lb/>
              que, & circulus
                <lb/>
              horæ à mer. uel
                <lb/>
              med. noc. ſecans
                <lb/>
              eundem paral-
                <lb/>
              lela
                <unsure/>
              m in pun-
                <lb/>
              cto æqualiter à
                <lb/>
              punctis conta-
                <lb/>
              ctuum diſtante,
                <lb/>
              ſe mutuo ſecãt
                <lb/>
              in vno eodéq́ue
                <lb/>
              puncto.</note>
            ęquales erunt. </s>
            <s xml:id="echoid-s3892" xml:space="preserve">Quare duo latera E C, E G, triangu-
              <lb/>
            li ſphærici C E G, duobus lateribus E D, E G, trian-
              <lb/>
            guli ſphęrici D E G, ęqualia erunt: </s>
            <s xml:id="echoid-s3893" xml:space="preserve">Sunt autem & </s>
            <s xml:id="echoid-s3894" xml:space="preserve">
              <lb/>
            baſes C H G, D I G, æquales. </s>
            <s xml:id="echoid-s3895" xml:space="preserve">Igitur per propoſ. </s>
            <s xml:id="echoid-s3896" xml:space="preserve">35.
              <lb/>
            </s>
            <s xml:id="echoid-s3897" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3898" xml:space="preserve">3. </s>
            <s xml:id="echoid-s3899" xml:space="preserve">Ioan. </s>
            <s xml:id="echoid-s3900" xml:space="preserve">Regiom. </s>
            <s xml:id="echoid-s3901" xml:space="preserve">de triangulis, vel per propoſ. </s>
            <s xml:id="echoid-s3902" xml:space="preserve">
              <lb/>
            18. </s>
            <s xml:id="echoid-s3903" xml:space="preserve">nostrorum triangulorum ſphęricorum, anguli
              <lb/>
            C E G, D E G, ęquales erunt, ac proinde arcus E G, angulum C E D, diuidet bifariam. </s>
            <s xml:id="echoid-s3904" xml:space="preserve">Quoniam verò
              <lb/>
            circulus maximus per polum E, & </s>
            <s xml:id="echoid-s3905" xml:space="preserve">punctum F, deſcriptus diuidit eundem angulum CED, bifariam, vt
              <lb/>
              <note position="left" xlink:label="note-0080-06" xlink:href="note-0080-06a" xml:space="preserve">20</note>
            mox demonſtrabimus, perſpicuum eſt, circulum maximum E G, productum per punctum F, tranſire:
              <lb/>
            </s>
            <s xml:id="echoid-s3906" xml:space="preserve">alias duo maximi circuli, nempe E G, & </s>
            <s xml:id="echoid-s3907" xml:space="preserve">ille qui ex E, per F, ducitur, diuiderent eundem angulum
              <lb/>
            C E D, bifariam, quod est
              <unsure/>
            abſurdum. </s>
            <s xml:id="echoid-s3908" xml:space="preserve">Quòd autem circulus maximus per E, & </s>
            <s xml:id="echoid-s3909" xml:space="preserve">F, deſcriptus di-
              <lb/>
            uidat bifariam angulum C E D, ita demonſtrabimus. </s>
            <s xml:id="echoid-s3910" xml:space="preserve">Intelligatur per polum E, & </s>
            <s xml:id="echoid-s3911" xml:space="preserve">per F, deſcri-
              <lb/>
            ptus circulus maximus E F H. </s>
            <s xml:id="echoid-s3912" xml:space="preserve">Dico angulum C E F, æqualem eſſe angulo D E F. </s>
            <s xml:id="echoid-s3913" xml:space="preserve">Quoniam enim cir-
              <lb/>
            culi maximi E C, E D, tranſeunt, per propoſ. </s>
            <s xml:id="echoid-s3914" xml:space="preserve">5. </s>
            <s xml:id="echoid-s3915" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3916" xml:space="preserve">2. </s>
