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-s20683" xml:space="preserve">
              <pb o="307" file="0323" n="323" rhead="LIBER TERTIVS."/>
            la erit ipſi S T, ex ſcholio propoſ. </s>
            <s xml:id="echoid-s20684" xml:space="preserve">27. </s>
            <s xml:id="echoid-s20685" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s20686" xml:space="preserve">3. </s>
            <s xml:id="echoid-s20687" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s20688" xml:space="preserve">ob æquales arcus inter puncta P, V, & </s>
            <s xml:id="echoid-s20689" xml:space="preserve">diametrũ S T,
              <lb/>
            productam interceptos) & </s>
            <s xml:id="echoid-s20690" xml:space="preserve">per punctum Y, horæ 12. </s>
            <s xml:id="echoid-s20691" xml:space="preserve">in minori circulo agatur maiori Ellipſis diametro
              <lb/>
            R N, par allela Y Z, (quæ eadem facilitate ducetur, ſi arcui S Y, æqualis accipiatur arcus S X. </s>
            <s xml:id="echoid-s20692" xml:space="preserve">Recta
              <lb/>
            euim X Y, ex epdem ſcholio propoſ. </s>
            <s xml:id="echoid-s20693" xml:space="preserve">27. </s>
            <s xml:id="echoid-s20694" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s20695" xml:space="preserve">3. </s>
            <s xml:id="echoid-s20696" xml:space="preserve">Eucl. </s>
            <s xml:id="echoid-s20697" xml:space="preserve">parallela erit ipſi R N, ob ęquales arcus inter puncta
              <lb/>
            Y, X, & </s>
            <s xml:id="echoid-s20698" xml:space="preserve">diametrum R N, interiectos) ſecans P Z, in Z. </s>
            <s xml:id="echoid-s20699" xml:space="preserve">Erit ergo Z, punctum Ellipſis, cuius diametri
              <lb/>
            R N, S T, vt conſtat ex coroll. </s>
            <s xml:id="echoid-s20700" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s20701" xml:space="preserve">26. </s>
            <s xml:id="echoid-s20702" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s20703" xml:space="preserve">1. </s>
            <s xml:id="echoid-s20704" xml:space="preserve">Dico per idem ducendam eſſe lineam meridianam, ſeu
              <lb/>
            horę 12. </s>
            <s xml:id="echoid-s20705" xml:space="preserve">Intelligatur enim (poſito triangulo C H N, ad planum horologii recto, ita vt cum Meridiano
              <lb/>
            proprio plani declinantis ſit coniunctum) in plano Aequatoris ex H, centro mundi, Aequatorisve de-
              <lb/>
            ſcriptus ad interuallum H N, circulus N h ſ m, circulo O S T, æqualis, qui baſis ſit cylindri recti, cu-
              <lb/>
            ius axis idem qui axis mundi C H, & </s>
            <s xml:id="echoid-s20706" xml:space="preserve">ſemicirculus N h ſ, vergat verſus punctum P, ſeu lineam meri-
              <lb/>
              <note position="left" xlink:label="note-0323-01" xlink:href="note-0323-01a" xml:space="preserve">10</note>
            dianam, in quo ſumatur arcus N h, arcui O Y, æqualis, vel ipſi N P, ſimilis. </s>
            <s xml:id="echoid-s20707" xml:space="preserve">Ducetur per h, Meridia-
              <lb/>
            n{us} Horizontis, quandoquidem arcus N h, ſimilis eſt arcui Aequatoris inter Meridianum Horizontis,
              <lb/>
            & </s>
            <s xml:id="echoid-s20708" xml:space="preserve">proprium Meridianum plani declinantis poſito. </s>
            <s xml:id="echoid-s20709" xml:space="preserve">Igitur latus cylindri ex puncto h, ductum, per quod
              <lb/>
            planum Meridiani Horizontis, vel circuli horę 12. </s>
            <s xml:id="echoid-s20710" xml:space="preserve">ducitur, cadet in punctum Ellipſis Z, ob ſimilitudi-
              <lb/>
            nem arcuum N h, N P, vt constat ex demonſtratis in ſcholio propoſ. </s>
            <s xml:id="echoid-s20711" xml:space="preserve">1. </s>
