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

Table of contents

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[141.] GNOMONICES LIBER SECVNDVS.
Page: 161 (141)
[142.] AVCTORE CHRISTOPHORO CLAVIO BAMBER GENSI SOCIETATIS IESV.
Page: 161 (141)
[143.] DE HOROLOGIIS HORIZONTALIBVS. PROBLEMA 1. PROPOSITIO 1.
Page: 162 (142)
[144.] SCHOLIVM.
Page: 165 (145)
[145.] PROBLEMA 2. PROPOSITIO 2.
Page: 177 (157)
[146.] SCHOLIVM.
Page: 186 (166)
[147.] PROBLEMA 3. PROPOSITIO 3.
Page: 189 (169)
[148.] SCHOLIVM.
Page: 190 (170)
[149.] PROBLEMA. 4. PROPOSITIO 4.
Page: 190 (170)
[150.] SCHOLIVM.
Page: 191 (171)
[151.] PROBLEMA 5. PROPOSITIO 5.
Page: 191 (171)
[152.] SCHOLIVM.
Page: 192 (172)
[153.] PROBLEMA 6. PROPOSITIO 6.
Page: 192 (172)
[154.] SCHOLIVM.
Page: 193 (173)
[155.] PROBLEMA 7. PROPOSITIO 7.
Page: 194 (174)
[156.] SCHOLIVM.
Page: 194 (174)
[157.] PROBLEMA 8. PROPOSITIO 8.
Page: 195 (175)
[158.] COROLLARIVM.
Page: 197 (177)
[159.] SCHOLIVM.
Page: 197 (177)
[160.] PROBLEMA. 9. PROPOSITIO 9.
Page: 197 (177)
[161.] I. Sole exiſtente in principio ♈.
Page: 198 (178)
[162.] II. Sole exiſtente in principio ♎.
Page: 198 (178)
[163.] III. Sole exiſtente in principio ♋.
Page: 198 (178)
[164.] IIII. Sole exiſtente in principio ♑.
Page: 198 (178)
[165.] Arcus ſemidiurni in initijs ſignorum, ad latitudinem grad. 42.
Page: 200 (180)
[166.] VI. Mediationes cœli, & anguli terræ, eorumq́; declinationes, orientibus 12. ſignorum Zodiaci initiis, ad latitudinem grad. 42.
Page: 201 (181)
[167.] VII. Puncta Eclipticæ in circulo horę 6. conſtituta, eorumq́ue declinationes, orientibus 12. ſignorum Zodiaci principijs, ad latitudinem grad. 42.
Page: 202 (182)
[168.] VIII. Puncta Eclipticæ in circulo horę 11. exiſtentia, eorumq́; declinationes, cum principia 12. ſignorum Zodiaci oriuntur, ad latitudinem grad. 42.
Page: 204 (184)
[169.] SCHOLIVM.
Page: 206 (186)
[170.] SEQVVNTVR TABELLÆ.