            <s xml:id="echoid-s3917" xml:space="preserve">Theodoſii, per polos circulorum C F, D F, quòd per
              <lb/>
            contactus C, D, & </s>
            <s xml:id="echoid-s3918" xml:space="preserve">per polum E, circuli C D, ducti ſint; </s>
            <s xml:id="echoid-s3919" xml:space="preserve">Tranſeunt autem & </s>
            <s xml:id="echoid-s3920" xml:space="preserve">per polum Aequatoris
              <lb/>
            A B; </s>
            <s xml:id="echoid-s3921" xml:space="preserve">fit vt per propoſ. </s>
            <s xml:id="echoid-s3922" xml:space="preserve">9. </s>
            <s xml:id="echoid-s3923" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3924" xml:space="preserve">2. </s>
            <s xml:id="echoid-s3925" xml:space="preserve">Theodoſii, circulus E C, ſecet ſegmenta circulorum C B, A B, ſe ſe in
              <lb/>
            B, ſe
              <unsure/>
            cantium, quæ quidem, per propoſ. </s>
            <s xml:id="echoid-s3926" xml:space="preserve">11. </s>
            <s xml:id="echoid-s3927" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3928" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3929" xml:space="preserve">Theodoſii, ſemicirculi ſunt, bifariam: </s>
            <s xml:id="echoid-s3930" xml:space="preserve">Ac propterea ar-
              <lb/>
            cus C B, ſit quadrans. </s>
            <s xml:id="echoid-s3931" xml:space="preserve">Eadem{q́ue} ratione quadrans erit arcus D A. </s>
            <s xml:id="echoid-s3932" xml:space="preserve">Quia vero per Theorema 1. </s>
            <s xml:id="echoid-s3933" xml:space="preserve">ſcholij
              <lb/>
              <note position="left" xlink:label="note-0080-07" xlink:href="note-0080-07a" xml:space="preserve">30</note>
            propoſ. </s>
            <s xml:id="echoid-s3934" xml:space="preserve">21. </s>
            <s xml:id="echoid-s3935" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3936" xml:space="preserve">2. </s>
            <s xml:id="echoid-s3937" xml:space="preserve">Theodoſii, circuli maximi C B, D N, eundem parallelum C D, tangentes, æqualiter in-
              <lb/>
            clinantur ad A B, maximum parallelorum, æquales erunt anguli ſphærici C B A, D N K. </s>
            <s xml:id="echoid-s3938" xml:space="preserve">Cum ergo an-
              <lb/>
            gulo D N K, æqualis quoque ſit angulus D A K, per propoſ. </s>
            <s xml:id="echoid-s3939" xml:space="preserve">13. </s>
            <s xml:id="echoid-s3940" xml:space="preserve">noſtrorum triangulorum ſphæricorum,
              <lb/>
            quòd A D N, ABN, per propoſ. </s>
            <s xml:id="echoid-s3941" xml:space="preserve">11. </s>
            <s xml:id="echoid-s3942" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3943" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3944" xml:space="preserve">Theod. </s>
            <s xml:id="echoid-s3945" xml:space="preserve">ſemicirculi ſint; </s>
            <s xml:id="echoid-s3946" xml:space="preserve">æquales erunt anguli ſphærici F B A,
              <lb/>
            F A B; </s>
            <s xml:id="echoid-s3947" xml:space="preserve">ac proinde & </s>
            <s xml:id="echoid-s3948" xml:space="preserve">arcus B F, A F, æquales erunt, per propoſ. </s>
            <s xml:id="echoid-s3949" xml:space="preserve">4. </s>
            <s xml:id="echoid-s3950" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3951" xml:space="preserve">1. </s>
            <s xml:id="echoid-s3952" xml:space="preserve">Menelai, vel per propoſ. </s>
            <s xml:id="echoid-s3953" xml:space="preserve">40.