            <s xml:id="echoid-s20712" xml:space="preserve">ſuperioris lib. </s>
            <s xml:id="echoid-s20713" xml:space="preserve">Ibi enim, quia ar-
              <lb/>
            cus B P, B F, ſimiles erant, demonſtrauimus latus cylindri ex P, ductũ cadere in punctũ Ellipſis L, & </s>
            <s xml:id="echoid-s20714" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s20715" xml:space="preserve">Eadem ratione ſi per puncta Q, u, oppoſita punctis P, Y, ducamus duas lineas duabus diametris Elli-
              <lb/>
            pſis parallelas, reperiemus aliud punctum δ, per quod eadem linea meridiana ducenda eſt. </s>
            <s xml:id="echoid-s20716" xml:space="preserve">Nam in illud
              <lb/>
            punctum cadet latus cylindri ductum ex puncto m, quod opponitur puncto h, per quod nimirum ducitur
              <lb/>
            Meridianus Horizontis, propter ſimi itudinem arcuum N m, N Q. </s>
            <s xml:id="echoid-s20717" xml:space="preserve">Sic etiam ex punctis a, b, horæ 6. </s>
            <s xml:id="echoid-s20718" xml:space="preserve">
              <lb/>
              <note position="left" xlink:label="note-0323-02" xlink:href="note-0323-02a" xml:space="preserve">20</note>
            inuenietur punctum f, per quod hora 6. </s>
            <s xml:id="echoid-s20719" xml:space="preserve">ducenda eſt, & </s>
            <s xml:id="echoid-s20720" xml:space="preserve">ex puncto t, oppoſito in maiore circulo, eius{q́ue} re-
              <lb/>
            ſpondente in minori circulo, aliud punctũ oppoſitum, per quod eadem hora 6. </s>
            <s xml:id="echoid-s20721" xml:space="preserve">ducẽda eſt. </s>
            <s xml:id="echoid-s20722" xml:space="preserve">Atque it a de cæ-
              <lb/>
            teris. </s>
            <s xml:id="echoid-s20723" xml:space="preserve">Vides igitur puncta Z, δ, cadere in lineam meridianam in initio ductam, & </s>
            <s xml:id="echoid-s20724" xml:space="preserve">lineam horæ 6. </s>
            <s xml:id="echoid-s20725" xml:space="preserve">du-
              <lb/>
            ctam per f, eius{q́ue} punctum oppoſitum, tranſire per punctum α, per quod in præcedenti deſcriptione du-
              <lb/>
            cendam eſſe horam 6. </s>
            <s xml:id="echoid-s20726" xml:space="preserve">oſtendimus. </s>
            <s xml:id="echoid-s20727" xml:space="preserve">Denique vides lineam horæ 3. </s>
            <s xml:id="echoid-s20728" xml:space="preserve">quæ vix ſine errore in precedenti deſcri-
              <lb/>
            ptione duci poterat, exquiſitiſſime duci poſſe beneficio huius ellipſis, & </s>
            <s xml:id="echoid-s20729" xml:space="preserve">c. </s>
            <s xml:id="echoid-s20730" xml:space="preserve">In noſtro exemplo, quoniam li-
              <lb/>
            nea ſtyli inęqualiter diſtat à binis horis hinc inde poſitis, inæqualiter etiam diſtabunt diametri Ellipſis à
              <lb/>
            binis punctis ex vtraque parte inuentis. </s>
            <s xml:id="echoid-s20731" xml:space="preserve">Vnde (quod ad demonſtrationem attinet) in circulo Aequatori
              <lb/>
            concentrico N h ſ m, à ſingulis horis ducendę erunt ad diametrũ ſ N, perpendiculares ſingulæ, per quas
              <lb/>
            ducenda ſunt plana axi parallela, vt fiant parallelogramma, quorum latera ex horis ducta cadunt in pun
              <lb/>
              <note position="left" xlink:label="note-0323-03" xlink:href="note-0323-03a" xml:space="preserve">30</note>
            cta Ellipſis, vt in ſch olio dicto propoſ. </s>
            <s xml:id="echoid-s20732" xml:space="preserve">1. </s>
            <s xml:id="echoid-s20733" xml:space="preserve">ſuperioris lib. </s>
            <s xml:id="echoid-s20734" xml:space="preserve">demonstratum eſt. </s>
            <s xml:id="echoid-s20735" xml:space="preserve">In figura propoſita perpendi.