Page: 207 (187)
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          <p style="it">
            <s xml:id="echoid-s9847" xml:space="preserve">
              <pb o="153" file="0173" n="173" rhead="LIBER SECVNDVS."/>
            P, demiſſum per punctum L. </s>
            <s xml:id="echoid-s9848" xml:space="preserve">Quamobrem, cum per hoc latus cylindri ducatur circulus horę 4. </s>
            <s xml:id="echoid-s9849" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9850" xml:space="preserve">vel
              <lb/>
            med. </s>
            <s xml:id="echoid-s9851" xml:space="preserve">noc. </s>
            <s xml:id="echoid-s9852" xml:space="preserve">vt ſupra diximus, occurret latus parallelogrammi ſacti à circulo horę 4. </s>
            <s xml:id="echoid-s9853" xml:space="preserve">ex P, demiſſurn pla-
              <lb/>
            no horologij in puncto L. </s>
            <s xml:id="echoid-s9854" xml:space="preserve">Eademq́, ratione alterum latus eiuſdem parallelogrammi ex ε, demiſſum plano
              <lb/>
            horologij occurret in puncto α Ac propterea circulus ipſe horæ 4. </s>
            <s xml:id="echoid-s9855" xml:space="preserve">planum horologij ſecabit in punctis L,
              <lb/>
            a. </s>
            <s xml:id="echoid-s9856" xml:space="preserve">Cum ergo tranſeat quoque per centrum H, dabit recta α H L, horam 4. </s>
            <s xml:id="echoid-s9857" xml:space="preserve">à mer. </s>
            <s xml:id="echoid-s9858" xml:space="preserve">vel med. </s>
            <s xml:id="echoid-s9859" xml:space="preserve">noc. </s>
            <s xml:id="echoid-s9860" xml:space="preserve">Eadem
              <lb/>
            ratione demonſtrabimus reliqua latera cylindri à circulis horarijs facta cadere in reliqua puncta Ellipſis
              <lb/>
            deſcriptæ in plano horologij, & </s>
            <s xml:id="echoid-s9861" xml:space="preserve">c.</s>
            <s xml:id="echoid-s9862" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9863" xml:space="preserve">QVAMVIS autem ſatis ſit ad deſcriptionem horologij, ſi dimidiata duntaxat Ellipſis R L B X,
              <lb/>
            deſcribatur, accur atius tamen horologium delineabitur, ſi tota Ellipſis deſcribatur; </s>
            <s xml:id="echoid-s9864" xml:space="preserve">vt ſingulę horæ ha-
              <lb/>
            beant terna puncta, per quæ ducantur.</s>
            <s xml:id="echoid-s9865" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">10</note>
          <p style="it">
            <s xml:id="echoid-s9866" xml:space="preserve">EX demonſtratis colligitur, ſi cylindrus rectus ſecetur plano, quod neque per axem ducatur, neque
              <lb/>
              <note position="right" xlink:label="note-0173-02" xlink:href="note-0173-02a" xml:space="preserve">Planum ſecans
                <lb/>
              conum rectum,
                <lb/>
              per cuius axem
                <lb/>
              non ducitur, ne
                <lb/>
              que axi æquidi
                <lb/>
              ſtat, facit Elli-
                <lb/>
              pſim.</note>
            axi æquidiſtet, ſectionẽ factã eſſe Ellipſim. </s>
            <s xml:id="echoid-s9867" xml:space="preserve">Quẽadmodum enim oſtendimus, planũ horologij noſtri horizon
              <lb/>
            talis ad latitudinem grad. </s>
            <s xml:id="echoid-s9868" xml:space="preserve">42. </s>
            <s xml:id="echoid-s9869" xml:space="preserve">fabricati ſecans cylindrum rectum, cuius axis eſt axis mundi, facere El-
              <lb/>
            lipſim, propterea quòd omnia later a cylindri cadant in puncta Ellipſis, ita eodem modo demonſtrabimus
              <lb/>
            idem contingere, ſi maior fuerit, aut minor altitudo poli, quàm grad. </s>
            <s xml:id="echoid-s9870" xml:space="preserve">42. </s>
            <s xml:id="echoid-s9871" xml:space="preserve">ita vt planum horologij horizon
              <lb/>
            talis quomodocunque inclinatum ad axem, ſeu dictum cylindrum rectum, ſemper faciat Ellipſim. </s>
            <s xml:id="echoid-s9872" xml:space="preserve">Id quod
              <lb/>
            Serenus lib. </s>
            <s xml:id="echoid-s9873" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9874" xml:space="preserve">de ſectione cylindri in omni cylindro demonſtrat, quando planum ſecans neque ęquidiſtat
              <lb/>
            baſibus cylindri, aut axi, neque per axem tranſit, aut ſubcontrarie ponitur.</s>
            <s xml:id="echoid-s9875" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9876" xml:space="preserve">IAM verò, quando altitudo poli ſupra Horizontem perexigua eſt, puta grad. </s>