              <lb/>
            </s>
            <s xml:id="echoid-s3954" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3955" xml:space="preserve">3. </s>
            <s xml:id="echoid-s3956" xml:space="preserve">Ioan. </s>
            <s xml:id="echoid-s3957" xml:space="preserve">Regiom. </s>
            <s xml:id="echoid-s3958" xml:space="preserve">de triangulis, vel certè per propoſ. </s>
            <s xml:id="echoid-s3959" xml:space="preserve">9. </s>
            <s xml:id="echoid-s3960" xml:space="preserve">noſtrorum triangulorũ ſphæricorum. </s>
            <s xml:id="echoid-s3961" xml:space="preserve">Cum ergo
              <lb/>
            & </s>
            <s xml:id="echoid-s3962" xml:space="preserve">toti arcus B C, A D, æquales ſint, nempe quadrantes, vt oſtendimus; </s>
            <s xml:id="echoid-s3963" xml:space="preserve">erunt & </s>
            <s xml:id="echoid-s3964" xml:space="preserve">reliqui arcus F C, F D,
              <lb/>
            æquales. </s>
            <s xml:id="echoid-s3965" xml:space="preserve">Et quoniam ostenſi ſunt arcus E C, E D, æquales, erunt duo latera E C, E F, trianguli ſphærici
              <lb/>
            C E F, æqualia duobus lateribus E D, E F, trianguli ſphærici D E F. </s>
            <s xml:id="echoid-s3966" xml:space="preserve">Cum ergo habeant & </s>
            <s xml:id="echoid-s3967" xml:space="preserve">baſes C F,
              <lb/>
            D F, æquales, vt oſtenſum eſt, erunt per propoſ. </s>
            <s xml:id="echoid-s3968" xml:space="preserve">35. </s>
            <s xml:id="echoid-s3969" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s3970" xml:space="preserve">3. </s>
            <s xml:id="echoid-s3971" xml:space="preserve">Ioan. </s>
            <s xml:id="echoid-s3972" xml:space="preserve">Regiom. </s>
            <s xml:id="echoid-s3973" xml:space="preserve">de triangulis, vel per propoſ. </s>
            <s xml:id="echoid-s3974" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-0080-08" xlink:href="note-0080-08a" xml:space="preserve">40</note>
            18. </s>
            <s xml:id="echoid-s3975" xml:space="preserve">noſtrorum triangulorum ſphæricorum, anguli C E F, D E F, æquales. </s>
            <s xml:id="echoid-s3976" xml:space="preserve">Diuidit ergo arcus E F, cir-
              <lb/>
            culi maximi angulum C E D, bifariam; </s>
            <s xml:id="echoid-s3977" xml:space="preserve">Ac propterea cum eundem bifariam ſecet arcus E G, vt de-
              <lb/>
            monſtratum eſt, tranſibit omnino arcus E G, productus per F, adeo vt ab arcu E F, non differat, ne
              <lb/>
            duo arcus concedantur eundem angulum C E D, bifariam ſecare. </s>
            <s xml:id="echoid-s3978" xml:space="preserve">Quapropter tres circuli horarij C F,
              <lb/>
            D F, E F, vnam eandem{q́ue} ſectionem habent communem. </s>
            <s xml:id="echoid-s3979" xml:space="preserve">Quod eſt propoſitum: </s>
            <s xml:id="echoid-s3980" xml:space="preserve">Acproinde in quo pun-
              <lb/>
              <note position="left" xlink:label="note-0080-09" xlink:href="note-0080-09a" xml:space="preserve">Quænam horæ
                <lb/>
              ab or. vel occ.