              <lb/>
            </s>
            <s xml:id="echoid-s20736" xml:space="preserve">culares h g, m l, ducuntur ab hora 12. </s>
            <s xml:id="echoid-s20737" xml:space="preserve">tam meridiei, quàm mediæ noctis, perpendiculares autem p n
              <unsure/>
            , q r,
              <lb/>
            ab hora 6. </s>
            <s xml:id="echoid-s20738" xml:space="preserve">tam à mer. </s>
            <s xml:id="echoid-s20739" xml:space="preserve">quàm à med. </s>
            <s xml:id="echoid-s20740" xml:space="preserve">noc.</s>
            <s xml:id="echoid-s20741" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s20742" xml:space="preserve">PORRO cum declinatio plani horologij à Verticali circulo tanta eſt, vt parum à grad. </s>
            <s xml:id="echoid-s20743" xml:space="preserve">90. </s>
            <s xml:id="echoid-s20744" xml:space="preserve">diffe-
              <lb/>
              <note position="right" xlink:label="note-0323-04" xlink:href="note-0323-04a" xml:space="preserve">Quando horo-
                <lb/>
              logii planum
                <lb/>
              parum à Meri-
                <lb/>
              diano c@culo
                <lb/>
              abeſt, difficul-
                <lb/>
              ter horologium
                <lb/>
              per tradita præ
                <lb/>
              cepta deſcribi
                <lb/>
              potcſt.</note>
            rat, ac proinde planum ipſum ferè cum Meridiano circulo coniungatur, vix per præcepta tradita con-
              <lb/>
            ſtruere quis poterit horologium, quin in difficultatem aliquam, eamq́, non leuem, incurrat. </s>
            <s xml:id="echoid-s20745" xml:space="preserve">Nam tunc li-
              <lb/>
            nea declinationis E F, valde prope accedit ad rectam A B, propterea{q́ue} nimis breuis efficitur recta F G,
              <lb/>
            & </s>
            <s xml:id="echoid-s20746" xml:space="preserve">illi æqualis G H: </s>
            <s xml:id="echoid-s20747" xml:space="preserve">ex quo fit, axem C H, parum à linea ſtyli C H, diſtare, atque adeo ſtylum I K, fo-
              <lb/>
            re breuiſſimum, ita vt quaſi inutile horologium ipſum reddatur. </s>
            <s xml:id="echoid-s20748" xml:space="preserve">Quòd ſi velimus rectam F G, eſſe maio
              <lb/>
            ris cuiuſdam, & </s>
            <s xml:id="echoid-s20749" xml:space="preserve">aptę, conuenientis{q́ue} magnitudinis, quis non videt, punctum F, futurum à puncto E, ni-
              <lb/>
              <note position="left" xlink:label="note-0323-05" xlink:href="note-0323-05a" xml:space="preserve">40</note>
            mis remotum? </s>
            <s xml:id="echoid-s20750" xml:space="preserve">ideo{q́ue} & </s>
            <s xml:id="echoid-s20751" xml:space="preserve">puncta G, β, C, ab eodem puncto E, ptuſquam deceat, excurſura? </s>
            <s xml:id="echoid-s20752" xml:space="preserve">Quæ ſanè res
              <lb/>
            non parum difficultatis, ac laboris in ſe continet, quòd tunc immenſum ferè planum ad horologii deſcri-
              <lb/>
            ptionem ſit neceſſarium. </s>
            <s xml:id="echoid-s20753" xml:space="preserve">Quod inde etiam constare poteſt, quòd lineæ horariæ (quę in horologio Meridia-
              <lb/>
            no parallelę ſunt, vt iamdudum propoſ. </s>
            <s xml:id="echoid-s20754" xml:space="preserve">25. </s>
            <s xml:id="echoid-s20755" xml:space="preserve">ſuperioris libri demonſtrauimus) in plano horologii, quod pa-
              <lb/>
            rum à Meridiano circulo diſtat, parallelæ inter ſe quaſi videantur; </s>
            <s xml:id="echoid-s20756" xml:space="preserve">præſertim quando declinatio fueri@
              <lb/>
            grad. </s>
            <s xml:id="echoid-s20757" xml:space="preserve">89. </s>