            <s xml:id="echoid-s9877" xml:space="preserve">1. </s>
            <s xml:id="echoid-s9878" xml:space="preserve">2. </s>
            <s xml:id="echoid-s9879" xml:space="preserve">3. </s>
            <s xml:id="echoid-s9880" xml:space="preserve">vel 4. </s>
            <s xml:id="echoid-s9881" xml:space="preserve">& </s>
            <s xml:id="echoid-s9882" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s9883" xml:space="preserve">
              <note position="right" xlink:label="note-0173-03" xlink:href="note-0173-03a" xml:space="preserve">Quando altitue
                <lb/>
              do poli ſupra
                <lb/>
              Horizonté val-
                <lb/>
              de exigua eſt,
                <lb/>
              difficilior red-
                <lb/>
              ditur deſcripero
                <lb/>
              horologii hori-
                <lb/>
              aontalis.</note>
            difficilis aliquantulum, & </s>
            <s xml:id="echoid-s9884" xml:space="preserve">laborioſa efficitur deſcriptio horologij horizontalis, propterea quòd tunc axis
              <lb/>
              <note position="left" xlink:label="note-0173-04" xlink:href="note-0173-04a" xml:space="preserve">20</note>
            mundi E D, in portione Analemmatis huius propoſ. </s>
            <s xml:id="echoid-s9885" xml:space="preserve">nimis prope ad diamctrum Horizontis B C, accedit.
              <lb/>
            </s>
            <s xml:id="echoid-s9886" xml:space="preserve">Ex quo fit, ſtylum D G, admodum breuem fore, niſi velimus punctum H, à puncto G, atque adeo à pun-
              <lb/>
            cto I, plus æquo recedere, quod incommodum ſanè eſt, tum quia nimis amplum planum ad deſcriptionem
              <lb/>
            horologij requireretur, propterea quòd centrum horologij H, vltra quàm ſatis est, ab ęquinoctiali linea
              <lb/>
            remoueretur, (Nam lineę horarię in remotiſſimo puncto conueniant, neceſſe eſt, cum in eo plano, quod
              <lb/>
            axi mundi æquidistat, à quo planum horologij parum abeſſe ponitur ſint parallelę, vt in coroll. </s>
            <s xml:id="echoid-s9887" xml:space="preserve">pro-
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s9888" xml:space="preserve">22. </s>
            <s xml:id="echoid-s9889" xml:space="preserve">ſuperioris lib. </s>
            <s xml:id="echoid-s9890" xml:space="preserve">demonſtrauimus, ac proinde in propoſito horologio horizontali ferè etiam paralle-
              <lb/>
            læ videantur) tum etiam, quia difficile admodum eſt, in tam remoto ſpatio diſcernere, atque diſtinguere
              <lb/>
            punctum H, ſine aliqua erroris ſuſpitione, eo qòd angulus D H I, acutiſſimus tunc efficitur.</s>
            <s xml:id="echoid-s9891" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9892" xml:space="preserve">IDEM quodammodo contingit, quando maxima eſt altitudo poli ſupra Horizontem, vt grad. </s>
            <s xml:id="echoid-s9893" xml:space="preserve">89.
              <lb/>
            </s>
            <s xml:id="echoid-s9894" xml:space="preserve">
              <note position="left" xlink:label="note-0173-05" xlink:href="note-0173-05a" xml:space="preserve">30</note>
              <note position="right" xlink:label="note-0173-06" xlink:href="note-0173-06a" xml:space="preserve">Quando altitu-
                <lb/>
              do poli ſupra
                <lb/>
              Horizontem n@
                <lb/>
              mis magna eſt,
                <lb/>
              difficilior etiã
                <lb/>
              efficitur deli
                <lb/>
              neatio horolo-
                <lb/>
              gii horizonia-
                <lb/>
              lis.</note>
            88. </s>
            <s xml:id="echoid-s9895" xml:space="preserve">87. </s>
            <s xml:id="echoid-s9896" xml:space="preserve">86. </s>
            <s xml:id="echoid-s9897" xml:space="preserve">&</s>
            <s xml:id="echoid-s9898" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9899" xml:space="preserve">ita vt polus parum à vertice abſit: </s>
            <s xml:id="echoid-s9900" xml:space="preserve">quoniam tunc diameter Aequatoris F D, acutiſ-
              <lb/>
            ſimum quoque angulum conſtituit in I, cum recta H I, & </s>
            <s xml:id="echoid-s9901" xml:space="preserve">cum eadem in puneto remotiſſimo conuenit, vt
              <lb/>
            non facile ſit diiudicare, vbi rectę F D, H I, ſe mutuo interſecent, propter anguſtiam anguli acuti HID.