                <lb/>
              & à mer. vel
                <lb/>
              med. noc. ſe mu
                <lb/>
              tuo ſecẽt in eo-
                <lb/>
              dem puncto.</note>
            cto planum horologij communem hanc ſectionem ſecat, per idem communes ſectiones eorundem circulo-
              <lb/>
            rum, & </s>
            <s xml:id="echoid-s3981" xml:space="preserve">plani horologij tranſibunt, per propoſ. </s>
            <s xml:id="echoid-s3982" xml:space="preserve">18. </s>
            <s xml:id="echoid-s3983" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s3984" xml:space="preserve">adeo vt in eodem puncto horologij ſe inter-
              <lb/>
            ſecent lineæ horariæ illorum circulorum. </s>
            <s xml:id="echoid-s3985" xml:space="preserve">Quocirca in quo puncto horaria linea circuli C F, horariam li-
              <lb/>
            neam circuli E F, ſecat, per idem ducenda erit linea horaria circuli D F, & </s>
            <s xml:id="echoid-s3986" xml:space="preserve">e contrario. </s>
            <s xml:id="echoid-s3987" xml:space="preserve">Quibus autem
              <lb/>
            horis deputentur circuli C F, D F, E F, docebit nos figura, quam in propoſ. </s>
            <s xml:id="echoid-s3988" xml:space="preserve">9. </s>
            <s xml:id="echoid-s3989" xml:space="preserve">huius libri poſuimus. </s>
            <s xml:id="echoid-s3990" xml:space="preserve">Si
              <lb/>
              <note position="left" xlink:label="note-0080-10" xlink:href="note-0080-10a" xml:space="preserve">50</note>
            enim alter circulorum C F, D F, nempe C F, tribuatur, verbi gratia, horæ 12. </s>
            <s xml:id="echoid-s3991" xml:space="preserve">ab ortu, vel occaſu; </s>
            <s xml:id="echoid-s3992" xml:space="preserve">& </s>
            <s xml:id="echoid-s3993" xml:space="preserve">
              <lb/>
            alter D F, exempli cauſa, horæ 20. </s>
            <s xml:id="echoid-s3994" xml:space="preserve">ab ortu, vel occaſu, erit E F, circulus horæ quartæ à meridie, vel me-
              <lb/>
            dia nocte, cum hæc hora in medio illarum ſit poſita in figura dicta propoſitionis 9. </s>
            <s xml:id="echoid-s3995" xml:space="preserve">huius libri, quemad-
              <lb/>
            modum & </s>
            <s xml:id="echoid-s3996" xml:space="preserve">punctum G, in medio punctorum C, & </s>
            <s xml:id="echoid-s3997" xml:space="preserve">D, poſitum eſt. </s>
            <s xml:id="echoid-s3998" xml:space="preserve">Sic etiam, ſi E F, ponatur eſſe
              <lb/>
            circulus horæ 1 {1/2}. </s>
            <s xml:id="echoid-s3999" xml:space="preserve">à meridie, vel media nocte, & </s>
            <s xml:id="echoid-s4000" xml:space="preserve">E C, horæ tertiæ ab ortu, vel occa-
              <lb/>
            ſu; </s>
            <s xml:id="echoid-s4001" xml:space="preserve">erit E D, circulus horæ 24. </s>
            <s xml:id="echoid-s4002" xml:space="preserve">ab ortu, vel occaſu, quòd hæ duę horę ab
              <lb/>
            illa hinc inde ęqualiter abſint, ſicuti & </s>
            <s xml:id="echoid-s4003" xml:space="preserve">puncta C, D, à puncto
              <lb/>
            G, ęqualibus interuallis diſiunguntur, &</s>
            <s xml:id="echoid-s4004" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4005" xml:space="preserve">Ex his
              <lb/>
              <note position="left" xlink:label="note-0080-11" xlink:href="note-0080-11a" xml:space="preserve">Compoſitio ſu-
                <lb/>
              periorum qua-
                <lb/>
              tuot, & ſequen-
                <lb/>
              tium duarum
                <lb/>
              & triginta ta-
                <lb/>
              bularum.</note>
            autem nullo negotio conficiemus trigin-
              <lb/>
            ta ſex illas tabulas, quas in hoc
              <lb/>
            ſcholio conſcripſimus.</s>
            <s xml:id="echoid-s4006" xml:space="preserve"/>
          </p>
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