            <s xml:id="echoid-s20758" xml:space="preserve">88. </s>
            <s xml:id="echoid-s20759" xml:space="preserve">vel 87. </s>
            <s xml:id="echoid-s20760" xml:space="preserve">& </s>
            <s xml:id="echoid-s20761" xml:space="preserve">c. </s>
            <s xml:id="echoid-s20762" xml:space="preserve">Verum incommodo huic remedium afferemus ſimile ferè illi, quod in ſcholio
              <lb/>
            propoſ. </s>
            <s xml:id="echoid-s20763" xml:space="preserve">1. </s>
            <s xml:id="echoid-s20764" xml:space="preserve">huius lib. </s>
            <s xml:id="echoid-s20765" xml:space="preserve">pro horologio horizontali ſcripſimus, ita vt per præcepta ferè hactenus tradita horo-
              <lb/>
            logium declinans deſcribamus, quantumuis parum à circulo Meridiano diſtans, licet centrum horologii
              <lb/>
            (quod tunc plus æquo à puncto E, diſtet, neceſſe eſt, ſi cupimus habere ſtylum tantæ magnitudinis, vt
              <lb/>
            commodè horas indicare poſſit) nobis prorſus ſit ignotum, quippe quod in planum (niſi illud immenſam
              <lb/>
              <note position="left" xlink:label="note-0323-06" xlink:href="note-0323-06a" xml:space="preserve">50</note>
            quodammodo habeat magnitudinem) cadere non poſſit, ſed extra ipſum exiſtat. </s>
            <s xml:id="echoid-s20766" xml:space="preserve">Hoc autem artifi-
              <lb/>
            cio vtemur.</s>
            <s xml:id="echoid-s20767" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s20768" xml:space="preserve">IN plano aliquo ductis duabus rectis A B, C D, ſe mutuo ad angulos rectos ſecantibus in E, puncto,
              <lb/>
              <note position="right" xlink:label="note-0323-07" xlink:href="note-0323-07a" xml:space="preserve">Quo pacto ho-
                <lb/>
              rologiũ deſcri-
                <lb/>
              ba@u@ in plano,
                <lb/>
              quod parum à
                <lb/>
              Meridiano cir-
                <lb/>
              culo abeſt, licet
                <lb/>
              centrum horo-
                <lb/>
              log@ non ha-
                <lb/>
              beatur.</note>
            constituatur in E, ad rectam C D, angulus declinationis infra, vel ſupra rectam A B, & </s>
            <s xml:id="echoid-s20769" xml:space="preserve">ad dextram,
              <lb/>
            vel ſiniſtram rectę C D, prout planum à meridie, vel borea in ortum, occaſumve declinat, vt ad initium
              <lb/>
            huius propoſ. </s>
            <s xml:id="echoid-s20770" xml:space="preserve">ſcripſimus. </s>
            <s xml:id="echoid-s20771" xml:space="preserve">In hoc propoſito exemplo ponimus declinationem plani à circulo Verticali in
              <lb/>
            ortum grad. </s>
            <s xml:id="echoid-s20772" xml:space="preserve">60. </s>
            <s xml:id="echoid-s20773" xml:space="preserve">& </s>
            <s xml:id="echoid-s20774" xml:space="preserve">à Meridiano grad. </s>
            <s xml:id="echoid-s20775" xml:space="preserve">30. </s>
            <s xml:id="echoid-s20776" xml:space="preserve">propterea angulus declinationis D E F, conſtitutus eſt infra
              <lb/>
            rectam A B, verſus A. </s>
            <s xml:id="echoid-s20777" xml:space="preserve">Non aſſumimus autem maiorem declinationem, vt magis perſpicua reddatur
              <lb/>
            deſcriptio hæc noſtra, eius demonſtratio planior fiat.</s>
            <s xml:id="echoid-s20778" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s20779" xml:space="preserve">POST hæc in recta A B, ſumpto quocunque puncto G, ſiue verſus eam partem, in quam linea de-
              <lb/>
            clinationis vergit à recta C D, ſiue in partem contrariam, quod magis placet, (quamuis in exemplo </s>
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