              <lb/>
            </s>
            <s xml:id="echoid-s9902" xml:space="preserve">Quamobrem ad duplex hoc incommodum vitandum, duplex etiam remcdium excogitauimus. </s>
            <s xml:id="echoid-s9903" xml:space="preserve">Priore
              <lb/>
            deſcribemus horarias lineas, etiamſi centrum, vbi omnes coeunt, non habeamus: </s>
            <s xml:id="echoid-s9904" xml:space="preserve">Poſteriore reperiemus
              <lb/>
            punctum in meridiana linea, per quod ęquinoctialis linea ducenda eſt, licet rectam D I, in portione Ana
              <lb/>
            lemmatis huius propoſ. </s>
            <s xml:id="echoid-s9905" xml:space="preserve">quę in illud punctum cadere debet, non ducamus.</s>
            <s xml:id="echoid-s9906" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9907" xml:space="preserve">SIT ergo deſcribendum horologium horizontale ad latitudinem grad. </s>
            <s xml:id="echoid-s9908" xml:space="preserve">20. </s>
            <s xml:id="echoid-s9909" xml:space="preserve">(tantam autem latitudi-
              <lb/>
              <note position="right" xlink:label="note-0173-07" xlink:href="note-0173-07a" xml:space="preserve">Deſcriptio ho-
                <lb/>
              rologii horizon
                <lb/>
              talis, cum poli
                <lb/>
              altitudo perexi
                <lb/>
              gua eſt.</note>
            nem eligimus, vt ratio deſcriptionis fiat planior, quætamen in alias latitudines quamuis minimas qua-
              <lb/>
            drat) boc eſt, in ea regione, ſupra cuius Horizontem polus attollitur grad. </s>
            <s xml:id="echoid-s9910" xml:space="preserve">20. </s>
            <s xml:id="echoid-s9911" xml:space="preserve">In plano aliquo ducatur
              <lb/>
              <note position="left" xlink:label="note-0173-08" xlink:href="note-0173-08a" xml:space="preserve">40</note>
            pro linea meridiana, recta vtcunque A B; </s>
            <s xml:id="echoid-s9912" xml:space="preserve">& </s>
            <s xml:id="echoid-s9913" xml:space="preserve">C D, eam ad angulos rectos ſecans in E, referat æquino ctia-
              <lb/>
            lem lineam. </s>
            <s xml:id="echoid-s9914" xml:space="preserve">Deinceps in E, puncto, ad rectam E D, vel E C, conſtituatur angulus altitudinis poli D E F,
              <lb/>
            atque in recta E F, aſſumpto quoquis puncto F, (quod quo remotius fuerit ab E, eo maius delineabitur ho
              <lb/>
            rologium) excitetur in F, ad rectam E F, perpendicularis F G. </s>
            <s xml:id="echoid-s9915" xml:space="preserve">Poſtea in recta A E, accepto quolibet pun
              <lb/>
            cto H, ducatur per illud recta H G, ipſi E F, parallela, vel ad F G, perpendicularis ſecans F G, in G; </s>
            <s xml:id="echoid-s9916" xml:space="preserve">& </s>
            <s xml:id="echoid-s9917" xml:space="preserve">
              <lb/>
            rurſus per idem punctum H, recta I K, ad A B, perpendicularis. </s>
            <s xml:id="echoid-s9918" xml:space="preserve">Poſtremò ſumptis in recta A B, rectis
              <lb/>
            E B, H L, quæ rectis E F, H G, ſint æquales, deſcribantur ex B, L, circuli, vel potius quadrantes circulo
              <lb/>
            rum; </s>
            <s xml:id="echoid-s9919" xml:space="preserve">(quod ſatis eſt, ſi puncta per hos quadrantes in rectis E D, H K, inuenta transferantur in rectas
              <lb/>
            E C, H I) & </s>
            <s xml:id="echoid-s9920" xml:space="preserve">circuli quidem, ſi deſcripti ſint, in partes 24. </s>
            <s xml:id="echoid-s9921" xml:space="preserve">quadrantes vero, ſi forte quadrantes tantum
              <lb/>
            ſint deſcripti, in 6. </s>
            <s xml:id="echoid-s9922" xml:space="preserve">partes ęquales diſtribuantur, initio ſemper facto à linea meridiana A B. </s>
            <s xml:id="echoid-s9923" xml:space="preserve">Nam rectæ
              <lb/>
              <note position="left" xlink:label="note-0173-09" xlink:href="note-0173-09a" xml:space="preserve">50</note>
            ductæ, occultè tamen, per centra B, L, & </s>
            <s xml:id="echoid-s9924" xml:space="preserve">per puncta diuiſionum circulorum, quadrantumve ſecabunt re
              <lb/>
            ctas C D, I K, vel ipſas E D, H K, ſi quadrantes tantum ſint deſcripti, in punctis, per quæ eductæ rectæ
              <lb/>
            lineæ (ſumendo ſemper bina puncta reſpondentia inter ſe, hoc eſt, primùm duo proxima punctis E, H,
              <lb/>
            deinde duo ſequentia, & </s>
            <s xml:id="echoid-s9925" xml:space="preserve">c.) </s>
            <s xml:id="echoid-s9926" xml:space="preserve">dabunt lineas horarias, quas deſideramus; </s>
            <s xml:id="echoid-s9927" xml:space="preserve">ita tamen, vt ſi quadrantes dun-
              <lb/>
            taxat ſint deſcripti, puncta rectarum E D, H K, transfer antur prius in rectas E C, H I, & </s>
            <s xml:id="echoid-s9928" xml:space="preserve">poſtea lineæ
              <lb/>
            ducantur, vt dictum eſt. </s>
            <s xml:id="echoid-s9929" xml:space="preserve">Has autem producemus, quantum libuerit. </s>
            <s xml:id="echoid-s9930" xml:space="preserve">Nam arcus ſignorum, quos ſequenti
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s9931" xml:space="preserve">deſcribemus, terminabunt earum longitudines. </s>
            <s xml:id="echoid-s9932" xml:space="preserve">Longitude gnomonis erit F M, perpendicularis ad
              <lb/>
            A B, eiuſ{q́ue} locus in M, puncto. </s>
            <s xml:id="echoid-s9933" xml:space="preserve">Quod ita demonſtr abimus.</s>
            <s xml:id="echoid-s9934" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9935" xml:space="preserve">QVONIAM angulus D E F, altitudini poli æqualis eſt, erit angulus A E F, complemento eiuſ-
              <lb/>
              <note position="right" xlink:label="note-0173-10" xlink:href="note-0173-10a" xml:space="preserve">Demonſtr@ti@
                <lb/>
              proximæ deſ@c@
                <lb/>
              ptionis.</note>
            dem altitudinis æqualis; </s>
            <s xml:id="echoid-s9936" xml:space="preserve">ac proinde duo anguli A E F, E F G, duobus rectis erunt minores, ideo{q́ue} recta
              <lb/>
            F G, cum recta E A, ad partes A, tandem conueniet, vt in A, puncto, quod etiamſi in plano propter </s>
          </p>
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